CHAPTER 6 APPLICATION OF NUMERICAL SIMULATION TO VENTILATED FLOW PROBLEMS

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2 in the direction of the experiment data from Restivo (1979). The same situation was also found by Davidson and Nielsen (1998) for Large Eddy Simulation. However, additional simulations could be performed in order to rule out problems with grid density, inadequate subgrid scale models and improved boundary conditions at the inlet. But there could also be inaccuracy in the experimental data, so further experiments would be needed to investigate the discrepancies between the Large Eddy Simulation and the experimental data from Restivo (1979). However, the simulations concur well with the findings made by Armaly et al. (1983), for all the four different Reynolds numbers both within the transitional and the fully turbulent regions. This further indicates that the data by Restivo could be subjected to uncertainty. Further results about the backward facing geometry and the modelling of transitional flows is given in Bennetsen (C) (2001). Next, a test from the Air Physic Lab at Research Centre Bygholm, The Danish Institute of Agricultural Sciences is studied. The geometry of the room is changed to a different room width, a different slot height and a different outlet location. This has an effect upon the flow geometry which changes from two-dimensional to fully three-dimensional. 6.4 The SJVF-Room Figure 6.89: Geometry of the SJVF-room; heigth = 3 m; length = 8.5 m; width = 5 m; height of slot inlet is m, length of slot outlet is 0.11m and width of 4.8 m. The last room geometry studied within this thesis is based on an experimental room in the Air Physics Laboratory, at Research Centre Bygholm. The length of the room is 8.5 m, the heigh is 3 m and the width is 5.0 m. The inlet which consists of a slot opening in the full width of the room and is placed just below the ceiling. The height of the inlet slot was 0.1 m when fully open. The actual opening height could be adjusted within the range from 0 to 0.1 m with a 0.24 m bottom-hinged flap. But in the currently studied case the inlet heigh was set at m. The outlet was placed in the floor below the ceiling, but it was only 4.8 m wide and 0.11 m long. 168

3 This, however, is small modification of the original geometry where the outlet is though a slatted floor located at the same place. The room was originally designed as extension of the previously studied geometry the Annex 20 test case and was designed for the creation of a two-dimensional recirculating flow. This room has previously been studied by Bjerg et al. (1999) using the commercial CFD code CFX with the k-i turbulence model with different boundary conditions (see Bjerg et al. for further information about experimental setup etc.). They did find a three-dimensional air flow pattern with the given geometry. The Reynolds number based on the inlet height was 4800, which gives an inlet velocity of 3.8 m/s. At first a steady state solution was computed by using Solve4kI, where the two-layer k-i turbulence model was employed. The iso-surface of velocity depicting the three-dimensional flow pattern is shown in Figure 6.90, where also the vector vectors in the return flow close to the floor is depicted. A hyperbolic tangent function was used to calculate a stretched grid. The minium grid cell dimension was selected as (Fx min /H, Fy min /H, Fz min /H) = (0.002, , 0.004), respectively. The number of grid points was The numerical setup and parameters were similar to the computational setup of the Annex 20 test. The kinetic energy at the inlet was prescribed to 4%. The three-dimensional flow pattern is clearly visible and the simulated direction of the return flow was 215 compared to measurement of 214 (Bjerg et al. (1999)) at (x, y, z) = (6.5, 0.2, 2.5). A return direction of 180 would have indicated a twodimensional return flow direction. Further grid refinement did not change the return flow direction. The inlet profile is assumed uniform and a 4% random fluctuation is superimposed. By mean of the computed k-i solution, a Large Eddy Simulation was started from here by using the implicit Solve4LES with the previously described dynamic one-equation subgrid scale model. The CFLnumber was kept at 1. One Large Eddy Turn Over Time (LETOT) was set to 2 L / U in Z5 sec. Figure 6.90: Iso-surface of velocity computed by Solve4kI by using the two layer k-i turbulence model with grid points. Light Green: 0.16 U in. The slice with velocity vector is located 1/15 H above the floor. Re =

