[{"content":" ExperimentSimulation \u0026ldquo;Dye shows the streamlines in water flowing at 1 mm per second between glass plates spaced 1 mm apart. It is at first sight paradoxical that the best way of producing the unseparated pattern of plane potential flow past a bluff object, which would be spoiled by separation in a real fluid of even the slightest viscosity, is to go to the oposite of extreme of creeping flow in a narrow gap, which is dominated by viscous forces.\u0026rdquo; Photograph by D. H. Peregrine\nTheory\u003e Theory # Hele-Shaw flow\u003e Hele-Shaw flow # Hele-Shaw flow is characterized as flow through the narrow gap between two parallel plates. The wall-contact induced viscous shear stresses lead to flow patterns that may be described as if the flow were \u0026lsquo;potential\u0026rsquo;. As the caption of this figure says, this may seem paradoxical: potential flows are those that are free of viscous stresses.\nIn the next two subsections we describe this effect mathematically. We start with the equations of viscosity dominated incompressible, creeping, flow (the Stokes equations), and show that they reduce to the potential flow equations for an inviscid fluid (the Darcy equations) in the case of fluid forced in between two parallel plates.\nStokes equations: from 3D to 2D\u003e Stokes equations: from 3D to 2D # When an incompressible fluid is characterized by a sufficiently low Reynolds number, the advective effects may be neglected and we retrieve the Stokes equations: $$ \\begin{cases} \\mu \\Delta \\mathbf{u} - \\nabla p = \\mathbf{0} \\\\ \\nabla \\cdot \\mathbf{u} = 0 \\end{cases} $$ where the time-derivative terms is dropped as we assumed that steady state is reached.\nThe \u0026ldquo;flattened\u0026rdquo; appearance of the flow in a narrow slit may lead one to naïvely conclude that Hele-Shaw flow should be modeled as a 2D Stokes goverened system. This is quite untrue: it is precisely in that third dimension that high velocity gradients occur, leading to high viscous drag forces. We thus ought to start our analysis from the three-dimensional Stokes equations: $$ \\begin{cases} \\mu ( \\frac{\\partial^2}{\\partial x^2} u_x + \\frac{\\partial^2}{\\partial y^2} u_x + \\frac{\\partial^2}{\\partial z^2} u_x ) - \\frac{\\partial}{\\partial x} p = 0 \\\\ \\mu ( \\frac{\\partial^2}{\\partial x^2} u_y + \\frac{\\partial^2}{\\partial y^2} u_y + \\frac{\\partial^2}{\\partial z^2} u_y ) - \\frac{\\partial}{\\partial y} p = 0 \\\\ \\mu ( \\frac{\\partial^2}{\\partial x^2} u_z + \\frac{\\partial^2}{\\partial y^2} u_z + \\frac{\\partial^2}{\\partial z^2} u_z ) - \\frac{\\partial}{\\partial z} p = 0 \\\\ \\frac{\\partial}{\\partial x}u_x + \\frac{\\partial}{\\partial y}u_y + \\frac{\\partial}{\\partial z} u_z = 0 \\end{cases} $$\nFocusing first on the velocity dependence on the third dimension, we note that such wall-bounded channel flow has its own name: Poiseuille flow. The well-known corresponding velocity profile is a quadratic function that is zero at the two walls, and achieves its maximum at the center. Schematically, the velocity profile through the gap width looks like:\nMaking use of this knowlegde, we can write our full three-dimensional velocity field \\(\\mathbf{u} = \\mathbf{u}(x,y,z)\\) just in terms the two-dimensional center velocity \\(\\hat{\\mathbf{u}} = (\\hat{u}_x , \\hat{u}_y , 0 )^\\text{T} = \\hat{\\mathbf{u}}(x,y) \\): $$ \\mathbf{u}(x,y,z) = (1 - \\frac{2}{\\delta} z ) \\, ( 1 + \\frac{2}{\\delta} z ) \\, \\hat{\\mathbf{u}}(x,y) \\,, $$ where \\( \\delta \\) is the width of the gap. By construction, this assumed flow profile satisfies the no-slip boundary conditions at the two parallel plates: \\( \\mathbf{u}(x,y,\\delta/2) = \\mathbf{u}(x,y,-\\delta/2) = \\mathbf{0} \\), and simply equals the center velocity at the midpoint: \\( \\mathbf{u}(x,y,0) = \\hat{\\mathbf{u}}(x,y) \\).\nWhen this assumed velocity profile is substituted into the three-dimensional Stokes equations, they produce: $$ \\begin{cases} \\mu ( \\frac{\\partial^2}{\\partial x^2} \\hat{u}_x + \\frac{\\partial^2}{\\partial y^2} \\hat{u}_x - \\frac{8}{\\delta^2} \\hat{u}_x ) - \\frac{\\partial}{\\partial x} p = 0 \\\\ \\mu ( \\frac{\\partial^2}{\\partial x^2} \\hat{u}_y + \\frac{\\partial^2}{\\partial y^2} \\hat{u}_y - \\frac{8}{\\delta^2} \\hat{u}_y ) - \\frac{\\partial}{\\partial y} p = 0 \\\\ \\frac{\\partial}{\\partial z} p = 0 \\\\ \\frac{\\partial}{\\partial x}\\hat{u}_x + \\frac{\\partial}{\\partial y}\\hat{u}_y = 0 \\end{cases} $$ or, in vector notation: $$ \\begin{cases} \\mu \\Delta \\hat{\\mathbf{u}} - \\nabla \\hat{p} = \\frac{8 \\mu}{\\delta^2} \\hat{\\mathbf{u}} \\\\ \\nabla \\cdot \\hat{\\mathbf{u}} = 0 \\end{cases} $$ where \\( \\hat{p} = \\hat{p}(x,y) \\), and the Laplace and gradient operators can now be understood as two-dimensional differential operators.\nDarcy equations\u003e Darcy equations # As may be observed from this dimensionally-reduced Stokes-like equation, the effect of the viscous wall-shear stress manifests itself as a drag force that scales linearly with the center velocity. In its current form, the equations are called the Darcy-Brinkman equations. However, if \\( \\mu \\Delta \\hat{\\mathbf{u}} \\ll \\frac{8 \\mu}{\\delta^2} \\hat{\\mathbf{u}} \\), as would be the case when \\(\\delta\\) is much smaller than the other geometric features, then the remaining viscous stress may be neglected, and we find: $$ \\begin{cases} \\hat{\\mathbf{u}} = - \\frac{\\delta^2}{8 \\mu} \\nabla \\hat{p} \\\\ \\nabla \\cdot \\hat{\\mathbf{u}} = 0 \\end{cases} $$ which are the Darcy equations.\nThe Darcy and Stokes equations leads to quite different solution fields. This can be observed in the figures below, for which the same streamline inflow locations have been chosen. In case of Stokes flow, the effect of the obstructing cylinder is noticible much further upstream.\nStokesDarcy Simulation\u003e Simulation # Finite element formulation\u003e Finite element formulation # The focus of the current post is on the Hele-Shaw theory section. As such, the section of the mathematics behind the finite element approximation is kept relatively short, and may seem overly concise to the uninitiated. Other posts will focus more on this aspect.\nTo obtain a finite element approximation, we require a weak formulation of the partial differential equation at hand. The Sokes and Darcy equations are both systems of differential equations. In weak form, these lead to so-called mixed formulations. For the Darcy equations, we obtain: $$ \\text{Find } (\\hat{\\mathbf{u}}, \\hat{p} ) \\in \\mathbf{H}_{h}(\\text{div}) \\times L^2 \\text{ s.t.}: \\qquad\\qquad\\quad\\qquad\\quad\\qquad\\quad\\\\ \\begin{cases} \\int \\big( \\frac{8 \\mu}{\\delta^2} \\hat{\\mathbf{u}} \\cdot \\mathbf{v} - \\hat{p} \\, \\nabla \\cdot \\mathbf{v} \\big) \\text{d}S = - \\int g \\, \\mathbf{v}\\cdot\\mathbf{n} \\, \\text{d}L \\quad \\forall\\, \\mathbf{v} \\in \\mathbf{H}_0 (\\text{div}) \\\\ \\int q\\,\\nabla \\cdot \\hat{\\mathbf{u}} \\text{d}S = 0 \\qquad\\qquad\\qquad\\qquad\\qquad\\quad\\,\\,\\, \\forall\\, q \\in L^2 \\end{cases} $$ where \\( L^2 \\) is the space of square integrable functions and \\( \\mathbf{H}(\\text{div}) \\) is the space of vector-valued functions whose divergence\u0026rsquo;s are functions in \\( L^2 \\). Members of \\( \\mathbf{H}(\\text{div}) \\) permit evaluation of their normal component on domain boundaries, whereby \\( \\hat{\\mathbf{u}}\\cdot\\mathbf{n} = h \\) represents an essential condition which is thus imposed on all functions in the space \\( \\mathbf{H}_{h} (\\text{div}) \\) (c.q. \\( \\mathbf{v}\\cdot\\mathbf{n} = 0 \\) on \\( \\mathbf{H}_{0} (\\text{div}) \\)). The natural condition is the imposed pressure \\( \\hat{p}=g \\), which is already substituted in the right-hand-side integral in the above equation. The chosen values for all boundary conditions are illustrated in the figure below.\nComputational domain, boundary conditions and mesh. The bottom of the computational domain is chosen slightly larger than the figure frame, as the streamlines in the original figure slightly curve out of the frame. This figure also shows the relative dimensions used for the computational domain. Units are purposely left out: the qualitative nature of the solutionfield (i.e., the streamslines) is not affected by the scale of the domain, nor by the magnitude of the inflow, or even by the value of the effective permeability \\( \\frac{8 \\mu}{\\delta^2} \\).\nWhile devising a finite element approximation of the above weak formulation, one must take care to choose an appropriate pair of discrete function spaces. Not every choice of vector-valued velocity and scalar-valued pressure finite element spaces is \u0026ldquo;stable\u0026rdquo; and produces accurate results, or even produces a solveable system of equations. Typical compatible spaces for the Darcy equations, are a \\( P \\)-th order Brezzi-Douglas-Marini (BDM) space for the velocity and a \\( ( P-1 ) \\)-th discontinuous Galerkin (DG) space for pressure, or \\( P \\)-th order Raviart-Thomas (RT) space paired with a \\( ( P-1 ) \\)-th order DG space. In the following implementation, we make use of BDM elements. The mesh size used is illustrated in the figure above.\nCode implementation\u003e Code implementation # This simulation is performed with the finite element method, with the library FEniCS. The following code outputs the required data:\nfrom dolfin import * from mshr import * # Physical parameters D = 1 L = 6 Ht = 1.62 # height till top Hb = 1.9 # height till bottom # Numerical parameters resolution = 120 # Create mesh domain = Rectangle(Point(-L/2,-Hb),Point(L/2,Ht)) - Circle(Point(0,0), D/2) mesh = generate_mesh(domain, resolution) # Create finite element spaces BDM = FiniteElement(\u0026#34;BDM\u0026#34;, mesh.ufl_cell(), 1) DG = FiniteElement(\u0026#34;DG\u0026#34;, mesh.ufl_cell(), 0) V = FunctionSpace(mesh, BDM * DG) # Define test, trial and solution function u, p = TrialFunctions(V) v, q = TestFunctions(V) u_sol = Function(V) # Define boundary conditions def boundary_noflow(x): return x[1] \u0026lt; -Hb+DOLFIN_EPS or x[1] \u0026gt; Ht - DOLFIN_EPS \\ or x[0]**2+x[1]**2 \u0026lt; (D/2)**2+DOLFIN_EPS def boundary_inflow(x): return x[0] \u0026lt; -L/2+DOLFIN_EPS bcs = [DirichletBC(V.sub(0), Constant((0,0)), boundary_noflow),\\ DirichletBC(V.sub(0), Constant((1,0)), boundary_inflow),] # Weak formulation a = (dot(u,v) + div(v)*p + div(u)*q)*dx Lrhs = - Constant(0)*q*dx # Solve and export solve(a == Lrhs, u_sol, bcs) (u,p) = u_sol.split() u.rename(\u0026#39;u\u0026#39;,\u0026#39;u\u0026#39;) File(\u0026#34;sol.pvd\u0026#34;) \u0026lt;\u0026lt; u Visualization\u003e Visualization # Visualization is performed in Paraview, with the streamline and tube filters. The seeds of the streamlines are manually placed at the correct inflow points. The tube filter permits a variable tube width, as specified by a minimum value, a maximum scaling w.r.t that minimum value, and a scalar-field between whose minimal and maximal values is interpolated. For the latter, we specify a simple \\(\\frac{1}{\\sqrt{|\\mathbf{u}|}}\\) function, representing the streching and broadening of the paint streak with increasing resp. decreasing flow velocity. Below follow screenshots of the visualization settings, and some further visualizations of the qualitative behavior of the solution.\nPrevious Next ","date":"8 April 2023","permalink":"/chapters/01-creeping/fig1/","section":"Chapters","summary":"ExperimentSimulation \u0026ldquo;Dye shows the streamlines in water flowing at 1 mm per second between glass plates spaced 1 mm apart. It is at first sight paradoxical that the best way of producing the unseparated pattern of plane potential flow past a bluff object, which would be spoiled by separation in a real fluid of even the slightest viscosity, is to go to the oposite of extreme of creeping flow in a narrow gap, which is dominated by viscous forces.","title":"Fig 1. Hele-Shaw flow past a circle"},{"content":" ExperimentSimulation \u0026ldquo;At this Reynolds number the streamline pattern has clearly lost the fore-and-aft symmetry of figure 6. However, the flow has not yet separated at the rear. That begins at about R=5, though the value is not known accurately. Streamlines are made visible by aluminum powder in water.\u0026rdquo; Photograph by Sadathoshi Taneda\nGeneral Info\u003e General Info # This post is part of a series on flow separation, studied for the case of flow past a circular cylinder at different Reynolds numbers. In the current figure, it is clearly visible that the flow is still attached to the cylinder and therefore separation did not occur yet.\nThe main theory and simulation and visualization set-up are discussed in the web post from Figure 42. The full series is:\nFigure 24: Circular cylinder at R=1.54. Figure 40: Circular cylinder at R=9.6. Figure 41: Circular cylinder at R=13.1. Figure 42: Circular cylinder at R=26. Figure 45: Circular cylinder at R=28.4. Figure 46: Circular cylinder at R=41. Figure 96: Kármán vortex street behind a circular cylinder at R=105. Figure 94: Kármán vortex street behind a circular cylinder at R=140. An overview of these posts can be viewed here:\n","date":"18 August 2023","permalink":"/chapters/02-laminar/fig24/","section":"Chapters","summary":"ExperimentSimulation \u0026ldquo;At this Reynolds number the streamline pattern has clearly lost the fore-and-aft symmetry of figure 6. However, the flow has not yet separated at the rear. That begins at about R=5, though the value is not known accurately.","title":"Fig 24. Circular cylinder at R=1.54"},{"content":" ExperimentSimulation Air bubbles in water show the turbulent flow field and laminar reattachment on an inclined plate at Reynolds number \\(Re = 10000\\). At \\(2.5^{\\circ}\\) inclination relative to the oncoming flow, the flow briefly separates from the upper surface at the leading edge before it reattaches to the inclined plate. The separation region creates a recirculating turbulent boundary layer. For comparison, a transient simulation was created in Simcenter StarCCM+ using the SST turbulence model. Photograph by ONERA photograph, Werlé 1974\nTheory\u003e Theory # The flow field over a \\(2.5^{\\circ}\\) inclined plate shows a large transition region between laminar and turbulent flow. This is an ideal case for putting turbulence models to the test. For the comparison figure between the experiment picture and the simulation visualization, the SST turbulence model was used. However, this simulation was also run with the k-epsilon and k-omega turbulence models, with a model comparison given below. A detailed overview about RANS turbulence models is given in this post and all of the theory referring to k-epsilon, k-omega and SST turbulence models can be found there.\nSimulation\u003e Simulation # All of the simulations of the inclined plate experiment were carried out in Simcenter StarCCM+ using the k-epsilon, k-omega and SST turbulence models. The differences in results of each of the turbulence models will be discussed in the \u0026lsquo;Visualization\u0026rsquo; section of this post, since the differences become apparent during post-processing.\nComputational domain\u003e Computational domain # As with all replications of the figures from the book, the computational domain, boundary conditions and finite volume mesh are created with the highest possible fidelity to the information available in the caption of the original picture. In this case it is known that the flow field is visualized with air bubbles in water and that the flow is characterized by the Reynolds number \\(Re = 10000\\) based on the length of the plate. The length of the plate can be calculated to be \\(0.1\\ m\\) and the inlet velocity to be \\(0.1\\ m/s\\) with the incidence angle set to \\(2.5^{\\circ}\\). The walls of the domain have a no-slip condition applied to them, the inlet is specified as a velocity inlet and the outlet as a pressure outlet. Since this experiment closely resembles the setup of an airfoil simulation, the domain size was set accordingly an can be seen in the figure below. The beveled edges were recreated as close as possible and the plate thickness was set to \\(2\\)% of the length, as is stated in the caption of the original figure.\nComputational domain, boundary conditions and mesh. The base size of the quadrilateral FVM mesh is set to \\(0.001\\ m\\). To save on computational expenses, this base size is only applied in the region near the inclined plate. The custom mesh controls available in StarCCM+ are used to coarsen the mesh in the far-field parts of the domain. The domain walls have an element size of \\(0.3\\ m\\) (\\(30000\\)% of base size) and the global growth rate is set to \\(1.1\\). On top of this, further refinement is applied in the boundary layer region of the plate. Namely the element size in this region is set to \\(10\\)% of the base size. The total cell count of this FVM mesh is \\(102617\\). Visualizations of this mesh can be seen below.\nPrevious Next StarCCM+ setup\u003e StarCCM+ setup # The simulation setup in StarCCM+ is the exact same as mentioned in this post about flow over a circular cylinder. More details and information about the simulation setup can be found there. What\u0026rsquo;s important to note is that three simulations were run in a transient environment, one using the low Reynolds number k-epsilon model, one using Wilcox\u0026rsquo;s (2008) k-omega model and a last one using Menter\u0026rsquo;s SST model.\nVisualization\u003e Visualization # Post processing and visualization in done in Paraview. Compared to the visualizations discussed in this post about flow over a circular cylinder and this post about flow over a large angle plate, the \\(2.5^{\\circ}\\) inclined plate has less chaotic and small scale turbulent phenomena. Therefore this visualization is mostly based on the streamlines of the flow field and allows for a better quantitative comparison between different turbulent models. As with other visualizations, the output data must be transformed from cell data to point data using the CellDatatoPointData filter in ParaView. From here, the velocity vector can be computed with the Calculator filter, adding the X and Y velocities with their respective unit vectors. The StreamTracer filter can now be applied to the velocity field and can be visualized with the Tube filter on top of it. However, the stream lines don\u0026rsquo;t show the recirculating bubble boundary layer at the leading edge of the plate, thus some particles can be injected here. For this, the Line source is used, to which the ParticleTracer and TemporalParticlestoPathlines filter can be added on top of. The full ParaView pipeline is shown below.\nParaView pipeline The final visualization can be seen in the slider at the top of the page for the comparison between the closest result of the numerical simulation and the actual experiment picture. For the final visualizations, the timesteps were matched to that of SST model with the most accurate visualization compared to the original figure. It should be noted that the best matching timestep is not in the \u0026lsquo;steady state\u0026rsquo; of the transient simulation. All of the simulations show a more elongated separation bubble at the leading edge of the plate after a long enough physical time period, which can be assumed to show the steady state. There could be multiple reasons for this, the most probable being a slightly inaccurate simulation geometry model compared to the plate used in the experiment. There are no specific details about the beveled edges of the real plate, so these were simply modeled as close as possible to the actual plate. The visualizations of the all of the turbulence model results can be seen in the slider figures below.\nExperiment pictureSteady state SST model SST modelk-omega model k-omega modelk-epsilon model As can be seen from the second slider, the difference in flow field prediction between the k-omega and SST model are barely noticeable. This is to be expected, since in the SST model, the blending function activated the k-omega model in the near wall region, thus both models should predict a similar flow field close to the wall boundary. However, there is a noticeable difference between the k-epsilon and k-omega model when comparing the height of the recirculating separation bubble at the leading edge of the plate.\nTurbulence model comparison\u003e Turbulence model comparison # To further highlight the difference in boundary layer height prediction between the two k-omega models (k-omega and SST) and k-epsilon model, some post processing filters can be applied to the velocity field of the simulation outputs at equal timesteps. First, one can add a grid to the background of the graphics window in order to quantify the actual boundary layer height. However, the grid (or the data) must be rotated 2.5 degrees using the Transform filter in order to turn the plate horizontally and be able to measure the boundary layer height normal to the plate. Then, a Contour filter can be added with a customized value range for different definitions of the boundary layer thickness. In general, the boundary layer thickness is defined as the height \\(\\delta\\) where $$ u(x, \\delta_{99}) = 0.99 \\ u_{\\infty} $$ with \\(\\delta_{99}\\) the normal distance to the plate surface where the velocity magnitude is \\(99\\)% that of the freestream velocity. To better highlight the boundary layer height in these simulations, multiple different velocity definitions of the boundary layer thickness are taken. Namely the normal distance to \\(50\\)%, \\(60\\)%, \\(70\\)%, \\(80\\)%, \\(90\\)% and \\(95\\)% of the freestream velocity. The result of these post processing steps and the difference between the boundary layer prediction of the k-epsilon and k-omega models can be seen in the slider below.\nk-epsilon boundary layerk-omega boundary layer Although the simulation results can only be compared to the picture of the experiment results and not actual data, the SST and k-omega turbulence model seem to be the most accurate, as can be seen in the slider figure at the very top of the page. If the k-omega and SST model are taken as reference, the slider figure just above shows that the k-epsilon model under-predicts the boundary layer height. The graphs above actually show that at \\(15\\)% of the plate length (\\(x=0.015\\ m\\)), the k-epsilon model predicts a \\(\\delta_{70}\\) boundary layer height where the k-omega and SST models predict a \\(\\delta_{50}\\) boundary layer height. As for the \\(\\delta_{95}\\) boundary layer height, the k-epsilon model predicts a \\(10\\)% lower boundary layer height than the two k-omega models.