3.2 Aspects of numerical-diffusion regulation for 1D and quasi-1D flows 52
3.2.5 Effects of numerical diffusion in 1D and quasi-1D flows
All the numerical solutions for Sod’s 1st problem are obtained with 200 cells and a CFL number of 0.2. Figure 3.2 shows the plot for the variations of density for Sod’s 1st problem computed by using the DRLLF scheme for different values of δ (see equation 2.83) with increments of 0.1. The figure shows that as the numerical diffusion is reduced the numerical oscillations grow, especially near dis-
continuities1. Accordingly numerical oscillations are the maximum forδ = 0.1 and as the numerical-diffusion level is raised these oscillations get reduced. There is only slight difference in the smearing of the shock and the contact discontinuity captured with δ=0.7 and δ=0.9. However the numerical oscillations are greatly reduced forδ= 0.9. It may be noted that for δ= 1.0 the DRLLF scheme becomes equivalent to the fully diffusive LLF scheme. Thus for the computation of the 1D shock tube problem using DRLLF scheme, the value of δ = 0.9 is found to be the most appropriate in terms of reduction in spurious numerical oscillations with slight compromise in the resolution of the captured discontinuities. The reason for oscillations produced near the shock using the DRLLF scheme even at higher values of δ is that the jump in absolute Mach number across the shock is much less than unity for Sod’s 1st problem since the absolute flows are subsonic.
Fig. 3.2. Comparison of densities for the shock tube problem given by the DRLLF scheme using different values of δ.
Figure 3.3 shows the comparison of the density-variation plots with the LLF, DRLLF (δ= 0.9) and DDRLLF schemes for the same problem. For the DDRLLF scheme, since the DDR assumes the value of unity for values ofµg ≥1 and for 1D flowµg ≥1, henceΦd will always be close to unity. Thus this method works as an unregulated method for 1D problems. This is demonstrated in figure 3.3, where the density-variation plots for LLF and DDRLLF schemes are very close to each
1This work has been published in the proceedings of the 40th National Conference on Fluid Mechanics and Fluid Power (2013), National Institute of Technology Hamirpur, India
other. It is evident that among the LLF, DRLLF (δ= 0.9) and DDRLLF schemes the DRLLF (δ = 0.9) scheme outperforms the other two in computing Sod’s 1st problem.
Fig. 3.3. Comparison of densities for the shock tube problem given by the LLF, DRLLF (δ=0.9) and DDRLLF schemes.
Fig. 3.4. Comparison of densities for the shock tube problem given by van Leer’s FVS and AUSM schemes.
At this point it makes sense to make a comparison between the robust van Leer’s FVS scheme [14] with the accurate AUSM [7] scheme. Figure 3.4 plots the density variations for Sod’s 1st problem using the two schemes. It shows that the AUSM scheme, being less dissipative, resolves the expansion fan, contact discontinuity and the normal shock wave better than van Leer’s FVS scheme.
With these observations we compare the AUSM and DRLLF (δ = 0.9) schemes with the analytical solution for Sod’s 1st problem in the density-variation plot shown in figure 3.5. The smearing of the shock captured by the DRLLF scheme is found to be less than that by AUSM. The resolutions of the contact discontinuity and the expansion fan captured by the two schemes are very much comparable to each other. Thus the DRLLF scheme appears to be a promising alternative for computation of inviscid-flux terms in the computation of high-speed flows.
Fig. 3.5. Comparison of densities for the shock tube problem given by the AUSM scheme, DRLLF (δ=0.9) scheme and the analytical method.
Fig. 3.6. Comparison of velocities for the shock tube problem given by the AUSM scheme, DRLLF (δ=0.9) scheme and the analytical method.
Figure 3.6 compares the computed velocity profiles for Sod’s 1st problem us- ing the AUSM and DRLLF (δ = 0.9) schemes with the analytical results. The
continuity of the velocity field across the contact discntinuity is maintained by both the schemes. Along the flow-direction the downstream end of the expansion wave is marginally smeared by the DRLLF scheme. However the shock smear- ing is less with the DRLLF scheme compared with AUSM. At the same time the small-numerical oscillations near the shock produced by the DRLLF scheme leave a scope for further improvement of the scheme.