4 A total of 200 LETOT were simulated to collect averaged values. This was done to ensure that any predominant effect from the k-i solution had diminished. In the next four figures different iso-surfaces of instantaneous velocity are depicted, thereby showing the fluctuation and separation of the inlet jet, at first closely to the side walls, and later the evolution of the three-dimensional flow pattern is shown. The flow begins to separate the side wall after only 1/3 of the room length. Figure 6.91: Instantaneous iso-surface of velocity = 0.37 computed by use of Solve4LES and the dynamic one-eqn. model sgs. Figure 6.92: Instantaneous iso-surface of velocity = 0.26 computed by use of Solve4LES and the dynamic one-eqn. model sgs. Figure 6.93: Instantaneous iso-surface of velocity = 0.21 computed by use of Solve4LES and the dynamic one-eqn. model sgs. Figure 6.94: Instantaneous iso-surface of velocity = 0.16 computed by use of Solve4LES and the dynamic one-eqn. model sgs. 170

5 Figure 6.95: Return flow direction of the monitor point in the symmetrical plane of the room (x, y, z) = (6.5, 0.2, 2.5). Direction of 180 indicate two-dimensional flow. In Figure 6.95 the instantaneous direction of the return flow is depicted 0.2 m above the floor. The averaged direction of the return flow was 217, which is close to the experimental data and the previous k-i solution. The fluctuation of the flow in the return direction indicates that the flow is unsteady. Although a switch-over of the inlet jet from the right downstream corner (right side wall) to the left downstream corner (left side wall) was captured during one measurement 1) it could not be reproduced during the Large Eddy Simulation. The change in the experiments from one side wall to the other one occurred without any obvious disturbances within the room. The period of switch-over was measured to sec. and compared to the time spent within the Large Eddy Simulation which represented less than 10%, this was not enough to capture any switch-over to occur. In Figure 6.96, the instantaneous velocity vectors for different planes perpendicular to the streamwise direction are depicted and coloured with the instantaneous streamwise velocity component. Vortical structures are visible just as in the Annex 20 test case. These vortical structures are generated due to the shear layer between the fluid within the inlet jet and the fluid outside the jet. Also, the friction at the wall and at the ceiling contributes to the generation of these secondary motions. Although this has not been identified, these secondary motions could contribute to the generation of the three-dimensional flow pattern within the room, which differs in geometry from the Annex 20 test case in form of a different 1 ) Personal communication with Senior Scientist Guoqiang Zhang, Danish Institute of Agricultural Sciences, Research Centre Bygholm. 171

6 inlet heigh, a larger width and a higher air flow rate. But also the difference in inlet height and the large width could contribute to the generation of the three-dimensional flow pattern. Figure 6.96: Instantaneous velocity vector plot at different locations in the SJVF-room: x/h = 1/3, 2/3, 4/3 and 15/6 computed using the implicit Solve4LES. The velocity vector has been projected on the yz-plane to illustrate secondary flow structures within the flow. Re = The velocity vectors are coloured with the streamwise instantaneous velocity component ( ). Figure 6.97: SJVF-room: Instantaneous velocity vectors coloured with normalized helicity within the xy-plane at z/h = 1/30 from left side wall. 172

7 In the Figures , the instantaneous velocity vectors in the xy-plane at different distances from the side wall are depicted. These velocity vectors are coloured with the normalized helicity. Figure 6.98: SJVF-room: Instantaneous velocity vectors coloured with normalized helicity within the xy-plane at z/h = 0.1 from left side wall. Figure 6.99: SJVF-room: Instantaneous velocity vectors coloured with normalized helicity within the xy-plane at z/h = 0.5 (symmetrical plane) from left side wall. Figure 6.100: SJVF-room: Instantaneous velocity vectors coloured with normalized helicity within the xy-plane at z/h = 0.9 from left side wall. 173

