Backward facing step Homework Department of Fluid Mechanics Budapest University of Technology and Economics Budapest, 2010 autumn Updated: October 26, 2010
CONTENTS i Contents 1 Introduction 1 2 The problem 2 3 The flow 2 3.1 Geometry.............................. 2 3.1.1 α.............................. 2 3.2 Reynolds number.......................... 3 4 Meshing 4 5 Boundary condition 5 6 Quality Issues 6 6.1 Iteration error (2p)......................... 6 6.2 Validation of the results (7p).................... 6 6.2.1 Skin friction coefficient (1p)................ 6 6.2.2 Reattachment position (1p)................. 6 6.2.3 Pressure coefficient (1p).................. 7 6.2.4 Velocity and Reynold stress profiles (4p)......... 7 6.3 Mesh dependency of the solution (3p)............... 7 6.4 Influence of the scheme (2p).................... 8 6.5 Effect of inlet profiles and outlet condition and geometrical uncertainty (3p).............................. 8 6.6 Effect of the turbulence model (3p)................. 8 A Appendix 9 A.1 Mesh requirements......................... 9 A.2 Resolution of gradients....................... 9 A.3 Skewness.............................. 9 A.4 Cell size variation.......................... 9 A.5 Wall boundary layer resolution................... 9
NOMENCLATURE 1 1 Introduction The aim of this homework is to have and extensive experience in doing CFD simulation preciously considering a number of the uncertainties associated with CFD. Since the goal is to carry out a detailed investigation, scores will be only given for more advanced topic, and not for the basic steps of the simulation.
2 THE PROBLEM 2 2 The problem Figure 1: The geometry and main flow features In engineering problems strong change in the cross section perpendicularly to the flow direction is an often occurring situation. The most typical example is being the Borda-Carnot sudden flow expansion. 3 The flow 3.1 Geometry Channel width in the spanwise direction (perpendicular to the Fig. 1) is constant, side boundary layer thickening effect can be neglected and 2D flow can be assumed. The characteristic size of the flow is the step height (H).The channel height upstream from the step is Y 0 = 8H. The coordinate system is placed in the bottom corner of the step. Inlet boundary of the geometrical model is placed at the position x = 4H (upstream from the step). Draw the geometry using H = 1m to fit with the provided inlet profile, and with the validation datas. In the first attempt you can place the outlet boundary at the x = 15H position. 3.1.1 α In order to control the pressure gradient the upper wall of the channel can be turned around the point x = 0, y = 9H by α angle. Value of α is dependent on the first letter of your family name, for A-K α = 0 and for L-Z α = 6.
3 THE FLOW 3 3.2 Reynolds number The Reynolds number based on the reference velocity U ref and the step height (H) is Re H = 37423. For H = 1m it is most easily satisfied by ρ = 1 and ν = 1/Re H = 2.672153e 05.
4 MESHING 4 4 Meshing Create a simple block structured mesh (L type), where you try to resolve the wall boundary layers and the free shear layer downstream the corner.
5 BOUNDARY CONDITION 5 5 Boundary condition Use the provided profiles Inlet Profile as inlet condition for the required quantities. Check their shape in Fluent.
6 QUALITY ISSUES 6 6 Quality Issues This is the section where you will be able to receive scores for your homework. To be able to evaluate your work you have to write a detailed report describing the mesh and further numerical and physical setting of your simulation. You report should enable a complete reproduction of the simulation when your mesh is provided. But give also some information on the mesh as well, typical and smallest, largest cell sizes, volume changes in critical regions adn so on. 6.1 Iteration error (2p) Show the appropriate reduction of the iteration error, i.e. the convergence of the solution. Plot the variation of some (at least 3) integral result of the simulation with the iteration number. Select an appropriate scale for the abscissa be able to recognise errors in the order of 0.1% of the final result. This is most easily done when you write the results to a file, and post-process it a-posteriori. 6.2 Validation of the results (7p) For making your work easier, the measurement results are collected in a spreadsheet for each flow case. You can download it for the 0 case and for the 6 case. 6.2.1 Skin friction coefficient (1p) Plot the skin friction coefficient (c f ) on the bottom wall and compare it to the measurement (including uncertainty). c f def = 2τ w For the proposed non-dimensional setup you only need to multiply the wall shear stress by 2. The wall shear stress (τ w ) is the x component of the shear stress (τ w,x ). You can use (Plot -> XY Plot...) tool in Fluent with the write to file option. For the comparison you can copy-paste your result in the provided spreadsheet file. ρu 2 r 6.2.2 Reattachment position (1p) The reattachment position is defined at the point where the skin friction changes its sign from negative to positive (since the flow reattaches back to the wall and the (1)
