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Chapter 5

Assessment of the numerical solverIn this chapter the numerical methods described in the previous chapter are validated andbenchmarked by applying them to some relatively simple test cases for which detailedexperiments, LES simulation data or Direct Numerical Simulation data exist. Thesedifferent test cases have been chosen for independent testing of especially the Solve4keand Solve4LES modules of the Solver code.

5.1 IntroductionThe numerical method outlined in the previous chapter has been implemented into the code Solver andthe modification of the Solver code to perform either Reynolds Averaged Simulation with the k-Iturbulence model (Solve4kI) or LES (Solve4LES) with different subgrid scale models. This code hasbeen written in order to meet the needs of this project, due to the fact that no initial documentation wasprovided for the LESROOM code, and that the computational speed of the CFX code was generally tooslow when LES were used.

The verification of the CFD code is a very important point, nowadays. In Freitas (1995) severalcommercial CFD codes are tested and benchmarked against each other in different cases, such as thebackward facing-step, three-dimensional shear driven cavity flow, turbulent flow around a square cross-section cylinder and a turbulent flow in a duct with a 180° bend. Though the codes do not use the samenumerical methods, such as discretization, grid resolution, turbulence models etc., the dispersion of theresults is not encouraging, although some of the errors were introduced by more or less improper useof the CFD codes. This would furthermore encourage the need for validation of the code described beforeapplying it to ventilation problems.

In the following section some test cases have been calculated by using the Solver code and itsdifferent modules. This was done to evaluate the performance of the code and its stability and reliability.

CHAPTER 5 ASSESSMENT OF THE NUMERICAL SOLVER

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(5.1)

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 Cpu @133 Mhz, a SGI Onyx workstation (R8000CPU at 75 Mhz) or SGI Origin 200 (R10000 CPU @ 180 Mhz). But the CPU time was not recorded forthese test cases. For the laminar flow case the simulation was stopped when all the residuals were below10-4. 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

5.2.1 Laminar flow in a square driven cavityThe laminar square driven cavity problem was selected as the first test case. It is one of the simplest fluidflow configurations, yet it manifests the full non-linearity of the Navier-Stokes equations. The separationand reattachment of turbulent flows occur in many practical engineering applications, both in internalsystems such as diffusers, combustors and channels with sudden expansions, and in external ones likethose 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 fromthe solid surface. The flow subsequently reattaches downstream forming a recirculating bubble. Theconfiguration 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 forvalidation of the incompressible flow solver because its solution is available in a number of sources inliterature. The driven cavity flow problem has been solved by a vorticity and stream function method byGhia et al.,(1982) and by a primitive variable method by Hortman et al. (1990) including many otherresearchers (for instand Agarwal, 1981 and Rubin and Khosla, 1981). It has been shown that the higherthe Reynolds number the more pronounced is the effect of the numerical error introduced by the solutionmethods. In this particular test a Reynolds number of 1000 has been used for the validations. Where theReynolds number is calculated as:

A uniform grid has been used on the square flow domain and a set of grids ranging in sizes from32x32 to 256x256 has been used to check the computed results and to illustrate grid independence. Asecond order centre/upwind differencing scheme was used. Together with a no-slip condition at the walls.


71

Figure 5.3: Driven cavity flow; calculatedhorizontal centre line velocity profiles comparedto benchmark results of Ghia et.al. (1982) fordifferent grid refinement.

Figure 5.1:Configurationof a laminar square drivencavity.

Figure 5.2: Driven cavity flow; calculated verticalcentre line velocity profiles compared tobenchmark results of Ghia et.al. (1982) fordifferent grid refinement.

The simulation was performed in two dimensions to verify the correct implementation of the Navier-Stokesequations, the second order accuracy of the convection-diffusion discretization scheme, the boundaryconditions and the Rhie-Chow interpolation.

In Figures 5.2 and 5.3, the calculated velocity profile along the vertical and horizontal centre-line ofthe cavity is depicted and compared with the results of Ghia et. al. (1982). The centre line profile showsa recirculating flow in the cavity which is more apparent in some of the streamline plots (Figure 5.5)


72

Figure 5.5: Streamline for a driven cavityflow from the present code.