8 Helicity is a scalar quantity defined as an inner (dot) product of velocity and vorticity vectors. The vorticity q is defined as q = i. The normalized helicity is the cosine of the angle between the velocity and the vorticity vector. The extreme values of the normalized helicity, e.g. -1 and 1, indicate where the flow is highly three-dimensional, since there is alignment between the velocity and the vorticity vectors (i.e. the vorticity and the velocity vectors in the streamwise direction). The value zero means that the flow is basically two-dimensional, since the vectors are not aligned (vorticity is normal to the velocity). As a result, the normalized helicity value will be very close to 1.0 or -1.0 at the core of the secondary vortices perpendicular to the streamwise direction of the flow. This can also be utilized to locate the core of those streamwise vortices. When velocity vectors are coloured with normalized helicity, an indication of the three-dimensionality of the flow field can be studied. In the upper right corner a recirculating zone is clearly visible and also within the return flow there is a significantly high level of helicity that will indicate an unsteady flow pattern. Very close to the left side wall the angle between the velocity and vorticity vectors are large. In the symmetrical plane of the room extreme values of the angle can furthermore be observed both within the inlet jet and in the return flow area. Figure 6.101: Instantaneous iso-surface of velocity = 0.3 coloured with the normalized helicity to indicate the three-dimensionality of the flow for the Annex 20 test case. Implicit Solve4LES was used with grid 3 (Table 6.3) and the dynamic one-equation subgrid scale model. Red: indicates high level of helicity ( > 0.9); Green: middle level of helicity ( = 0.5); Blue: low level of helicity (< 0.1). 174

9 The level of helicity is compared to the Annex 20 test case depicted in Figure 6.101, which shows the instantaneous iso-surface of the streamwise velocity, coloured with the absolute value of the normalized helicity. For the inlet jet the level of helicity is at a moderate level, and as the back wall is approached the level of helicity is increased thus indicating that the flow is becoming more and more threedimensional and chaotic. The first part of the Annex 20 room also indicates that the flow pattern is almost two-dimensional. This was also confirmed by comparing the probability density for points in the symmetrical plane and close to the side walls. Finally in Figures show the velocity profile comparing the two-layer k-i turbulence within Solve4kI and the time averaged velocity profile from the Large Eddy Simulation with the dynamic one-equation subgrid scale model to the measured data (Bjerg et al. (1999)). Figure 6.102: Comparison between computed velocity profile by the two-layer k-i turbulence model and the time-averaged velocity from LES by using the dynamic one-equation sgs model close to the left side wall. (x/h=1.5; z/h = 1/6). Red: twolayer k-i turbulence mode; Blue: LES with Dynamic One-eqn. sgs. Markes: Measurement: Bjerg et al.(1999). Figure 6.103: Comparison between computed velocity profile by the two-layer k-i turbulence model and the time-averaged velocity from LES by using the dynamic one-equation sgs model at the symmetry plane. (x/h=1.5; z/h = 5/6). Red: twolayer k-i turbulence mode; Blue: LES with Dynamic One-eqn. sgs. Markes: Measurement: Bjerg et al.(1999). Just below the ceiling at y/h = 0.99 and x/h = 1.5, both types of simulation agree well with the measurement. Although for z/h = 3/2, the LES with the dynamic one-equation subgrid scale model predicts the jet peak velocity a little closer to the measurement. In the return flow the maximum velocity is also predicted quite well, compared to the experiments. The differences between the two-layer k-i turbulence model and the LES is observed in the middle of the room, where the low velocity region exists. Unfortunately, no experimental data exist for that area of the room. A very interesting point remains from these simulations when LES is used: Can LES predict the switch-over of the inlet jet? Since the present LES was only run for 1000 sec. and the measurements were more than 10 times longer, this was not attempted in this project. 175

