layer interaction
This section presents the problem statement for a ramp-induced hypersonic SWBLI problem and the grid used along with the boundary conditions. This is followed by the results and discussion on the effects of numerical diffusion in the computation of hypersonic SWBLI with laminar separation. The level of numerical diffusion in the DRLLF scheme is regulated by changing the parameterδ in equation 2.83 and a qualitative analysis is offered on its effects on the computed-skin-friction profile, laminar-separation-bubble (LSB) size, pressure-coefficient profile and convergence behaviour. Comparisons are made with the AUSM and van Leer’s FVS schemes as well.
4.3.1 The problem statement and the grid used with the boundary conditions
A ramp-induced hypersonic SWBLI problem as discussed in section 4.1 is consid- ered. The length of the flat surface from the ramp upto the compression-corner is 0.05 m. The total length of the ramped passage is 0.12 m. The unit Reynolds number Re∞ = ρ∞µU∞
∞ is 8×105m−1. The free-stream Mach number M∞ is 6.
The free-stream stagnation temperature (T0)∞ is 1080 K. The wall is maintained isothermally at 300 K. The ramp angle is 150, which, under the given free-stream and wall conditions, is greater than the incipient-separation angle suggested by Needham ans Stollery [96]. Table 4.3 lists the geometric and free-stream parame- ters for this problem [98].
The problem statement is given again in table 4.3. Computations are carried out on a stretched grid with clustering near the solid wall and the ramp junction.
For the mesh generation, the grid-stretching function given by equation 4.9 is used in this problem also. Figure 4.18 shows a typical coarse grid with 120×80 cells along with the boundary conditions for the computation of the hypersonic SWBLI problem. The height of the computational domain is taken as 1.2 times the length of the flat-plate portion from the leading edge up to the ramp junction [102]. This is done to ensure that at the top the free-stream boundary condition can be used without causing significant error.
Table. 4.3. The geometric and free-stream parameters for the hypersonic SWBLI problem.
Parameter Value
Length of the plate upto the compression-corner (Lc) 0.05 m Total length of the ramped surface 0.12 m
Ramp angle θ 150
free-stream stagnation temperature (T0)∞ 1080 K
Unit Reynolds number Re∞ 8×105m−1
free-stream Mach number M∞ 6
Wall-temperature Tw 300 K
Fig. 4.18. A typical coarse grid for computing the hypersonic SWBLI problem.
Since the problem involves non-orthogonal grids, the method of central differ- encing for gradient-computation will not work in this case. Therefore, the gradient- terms in the viscous fluxes are computed by using the Green’s theorem. First-order Euler explicit time-integration method with a CFL number of 0.2 is used for all the computations. Figure 4.19 shows the grid-independence test for the varia- tion of skin friction coefficient along the ramped surface computed with the first- order-accurate AUSM scheme. The plot shows that grid-independent solutions are achieved for the 192×140 mesh. Hence subsequent results are shown for this grid only.
Fig. 4.19. Grid-independence test for the hypersonic SWBLI problem with the first-order AUSM scheme.
4.3.2 Effects of numerical diffusion on the skin-friction pro- file
Figure 4.20 shows the skin-friction-coefficient profiles for the problem using the DRLLF scheme2 with δ = 0.1, 0.3 and 0.5. Numerical experiments suggest that the DRLLF scheme becomes numerically unstable for δ < 0.1. It is seen that in the attached region, the computed-skin-friction coefficient is the maximum for δ = 0.1 and its value decreases with increasing δ. Since the numerical diffusion decreases with the reduction ofδ, therefore the computed-boundary-layer thickness also decreases, leading to increased velocity gradients at the solid wall. As a result the computed-wall-shear stress increases and hence the skin friction coefficient also increases.
Fig. 4.20. skin-friction profiles for the hypersonic SWBLI problem using the DRLLF scheme with different values of δ.
Figure 4.21 shows the skin-friction-coefficient profiles computed with the AUSM, DRLLF (δ = 0.1), DRLLF (δ = 0.2), DRLLF (δ = 0.5) and van Leer’s FVS schemes. The DRLLF(δ = 0.5) scheme highly underpredicts the skin-friction- coefficient profile similar to van Leer’s FVS scheme, indicating comparable levels of numerical diffusion of the two schemes. The DRLLF scheme with (δ = 0.1) provides a much crisper resolution of the skin-friction-coefficient profile compara-
2This work is published in the proceedings of the International Symposium on Aspects of Mechanical Engineering and Technology for Industry (2014), NERIST, Arunachal Pradesh, India
ble with the AUSM scheme. The performance of the DRLLF (δ = 0.2) is close to the DRLLF (δ = 0.1) scheme upstream of the separation point. However, in the post re-attachement zone the former scheme underpredicts the skin-friction profile compared with the latter. It is evident that in the presence of physical viscosity, there is ample scope for reducing the numerical diffusion of the DRLLF scheme compared with the level originally suggested for the Euler solvers by Jaisankar and Raghurama Rao [40].
Fig. 4.21. skin-friction profiles for the hypersonic SWBLI problem using the AUSM, van Leer’s FVS, DRLLF (δ = 0.1), DRLLF (δ = 0.2) and DRLLF (δ= 0.5) schemes.
4.3.3 Effects of numerical diffusion on the size of laminar separation bubble (LSB)
A study of the effect of numerical diffusion on the computed LSB length is done.
Table 4.4 lists the LSB lengths obtained by the AUSM, van Leer’s FVS and DRLLF schemes. For the DRLLF scheme, the values are reported for different values ofδ by decreasing the parameter from 0.5 to 0.1 in steps of 0.1. The length of the LSB is calculated as the difference between the extrema of thex-coordinates where the skin friction coefficient remains negative. It can be seen that the computed LSB
length increases with the increase in the numerical-diffusion level.
Table. 4.4. Comparison of LSB length for the hypersonic SWBLI problem Scheme LSB length (mm)
AUSM 5.24
van Leer’s FVS 14.42
DRLLF (δ=0.5) 15.19
DRLLF (δ=0.4) 12.18
DRLLF (δ=0.3) 9.46
DRLLF (δ=0.2) 7.26
DRLLF (δ=0.1) 5.56
4.3.4 Effects of numerical diffusion on the pressure-coefficient profile
Figure 4.22 shows the wall-pressure-coefficient profiles with the AUSM, DRLLF (δ = 0.1), DRLLF (δ = 0.2), DRLLF (δ = 0.5) and van Leer’s FVS schemes.
It is seen that with increased level of numerical diffusion, the separation zone (encircled) in the pressure-coefficient profile increases in size.
Fig. 4.22. pressure-coefficient profiles for the hypersonic SWBLI problem using the AUSM, DRLLF (δ= 0.1), DRLLF (δ = 0.2), DRLLF (δ = 0.5) and van Leer’s FVS schemes.
4.3.5 Effects of numerical diffusion on convergence
Figure 4.23 shows the residual-history plots for computing the SWBLI problem with the AUSM, DRLLF (δ = 0.1) and van Leer’s FVS schemes. It can be noted that the residual-history plots for the DRLLF scheme with δ = 0.2, 0.3, 0.4 and 0.5 are almost the same and therefore the residual-history plots for the DRLLF scheme with these values ofδ are not shown for clarity of the figure. The schemes have markedly similar convergence behaviour.
Fig. 4.23. Residual-history plots for the hypersonic SWBLI problem with the AUSM, DRLLF (δ= 0.1) and van Leer’s FVS schemes.