Assessment of the numerical solver

Transcription

1 Chapter 5 Assessment of the numerical solver In this chapter the numerical methods described in the previous chapter are validated and benchmarked by applying them to some relatively simple test cases for which detailed experiments, LES simulation data or Direct Numerical Simulation data exist. These different test cases have been chosen for independent testing of especially the Solve4ke and Solve4LES modules of the Solver code. 5.1 Introduction The numerical method outlined in the previous chapter has been implemented into the code Solver and the modification of the Solver code to perform either Reynolds Averaged Simulation with the k-i turbulence model (Solve4kI) or LES (Solve4LES) with different subgrid scale models. This code has been written in order to meet the needs of this project, due to the fact that no initial documentation was provided for the LESROOM code, and that the computational speed of the CFX code was generally too slow when LES were used. The verification of the CFD code is a very important point, nowadays. In Freitas (1995) several commercial CFD codes are tested and benchmarked against each other in different cases, such as the backward facing-step, three-dimensional shear driven cavity flow, turbulent flow around a square crosssection cylinder and a turbulent flow in a duct with a 180 bend. Though the codes do not use the same numerical methods, such as discretization, grid resolution, turbulence models etc., the dispersion of the results is not encouraging, although some of the errors were introduced by more or less improper use of the CFD codes. This would furthermore encourage the need for validation of the code described before applying it to ventilation problems. In the following section some test cases have been calculated by using the Solver code and its different modules. This was done to evaluate the performance of the code and its stability and reliability. 69

2 The five test cases were: ± Laminar flow in a square driven cavity (basic solver). ± Laminar flow over a backward-facing step (basic solver). ± Turbulent flow over a backward-facing step (basic solver and k-i implementation). ± Turbulent flow around a square cube in a channel (turbulent flow and LES). ± Turbulent flow around a surface mounted obstacle in a channel (turbulent flow and LES). The calculations were performed on a SGI Indy (R4600 Mhz, a SGI Onyx workstation (R8000 CPU at 75 Mhz) or SGI Origin 200 (R Mhz). But the CPU time was not recorded for these test cases. For the laminar flow case the simulation was stopped when all the residuals were below The performance ratio between the different computer in terms of cpu-time are roughly 6:2:1, respectively. 5.2 Validations of Solve4k and Solve4LES Laminar flow in a square driven cavity The laminar square driven cavity problem was selected as the first test case. It is one of the simplest fluid flow configurations, yet it manifests the full non-linearity of the Navier-Stokes equations. The separation and reattachment of turbulent flows occur in many practical engineering applications, both in internal systems such as diffusers, combustors and channels with sudden expansions, and in external ones like those around air foils and buildings. In these situations the flow experiences an adverse pressure gradient, i.e. the pressure increases in the direction of the flow, which causes the boundary layer to separate from the solid surface. The flow subsequently reattaches downstream forming a recirculating bubble. The configuration consists of a square cavity with one of the walls moving in its plane at a constant speed (Figure 5.1). This will introduce a vortical flow inside the cavity. The problem has been widely used for validation of the incompressible flow solver because its solution is available in a number of sources in literature. The driven cavity flow problem has been solved by a vorticity and stream function method by Ghia et al.,(1982) and by a primitive variable method by Hortman et al. (1990) including many other researchers (for instand Agarwal, 1981 and Rubin and Khosla, 1981). It has been shown that the higher the Reynolds number the more pronounced is the effect of the numerical error introduced by the solution methods. In this particular test a Reynolds number of 1000 has been used for the validations. Where the Reynolds number is calculated as: (5.1) A uniform grid has been used on the square flow domain and a set of grids ranging in sizes from 32x32 to 256x256 has been used to check the computed results and to illustrate grid independence. A second order centre/upwind differencing scheme was used. Together with a no-slip condition at the walls. 70

3 The simulation was performed in two dimensions to verify the correct implementation of the Navier- Stokes equations, the second order accuracy of the convection-diffusion discretization scheme, the boundary conditions and the Rhie-Chow interpolation. Figure 5.1:Configuration of a laminar square driven cavity. In Figures 5.2 and 5.3, the calculated velocity profile along the vertical and horizontal centre-line of the cavity is depicted and compared with the results of Ghia et. al. (1982). The centre line profile shows a recirculating flow in the cavity which is more apparent in some of the streamline plots (Figure 5.5) Figure 5.2: Driven cavity flow; calculated vertical Figure 5.3: Driven cavity flow; calculated centre line velocity profiles compared to horizontal centre line velocity profiles compared benchmark results of Ghia et.al. (1982) for to benchmark results of Ghia et.al. (1982) for different grid refinement. different grid refinement. 71

