Basic Features of the Fluid Dynamics Simulation Software FrontFlow/Blue

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1 11 Basic Features of the Fluid Dynamics Simulation Software FrontFlow/Blue Yang GUO*, Chisachi KATO** and Yoshinobu YAMADE*** 1 FrontFlow/Blue 1) is a general-purpose finite element program that calculates incompressible unsteady flows in arbitrarilyshaped geometries that may involve a moving boundary interface. This paper will introduce the basic features of this software and discuss the results computed by this code for some basic turbulent flows. The test cases include homogeneous isotropic turbulence and turbulent flow around a circular cylinder. The objective of the current computations is to validate the basic performance of the code and check the accuracy of the implemented numerical methods such as pressure algorithm, time integration scheme and boundary condition. Particularly, the accuracy of the energy spectrum and the sound pressure level will be checked. 2 Collaborative Research Center of FSIS, Institute of Industrial Science, The University of Tokyo Department of Mechanical and Biofunctional Systems, Institute of Industrial Science, The University of Tokyo Mizuho Information&Research Institute, Inc. FrontFlow/Blue is based on Large-Eddy Simulation (LES) that can compute unsteady flow accurately. The governing equations of LES for an incompressible flow are spatially filtered continuity equation and Navier-Stokes equations. In the current code, Standard Smagorinsky Model (SSM) 2) and Dynamic Smagorinsky Model (DSM) 3) with modification by Lilly 4) are implemented to account for the effects of subgrid-scale (SGS) stresses. Both an explicit time-accurate streamline upwind scheme 5,6) and an implicit Crank-Nicolson (CN) scheme have been implemented in the code for integrating the momentum equations with respect to time. For the CN implicit scheme two to three Newton iterations are made within a time step in order to maintain the second-order accuracy in time. The spatial discretization is based on the Finite Element Method (FEM). The implemented algorithms have second-order accuracy both in time and space and we believe that the second-order scheme is most appropriate in engineering applications of LES. Both Arbitrary Boundary Marker and Cell (ABMAC) 7) method and Fractional-Step (FS) method are employed to solve the pressure equation. The global linear system of equations that results from the pressure Poisson equation in FS method is solved by Bi-CGSTAB 8) method combined with Residual Cutting Method (RCM) 9) as its outer loop. Bi-CGSTAB method is also used as the matrix solver of the global linear equations that result from the CN scheme. By multi-frame overset function 6) in the code, it is possible to compute internal flows in fluid machinery such as fans or pumps. Sound radiated from a turbulent flow can also be computed. In Section 3, the computation of the sound generated from turbulent flow around a circular cylinder will be presented. The present LES code has been implemented as a parallel program by using the domain-decomposition technique. The ratio of the sustained performance to the peak performance is about 7 % to 15 % on various hardware platforms. 3 Homogeneous isotropic turbulence and turbulent flow around a circular cylinder are typical cases of the benchmark flows. The purpose of the computations in this section is to validate the basic performance and the accuracy of our code. In addition, the effects of numerical parameters will be discussed. 3 1 At first, homogenous isotropic turbulence was computed. The Taylor micro-scale Reynolds number (Re apple ), based on r.m.s. velocity and Taylor micro-scale (apple), is 30.5 at non-dimensional time t= The computational domain is a periodic box (2Û) 3 and the number of grid nodes is The non-dimensional time increment is set to»t= DSM is used as the SGS model in this case unless otherwise stated. The LES results are compared with DNS of Tanahashi et al. 10). Figure 1(a) shows the convergence rate of the pressure equation with two pressure algorithms: ABMAC method and FS method. Here the temporal algorithm is fixed as the explicit time-accurate 11

2 SEISAN-KENKYU Fig. 1 Effects of the pressure algorithm on the energy spectrum in homogenous isotropic turbulence of Re apple = Fig. 2 Effects of SGS model (top) and the time increment (bottom)on the energy spectrum at t = in homogenous isotropic turbulence of Re apple = streamline upwind scheme. The maximum divergence of the velocity using FS method is much smaller than that of ABMAC method because of the higher convergence rate of FS method. However, the pressure algorithm has almost no effect on the energy spectrum at t=3.792 as shown in Fig. 1(b). Furthermore, both ABMAC method and FS method give very close results to DNS 10) except in the highest wave number region. The reason is that in free turbulence the pressure does not play an important role on the energy spectrum. However, this is not true in the case of wall-bounded flows, as will be shown in turbulent flow around a circular cylinder. Figure 2 shows effects of the subgrid-scale (SGS) model (top) and the time increment (bottom) on the energy spectra at t= DSM gives very close results to DNS 10) while Standard Smagorinsky Model (SSM) gives a rather dissipative energy spectrum in the entire wave space. To investigate the effect of time increment, we calculated the homogenous isotropic turbulence using»t = ,»t = ,»t = ,»t = ,»t = and»t = The corresponding CFL numbers are 0.2, 0.4, 1.0, 4.0, 10 and 20, where the CFL number is defined as CFL = max ( u /»x+ v /»y+ w»z)»t. The pressure algorithm is fixed as FS method and we use the implicit CN scheme as the temporal algorithm. Figure 2 (bottom) presents the energy spectrum of different time increment. It can be concluded that the agreement of LES and DNS data 10) is quite good with CFL less than 10. From DNS 10), we can estimate that the physical Kolmogorov time scale is t k =(v/è) 0.1, where È is the initial dissipation rate of the turbulence energy. From Fig. 2 (bottom), the result of LES is reasonable if the time increment is less than the Kolmogorov time scale that represents the smallest time scale of the vortical motion. This conclusion is similar to that of Choi and Moin 11) in their fully developed channel flow. With the implicit CN scheme, we can obtain a reduction of total computational cost, as compared with explicit methods. In the benchmark test, the total speed-up ratio can be 3~3.5 if the time increment of the implicit method is five times as much as that of explicit methods. 12