Figure 5.4: Vector plot of a driven cavityflow from the present code.

The streamline plot shows a primary vortex at the centre of the cavity with two secondary vorticesat the lower left and right corners. In figures 5.2 and 5.3 the different solutions with a range of differentgrid sizes from 32x32 to 256x256 are also depicted. The essential flow characteristics are captured witha low grid resolution. Further grid refinement does not change the solution substantially.

Grid size Error FFFFx ( = FFFFy )

32x32 0.0102 0.03125

64x64 0.00259 0.01563

128x128 0.000717 0.00781

256x256 0.000281 0.00391

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 reductionin 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 effectsclose to the wall. Using uniform grids would have been rather uneconomical for most flows, especiallywall-bounded flows. This is due to large wall-normal gradients, which require a very high spatialresolution in the wall normal direction. The non-equidistant grid was generated by using a hyperbolictangent function


73

(5.2)

y

y

~

Figure 5.6: Equidistant grid before thetransformation by the hyperbolic function.

y~

y

Figure 5.7: The non-equidistant grid after thetransformation by the hyperbolic function. The slopeof the red curve is given by E and the inflectionpoint (here at zero) by I.

which transforms an equidistant grid in into a non-uniform grid in the y-direction.

Here E determines the slope and [ the inflection point of the function.

5.2.2 Laminar flow over a backward-facing stepThe laminar flow over a backward-facing step is one of the classical problems for testing the capabilitiesof a solution scheme to solve recirculating flow problems. The flow geometry consists of a long channelwhere a fully developed flow enters the domain, and goes through a sudden expansion. The channelexpansion is 1.94 (fig 5.8). The channel is long enough to make the flow exciting the domain as a fullydeveloped flow. The sudden expansion gives rise to one or more regions of recirculations depending uponthe Reynolds number of the entering flow. Both experimental and numerical solutions to this test caseare 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 (Akselvolland Moin,1995) , and DNS (Le and Moin, 1994; Le et al., 1997) to name a few. In this current test casethe setup resembles that of Armaly et al. (1983). The entrance region of height h, where the flow becomesfully developed before going through the sudden expansion was at first not considered as it is done byArmaly et al.(1983). The experimental configuration and condition of Armaly et.al.(1983) was that theflow originates in a long pipe which has a sudden expansion. Here the entrance pipe was at firstneglected.


74

(5.3)

x

x xdetach re-attach

sep

u(y)h

sfully developedflow

L

Blockage applied to square

Uin

Figure 5.8: Configuration of the laminar flow over the backward-facing step.

Figure 5.9: Grid configuration for the laminar back-facing step.

And instead, a boundary profile of a fully-developed parabolic profile at the inlet as given below has beenintroduced:

Afterwards a small inlet section was added, since the results were in better agreement with theexperimental data of Armaly et al. (1983). The inlet section was 6 times the step height (6]h). Thefollowing material property is used in the simulation, s is the step height (=00049m) and the height of theentering flow, h, is 0.0051m. 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.

The problem is solved and recorded for three different Reynolds numbers 200, 450 and 1000 and thesolution is compared with the experimental results obtained by Armaly et al. (1983). The second-orderQuick differencing is used for the convective parts. This was because the central differencing did notperform very well for this flow. The simulation was at first performed in two dimensions. And the meshwas stretched using the hyperbolic function towards the wall and the centre of the channel (figure 5.9)

A recirculating zone is observed after the step for all Reynolds numbers. The reattachment lengths of thisand 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 thebottom of the step where the x-velocity changes sign. A good agreement between the experiment andsimulations for Reynolds numbers until 450 is found. But beyond that the size of the recirculating zone


75

Figure 5.10: Normalized primary re-attachment length versus Reynolds numberfor both two- and three-dimensional simulation and experiments.

declines compared to the experimental observation made by Armaly et al.