\nKnowing how these models behave, this behaviour is to be expected. As mentioned in the \u0026lsquo;Theory\u0026rsquo; section in this post, the low Reynolds number k-epsilon model uses damping functions on the coefficients in the model for the energy dissipation rate \\(\\epsilon\\). Although these damping functions greatly increase the near-wall flow prediction compared to the standard k-epsilon model, they are empirical functions and are simply not as accurate as a direct modelling approach found in the k-omega and SST model. These models are based on the specific energy dissipation rate \\(\\omega = \\frac{\\epsilon}{k}\\), which is the energy dissipation rate per unit kinetic turbulent energy and thus do not need empirical damping functions on the model coefficients.\nAnother turbulence model comparison with an almost identical simulation and experiment setup to this one can be found on this page. This post also contains more information about the low Reynolds number k-epsilon model.\n","date":"14 June 2023","permalink":"/chapters/03-separation/fig35/","section":"Chapters","summary":"ExperimentSimulation Air bubbles in water show the turbulent flow field and laminar reattachment on an inclined plate at Reynolds number \\(Re = 10000\\). At \\(2.5^{\\circ}\\) inclination relative to the oncoming flow, the flow briefly separates from the upper surface at the leading edge before it reattaches to the inclined plate.","title":"Fig 35. Leading-edge separation on a plate with laminar reattachment"},{"content":" ExperimentSimulation Air bubbles in water show the turbulent flow field and turbulent reattachment on an inclined plate at Reynolds number \\(Re = 50000\\). At 2.5\\(^{\\circ}\\) inclination relative to the oncoming flow, the flow separates from the upper surface, creating a turbulent boundary layer at the leading edge before reattaching to the inclined plate. The separation region creates a recirculating turbulent boundary layer. For comparison, a transient simulation was created in Simcenter StarCCM+ using the SST turbulence model. Photograph by ONERA photograph, Werlé 1974\nTheory\u003e Theory # In general, this experiment and simulation setup is the same as described in this post, except that the characteristic Reynolds number is increased by a factor of 5. Practically speaking this means that the inlet velocity is increased by a factor of 5, while all other parameters remain the same. With the Reynolds number being the ratio between inertial and viscous forces, higher Reynolds number flows tend to display higher turbulence and chaos in the flow field, which is what can be observed in this experiment picture. As for the recirculating boundary layer at the leading edge of the plate, it is much less visible than in the picture of the same experiment at Reynolds number \\(Re = 10000\\). However, the streamlines of this flow field, visualized with the air bubbles in the water and the shutter speed of the camera, clearly show a recirculating bubble as a boundary layer at the leading edge.\nSimulation\u003e Simulation # All of the simulations of the inclined plate experiment were carried out in Simcenter StarCCM+ using the k-epsilon, k-omega and SST turbulence models. Since this simulation setup closely resembles that of this post, more detailed information about the computational domain and finite volume mesh setup can be found there. This post will focus mainly on the differences in results of each of the turbulence models and will be discussed in the \u0026lsquo;Visualization\u0026rsquo; section of this post. More details and information about RANS turbulence models, can be found in this post.\nComputational domain\u003e Computational domain # As mentioned above, the experiment setup for this picture, and thus the simulation setup are identical to the one mentioned in the previous post about a \\(2.5^{\\circ}\\) inclined plate with the only change in the simulation setup being the increase from \\(0.1\\ m/s\\) to \\(0.5\\ m/s\\) for the inlet velocity boundary condition. This changes the characteristic Reynolds number from \\(Re = 10000\\) to \\(Re = 50000\\) as is the case for this experiment setup. For visualization purposes, the computational domain is shown again below.\nComputational domain, boundary conditions and mesh. StarCCM+ setup\u003e StarCCM+ setup # The simulation setup in StarCCM+ is the exact same as mentioned in this post about flow over a circular cylinder. More details and information about the simulation setup can be found there. What\u0026rsquo;s important to note is that three simulations were run in a transient environment, one using the low Reynolds number k-epsilon model, one using Wilcox\u0026rsquo;s (2008) k-omega model and a last one using Menter\u0026rsquo;s SST model.\nVisualization\u003e Visualization # Again, the detailed steps to obtaining the final visualizations of this simulation setup in Paraview can be found in the previous post. The main visualization feature is the StreamTracer filter, since away from the turbulent boundary layer, the air bubbles in the water mainly show the streamline of this flow field. All of the visualizations are taken at the same time step with respect to the most accurate visualization compared to the original picture, which again is the SST model simulation. All of the visualizations can be seen in the slider figures below and the comparison between the SST model and the actual experiment picture can be seen at the very top of the page.\nSST modelk-omega model k-omega modelk-epsilon model Looking at the comparison between the visualization of the SST model and the original experiment figure at the top of the page, it becomes apparent that the small scale turbulent phenomena is not really captured. Although the boundary layer height of the recirculating bubble seems to be accurately predicted, the actual turbulence and chaos of the turbulent boundary layer is not really displayed. Comparing the visualization of the k-omega and SST model, the boundary layer height prediction is almost identical, but the SST model seems to depict slightly more turbulence along the top and the wake of the plate. However, when looking at the k-epsilon model visualization, the recirculating boundary layer at the leading edge seems to be completely missing. A likely explanation for this is that the low Reynolds number k-epsilon model is not well suited for this simulation. A deeper analysis of this will be given in the next section.\nTurbulence model comparison\u003e Turbulence model comparison # To give a more quantitative comparison and evaluation between the different turbulence models, the boundary layer height in the recirculating bubble can be measured. This is done the same way as described in the \u0026lsquo;Turbulence model comparison\u0026rsquo; section of this post. The slider figures below show the boundary layer height \\(\\delta\\) for different boundary layer definitions. Namely $$ u(x, \\delta) = a \\cdot u_{\\infty} $$ with \\(a\\) the percentage of the freestream velocity \\(u_{\\infty}\\) (\\( = 0.5\\ m/s\\) in this case) between \\(50\\)% and \\(95\\)%.\nk-omega boundary layerSST boundary layer SST boundary layerk-epsilon boundary layer As can be seen in the slider figures above, the k-omega and SST model show very similar results for the boundary layer height over the first \\(15\\)% of the plate length. After this, the SST model seems to predict a flatter boundary layer profile. Since the simulation results can only be compared to the picture of the experiment and not to actual experimental data, it is difficult to say which model is the most accurate, especially since the simulation visualizations do not match the actual picture that well. However, the simulations for flow around a cylinder at \\(Re = 2000\\) and for flow around a \\(2.5^{\\circ}\\) inclined plate at \\(Re = 10000\\) have shown that the SST model consistently provides the best visual results between the three RANS models tested for these simulations. It is still interesting to highlight the differences between the models, even without reference data.\nHowever, comparing the k-epsilon model to the SST model reveals a large discrepancy between the two. As could already be seen in the visualizations above, the k-epsilon model shows a very unexpected boundary layer shape. Although the boundary layer height \\(\\delta_{95}\\) (at \\(95\\)% of the freestream velocity) doesn\u0026rsquo;t seem that far off the results of the SST model, the height prediction of other velocities are largely different.\nOne of the hypotheses for these seemingly faulty results of the k-epsilon model is that the low Reynolds number formulation of this model was used. This means that compared to the standard k-epsilon model, empirical damping functions (\\(f_1\\), \\(f_2\\), \\(f{\\mu}\\)) are applied to the coefficients \\(C_{1\\epsilon}\\) and \\(C_{2\\epsilon}\\) $$ \\begin{align*} \\frac{\\partial \\rho k}{\\partial t} + \\frac{\\partial \\rho k u_i}{\\partial x_i} \u0026amp;= \\frac{\\partial}{\\partial x_j} \\left( \\frac{\\mu_t}{\\sigma_k} \\frac{\\partial k}{\\partial x_j} \\right) + P_k - \\rho \\epsilon \\\\ \\frac{\\partial \\rho \\epsilon}{\\partial t} + \\frac{\\partial \\rho \\epsilon u_i}{\\partial x_i} \u0026amp;= \\frac{\\partial}{\\partial x_j} \\left( \\frac{\\mu_t}{\\sigma_{\\epsilon}} \\frac{\\partial \\epsilon}{\\partial x_j} \\right) + f_1 C_{1\\epsilon} \\frac{\\epsilon}{k} P_k - f_2 C_{2\\epsilon} \\frac{\\rho \\epsilon^2}{k} \\end{align*} $$ and the eddy viscosity \\(\\mu_t\\) also adjusted with $$ \\mu_t = f_{\\mu} C_{\\mu} \\frac{k^2}{\\epsilon} $$ However, the general guideline for this low Reynolds number k-epsilon is that it should only be used for \\(y^+\\) values below 30, with \\(y^+\\) known as the dimensionless wall distance. This value can be interpreted as a local Reynolds number for near wall turbulence and states that velocity distributions in the near-wall region are very similar for almost all turbulent flows. This is also known as the universal law of the wall [1]. For \\(y^+\\) values below 30, the mesh inflation layer on wall boundaries also plays a large role for accurate simulation results. Unfortunately, these parameters were not considered when setting up this simulation and the low Reynolds number k-epsilon model was chosen, just as it was used for the \\(Re = 10000\\) inclined plate simulation. Also, as mentioned above, the same mesh settings were used for both of these simulations.\nOne further metric that can be compared between the different turbulence model results (although the k-epsilon model shouldn\u0026rsquo;t necessarily be considered because of the outlying results) is the lift coefficient produced by the inclined plate. For this, a report monitor was set up in StarCCM+ and the data was exported at each timestep. To properly visualize the data, the following MATLAB script was used, with the resulting plot shown below.\nkepsilon = readtable(\u0026#34;Cl data\\HighReInclinedPlate_Cl_data_kepsilon.csv\u0026#34;); iters1 = kepsilon.Iteration; t1 = iters1/(20000/15); cl_kepsilon = kepsilon.ClMonitor_ClMonitor; komega = readtable(\u0026#34;Cl data\\HighReInclinedPlate_Cl_data_komega.csv\u0026#34;); iters2 = komega.Iteration; t2 = iters2/(20000/15); cl_komega = komega.ClMonitor_ClMonitor; sst = readtable(\u0026#39;Cl data\\HighReInclinedPlate_Cl_data_SST.csv\u0026#39;); iters3 = sst.Iteration; t3 = iters3/(20000/15); cl_sst = sst.ClMonitor_ClMonitor; figure(1) hold on plot(t1,cl_kepsilon) plot(t2,cl_komega) plot(t3,cl_sst) hold off xlim([0 15]) ylim([0.15 0.3]) grid on grid minor title(\u0026#39;Lift coefficient using different turbulence models\u0026#39;) xlabel(\u0026#39;Physical time [s]\u0026#39;) ylabel(\u0026#39;Lift coefficient [-]\u0026#39;) legend(\u0026#39;k-epsilon\u0026#39;,\u0026#39;k-omega\u0026#39;,\u0026#39;SST\u0026#39;) Lift coefficient over time for three different turbulence models As can be seen from the graph, the k-epsilon results can definitely be considered to be outliers. However, it is interesting to note the slight difference of about \\(3.5\\)% between the steady-sate lift coefficient of the SST model and that of the k-omega model. This graph also shows that the SST model and k-omega model show an almost identical time at which steady-state is reached.\n","date":"18 June 2023","permalink":"/chapters/03-separation/fig36/","section":"Chapters","summary":"ExperimentSimulation Air bubbles in water show the turbulent flow field and turbulent reattachment on an inclined plate at Reynolds number \\(Re = 50000\\). At 2.5\\(^{\\circ}\\) inclination relative to the oncoming flow, the flow separates from the upper surface, creating a turbulent boundary layer at the leading edge before reattaching to the inclined plate.","title":"Fig 36. Leading-edge separation on a plate with turbulent reattachment"},{"content":" ExperimentSimulation Air bubbles in water show the turbulent flow field around an inclined plate at Reynolds number \\(Re = 10000\\). At 20\\(^{\\circ}\\) inclination relative to the oncoming flow, the flow fully separates from the entire upper surface of the plate and creates a turbulent wake. The turbulent boundary layer is of highly chaotic and unsteady nature. For comparison, a transient simulation was created in Simcenter StarCCM+ using the SST turbulence model. Photograph by ONERA photograph, Werlé 1974\nTheory\u003e Theory # Turbulent flow is characterized by its chaotic and irregular behaviour. Although turbulent flow is often encountered and widely relevant to engineers, accurately modelling and predicting it has been the subject of intense scientific research over the past few decades. In contrast to the predictable and easy to model laminar flow, turbulent flow requires complex models to solve the million dollar Navier-Stokes equations. $$ \\rho (\\frac{\\partial\\textbf{u}}{\\partial t} + \\textbf{u}\\cdot \\nabla \\textbf{u}) = -\\nabla p + \\nabla \\cdot (\\mu (\\nabla \\textbf{u} + (\\nabla \\textbf{u}^T))-\\frac{2}{3}\\mu(\\nabla \\cdot \\textbf{u})\\textbf{I}) + \\textbf{F} $$ Numerically solving the Navier-Stokes equations is extremely computationally expensive because of the largely different mixing-length scales present in turbulent flow. From modelling planet sized meteorological effects such as rotating tropical cyclones to modelling microscopic effects of vortex energy dissipation due to viscous losses, the Navier-Stokes equations need to be adjusted for numerical computations.\nMore in-depth information on modelling turbulence using the so-called Reynolds-averaged Navier-Stokes equations (RANS) can be found on this page. Details about deriving different types of turbulence models, such as k-epsilon, k-omega and SST, is also explained there. As was done in the posts for Fig. 35, Fig. 36 and Fig. 47, this post serves as a comparison and evaluation of different turbulence models for different simulations setups of the equivalent experiment setup from the original figures in the book.\nSimulation\u003e Simulation # All of the simulations of the inclined plate experiment were carried out in Simcenter StarCCM+ using the k-epsilon, k-omega and SST turbulence models. The differences in results of each of the turbulence models will be discussed in the \u0026lsquo;Visualization\u0026rsquo; section of this post, since the differences become apparent during post-processing.\nComputational domain\u003e Computational domain # Considering the information about the experiment setup given in the caption of Figure 37 and the experiment being very similar to the two previous pictures, the computational domain, boundary conditions and finite volume mesh can be set up. The visualization of the flow field in this experiment is done with air bubbles in water. Knowing that the characteristic Reynolds number is \\(Re = 10000\\) based on the length of the plate, the length can be calculated to be \\(0.1\\ m\\) and the inlet velocity \\(0.1\\ m/s\\). As for the two previous posts, this experiment closely resembles that of flow over an airfoil, with the angle of attack increased to \\(20^{\\circ}\\) for this experiment. The plate geometry was kept the same, with beveled edges and \\(2\\)% thickness.\nComputational domain, boundary conditions and mesh. The domain size is given in the figure above. The base size of the quadrilateral FVM mesh is set to \\(0.001\\ m.\\) To save on computational expenses, this base size is only applied in the region near the inclined plate. The custom mesh controls available in StarCCM+ are used to coarsen the mesh in the far-field parts of the domain. The domain walls have an element size of \\(0.3\\ m\\) (\\(30000\\)% of base size) and the global growth rate is set to \\(1.1\\). Further refinement is added on both the leading and trailing edge of the plate, where separation is expected to occur. In this area, the mesh is refined to \\(25\\)% of the base size. The turbulent separation region shown in the original figure also has \\(50\\)% refinement applied to it, while the wall boundary and near wall region is refined with an inflation layer. This layer consists of \\(24\\) prism layers with a stretch factor of \\(1.2\\) and a total thickness of \\(0.002\\ m\\) (\\(200\\)% of base size). The total cell count is \\(75077\\) and some figures of this mesh and the refinement region can be seen below.\nPrevious Next StarCCM+ setup\u003e StarCCM+ setup # The simulation setup in StarCCM+ is generally the same as for the two other inclined plate simulations and are already referenced above. The simulation is run a total of three times, using the low Reynolds number k-epsilon model, one using Wilcox\u0026rsquo;s (2008) k-omega model and a last one using Menter\u0026rsquo;s SST model. These simulations are run in a transient environment in order to capture the instantaneous flow field and turbulence.\nVisualization\u003e Visualization # The visualization setup for Paraview is very similar to the one mentioned in the post for Fig. 47 about flow around a circular cylinder and more details can be found there. Since a highly chaotic and turbulent flow field is expected, random particle injection and particle tracing is used to visualize the velocity field of the simulation results. The ParaView pipeline used for this visualization can be seen below.\nParaView pipeline Turbulence model comparison\u003e Turbulence model comparison # Accurately comparing the simulation results of the different turbulence models to the actual experiment and between each other is hard to do because of the highly chaotic and turbulent nature of the instantaneous flow field captured in the original picture. Having considered this, the SST turbulence model seems to give the most accurate representation of the flow field, at least in terms or correctly predicting the relevant length scales. This comparison can be seen in the slider figure at the very top of the page. As for the visualization results provided by the k-epsilon and k-omega turbulence models, these are shown below and can be compared using the slider.\nk-omega modelk-epsilon model SST modelk-omega model From the comparisons above, the expected behaviours of the different turbulence models are observed. As explained in the posts about flow over a low angle inclined plate ( Fig. 35 and Fig. 36) and considering the turbulence model derivations explained in the post for Fig. 47, the k-epsilon shows a clear lack of near-wall turbulence prediction. On the other hand, the k-omega predicts a higher amount of eddies forming both on the top side of the plate as well as in the wake of the plate. Comparing this to the results of the SST simulation and the real experiment picture, these do not seem to show any vorticity in the wake of the plate. Although the SST model predictions best match those of the actual experiment picture, the small scale turbulence and chaos is not really captured. The SST model only seems to depict large scale eddies. Although the k-omega model predicts a larger amount of largely differing eddy sizes, there is a still clear presence of recurring large scale eddies, as can be seen halfway down the length of the plate and in the wake of the inclined plate.\nIn all three cases, this simulation setup starts showing the general limitations of Reynolds-averaged Navier-Stokes (RANS) models. As the name suggests, these models are based on the average velocities of the flow field. More information about the derivation of the RANS equations can be found in the theory section of this post. Because RANS models aim to model turbulent phenomena on the mean velocity scale, they struggle to show small scale chaos and turbulence that would be expected in any instantaneous flow field. Also, because of their average velocity nature, RANS models are better suited for steady state simulations of high Reynolds number flows, where small fluctuations in the velocity field are not of high importance.\nTo better understand the limitation of RANS models for the simulations shown above, the figure below gives a representation of where RANS models are situated on the eddy scale spectrum compared to other types of numerical simulation approaches.\nModeled and resolved eddy size spectrum This figure mentions DNS, which stands for Direct Numerical Simulation. This is in fact not a turbulence model, rather a direct simulation of the Navier-Stokes equations at the entire temporal and spatial turbulence scales. While resolving the Navier-Stokes equations numerically still requires a computational grid, it is extremely fine and makes this method massively computationally expensive, which is simply not a viable simulation method for most cases. DNS is reserved for very small computational domains and trivial flow situations, but is accurate enough to serve as validation data for CFD models.\nNext, this figure mentions LES models, which stands for Large Eddy simulation. These models also fully resolve the Navier-Stokes equations, however only down to a certain scale. Below this scale, a type of low-pass filter is applied to the Navier-Stokes equations, which spatially and temporally averages them. This greatly saves on computation costs while still providing highly accurate results.\nLES models still have very high computation costs compared to RANS models, which model the entire eddy size spectrum. The low computational costs are what make RANS models so popular. However, there is one more popular type of turbulence model, namely DES models. DES stands for Detached Eddy Simulation. These models can be considered hybrids between RANS models ans LES models.