10 Next, a closer look at the performance of the Solve4LES program is taken. Figure 6.104: Comparison between computed velocity profile by the two-layer k-i turbulence model and the time-averaged velocity from LES by use of the dynamic one-equation sgs model at the symmetry plane. (x/h = 1.5; z/h = 3/2). Red: two-layer k-i turbulence mode; Blue: LES with dynamic one-equation subgrid scale model. Markes: Measurement by Bjerg et al.(1999). 3.5 The performance of the explicit and implicit version of Solve4LES In the previous sections results from the use of different versions of the code Solve4LES are applied to compute transitional and fully turbulent flows inside ventilated enclosures by use of the Large Eddy Simulation. Since LES is a quite time-consuming method of dealing with turbulence modelling, because it computes the evolution of the flow field in time, an efficient numerical solver will be preferable in order to reduce the overall time demand for computing the solution. The basic numerical method in Solve4LES is the same as the Calc-LES code developed by Prof. Lars Davidson and others at Chalmers University, Sweden. But in the current project other numerical methods were tested to evaluate the overall performance of these methods and also in the search for the better numerical methods when using LES. Davidson & Nielsen (1996) and Emvin (1997) mention that around 80% of the total cpu-time required was spent on solving the pressure Poisson equations at each time-step when using an explicit formulation (see Chapter 4). As mentioned earlier in the present codes, the velocity component in the momentum equation can be solved by using either the Gauss-Seidel method, the Strongly Implicit Procedure method (SIP) (Stone, 1968) or the Modified Strongly Implicit Procedure (MSIP) (Schneider and Zedan, 1981) method. Davidson & Nielsen (1996) use the conjugate gradient (CG) method with incomplete Choleski preconditioning when solving the pressure Poisson equations within the explicit method. Preconditioning is a very important task for solving a system of linear equations by an iterative method, like CG and 176

11 other Krylov subspace methods in other to accelerate the convergence rate. During the implementation of the explicit version of Solve4LES the same Sparse Linear Algebra Package (SLAP) 2) as Davidson and Nielsen (1996) were tested briefly, simulating the Annex 20 test case. The CFL number was kept below 0.4, and only the Smagorinsky model was applied. The cpu-time per time step/grid point was found to (sec./grid point). The number of grid points used for this test were only in the x-, y- and z-directions, respectively, and the cpu-time increased dramatically when larger number of grid points were used. This means that 95% of the total cpu-time per time step will be required to solve the pressure Poisson equations, although this explicit method has the advantage that the matrix arising from the discretion of the pressure Poisson equations will have the same coefficients, which implies that the calculation of the preconditioning matrix only has to be carried out once within the first time step, by using the SLAP package for the calculation of the incomplete Choleski preconditioner required approximably the same time as 12 time steps during the Large Eddy Simulation. However, since this calculation will only be needed once, it represents only a fraction of the total computational time when using Large Eddy Simulation. In order to test the performance of other preconditioners, a separate implementation of the Conjugate Gradient method together with an incomplete Choleski (IC) preconditioner, the ILUT (Saad, 1996) which combines incomplete factorization and a threshold strategy for dropping the numerical fill-in values in the preconditioning matrix was implemented and tested. The dropping residual was set at 10-6 and number of fill-in s were set at 7 (see Saad (1996) for further information). Moreover, the multilevel-like incomplete factorization method by Botta &Wubs (1999) named, the Matrix-Renumbering Incomplete LU factorization method (MRILU), which is able to achieve near grid independent convergence rate as also the multigrid method is know for was implemented and tested. Finally, a multigrid method (Llorente and Melson, 1998) was implemented and tested for both the explicit and implicit versions of Solve4LES. Some other important factors to be considered when different numerical methods are implemented are the computer architectural issues. The testing of different storage formats for the sparse banded matrix arising from the pressure Poisson equations like those mentioned in the SPARSKIT 3) package can give performances between 4 and 52 Mflops on the currently used PC workstation. A technique to be used for improving performance on most workstations may be the re-use of the cache-memory, which will often require some sort of re-ordering of the storage-structure or the sparse-matrix bandwidth minimization. This, however, has not been utilized in the current code. Improvements to the code were made by carefull loop unrolling (the creation of fat loop), reduction of procedure, function calls within loops (by in-lining functions and even procedure calls), and the reduction of branches (conditional statements like if-then) within loops. The original matrix is stores as a banded sparse matrix and the preconditioning matrix using compressed row format (Saad, 1996). The same code running on vector supercomputers and cache-based RISC workstations will usually involve severe problems regarding the overall performance. This was also 2 SLAP: available from 3 ) SPARSKIT II available from 177

