REVISITING THE SHARP INTERFACE IMMERSED BOUNDARY FOR VISCOUS FLOWS
6.5 Dependence on freestream and wall conditions
and Reynolds numbers have on the differences between the computed estimates using the two approaches for the same grid resolution.
θo
Temperature(K)
0 15 30 45 60 75 90
300 305 310
315 FV (M∝= 2)
Conformal IB (M∝= 2)
θo
Temperature(K)
0 15 30 45 60 75 90
300 310 320 330 340 350
FV (M∝= 3.5) Conformal IB (M∝= 3.5)
θo
Temperature(K)
0 15 30 45 60 75 90
300 325 350 375 400 425
FV (M∝= 5) Conformal IB (M∝= 5)
Figure 6.11: Comparison of near wall temperature distribution for freestream Mach number (a) 2.0 (b) 3.5 (c) 5.0
We first study the influence of freestream Mach number for a fixed Reynolds number Re∞ = 500 by considering three different Mach numbers. The surface skin friction distribution are shown forM∞= 2, 3.5 and 5 respectively in Figure6.10where one can see that the estimates from FV and conformal IB show a qualitative agreement and the deviation is relatively more prominent at the lowest and highest values of Mach number. While the results from the two approaches for the wall shear are not expected to agree, the deviation in the results are not too significant and the results from the conformal IB approach can be improved through adaptive mesh refinement, which is however not shown herein. The near-wall temperature distribution for the three Mach numbers are shown in Figure6.11and one can clearly discern the differences. Noticing that the temperature scales differ in these plots, one can see that the disagreement between the computed estimates from the approaches tend to increase as the Mach
number increases, which would consequently appear as deviation in wall heat fluxes.
Therefore, one can conclude that the heat flux estimates of the conformal IB solver tend to be further off from those predicted using the conservative FV approach as the freestream Mach number increases.
θo Skin-frictioncoefficient(Cf)
0 15 30 45 60 75 90
0 0.02 0.04 0.06 0.08
FV (Re∝= 500) Conformal IB (Re∝= 500)
θo Skin-frictioncoefficient(Cf)
0 15 30 45 60 75 90
0 0.003 0.006 0.009 0.012 0.015
FV (Re∝= 5000) Conformal IB (Re∝= 5000)
Figure 6.12: Comparison of skin-frictionCf along the cylinder for freestream Reynolds number (a) 500 (b) 5000
θo
Temperature(K)
0 15 30 45 60 75 90
300 310 320 330 340 350
FV (Re∝= 500) Conformal IB (Re∝= 500)
θo
Temperature(K)
0 15 30 45 60 75 90
300 350 400 450
FV (Re∝= 5000) Conformal IB (Re∝= 5000)
Figure 6.13: Comparison of near wall temperature distribution for freestream Reynolds number (a) 500 (b) 5000
θo
Pressure(Pa)
0 15 30 45 60 75 90
0 10 20 30 40 50 60
FV
Conformal IB, Quadratic Conformal IB, Non-polynomial
θo Skin-frictioncoefficient(Cf)
0 15 30 45 60 75 90
0 0.025 0.05 0.075 0.1 0.125 0.15 0.175
FV
Conformal IB, Quadratic Conformal IB, Non-polynomial
Figure 6.14: Comparison of (a) pressure (b) skin-friction coefficient Cf, along the cylinder
The second study we carry out is to understand the role of freestream Reynolds number for high-speed flows for which we choose two Reynolds numbers viz. Re∞
= 500 and 5000 respectively. We keep the freestream Mach number constant at 3.5 for these computations and the skin friction and near-wall temperature distributions for the two cases are shown in Figures 6.12 and 6.13. While one can notice that the discrepancy between the FV and conformal IB solutions for skin friction distribution is not considerable at either Reynolds numbers, the near-wall temperatures are dif- ferent between the two methods with the differences being significantly higher for the case of higher Reynolds number (see Figure 6.13). The near-wall temperatures for the isothermal cases are directly linked to the wall heat flux estimates and the lesser estimates of these temperatures atRe∞= 5000 would translate into under-prediction of wall heat fluxes as well when compared to the estimates from the FV approach.
θo
Temperature(K)
0 15 30 45 60 75 90
800 850 900 950 1000 1050
FV
Conformal IB, Quadratic Conformal IB, Non-polynomial
Figure 6.15: Distribution of skin temperature
To assess the influence of wall conditions, we consider the same test case of lam- inar hypersonic flow past the cylinder except that the wall is considered adiabatic instead of isothermal. The flow conditions correspond to M∞ = 3.5, Re∞ = 500 and T∞ = 300 K, for a cylinder with radius of 0.1 m. A naive argument would be that since the wall heat flux is zero by definition, the IB approaches would work well for the case of adiabatic surfaces. It may be noted that the temperature variation is assumed to be either quadratic (see Section 5.9) or one could employ the Walz correlation (see Section6.4). One can see from Figures6.14(a) that the wall pressure distribution from the conformal IB approach using either temperature reconstructions agrees excellently with those computed using the FV solver. The agreement of the skin friction distri- bution is fair (Figure 6.14(b)), although one would require a greater near-wall grid resolution when the conformal IB is employed for the estimates to agree excellently with those obtained from the FV approach. Although the wall heat flux is zero, the estimate of skin temperatures (or the wall temperatures) clearly highlight the inac- curacies inherent in the solution reconstruction as can be observed from Figure 6.15.
The largest difference between the FV and conformal IB estimates is around 110 K when the quadratic interpolant is employed while the non-polynomial interpolation based on Walz correlation shows a better agreement with a maximum difference of 30 K. Moreover, while the non-polynomial interpolation over-predicts the skin tempera- tures compared with the FV benchmark, the use of quadratic interpolation leads to an under-prediction. It must be emphasised that the adiabatic test case considered here is however at a relatively low Reynolds number and investigations at higher Reynolds numbers are necessary to make substantial conclusions related to the performance of IB approaches for hypersonic viscous flows with adiabatic walls.
These studies show that the non-conservative approach inherent to the sharp- interface IB approach could lead to smaller heat flux estimates (for isothermal sur- faces) than those from the conservative FV approach with the differences being more prominent at higher Mach and Reynolds numbers. The solution reconstruction is also found to influence the skin temperatures in case of adiabatic walls. These studies may also be construed as evidence in support of the under-prediction of stagnation point heat transfer observed for the studies in Sections5.2.8 and5.2.9 wherein the flows are at high Reynolds numbers (of the order of 106) and hypersonic Mach numbers (greater than 5).