\nThe figure below ( link ) shows the difference in instantaneous flow field results between the above mentioned types of numerical simulation approaches.\nInstantaneous velocity field of different numerical simulation approaches ","date":"30 June 2023","permalink":"/chapters/03-separation/fig37/","section":"Chapters","summary":"ExperimentSimulation Air bubbles in water show the turbulent flow field around an inclined plate at Reynolds number \\(Re = 10000\\). At 20\\(^{\\circ}\\) inclination relative to the oncoming flow, the flow fully separates from the entire upper surface of the plate and creates a turbulent wake.","title":"Fig 37. Global separation over an inclined plate"},{"content":" ExperimentSimulation \u0026ldquo;Here, in contrast to figure 24, the flow has clearly separated to form a pair of recirculating eddies. The cylinder is moving through a tank of water containing aluminum powder, and is illuminated by a sheet of light below the tree surface. Extrapolation of such experiments to unbounded flow suggests separation at R=4 or 5, whereas most numerical computations give R=5 to 7.\u0026rdquo; Photograph by Sadathoshi Taneda\nGeneral Info\u003e General Info # This post is part of a series on flow separation, studied for the case of flow past a circular cylinder at different Reynolds numbers. The current figure is the first figure of this series displaying flow seperation, although the onset of separation should occur for a Reynolds number around 5.\nThe main theory and simulation and visualization set-up are discussed in the web post from Figure 42. The full series is:\nFigure 24: Circular cylinder at R=1.54. Figure 40: Circular cylinder at R=9.6. Figure 41: Circular cylinder at R=13.1. Figure 42: Circular cylinder at R=26. Figure 45: Circular cylinder at R=28.4. Figure 46: Circular cylinder at R=41. Figure 96: Kármán vortex street behind a circular cylinder at R=105. Figure 94: Kármán vortex street behind a circular cylinder at R=140. An overview of these posts can be viewed here:\n","date":"18 August 2023","permalink":"/chapters/03-separation/fig40/","section":"Chapters","summary":"ExperimentSimulation \u0026ldquo;Here, in contrast to figure 24, the flow has clearly separated to form a pair of recirculating eddies. The cylinder is moving through a tank of water containing aluminum powder, and is illuminated by a sheet of light below the tree surface.","title":"Fig 40. Circular cylinder at R=9.6"},{"content":" ExperimentSimulation \u0026ldquo;The standing eddies become elongated in the flow direction as the speed increases. Their length is found to increase linearly with Reynolds number until the flow becomes unstable above R=40.\u0026rdquo; Photograph by Sadathoshi Taneda\nGeneral Info\u003e General Info # This post is part of a series on flow separation, studied for the case of flow past a circular cylinder at different Reynolds numbers. In the current figure, it is clearly visible that the vortices behind the cylinder, which appear due to separation, grew in comparison to Figure 40. Also, one can see that the flow separates at an earlier point on the cylinder.\nThe main theory and simulation and visualization set-up are discussed in the web post from Figure 42. The full series is:\nFigure 24: Circular cylinder at R=1.54. Figure 40: Circular cylinder at R=9.6. Figure 41: Circular cylinder at R=13.1. Figure 42: Circular cylinder at R=26. Figure 45: Circular cylinder at R=28.4. Figure 46: Circular cylinder at R=41. Figure 96: Kármán vortex street behind a circular cylinder at R=105. Figure 94: Kármán vortex street behind a circular cylinder at R=140. An overview of these posts can be viewed here:\n","date":"18 August 2023","permalink":"/chapters/03-separation/fig41/","section":"Chapters","summary":"ExperimentSimulation \u0026ldquo;The standing eddies become elongated in the flow direction as the speed increases. Their length is found to increase linearly with Reynolds number until the flow becomes unstable above R=40.","title":"Fig 41. Circular cylinder at R=13.1"},{"content":" ExperimentSimulation \u0026ldquo;The downstream distance to the cores of the eddies also increases linearly with Reynolds number. However, the lateral distance between the cores appears to grow more nearly as the square root.\u0026rdquo; Photograph by Sadathoshi Taneda\nTheory\u003e Theory # This post is part of a series on flow separation, studied for the case of flow past a circular cylinder at different Reynolds numbers. The full series is:\nFigure 24: Circular cylinder at R=1.54. Figure 40: Circular cylinder at R=9.6. Figure 41: Circular cylinder at R=13.1. Figure 42: Circular cylinder at R=26. Figure 45: Circular cylinder at R=28.4. Figure 46: Circular cylinder at R=41. Figure 96: Kármán vortex street behind a circular cylinder at R=105. Figure 94: Kármán vortex street behind a circular cylinder at R=140. An overview of these posts can be viewed here:\nReynolds number dependency\u003e Reynolds number dependency # Given its elementary nature, the flow past a circular cylinder is a well-studied case in the field of fluid dynamics. The characteristics of the flow pattern changes with the Reynolds number, resulting in wildly different flow patterns. Representative illustrations of all types of flow behavior are collected in the figure below. Examples of every such flow pattern can be found in van Dyke\u0026rsquo;s book and on this website. Take for example Figure 24 for attached flow (subfigure 1), Figures 40, 41, 42, 45 and 46 for separated flow with different sizes of separation regions (subfigure 2), Figures 94 and 96 for laminar flow with vortex shedding; von Kármán streets (subfigure 3), Figure 47 for flow shedding turbulent vortices (subfigure 4), and Figure 147 for a fully developed turbulent wake (subfigure 5).\nThe change in flow behavior when increasing the Reynolds number (Banerjee and Galtier, 2014) To understand why the nature of the flow changes so drastically with the change in Reynolds number, we must take a look at the governing equations; the Navier-Stokes equations. For an incompressible fluid and with gravitational effects neglected, these read:\n$$ \\begin{cases} \\rho \\frac{\\partial \\mathbf{u}}{\\partial t} +\\rho (\\mathbf{u} \\cdot \\nabla)\\mathbf{u} = \\mu \\nabla^2 \\mathbf{u} - \\nabla p \\\\ \\nabla\\cdot \\mathbf{u} = 0 \\end{cases} $$\nA derivation of these equations can be found here. The incompressible Navier-Stokes equation are a set of coupled differential equations that describe the conservation of mass and momentum of a viscous fluid. The dependent fields (i.e., the sought after solution functions) are the vector-valued velocity field \\(\\mathbf{u}\\) and the scalar-valued pressure field \\( p\\). General forms of the conservation of momentum and mass in partial differential form are: $$ \\begin{cases} \\frac{\\partial}{\\partial t}(\\rho \\mathbf{u} ) + \\nabla \\cdot ( \\rho \\mathbf{u} \\otimes \\mathbf{u} ) = \\nabla \\cdot \\sigma + \\rho\\mathbf{f} \\\\ \\frac{\\partial}{\\partial t}\\rho + \\nabla \\cdot ( \\rho \\mathbf{u} ) = 0 \\end{cases} $$ with \\( \\rho \\) the given material density, \\(\\mathbf{f}\\) the specified body acceleration, and \\(\\mathbf{\\sigma}\\) the Cauchy stress tensor.\nThe left-hand-side of the momentum equation can be simplified by expanding the differentiation terms with appropriate chain-rules and by subsequently substituting the mass conservation equation: $$ \\frac{\\partial}{\\partial t}(\\rho \\mathbf{u} ) + \\nabla \\cdot ( \\rho \\mathbf{u} \\otimes \\mathbf{u} ) = \\mathbf{u} \\frac{\\partial}{\\partial t}\\rho + \\rho \\frac{\\partial}{\\partial t} \\mathbf{u} + \\mathbf{u} \\nabla \\cdot ( \\rho \\mathbf{u} ) + (\\rho \\mathbf{u}\\cdot \\nabla )\\mathbf{u} \\\\ = \\rho \\Big( \\frac{\\partial}{\\partial t}\\mathbf{u} + (\\mathbf{u}\\cdot \\nabla )\\mathbf{u} \\Big) \\qquad $$\nFor an incompressible medium, the density of each moving fluid element remains constant in time. Mathematically, this means that the material derivative of the density is zero: \\( \\frac{D}{D t}\\rho = \\frac{\\partial}{\\partial t}\\rho + \\nabla \\rho \\cdot \\mathbf{u} = 0 \\). This assumption may be used in the earlier mass conservation law to find: $$ \\frac{\\partial}{\\partial t}\\rho + \\nabla \\cdot ( \\rho \\mathbf{u} ) = \\frac{\\partial}{\\partial t}\\rho + \\nabla \\rho \\cdot \\mathbf{u} + \\rho \\nabla \\cdot \\mathbf{u} = \\rho \\nabla \\cdot \\mathbf{u} = 0 $$ Or, in short, \\( \\nabla\\cdot\\mathbf{u} = 0 \\).\nLastly, a constitutive relation must be substituted in place of the Cauchy stress tensor. First we separate it into its isotropic (\\( - p \\mathbf{I} \\)) and deviatoric (\\( \\mathbf{\\tau}\\)) components: $$ \\mathbf{\\sigma} = \\mathbf{\\tau} - p \\mathbf{I} $$\nThe incompressible Navier-Stokes equations in their familiar form are then the result of the linear Stokes constitutive relation for the deviatoric stress: $$ \\mathbf{\\tau} = \\mu \\Big(\\nabla \\mathbf{u} + (\\nabla\\mathbf{u})^T - \\frac{2}{3}( \\nabla\\cdot \\mathbf{u})\\mathbf{I} \\Big) $$ where the last term ensures that \\( \\mathbf{\\tau}\\) is deviatoric, but in the case of an incompressible medium it is simply zero. When it may be assumed that the dynamic viscosity \\( \\mu \\) is constant in space, the following holds: $$ \\nabla \\cdot \\mathbf{\\tau} = \\mu \\nabla \\cdot \\big(\\nabla \\mathbf{u} + (\\nabla \\mathbf{u})^T \\big) = \\mu \\nabla \\cdot \\nabla \\mathbf{u} + \\mu \\nabla (\\nabla \\cdot \\mathbf{u}) = \\mu \\nabla \\cdot \\nabla \\mathbf{u} = \\mu \\nabla^2 \\mathbf{u} $$ where, again, the divergence-free nature of the velocity field is made use of.\nCombining all the above equations results in the incompressible Navier-Stokes equations: $$ \\begin{cases} \\rho \\frac{\\partial \\mathbf{u}}{\\partial t} +\\rho (\\mathbf{u} \\cdot \\nabla)\\mathbf{u} = \\mu \\nabla^2 \\mathbf{u} - \\nabla p + \\rho\\mathbf{f} \\\\ \\nabla\\cdot \\mathbf{u} = 0 \\end{cases} $$\nThe first equation represents (in vector form) conservation of linear momentum, and the second equation conservation of mass. While it may seem that there are a multitude of physical parameters that define the solution field (density, viscosity, length and time-scales), there is in effect only a single one: the Reynolds number. This becomes apparent after non-dimensionalizing all quantities in terms of the material parameters (density and viscosity), a characteristic length and a characteristic velocity: \\( \\hat{L} \\) and \\( \\hat{U} \\). Considering each term separately:\nVelocity: \\( \\mathbf{u} = \\hat{U}\\, \\hat{\\mathbf{u}} \\) Pressure: \\( p = \\rho \\hat{U}^2 \\, \\hat{p} \\) Temporal coordinate and time-derivative: \\( t = \\frac{\\hat{L}}{\\hat{U}}\\, \\hat{t} \\) so that \\( \\frac{\\partial}{\\partial t} = \\frac{\\hat{U}}{\\hat{L}} \\frac{\\partial}{\\partial \\hat{t}} \\) Spatial coordinate and gradient: \\( \\mathbf{x} = \\hat{L}\\, \\hat{\\mathbf{x}} \\) so that \\( \\nabla = \\frac{1}{\\hat{L}}\\, \\hat{\\nabla} \\) Substitution of these non-dimensionalizations into the earlier Navier-Stokes equations gives the non-dimensionalized Navier-Stokes equations:\n$$ \\begin{cases} \\frac{\\partial \\hat{\\mathbf{u}}}{\\partial \\hat{t}} + (\\hat{\\mathbf{u}} \\cdot \\hat{\\nabla})\\hat{\\mathbf{u}} = \\frac{\\mu}{\\rho \\hat{U}\\hat{L} } \\hat{\\nabla}^2 \\hat{\\mathbf{u}} - \\hat{\\nabla} \\hat{p} \\\\ \\nabla\\cdot \\hat{\\mathbf{u}} = 0 \\end{cases} $$\nThese equations illustrate that (for a domain, initial and boundary conditions fixed in terms of the characteristic quantities) it is really only the Reynolds number \\( Re = \\frac{\\rho \\hat{U}\\hat{L} }{\\mu} \\) that defines the solution.\nFor a small Reynolds number, the viscous term dominates, effectively linearizing the equations (eventually reducing them to the Stokes equations), and for large Reynolds number the viscous term becomes negligible whereby the non-linear inertial term dominates. The transition from small to large Reynolds number thus fundamentally changes the nature of the equations, causing the radical changes of the behavior of the flow illustrated in the earlier figure.\nSeparation\u003e Separation # At a sufficiently small Reynolds numbers, in literature typically stated as between 4 and 5, the fluid follows the curvature of the cylinder without separating from its surface. This can be seen in the figure below, which considers the near-limit case of \\(Re=4\\).\nFlow around a circular cylinder at \\(Re=4\\), just before the onset of separation Past this Reynolds number, separation at the rear of the cylinder begins to take place. Then, the boundary layer detaches from the cylinder surface. At the location of the detachment, the \u0026ldquo;separation point\u0026rdquo;, the flow velocity is zero, and past that point the fluid particles have reversed their direction. This flow reversal starts to occurs when the negative pressure gradient required to curve the flow around the surface of the cylinder, i.e., the pattern shown in the above figure at \\(Re = 4\\), becomes too large. Later downstream, the separated flow will reattach at the \u0026ldquo;reattachment point\u0026rdquo;, resulting in pockets of circulating fluid (vortices) that are trapped by the surrounding flow. After a Reynolds number of \\(Re = 46\\) these vortices begin to shed, creating a von Kármán vortex street (see Figure 96 for more detail).\nFrom \\(Re = 4\\) to \\(Re = 46\\) the trapped vortex, or separation bubble, grows in length and width. The streamwise length of the separation bubble scales linearly with Reynolds number. The Reynolds number dependency of the location of the separation point, characterized by the \u0026ldquo;separation angle\u0026rdquo; as the angle along the cylinder surface, is somewhat more elusive. It has been studied extensively, both experimentally and numerically, as can be seen in the following figure:\nRelationship Re and the separation angle (Jiang 2020) Fitting the data provides an empirical relationship between the separation angle and Reynolds number, as described by the following formula: $$ \\theta_S = 95.7 + 267.1Re^{-\\frac{1}{2}} - 625.9Re^{-1} +1046.6Re^{\\frac{-3}{2}} $$ The current figure, Figure 42, involves a Reynolds number of \\(26\\). This formula then predicts a separation angle equal to \\(132\\) degrees, corresponding well with the experiment and the simulation.\nSimulation\u003e Simulation # The CFD simulation program used to perform the above simulations is Simcenter StarCCM+. The two-dimensional nature of the flow permits the use of 2D flow models to decrease the computational time.\nBoundary conditions and computational domain\u003e Boundary conditions and computational domain # With the Reynolds number (\\(Re=26\\)) given in the caption of the original figure the entire flow field is determined in terms of non-dimensionalized quantities. StarCCM+ requires dimensionalized quantities, however. The caption also indicates that the photograph is made by Sadathoshi Taneda, who provided multiple pictures from the original book. In other pictures, Mr. Taneda uses a cylinder with a diameter of \\(1 \\)cm, and it seems to be the case that he uses the same experimental setup for all photographs. With the length dimension and the material parameters determined, the velocity can be calculated: \\(Re = \\frac{\\rho u D}{\\mu}\\), or \\( u = \\frac{\\mu \\, Re}{\\rho \\, D} = 2.6 \\)mm/s. Based on these quantities, the computational domain and boundary conditions are defined, as illustrated in the figure below.\nComputational domain The total domain has a length of \\(5 \\)cm and the distance from the centre of the cylinder to the end of the domain is \\(3 \\)cm, leaving \\(2 \\)cm in front of the circle centre. The height of the domain is \\(3 \\)cm, with the centre of the cylinder at \\(1.5 \\)cm. The total domain of the Figure was not given in the caption but derived from the original figure. To avoid too disruptive wall-influences from the top and bottom wall, these are set as symmetry planes. The cylinder is set as a wall with a no-slip boundary condition applied to it. The inflow is set as a velocity inlet with a velocity of \\(2.6\\)mm/s in the normal direction and the outlet is set as a no-traction outlet.\nMeshing\u003e Meshing # The mesh created for this simulation is based on a quadrilateral 2D mesh, which is one of many meshing options from STARCCM+. After selecting the quadrilateral 2D mesh, the polygonal mesh option is used. This mesh consists of polyhedral-shaped cells. In comparison to an equivalent tetrahedral mesh, a polygonal mesh contains five times fewer cells and is documented to be more accurate, more stable and less diffusive. The base size of the mesh is set to \\(0.02 \\)m and the growth rate to 1.004. As the near-boundary behavior of the flow is of crucial importance for accurate prediction of flow separation, a highly refined prism boundary layer mesh is created. A total amount of 34386 cells were created. Some representations of the mesh can be found below.\nPrevious Next Fluid model\u003e Fluid model # The simulation was run with a laminar, steady model with water as the flowing fluid. For the simulation, a total of 3500 iterations were simulated, of which the velocity data was exported to an Ensight Gold case file, so it could be processed in Paraview. All models used in STARCCM+ and the STARCCM+ result for Figure 40 after 3500 iterations are shown below.\nPrevious Next Visualization\u003e Visualization # Paraview is used for the post-processing and visualization of the simulation. After importing the velocity data, a calculator filter is used to compute the velocity vector from the exported x-velocity and y-velocity data. Next, the circumference of the cylinder is covered in particles with the code supplied at the bottom of this page. In combination with a TableToPoints filter, this is used as an input seed for the Particletracer filter. The Particletracer filter together with the Streamtracer filter is used to produce the final visualization shown at the start of this web post. The complete pipeline looks as follows:\nParaview pipeline And the code used for creating the particles at the circumference of the cylinder can be found below.\nn = 720; angles = linspace(0, 2*pi, n); radius = 0.00501; xCenter = 0; yCenter = 0; X = -radius * cos(angles) + xCenter; Y = -radius * sin(angles) + yCenter; P = zeros(n+1,3); P(1,:) = [1,2,3]; P(2:end,1) = X; P(2:end,2) = Y; writematrix(P,\u0026#39;circumference.csv\u0026#39;) ","date":"18 August 2023","permalink":"/chapters/03-separation/fig42/","section":"Chapters","summary":"ExperimentSimulation \u0026ldquo;The downstream distance to the cores of the eddies also increases linearly with Reynolds number. However, the lateral distance between the cores appears to grow more nearly as the square root.","title":"Fig 42. Circular cylinder at R=26"},{"content":" ExperimentSimulation \u0026ldquo;Here just the boundary of the recirculating region has been made visible by coating the cylinder with condensed milk and setting it in motion through water.\u0026rdquo; Photograph by Sadathoshi Taneda\nGeneral Info\u003e General Info # This post is part of a series on flow separation, studied for the case of flow past a circular cylinder at different Reynolds numbers. In the current figure, it is clearly visible that the vortices behind the cylinder, which appear due to separation, grew in comparison to earlier figures in this series. Also, one can see that the flow separates at an earlier point on the cylinder.\nThe main theory and simulation and visualization set-up are discussed in the web post from Figure 42. The full series is:\nFigure 24: Circular cylinder at R=1.54. Figure 40: Circular cylinder at R=9.6. Figure 41: Circular cylinder at R=13.1. Figure 42: Circular cylinder at R=26. Figure 45: Circular cylinder at R=28.4. Figure 46: Circular cylinder at R=41. Figure 96: Kármán vortex street behind a circular cylinder at R=105. Figure 94: Kármán vortex street behind a circular cylinder at R=140. An overview of these posts can be viewed here:\n","date":"18 August 2023","permalink":"/chapters/03-separation/fig45/","section":"Chapters","summary":"ExperimentSimulation \u0026ldquo;Here just the boundary of the recirculating region has been made visible by coating the cylinder with condensed milk and setting it in motion through water.\u0026rdquo; Photograph by Sadathoshi Taneda","title":"Fig 45. Circular cylinder at R=28.4"},{"content":" ExperimentSimulation \u0026ldquo;This is the approximate upper limit for steady flow. Far downstream the wake has already begun to oscillate sinusoidally. Tiny irregular gathers are appearing on the boundary of the recirculating region, but dying out as they reach its downstream end.\u0026rdquo; Photograph by Sadathoshi Taneda\nGeneral Info\u003e General Info # This post is part of a series on flow separation, studied for the case of flow past a circular cylinder at different Reynolds numbers. The current figure, roughly represents the upper limit for steady flow. When increasing the Reynolds number, the flow will start to oscillate and develop into a periodic state. More information on such a \u0026ldquo;von Kármán vortex street\u0026rdquo; can be found in the web post from Figure 94.\nThe main theory and simulation and visualization set-up are discussed in the web post from Figure 42. The full series is:\nFigure 24: Circular cylinder at R=1.54. Figure 40: Circular cylinder at R=9.6. Figure 41: Circular cylinder at R=13.1. Figure 42: Circular cylinder at R=26. Figure 45: Circular cylinder at R=28.4. Figure 46: Circular cylinder at R=41. Figure 96: Kármán vortex street behind a circular cylinder at R=105. Figure 94: Kármán vortex street behind a circular cylinder at R=140. An overview of these posts can be viewed here:\n","date":"18 August 2023","permalink":"/chapters/03-separation/fig46/","section":"Chapters","summary":"ExperimentSimulation \u0026ldquo;This is the approximate upper limit for steady flow. Far downstream the wake has already begun to oscillate sinusoidally. Tiny irregular gathers are appearing on the boundary of the recirculating region, but dying out as they reach its downstream end.","title":"Fig 46. Circular cylinder at R=41"},{"content":" ExperimentSimulation \u0026ldquo;Air bubbles in water show the velocity field of a flow around a circular cylinder at Reynolds number \\(Re = 2000\\). At this Reynolds number, there is a clear boundary layer separation followed by an oscillating turbulent wake. For comparison, a transient simulation was created in Simcenter StarCCM+ using the SST turbulence model.