3 This section will present the predictions of the far-field sound generated from turbulent flow around a circular cylinder with the emphasis on the effects of the pressure algorithm and matrix solver on the predicted sound spectra. In addition, the non-reflecting boundary conditions (NRBC) will be tested in the case with low- Mach-number assumption. The far-field sound generated from a circular cylinder with a diameter of D = 0.05 m perpendicularly placed in flow with a uniform velocity of U = m/s was computed. At first, we calculated the flow as an incompressible case. The Reynolds number based on the velocity U and the cylinder s diameter D is The far-field sound was computed by Lighthill-Curle s equation 12,13) at a location 2 m away from the cylinder in the lateral direction and compared with that measured at the same location by Takaishi et al. 14). The computational domain is [-11D, 19D] [-11D, 11D] [0, 4D] in x, y and z directions, where x, y, and z respectively denote the streamwise, lateral, and spanwise coordinates with its origin located at the center of one spanwise end surface of the cylinder. Figure 3 shows the computational mesh in x-y plane. Twenty elements are used in the spanwise direction with a constant spacing of»z = 0.2D. The total number of the computational elements is approximately 130,000. A uniform velocity was specified at the inlet. The fluid traction was assumed to zero at the outlet and symmetric boundary condition was used in the spanwise end surfaces. Non-slip condition was used on the wall. The explicit time-accurate streamline upwind scheme was used with a nondimensional time increment of and SSM was selected as the SGS model. The far-field sound spectrum was computed by using pressure fluctuation on the cylinder during 180 non-dimensional time after the flow field reached a fully equilibrium state. Three different methods were used to solve the pressure Poisson equation: ABMAC method, FS method with Bi-CGSTAB solver and FS method with RCM solver combined with Bi-CGSTAB method. Figure 4 shows the L 2 -norm residual of the pressure equation (RESP) during the inner iterations of the matrix solver at nondimensional time t = 510. The Bi-CGSTAB solver is unstable in this case. With that solver the L 2 -norm residual of the pressure equation suddenly jumps during the iterations, which results in a sudden jump of the computed sound pressure in the time domain and flat sound pressure level in the high frequency domain (f 1 khz), as is shown in Fig. 5. On the other hand monotonous convergence is obtained with RCM solver. This demonstrates superiority of RCM solver to Bi-CGSTAB solver although the rates of convergence are almost same for both solvers. When compared with that predicted by FS method combined Fig. 3 Computational mesh in x-y plane for flow around a circular cylinder at Re = Fig. 4 L 2 -norm residual of pressure equation during inner iterations at t = 510 for flow around a circular cylinder at Re = with RCM solver, frequency spectrum of the sound predicted by ABMAC method has relatively large oscillation in a frequency range of 400~1000 Hz, as is shown in Fig. 5 (bottom). This large oscillation is unphysical and is almost certainly attributed to inaccurate prediction of turbulence energy cascade due to the slow rate of convergence of ABMAC method. FS method combined with RCM solver can overcome this defect and provides the best frequency spectrum of sound among these three pressure algorithms. From this case, we can conclude that a stable solver with high convergence rate is needed to predict the sound accurately. The above case is an incompressible case. In some situations, low-mach-number assumption is necessary for a stable computation of a low-speed flow, such as in the case of cavitating flows. Weak compressibility is allowed in such cases and as a result the pressure wave is generated. When the pressure wave reaches the inlet or outlet boundary, it may cause unphysical reflection if the boundary conditions are not appropriate. To deal with these problems, the non-reflecting boundary condition (NRBC) was 13