4 Figure 5.4: Vector plot of a driven cavity flow from the present code. Figure 5.5: Streamline for a driven cavity flow from the present code. The streamline plot shows a primary vortex at the centre of the cavity with two secondary vortices at the lower left and right corners. In figures 5.2 and 5.3 the different solutions with a range of different grid sizes from 32x32 to 256x256 are also depicted. The essential flow characteristics are captured with a low grid resolution. Further grid refinement does not change the solution substantially. Grid size Error Fx ( = Fy ) 32x x x x Table 5.1: Error reduction measured as the difference between the benchmark results by Ghia et al. (1982) at the center of the cavity and the computed velocity for different grid resolutions. In table 5.1 an error analysis has been performed by taking the benchmark results by Ghia et al. (1982) as the reference and computing the error of the velocity predicted at the centre. This shows a reduction in the error as the grid is refined. Error reduces by close to a factor of four when the grid size is halved. This indicates that the second order scheme used for discretization is accurate. In the following sections a non-equidistant grid has been used in order to capture important effects close to the wall. Using uniform grids would have been rather uneconomical for most flows, especially wall-bounded flows. This is due to large wall-normal gradients, which require a very high spatial resolution in the wall normal direction. The non-equidistant grid was generated by using a hyperbolic tangent function 72

5 which transforms an equidistant grid in into a non-uniform grid in the y-direction. (5.2) Here E determines the slope and [ the inflection point of the function. y y Figure 5.6: Equidistant grid before the transformation by the hyperbolic function. ~ y y ~ Figure 5.7: The non-equidistant grid after the transformation by the hyperbolic function. The slope of the red curve is given by E and the inflection point (here at zero) by I Laminar flow over a backward-facing step The laminar flow over a backward-facing step is one of the classical problems for testing the capabilities of a solution scheme to solve recirculating flow problems. The flow geometry consists of a long channel where a fully developed flow enters the domain, and goes through a sudden expansion. The channel expansion is 1.94 (fig 5.8). The channel is long enough to make the flow exciting the domain as a fully developed flow. The sudden expansion gives rise to one or more regions of recirculations depending upon the Reynolds number of the entering flow. Both experimental and numerical solutions to this test case are available in literature for a range of Reynolds numbers, e.g. Armaly et al. (1983) and Kim and Moin (1985). The turbulent case has also been considered by a number of authors using either LES (Akselvoll and Moin,1995), and DNS (Le and Moin, 1994; Le et al., 1997) to name a few. In this current test case the setup resembles that of Armaly et al. (1983). The entrance region of height h, where the flow becomes fully developed before going through the sudden expansion was at first not considered as it is done by Armaly et al.(1983). The experimental configuration and condition of Armaly et.al.(1983) was that the flow originates in a long pipe which has a sudden expansion. Here the entrance pipe was at first neglected. 73

6 And instead, a boundary profile of a fully-developed parabolic profile at the inlet as given below has been introduced: (5.3) Afterwards a small inlet section was added, since the results were in better agreement with the experimental data of Armaly et al. (1983). The inlet section was 6 times the step height (6]h). The following material property is used in the simulation, s is the step height (=00049m) and the height of the entering flow, h, is m. The channel length L is 0.6m, which is close to 60 times the channel width. The Reynolds number is based on the hydraulic diameter of the entering flow, i.e. 2h. U in h Blockage applied to square s u(y) x detach x sep L x re-attach fully developed flow Figure 5.8: Configuration of the laminar flow over the backward-facing step. The problem is solved and recorded for three different Reynolds numbers 200, 450 and 1000 and the solution is compared with the experimental results obtained by Armaly et al. (1983). The second-order Quick differencing is used for the convective parts. This was because the central differencing did not perform very well for this flow. The simulation was at first performed in two dimensions. And the mesh was stretched using the hyperbolic function towards the wall and the centre of the channel (figure 5.9) Figure 5.9: Grid configuration for the laminar back-facing step. A recirculating zone is observed after the step for all Reynolds numbers. The reattachment lengths of this and other recirculating zones have been reported in (the literature) Freitas,(1995) and references in there. Table 5.2 shows the reattachment length of the primary recirculating zone on the side of the step normalized by the step height. These are computed as being the first boundary grid points along the bottom of the step where the x-velocity changes sign. A good agreement between the experiment and simulations for Reynolds numbers until 450 is found. But beyond that the size of the recirculating zone 74

7 declines compared to the experimental observation made by Armaly et al. Armaly et al(1983) Experimental Armaly et al. (1983) Numerical Solver (Present Simulation) Re = Re = Re = Table 5.2: Normalized reattachment length x sep /s for the primary recirculating zone. The large deviation of the results in the case of Re = 1000 from the experimental results can be explained by the fact that the flow does not remain strictly two-dimensional beyond Re = 400 as noted by Armaly et al. (1983). Not only does the flow become three-dimensional and unsteady at higher Reynolds numbers, but it is also influenced by the transition to turbulence which begins at about Re = It has already been shown that this would cause deviation which is not due to numerical errors (Kim and Moin, 1985). This would also pose difficulties in getting a grid independent steady solution in that case. Figure 5.10: Normalized primary re-attachment length versus Reynolds number for both two- and three-dimensional simulation and experiments. 75