\u0026rdquo; Photograph by ONERA photograph, Werlé \u0026amp; Gallon 1972\nTheory\u003e Theory # Turbulent flow\u003e Turbulent flow # Turbulent flow is characterized by its chaotic and irregular behaviour. Although turbulent flow is often encountered and widely relevant to engineers, accurately modelling and predicting it has been the subject of intense scientific research over the past few decades. In contrast to the predictable and easy to model laminar flow, turbulent flow requires complex models to solve the million dollar Navier-Stokes equations. $$ \\rho (\\frac{\\partial\\textbf{u}}{\\partial t} + \\textbf{u}\\cdot \\nabla \\textbf{u}) = -\\nabla p + \\nabla \\cdot (\\mu (\\nabla \\textbf{u} + (\\nabla \\textbf{u}^T))-\\frac{2}{3}\\mu(\\nabla \\cdot \\textbf{u})\\textbf{I}) + \\textbf{F} $$ Numerically solving the Navier-Stokes equations is extremely computationally expensive because of the largely different mixing-length scales present in turbulent flow. From modelling planet sized meteorological effects such as rotating tropical cyclones to modelling microscopic effects of vortex energy dissipation due to viscous losses, the Navier-Stokes equations need to be adjusted for numerical computations.\nReynolds-averaged-Navier-Stokes\u003e Reynolds-averaged-Navier-Stokes # The Reynolds-averaged Navier-Stokes equations (RANS) choose to model turbulence effects on the mean flow scale. This is done with the so-called Reynolds decomposition, which decomposes a quantity \\(u(x,y,z,t)\\) into its mean and fluctuating part. $$ u(x,y,z,t) = \\overline{u(x,y,z)} + u\u0026rsquo;(x,y,z,t) $$ Deriving the RANS equations for stationary, incompressible fluid flow starts with the standard Navier-Stokes equations for stationary incompressible fluid flow. Using the Einstein summation convention and substituting the dynamic viscosity \\(\\mu\\) with the kinematic viscosity \\(\\nu = \\mu / \\rho\\), these Navier-Stokes equations can be written as $$ \\frac{\\partial u_i}{\\partial t} + u_j \\frac{\\partial u_i}{\\partial x_j} = -\\frac{1}{\\rho} \\frac{\\partial p}{\\partial x_i} + \\nu \\frac{\\partial ^2 u_i}{\\partial x_j \\partial x_j} $$ Applying the Reynolds decomposition to the quantities \\(u\\) and \\(p\\) and time-averaging the resulting equation, one is left with the following equation. $$ \\frac{\\partial \\overline{u}_i}{\\partial t} + \\overline{u}_j \\frac{\\partial \\overline{u}_i}{x_j} + \\overline{u_j\u0026rsquo; \\frac{\\partial u_i\u0026rsquo;}{\\partial x_j}} = -\\frac{1}{\\rho} \\frac{\\partial \\overline{p}}{\\partial x_i} + \\nu \\frac{\\partial^2 \\overline{u}_i}{\\partial x_j \\partial x_j} $$ \\(\\overline{u_j\u0026rsquo; \\frac{\\partial u_i\u0026rsquo;}{\\partial x_j}}\\) is the only term that cannot be directly time-averaged. However, by applying the chain rule and the law of conservation of mass, it can be shown that $$ \\overline{\\frac{\\partial u_i\u0026rsquo;}{\\partial x_j} u_j\u0026rsquo;} = \\frac{\\partial}{\\partial x_j} \\overline{u_i\u0026rsquo; u_j\u0026rsquo;} $$ The detailed derivation can be found here. Applying the chain rule to \\(\\frac{\\partial}{\\partial x_j} \\overline{u_i\u0026rsquo; u_j\u0026rsquo;}\\) leads to $$ \\frac{\\partial}{\\partial x_j} \\overline{u_i\u0026rsquo; u_j\u0026rsquo;} = \\overline{\\frac{\\partial u_i\u0026rsquo;}{\\partial x_j} u_j\u0026rsquo;} + \\overline{\\frac{\\partial u_j\u0026rsquo;}{\\partial x_j} u_i\u0026rsquo;} $$ Yet the law of conservation of mass says that \\(\\frac{\\partial u_j}{\\partial x_j} = 0\\) and thus the term above can be simplified to $$ \\frac{\\partial}{\\partial x_j} \\overline{u_i\u0026rsquo; u_j\u0026rsquo;} = \\overline{\\frac{\\partial u_i\u0026rsquo;}{\\partial x_j} u_j\u0026rsquo;} $$ Multiplying \\(\\overline{u_i\u0026rsquo; u_j\u0026rsquo;}\\) by \\(\\rho\\) is known as the Reynolds stress term and can be substituted in the Reynolds-averaged equation above. Rearranging and factoring the right-hand side of the aforementioned equation results in the so-called Reynolds-averaged Navier–Stokes (RANS) equation. $$ \\overline{u}_j \\frac{\\partial \\overline{u}_i}{x_j} = -\\frac{1}{\\rho} \\frac{\\partial \\overline{p}}{\\partial x_i} + \\frac{\\partial}{\\partial x_j} \\left(\\mu \\frac{\\partial \\overline{u}_i}{\\partial x_j} - \\rho \\overline{u_i\u0026rsquo; u_j\u0026rsquo;} \\right) $$ The RANS equations are the basis for all RANS turbulence models including the shear stress transport (SST), k-omega and k-epsilon models. More specifically, these models are based on the Reynolds stresses \\(\\rho \\overline{u_i\u0026rsquo; u_j\u0026rsquo;}\\) from the above RANS equation.\nIn 1945, P.Y. Chou gave a very complicated closure equation for the time evolution of the Reynolds stress [1]. Tracing \\(\\overline{u_i\u0026rsquo;u_j\u0026rsquo;}\\) in the said equation would lead to the turbulent kinetic energy \\(k\\) and \\(\\nu \\overline{\\frac{\\partial u_i\u0026rsquo;}{\\partial x_k} \\frac{\\partial u_j\u0026rsquo;}{\\partial x_k}}\\) is known as the turbulent dissipation rate \\(\\epsilon\\). Most RANS models are based on the above mentioned equation or on the simpler Boussinesq eddy viscosity hypothesis expressed below [2]. $$ \\rho \\overline{u_i\u0026rsquo; u_j\u0026rsquo;} = \\frac{2}{3} \\rho k \\delta_{ij} - \\mu_t \\left( \\frac{\\partial \\overline{u_i}}{\\partial x_j} + \\frac{\\partial \\overline{u_j}}{\\partial x_i} \\right) $$\nk-epsilon turbulence model\u003e k-epsilon turbulence model # As the name suggests, the k-epsilon turbulence model is based on the turbulent kinetic energy \\(k\\) and the turbulent dissipation rate \\(\\epsilon\\). These two quantities are based on the two ends of the eddy size spectrum. \\(k\\) being the turbulent kinetic energy conserved in large, energy dense vortices and \\(\\epsilon\\) being the conversion rate from kinetic energy to thermal energy through viscous forces in microscopic eddies. The original model developed by Jones and Launder [3] prescribes the following transport equations for \\(k\\) and \\(\\epsilon\\):\n$$ \\begin{align*} \\frac{\\partial \\rho k}{\\partial t} + \\frac{\\partial \\rho k u_i}{\\partial x_i} \u0026amp;= \\frac{\\partial}{\\partial x_j} \\left( \\frac{\\mu_t}{\\sigma_k} \\frac{\\partial k}{\\partial x_j} \\right) + P_k - \\rho \\epsilon \\\\ \\frac{\\partial \\rho \\epsilon}{\\partial t} + \\frac{\\partial \\rho \\epsilon u_i}{\\partial x_i} \u0026amp;= \\frac{\\partial}{\\partial x_j} \\left( \\frac{\\mu_t}{\\sigma_{\\epsilon}} \\frac{\\partial \\epsilon}{\\partial x_j} \\right) + C_{1\\epsilon} \\frac{\\epsilon}{k} P_k - C_{2\\epsilon} \\frac{\\rho \\epsilon^2}{k} \\end{align*} $$\nwith the eddy viscosity \\(\\mu_t\\) modelled as $$ \\mu_t = \\rho C_{\\mu} \\frac{k^2}{\\epsilon} $$ and the production of \\(k\\) modelled as $$ P_k = - \\rho \\overline{u_i\u0026rsquo;u_j\u0026rsquo;} \\frac{\\partial u_j}{\\partial x_i} $$ Being able to model the eddy viscosity \\(\\mu_t\\) with \\(k\\) and \\(\\epsilon\\), provides a closure equation to the RANS equations, since \\(\\mu_t\\) is needed in the Boussinesq eddy viscosity hypothesis for the Reynolds stresses. Thus, the RANS equations can be solved.\nAs is well known from literature [4], the main drawback of the k-epsilon model is its poor prediction of energy dissipation in viscous sublayers and adverse pressure gradients. More specifically, the k-epsilon model struggles to determine energy dissipation in scenarios where viscous forces are dominant. A potential solution to this is the so-called low Reynolds number k-epsilon model (low-Re k-epsilon), which uses damping functions on the empirical coefficients \\(C_{1 \\epsilon}\\), \\(C_{2 \\epsilon}\\) and \\(C_{\\mu}\\) in near wall regions. However, these empirical damping functions are based on a simple experiment with flow over a spinning disc [5], which inherently makes this model less accurate in other situations. Unfortunately, most engineering industries encounter many different types of turbulent flows, which is where the need for a better suited model arises.\nk-omega turbulence model\u003e k-omega turbulence model # The motivation behind the development of recent k-omega models is to make up for what even the low-Re k-epsilon model lacks in near-wall and adverse pressure gradient prediction accuracy. Although the k-omega and k-epsilon models are very similar and based on the same assumptions for the RANS equations, the key difference is the need for empirical damping functions on the coefficients of the low-Re k-epsilon model and the absence of these for the k-omega model.\nThe k-omega and k-epsilon model are closely related through the following expression. $$ \\omega = \\frac{\\epsilon}{C_{\\mu}k} \\quad \\quad C_{\\mu} = 0.09 $$ While the transport equation for \\(k\\) remains practically the same for both models, plugging the above expression into the transport equation for \\(\\epsilon\\) yields the k-omega model. $$ \\begin{align*} \\frac{\\partial k}{\\partial t} + \\frac{\\partial k u_i}{\\partial x_i} \u0026amp;= \\frac{1}{\\rho} \\frac{\\partial}{\\partial x_j} \\left[ \\left( \\mu + \\frac{\\mu_t}{\\sigma_k} \\right) \\frac{\\partial k}{\\partial x_j} \\right] + P_k - \\beta^* \\omega k \\\\ \\frac{\\partial \\omega}{\\partial t} + \\frac{\\partial \\omega u_i}{\\partial x_i} \u0026amp;= \\frac{1}{\\rho} \\frac{\\partial}{\\partial x_j} \\left[ \\left( \\mu + \\frac{\\mu_t}{\\sigma_{\\omega}} \\right) \\frac{\\partial \\omega}{\\partial x_j} \\right] + \\frac{\\gamma \\omega}{k} P_k - \\beta \\omega^2 \\end{align*} $$ Although new empirical coefficients need to be computed for this model, there is no need for empirically developed damping functions for the coefficients, which makes this model more generally applicable.\nk-omega SST turbulence model\u003e k-omega SST turbulence model # The simulation that provided the data for the comparison figure at the top of the page was run using Menter\u0026rsquo;s SST turbulence model [6]. Also known as the k-omega SST model, this turbulence model combines the strengths of both the k-epsilon and k-omega model by making use of both models at the same time. Using a blending function the k-epsilon model is activated in the freestream areas of the domain and the k-omega model is activated in near-wall and adverse pressure gradient regions.\nTo start things off, the k-omega model can be left in it\u0026rsquo;s original form as developed by Wilcox, but the k-epsilon model needs to be expressed in terms of \\(\\omega\\) using \\(\\epsilon = \\omega k\\). Using the Lagrangian derivative on the left-hand side and expressing the production term \\(P_k\\) with the Reynolds stress \\(P_k = \\tau_{ij} \\frac{\\partial u_i}{\\partial x_j}\\), the transport equation of \\(\\epsilon\\) in terms of \\(\\omega\\) is expressed below. $$ \\rho \\frac{D k}{D t} = \\frac{\\partial}{\\partial x_j} \\left( \\frac{\\mu_t}{\\sigma_k} \\frac{\\partial k}{\\partial x_j} \\right) + \\tau_{ij}\\frac{\\partial u_i}{\\partial x_j} - \\rho \\omega k $$ The full derivation including a double chain rule expansion and the substitution of the transport equation for \\(k\\) can be found here. The next step in the transformed k-epsilon model derivation is to express the original transport equation for \\(\\epsilon\\) in terms of \\(\\omega\\). Applying the chain rule on the left-hand side and in the diffusion term leads to $$ \\rho \\frac{D \\omega}{D t} k + \\rho \\frac{D k}{D t} \\omega = \\frac{\\partial}{\\partial x_j}\\left( \\frac{\\mu_t}{\\sigma_{\\epsilon}} \\frac{\\partial \\omega}{\\partial x_j} k \\right) + \\frac{\\partial}{\\partial x_j}\\left( \\frac{\\mu_t}{\\sigma_{\\epsilon}} \\frac{\\partial k}{\\partial x_j} \\omega \\right) + C_1 \\omega \\tau_{ij}\\frac{\\partial u_i}{\\partial x_j} - C_2 \\rho \\omega^2 k $$ The chain rule still needs to be applied to the following terms. $$ \\begin{split} \\frac{\\partial}{\\partial x_j}\\left( \\frac{\\mu_t}{\\sigma_{\\epsilon}} \\frac{\\partial \\omega}{\\partial x_j} k \\right) + \\frac{\\partial}{\\partial x_j}\\left( \\frac{\\mu_t}{\\sigma_{\\epsilon}} \\frac{\\partial k}{\\partial x_j} \\omega \\right) \u0026amp;= \\frac{\\partial}{\\partial x_j}\\left( \\frac{\\mu_t}{\\sigma_{\\epsilon}} \\frac{\\partial \\omega}{\\partial x_j} \\right) k + \\frac{\\mu_t}{\\sigma_{\\epsilon}}\\left( \\frac{\\partial k}{\\partial x_j} \\frac{\\partial \\omega}{\\partial x_j} \\right) \\\\ \u0026amp;+\\frac{\\partial}{\\partial x_j}\\left( \\frac{\\mu_t}{\\sigma_{\\epsilon}} \\frac{\\partial k}{\\partial x_j} \\right) \\omega + \\frac{\\mu_t}{\\sigma_{\\epsilon}}\\left( \\frac{\\partial k}{\\partial x_j} \\frac{\\partial \\omega}{\\partial x_j} \\right) \\end{split} $$ Regrouping these terms before plugging them into the equation above and isolating \\(\\frac{D \\omega}{D t}\\) leads to $$ \\begin{split} \\rho \\frac{D \\omega}{D t} \u0026amp;= \\frac{\\partial}{\\partial x_j}\\left( \\frac{\\mu_t}{\\sigma_{\\epsilon}} \\frac{\\partial \\omega}{\\partial x_j} \\right) + \\frac{2}{k} \\frac{\\mu_t}{\\sigma_{\\epsilon}}\\left( \\frac{\\partial k}{\\partial x_j} \\frac{\\partial \\omega}{\\partial x_j} \\right) + \\frac{\\omega}{k} \\frac{\\partial}{\\partial x_j}\\left( \\frac{\\mu_t}{\\sigma_{\\epsilon}} \\frac{\\partial k}{\\partial x_j} \\right) + C_1 \\frac{\\omega}{k} \\tau_{ij}\\frac{\\partial u_i}{\\partial x_j} \\\\ \u0026amp;- C_2 \\rho \\omega^2 - \\rho \\frac{D k}{D t} \\frac{\\omega}{k} \\end{split} $$ The transformed transport equation for \\(k\\) can now be substituted into the equation above. $$ \\begin{split} \\rho \\frac{D \\omega}{D t} \u0026amp;= \\frac{\\partial}{\\partial x_j}\\left( \\frac{\\mu_t}{\\sigma_{\\epsilon}} \\frac{\\partial \\omega}{\\partial x_j} \\right) + \\frac{2}{k} \\frac{\\mu_t}{\\sigma_{\\epsilon}}\\left( \\frac{\\partial k}{\\partial x_j} \\frac{\\partial \\omega}{\\partial x_j} \\right) + \\frac{\\omega}{k} \\frac{\\partial}{\\partial x_j}\\left( \\frac{\\mu_t}{\\sigma_{\\epsilon}} \\frac{\\partial k}{\\partial x_j} \\right) + C_1 \\frac{\\omega}{k} \\tau_{ij}\\frac{\\partial u_i}{\\partial x_j} - C_2 \\rho \\omega^2 \\\\ \u0026amp;- \\frac{\\omega}{k} \\frac{\\partial}{\\partial x_j}\\left( \\frac{\\mu_t}{\\sigma_k} \\frac{\\partial k}{\\partial x_j} \\right) - \\frac{\\omega}{k} \\tau_{ij}\\frac{\\partial u_i}{\\partial x_j} + \\rho \\omega^2 \\end{split} $$ Regrouping these terms leads to a very similar equation that can be found in this paper [7, eq (26)]. $$ \\begin{split} \\rho \\frac{D \\omega}{D t} \u0026amp;= \\frac{\\partial}{\\partial x_j}\\left( \\frac{\\mu_t}{\\sigma_{\\epsilon}}\\frac{\\partial \\omega}{\\partial x_j} \\right) + \\frac{\\omega}{k} \\frac{\\partial}{\\partial x_j} \\left[ \\left( \\frac{1}{\\sigma_{\\epsilon}} - \\frac{1}{\\sigma_k} \\right) \\mu_t \\frac{\\partial k}{\\partial x_j} \\right] + (C_1 -1) \\frac{\\omega}{k} \\tau_{ij} \\frac{\\partial u_i}{\\partial x_j} - (C_2 -1) \\rho \\omega^2 \\\\ \u0026amp;+ \\frac{2}{k} \\frac{\\mu_t}{\\sigma_{\\epsilon}} \\left( \\frac{\\partial k}{\\partial x_j} \\frac{\\partial \\omega}{\\partial x_j} \\right) \\end{split} $$ However, for simplification purposes, it can be assumed that \\(\\sigma_{\\epsilon} = \\sigma_k\\). On top of this, it is known that $$ \\mu_t = \\rho \\frac{k}{\\omega} \\quad \\quad \\nu_t = \\frac{k}{\\omega} $$ Thus, \\(\\frac{\\omega}{k}\\) can be replaced with \\(\\frac{1}{\\nu_t}\\) and \\(\\frac{\\mu_t}{k}\\) can be replaced with \\(\\frac{\\rho}{\\omega}\\). The following equation is very similar to the one found in Pope\u0026rsquo;s book \u0026ldquo;Turbulent Flows\u0026rdquo; [8, eq (10.94)]. $$ \\rho \\frac{D \\omega}{D t} = \\frac{\\partial}{\\partial x_j}\\left( \\frac{\\mu_t}{\\sigma_{\\epsilon}} \\frac{\\partial \\omega}{\\partial x_j} \\right) + (C_1 -1) \\frac{1}{\\nu_t} \\tau_{ij}\\frac{\\partial u_i}{\\partial x_j} - (C_2 -1) \\rho \\omega^2 + \\rho \\frac{2}{\\omega \\sigma_{\\epsilon}} \\frac{\\partial k}{\\partial x_j} \\frac{\\partial \\omega}{\\partial x_j} $$ The final form of the transformed k-epsilon model as can be found in Menter\u0026rsquo;s original paper is expressed below. $$ \\rho \\frac{D \\omega}{D t} = \\frac{\\partial}{\\partial x_j} \\left[ \\left( \\mu + \\sigma_{\\omega 2} \\mu_t \\right) \\frac{\\partial \\omega}{\\partial x_j} \\right] + \\gamma_2 P_{\\omega} - \\rho \\beta_2 \\omega^2 + 2 \\rho \\sigma_{\\omega} \\frac{1}{\\omega} \\frac{\\partial k}{\\partial x_j} \\frac{\\partial \\omega}{\\partial x_j} $$ This equation closely resembles the transport equation of \\(\\omega\\) in the k-omega model, except for the additional cross-diffusion term. Finally, the standard k-omega model can be multiplied with the blending function \\(F_1\\) and the transformed k-epsilon model can be multiplied with the blending function \\((1-F_1)\\). The corresponding equations can be added together to give what is known as the baseline model (BSL). $$ \\begin{split} \\rho \\frac{D k}{D t} \u0026amp;= \\frac{\\partial}{\\partial x_j} \\left[ \\left( \\mu + \\sigma_k \\mu_t \\right) \\frac{\\partial k}{\\partial x_j} \\right] + \\tau_{ij}\\frac{\\partial u_i}{\\partial x_j} - \\rho \\omega k \\\\ \\rho \\frac{D \\omega}{D t} \u0026amp;= \\frac{\\partial}{\\partial x_j} \\left[ \\left( \\mu + \\sigma_{\\omega} \\mu_t \\right) \\frac{\\partial \\omega}{\\partial x_j} \\right] + \\frac{\\gamma}{\\nu_t} P_{\\omega} - \\beta \\rho \\omega^2 + 2 \\rho (1 - F_1) \\sigma_{\\omega 2} \\frac{1}{\\omega} \\frac{\\partial k}{\\partial x_j} \\frac{\\partial \\omega}{\\partial x_j} \\end{split} $$ Since this model is a blend of two models, so are the constants. If \\(\\phi_1\\) represents the constants of the original k-omega model and \\(\\phi_2\\) represents the constants of the transformed k-epsilon model, then the constants \\(\\phi\\) of the BSL model are represented by $$ \\phi = F_1 \\phi_1 + (1-F_1) \\phi_2 $$ All of the constants and exact definition of the blending function \\(F_1\\) are given in the appendix of the original paper by Menter. What is important to know is that the blending function\\(F_1\\) is designed such that it is equal to one in the near-wall region and zero away from the surface.\nAccording to Menter, the BSL model behaves very similarly to the standard k-omega model, without the freestream turbulence dependence, which is a considerable improvement. However, the seemingly small difference between the BSL model and the shear stress transport model is what makes the SST model far superior in predicting adverse pressure gradient boundary layer flows. The difference between these two models lies in the definition of the eddy viscosity \\(\\nu_t\\). Namely, the BSL model does not account for the transport of the principal turbulent shear stress, otherwise known as the Reynolds stress. In the SST model, the eddy viscosity is modelled as $$ \\nu_t = \\frac{a_1 k}{\\max(a_1 \\omega ; \\Omega F_2)} $$ with \\(\\Omega = \\frac{\\partial u_i}{\\partial x_j}\\) and \\(F_2\\) being one for boundary-layer flows and zero for freestream flows. This simply guarantees that \\(\\nu_t = \\frac{k}{\\omega}\\) in boundary-layer flows and that the Reynolds-stresses are not over predicted in freesteam conditions.\nTurbulence model comparisons\u003e Turbulence model comparisons # Directly comparing the k-omega and low-Re k-epsilon models in the same transient simulation environment reveals that they lead to very different solution fields. The k-omega model clearly shows a (potentially over-exagerated) oscillating turbulent wake with many differently sized eddies forming right behind the cylinder, whereas the low-Re k-epsilon model struggles to reach any kind of turbulent oscillation and seems to be depicting stable recirculating eddies, even after a considerable number of calculation iterations. It is important to note that in this particular simulation, the residuals of the standard k-epsilon model did not converge, which is why the low-Re k-epsilon model was used. Comparing the k-epsilon and k-omega models to the SST model used for the visualization at the top of the page, it becomes clear that the SST model is able to provide the closest replication to the experiment.\nk-omegalow-Re k-epsilon To further highlight the difference between all of the turbulence models used, the lift coefficient of the cylinder was plotted while running the simulations. These plots are shown below.\nPrevious Next A more in-depth comparison and analysis between the different turbulence models using a different experiment and simulations setups can be found in this post and in this post about flow over a \\(2.5^{\\circ}\\) inclined plate at \\(Re = 10000\\) and \\(Re = 50000\\) respectively. More results can also be found on this page about flow over a larger angle inclined plate.\nSimulation\u003e Simulation # All of the above simulations were carried out in Simcenter StarCCM+. RANS equations and turbulence models based on them can be derived and are valid for 2D cases. This allowed the simulations to be run in 2D and save on computational expenses.\nComputational domain\u003e Computational domain # The computational domain, boundary conditions and finite volume mesh were created based on the information available in the caption of the original figure. It is know that the Reynolds number \\(Re=2000\\) and that the experiment is set up with water flowing around a circular cylinder. From \\(Re = \\frac{\\rho u D}{\\mu}\\) the cylinder diameter and inlet velocity can be calculated to be \\(D = 0.2\\ m\\) and \\(u = 0.1\\ m/s\\). The computational domain can be seen in the figure below. The top and bottom walls are located \\(8D\\) away from the center of the cylinder, as is mentioned in the caption of Figure 44 from the book, which has the same experimental setup as this figure. In the computational domain, the walls and the cylinder have a no-slip boundary condition applied to them, while the inlet is specified as a velocity inlet and the outlet a pressure outlet.\nComputational domain, boundary conditions and mesh. StarCCM+ has quite extensive meshing options and allows for very precise control over the finite volume mesh. The mesh created for this simulation is a quadrilateral mesh with a base size of \\(0.2\\ m\\). However, this base size is only applied to the outermost parts of the domain, as there is a large area around the cylinder and \\(6D\\) behind the cylinder where mesh refinement with a base size of \\(2.5 \\cdot 10^{-4}\\ m\\) and growth rate of \\(1.1\\) is applied. The total cell count is 25576. Some visual representations can be seen below.