The numerical computations for the quasi-1D nozzle-flow problem are carried out using 60 cells with a CFL number of 0.2. In order to assess the effects of numerical diffusion in computing quasi-1D flows, figure 3.7 presents a comparison of the density variation along the nozzle, computed with the DRLLF scheme using various values ofδ. Forδ= 0.1 andδ= 0.3, although the smearing of the captured normal shock is less, the numerical oscillations near the shock are more. For δ = 0.5, with slight reduction in the shock resolution these oscillations are almost eliminated. At higher values ofδ, the smearing of the shock increases. Thus from figure 3.7 it appears that for the computation of quasi-1D converging-diverging nozzle-flow using the DRLLF scheme, the value δ = 0.5 produces a reasonably accurate result.
Fig. 3.7. Comparison of densities for the quasi-1D nozzle-flow problem given by the DRLLF scheme using different values ofδ.
Figure 3.8 compares the density variations for the quasi-1D nozzle-flow com- puted with the LLF, DRLLF (δ= 0.5) and DDRLLF schemes. As observed for the shock tube problem, in this case also the DDRLLF scheme acts like an unregulated
scheme. The DRLLF (δ = 0.5) scheme outperforms the other two in resolving the shock. Thus it gets reinforced as reported in [41] that there is no scope to regulate the numerical diffusion with the DDRLLF scheme for 1D and quasi-1D flows. On the other hand the DRLLF scheme has a scope for improvement over the LLF scheme for this class of problems.
Fig. 3.8. Comparison of densities for the quasi-1D nozzle-flow problem given by the LLF, DRLLF (δ=0.5) and DDRLLF schemes.
Fig. 3.9. Comparison of densities for the quasi-1D nozzle-flow problem given by van Leer’s FVS and AUSM schemes.
Figure 3.9 shows the comparison of density variations for the quasi-1D nozzle- flow problem computed by using the AUSM and van Leer’s FVS schemes and the curves are in close agreement. AUSM, being less diffusive, captures the normal shock with marginally less smearing. The density profiles for the problem using the
AUSM and DRLLF (δ = 0.5) are compared with the analytical solution in figure 3.10. In this case since the flow upstream of the shock is supersonic hence the jump in Mach number is larger compared with Sod’s 1stproblem. As a result near the shock the DRLLF scheme is more diffusive and therefore the shock captured by the scheme gets more smeared compared with the AUSM scheme. Similar observations can be made from the Mach-number profile comparison of the AUSM and DRLLF (δ = 0.5) schemes and the analytical results for the quasi-1D flow problem as shown by figure 3.11.
Fig. 3.10. Comparison of densities for the quasi-1D nozzle-flow problem given by the AUSM and DRLLF (δ=0.5) schemes and the analytical method.
Fig. 3.11. Comparison of axial variations of Mach number for the quasi-1D nozzle-flow problem given by the AUSM and DRLLF (δ=0.5) schemes and the analytical method.
A study on the computational time taken by van Leer’s FVS, AUSM, DRLLF and DDRLLF schemes for the 1D shock tube and quasi-1D nozzle-flow problems is done. The same machine and compiler are used for the comparison exercise.
The processor is Intel(R) Core (TM) 2 Duo T5550 with processor speed of 1.83 GHz, and having 1GB RAM. The Microsoft Visual C++ 6 compiler is used to compile all the codes. Table 3.1 lists the CPU-time for these test problems taken by the four schemes. For Sod’s 1stproblem apparently there is no difference among the schemes in terms of the CPU-time. For the quasi-1D flow case the DRLLF scheme competes well with the AUSM and van Leer’s FVS schemes in terms of the computational time. The DDRLLF scheme falls in between the DRLLF and AUSM schemes in terms of computational time. However there is hardly any point in focusing on this observation since the DDRLLF scheme does not actually control the numerical diffusion of the LLF scheme for the 1D and quasi-1D flows.
Table. 3.1. Comparison of CPU-time for 1D and quasi-1D flows Test Problem Scheme CPU-time (s)
Sod’s 1st problem
AUSM 0.281
van Leer’s FVS 0.281
DRLLF 0.281
DDRLLF 0.281
Quasi-1D flow
AUSM 7.187
van Leer’s FVS 7.406
DRLLF 6.890
DDRLLF 7.359