Laminar supersonic flow with I-SWBLI 61
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Billig Correlation Present numerical
FIGURE3.12: Comparison of shock shape for freestream Mach number 8
Laminar supersonic flow with I-SWBLI 62 are scaled with flat plate length upstream of interaction location, ie. 49.6 mm. In the domain, the first1/3rdof the bottom wall is set as inviscid wall whereas rest2/3rdis set as viscous wall to represent the flat plate. The various boundary conditions of the computational domain are marked in figure 3.14. The top inclined wall of the computational domain represents the inviscid surface of the wedge (shock generator). The inclination of the top wall has been selected to match the flow deflection angle of a wedge that could generate a shock at an angle of32.580. The domain is meshed with quadrilateral grids. Three meshes, viz. 240×80,360×120and480×160 are used to obtain grid independent solution. Grid clustering is carried out near the bottom wall for capturing the boundary layer and other viscous wall parameters for all meshes. Grid points are again clustered along the span wise direction locally at leading edge region and interaction location for better prediction of flow features. Explicit time stepping has been chosen and CFL of 0.2 is selected for this case. Time marching is performed until the density residue falls to10−6, which is assumed as the steady state for this case. In addition to simple explicit scheme, implicit formulation has also been considered for simulation to compare its performance in viscous flow simulation. For implicit simulation under relaxation factor of 0.6 along with starting CFL of 0.2 and linear CFL ramping is used. The number of sweeps are kept as 16 for this viscous simulation. The residual falls of these two time marching schemes are therefore compared in figure 3.15. This comparison clearly portrays considerably large computational cost for explicit scheme based computations as compared to implicit formulation. Explicit formulation has taken more than 400000 iterations to reach steady state, whereas implicit formulation takes only 3350 iterations for steady state solution. For a grid level of360×120, the implicit formulation took 1.45 sec per iteration, which results in total simulation time of 4865 sec (around 1 hr 21 min).
On the other hand, explicit formulation took only 0.3534 sec per iteration. However with the rate of 0.3534 sec/iteration, the total simulation time extended to 39 hrs. This is the clear evidence of the computational efficiency of implicit scheme in steady flow simulations.
The pressure distributions obtained with three different grid levels are compared along with experimental measurements [19] in figure 3.16(a). The numerical predictions are in good agree- ment with experimental measurements. The present simulations could predict almost all the expected flow features of this particular case. Although the distributions of different grid levels
Laminar supersonic flow with I-SWBLI 63
FIGURE3.13: Schematic of physical situation of shock impingement caused SWBLI
FIGURE3.14: Meshed computational domain for shock impingement SWBLI study
have slight deviations, the difference between the distributions of320×160 and480×160 is very minor . Therefore results on 320×160grid level is used for the further discussions. The Mach contours of the present study obtained with this grid is therefore shown in figure 3.18.
The Mach contours of implicit simulation were observed to be the same. Hence, for the sake of brevity, those contours are not merged with Mach contours of explicit scheme in this figure 3.18. Although the Mach contours show the interaction of oblique shock emerging from the compression corner with the lip shock before hitting the viscous wall, it is observed that this interaction does not alter the strength and angle of incident shock considerably. The comparison of predicted skin friction coefficient (Cf) distribution with reported experimental [19] as well as numerical results [41, 84], given in figure 3.17(a), confirm this argument. The skin friction
Laminar supersonic flow with I-SWBLI 64
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FIGURE3.15: Comparison of residual fall of time marching schemes on360×120grid level
FIGURE3.16: Comparison of surface pressure distributions of viscous flow test case
Laminar supersonic flow with I-SWBLI 65
FIGURE3.17: Comparison of skin friction distributions of viscous flow test case
FIGURE3.18: Mach contours of incident shock-boundary layer interaction study
Summary 66 distributions of both explicit and implicit simulations can be observed to be the same in figure 3.17(a). It shows that, the accuracy is same for both the formulations. Moreover, it is clear from the Mach contours and skin friction coefficient distribution that, the boundary layer thickens and separates due to the impingement of shock wave on the boundary layer. This flow separation leads to the formation of separation shock emanating from the separation point. The separated recirculating flow region extends from separation point to reattachment point. At the reattach- ment point a re-compression shock can be seen. The separation bubble size is thus the distance between separation point and reattachment point. Skin friction distribution is being utilized for this quantification. All numerical distributions are slightly over predicting the separation bubble size as compared to corresponding experimental value. This mismatch may be attributed to slight variation in assumed shock impingement location in the experimental study or the uncertainties in the skin friction measurements. The pressure distribution (figure 3.16(a)) have revealed that, in the region of separation, pressure rises and attains a plateau before the further increment asso- ciated with reattachment. On the other handCf is observed to be reducing and attaining negative values, which attribute to the flow separation and recirculation. Although numerically obtained Cf distributions has mismatch with experimental measurements in the separation region, the pressure distribution predictions of both experimental and numerical are closely matching each other. Since the present simulation results are observed to be falling very close to earlier nu- merical and experimental measurements, the solver accuracy in viscous flow simulations is also very clear.