\nPrevious Next StarCCM+ setup\u003e StarCCM+ setup # The simulation was run as a transient simulation for a total of three times; once using Menter\u0026rsquo;s SST turbulence model, once using the low Reynolds number k-epsilon turbulence model and a final time using Wilcox\u0026rsquo;s (2008) k-omega turbulence model. The turbulence model parameters were left in their default configuration and the exact simulation parameters are shown in the figures below. For each of these, a second order implicit unsteady solver with a time-step of \\(0.025\\ s\\) was used. The stopping criteria was set to 50 maximum inner iterations and \\(20\\ s\\) maximum physical time. This means a total of 800 timesteps and 40.000 iterations. The velocity and pressure data was exported to Ensight Gold case files for further post processing.\nVisualization\u003e Visualization # Post processing and visualization is done in Paraview. The ParaView pipeline is shown in the figures below. After importing the simulation data, it first needs to be processed a bit. Namely a CellDataToPointData filter as well as the Calculator filter need to be applied in order to compute the velocity vector from the x-velocity and y-velocity data. Next, in order to achieve a similar visualization as the original figure, particles need to be injected into the velocity field. For this, a point coordinate matrix is created in MATLAB:\nk = 0.6; % parameter for increasing point concentration %% General point cloud\rP1 = zeros((k*12000)+1,3); % allocating matrix (3D)\rP1(1,:) = [1,2,3]; % First row description for ParaView\rX1 = randi([-1000,1000],k*12000,1); % 7200 random x-coordinates between 300 and 1200\rX1 = X1/10000; % dividing by 1e04 to scale coordinates\rY1 = randi([-500,500],k*12000,1); % 7200 random x-coordinates between 800 and 1200\rY1 = Y1/10000; % dividing by 1e04 to scale coordinates\rP1(2:end,1) = X1; % adding x-coordinates to P matrix\rP1(2:end,2) = Y1; % adding y-coordinated to P matrix\r%% Fine point cloud behind cylinder\rP2 = zeros((k*10000)+1,3); % allocating matrix (3D)\rP2(1,:) = [1,2,3]; % First row description for ParaView\rX2 = randi([-500,500],k*10000,1); % 6000 random x-coordinates between 300 and 1200\rX2 = X2/10000; % dividing by 1e04 to scale coordinates\rY2 = randi([-500,500],k*10000,1); % 6000 random x-coordinates between 800 and 1200\rY2 = Y2/10000; % dividing by 1e04 to scale coordinates\rP2(2:end,1) = X2; % adding x-coordinates to P matrix\rP2(2:end,2) = Y2; % adding y-coordinated to P matrix\r%% Points around cylinder (back half)\rn = 180; % number of points\rangles = linspace(0.5*pi, 1.5*pi, n); % points equally distributed\rradius = 0.0101; % radius of cylinder + margin\rxCenter = 0; % Coordinates of center point\ryCenter = 0;\rX3 = -radius * cos(angles) + xCenter; % converstion from polar coordinates\rY3 = -radius * sin(angles) + yCenter;\rP3 = zeros(n+1,3); % allocating matrix\rP3(1,:) = [1,2,3]; % First row description for ParaView\rP3(2:end,1) = X3; % adding x-coordinates to P matrix\rP3(2:end,2) = Y3; % adding y-coordinates to P matrix\r%% combining points into one matrix\rP = [P1 ; P2(2:end,:) ; P3(2:end,:)];\rwritematrix(P,\u0026#39;P_Final_StarCCM.csv\u0026#39;) These points are imported into ParaView to serve as the input seed for the ParticleTracer filter. The generated point cloud can be seen in the figures below.\nLocations of the particles as outputted by the Matlab script. Finally, some post processing filter such as TemporalParticlesToPathlines and Tube filters can be applied to achieve the details of the final visualization.\nParaview pipeline. ","date":"11 May 2023","permalink":"/chapters/03-separation/fig47/","section":"Chapters","summary":"ExperimentSimulation \u0026ldquo;Air bubbles in water show the velocity field of a flow around a circular cylinder at Reynolds number \\(Re = 2000\\). At this Reynolds number, there is a clear boundary layer separation followed by an oscillating turbulent wake.","title":"Fig 47. Circular cylinder at R=2000 "},{"content":" ExperimentSimulation \u0026ldquo;Water is flowing at 1.4 cm/s past a cylinder of diameter 1cm. Integrated streamlines are shown by electrolytic precipitation of white colloidal smoke, illuminated by a sheet of light. The vortex street is seen to grow in width downstream for some diameters.\u0026rdquo; Photograph by Sadathoshi Taneda\nTheory\u003e Theory # This post is part of a series on flow separation, studied for the case of flow past a circular cylinder at different Reynolds numbers. The full series is:\nFigure 24: Circular cylinder at R=1.54. Figure 40: Circular cylinder at R=9.6. Figure 41: Circular cylinder at R=13.1. Figure 42: Circular cylinder at R=26. Figure 45: Circular cylinder at R=28.4. Figure 46: Circular cylinder at R=41. Figure 96: Kármán vortex street behind a circular cylinder at R=105. Figure 94: Kármán vortex street behind a circular cylinder at R=140. An overview of these posts can be viewed here:\nLaminar, unsteady flow\u003e Laminar, unsteady flow # When the Reynolds number of the flow past the circular cylinder exceeds the \u0026ldquo;critical Reynolds number\u0026rdquo;, which is roughly \\(Re=46 \\), the trapped vortices shown in the earlier figures of this series detach from the cylinder and are advected with the flow. This behavior is oscillatory; shedding occurs successively from the top and bottom of the cylinder. The resulting flow pattern is called a \u0026ldquo;von Kármán vortex street\u0026rdquo;. Despite its unsteady nature, the flow is still laminar; the advected vortices of layers of fluid circulating around each other, as is nicely shown in Figure 98. The vortices become turbulent when the Reynolds number exceeds roughly \\(Re=189 \\), see e.g. Figure 47.\nHopf Bifurcation\u003e Hopf Bifurcation # The point at which a system switches from a stable stationary state to a stable oscillatory state is called a Hopf Bifurcation. It is a fixed critical point in a system where the system loses its primary stability. Indeed, for flow behind a circular cylinder, the first Hopf bifurcation arises when the Reynolds number reaches its critical point, \\(Re = 46\\), and a second bifurcation point can be identified at a Reynolds number of approximately \\(Re = 189\\). This second Bifurcation point marks the onset of turbulent motions, and with increasing Reynolds number the flow patterns disappear into a chaotic flow.\nStrouhal number\u003e Strouhal number # The periodic nature of the flow that arises after a Reynolds number of \\(46\\) can be characterized with a dimensionless number, the Strouhal Number. The Strouhal number is defined as follows:\n$$ St = \\frac{f\\,\\hat{L}}{\\hat{U}} $$\nwhere \\(f\\) is the frequency of vortex shedding, \\(\\hat{L}\\) is a characteristic length scale (e.g. the diameter of the cylinder) and \\(\\hat{U}\\) is a characteristic velocity.\nThe Strouhal number has a dependency on the Reynolds number but is for a large Reynolds number interval (\\(250 \u0026lt; Re \u0026lt; 2 \\cdot 10^5\\)) approximately constant and equal to \\(0.2\\) as can be seen in the figure below.\nStrouhal number vs Reynolds number for flow behind a circular cylinder (Blevins, 1990) Simulation\u003e Simulation # The CFD simulation program used to perform the above simulations is Simcenter StarCCM+.\nBoundary conditions and computational domain\u003e Boundary conditions and computational domain # The velocity (\\(u = 1.4 cm/s = 1.4 \\))cm/s and cylinder diameter (\\(D = 1 \\))cm are given in the caption of the figure. The domain size of the experiment itself is not given in the caption, so reasonable estimates need to be made. The choices made for the computational domain and boundary conditions can be seen in the figure below. The total domain has a length of \\(19\\)cm to accommodate a number of travelling vortices. The distance fro the center of the cylinder to the end of the domain is \\(16 \\)cm, leaving \\(3 \\)cm in front of the center. The height of the domain is \\(10 \\)cm, with the center of the cylinder at \\(5 cm\\). To avoid major influences of the top and bottom boundary on the flow around the vortex street, the top and bottom of the domain are set as symmetry planes. The cylinder is set as a wall with a no-slip boundary condition applied to it. The inflow is set as a velocity inlet with a velocity of \\(1.4\\)cm/s, and the outlet is set as a no-traction outlet.\nComputational domain, boundary conditions and mesh. Meshing\u003e Meshing # The mesh is created in the same fashion as described in Figure 42. For this larger domain, this results in a mesh with a total amount of 135212 cells. Some representations of the mesh can be found below.\nPrevious Next Fluid model\u003e Fluid model # The simulation was run with a laminar, unsteady model with water as the flowing fluid. For the simulation, a second-order implicit unsteady solver with a time-step of \\(0.025 sec\\) was used. A total of 1400 time steps were run with 50 iterations per time step, resulting in 70000 iterations and a physical time of 35 sec. The velocity data of all time steps was exported to an Ensight Gold case file, so it could be processed in Paraview. All models used in STARCCM+ and the STARCCM+ result after 1400 time steps can be found below.\nPrevious Next Visualization\u003e Visualization # Details of the visualization procedure may be found in the post of Figure 42.\n","date":"18 August 2023","permalink":"/chapters/04-vortices/fig94/","section":"Chapters","summary":"ExperimentSimulation \u0026ldquo;Water is flowing at 1.4 cm/s past a cylinder of diameter 1cm. Integrated streamlines are shown by electrolytic precipitation of white colloidal smoke, illuminated by a sheet of light.","title":"Fig 94. Kármán vortex street behind a circular cylinder at R=140"},{"content":" ExperimentSimulation \u0026ldquo;The initially spreading wake shown opposite develops into the two parallel rows of staggered vortices that von Kármàn\u0026rsquo;s inviscid theory shows to be stable when the ratio of width to streamwise spacing is 0.28. Streaklines are shown by electrolytic precipitation in water.\u0026rdquo; Photograph by Sadathoshi Taneda\nGeneral Info\u003e General Info # This post is part of a series on flow separation, studied for the case of flow past a circular cylinder at different Reynolds numbers. In the current figure, one finds a longer version of the von Kármán street as that shown in Figure 94. In comparison to Figure 94, the photograph of this street is taken at a later time step. The vortices in the current figure all appear at the same height compared to the cylinder, whereas the vortices in Figure 94 succesively increase in height; an artifact of the start-up of the vortex shedding. When a later time step of the flow in Figure 94 were to be taken, the vortices also even out and appear at the same height.\nThe main theory on flow separation and visualization set-up are discussed in the web post from Figure 42. More information on the von Kármán vortex street can be found in the web post from Figure 94. The full series is:\nFigure 24: Circular cylinder at R=1.54. Figure 40: Circular cylinder at R=9.6. Figure 41: Circular cylinder at R=13.1. Figure 42: Circular cylinder at R=26. Figure 45: Circular cylinder at R=28.4. Figure 46: Circular cylinder at R=41. Figure 96: Kármán vortex street behind a circular cylinder at R=105. Figure 94: Kármán vortex street behind a circular cylinder at R=140. An overview of these posts can be viewed here:\nThe computational domain used to simulate the flow for the current figure differs in length and height from that of Figure 94, and is illustrated in the Figure below.\nComputational domain, boundary conditions and mesh. ","date":"18 August 2023","permalink":"/chapters/04-vortices/fig96/","section":"Chapters","summary":"ExperimentSimulation \u0026ldquo;The initially spreading wake shown opposite develops into the two parallel rows of staggered vortices that von Kármàn\u0026rsquo;s inviscid theory shows to be stable when the ratio of width to streamwise spacing is 0.","title":"Fig 96. Kármán vortex street behind a circular cylinder at R=105"},{"content":" ExperimentSimulation \u0026ldquo;Osborne Reynolds\u0026rsquo; celebrated 1883 investigation of stability of flow in a tube was documented by sketches rather than photography. However the original apparatus has survived at the University of Manchester. Using it a century later, N.H. Johannesen and C. Lowe have taken this sequence of photographs. In laminar flow a filament of colored water introduced at a bell-shaped entry extends undisturbed the whole length of the glass tube. Transition is seen in the second of the photographs as the speed is increased; and the last two paragraphs show fully turbulent flow. Modern traffic in the streets of Manchester made the critical Reynolds number lower than the value 13,000 found by Reynolds.\u0026rdquo;\nTheory\u003e Theory # Reynolds number and (in)stability\u003e Reynolds number and (in)stability # Reynolds\u0026rsquo; 1883 experiment of pipe flow eventually lead to the formulation of the Reynolds number as \\( Re = \\frac{\\rho \\hat{U}\\hat{L} }{\\mu} \\), with \\( \\rho \\) and \\( \\mu \\) the fluid\u0026rsquo;s density and dynamic viscosity, and \\( U \\) and \\( L \\) a reference velocity and length. This dimensionless number assigns a value to a flow system so that a similar system operating at a different velocity, with a geometric scaling and/or with different fluid properties will exhibit the same flow phenomena if these values are changed in such a way that the Reynolds number remains the same. This concept forms the foundation for wind tunnel and tow-tank experiments. The mathematical background of the Reynolds number is detailed in Figure 42, where it is indeed shown that, for equal Reynolds number, the non-dimensionalized governing equations are identical.\nHowever, the 1883 experiment primarily identified \\( Re \\) as an indicator of the stability of the flow. Here, `stability\u0026rsquo; should be understood as the tendency of flow to, after some perturbation, either return to a laminar state or develop further into a turbulent state. The images show four instances with increasing Reynolds number from top to bottom. The top image is a stable laminar flow as shown with dye following a straight line. The second image shows the transition from laminar to turbulent flow which was given a value for the Reynolds number of around \\(13,000\\), the critical Reynolds number. The next two images then show fully turbulent, unstable flows corresponding to a Reynolds number greater than \\(13,000\\).\nFor turbulent flows, the Reynolds number has one more interpretation of importance to this study: it dictates the range of scales involved in the flow. Here, \u0026ldquo;scales\u0026rdquo; can loosely be interpreted as the differently sized vortical structures (eddies). As the Reynolds number increases, the difference in size between the largest eddies and smallest eddies also increases. The distribution of these eddies in terms of how much kinetic energy is housed in eddies of a certain length follows a particular pattern:\nTurbulent energy cascade The larger vortices are represented on the left side of the graph and the size of vortices is smaller as you move to the right. As the graph indicates, the larger vortices carry most of the kinetic energy. The vortices interact, and kinetic energy is on average transferred from larger length scales to smaller length scales, where the smallest vortices then dissipate the kinetic energy as heat. This is known as the energy cascade. Kinetic energy can also travel up the stream, which is called \u0026ldquo;backscatter\u0026rdquo;. This is how small perturbations to a steady flow can trip turbulence.\nLarge Eddy Simulation (LES)\u003e Large Eddy Simulation (LES) # Computationally reproducing this experiment requires careful consideration of the just described physical processes. Clearly, a turbulence model is required. The two available paradigms are (Unsteady) Reynolds-averaged Navier-Stokes ((U)RANS, discussed in Figure 47) and Large Eddy Simulation (LES). (U)RANS involves averaging and thus aims at resolving the left most scales in the above diagram. The corresponding turbulence models are highly dissipative, dampening small perturbations. It will thus not suffice for capturing the sensitive transition phase from laminar to turbulent flow, nor for resolving the flow up to the level of detail required to reproduce the image. For this reason, this study focuses on Large Eddy Simulation.\nLarge Eddy Simulation (LES) provides the means to simulate a range of the scales that form in the turbulent fluid flow. It is just below direct numerical simulation (DNS) in terms of computational expense, and the key difference between the two is the use of a filter to remove the vortices that are too small to resolve on the computational mesh. The effect of these eliminated small vortices must be incorporated in the model for the large vortices in the form of a subgrid-scale model. This process greatly decreases the computational expense in comparison to DNS, which resolves the full range of scales.\nThe quality and level of detail provided by LES is highly dependent on the mesh size as this dictates the permissible filter width. The finer the mesh, the more eddies can be resolved and the less must be modeled with the subgrid-scale models. It is generally accepted that an LES simulation should solve around 80% of the turbulent kinetic energy of the flow, leaving 20% to be represented by the subgrid-scale model. As a consequence, LES models require refinement with increasing Reynolds number, as we\u0026rsquo;ve seen that this widenes the complete scale range. Techniques for estimating the required mesh-size to capture 80% of the kinetic energy exist and are the topic of a later post.\nFiltered Navier-Stokes equations\u003e Filtered Navier-Stokes equations # LES is based on a modified version of the Navier-Stokes equations, which are the equations that describe mass and momentum (and sometimes energy) balance. In the incompressible case, they read:\n$$ \\begin{cases} \\rho \\frac{\\partial \\boldsymbol{u}}{\\partial t} +\\nabla \\cdot ( \\rho \\boldsymbol{u} \\oplus \\boldsymbol{u} ) = \\mu \\nabla^2 \\boldsymbol{u} - \\nabla p \\\\ \\nabla\\cdot \\boldsymbol{u} = 0 \\end{cases} $$\nIf these equations were to be solved exactly (e.g., by DNS), one obtains the velocity and pressure fields in full detail (i.e., with their full range of scales).\nTo compute only the larger scales, one needs a way to formally separate them from the small scales. In LES, this is performed with a filter:\n$$ \\mathcal{F} \\boldsymbol{u}(\\boldsymbol{x},t) = \\bar{\\boldsymbol{u}}(\\boldsymbol{x},t) $$\nwhere \\( \\mathcal{F} \\) is the filtering operator and \\( \\bar{\\boldsymbol{u}} \\) the filtered solution. Many types of filters exist, each with their own properties, but in general you can think of them as the blurring operation known from image editing. The evolution equations governing the filtered velocity and pressure field \\( ( \\boldsymbol{\\bar{u}}(x,t), \\bar{p}(x,t) ) \\) are then obtained by filtering the Navier-Stokes equations [1]:\n$$ \\begin{cases} \\rho \\frac{\\partial \\bar{\\boldsymbol{u}} }{\\partial t} +(\\nabla \\cdot \\rho \\bar{\\boldsymbol{u}} \\oplus \\bar{\\boldsymbol{u}} ) = \\mu \\nabla^2 \\bar{\\boldsymbol{u}} - \\nabla \\bar{p} -\\nabla\\cdot\\boldsymbol{\\tau} \\\\ \\nabla\\cdot \\bar{\\boldsymbol{u}} = 0 \\end{cases} $$\nThe crucial part of these equations is the occurence of the tensor \\(\\tau_{ij} = \\rho (\\overline{u_iu_j} - \\bar{u}_i \\bar{u}_j)\\), called the subgrid-scale stress. Mind you, it is not really a stress (rather, it is the difference between the convective transport of the filtered solution and the filtered convective transport of the solution), but in the way that the tensor manifests itself in the equations it is indistinguishable from a stress. Once an expression for this tensor is substituted in terms of the filtered quantities, the filtered equations can be solved for the pair \\( ( \\boldsymbol{\\bar{u}}(x,t), \\bar{p}(x,t) ) \\). The chosen expression for \\( \\boldsymbol{\\tau} \\) is called the subgrid-scale model.\nSubgrid-scale models\u003e Subgrid-scale models # The particular form of the subgrid-scale (SGS) model inevitably relates to the filter width \\( \\Delta \\), and hence effectively to the cell size of the mesh. Conceptually, it takes at least 4 cells to capture an eddie as illustrated in the below images. As such, SGS models should only model eddies of length scales smaller than the individual cells.\nMost LES subgrid-scale stress models are based on the eddy-viscosity assumption. Proposed by Boussinesq in 1877, it hypothesizes that the dissipative nature of the smallest eddies in the flow effectively increases the viscosity of the fluid. This naturally leads to:\n$$ \\tau_{ij} = \\frac{2}{3} k \\delta_{ij} - 2 \\mu_{SGS} \\bar{S} _{ij} $$\nwhere \\(k\\) is the turbulent kinetic energy, \\(\\mu_{SGS}\\) is the SGS eddy-viscosity and \\(k\\) is the filtered strain-rate tensor \\( \\bar{S} _{ij} = \\frac{1}{2} (\\frac{\\partial \\bar{u}_i}{\\partial x_j} + \\frac{\\partial \\bar{u}_j}{\\partial x_i}) \\).\nAlso within the class of eddy-viscosity models there exist many variations. The models that we focus on are the Smagorinsky model and the Wall-Adapting Local Eddy-viscosity (WALE) model.\nSmagorinsky:\n$$ \\mu_{SGS} = (C_S \\Delta)^2 \\sqrt{ 2 \\bar{S} _{ij} \\bar{S} _{ij} } $$\nWALE:\n$$ \\mu_{SGS} = (C_W \\Delta)^2 \\sqrt{ f(\\bar{\\boldsymbol{S}}) } $$ Refer to [2] for the exact definition of \\( f(\\bar{\\boldsymbol{S}}) \\).\nThe Smagorinsky model is the base model from which all other models are adapted. The WALE model is a particular version that exhibits almost zero eddy-viscosity when the flow is laminar. This feature makes it well suited for modeling the laminar to turbulent transition, the focus of this work. Additionally, it provides a more accurate decay of eddy-viscosity at near wall regions, hence being wall-adapting [2]. Pipe flow is highly wall bounded, making this an important factor.