12 illustrated with the LESROOM code since this was optimized for the Cray vector supercomputer by means of a data structure that favours long vector lengths that could easily be pipelined by this kind of processor. Comparing the performance of LESROOM between the Cray and the PC based on the Mflops values, the ratio was between in favour of the Cray. Even if the Linpack 4) benchmark shows a ratio between the Cray and the currently used PC workstation to be 3.6 and 3, respectively, when the sizes of the dense system of linear equations are and For the explicit version of Solve4LES: in Table 6.8, the cpu-time per time step and grid point is displayed by using the different preconditioning methods, previously described. The Conjugate Gradient method with the MRILU preconditioner which was provided by Ass. Prof. Frank Wubs (see Acknowledgment) are one of the better methods including the multigrid method. The reason why the multigrid method is not the fastest, is that a the low number of grid points are used and that the number of coarsing levels within the multigrid method is limited, due to the number of grid points. Another major advantage of the last two mentioned methods are that they do not provide the same increase in cpu-time as the number of grid points is increased. An comparison between the different preconditioning methods used with the Conjugate Gradient solver and the multigrid method is depicted in Figure This clearly shows the advantage of using the CG with MRILU preconditioning and the multigrid method, because the convergence rate will be nearly grid independent. Furthermore, if the implementation of the currently used conjugate gradient is compared with the incomplete Choleski preconditioning to the same numerical method within the SLAP implementation, the first implementation will turn out to be more than 6 times faster on the same computer. Method of preconditioning for the pressure Poisson equations CPU Time / grid point [sec.] PC[Pentium II Xeon] Incomplete Choleski 4.01 l 10-5 ILUT 3.30 l 10-5 MRILU 2.11 l 10-5 Multigrid 2.14 l 10-5 Table 6.8: Comparison of cpu time per grid point for each time step the explicit Solve4LES when using different versions of preconditioning for the Conjugate Gradient solver and the multigrid solver. The second column represents a pc workstation with a Pentium II Xeon 450 Mhz and Windows NT 4.0 when using Digital Visual Fortran compiler version 6.0 and full compiler optimization. All computations were carried out by use of using grid points, and the CFL-number was kept at 0.4. The residual was set to Linpack benchmark: is used to solve a dense system of linear equations. It reflects the perfor-mance of a computer for solving a dense system of linear equations, and since the problem is very regular, the performance achieved will be quite high, and the performance numbers will give a good representation of the peak performance The sizes of the dense system of linear equations will usually be and See for additional information about the Linpack benchmark program and software. 178

13 Figure 6.105: Comparison between Conjugate Gradient solver with different preconditioning methods and a multigrid method in terms of total cpu-time as a function of number of grid points. Explicit version of SolveLES used on a PC workstation Pentium II 450 Mhz with the Smagorinsky sgs model The ratio between solving the pressure Poisson equation, the velocity components and the Smagorinsky sgs model was 80%, 15% and 5% of the total cpu time, respectively, obtained by using 64x32x32 grid points. Large Eddy Simulation Time / grid point [sec.] CPU requirements Subgrid Scale Model PC[Pentium II Xeon] relative to Smagrinsky sgs model Smagorinsky Dynamic with plane averaging Dynamic no averaging Dynamic One-Eqn Table 6.9: Comparison of cpu time per grid point for the explicit Solve4LES obtained by using different version of subgrid scale models.the second column represents an pc workstation with a Pentium II Xeon 450 Mhz and Windows NT 4.0 and using Digital Visual Fortran compiler version 6.0 and full compiler optimization. All computations were carried out by using grids points and the CFL-number was kept at 0.4. The residual was set to

