LEE- Uçak ve Uzay Mühendisliği Lisansüstü Programı
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ÖgeA high-order finite-volume solver for supersonic flows(Lisansüstü Eğitim Enstitüsü, 2022) Spinelli, Gregoria Gerardo ; Çelik, Bayram ; 721738 ; Uçak ve Uzay MühendisliğiNowadays, Computational Fluid Dynamics (CFD) is a powerful tool in engineering used in various industries such as automotive, aerospace and nuclear power. More than ever the growing computational power of modern computer systems allows for realistic modelization of physics. Most of the open-source codes, however, offer a second-order approximation of the physical model in both space and time. The goal of this thesis is to extend this order of approximation to what is defined as high-order discretization in both space and time by developing a two-dimensional finite-volume solver. This is especially challenging when modeling supersonic flows, which shall be addressed in this study. To tackle this task, we employed the numerical methods described in the following. Curvilinear meshes are utilized since an accurate representation of the domain and its boundaries, i.e. the object under investigation, are required. High-order approximation in space is guaranteed by a Central Essentially Non-Oscillatory (CENO) scheme, which combines a piece-wise linear reconstruction and a k-exact reconstruction in region with and without discontinuities, respectively. The usage of multi-step methods such as Runge-Kutta methods allow for a high-order approximation in time. The algorithm to evaluate convective fluxes is based on the family of Advection Upstream Splitting (AUSM) schemes, which use an upwind reconstruction. A central stencil is used to evaluate viscous fluxes instead. When using high-order schemes, discontinuities induce numerical problems, such as oscillations in the solution. To avoid the oscillations, the CENO scheme reverts to a piece-wise linear reconstruction in regions with discontinuities. However, this introduces a loss of accuracy. The CENO algorithm is capable of confining this loss of accuracy to the cells closest to the discontinuity. In order to reduce this accuracy loss Adaptive Mesh Refinement (AMR) is used. This algorithm refines the mesh near the discontinuity, confining the loss of accuracy to a smaller portion of the domain. In this study, a combination of the CENO scheme and the AUSM schemes is used to model several problems in different compressibility regimes, with a focus on supersonic flows. The scope of this thesis is to analyze the capabilities and the limitations of the proposed combination. In comparison to traditional implementations, which can be found in literature, our implementation does not impose a limit on the refinement ratio of neighboring cells while utilizing AMR. Due to the high computational expenses of a high-order scheme in conjunction with AMR, our solver benefits from a shared memory parallelization. Another advantage over traditional implementations is that our solver requires one layer of ghost cells less for the transfer of information between adjacent blocks. The validation of the solver is performed in different steps. We assess the order of accuracy of the CENO scheme by interpolating a smooth function, in this case the spherical cosine function. Then we validate the algorithm to compute the inviscid fluxes by modeling a Sod shock tube. Finally, the Boundary Conditions (BCs) for the inviscid solver and its order of accuracy are validated by modeling a convected vortex in a supersonic uniform flow. The curvilinear mesh is validated by modeling the flow around a NACA0012 airfoil. The computation of the viscous fluxes is validated by modeling a viscous boundary layer developing on a flat plate. The BCs for viscous flows and the curvilinear implementation are validated by modeling the flow around a cylinder and a NACA0012 airfoil. The AUSM schemes are tested for shock robustness by modeling an inviscid hypersonic cylinder at a Mach number of 20 and a viscous hypersonic cylinder at a Mach number of 8.03. Then, we validate our AMR implementation by modeling a two-dimensional Riemann problem. All the validation results agree well with either numerical or experimental results available in literature. The performance of the code, in terms of computational time required by the different orders of approximation and the parallel efficiency, is assessed. For the former a supersonic vortex convection served as an example, while the latter used a two-dimensional Riemann problem. We obtained a linear speed-up until 12 cores. The highest speedup value obtained is 20 with 32 cores. Furthermore, the solver is used to model three different supersonic applications: the interaction between a vortex and a normal shock, the double Mach reflection and the diffraction of a shock on a wedge. The first application resembles a strong interaction between a vortex and a steady shock wave for two different vortex strengths. In both cases our results perfectly match the ones obtained by a Weighted Essentially Non-Oscillatory (WENO) scheme documented in literature. Both schemes are approximating the solution with the same order of accuracy in both, time and space. The second application, the double Mach reflection, is a challenging problem for high-order solvers because the shock and its reflections interact strongly. For this application, all AUSM-schemes under investigation fail to obtain a stable result. The main form of instability encountered is the Carbuncle phenomenon. Our implementation overcomes this problem by combining the AUSM+M scheme with the formulation of the speed of sound of the AUSM+up scheme. This combination is capable of modeling this problem without instabilities. Our results are in agreement with those obtained with a WENO scheme. Both, the reference solutions and our results, use the same order of accuracy in both, time and space. Finally, the third example is the diffraction of a shock past a delta wedge. In this configuration the shock is diffracted and forms three different main structures: two triple points, a vortex at the trailing edge of the wedge and a reflected shock traveling upwards. Our results agree well with both, numerical and experimental results available in literature. Here, a formation of a vortex-let is observed along the vortex slip-line. This vorticity generation under inviscid flow condition is studied and we conclude that the stretching of vorticity due to compressibility is the reason. The same formation is observed when the angle of attack of the wedge is increased in the range of 0-30. In general, the AUSM+up2 scheme performed best in terms of accuracy for all problems tested here. However, for configurations, in which the Carbuncle phenomenon may appear, the combination of the AUSM+M scheme and the computation of the speed of sound formula of the AUSM+up scheme is preferable for stability reasons. During our computations, we observe a small undershooting right behind shocks on curved boundaries. This is imputable to the curvilinear approximation of the boundaries, which is only second-order accurate. Our experience shows that the smoothness indicator formula in its original version, fails to label uniform flow regions as smooth. We solve the issue by introducing a threshold for the numerator of the formula. When the numerator is lower than the threshold, the cell is labeled as smooth. A value higher than 10^-7 for the threshold might force the solver to apply high-order reconstruction across shocks, and therefore will not apply the piece-wise linear reconstruction which prevents oscillations. We observe that the CENO scheme might cause unphysical states in both inviscid and viscous regime. By reconstructing the conservative variables instead of the primitive ones, we are able to prevent unphysical states for inviscid flows. For the viscous flows, temporarily reverting to first-order reconstruction in the cells where the temperature is computed as negative, prevents unphysical states. This technique is solely required during the first iterations of the solver, when the flow is started impulsively. In this study the CENO, the AUSM and the AMR methods are combined and applied successfully to supersonic problems. When modeling supersonic flow with high-order accuracy in space, one should prefer the combination of the AUSM schemes and the CENO scheme. While the CENO scheme is simpler than the WENO scheme used in comparison, we show that it yields results of comparable accuracy. Although it was beyond the scope of this study, the AUSM can be extended to real gas modeling which constitutes another advantage of this approach.