6 QUALITY ISSUES 7 wall boundary layer profile develops again). In the simulation you can have multiple singular points on the bottom wall (where the skin friction is zero) because of small corner vortices at the corner of the step. For the evaluation of the reattachment point first you should evaluate the singular point of the skin friction. You have to create a iso-surface of zero t x on the bottom wall (you should visualise them to make sure of your result). The reattachment position will be the maximum x coordinate of the this set of points. You can evaluate it as Surface Integral... -> Vertex Maximum... -> Grid... -> X-Coord. To check your evaluation plot the stream lines and guess the position of the reattachment. 6.2.3 Pressure coefficient (1p) Carry out the same procedure for the pressure coefficient (c p ) on the bottom wall. c p def = 2(p w p ref ) Since the reference pressure is not know form the database, you have to match c p at a given point to the measurement. ρu 2 r 6.2.4 Velocity and Reynold stress profiles (4p) The comparison of the mean velocity components (u, v ) and the most dominant, the Reynolds shear stress component u v should be exported at 5 different streamwise position (for the exact position see the spreadsheet). You can create the profiles by creating an Iso-Surface of constant x coordinate. For the evaluation of the Reynolds stress you should use the Custom Field Function in Fluent (Define... -> Custom Field Function...). Since we use eddy viscosity models the expression is the following: (2) u i u j = 2 3 kδ i,j 2ν t S ij (3) For our actual problem i = 1 (x) and j = 2 (y). It means: u v = 2 3 kδ 1,2 2ν t S xy = ν t ( x v + y u) (4) 6.3 Mesh dependency of the solution (3p) First make sure that your mesh fulfil most of the mesh requirements (A.1) including the issues related to wall resolution. The upper wall should be Hi-Re model, and the other walls Low-Re (2p). Make a mesh dependency study to show the
6 QUALITY ISSUES 8 mesh dependency. Prepare 2 further meshes with remarkable refinement (the cell number increase should be at least 1.5-times in each direction), and show the dependency of the result on a Richardson plot (some values versus 1/N, where N is the number of cells). 6.4 Influence of the scheme (2p) Change the convection flux scheme and show its results on the solution (1p). Change the type of the pressure velocity coupling (the largest difference should exist between segregated and coupled approach) and show the difference in the convergence behaviour (1p). 6.5 Effect of inlet profiles and outlet condition and geometrical uncertainty (3p) Change the inlet turbulence or velocity profile to constant values to see its effect on the solution (1.5p). (You can choose between changing the velocity or the turbulence profile) Those of you having α = 6 use two different outlet boundary conditions to investigate its effect (1.5p). Those of you having α = 0 modify the downstream extent of the domain and investigate its effect (1.5p). Alternatively instead of investigating the outlet effect you can investigate the uncertainty of the geometry by running a simulation of perturbed α textp = 0.5. 6.6 Effect of the turbulence model (3p) Change the turbulence model (among two equation models) and show its effect on the results. Obviously both method should be reasonable for the actual simulation (Low-Re and pressure gradient effect should be incorporated, use standard k ε only as benchmark as third model if you wish).
A APPENDIX 9 A Appendix A.1 Mesh requirements The most important mesh properties are summarised which are to be fulfilled for a generic finite volume solution. Beside these requirements which can be most easily fulfilled by high amount of cells, you have to keep the number of cells as low as possible to have a feasible computational time. A.2 Resolution of gradients The major rule in meshing is to create a mesh which is able to resolve the gradients of the flow appropriately. This means that without the knowledge of the flow the mesh can not be created. Conclusion: The meshing and simulation the flow is a iterative process. A.3 Skewness ff The distortion of the cells has to be reduced as much as possible, the mesh faces has to be close to perpendicular and placed in the middle to the line connecting the center of the two neighbouring cells. This property used to be measured by the normalised angles compared to the optimal angles of a cell. You can check this property in Fluent for example by plotting the contours of the Equiangle Skewness scalar. A.4 Cell size variation The volume of neighbouring cells should be close to each-other, value of 5% is optimal but values below 5%0 are required even for complex geometries. You can check this property in Fluent by (Adapt -> Volume -> Change). A.5 Wall boundary layer resolution The appropriate resolution of the wall boundary layer can be checked by the number of cells included in the boundary layer perpendicularly to the wall. At least 20 cells should be used. To check what type of wall resolution you use for turbulence modelling you can plot the y + distribution in Fluent form the menu Turbulence -> Y Plus.