8 To further illustrate the importance of the third dimension, two simulations were performed with the Reynolds numbers 450 and 800. In these simulation no-slip wall conditions were applied in the spanwise dimension, thereby allowing the flow to become three-dimensional. The results from both the two and three dimensional simulation of the backward facing step are depicted in fig and compared with results from other authors. Clearly the two-dimensional numerical simulation on the current geometry underestimate the experimental length of the reattachment for Reynolds numbers greater than 450. Furthermore the onset of three-dimensionality can be captured if the third dimension is included in the simulation.the recirculating zone near the top wall is observed for Re = 450 and The recirculating bubble that can been seen in Figures 5.12, 5.13 and 5.14 grows in length and moves downstream as the Reynolds number increases and the corresponding reattachment length increases (Table 5.2). This was also reported by Armaly et. al.(1983) and Kim and Moin (1985). As depicted in Figures a small second vortex in the lower corner becomes visible when the Reynolds number is further increased to 1000 and beyond. This is also observed by Armaly et.al, but was not measured. The point of detachment and point of re-attachment for the recirculating zone next to the top wall are depicted in the following table. Armaly, et al. (1983) Armaly, et al. (1983) Solver Solver x detach /s x re-attach /s x detach /s x re-attach /s Re = Re = Table 5.3: Normalized upper detachment x detach /s and re-attachment length x re-attach /s of the current solver compared to Armaly et al. (1983). It has been shown that the current code does compute the highly elliptic recirculating flows with good accuracy for Reynolds numbers at 200 and 450. At Re = 1000 the assumption of two-dimensionality is no longer justified and the flow also comes close to the transitional area. But still the essential flow characteristics, e.g multiple recirculating zones at the top and bottom walls have been captured. Next we move on to the turbulent case for the backward facing step. 76

9 Figure 5.11: Streamline for the back facing step with a primary recirculating zone at Re = 200. Figure 5.12: Streamline for the back facing step with primary and secondary (at the upper wall) recirculating zones at Re = 450. Figure 5.13:Closeup of streamline for the back facing step with a primary recirculating zone at Re = A small vortex is visible in the corner of the step. Figure 5.14:Streamline for the back facing step with primary and secondary (at the upper wall) recirculating zones at Re = Figure 5.15: Streamline and velocity vector at the back facing step at begin of transition to turbulent flow (Re=1200). 77

10 5.2.3 Turbulent flow over a backward facing step In this case the condition has been changed slightly from the previous example. In Figure 5.16 the geometry of the back facing step is depicted for the turbulent case. The computational domain consists of a streamwise length L x = 30H, including an inlet section L inlet = 5H prior to the sudden expansion, vertical height L y = 6H, where H is the step height. The coordinate system is placed at the lower step corner as shown in Figure The mean inlet velocity profile, U(y), imposed at the left boundary x = L i is the same as in the case of the laminar case of the back facing step. U(y) L inlet L y Fully developed flow H X Sep Blockage has been applied to square L x Figure 5.16: Configuration of the turbulent case of back facing step. The inlet profiles for the velocities and turbulence quantities are specified 5H (=L inlet ) upstream. Le et.al (1997) specified the inlet condition 10H for the use in Direct Numerical Simulation. But initial simulation did not give any further improvement of either the re-attachment length or the velocity profile calculated. (5.4) The velocity profile upstream has been prescribed using the power-law equations. The turbulent intensity was set to the values found at the ERCOFTAC 1 database which holds data sets from the DNS by Le and Moin (1994). The expansion factor (ER=L y /(L y -h)) was 1.2. The effects of the expansion factor on the reattachment length were studied by Kuhn (1980), Durst & Tropea (1981), and Ra & Chang (1990). The reattachment length was found to increase with ER in these studies. As already mentioned Armaly et al. (1983) studied the effect of increasing the Reynolds number and the corresponding reattachment length. They did find that reattachment length x sep increases with the Reynolds number up to Re = 1200 (Reynolds number based on the step height h and inlet velocity u in ). This was also simulated in the previous test case. The reattachment length decreases in the 1 ) Details on this and others test cases can be found at the WWW ERCOFTAC Database, which can be accessed at: (Test case 31) 78