\nSimulation\u003e Simulation # Perturbation of equilibrium state\u003e Perturbation of equilibrium state # To reproduce the Reynolds dye experiment, we perturb the equilibrium state at an increasingly higher Reynolds number. Past the critical Reynolds number, the perturbation does not damp out (= steady equilibrium), but rather grows and leads to turbulent flow (= unsteady equilibrium). The equilibrium velocity profile of pipe flow does not depend on the Reynolds number: it is the parabolic Poiseuille flow profile:\n$$ U(r) = \\frac{1}{4 \\mu} \\frac{dP}{dx}(r^2-R^2) $$ where \\(\\mu\\) is the dynamic viscosity, \\( \\frac{dP}{dx} \\) is the mean pressure gradient, and \\(R\\) the pipe radius.\nEquilibrium state In the experiment, the fluid is driven through the pipe by a pressure gradient. This pressure gradient can be modelled as a momentum source, a vector quantity that gives the magnitude and direction of the body force acting on the fluid. With the above expression for the Poiseuille flow, the required body force to hit a target Reynolds number can be derived as:\n$$ \\boldsymbol{f} = \\frac{dP}{dx} \\boldsymbol{e}_x = Re \\frac{2\\mu^2}{\\rho R^3} \\boldsymbol{e}_x $$\nwhere it must be noted that the Reynolds number is then based on the center velocity of the laminar Poiseuille flow, which is anticipated to be a severe overprediction compared to the (unspecified) reference velocity taken in the experimental photographs.\nThe perturbation to this equilibrium state is introduced in the form of a synthetic eddy method. As parameters, this takes values for the turbulence intensity and turbulent length scale, for which we use the following expressions:\n$$ I = 0.7 \\sqrt{\\frac{\\frac{3}{2} v_t^2}{U^2}} $$\n$$ L_s = 0.07 D $$\nwhere \\(v_t\\) is the turbulent velocity scale which is 10% of the free stream velocity (\\(U\\)). The intensity is multiplied by a factor \\(0.7\\) to reduce the intensity of the perturbation and uncover the intrinsic stability characteristics of the flow.\nThe simulations are then run for 90 seconds of real time to allow the flow to reach a statistically steady state where it is no longer affected by the initial perturbation. Instead, it either returns to the laminar state or remains in a turbulent state depending on the system\u0026rsquo;s stability.\nComputational set-up\u003e Computational set-up # To model this system, we define the pipe domain as having a diameter of \\(100 mm\\) and a length of \\(500 mm\\). The open ends of the pipe are connected with periodicity conditions. In this way, there is no real inlet and outlet, but instead the flow moving through the outlet is reintroduced at the inlet. It can be interpreted as stacking many small pipe segments next to one another to create an infinitely long pipe. This ensures that the flow can fully develop, and the progression of turbulence is not limited by the pipe length.\nThis domain is then meshed with a trimmed cell mesher. The geometry is divided into equally sized cubes of \\(2 mm\\), which are trimmed and subdivided to fit the domain boundary and provide a moderately well refined boundary layer mesh. This leads to a total of 400,000 grid cells.\nPrevious Next On top of the mesh resolution, the quality of LES also depends on the temporal resolution. The accuracy of the results are dependent on time-step size, as characterized by the so-called Courant number. The Courant number is a dimensionless number indicating how many cells of distance a fluid particle travels per time-step. It is defined as:\n$$ C = |\\boldsymbol{u}| \\frac{\\Delta t}{\\Delta x} $$\nwhere \\(\\boldsymbol{u}\\) is the fluid velocity, \\(\\Delta t\\) is the timestep size, and \\(\\Delta x\\) is the mesh cell size.\nFor LES, good practice is to keep the Courant number below one. Then, during each time-step, the flow will at most move one cell length across. This ensures that there is sufficient amount \u0026lsquo;communication time\u0026rsquo; between the cells in the mesh; if the Courant number exceeds a value of one the fluid would be \u0026ldquo;skipping over\u0026rdquo; cells. For this study, we use a Courant number of \\(0.9\\).\nVisualization\u003e Visualization # The flow is visualized in Paraview. Since a limited length of the pipe is simulated, the first step is to extend the pipe during post-processing. This is done by opening multiple of the exported Ensight Gold file and then using the transform filter to translate these datasets to be next to one another to form a longer pipe. The data must be shared across these datasets, so the group datasets, merge blocks, and append datasets filters are used. This essentially makes the datasets uniform so that, as the flow passes through the periodic boundary, it is displayed on the next dataset in line.\nThe next step is to visualize a stream of dye moving through the flow. Since data is exported as cell data when simulating, the cell to point data filter is utilized. The point data can then be used to calculate the velocity vectors of the flow within the calculator. To simulate the dye, the particle tracer filter is given a point source from which the dye flow is initiated, and propagated by the velocity values found with the calculator. The Paraview pipeline is shown below:\nParaview visualization filters With this visualization, we are able to get results which are a qualitative match with the images in the experiment. Though the turbulent cases are naturally not identical to the experiment due to the unpredictability of turbulence, the characteristics are similar. The key similarities are in the frequency and amplitude of the vortices, and the thinning of the dye.\nThe main discrepancy is that our LES simulation of the transitioning flow exhibits stronger turbulent features than the flow in the photograph. This is most likely due to our conservative estimate of the Reynolds number, as described before. The actual Reynolds number of the LES simulation would have to be determined a-posteriori and iterated upon.\nReferences\u003e References # [1] Stephen B Pope. Turbulent Flows. Cambridge University Press (2020).\n[2] Minwoo Kim et al. “Assessment of the wall-adapting local eddy-viscosity model in transitional boundary layer”. In: Computer Methods in Applied Mechanics and Engineering 371 (2020), p. 113287. doi: 10.1016/J.CMA.2020.113287.\n","date":"4 February 2024","permalink":"/chapters/05-instability/fig103/","section":"Chapters","summary":"ExperimentSimulation \u0026ldquo;Osborne Reynolds\u0026rsquo; celebrated 1883 investigation of stability of flow in a tube was documented by sketches rather than photography. However the original apparatus has survived at the University of Manchester.","title":"Fig 103. Repetition of Reynolds' dye experiment"},{"content":" ExperimentSimulation \u0026ldquo;The plate is uniformly heated in air, producing a steady laminar flow. An interferogram shows lines of constant density which, at nearly constant pressure, are also isotherms. The Grashof number is approximately five million at a distance of 0.1 m from the lower end of the plate, so that the thermal boundary layer is rather thick.\u0026rdquo; Eckert \u0026amp; Soehngen 1948\nIntroduction\u003e Introduction # In the above experiment, a hot plate is placed inside a room of cooler air, causing heat to transfer from the plate to the air. More specifically, conduction takes place at the interface between the plate and the air, causing the air to heat up, and buoyancy effects then cause the warm air to rise. Buoyancy thus introduces advective heat transport to the problem. Then, what ends up being the dominant heat transport mechanism? And by which physical parameters is this affected? To answer such questions for complex systems (i.e., more complex than a plate in a room), one requires a mathematical model of the flow. Typically, the Boussinesq model is used for simulating temperature-induced buoyant flows. Which assumptions then underlie this model?\nIn the following, these questions are addressed. We divide this post into two sections: first, we explore the Richardson and Grashof numbers as the dimensionless numbers that capture the relative magnitude of the different heat transfer mechanisms present in the system. Then, we derive the Boussinesq approximation as a simplification of the compressible Navier-Stokes equations. We explore the severity of the underlying approximations by modeling the above system at different temperatures with and without the Boussinesq approximation.\nDimensionless heat transport: Conduction, Convection \u0026amp; Radiation\u003e Dimensionless heat transport: Conduction, Convection \u0026amp; Radiation # Three basic principles govern heat transfer: conduction, convection, and radiation. In the experimental photograph above, heat transfer takes place in a fluid (air). Typically, this hints at convection being the largest contributor to the transfer. Nevertheless, conduction remains an important factor as it describes both the transfer from solid to fluid and the mixing of the advecting flow with its surroundings. Radiation can be neglected as its contribution becomes significant only at temperatures above 1000 K [1].\nConvective heat transfer can be subdivided into two categories: \u0026ldquo;forced\u0026rdquo; and \u0026ldquo;free\u0026rdquo; (or \u0026ldquo;natural\u0026rdquo;). Simply put, free convection refers to cases where the bulk fluid is initially at rest and buoyancy effects cause the fluid motion, whereas for forced convection, the transporting fluid is driven by external means [2]. However, this distinction becomes blurry when the rising air of a lower segment of the system reaches a significant velocity by the time it interacts with a higher segment of the system. Can we still consider this convection free, or should it be considered forced? The Richardson number provides a handle on estimating which type of convection is at play.\nRichardson number (Ri)\u003e Richardson number (Ri) # The (forced) convective momentum transport is described by the term \\( \\boldsymbol{u}\\cdot \\nabla \\boldsymbol{u} \\) in the Navier-Stokes equations and thus scales with \\( u^2 / L \\). The buoyancy term scales with the gravitational acceleration times the volumetric expansion of the fluid. The Richardson number is then defined as their ratio:\n$$ \\frac{Buoyancy force}{Inertial force} \\propto \\frac{g \\beta \\Delta T }{ u^2 / L } = \\frac{g \\beta \\Delta T L }{ u^2 } = Ri $$\nWherein\n\\(u = \\) A characteristic velocity of the surrounding fluid \\([\\frac{m}{s}]\\) \\(L = \\) A reference length, e.g., the vertical length of the heated surface \\([m]\\) \\(g = \\) The gravitational acceleration \\([\\frac{m}{s^2}]\\) \\(\\beta = \\) The coefficient of thermal expansion (approximately \\(\\frac{1}{T}\\) for an ideal gas, as shown later) \\(\\Delta T = \\) A reference temperature difference, e.g., between the heated surface and the surrounding \\([K]\\) If the Richardson number is substantially larger than one, buoyancy dominates inertia, indicating a case of free convection. Conversely, if the Richardson number is substantially smaller than one, the system may be considered a case of forced convection. If it is almost equal to one (in order of magnitude), both free and forced convection play an important role [3].\n\\(Ri \\gg 1\\) : Free convection (ignore forced convection). \\(Ri \\approx 1\\) : Free and forced convection. \\(Ri \\ll 1\\) : Forced convection (ignore free convection). In earlier posts, we have seen that the Reynolds number is an important dimensionless number that characterizes the nature and stability of a flow. An insightful representation of the Richardson number follows from factoring out the Reynolds number (\\(Re \\)). This gives:\n$$ Ri = \\frac{1}{Re^2}(\\frac{g \\beta \\Delta T L^3}{\\nu^2}) = \\frac{Gr}{Re^2} $$\nWith \\(\\nu \\) the kinematic viscosity of the fluid \\( [\\frac{m^2}{s}]\\). This form of the equation leads to another dimensionless number: the remainder is called the Grashof number. In the earlier experiment, the Grashof number is stated to equal roughly five million. Additionally, we may assume that the velocity of the fluid remains quite small, and thus that \\(Re\\) is relatively small. It then follows that \\(Ri \\) is substantially larger than 1, indicating a case of free convection.\nGrashof number (Gr)\u003e Grashof number (Gr) # The Grashof number is often described as the ratio between the buoyancy force and viscous forces acting on the fluid. In that sense, the Grashof number is indicative of the stability of the flow, in much the same way as the Reynolds number. Studies show that for \\(Gr \u0026lt; 10^8\\), the flow typically remains laminar, whereas for \\(Gr \u0026gt; 10^9\\) turbulence occurs [4]. For this interpretation to work, though, the velocity measure in the scaling of the viscous forces (i.e., the \\( u \\) in \\( Viscous force \\propto \\nu u /L^2 \\) ) is replaced by \\( u \\propto \\nu/L \\). While this is dimensionally consistent, the implied scaling is rather questionable. Nevertheless, this does motivate the definition of the Grashof number as:\n$$ \\frac{Buoyancy force}{Viscous forces} \\propto \\frac{g \\beta (T_s - T_{\\infty}) }{\\nu u / L^2 } \\propto \\frac{g \\beta (T_s - T_{\\infty}) L^3 }{\\nu^2 } = Gr $$\nThe Grashof number is reported to be \\( Gr \\approx 5 \\cdot 10^6\\) at a distance of 0.1 m from the lower end of the plate. Filling in typical office-room values for \\(g\\), \\(\\beta=1/T_{\\infty}\\), \\(T_{\\infty}\\) and \\(\\nu \\) gives an estimated plate temperature of 318.2 K. The value of \\( 5 \\cdot 10^6\\) is also well below the earlier mentioned stability threshold, so that laminar flow can be assumed.\nModeling natural convection\u003e Modeling natural convection # Conduction causes the air surrounding the plate to heat up. The increase in temperature leads to thermal expansion of the fluid. Since the mass of a heated fluid parcel stays the same, the volume increase means density decreases. As gravity pulls harder on fluids/objects that are denser, the colder air (with higher density) will experience a higher gravitational pull than the hotter air, causing cold air to drop and hot air to rise. In order of the just described physical processes, we are dealing with the following conservation laws:\nConservation of energy Conservation of mass Conservation of momentum The compressible Navier-Stokes equations\u003e The compressible Navier-Stokes equations # As the compressibility of the fluid is the driving factor behind buoyancy, the relevant conservation equations are the compressible Navier-Stokes equations. They read as follows:\nMass (also named continuity): $$ \\frac{1}{\\rho} \\frac{D \\rho}{D t} + \\nabla \\cdot \\boldsymbol{u} = 0 $$ Momentum $$ \\rho (\\frac{\\partial\\boldsymbol{u}}{\\partial t} + (\\boldsymbol{u} \\cdot \\nabla) \\boldsymbol{u} ) = - \\nabla p + \\nabla \\cdot \\boldsymbol{\\tau} + \\rho \\boldsymbol{g} $$ Energy $$ \\frac{\\partial}{\\partial t} [\\rho (e + \\frac{1}{2}\\boldsymbol{u}\\cdot \\boldsymbol{u})] + \\nabla \\cdot [\\rho \\boldsymbol{u} (e + \\frac{1}{2}\\boldsymbol{u}\\cdot\\boldsymbol{u})] = - \\nabla \\cdot \\boldsymbol{q} + \\nabla \\cdot (\\boldsymbol{\\tau} \\cdot \\boldsymbol{u} - p\\boldsymbol{u}) + \\rho \\boldsymbol{u} \\cdot \\boldsymbol{g} $$ With\n\\(e\\) the internal energy per unit mass \\(\\boldsymbol{q}\\) the conductive heat flux vector, for which typically \\(\\boldsymbol{q} = -k\\nabla T \\) where \\(k\\) is the thermal conductivity \\( \\boldsymbol{\\tau} \\) the viscous stress tensor, for which typically: \\(\\boldsymbol{\\tau} = \\mu (\\nabla \\boldsymbol{u} + (\\nabla \\boldsymbol{u})^T) - \\frac{2}{3} \\mu (\\nabla \\cdot \\boldsymbol{u}) \\boldsymbol{I} \\) where \\( \\mu \\) is the fluid dynamic viscosity As buoyancy is temperature-driven, it is more natural to rewrite the energy equation into an equation for temperature transport.\n$$ \\rho C_p \\frac{\\partial T}{\\partial t} + \\rho C_p \\boldsymbol{u} \\cdot \\nabla T = \\nabla \\cdot (k \\nabla T) + (\\frac{\\partial p}{\\partial t} + \\boldsymbol{u} \\cdot \\nabla p) + \\boldsymbol{\\tau} : \\nabla \\boldsymbol{u} $$ with \\( T \\) the temperature and \\(C_p\\) the heat capacity at constant pressure.\nThe derivation of the temperature transport equation from the equation conservation energy is quite involved, and can be found by expanding the following box. Derivation of the temperature transport equation. Here the equation for conservation of energy will be rewritten to the equation for conservation of enthalpy [5], and subsequently into the temperature transport equation.\nConservation of energy: $$ \\frac{\\partial}{\\partial t} [\\rho (e + \\frac{1}{2}\\boldsymbol{u}\\cdot \\boldsymbol{u})] + \\nabla \\cdot [\\rho \\boldsymbol{u} (e + \\frac{1}{2}\\boldsymbol{u}\\cdot\\boldsymbol{u})] = - \\nabla \\cdot \\boldsymbol{q} + \\nabla \\cdot (\\boldsymbol{\\tau} \\cdot \\boldsymbol{u} - p\\boldsymbol{u}) + \\rho \\boldsymbol{u} \\cdot \\boldsymbol{g} $$\nThe stress term on the right hand side can be expanded as: $$ \\nabla \\cdot (\\tau \\cdot \\boldsymbol{u}) = \\tau : \\nabla \\boldsymbol{u} +\\boldsymbol{u} \\cdot (\\nabla \\cdot \\tau) $$ $$ - \\nabla \\cdot (p \\boldsymbol{u}) = - p \\nabla \\cdot \\boldsymbol{u} - \\boldsymbol{u} \\cdot \\nabla p $$\nTTo reformulate the above conservation of total energy into a conservation equation for the internal energy alone, we subtract the conservation equation of the kinetic energy. This latter equation is obtained from the momentum equation by multiplying it with the velocity \\(\\boldsymbol{u}\\). Some algebraic manipulation gives: $$ \\frac{\\partial}{\\partial t} [\\rho \\frac{1}{2}\\boldsymbol{u}\\cdot \\boldsymbol{u}] + \\nabla \\cdot (\\rho \\boldsymbol{u} \\frac{1}{2}\\boldsymbol{u}\\cdot \\boldsymbol{u}) = - \\boldsymbol{u} \\cdot \\nabla p +\\boldsymbol{u} \\cdot (\\nabla \\cdot \\tau) + \\rho \\boldsymbol{u} \\cdot \\boldsymbol{f} $$\nPerforming the subtraction and earlier substitutions then leaves: $$ \\frac{\\partial}{\\partial t} (\\rho e) + \\nabla \\cdot (\\rho \\boldsymbol{u} e) = - \\nabla \\cdot q + \\tau : \\nabla \\boldsymbol{u} - p \\nabla \\cdot \\boldsymbol{u} $$\nNext, we introduce the enthalpy in place of the internal energy. Their relation is as follows: $$ e = h - \\frac{p}{\\rho} $$\nSubstituting the enthalpy relation in the equation for conservation of internal energy (and rearranging it) gives the equation for conservation of enthalpy: $$ \\frac{\\partial}{\\partial t} (\\rho h) + \\nabla \\cdot (\\rho \\boldsymbol{u} h) = - \\nabla \\cdot q + \\frac{\\partial p}{\\partial t} + \\boldsymbol{u} \\cdot \\nabla p + \\tau : \\nabla \\boldsymbol{u} $$\nThe left side of the equation above can be expanded: $$ \\frac{\\partial}{\\partial t} (\\rho h) + \\nabla \\cdot (\\rho \\boldsymbol{u} h) = h [\\frac{\\partial \\rho}{\\partial t} + \\nabla \\cdot (\\rho \\boldsymbol{u})] + \\rho \\frac{\\partial h}{\\partial t} + \\rho \\boldsymbol{u} \\cdot \\nabla h = \\rho \\frac{\\partial h}{\\partial t} + \\rho \\boldsymbol{u} \\cdot \\nabla h $$\nWhere the second equality is due to mass conservation.\nInserting the equation above into the equation for conservation of enthalpy gives: $$ \\rho \\frac{\\partial h}{\\partial t} + \\rho \\boldsymbol{u} \\cdot \\nabla h = - \\nabla \\cdot q + \\frac{\\partial p}{\\partial t} + \\boldsymbol{u} \\cdot \\nabla p + \\tau : \\nabla \\boldsymbol{u} $$\nFrom here, the equation can be rewritten in terms of temperature. The relation between enthalpy, temperature, and pressure is: $$ dh = C_p dT + \\frac{1}{\\rho} [1 + \\frac{T}{\\rho} \\frac{\\partial \\rho}{\\partial T} ] dp = C_p dT + \\frac{1}{\\rho} [1 - T \\beta] dp $$\nWherein\n\\(\\beta = \\frac{1}{\\rho} \\frac{\\partial \\rho}{\\partial T} \\) is the coefficient of thermal expansion \\(C_p\\) is the heat capacity at constant pressure Since the velocity of the flow is small, \\(C_p\\) can be used in this equation [6].\nSubstituting the relation above into the equation for conservation of enthalpy gives the equation for conservation of temperature: $$ \\rho C_p \\frac{\\partial T}{\\partial t} + \\rho C_p \\boldsymbol{u} \\cdot \\nabla T = - \\nabla \\cdot q + \\beta T (\\frac{\\partial p}{\\partial t} + \\boldsymbol{u} \\cdot \\nabla p) + \\tau : \\nabla \\boldsymbol{u} $$\nInserting Fourier\u0026rsquo;s law: \\(q = - k \\nabla T\\) and assuming an ideal gas for which \\(\\beta = \\frac{1}{T}\\) results in: $$ \\rho C_p \\frac{\\partial T}{\\partial t} + \\rho C_p \\boldsymbol{u} \\cdot \\nabla T = \\nabla \\cdot (k \\nabla T) + (\\frac{\\partial p}{\\partial t} + \\boldsymbol{u} \\cdot \\nabla p) + \\tau : \\nabla \\boldsymbol{u} $$\nCurrently, there are four unknowns (velocity, pressure, temperature (energy), and density), but only three equations. One more closure relation is required. For air, we can make use of the ideal gas law as a final closure relation between dependent variables: $$ \\rho(p,T) = \\frac{p}{R_s T} $$\nwith \\(R_s\\) the specific gas constant.\nThe Boussinesq approximation\u003e The Boussinesq approximation # For buoyancy-driven flows, small density changes have a significant impact on the behavior of the flow. Consequently, numerical approximation is challenging: small errors in the density field majorly affect the results. To remedy this, one can take a different route. The fluid may be assumed incompressible if the effect of the temperature-induced compressibility is introduced explicitly in the momentum balance equations as a body-force term. This is called the Boussinesq approximation. In the following, we derive the governing equations from the earlier compressible Navier-Stokes equations.\nConsider first the equation for conservation of mass. As the density change in the equation for conservation of mass is neglected, i.e., \\( \\frac{1}{\\rho} \\frac{D \\rho}{D t} = 0\\), the velocity field is divergence-free (solenoidal):\n$$ \\nabla \\cdot \\boldsymbol{u} = 0 $$\nThis result can be substituted into the temperature transport equation. Additionally, the velocity gradients can be assumed small, rendering the viscous heating a negligible contribution. Together with a constant density,\\( \\rho = \\rho_0 \\), this gives:\n$$ \\rho_0 C_p \\frac{\\partial T}{\\partial t} + \\rho_0 C_p \\boldsymbol{u} \\cdot \\nabla T = \\nabla \\cdot (k \\nabla T) $$\nThe momentum balance equation requires further elaboration. The Boussinesq approximation follows from writing this momentum balance for the temperature-induced perturbation around a constant-temperature stagnant state. For the constant-temperature stagnant state, the velocity is zero, but the pressure follows the hydrostatic profile. Writing the true pressure field as a perturbation around this hydrostatic state, we get: \\( p = - \\rho_0 g z + \\hat{p} \\).