Armaly etal(1983)

Experimental

Armaly et al.(1983)

Numerical

Solver(Present

Simulation)

Re = 200 5.29 5.12 5.2

Re = 450 9.43 8.4 8.7

Re = 1000 16.43 7.4 11.6

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 explainedby the fact that the flow does not remain strictly two-dimensional beyond Re = 400 as noted by Armalyet 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 = 1200. It has alreadybeen 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.


76

To further illustrate the importance of the third dimension, two simulations were performed with theReynolds numbers 450 and 800. In these simulation no-slip wall conditions were applied in the spanwisedimension, thereby allowing the flow to become three-dimensional. The results from both the two andthree dimensional simulation of the backward facing step are depicted in fig. 5.10 and compared withresults from other authors. Clearly the two-dimensional numerical simulation on the current geometryunderestimate 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 thesimulation.The recirculating zone near the top wall is observed for Re = 450 and 1000. The recirculatingbubble that can been seen in Figures 5.12, 5.13 and 5.14 grows in length and moves downstream as theReynolds number increases and the corresponding reattachment length increases (Table 5.2). This wasalso reported by Armaly et. al.(1983) and Kim and Moin (1985). As depicted in Figures 5.13 - 5.15 asmall second vortex in the lower corner becomes visible when the Reynolds number is further increasedto 1000 and beyond. This is also observed by Armaly et.al, but was not measured. The point ofdetachment and point of re-attachment for the recirculating zone next to the top wall are depicted in thefollowing table.

Armaly, et al.(1983)

Armaly, et al.(1983)

Solver Solver

xdetach/s xre-attach/s xdetach/s xre-attach/s

Re = 450 7.86 11.3 7.7 11.6

Re = 1000 13.5 21.9 9.4 21.7

Table 5.3: Normalized upper detachment xdetach/s and re-attachment length xre-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 goodaccuracy for Reynolds numbers at 200 and 450. At Re = 1000 the assumption of two-dimensionality isno longer justified and the flow also comes close to the transitional area. But still the essential flowcharacteristics, e.g multiple recirculating zones at the top and bottom walls have been captured. Next wemove on to the turbulent case for the backward facing step.


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Figure 5.11: Streamline for the back facing stepwith a primary recirculating zone at Re = 200.

Figure 5.14:Streamline for the back facing step with primary and secondary (at the upper wall) recirculating zones at Re = 1000.

Figure 5.12: Streamline for the back facing step with primary and secondary (at the upper wall) recirculating zones at Re = 450.

Figure 5.15: Streamline and velocity vector at the back facing step at begin of transition to turbulentflow (Re=1200).

Figure 5.13:Closeup of streamline for the back facing step with a primary recirculating zone at Re = 1000. A small vortex is visible in the corner of the step.


1) Details on this and others test cases can be found at the WWW ERCOFTAC Database ,which can be accessed at: http://fluidigo.mech.surrey.ac.uk. (Test case 31)

78

XSep

U(y)

Fully developedflow

LxBlockage has been applied to square

inlet

H

LLy

Figure 5.16: Configuration of the turbulent case of back facing step.

(5.4)

5.2.3 Turbulent flow over a backward facing stepIn this case the condition has been changed slightly from the previous example. In Figure 5.16 thegeometry of the back facing step is depicted for the turbulent case. The computational domain consistsof a streamwise length Lx = 30H, including an inlet section Linlet = 5H prior to the sudden expansion,vertical height Ly = 6H, where H is the step height. The coordinate system is placed at the lower stepcorner as shown in Figure 5.16. The mean inlet velocity profile, U(y), imposed at the left boundary x =Li is the same as in the case of the laminar case of the back facing step.

The inlet profiles for the velocities and turbulence quantities are specified 5H (=Linlet) upstream. Le et.al(1997) specified the inlet condition 10H for the use in Direct Numerical Simulation. But initialsimulation did not give any further improvement of either the re-attachment length or the velocity profilecalculated.