14 Large Eddy Simulation. Time / grid point [sec.] CPU requirements relative to Smagorinsky Subgrid Scale Model PC[Pentium II Xeon] sgs model Smagorinsky Dynamic with plane averaging Dynamic no averaging Dynamic One-Eqn Table 6.10: Comparison of cpu time per grid point for the implicit Solve4LES by using different versions of subgrid scale models.the second column represents an pc workstation with a Pentium II Xeon 450 Mhz and Windows NT 4.0 and using Digital Visual Fortran compiler version 6.0and full compiler optimization. All computations were carried out by using grid points, and the CFL-number was kept at 1.0. The residual was set to 10-3, defined by the L 2 - normwhich requires 2-3 iterations within each time step. The cpu-time per grid point is taken as an average between 2 and 3 iterations. In Tables 6.9 and 6.10, the different subgrid scale models are compared in terms of cpu-time used in the explicit and implicit code. The reason why the more advanced dynamic subgrid scale models seem less expensive when the implicit code is used, is that more time is spent on solving the pressure and velocity component because of iterations within the implicit method. The subgrid scale model only needs to be solved once every time step. In the next 6 figures the cpu-time obtained by using the explicit and implicit Figure 6.106: CPU-time per time step obtained by using explicit Solve4LES with grid points and the Smagorinsky subgrid model. MSIP is used for velocity component and MG for pressure Poisson equations. Figure 6.107: CPU-time per time step obtained by using explicit Solve4LES with grid points and the dynamic subgrid model with plane averaging. MSIP is used for velocity component and MG for pressure Poisson equations. 180

15 Figure 6.108: CPU-time per time step obtained by Figure 6.109: CPU-time per time step obtained using explicit Solve4LES with grid by using implicit Solve4LES with grid points and the dynamic one-equation subgrid points and the Smagorinsky subgrid model. SIP is model. MSIP is used for velocity component and used for velocity component and MG for pressure MG for pressure Poisson equations. Poisson equations. versions of Solve4LES is depicted for each component, i.e. pressure, velocity and subgrid scale model. In the Figures , the total cpu-time per time step and for the other components only the cpu-time per implicit iteration are depicted. Therefore, variations in the implicit iteration are clearly depicted within the curve for the total cpu-time. Also the cpu-time for the u-velocity component is larger, due to some setup calculation that has to be performed before all the velocity component can be computed. The variations between the three different solvers for the velocity component were small. The SIP and MSIP were 1% and almost 3% faster, respectively, than the Gauss-Seidel method. Figure 6.110: CPU-time per time step obtained by using implicit Solve4LES with grid points and the dynamic subgrid model with plane averaging. SIP is used for velocity component and MG for pressure Poisson equations. Figure 6.111: CPU-time per time step obtained by using implicit Solve4LES with grid points and the dynamic one-eqn. subgrid model. SIP is used for velocity component and MG for pressure Poisson equations. 181

16 From the depicted results on the requirement of cpu-time when using the currently described implementation featuring either an explicit or an implicit method of time advancement, Large Eddy Simulation is still quite time-consuming compared to other methods of turbulence modelling where the flow field is decomposed into a steady and a fluctuating part, the Reynolds Averaging, although the implicit method has the potential to even larger time steps than the currently used limits of one. But it could be questionable to use large time steps in a transitional evolving flow field. This has so far not been tested. But LES is capable of providing much more information about the flow field, although not superior when mean averaging velocity profiles are compared. Furthermore, LES is capable of computing a solution to a flow field that is not fully developed, thus making it very attractive for many ventilation problems where the Reynolds number is often below 5000 and for many cases within the transitional flow range. Finally, the reported cpu measurement on the PC workstation with a Pentium II, Xeon processor, which is approximable one year behind the latest processor trend, could be reduced by upgrading to the latest processor. This modern processor will be nearly capable of reducing the cpu-time by a factor of two and a factor of almost four when the latest UNIX-based workstation with the Alpha processor is used. This assumption is based on the currently available specmark 5) ( Spec_fp95 ) numbers for floating point speeds. The additional use of parallel computer would certainly make Large Eddy Simulations attractive. But the interaction between implementation, the numerical methods, the architectural issues, and the compiler technology remains a challenge. The easy gains in computational speed are over. 5 SPECMARK: see for further information. 182