11 transitional range 1200 < Re < 6600, and it remains close to constant when the flow becomes fully turbulent at Re > Their findings agreed well with experiments by Durst & Tropea (1981). Here only the case where Re = 5100 has been simulated. Due to the need to prescribe the inlet condition prior to the expansion, part of the computational domain was blocked in order to be able to simulate the enhanced flow over the expansion (Figure 5.17). Figure 5.17: Grid configuration for the turbulence case of flow over a backward facing step at Re= The grid used prior to the step was: 64x64 and after the step: 96x128. The grid dependent was checked using an initial grid consisting of half the above grid points, and this indicates only a little change in the reattachment length compared to the results on the fine mesh. The figure below shows the computed streamlines. The calculated reattachment length was x sep /H = 5.05, which gives 24% under predictions of the reattachment length of x sep /H = 6.28 found with DNS (Le and Moin, 1994). In the lower corner a small secondary motion is observed. From the Figures , it is seen that the overall agreement is good between the DNS and the k-i results from the Solve4kI code. The deficiencies in the results are very typical for the standard k-i model. Now we move on to validate the large eddy simulation models in the Solve4LES code by using two test cases that have been extensively used for benchmarking other large eddy simulations and even direct numerical simulation. 79

12 Figure 5.18: Vector plot of U-velocity flow over backward-facing step using Solve4kI code. Figure 5.19: Computed streamline for flow over backward-facing step using the k-i turbulence model in Solve4kI code. 80

13 Figure 5.20: Streamwise velocity profile at x/h=4. Figure 5.21: Streamwise velocity profile at x/h=6. Figure 5.22: Streamwise velocity profile at x/h=10.figure 5.23: Streamwise velocity profile at x/h= Turbulent flow around a square cylinder in a channel The flow past a square cylinder at a Reynolds number of based on the upstream velocity U in and the cylinder side dimension D was studied experimentally by Lyn and Rodi (1994) and Lyn et al. (1995). The flow is interesting as a test case for Large Eddy Simulation since it involves what is known as semincoherent shedding of vortices from the square cylinder - which is mounted transverse to the flow (Figure 5.23). The reasons for the popularity of this test case are that it has a quite simple geometry and that detailed measurements have been conducted by Lyn and Rodi (1994) and Lyn et al. (1995). 81

14 Figure 5.24: Configuration of the Lyn test case; turbulent flow around a square cylinder in a channel. Although the geometry is simple a complex flow field which is unstable with both separation and recirculation after the cylinder is seen. This configuration has been selected by Rodi et al.(1996) as a test case for a workshop on LES of flows past bluff bodies. The same flow was adopted as a test case at the First ERCOFTAC Workshop on Direct and LargenEddy Simulation at the University of Surrey, 1994 (Voke et al.,1995) and again for the Second Workshop on Direct and Large-Eddy Simulation at Grenoble, 1996 (Chollet et al., 1997). Although it has been chosen for these workshops, there are still some differences between different LES and the measurements by Lyn et. al. (1995). One reason for the differences between the results could be the grid stretching of the mesh, especially in the Streamwise direction of the flow. The resolved eddies are convected downstream from the square cylinder onto a grid that cannot resolve them, since grid stretching has only been applied towards the cylinder. Thereby requiring either a higher order discretization scheme or an introduction of more grid points into the present simulation to preserve the accuracy. The domain for the flow around the cylinder is given by 4D in periodic direction, 14D in the lateral direction, 4.5D in front of the cylinder, and 15D after the back of the cylinder. The grids are equidistant in the spanwise direction of the cylinder. In the other two directions a non-equidistant grid was applied. Stretching of the grid was done by using a hyperbolic tangent function as previously described. The maximum distance from the wall is prescribed for the grid cells nearest to the square cylinder wall and the details of the grid are laid out in Table 5.4. Here and elsewhere all distances are normalised by the cylinder dimension D or by the upstream velocity U in. 82

15 Total no. of grid points grid cell length near wall front of cylinder behind cylinder top and bottom of cylinder 162 x 98 x 26 D/25 D/50 D/45 Table 5.4: The configuration of the grid for square cylinder flow (the Lyn test case). The distance is normalised by the cylinder dimension D. At the inflow plane constant velocity is imposed (no fluctuation is added). (5.5) A no-slip condition was applied to the cylinder surfaces. And at the top and bottom walls a free slip (frictionless wall) condition is prescribed. (5.6) Periodic boundary conditions are applied in the spanwise direction of the cylinder and a convective boundary condition is used at the outflow boundary. Here the inflow velocity U in is used as the convective velocity for the outflow boundary condition (Pauley et al., 1990; Dai et al., 1992; Sohankar, (1998); Sohankar et al.(1998)): (5.7) Size of time step Ft no. of computed shedding cycles for averaging no. of Ft for averaging Time averaging 0.01 sec sec. Table 5.5: Configuration of time step parameters for square cylinder flow (the Lyn test case). The finite volume solver for incompressible flow which was described in the previous chapter is used. The fractional step method which is implicit in time with a Crank-Nicholson method of second order was used for both convective, diffusive and pressure terms. All the terms were discretized using the secondorder central differencing method. In order to solve the Poisson equation for the pressure correction, the linear system of equations was solved by a SIP method which was accelerated by a multigrid technique with V-cycles. The convergence criterion between each time step was set to