The velocity profile upstream has been prescribed using the power-law equations. The turbulent intensitywas set to the values found at the ERCOFTAC1 database which holds data sets from the DNS by Le andMoin (1994). The expansion factor (ER=Ly /(Ly-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 inthese studies. As already mentioned Armaly et al. (1983) studied the effect of increasing the Reynoldsnumber and the corresponding reattachment length. They did find that reattachment length xsep increaseswith the Reynolds number up to Re = 1200 (Reynolds number based on the step height h and inletvelocity uin). This was also simulated in the previous test case. The reattachment length decreases in the


79

Figure 5.17: Grid configuration for the turbulence case of flow over a backward facing stepat Re=5.100.

transitional range 1200 < Re < 6600, and it remains close to constant when the flow becomes fullyturbulent at Re > 6600. 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 inletcondition prior to the expansion, part of the computational domain was blocked in order to be able tosimulate the enhanced flow over the expansion (Figure 5.17).

The grid used prior to the step was: 64x64 and after the step: 96x128. The grid dependent waschecked using an initial grid consisting of half the above grid points, and this indicates only a little changein the reattachment length compared to the results on the fine mesh. The figure below shows thecomputed streamlines. The calculated reattachment length was xsep /H = 5.05, which gives 24% underpredictions of the reattachment length of xsep /H = 6.28 found with DNS (Le and Moin, 1994). In thelower corner a small secondary motion is observed.

From the Figures 5.20-5.23, it is seen that the overall agreement is good between the DNS and thek-I results from the Solve4kIIII code. The deficiencies in the results are very typical for the standard k-Imodel.

Now we move on to validate the large eddy simulation models in the Solve4LES code by using twotest cases that have been extensively used for benchmarking other large eddy simulations and evendirect numerical simulation.


80

Figure 5.18: Vector plot of U-velocity flow over backward-facing step using Solve4kIIII code.

Figure 5.19: Computed streamline for flow over backward-facing step using the k-I turbulencemodel in Solve4kIIII code.


81

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=15.

5.2.3 Turbulent flow around a square cylinder in a channelThe flow past a square cylinder at a Reynolds number of 21400 based on the upstream velocity Uin andthe 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 assemincoherent 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 andthat detailed measurements have been conducted by Lyn and Rodi (1994) and Lyn et al. (1995).


82

Figure 5.24: Configuration of the Lyn test case; turbulent flow around a square cylinderin 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 testcase for a workshop on LES of flows past bluff bodies. The same flow was adopted as a test case at theFirst 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 atGrenoble, 1996 (Chollet et al., 1997). Although it has been chosen for these workshops, there are stillsome differences between different LES and the measurements by Lyn et. al. (1995). One reason for thedifferences between the results could be the grid stretching of the mesh, especially in the Streamwisedirection of the flow. The resolved eddies are convected downstream from the square cylinder onto a gridthat cannot resolve them, since grid stretching has only been applied towards the cylinder. Therebyrequiring either a higher order discretization scheme or an introduction of more grid points into thepresent simulation to preserve the accuracy.

The domain for the flow around the cylinder is given by 4D in periodic direction, 14D in the lateraldirection, 4.5D in front of the cylinder, and 15D after the back of the cylinder. The grids are equidistantin 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 cylinderwall and the details of the grid are laid out in Table 5.4. Here and elsewhere all distances are normalisedby the cylinder dimension D or by the upstream velocity Uin.


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(5.5)

(5.6)

(5.7)

Total no. of grid pointsgrid cell length near wall

front of cylinder behind cylinder

top and bottomof 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).

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.

Periodic boundary conditions are applied in the spanwise direction of the cylinder and a convectiveboundary condition is used at the outflow boundary. Here the inflow velocity Uin is used as the convectivevelocity for the outflow boundary condition (Pauley et al., 1990; Dai et al., 1992; Sohankar, (1998);Sohankar et al.(1998)):

Size of time step FFFFtno. of computedshedding cycles for averaging

no. of FFFFt foraveraging Time averaging

0.01 sec. 20 15 150 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 isused. The fractional step method which is implicit in time with a Crank-Nicholson method of secondorder 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, thelinear system of equations was solved by a SIP method which was accelerated by a multigrid techniquewith V-cycles. The convergence criterion between each time step was set to 10-3.