\nThen, the earlier momentum equation simplifies to: $$ \\frac{\\partial\\boldsymbol{u}}{\\partial t} + (\\boldsymbol{u} \\cdot \\nabla) \\boldsymbol{u} = - \\frac{1}{\\rho} \\nabla \\hat{p} + \\frac{\\mu}{\\rho} \\nabla^2 \\boldsymbol{u} + \\frac{ \\rho - \\rho_0 }{\\rho} \\boldsymbol{g} $$\nThe ideal gas law may then be employed to explicitly relate the temperature change to a density change. A first-order Taylor expansion around \\( \\rho_0 = \\rho(p_0,T_0) \\) yields: $$ \\rho \\approx \\rho_0 (1 - \\frac{T-T_0}{T_0} ) $$\nA detailed derivation on this relation The ideal gas law is: $$ \\rho = \\rho (p, T) = \\frac{p}{R_{s} T} $$\nWe define the initial density \\( \\rho \\) as \\( \\rho_0 \\), for which $$ \\rho_0 = \\rho(p_0, T_0) = \\frac{p_0}{R_{s} T_0} $$\nThe equation for some closely related density \\( \\rho \\) follows from a first-order taylor expansion:\n$$ \\rho (p+\\text{d} p, T+\\text{d} T ) = \\rho (p_0, T_0) + \\frac{\\partial \\rho}{\\partial p}\\Big|_{{p_0,T_0}} \\text{d}p + \\frac{\\partial \\rho}{\\partial T} \\Big|_{{p_0,T_0}} \\text{d} T $$\n$$ = \\rho_0 + \\frac{1}{R_{s} T_0} \\text{d} p - \\frac{p_0}{R_{s} T_0^2} \\text{d} T $$\n$$ = \\rho_0 + \\frac{\\rho_0}{p_0} \\text{d} p - \\frac{\\rho_0}{T_0} \\text{d} T $$\n$$ = \\rho_0 (1 + \\frac{\\text{d} p}{p_0} - \\frac{\\text{d} T}{T_0}) $$\nWe now make the assumption that: $$ \\frac{\\text{d} p}{p_0} = \\frac{p-p_0}{p_0} \\ll \\frac{T-T_0}{T_0} = \\frac{\\text{d} T}{T_0} $$\nSuch that we may simplify the approximation of the density \\(\\rho\\) as: $$ \\rho \\approx \\rho_0 (1 - \\frac{T-T_0}{T_0} ) $$\nThe above assumption is indeed valid for the considered experiment because:\nthe atmospheric pressure has an order of magnitude of \\(10^5\\) the change in pressure has an order of magnitude of \\(10^{-3}\\) (confirmed in Ansys) the change in temperature has an order of magnitude of 10\\(10\\) (between \\(10^{-1}\\) and \\(10^2\\)) the room temperature has an order of magnitude of \\(10^2\\) this gives for the assumption: \\( \\frac{10^{-3}}{10^5} = 10^{-8} \\ll \\frac{10}{10^2} = 10^{-1}\\) Substitution of this approximation into the momentum balance equation gives: $$ \\frac{\\partial\\boldsymbol{u}}{\\partial t} + (\\boldsymbol{u} \\cdot \\nabla) \\boldsymbol{u} = - \\frac{1}{ \\rho_0 (1 - \\frac{T-T_0}{T_0} )} \\nabla \\hat{p} + \\frac{\\mu}{ \\rho_0 (1 - \\frac{T-T_0}{T_0} )} \\nabla^2 \\boldsymbol{u} - \\frac{ \\frac{T-T_0}{T_0} }{ 1 - \\frac{T-T_0}{T_0} } \\boldsymbol{g} $$\nRecall that the fields \\( \\boldsymbol{u} \\) and \\(\\hat{p} \\) are interpreted as temperature-induced perturbations. As such, they are assumed to scale with the temperature perturbation, i.e., \\( T-T_0\\). Considering only leading-order terms, those of order \\( T-T_0 \\), the above simplifies to: $$ \\frac{\\partial\\boldsymbol{u}}{\\partial t} + (\\boldsymbol{u} \\cdot \\nabla) \\boldsymbol{u} = - \\frac{1}{ \\rho_0 } \\nabla \\hat{p} + \\nu_0 \\nabla^2 \\boldsymbol{u} - \\frac{T-T_0}{T_0} \\boldsymbol{g} $$\nIn summary, the equations for describing convective heat transport with the Boussinesq approximation are:\nConservation of mass: $$ \\nabla \\cdot \\boldsymbol{u} = 0 $$ Conservation of momentum $$ \\frac{\\partial\\boldsymbol{u}}{\\partial t} + (\\boldsymbol{u} \\cdot \\nabla) \\boldsymbol{u} = - \\frac{1}{ \\rho_0 } \\nabla \\hat{p} + \\nu_0 \\nabla^2 \\boldsymbol{u} - \\frac{T-T_0}{T_0} \\boldsymbol{g} $$ Temperature transport (following from conservation of energy). $$ \\rho_0 C_p \\frac{\\partial T}{\\partial t} + \\rho_0 C_p \\boldsymbol{u} \\cdot \\nabla T = \\nabla \\cdot (k \\nabla T) $$ The incompressible Boussinesq approximation significantly simplifies the heat transport equations compared to the compressible approximation. This makes it easier to use the equations in a simulation [7]. Since this approximation also represents reality reasonably well at low density changes, it is commonly used in heat transport studies like the present one. Another reason for using this incompressible Boussinesq approximation is that for the compressible approximation, convergence of the simulations is more important to obtain accurate results since a small deviation has a big impact.\nSimulation results\u003e Simulation results # Comparison between compressible Navier-Stokes and Boussinesq approximation\u003e Comparison between compressible Navier-Stokes and Boussinesq approximation # The incompressibility assumption simplifies all three conservation equations. For these simplifications to be appropriate, the density changes must be small, and therefore so must the temperature changes. The Boussinesq approximation itself also assumes small temperature variations. So, it is expected that at small temperature changes, the incompressible Boussinesq approximation gives a good representation of reality, while with larger temperature changes, results are less accurate. Now that the (theoretical) differences between both approximations are known, it is interesting to see how these manifest themselves in simulations. As a study case, we use the above experiment that Eckert and Soehngen executed in the late 1940s.\nIn the experiment, a Zehnder-Mach interferometer was used to photograph the isotherms. The experiment took place inside their laboratory, so general room conditions apply. The domain is modeled to be 2D, since the depth of the initial copper plate is large (around 609.6 mm) compared to the height and width (respectively 127 mm and 3.97 mm). In the experiment, the photograph was made shortly after the copper plate was placed inside the room (the researchers tried to do it immediately). To mimic this, we use a transient model to simulate the first few seconds of heat transport. The 2D model of the plate is put inside a much larger box so that the domain restrictions do not influence the solution field near the plate. The meshes used for the simulation are illustrated below.\nPrevious Next The sliders below then show the two approximations using three different initial temperature differences between the plate and the fluid (respectively: 0.3 K, 30.3 K, and 336.4 K, corresponding to Grashof numbers of \\(0.1\\cdot 10^6\\), \\(9\\cdot 10^6\\) and \\(100\\cdot 10^6\\), respectively). The gray-scale visualization is obtained by plotting the temperature field and then setting the bounds of a colorbar to the room temperature and plate temperature, and by choosing as many black-to-white intervals as there are bands in the experimental photograph.\nCompressibleBoussinesq \u0026ldquo;At Gr=100.000; Compressible vs Incompressible Boussinesq approximation (in simulation at t=3 seconds); the initial temperature difference is 0.3 K\u0026rdquo;\nCompressibleBoussinesq \u0026ldquo;At Gr=9.000.000; Compressible vs Incompressible Boussinesq approximation (in simulation at t=3 seconds); the initial temperature difference is 30.3 K\u0026rdquo;\nCompressibleBoussinesq \u0026ldquo;At Gr=100.000.000; Compressible vs Incompressible Boussinesq approximation (in simulation at t=3 seconds); the initial temperature difference is 336.4 K\u0026rdquo;\nAt a small temperature difference, one can see that both approximations give a similar result. When the temperature difference increases, the results of the two approximations are less aligned. At a d\\(T\\) (temperature change) of 30.3 K, the difference is relatively small, while at a d\\(T\\) of 336.4 K, one can see the differences well. At a d\\(T\\) of 0.3 K, the theory expected the incompressible Boussinesq approximation to be more reliable. The simulation results of both approximations show almost similar results, which confirms the theory and illustrates that the incompressible Boussinesq approximation is useful. When d\\(T\\) increases, the differences between the results of both approximations increase as well. This is also expected since the incompressible Boussinesq approximation becomes less reliable, and therefore the compressible approximation should be used at larger d\\(T\\).\nComparison between simulation and experiment\u003e Comparison between simulation and experiment # Finally, we compare the simulation results for the Boussinesq approximation to the experimental photograph. The researchers estimate a plate temperature of 318.2 K at the moment the picture was made (as derived from their stated value for the Grashof number: \\(Gr = 5 \\cdot 10^6\\)). However, a best simulation match is found for a temperature of 323.3 K (corresponding to \\(Gr = 9 \\cdot 10^6\\)). We consider this 5-degree temperature within uncertainty tolerances, as the researchers do not mention the plate temperature explicitly, nor do they state the equation and values they used to determine the Grashof number. For this plate temperature, we obtain:\nExperimentSimulation The simulation and experiment match very well. The main mismatch is the isotherm furthest away from the plate. In the experiment, this is not quite straight, probably being the result of some external air currents (forced convection).\nSummary\u003e Summary # When a hot plate is placed inside a room of air, the air directly surrounding the plate heats up due to conduction. As the air gets hotter, the volume increases, which means the density decreases. Air with a higher density is pulled onto harder by gravity so that it pushes the air with the lower density (the hot air) upwards. This net upwards force is called the buoyancy force. The moving air adds convection to the heat transport problem.\nWhether the convection can be considered \u0026ldquo;free\u0026rdquo; (the bulk fluid is not moving) or \u0026ldquo;forced\u0026rdquo; (the bulk fluid moving due to an external source) is determined by the Richardson Number (\\(Ri = \\frac{Gr}{Re^2}\\)). In this study, the Richardson number is substantially larger than 1, so that free convection is at play. This means the Grashof number (\\(Gr = g \\beta (T_s - T_{\\infty}) L^3 / \\nu^2 \\)) is the dimensionless number that predominantly describes the behavior of the flow.\nTo simulate the buoyancy-induced convective flow, the physics is described by three conservation laws: those for mass, momentum, and energy (temperature). Jointly, these are the compressible Navier-Stokes equations. Additionally, the Ideal Gas Law is used to relate pressure, temperature, and density. To simplify the equations, one may consider using instead the equations for incompressible flow while adding the Boussinesq approximation for a buoyancy force in the equation for conservation of momentum. While this approximation considerably simplifies the numerical approximation procedure, it is only valid for small temperature changes. The earlier example simulations confirm this difference between the two approximations.\nSources\u003e Sources # [1] https://www.sciencedirect.com/topics/engineering/radiation-heat-transfer [2] https://www.sfu.ca/~mbahrami/ENSC%20388/Notes/Forced%20Convection.pdf [3] https://en.wikipedia.org/wiki/Grashof_number [4] https://www.comsol.com/blogs/using-the-boussinesq-approximation-for-natural-convection/ [5] https://www.comsol.com/multiphysics/heat-transfer-conservation-of-energy [6] https://en.wikipedia.org/wiki/Pressure_coefficient [7] https://www.cambridge.org/core/books/abs/basic-aerodynamics/fundamentals-of-steady-incompressible-inviscid-flows/4F5672B432936FA6E9176571060C3822 ","date":"10 April 2024","permalink":"/chapters/08-convection/fig204/","section":"Chapters","summary":"ExperimentSimulation \u0026ldquo;The plate is uniformly heated in air, producing a steady laminar flow. An interferogram shows lines of constant density which, at nearly constant pressure, are also isotherms. The Grashof number is approximately five million at a distance of 0.","title":"Fig 204. Free convection from a vertical plate"},{"content":" ExperimentSimulation $$ \\vcenter{M = 0.840} $$ ExperimentSimulation $$ \\vcenter{M = 0.885} $$ ExperimentSimulation $$ \\vcenter{M = 0.900} $$ ExperimentSimulation $$ \\vcenter{M = 0.946} $$ ExperimentSimulation $$ \\vcenter{M = 0.971} $$ \u0026ldquo;The spark shadowgraphs on these two pages have been arranged to show the shock-wave pattern growing into the subsonic field around a model of an artillery shell as its Mach number is increased. The shell is in free flight through the atmosphere at less than \\(1.5^{\\circ}\\) incidence. These five photographs are from four different firings, in each of which the Mach number is gradually decreasing as the shell decelerates.\u0026rdquo; Photographs by A. C. Charters, in von Kármán 1947\nTheory\u003e Theory # The Euler equations\u003e The Euler equations # Flow is defined with the Navier-Stokes equations, which is a set of functions describing the conservation of mass, momentum and energy: $$ \\begin{cases} \\frac{\\partial \\rho}{\\partial t} + \\nabla \\cdot (\\rho \\mathbf{u}) = 0 \\\\ \\rho(\\frac{\\partial \\mathbf{u}}{\\partial t} + \\mathbf{u} \\cdot \\nabla \\mathbf{u})= - \\nabla p + \\nabla \\cdot { \\mu[\\nabla \\mathbf{u} + (\\nabla \\mathbf{u})^T - \\frac{2}{3}(\\nabla \\cdot u) I]+\\zeta(\\nabla \\cdot \\mathbf{u})I} + \\rho g \\\\ \\frac{\\partial e}{\\partial t} + \\mathbf{u} \\cdot \\nabla e = - \\frac{p}{\\rho} \\nabla \\cdot \\mathbf{u} \\end{cases} $$\nTo solve this set of equations is computationally heavy which is why simplifications are used, in some cases with high velocities it could be argued that the viscosity \\(\\mu\\) and bulk viscosity \\(\\zeta\\) are negligible and thus left out of the equation leaving the Euler equations: $$ \\begin{cases} \\frac{\\partial \\rho}{\\partial t} + \\nabla \\cdot (\\rho \\mathbf{u}) = 0 \\\\ \\rho(\\frac{\\partial \\mathbf{u}}{\\partial t} + \\mathbf{u} \\cdot \\nabla \\mathbf{u})= - \\nabla p + \\rho g \\\\ \\frac{\\partial e}{\\partial t} + \\mathbf{u} \\cdot \\nabla e = - \\frac{p}{\\rho} \\nabla \\cdot \\mathbf{u} \\end{cases} $$ These equations have four unknowns, thus to solve for a compressible gas the ideal gas law is used as an extra equation.\nSimulation\u003e Simulation # Euler equations compared to k-epsilon model\u003e Euler equations compared to k-epsilon model # The simulations above used the k-epsilon model but were also run without a viscosity model. The following results were obtained: k-epsilonEuler $$ \\vcenter{M = 0.840} $$ k-epsilonEuler $$ \\vcenter{M = 0.885} $$ k-epsilonEuler $$ \\vcenter{M = 0.900} $$ k-epsilonEuler $$ \\vcenter{M = 0.946} $$ k-epsilonEuler $$ \\vcenter{M = 0.971} $$ The first thing that can be seen is that with the Euler equation, the shockwaves are bigger at lower Mach. This is probably due to the no-slip wall and therefore higher flow velocity at the wall. Further, the wake downstream of the artillery shell is way smaller for the Euler simulations. This is due to the wake being a form of flow separation and turbulence, the last of which is not modelled in the Euler equations.\nSimulation set-up \u0026amp; Visualization\u003e Simulation set-up \u0026amp; Visualization # The simulation set-up and visualization can be read in figure 253.\n","date":"1 May 2024","permalink":"/chapters/09-subsonic/fig223/","section":"Chapters","summary":"ExperimentSimulation $$ \\vcenter{M = 0.840} $$ ExperimentSimulation $$ \\vcenter{M = 0.885} $$ ExperimentSimulation $$ \\vcenter{M = 0.900} $$ ExperimentSimulation $$ \\vcenter{M = 0.946} $$ ExperimentSimulation $$ \\vcenter{M = 0.971} $$ \u0026ldquo;The spark shadowgraphs on these two pages have been arranged to show the shock-wave pattern growing into the subsonic field around a model of an artillery shell as its Mach number is increased.","title":"Fig 223. Projectile at high subsonic speeds"},{"content":" ExperimentSimulation $$ \\vcenter{M = 0.978} $$ ExperimentSimulation $$ \\vcenter{M = 0.990} $$ \u0026ldquo;Still closer to the speed of sound, the shock-wave pattern of the preceding pages had spread laterally to great distances. These two photographs are from the same firing, so that the second one was actually taken earlier in the trajectory\u0026rdquo; Photographs by A. C. Charters, in von Kármán 1947\nTheory\u003e Theory # These simulations are part of a series together with figure 223 and figure 253 all done in transonic flow. The shockwaves and convex corner flow around the bullet make for switches between subsonic and supersonic flow. Here the main challenges of simulating transonic flow are discussed.\nThe Transonic flow problem\u003e The Transonic flow problem # For transonic flow the change in velocity around Mach 1 complicated early CFD computations. This can be seen in the non-linear perturbation velocity potential equation for transonic flow, which was used back in the day and follows from the definition of the velocity potential \\(\\phi\\) for 2D: $$ \\mathbf{V} = \\nabla \\phi \\\\ u = V_{\\infty} + \\hat{u} \\\\ v = \\hat{v} $$ With u and v being cartesian velocity components in x and y direction respectively, V being the velocity magnitude, \\(V_{\\infty}\\) flow field velocity and \\(\\hat{u}\\) and \\(\\hat{v}\\) being velocity perturbations (increments). Uniform to perturbed flow [1] Using \\(\\phi\\) to obtain one equation that represents a combination of the continuity, momentum and energy equations is useful to use one governing equation with one unknown. This creates the final equation:\n$$ (1-M_{\\infty}^2) \\frac{\\partial^2 \\hat{\\phi}}{\\partial x^2} + \\frac{ \\partial^2 \\hat{\\phi}}{\\partial y^2} = M_{\\infty}^2 [(\\gamma + 1)\\frac{\\partial \\hat{\\phi}}{\\partial x}\\frac{1}{V_{\\infty}}] \\frac{\\partial^2 \\hat{\\phi}}{\\partial x^2} $$ It is important to note here that this equation approximates the physics of a steady, compressible, inviscid 2D flow. The important point here is the factor (\\(1-M_{\\infty}^2\\)), which will be higher than 0 in subsonic flow and lower than 0 for supersonic flow and result in a different form of partial differential equation. In supersonic flow, it is a hyperbolic partial differential equation while in subsonic it is an elliptic partial differential equation, which is a big difference in math type. This shows a big difference in physics between subsonic and supersonic flow.\nThis equation was used in the earliest transonic CFD simulations. Later Euler equations were used, which predict shockwaves pretty well, but do not fare well in transonic flow due to shock wave interaction with the boundary layer, which needs viscosity. For a comparison a look can be taken at Figure 223. The current state of the art for transonic flow is solving the Navier-Stokes equations with the viscous term. Here the turbulence model frequently being the Achilles heel of these complicated calculations [1].\nTransonic buffet\u003e Transonic buffet # In transonic flow, the interactions between shock waves and separated shear layers result in self-sustained low-frequency oscillations of the shock wave also called transonic buffet. Some aspects of these oscillations are still incomprehensible [2]. A transient computational setup with LES is needed to show these oscillations and comes at a high cost. The steady-state simulations done estimate the average position.\nNumerical dissipation\u003e Numerical dissipation # CFD simulations currently in transonic flow are very sensitive to artificial dissipation characteristics of the computational fluid algorithm. With a bad algorithm problems such as difficult convergence, poor reliability and difficulty to accurately simulate shock position will occur [3].\nNumerical dissipation is an artificial effect that is introduced when solving the governing equations, which acts as an unnatural damping of the variables at play. This with as main goal to damp oscillations and smooth out high-frequency noise and thus stabilizing the solution ensuring convergence [4]. Turbulence models and mesh refinement around areas with steep gradients lower the amount of numerical dissipation needed.\nSimulation \u0026amp; Visualization\u003e Simulation \u0026amp; Visualization # About the simulation and visualization can be read in figure 253.\nReferences\u003e References # [1] Fundamentals-of-aerodynamics-6-Edition.pdf [2] https://www.sciencedirect.com/science/article/pii/S0376042117300271 [3] https://ieeexplore.ieee.org/document/10082199 [4] https://www.nas.nasa.gov/assets/nas/pdf/ams/2018/introtocfd/Intro2CFD_Lecture7_Zingg.pdf) ","date":"1 May 2024","permalink":"/chapters/09-subsonic/fig224/","section":"Chapters","summary":"ExperimentSimulation $$ \\vcenter{M = 0.978} $$ ExperimentSimulation $$ \\vcenter{M = 0.990} $$ \u0026ldquo;Still closer to the speed of sound, the shock-wave pattern of the preceding pages had spread laterally to great distances.","title":"Fig 224. Projectile at near-sonic speed "},{"content":" ExperimentSimulation \u0026ldquo;The model artillery shell of figures 223 and 224 is shown here still earlier in its trajectory, when it is flying at a slightly supersonic speed. A detached bow wave precedes it, and the distant field is quite different, but the pattern near the body is almost identical to that shown in figure 224 for a slightly subsonic speed. This illustrates how the near field is \u0026lsquo;frozen\u0026rsquo; as the free-stream Mach number passes through unity.\u0026rdquo; Photograph by A. C. Charters\nTheory\u003e Theory # Shockwaves\u003e Shockwaves # Shockwaves appear with flows that reach velocities higher than 1 Mach, the speed of sound through the medium. This is due to that pressure changes in a medium are diffused at the speed of sound of that medium. When a disturbance makes flow streamlines with a higher Mach number than 1 turn in on itself, the disturbance travels faster than the pressure diffusion and is unable to send a \u0026lsquo;signal\u0026rsquo; to the upstream flow. This is while in subsonic flow the particles in front of the object are already \u0026lsquo;informed\u0026rsquo; by pressure changes in front of the disturbance and move out of the way. In supersonic flow, the particles do not move out of the way and as the disturbance arrives nanoscale molecular collisions find place.\nWhen a body moves through a fluid at supersonic speed it creates a pressure front moving at supersonic speed pushing on the air. This pushing creates a high-pressure region resulting in a very thin shockwave with abrupt changes in flow properties, with an increase in potential energy at the cost of kinetic energy. The density, pressure, and temperature increase, while the velocity decreases [1].