4 SEISAN-KENKYU Fig. 5 Time history (top) and frequency spectra (bottom) of the farfield sound for flow around a circular cylinder at Re = Fig. 6 Effects of boundary conditions of pressure equation on instantaneous static pressure at z = 2D plane for flow around a circular cylinder at Re = and M = 0.2. implemented. The NRBC in the current code follows the procedures of Okita and Kajishima 15). We use the above cylinder flow as the test case for NRBC. The computational parameters are the same as the above case except the Mach number (M) is set to 0.2. FS method combined with RCM solver was used as the pressure algorithm. Two kinds of boundary conditions for pressure equation were adopted in the computation. In the first case, the normal pressure gradient was set to zero at the inlet and the pressure was set to constant at the outlet, for which there was no special mechanism to avoid the reflection of the pressure wave at these boundaries. In the other case, NRBC mentioned above were implemented both for the inlet and the outlet boundaries. Figure 6 shows instantaneous static pressure distribution at z = 2D plane. The computation has eventually diverged in the case where we do not use NRBC. In this case the divergence is due to the reflection of pressure wave, particularly the reflection at the inlet (see Fig. 6(a)). In the case of NRBC, this problem can be avoided, as is shown in Fig. 6(b). The static pressure shown in Fig. 6(b) demonstrates that NRBC works as expected in this case. The sound pressure level (SPL) in this case as well as SPL predicted by Fig. 7 Comparison of frequency spectra of the far-field sound for flow around a circular cylinder at Re = ABMAC method in the incompressible case is plotted in Fig. 7. There are some oscillations of the sound pressure level in the frequency range of 400~1000 Hz with M = 0.2 due to the numerical compressibility, which is similar to the result with ABMAC method in the incompressible case. This is inevitable because we 14

5 15 allow compressibility in the flow field if we set M = 0.2 and thus we cannot get a result as accurate as the one with FS method in the incompressible case. Overall, the SPL in the case of M = 0.2 with NRBC is reasonable, as compared with the experimental data of Takaishi et al. 14). 4 The basic features of flow simulation code FrontFlow/Blue were introduced. Two test cases were computed and results show that the agreement of LES results and DNS or experimental data is quite good if appropriate numerical methods are selected. The pressure algorithm shows little effect in the homogeneous isotropic turbulence while its effect is apparent in the turbulent flow around a cylinder. The results of the homogeneous isotropic turbulence show that the time increment should be set below some critical values that are determined by the physical Kolmogorov time scale rather than the CFL condition. The results of turbulent flow around a circular cylinder indicate that a stable solver with a high rate of convergence is needed for the pressure equation when predicting the sound. Computational results of the above cylinder flow with a Mach number of 0.2 demonstrate that non-reflecting boundary condition (NRBC) is effective in the flow computation with low-mach-number assumption. This work was supported by the Ministry of Education, Culture, Sports, Science and Technology (MEXT), Japan under an IT research program Frontier Simulation Software for Industrial Science and an IT research program Revolutionary Simulation Software. The benchmark in isotropic homogeneous turbeulence was initiated by Prof. A. Uddin of Shahjalal University of Science and Technology while he was staying as a visiting researcher in The University of Tokyo. The initial flow field for the isotropic homogeneous turbulence was generated by using a program written by Prof. M. Tanahashi of Tokyo Institute of Technology. The authors would like to thank Dr. K. Okita of The University of Tokyo for the beneficial discussion of non-reflecting boundary conditions. (Manuscript received, November 17, 2005) 1) IT research program Frontier Simulation Software for Industrial Science, 2) Smagorinsky, J., Mon. Weather Rev., Vol. 91, No. 3, pp (1963). 3) Germano, M., et al., Phys. Fluids A, Vol. 3, No. 7, pp (1991). 4) Lilly, D. K., Phys. Fluids A, Vol. 4, No. 3, pp (1992). 5) Kato, C. and Ikegawa, M., ASME-FED, Vol. 117, pp (1991). 6) Kato, C., et al., ASME Journal of Applied Mechanics, Vol. 70, pp (2003). 7) Viecelli, J. A., J. Comput. Phys., Vol. 8, pp (1971). 8) Van der Vorst, H. A., SIAM J. Sci. Statist. Comput., Vol. 13, No. 2, pp (1992). 9) Tamura, A., et al., J. Comput. Phys., Vol. 137, pp (1997). 10) Tanahashi, M., et al., Proc. of the 11th Symposium on Turbulent Shear Flows, Vol. 1, pp (1997). 11) Choi, H. and Moin, P., J. Comput. Phys., Vol. 113, pp. 1 4 (1994). 12) Lighthill, M. J., Proc. R. Soc. London A, Vol. 211, pp (1951). 13) Curle, N., Proc. R. Soc. London A, Vol. 231, pp (1955). 14) Takaishi, T., et al., Journal of Acoustical Society of America, Vol. 116, No. 3, pp (2004). 15) Okita, K. and Kajishima, T., Trans. JSME, Series B (in Japanese), Vol. 68, No. 667, pp (2002). 15