84

Figure 5.25: Two-dimensional slides of the grid around the square cylinder for the Lyntest case.

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 fourdifferent subgrid scale models, the Smagorinsky model with Van Driest damping (Cs = 0.1), the dynamicmodel which uses one homogenous direction to perform averaging of the coefficient C to ensurenumerical stability, the dynamic model with no averaging applied and finally the dynamic one-equationmodel.

To compare the computed results with the measurements by Lyn and Rodi (1994) phase-averaging wasapplied. The vortex shedding behind the square cylinder is accompanied by turbulent fluctuation. In orderto study this flow, it is split up in a large scale vortex shedding and a turbulent (small scale) part. Thesplitting is being done by phase-averaging (Figure 5.25).


85

tp

f''

f'

f~

time

f

Figure 5.26: Periodic and stochastic fluctuations in the turbulentvortex shedding flow. f l stochastic turbulent fluctuations; f k totalresolved fluctuation (periodic and stochastic); p phase averagedvalue; t time averaged value; periodic fluctuations. (Franke andRodi (1991), Rodi (1993)).

The same phase averaging procedure is used as in the experiments of Lyn et al.(1995) . The measuredpressure signal at the mid of the top surface is recorded and low-pass filtered. The resulting filtered signalis taken to be associated with the large scale vortex shedding. Thus, the phase of the large scale vortexshedding can be determined with the help of this filtered pressure signal. One problem in the phaseaveraging procedure is the fact that the vortex shedding is only quasi periodic, not fully periodic. Forfurther details of the phase averaging, see Lyn et al. (1995).

In order to study the vortex shedding the initial start up was overcome after 52 characteristic timeunits which were greater than 6 vortex shedding cycles. This is very similar to the results obtained bySohankar et al (1998).


86

Large Eddy Simulation Smag. sgsmodel

Dyn. sgs model(averaging)

Dyn. sgs model (No averag.)

Dyn. One -eqn.model

CPU time per time stepand grid point 2.1 ^̂̂̂ 10 - 4 3.5 ^̂̂̂ 10 - 4 3.0 ^̂̂̂ 10 - 4 3.4 ^̂̂̂ 10 - 4

CPU time normalized bythat used by the

Smagorinsky model 1 1.67 1.4 1.62

Number of iteration pertime step in the

Fractional Step method2 3 2 2

Table 5.6: Comparison of CPU time for the different subgrid scale models per time step andgrid points. The computations were performed on a SGI Origin 200 computer using one R10000processor. In the third row the CPU time requirements are normalized by that used by theSmagorinsky model. Finally the number of iterations of the fractional step method for velocity-pressure 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 needfor averaging in a homogenous direction the basic dynamic model requires the most time. Thecomputational requirements that have been found for the different models are not quite similar to theresults mentioned by Piomelli (1998). But the comparison between different subgrid scale models wasalso obtained with a pseudo-spectral code made for channel flow, which is different from using finitedifference or finite-volume methods as in the present case.

In order to compare the measurements the time averaged quantities which are found by averagingover the phase-averages were calculated. One quantity of particular interest is the time-averaged meanvelocity in the main flow direction on the centerline, which is depicted in Figure 5.25 for the simulationswith the different subgrid scale models and for the experiments. The velocity profiles for the dynamicmodel with and without the averaging of the coefficient C in one homogenous direction were so alike thatonly the dynamic version for non-homogenous flow is shown.


87

Figure 5.27: The time-averaged mean velocity in the meanflow direction on the centerline for three different subgrid scalemodels and measurements by Lyn et al. (1995).

More details are apparent from this figure. The recirculating length lre, (i.e. the region behind the cylinderwith time averaged < >t < 0 ) is rather well predicted when using the dynamic subgrid scale models. Butalso the difference between the simulations and experiment grows with the distance (x/D > 2) away fromthe cylinder. This indicates that the grid becomes too coarse to capture features of the flow that haveinfluence on the time averaged velocity in the mean flow direction.

Measurements(Lyn et al. (1995)

LES withSmagorinskyModel

LES with Dynamic Model

LES with Dynamic 1-eq. Model

1.38 1.23 1.41 1.35

Table 5.7: The recirculating length lre after the square cylinder.