\nThe angle of the shockwave of a particle is decided by the speed of sound as the pressure signal relative to the particle travels in all directions at sound velocity minus relative flight velocity depicted in the following figure: Point source moving in a compressible fluid. a, stationary. b, half the speed of sound. c, at the speed of sound. d, twice speed of sound [2] The leading curved shockwave is called a detached shock or bow shock. This happens as the change in flow direction through a shockwave is too small and the fluid cannot move out of the way, as is the case with a blunt body [3]. The distance between the body and the detached shock is based on the flow velocity and the body shape. A lower velocity and a wider body are factors that make the distance between the detached shock and the body larger [2]. Left oblique shock (figure 261), right bow shock The detached shockwave creates a subsonic flow in front of the body. Further from the object the detached shockwave gets a lower bending radius and becomes oblique.\nTransonic convex corner flow\u003e Transonic convex corner flow # Supersonic flow going around a convex corner is known as a Prandtl-Meyer expansion, which increases the dynamic properties and decreases the static properties as the \u0026ldquo;opposite\u0026rdquo; of a shockwave. In transonic flow, the flow region of this experiment, it is a bit different due to the flow arriving at the convex corner being subsonic. The flow accelerates around the corner to become supersonic and undergoes extra acceleration in the expansion Prandtl-Meyer fan, downstream shockwaves find place. Schematic drawing for biconvex-corner flows with \\({\\lambda}\\)-shock waves [4] This results in a so-called \\({\\lambda}\\)-shock structure, which is a prime example of shock wave boundary interaction. The first leg is where shock-induced boundary layer separation takes place, the second leg is near the reattachment location. Boundary layer separation can be attributed to the adverse pressure gradient, the static pressure increases over the boundary layer, due to the shockwave. The increase in potential energy is big enough to reduce the boundary velocity enough to reverse it creating flow separation. Schematic drawing of boundary layer separation interaction with \\({\\lambda}\\)-shock waves [5] For more information on detached flow visit Figure 42\nSimulation\u003e Simulation # Domain\u003e Domain # While making the 2D domain the shape was traced from the original image, and small details on the body were added based on the shock- and expansion waves. For example, a small increase in diameter at the middle of the shell is found based on expansion waves.\nPrevious Next Further downstream of the body, figures 223 and 224 show biconvex-corner flows for this shell model, which was tried to model correctly with iterations, where in the end a small curve showed the best result. There is suspected to be a hole based on images of different types of 155mm artillery shells and iterating over CFD simulations. CFD experiments have shown that the cutout significantly lowers the height of the back \\({\\lambda}\\)-shock in high subsonic velocities, mainly in Figure 223 and Figure 224. The detached shock distance is still to close to the shell, after iterations a small dimple in the nose seemed to move the shockwave a bit more upstream, however the real bow shock location was not achieved.\nThe domain is made for an axisymmetric simulation by using an axis boundary at the x-axis, a no-slip wall boundary condition for the object, and a far-field boundary for the other three boundaries. Gauge pressure and temperature have been found irrelevant to the shape of the waves, as the temperature influences the speed of sound and only the Mach number is used as input. Pressure\nMeshing\u003e Meshing # The start mesh is a quadrilateral hybrid mesh of 0.0055 m, which is quite small to have a better discretization of the problem before adaptive refinement. Hybrid meshes were easy to use while iterating a complex shape, but pay a price in memory, execution speed, and numerical convergence compared to a well-structured mesh. Another downside is that the mesh has quite a high boundary layer with a loss of detail in this region. After running the simulation, the mesh is refined by using ansys adaptive refinement tools around density changes, to capture the shockwaves with higher detail, and refined again around the shell body to better caption shock wave boundary layer interaction. More refinement was not possible at the risk of convergence as the Ansys tool only divides one quadrilateral element into four which is bad for the volume ratio between cells [6]. Previous Next Fluid model\u003e Fluid model # The solver type used is density-based, as it gives better accuracy for high-speed flow with shockwaves than pressure-based solvers. Because of the compressible flow used in calculations for shockwaves the ideal gas law is used for air, as an extra equation on top of the compressible Navier-Stokes equations to solve for four unknowns: Density, pressure, velocity, and energy or enthalpy.\nFor viscosity, the realizable k-epsilon model is used, as viscous forces are rarely dominant in the Navier-Stokes equations due to high velocity but are important to capture shock wave boundary layer interaction. More on viscosity models can be found in Figure 47. Further, Sutherland\u0026rsquo;s Law is used for the viscosity changes of the air due to temperature changes.\nVisualization\u003e Visualization # The original image is a spark photograph in which the different densities around the artillery shell act as lenses and refract the light. That is why on the original experiment the shock waves’ colour went from dark to light, due to the big density change and thus big change in refraction.\nParaview is used for visualization, using two gradient filters on density to create a second-order gradient in the horizontal direction. This is to create the dark-to-white shockwaves in the original image. The gradient filter has some opacity, and the colour scale values have been altered to fit the replicated image. The Transform filter is used to rotate the object around the z-axis to fit the original image.\nParaview filters References\u003e References # [1] https://www.americanscientist.org/article/high-speed-imaging-of-shock-waves-explosions-and-gunshots [2] https://arc.aiaa.org/doi/abs/10.2514/8.1394 [3] https://www.oxfordreference.com/display/10.1093/acref/9780198832102.001.0001/acref-9780198832102-e-1433 [4] https://arc.aiaa.org/doi/epdf/10.2514/1.J055430?src=getftr [5] https://www.sciencedirect.com/science/article/pii/B9780120864300500245?via%3Dihub [6] https://link.springer.com/chapter/10.1007/978-3-031-40594-5_7 ","date":"1 May 2024","permalink":"/chapters/11-supersonic/fig253/","section":"Chapters","summary":"ExperimentSimulation \u0026ldquo;The model artillery shell of figures 223 and 224 is shown here still earlier in its trajectory, when it is flying at a slightly supersonic speed. A detached bow wave precedes it, and the distant field is quite different, but the pattern near the body is almost identical to that shown in figure 224 for a slightly subsonic speed.","title":"Fig 253. Projectile at M=1.015"},{"content":"Welcome to this ongoing effort towards developing a CFD counterpart of the classic 1982 book \u0026ldquo;An Album of Fluid Motion\u0026rdquo; by professor Milton van Dyke. The original book comprises 278 photographs of fluid flow experiments, which professor van Dyke collected from colleagues around the world. In much the same way, the contributions to this website are made by many CFD enhusiasts. Care to contribute? Check out the contribute page, or let me know.\nIn his introduction, professor van Dyke writes: \u0026ldquo;Since 1958, \u0026hellip;, I have dreamed of someday assembling a collection of photographs of flow phenomena. To serve its purpose, however, such a collection needs to be inexpensive enough that it is readily accessible to students\u0026rdquo;. Keeping with this 1980s-style spirit of open access, this entire website is open-source.\n","date":"1 May 2024","permalink":"/","section":"","summary":"Welcome to this ongoing effort towards developing a CFD counterpart of the classic 1982 book \u0026ldquo;An Album of Fluid Motion\u0026rdquo; by professor Milton van Dyke. The original book comprises 278 photographs of fluid flow experiments, which professor van Dyke collected from colleagues around the world.","title":""},{"content":"","date":"1 May 2024","permalink":"/tags/ansys-fluent/","section":"Tags","summary":"","title":"Ansys Fluent"},{"content":" ","date":"1 May 2024","permalink":"/chapters/","section":"Chapters","summary":" ","title":"Chapters"},{"content":"Just like the original book, this \u0026ldquo;album of computational fluid motion\u0026rdquo; is made possible by the contributions of many CFD analysis colleagues. Below follows an overview. Would you like to contribute too? Have a look at the contribute page.\n","date":"1 May 2024","permalink":"/authors/","section":"Contributors","summary":"Just like the original book, this \u0026ldquo;album of computational fluid motion\u0026rdquo; is made possible by the contributions of many CFD analysis colleagues. Below follows an overview. Would you like to contribute too?","title":"Contributors"},{"content":"","date":"1 May 2024","permalink":"/tags/euler-equations/","section":"Tags","summary":"","title":"Euler equations"},{"content":"","date":"1 May 2024","permalink":"/tags/shockwaves/","section":"Tags","summary":"","title":"Shockwaves"},{"content":"","date":"1 May 2024","permalink":"/tags/subsonic/","section":"Tags","summary":"","title":"Subsonic"},{"content":"","date":"1 May 2024","permalink":"/tags/supersonic/","section":"Tags","summary":"","title":"Supersonic"},{"content":" ","date":"1 May 2024","permalink":"/tags/","section":"Tags","summary":" ","title":"Tags"},{"content":"","date":"1 May 2024","permalink":"/tags/transonic/","section":"Tags","summary":"","title":"Transonic"},{"content":"","date":"1 May 2024","permalink":"/authors/wouterlitjens/","section":"Contributors","summary":"","title":"Wouter Litjens"},{"content":"","date":"10 April 2024","permalink":"/tags/boussinesq/","section":"Tags","summary":"","title":"Boussinesq"},{"content":"","date":"10 April 2024","permalink":"/tags/compressible/","section":"Tags","summary":"","title":"Compressible"},{"content":"","date":"10 April 2024","permalink":"/authors/edgarverdijck/","section":"Contributors","summary":"","title":"Edgar Verdijck"},{"content":"","date":"10 April 2024","permalink":"/tags/grashof/","section":"Tags","summary":"","title":"Grashof"},{"content":"","date":"10 April 2024","permalink":"/tags/incompressible/","section":"Tags","summary":"","title":"Incompressible"},{"content":"","date":"10 April 2024","permalink":"/tags/natural-convection/","section":"Tags","summary":"","title":"Natural convection"},{"content":"","date":"10 April 2024","permalink":"/tags/richardson/","section":"Tags","summary":"","title":"Richardson"},{"content":"","date":"4 February 2024","permalink":"/tags/instability/","section":"Tags","summary":"","title":"instability"},{"content":"","date":"4 February 2024","permalink":"/tags/les/","section":"Tags","summary":"","title":"LES"},{"content":"","date":"4 February 2024","permalink":"/tags/paraview/","section":"Tags","summary":"","title":"ParaView"},{"content":"","date":"4 February 2024","permalink":"/tags/reynolds/","section":"Tags","summary":"","title":"Reynolds"},{"content":"","date":"4 February 2024","permalink":"/authors/sanderbooij/","section":"Contributors","summary":"","title":"Sander Booij"},{"content":"","date":"4 February 2024","permalink":"/tags/starccm+/","section":"Tags","summary":"","title":"StarCCM+"},{"content":"","date":"4 February 2024","permalink":"/tags/turbulence/","section":"Tags","summary":"","title":"Turbulence"},{"content":"","date":"4 February 2024","permalink":"/tags/wale/","section":"Tags","summary":"","title":"WALE"},{"content":"","date":"18 August 2023","permalink":"/tags/flow-past-cylinder/","section":"Tags","summary":"","title":"Flow past cylinder"},{"content":"","date":"18 August 2023","permalink":"/tags/fvm/","section":"Tags","summary":"","title":"FVM"},{"content":"","date":"18 August 2023","permalink":"/authors/jobsomhorst/","section":"Contributors","summary":"","title":"Job Somhorst"},{"content":"","date":"18 August 2023","permalink":"/tags/laminar/","section":"Tags","summary":"","title":"Laminar"},{"content":"","date":"18 August 2023","permalink":"/tags/separation/","section":"Tags","summary":"","title":"Separation"},{"content":"","date":"18 August 2023","permalink":"/tags/starccm/","section":"Tags","summary":"","title":"StarCCM"},{"content":"","date":"18 August 2023","permalink":"/tags/vortex-shedding/","section":"Tags","summary":"","title":"Vortex shedding"},{"content":"","date":"30 June 2023","permalink":"/tags/k-omega/","section":"Tags","summary":"","title":"k-omega"},{"content":"","date":"30 June 2023","permalink":"/authors/quentincarre/","section":"Contributors","summary":"","title":"Quentin Carré"},{"content":"","date":"30 June 2023","permalink":"/tags/sst/","section":"Tags","summary":"","title":"SST"},{"content":"","date":"18 June 2023","permalink":"/tags/k-epsilon/","section":"Tags","summary":"","title":"k-epsilon"},{"content":"","date":"8 April 2023","permalink":"/tags/creeping/","section":"Tags","summary":"","title":"Creeping"},{"content":"","date":"8 April 2023","permalink":"/tags/darcy/","section":"Tags","summary":"","title":"Darcy"},{"content":"","date":"8 April 2023","permalink":"/tags/fem/","section":"Tags","summary":"","title":"FEM"},{"content":"","date":"8 April 2023","permalink":"/tags/fenics/","section":"Tags","summary":"","title":"FEniCS"},{"content":"","date":"8 April 2023","permalink":"/authors/steinstoter/","section":"Contributors","summary":"","title":"Stein Stoter"},{"content":" ","date":"11 February 2023","permalink":"/chapters/01-creeping/","section":"Chapters","summary":" ","title":"1. Creeping flow"},{"content":"","date":"11 February 2023","permalink":"/series/creeping-flow/","section":"Series","summary":"","title":"Creeping flow"},{"content":"","date":"11 February 2023","permalink":"/series/","section":"Series","summary":"","title":"Series"},{"content":" ","date":"10 February 2023","permalink":"/chapters/02-laminar/","section":"Chapters","summary":" ","title":"2. Laminar flow"},{"content":"","date":"10 February 2023","permalink":"/series/laminar-flow/","section":"Series","summary":"","title":"Laminar flow"},{"content":" ","date":"9 February 2023","permalink":"/chapters/03-separation/","section":"Chapters","summary":" ","title":"3. Separation"},{"content":"","date":"9 February 2023","permalink":"/series/separation/","section":"Series","summary":"","title":"Separation"},{"content":" ","date":"8 February 2023","permalink":"/chapters/04-vortices/","section":"Chapters","summary":" ","title":"4. Vortices"},{"content":"","date":"8 February 2023","permalink":"/series/vortices/","section":"Series","summary":"","title":"Vortices"},{"content":" ","date":"7 February 2023","permalink":"/chapters/05-instability/","section":"Chapters","summary":" ","title":"5. Instability"},{"content":"","date":"7 February 2023","permalink":"/series/instability/","section":"Series","summary":"","title":"Instability"},{"content":" ","date":"6 February 2023","permalink":"/chapters/06-turbulence/","section":"Chapters","summary":" ","title":"6. Turbulence"},{"content":"","date":"6 February 2023","permalink":"/series/turbulence/","section":"Series","summary":"","title":"Turbulence"},{"content":" ","date":"5 February 2023","permalink":"/chapters/07-free-surface/","section":"Chapters","summary":" ","title":"7. Free-surface flow"},{"content":"","date":"5 February 2023","permalink":"/series/free-surface-flow/","section":"Series","summary":"","title":"Free-surface flow"},{"content":" ","date":"4 February 2023","permalink":"/chapters/08-convection/","section":"Chapters","summary":" ","title":"8. Natural convection"},{"content":"","date":"4 February 2023","permalink":"/series/natural-convection/","section":"Series","summary":"","title":"Natural convection"},{"content":" ","date":"3 February 2023","permalink":"/chapters/09-subsonic/","section":"Chapters","summary":" ","title":"9. Subsonic flow"},{"content":"","date":"3 February 2023","permalink":"/series/subsonic-flow/","section":"Series","summary":"","title":"Subsonic flow"},{"content":" ","date":"2 February 2023","permalink":"/chapters/10-shocks/","section":"Chapters","summary":" ","title":"10. Shock waves"},{"content":"","date":"2 February 2023","permalink":"/series/shock-waves/","section":"Series","summary":"","title":"Shock waves"},{"content":" ","date":"1 February 2023","permalink":"/chapters/11-supersonic/","section":"Chapters","summary":" ","title":"11. Supersonic flow"},{"content":"","date":"1 February 2023","permalink":"/series/supersonic-flow/","section":"Series","summary":"","title":"Supersonic flow"},{"content":"The progress made on this site relies largely on visitor contributions. Would you like to contribute? Here you find a how-to.\nBasic procedure\u003e Basic procedure As this site is automatically generated from the Pages Continuous Integration (CI) system of its gitlab repository, contributions to this site can be made directly through pull requests. The procedure is as follows:\nCreate a gitlab account. Fork the repository of this website. Make your changes (see details below). Create a pull-request. The last step requests a merge of your edited version of the website into the primary repository. Once your pull-request is accepted at the end of the primary repository, the changes automatically appear on the website.\nAdding a CFD analysis\u003e Adding a CFD analysis You can freely edit your fork (personal version) of the website. Additions to the different chapters are made by creating a new folder in the corresponding \u0026ldquo;content/chapters/\u0026hellip;\u0026rdquo; directory. This folder should be named \u0026ldquo;Fig\u0026hellip;\u0026rdquo;, with \u0026hellip; the figure number corresponding to the number in the original book. In this new folder, you add a .jpg screenshot of the figure from the book (not exceeding 80kb) called Featured.jpg. This automatically becomes the background of the new page, and the thumbnail of the new contribution in the chapters page. Additionally, you add an index.md markdown file, with a description of your CFD analysis. Have a look at the other examples to make use of all offered functionality.\nAdding an authors tag\u003e Adding an authors tag We\u0026rsquo;re all about giving credit. Adding your name to your contribution requires three things:\nA tag to your name in the \u0026ldquo;authors\u0026rdquo; variable in the index.md file of your contribution. A folder with the name of the chosen tag in the \u0026ldquo;authors\u0026rdquo; directory. In this folder you copy over an _index.md from any other author, and change the \u0026ldquo;title\u0026rdquo; variable to your name. A .json file with the name of the chosen tag in the \u0026ldquo;data/authors\u0026rdquo; directory. You can copy over any other author\u0026rsquo;s file and change the information. If you also add a .jpg image to your folder, this automatically becomes your thumbnail. Please ensure such an image is scaled down to a size of no more than 50kb.\nRunning a local version\u003e Running a local version While editing your work, you may want to run a local version of the website to ensure that everything looks good. To do so, you need to install Hugo, a static site generator. Be sure to install the latest version (i.e., at least v0.110.0). Then, after having navigated to the repository directory, you run the following command in the command-line interface:\nhugo server after which you can open localhost:1313/album in your browser to view the website.\n","date":"13 June 2022","permalink":"/contribute/","section":"Contribute","summary":"The progress made on this site relies largely on visitor contributions. Would you like to contribute? Here you find a how-to.\nBasic procedure\u003e Basic procedure As this site is automatically generated from the Pages Continuous Integration (CI) system of its gitlab repository, contributions to this site can be made directly through pull requests.","title":"Contribute"},{"content":"The 1982 book is no longer in print, and second-hand versions are hard to come by at the price point that prof. van Dyke would have had in mind when he wrote \u0026quot;\u0026hellip; needs to be inexpensive enough that it is readily accessible to students\u0026quot;. To make his work enjoyable for all, Parabolic press has made the below pdf version of the book freely available. Of course, it is not quite the same as a hard-copy that you can flip through, so I\u0026rsquo;d encourage you to take the opportunity if you can get your hands on one of those.\n","date":"13 June 2022","permalink":"/book/","section":"The original book","summary":"The 1982 book is no longer in print, and second-hand versions are hard to come by at the price point that prof. van Dyke would have had in mind when he wrote \u0026quot;\u0026hellip; needs to be inexpensive enough that it is readily accessible to students\u0026quot;.","title":"The original book"},{"content":"","date":"1 January 0001","permalink":"/categories/","section":"Categories","summary":"","title":"Categories"},{"content":"The (limitations to the) rights to reuse images from the 1982 book used to be described on the Parabolic press website, which is no longer active. It used to say \u0026ldquo;the images may be scanned and used in lectures, face-to-face instruction, and conference presentations. Inclusion of the images in publications is not allowed under Fair Use, and permission must be requested from the original photographers for those uses\u0026rdquo;, referring to these Fair Use guidelines.\nThis website is inteded as an educational tool. As such, I believe that I am adhering to the fair use guidelines. Please contact me for any comments in this regard.\nNaturally, the permissions decribed by Parabolic press carry over to the images that may be found on this website.\n","date":"1 January 0001","permalink":"/permissions/","section":"Permissions","summary":"The (limitations to the) rights to reuse images from the 1982 book used to be described on the Parabolic press website, which is no longer active. It used to say \u0026ldquo;the images may be scanned and used in lectures, face-to-face instruction, and conference presentations.","title":"Permissions"},{"content":"","date":"1 January 0001","permalink":"/allposts/","section":"Posts overview","summary":"","title":"Posts overview"},{"content":"","date":"1 January 0001","permalink":"/authors/roeljaspars/","section":"Contributors","summary":"","title":"Roel Jaspars"}]