16 Lowering the convergence criterion did not show any further improvement of the computed results. The unsteady and three dimensional simulation was started with the fluid at rest, so that no predominant initial flow field would disturb the simulation. The computations were performed for four different subgrid scale models, the Smagorinsky model with Van Driest damping (C s = 0.1), the dynamic model which uses one homogenous direction to perform averaging of the coefficient C to ensure numerical stability, the dynamic model with no averaging applied and finally the dynamic one-equation model. Figure 5.25: Two-dimensional slides of the grid around the square cylinder for the Lyn test case. To compare the computed results with the measurements by Lyn and Rodi (1994) phase-averaging was applied. The vortex shedding behind the square cylinder is accompanied by turbulent fluctuation. In order to study this flow, it is split up in a large scale vortex shedding and a turbulent (small scale) part. The splitting is being done by phase-averaging (Figure 5.25). 84

17 The same phase averaging procedure is used as in the experiments of Lyn et al.(1995). The measured pressure signal at the mid of the top surface is recorded and low-pass filtered. The resulting filtered signal is taken to be associated with the large scale vortex shedding. Thus, the phase of the large scale vortex shedding can be determined with the help of this filtered pressure signal. One problem in the phase averaging procedure is the fact that the vortex shedding is only quasi periodic, not fully periodic. For further details of the phase averaging, see Lyn et al. (1995). f f'' f' <f> t <f> p ~ f time Figure 5.26: Periodic and stochastic fluctuations in the turbulent vortex shedding flow. f l stochastic turbulent fluctuations; f k total resolved fluctuation (periodic and stochastic); <f> p phase averaged value; <f> t time averaged value; periodic fluctuations. (Franke and Rodi (1991), Rodi (1993)). In order to study the vortex shedding the initial start up was overcome after 52 characteristic time units which were greater than 6 vortex shedding cycles. This is very similar to the results obtained by Sohankar et al (1998). 85

18 Large Eddy Simulation Smag. sgs model Dyn. sgs model (averaging) Dyn. sgs model (No averag.) Dyn. One -eqn. model CPU time per time step and grid point 2.1 ^ ^ ^ ^ 10-4 CPU time normalized by that used by the Smagorinsky model Number of iteration per time step in the Fractional Step method Table 5.6: Comparison of CPU time for the different subgrid scale models per time step and grid points. The computations were performed on a SGI Origin 200 computer using one R10000 processor. In the third row the CPU time requirements are normalized by that used by the Smagorinsky model. Finally the number of iterations of the fractional step method for velocitypressure coupling is listed. The term averaging defines the need to stabilize the dynamic model, the model coefficient C was averaged in the spanwise direction and the total viscosity was clipped to be nonnnegative. In Table 5.6 the CPU time per time step and grid point is shown. It is quite clear that due to the need for averaging in a homogenous direction the basic dynamic model requires the most time. The computational requirements that have been found for the different models are not quite similar to the results mentioned by Piomelli (1998). But the comparison between different subgrid scale models was also obtained with a pseudo-spectral code made for channel flow, which is different from using finite difference or finite-volume methods as in the present case. In order to compare the measurements the time averaged quantities which are found by averaging over the phase-averages were calculated. One quantity of particular interest is the time-averaged mean velocity in the main flow direction on the centerline, which is depicted in Figure 5.25 for the simulations with the different subgrid scale models and for the experiments. The velocity profiles for the dynamic model with and without the averaging of the coefficient C in one homogenous direction were so alike that only the dynamic version for non-homogenous flow is shown. 86

19 Figure 5.27: The time-averaged mean velocity in the mean flow direction on the centerline for three different subgrid scale models and measurements by Lyn et al. (1995). More details are apparent from this figure. The recirculating length l re, (i.e. the region behind the cylinder with time averaged < > t < 0 ) is rather well predicted when using the dynamic subgrid scale models. But also the difference between the simulations and experiment grows with the distance (x/d > 2) away from the cylinder. This indicates that the grid becomes too coarse to capture features of the flow that have influence on the time averaged velocity in the mean flow direction. Measurements (Lyn et al. (1995) LES with Smagorinsky Model LES with Dynamic Model LES with Dynamic 1-eq. Model Table 5.7: The recirculating length l re after the square cylinder. One of the conclusions in Rodi et al. (1996) was that the good agreement between these simulations compared with results by many other contributors was due to the fine resolution of the wall region around the cylinder. The velocity far behind the cylinder recovers too quickly in comparison with the measured values. This may again be due to the lack of resolution: the grid becomes very coarse near the outflow boundary, because most grid points have been concentrated directly around the cylinder. Next a comparison between two different time averaging periods for the Smagorinsky model is made. 87

