3.3 Aspects of numerical-diffusion regulation for 2D supersonic flows
3.3.4 Performances of schemes in computing 2D supersonic flows . 72
mass has the same value as in the same zone to set the entropy. This condition is based on the assumption of a nearly steady flow near the corner. Similar boundary condition was applied by Woodward and Collela [11].
3.3.4 Performances of schemes in computing 2D super-
shock relatively better than the DDRLLF scheme. The reason for this observation is that the DR parameter approaches its maximum value of unity as the jump in normal Mach number across the cell interface, i.e., ∆M⊥ approaches unity (see equation 2.83). Similarly the DDR approaches its maximum value of unity as the value of µg in equation 2.88 exceeds a value of unity. Equation 2.85 suggests that µg ≥ 1 in the vicinity of a normal shock and hence the DDR also assumes its maximum value. As the DR parameter and DDR tend towards unity, the DRLLF and DDRLLF schemes behave like the fully diffusive LLF scheme. Therefore, it can be stated that the strong shocks are resolved better by the AUSM and van Leer’s FVS schemes than the DRLLF and DDRLLF schemes. This fact gets reinforced from figures 3.16(a)-(b) that plot the pressure and Mach-number variations along the nozzle axis, respectively, using the four schemes. Further figures 3.16(a)-(b) reaffirm the sonic-glitch problem exibited by the AUSM scheme, which is not present with the DRLLF and DDRLLF schemes, as shown by the encircled regions.
(a) (b)
Fig. 3.16. Axial variations using first-order-accurate AUSM, van Leer’s FVS, DRLLF and DDRLLF schemes: (a) Pressure and (b) Mach number.
Figures 3.17(a-d) compare the pressure contours for the supersonic flow through a forward-facing stepped channel presented in Table 3.2. A 120×40 grid is used for the computations. The CFL number is 0.2. Overall the detached strong shock upstream of the step is better resolved by the AUSM and van Leer’s FVS schemes.
The AUSM scheme, being less diffusive, produces marginally better results than van Leer’s FVS.
(a) (b)
(c) (d)
Fig. 3.17. Pressure contours for supersonic flow through a forward-facing-stepped channel using (a) AUSM, (b) van Leer’s FVS, (c) DRLLF, and (d) DDRLLF schemes.
(a) (b)
(c) (d)
Fig. 3.18. Pressure contours for supersonic flow past a 2D wedge using (a) AUSM, (b) van Leer’s FVS, (c) DRLLF and (d) DDRLLF schemes.
Figures 3.18(a-d) compare the steady-state pressure contours of the AUSM, van Leer’s FVS, DRLLF and DDRLLF schemes for supersonic flow over the 2D wedge problem (Table 3.3). The flow is symmetric about the axis as the wedge is considered at zero-incidence angle. Hence the flow is computed over the upper half of the body only. The gradual weakening of the leading-edge shock due to interaction with the expansion fan centered at the convex corner till it becomes a Mach wave is captured by all the schemes. Further it can be seen that the DRLLF scheme produces a much better resolution of the weak leading-edge-shock compared with rest of the schemes. It can be inferred that in the vicinity of weak shocks the DRLLF scheme offers less numerical diffusion to produce crisp resolution of grid-inclined shocks.
(a) (b)
(c) (d)
(e) (f)
Fig. 3.19. Pressure contours for supersonic flow through a ramped surface using first-order-accurate: (a) AUSM, (b) van Leer’s FVS, (c) DRLLF,(d) DDRLLF schemes, and higher-order-accurate: (e) AUSM, (f) van Leer’s FVS schemes.
The 2D ramp-in-a-channel problem (Table 3.4) is computed with the first- order-accurate AUSM, van Leer’s FVS, DRLLF and DDRLLF schemes and higher- order-accurate AUSM and van Leer’s FVS schemes with minmod limiter through the MUSCL approach. A 120×80 grid is used for all the methods with a CFL number of 0.2. Figures 3.19(a-f) show the pressure contours for this flow computed with different schemes2. For the first-order-accurate versions, the oblique shocks are best captured by the DRLLF scheme compared with the other three schemes.