One of the conclusions in Rodi et al. (1996) was that the good agreement between these simulationscompared with results by many other contributors was due to the fine resolution of the wall region aroundthe cylinder. The velocity far behind the cylinder recovers too quickly in comparison with the measuredvalues. This may again be due to the lack of resolution: the grid becomes very coarse near the outflowboundary, 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.


88

(5.8)

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 Smagorinskysubgrid scale model.

From the above figure it is clearly seen that the total of 20 shedding periods that have been chosenfor 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:

The conclusion based on these simulations is that the results are rather good, and that the large eddysimulation with the different subgrid scale models is working satisfactorily. For further information aboutthe results obtained with LES and the variation of parameters like grid density, and different sgs modelsetc. see Rodi et al. (1996), Chollet et al. (1997) and Sohankar (1998) to name a few.


89

UbIn flow Out flow

Upper wall

Floor x

y

Hz

H

Figure 5.30: Configuration of the turbulent flow around a surface mounted obstacle in achannel.

Figure 5.29: The RMS velocity profile after thesquare cylinder for the three different subgrid scalemodels. 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.

5.2.4 Turbulent flow around a surface mounted obstacle in a channelTo further increase the complexity of the geometry and the flow the next case for testing the subgrid scalemodel is the turbulent flow around a surface mounted obstacle in a channel. Detailed measurements areavailable 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 ofTechnology, 1997. The geometry of the computational domain is given in Figures 5.30 and 5.31.


90

y

x

b

x x

H

1 2In flow

Out flowLateral boundary

Lateral boundaryH

Figure 5.31: Top view of configuration for the surface mounted obstacle in a channel.

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 anupstream length of x1 /H = 3 and a downstream length of x2 /H = 6 was used, while the spanwise widthwas set to b/H = 7. The channel height was set to 2H. When comparing to RAS the computational domainis actually smaller due to the fact that LES needs a higher grid resolution of the flow field in the vicinityof the obstacle. And furthermore LES cannot take advantage of the symmetry of the time-averaged flowfield as RAS can do. The flow is fully turbulent with a high degree of complexity due to multipleseparation regions and vortices. Only the case for Reynolds numbers equal to Re=40.000 has beeninvestigated.

Only one grid was used to validate the implementation of the subgrid scale model and the Solvercode on this test case. The simulation was performed on a hyperbolically stretched grid with 160 x 64 x64 grid points for the x-, y- and z-directions respectively. In the streamwise direction 50 grid points wereused in front of the obstacle. On the surface of the obstacle 32 grid points were used in all directions. Dueto the cell-centred grid arrangement the smallest distance between the solid wall and the first grid pointwas located H/80 away from the wall. Here and elsewhere all distances are normalised by the obstacleheight H. In order to achieve homogeneity of the grid on the obstacle surfaces, 32 grid points were usedin all directions. The smallest distance in the vicinity of the solid walls has a size of 0.010 H.


91

(5.9)

(5.10)

Total no. of grid pointsgrid points in the streamwise direction

In front ofobstacle

on the surface ofthe obstacle

after theobstacle

160 x 64 x 64 58 32 70

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 thesame grid in the cross sectional plane. Thereby the velocity profile for a fully developed channel flow hasbeen described . The inlet velocity profile was used and superimposed with random fluctuation tosimulate a turbulent intensity level of 4 %.

Ub is therefore the maximum velocity in the channel and not as in the previous case a uniform velocityprofile. No-slip conditions were applied to all solid walls, and a convective boundary condition is usedat the outflow boundary. Here the inflow velocity Ub is used as the convective velocity for the outflowboundary condition (Pauley et al., 1990; Dai et al., 1992; Sohankar, (1998); Sohankar et al.(1998))as in the previous test case:

Reynolds Number Size of time step FFFFt Time averaging

40 0.0075 H/Ub 150 H/Ub 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 chapteris used. The fractional step method which is implicit in time with a Crank-Nicholson method of secondorder was used for both convective, diffusive and pressure terms. All the terms were discretized using thesecond-order central differencing method. In order to solve the Poisson equation for the pressurecorrection, the linear system of equations is solved by a SIP method which is accelerated by a multigridtechnique with a V-cycles technique. Due to the complexity of the flow only the dynamic subgrid scalemodel 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 ofcomputational time and the number of iterations to satisfy the convergence criterion of 10-3 is depicted.