20 Figure 5.28: Comparison between the time-averaged velocity < > t along the centerline for 2 different averaging periods (shedding periods behind the cylinder) for the Smagorinsky subgrid scale model. From the above figure it is clearly seen that the total of 20 shedding periods that have been chosen for the overall averaging is more than enough to create reproducible mean velocities. Finally a comparison between the different resolved velocity fluctuations is depicted in Figure 5.27, where the resolved fluctuation is defined as the RMS value: (5.8) The conclusion based on these simulations is that the results are rather good, and that the large eddy simulation with the different subgrid scale models is working satisfactorily. For further information about the results obtained with LES and the variation of parameters like grid density, and different sgs models etc. see Rodi et al. (1996), Chollet et al. (1997) and Sohankar (1998) to name a few. 88

21 Figure 5.29: The RMS velocity profile after the square cylinder for the three different subgrid scale models. LES Simulation: Blue line- Smagorinsky, Black line - Dynamic model and Green line - Dynamic 1-eq. Model. Measurement: Green points - u RMS and Red points - v RMS Turbulent flow around a surface mounted obstacle in a channel To further increase the complexity of the geometry and the flow the next case for testing the subgrid scale model is the turbulent flow around a surface mounted obstacle in a channel. Detailed measurements are available in Martinuzzi (1992), and Martinuzzi and Tropea (1993). The case was also used at the ''Workshop on LES of Flows past Bluff Bodies'', RotachnEgern, Tegernsee, 1995 (Rodi et al, (1995)), and the 6th ERCOFTAC/IAHR/COST Workshop on Refined Flow Modelling, Delft University of Technology, The geometry of the computational domain is given in Figures 5.30 and y Upper wall U b In flow Out flow z H H Floor Figure 5.30: Configuration of the turbulent flow around a surface mounted obstacle in a channel. x 89

22 A number of LES simulations have been carried out and have been reported in Krajnovic (1998), Breuer (1997), Breuer, and Pourquie, (1996) and Breuer et. al, (1996). In the simulation a domain with an upstream length of x 1 /H = 3 and a downstream length of x 2 /H = 6 was used, while the spanwise width was set to b/h = 7. The channel height was set to 2H. When comparing to RAS the computational domain is actually smaller due to the fact that LES needs a higher grid resolution of the flow field in the vicinity of the obstacle. And furthermore LES cannot take advantage of the symmetry of the time-averaged flow field as RAS can do. The flow is fully turbulent with a high degree of complexity due to multiple separation regions and vortices. Only the case for Reynolds numbers equal to Re= has been investigated. b y Lateral boundary Out flow H x In flow x H 1 2 Lateral boundary x Figure 5.31: Top view of configuration for the surface mounted obstacle in a channel. Only one grid was used to validate the implementation of the subgrid scale model and the Solver code on this test case. The simulation was performed on a hyperbolically stretched grid with 160 x 64 x 64 grid points for the x-, y- and z-directions respectively. In the streamwise direction 50 grid points were used in front of the obstacle. On the surface of the obstacle 32 grid points were used in all directions. Due to the cell-centred grid arrangement the smallest distance between the solid wall and the first grid point was located H/80 away from the wall. Here and elsewhere all distances are normalised by the obstacle height H. In order to achieve homogeneity of the grid on the obstacle surfaces, 32 grid points were used in all directions. The smallest distance in the vicinity of the solid walls has a size of H. 90

23 Total no. of grid points grid points in the streamwise direction In front of obstacle on the surface of the obstacle after the obstacle 160 x 64 x Table 5.8: The configuration of the grid for surface mounted obstacle. To avoid doing initial LES for detailed inflow data an initial k-i simulation was performed using the same grid in the cross sectional plane. Thereby the velocity profile for a fully developed channel flow has been described. The inlet velocity profile was used and superimposed with random fluctuation to simulate a turbulent intensity level of 4 %. U b is therefore the maximum velocity in the channel and not as in the previous case a uniform velocity profile. No-slip conditions were applied to all solid walls, and a convective boundary condition is used at the outflow boundary. Here the inflow velocity U b is used as the convective velocity for the outflow boundary condition (Pauley et al., 1990; Dai et al., 1992; Sohankar, (1998); Sohankar et al.(1998)) as in the previous test case: (5.9) (5.10) Reynolds Number Size of time step Ft Time averaging H/U b 150 H/U b Table 5.9: Configuration of time step for a surface mounted obstacle. Again the finite volume solver for incompressible flow which was described in the previous chapter is used. The fractional step method which is implicit in time with a Crank-Nicholson method of second order was used for both convective, diffusive and pressure terms. All the terms were discretized using the second-order central differencing method. In order to solve the Poisson equation for the pressure correction, the linear system of equations is solved by a SIP method which is accelerated by a multigrid technique with a V-cycles technique. Due to the complexity of the flow only the dynamic subgrid scale model for inhomogeneous flow and the dynamic one-equation model have been used. In the next table a comparison between the two different subgrid scale models in terms of computational time and the number of iterations to satisfy the convergence criterion of 10-3 is depicted. 91