The DDRLLF scheme also captures the oblique shocks with improved resolution compared with the normal shocks in the shock tube, quasi-1D and 2D nozzle-flow problems. From figures 3.19(a-f) it is also evident that the first-order-accurate DRLLF scheme resolves the oblique shocks almost as good as the second-order- accurate van Leer’s FVS and AUSM schemes. Moreover, in the contours obtained by the second-order-accurate AUSM and van Leer’s FVS schemes, oscillations are seen, especially near the shock. The first-order DRLLF scheme is free from such oscillations.
The reason for the improved performances of the DRLLF and DDRLLF schemes in resolving weak shocks over strong shocks is that the jump in nor- mal Mach number ∆M⊥ for the former type of shocks is less compared with that in case of the latter ones. As a result the DR parameter across an oblique shock assumes smaller values (see equation 2.83) and therefore the DRLLF scheme is more effective for oblique shocks. Similarly across oblique shocks, the gradients of flow variables across the interface are not as strong as that in case of normal shocks. Thus the value of µg (see equation 2.88) in the case of oblique shocks is also on the smaller side. Accordingly the value of the DDR is smaller and thus the DDRLLF scheme is more effective for oblique shock waves, especially when these are not aligned with the grid. Figures 3.19c and 3.19d also indicate that even in oblique-shock resolution the DRLLF scheme outperforms the DDRLLF scheme.
Thus the DRLLF scheme is found to be more effective for diffusion regulation in both 1D and 2D flow problems compared with the more recent DDRLLF scheme that is useful for 2D (or 3D) problems with grid-inclined shocks only.
2This 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
Table. 3.5. Comparison of CPU-time for 2D supersonic flows
Test Problem Scheme CPU-time (s)
Converging-diverging nozzle-flow
AUSM 365.48
van Leer’s FVS 339.17
DRLLF 322.84
DDRLLF 893.61
Forward-facing stepped channel
AUSM 68.64
van Leer’s FVS 62.34
DRLLF 60.39
DDRLLF 183.24
2D wedge
AUSM 1782.61
van Leer’s FVS 1690.74
DRLLF 1602.39
DDRLLF 3743.67
Ramped channel
AUSM 267.12
van Leer’s FVS 245.87
DRLLF 233.41
DDRLLF 1394.92
Table 3.5 lists the CPU-time for the 2D supersonic flow problems by the four schemes. The convergence behaviour of the schemes is found similar. Accord- ingly for a particular problem the CPU-time is reported for the same number of iterations for all the schemes. This data is a representation of the relative algorith- mic simplicity of the different schemes. The DRLLF scheme consumes the least CPU-time owing to its algorithmic simplicity. The CPU-time of the AUSM and van-Leer’s FVS schemes are close. The DDRLLF scheme consumes large CPU- time for 2D-flow computations owing to its increased programming complexity, since the flow-gradients have to be computed across more number of neighbour- ing points to compute the value of µg. The saving in computational time by the DRLLF scheme becomes more prominent as the number of grid points increases.
Thus the DRLLF scheme holds promise for computation of more complex flows involving a large number of grid points.
3.4 Aspects of the DRLLF scheme applied to hy- personic flows
The promising performance of the DRLLF scheme for computing 1D and quasi-1D flows, and 2D supersonic-flow computations provides the motivation for investigat- ing its performance in computing hypersonic flows. In this section some aspects of the DRLLF scheme applied to compute hypersonic flows over a semi-cylinder and a hemisphere are presented. A comparison of the performance of the DRLLF scheme with the robust van Leer’s FVS is made. The computations are done for the perfect-gas model and the high temperature equilibrium chemically reacting air model proposed by Tannehill and Mugge [73]. The ability of the DRLLF scheme to compute reacting gas models is demonstrated for the first time3.