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Figure 5.32: Illustration of the dynamics of the flow over a surface mounted obstacle in a channel at Re = 40.000. 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.

Grid: 162 x 66 x 66 LES with Dynamic model (No averaging)

LES with Dynamic 1 -eq. model

CPU time per time step and gridpoint 3.2 ^̂̂̂ 10 - 4 3.4 ^̂̂̂ 10 - 4

Number of iterations per time stepin 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 onlyone R10000 processor. Finally the number of iterations for the fractional step method to solve thevelocity-pressure coupling is listed. The term ‘averaging‘ defines the needs to stabilize the dynamicmodel, the model coefficient C was averaged in the spanwise direction and the total viscosity wasclipped to be nonnnegative.

As already mentioned, LES provides more insight into the time-dependent large-scale structure ofthe turbulent flow field. An instantaneous view of the velocity on a vertical plane through the centre ofthe square cube (fig. 5.32), displayed a very complicated flow field. This was simulated using the dynamic1-equation sgs model by Davidson (A) (1997).

The interaction of the different processes like the development of shear layer and the re-attachmentof the flow on the floor behind the cube, but also the re-circulation of turbulent flow and its re-entrainment into the shear layer takes place within a small section of the flow field.


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Figure 5.33: Shape shot of the velocity vectors at the centre plane through thesquare cube. Re = 40.000. Simulated using LES with the dynamic one-equationsubgrid scale model, Davidson (A) (1997).

Figure 5.34: Streamlines of the flow in the symmetry plane of the square cube at Re = 40.000 using RANS with the k-I turbulence model in Solve4kIIII.

In figure 5.33 the same vertical plane of the instantaneous velocity vectors is depicted. A small re-circulating zone on top of the square cube is clearly seen. This is very much in contrast to the RANS withthe 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 over-predicted by the RANS with the k-I model.


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Figure 5.35: Streamlines of the time-averaged flow on the vertical symmetry plane through thecentre of the square cube at Re = 40.000,- using LES with the dynamic 1-equation sgs model in Solve4LES.

ModelsExperimental Data(Martinuzzi andTropea, 1993)

Numerical Results,Breuer et. al.(1995)

SolveLES:present simulation

Experiments 1.612

RANS with the k-IIIImodel

2.2

RANS with the RNG k-IIII 2.08

SolvekIIII (RANS with thek-IIII), initial condition 2.2

LES with the Smagorinsky model 1.69

LES with the Dynamic model 1.43

LES with the Dynamicmodel (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 simulationsand the experimental data from Martinuzzi and Tropea, (1993). ‘Initial condition’ defined the startup flowfield for the Large Eddy Simulation. The term ‘no averaging‘ defines the dynamic subgrid model withthe 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 withthe k-I turbulence model and LES with a dynamic one-equations model. Although the overall flow


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feature


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Figure 5.36: Comparison of the meanvelocity profile U of the time-averaged flowin the symmetry plane of the cube at Re =40.000 at x/H = -1.0 with two differentdynamic sgs models.

Figure 5.37: Comparison of the meanvelocity profile U of the time-averaged flowin the symmetry plane of the cube at Re =40.000 at x/H = 0.5 with two differentdynamic sgs models.

is captured by the k-I model, the discrepancies are clearly visible in terms of the re-circulation lengthbehind 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 Simulationswith different subgrid scale models. The reattachment length behind the obstacle is only slightly over-predicted by the LES with the Smagorinsky model (Table 5.11) and under-predicted somewhat by theLES with the dynamic model using internal time averaging to stabilize the model. The dynamic one-equations model is in good agreement with experimental data indicating that it is capable of handling thevery complicated flows better than the dynamic model with internal time-averaging. Another importantresult is the extension of the separation bubble on the top of the cube. The one computed by LES agreesfairly well with the measurements too (Breuer et al. (1995) and Martinuzzi and Tropea, (1993)).