24 Grid: 162 x 66 x 66 LES with Dynamic model (No averaging) LES with Dynamic 1 -eq. model CPU time per time step and grid point 3.2 ^ ^ 10-4 Number of iterations per time step in the fractional step method 3 3 Table 5.10: Comparison of the CPU time for the two different subgrid scale models per time step and grid points. The computations were performed on a SGI Origin 200 computer using only one R10000 processor. Finally the number of iterations for the fractional step method to solve the velocity-pressure coupling is listed. The term averaging defines the needs to stabilize the dynamic model, the model coefficient C was averaged in the spanwise direction and the total viscosity was clipped to be nonnnegative. As already mentioned, LES provides more insight into the time-dependent large-scale structure of the turbulent flow field. An instantaneous view of the velocity on a vertical plane through the centre of the square cube (fig. 5.32), displayed a very complicated flow field. This was simulated using the dynamic 1-equation sgs model by Davidson (A) (1997). The interaction of the different processes like the development of shear layer and the re-attachment of the flow on the floor behind the cube, but also the re-circulation of turbulent flow and its reentrainment into the shear layer takes place within a small section of the flow field. Figure 5.32: Illustration of the dynamics of the flow over a surface mounted obstacle in a channel at Re = Flow direction from left to right. Time between snap shots is approx. 1.0 sec in real time. First snapshot is in upper left corner then we move right and down. Blue: indicates negative velocity, Red : indicates large positive velocities. Green : indicates lower levels of velocities. 92

25 Figure 5.33: Shape shot of the velocity vectors at the centre plane through the square cube. Re = Simulated using LES with the dynamic one-equation subgrid scale model, Davidson (A) (1997). In figure 5.33 the same vertical plane of the instantaneous velocity vectors is depicted. A small recirculating zone on top of the square cube is clearly seen. This is very much in contrast to the RANS with the k-i turbulence model which only gives very small or no re-circulating zones at all (Breuer et. al. (1996)). Furthermore the extension of the re-circulating zone behind the square cube is also overpredicted by the RANS with the k-i model. Figure 5.34: Streamlines of the flow in the symmetry plane of the square cube at Re = using RANS with the k-i turbulence model in Solve4kI. 93

26 Figure 5.35: Streamlines of the time-averaged flow on the vertical symmetry plane through the centre of the square cube at Re = ,- using LES with the dynamic 1-equation sgs model in Solve4LES. Models Experimental Data (Martinuzzi and Tropea, 1993) Numerical Results, Breuer et. al. (1995) SolveLES: present simulation Experiments RANS with the k-i model 2.2 RANS with the RNG k-i 2.08 SolvekI (RANS with the k-i), initial condition 2.2 LES with the Smagorinsky model 1.69 LES with the Dynamic model 1.43 LES with the Dynamic model (no averaging) 1.5 LES with the Dynamic 1-equation model 1.67 Table 5.11: Comparison of the re-circulating length X r between different kinds of numerical simulations and the experimental data from Martinuzzi and Tropea, (1993). Initial condition defined the startup flow field for the Large Eddy Simulation. The term no averaging defines the dynamic subgrid model with the time averaging applied to stabilize the model. In Figures 5.34 and 5.35, the streamlines in the plane of symmetry are depicted, comparing RANS with the k-i turbulence model and LES with a dynamic one-equations model. Although the overall flow 94

