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.
Summary 67 speed flows over a hemisphere. The close agreement of numerical predictions with literature reported analytical correlations for surface pressure distribution, shock stand-off distance and shock shape confirmed the accuracy of Euler solver. Finally the complete Navier-Stokes solver is validated through the simulation for I-SWBLI in supersonic flow. The solver results are in good agreement with experimental as well as earlier reported numerical results for this test case. Hence this study evidences the applicability of present solver in viscous flow simulations.
Additionally, the relative advantage of explicit and implicit time marching strategies of present solver is presented through the simulation of both inviscid and viscous flow test cases. It has been noticed that, although the implicit scheme require higher time per iteration, the use of this scheme ensures significant saving in total simulation time.
Chapter 4
Performance comparison of flux schemes for numerical simulation of high-speed inviscid flows
Overview
Numerical investigations are carried out to assess the performance of different inviscid flux computations schemes for the spatial discretisation of the Euler equations. Since the present solver is equipped with flux vector splitting, flux difference splitting and hybrid schemes, it is essential to identify the applicability of a particular scheme for a particular situation. In view of this, the solver is exposed to solve the problems ranging from lower supersonic to moderate hypersonic speeds. These problems involve various flow patterns like shock reflection, shock- shock interaction and interaction of shock with expansion fan and slip line. As a result of these studies, AUSM family of schemes are found to be the most accurate while the Rusanov scheme is observed to be the most robust for the range of Mach numbers considered in the study.
4.1 Background
Design and development of aircrafts for civil or military applications and space capsules for space missions require detailed knowledge of flow features of the flow regimes encountered by these vehicles. The relevant flow features provide the necessary inputs needed for efficient
Test case 1:supersonic flow through a ramp in a channel 70
aerodynamic design of these flight vehicles. In the high speed compressible flow regime, pre- cise prediction of the flow features and its effect on aerodynamic coefficients has been one of the prominent issues faced by aerodynamicists. Preliminary information about this design data can be achieved using the Euler solver for which considerable advancement in development of schemes for computation of the inviscid fluxes has been done. Most of those well established schemes are part of the present solver. Although such convective flux schemes were developed for compressible flows with subsonic, transonic speeds or lower supersonic speeds, some of these schemes have been sparingly used in problems involving higher supersonic flows or even in hypersonic flow regime. There has been very limited concerted effort to analyse the suit- ability and applicability of these schemes for such high-speed flows. Supersonic and hypersonic flows often involve several complex flow phenomena such as strong shocks and expansions, reg- ular or Mach reflections, shock-shock interactions and shock-expansion interactions. A recent study by Druguet et al. [85] demonstrated the effects of numerics on computations in hypersonic flows. Hence the focus of the present chapter is to evaluate the performance of several numerical flux computation schemes in predicting the complex physics associated with high Mach num- ber inviscid flows. Therefore the effect of numerical fluxes obtained using Van-Leer scheme (FVS)[86], Roe scheme [87], Rusanov scheme [88], HLLE scheme [89] (FDS), AUSM [90] and AUSM+ [91] (Hybrid) schemes on the solution will be analyzed for various flow problems with Mach numbers varying from 1 to 8. These studies would be beneficial to decide about the best suited scheme for SWBLI studies. A comparative study of two different variants of the AUSM family viz. AUSM+ [91] and a latter variant AUSMPW [92] are also performed on two selected test cases.
The suitability of different upwind schemes to capture the discontinuities present in the flow field is investigated using the Mach 2 flow past a ramp in a channel (Darren and Powell [93]). In the presence of the supersonic flow and the ramp, an attached shock is formed at the compression corner and reflects from the top wall, forming a small Mach stem. The reflected shock then further reflects from the bottom wall before exiting the channel. A weak slip line emanating from the triple point near the upper wall is also an integral part of the flowfield. The dimensions and boundary conditions for this test case are shown in figure 4.1. A180×90structured mesh is considered for the study.
This test case is simulated with all the seven schemes and steady state results are obtained for all simulations. Figure 4.2 and 4.3 show the pressure and Mach number contours while the surface pressure distribution is shown in figure 4.4, obtained using the Roe scheme. The surface pressure distribution is in good agreement with the results of Ganesh et al. [78] where unstructured adaptive meshing strategy was employed. Contours of flow quantities obtained using other schemes are similar and are not shown here for brevity. It was observed that while Roe and AUSM family of schemes indicate a crisp shock, the Rusanov scheme resulted in a more dissipative shock structure.
A comparison of the seven numerical schemes has been carried out for the pressure rise across the Mach stem. Figure 4.5 shows the pressure rise across the Mach stem computed with five dif- ferent schemes. The AUSM family of schemes predict similar flow structure. Predictions from
Test case 1:supersonic flow through a ramp in a channel 72
FIGURE4.2: Non-dimensional pressure contours of supersonic flow past ramp in a channel