Figures 5.36 - 5.40 show the simulated velocity versus the measured streamwise velocity profiles atfive different locations in the vertical symmetry plane for LES using either the dynamic model withinternal time-averaging or the dynamic one-equations model by Davidson (A) (1997). One obstacle heightin front of the cube the streamwise velocity profiles agree fairly well the experiment by Martinuzzi andTropea, (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. Comparedwith experiment the two different subgrid scale models provide nearly identical velocity profiles. Movingfurther downstream, the effect of the different subgrid scale models is quite small. Although in the wakeregion (x/H = 2.0) the computed velocity magnitudes are underestimated.


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Figure 5.39: Comparison of the mean velocityprofile U of the time-averaged flow in thesymmetry plane of the cube at Re = 40.000 atx/H = 2.0 with two different dynamic sgs models.



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Figure 5.41: Comparison of the Reynolds stresses of the time-averaged flow inthe symmetry plane of the cube at x = 0.5 and 1.0 in the vertical cross sections and usingtwo different dynamic subgrid scale models. Re=40.000.

Figure 5.42: Comparison of the Reynolds stresses of the time-averaged flow inthe symmetry plane of the cube at x = 2 and 4 in the vertical cross sections and using twodifferent dynamic subgrid scale models. Re=40.000.

Figures 5.41 - 5.44 display profiles of the mean Reynolds stresses t and t at threelocations as before. It should be noted that for LES only the resolved part of the turbulent stress isincluded. At x/H = 0.5 all simulations give similar peak values for t and t. At x/H = 1.0the peak in the experimental value is captured although the maximum value is not reached. Furtherdownstream (x/H = 2.) some differences in the simulated data for t and also for t could bedetected.


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Figure 5.43: Comparison of the Reynolds stresses of the time-averaged flow inthe symmetry plane of the cube at x = 0.5 and 1.0 in the vertical cross sections and usingtwo different dynamic subgrid scale models. Re=40.000.

Figure 5.44: Comparison of the Reynoldsstresses of the time-averaged flowin the symmetry plane of the cube at x = 2.0in the vertical cross sections and using twodifferent dynamic subgrid scale models. Re=40.000.


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It is not clear what these differences are caused by but one explanation could be that it is due to gridcoarseness. The grid is not able to capture and resolve the Reynolds stresses in a proper manner. But alsothe lack of proper inflow boundary conditions which include information of coherent structures couldcontribute to better Reynolds stress profiles further downstream the cube. In general the simulated profilesespecially for the Reynolds stresses are not very smooth which may be explained by too short anaveraging period to capture these higher order statistical moments of the flow.

The small discrepancies between the simulation data and the experiments would in many situationsbe insignificant. It could therefore be concluded that these simulations emphasize that the two differentdynamic 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 pasta three-dimensional cube mounted in a channel has been conducted - such as grid refinement, differentsubgrid scale models and the effect of wall functions. And a detailed comparison to RANS simulationwith different turbulence models has been carried out. These papers emphasize that the price for betteragreement 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 two-layer version of the k-I model and the Large Eddy Simulation was 1 : 25 : 200 , and up to 400 timesspent in order to provide sufficiently data for stable second-order statistics (Reynolds stresses). Thisemphasizes that LES although very capable is not quite ready for general engineering problems. But moreabout this later.

The numerical method described in the previous chapter was assessed by applying it to a number ofsimple test cases for which the solution is known either by experiment, Direct Numerical Simulationor other Large Eddy Simulations. It was later discovered that the implementation for the Solve4LEScode could be optimized, so further reduction in the requirement of computational time could beperformed.This modification of the code was only applied to the simulation done in the next chapter, where flowproblems relevant to ventilation are studied. Furthermore, the overall accuracy and efficiency of theimplemented methods may be considered as satisfactory. This will also be supported by the results of theapplication of the methods/codes to flow problems, which are presented later.


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