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28 is captured by the k-i model, the discrepancies are clearly visible in terms of the re-circulation length behind the cube and the re-circulating zone at the top of the cube. This can also be seen in Table 5.11: a comparison between RANS simulations with different turbulent models and Large Eddy Simulations with different subgrid scale models. The reattachment length behind the obstacle is only slightly overpredicted by the LES with the Smagorinsky model (Table 5.11) and under-predicted somewhat by the LES with the dynamic model using internal time averaging to stabilize the model. The dynamic oneequations model is in good agreement with experimental data indicating that it is capable of handling the very complicated flows better than the dynamic model with internal time-averaging. Another important result is the extension of the separation bubble on the top of the cube. The one computed by LES agrees fairly well with the measurements too (Breuer et al. (1995) and Martinuzzi and Tropea, (1993)). Figures show the simulated velocity versus the measured streamwise velocity profiles at five different locations in the vertical symmetry plane for LES using either the dynamic model with internal time-averaging or the dynamic one-equations model by Davidson (A) (1997). One obstacle height in front of the cube the streamwise velocity profiles agree fairly well the experiment by Martinuzzi and Tropea, (1993). However, small differences can be observed for the profile in the middle of the top of the cube at x/h = 0.5. Indicating that the re-circulating zone on top of the cube is not correctly captured. Compared with experiment the two different subgrid scale models provide nearly identical velocity profiles. Moving further downstream, the effect of the different subgrid scale models is quite small. Although in the wake region (x/h = 2.0) the computed velocity magnitudes are underestimated. Figure 5.36: Comparison of the mean velocity profile U of the time-averaged flow in the symmetry plane of the cube at Re = at x/h = -1.0 with two different dynamic sgs models. Figure 5.37: Comparison of the mean velocity profile U of the time-averaged flow in the symmetry plane of the cube at Re = at x/h = 0.5 with two different dynamic sgs models. 96

29 Figure 5.38: Comparison of the mean velocity profile U of the time-averaged flow in the symmetry plane of the cube at Re = at x/h = 1.0 with two different dynamic sgs models. Figure 5.39: Comparison of the mean velocity profile U of the time-averaged flow in the symmetry plane of the cube at Re = at x/h = 2.0 with two different dynamic sgs models. Figure 5.40: Comparison of the mean velocity profile U of the time-averaged flow in the symmetry plane of the cube at Re = at x/h = 4.0 with two different dynamic sgs models. 97

30 Figures display profiles of the mean Reynolds stresses <ukuk> t and <ukvk> t at three locations as before. It should be noted that for LES only the resolved part of the turbulent stress is included. At x/h = 0.5 all simulations give similar peak values for <ukuk> t and <ukvk> t. At x/h = 1.0 the peak in the experimental value is captured although the maximum value is not reached. Further downstream (x/h = 2.) some differences in the simulated data for <ukuk> t and also for <ukvk> t could be detected. Figure 5.41: Comparison of the Reynolds stresses of the time-averaged flow in the symmetry plane of the cube at x = 0.5 and 1.0 in the vertical cross sections and using two different dynamic subgrid scale models. Re= Figure 5.42: Comparison of the Reynolds stresses of the time-averaged flow in the symmetry plane of the cube at x = 2 and 4 in the vertical cross sections and using two different dynamic subgrid scale models. Re=

31 Figure 5.43: Comparison of the Reynolds stresses of the time-averaged flow in the symmetry plane of the cube at x = 0.5 and 1.0 in the vertical cross sections and using two different dynamic subgrid scale models. Re= Figure 5.44: Comparison of the Reynolds stresses of the time-averaged flow in the symmetry plane of the cube at x = 2.0 in the vertical cross sections and using two different dynamic subgrid scale models. Re=

32 It is not clear what these differences are caused by but one explanation could be that it is due to grid coarseness. The grid is not able to capture and resolve the Reynolds stresses in a proper manner. But also the lack of proper inflow boundary conditions which include information of coherent structures could contribute to better Reynolds stress profiles further downstream the cube. In general the simulated profiles especially for the Reynolds stresses are not very smooth which may be explained by too short an averaging period to capture these higher order statistical moments of the flow. The small discrepancies between the simulation data and the experiments would in many situations be insignificant. It could therefore be concluded that these simulations emphasize that the two different dynamic subgrid scale models are implemented in a correct manner and the code performes well. In Breuer, (1997), Breuer and Pourquie, (1996), Breuer et al. (1995) further analyses of the flow past a three-dimensional cube mounted in a channel has been conducted - such as grid refinement, different subgrid scale models and the effect of wall functions. And a detailed comparison to RANS simulation with different turbulence models has been carried out. These papers emphasize that the price for better agreement between the simulations and the experimental data is high in demands of CPU-time. Breuer (1995) stated that the ratios between RANS with the standard k-i turbulence model, RANS with a twolayer version of the k-i model and the Large Eddy Simulation was 1 : 25 : 200, and up to 400 times spent in order to provide sufficiently data for stable second-order statistics (Reynolds stresses). This emphasizes that LES although very capable is not quite ready for general engineering problems. But more about this later. The numerical method described in the previous chapter was assessed by applying it to a number of simple test cases for which the solution is known either by experiment, Direct Numerical Simulation or other Large Eddy Simulations. It was later discovered that the implementation for the Solve4LES code could be optimized, so further reduction in the requirement of computational time could be performed. This modification of the code was only applied to the simulation done in the next chapter, where flow problems relevant to ventilation are studied. Furthermore, the overall accuracy and efficiency of the implemented methods may be considered as satisfactory. This will also be supported by the results of the application of the methods/codes to flow problems, which are presented later. 100

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