CONCLUSIONS AND FUTURE SCOPE
8.1 Conclusions
The accuracy of the solution obtained using FV approaches is determined by the dis- cretisation of the convective and diffusive fluxes, which themselves are directly affected by the accuracy of the solution gradients. We have proposed a novel cell-centered gradient reconstruction method referred to as Modified Green Gauss reconstruction (MGG) on arbitrary polygonal meshes. This method employs a variant of the Gauss divergence theorem that is devoid of any interpolations on purely orthogonal meshes (like uniform or non-uniform non-orthogonal Cartesian grids). However, the methodol- ogy requires interpolation and an iterative approach on non-orthogonal meshes, which
could be computationally expensive for time marching problems. In order to cut down the cost and its applicability to adaptive Cartesian meshes typical of IB-FV frame- work, we have also employed a non-iterative variant that is accurate and robust. The numerical investigations show that the MGG reconstruction is less dissipative than the Standard Green Gauss (SGG) reconstruction and allows for the use of a larger Courant number than the SGG reconstruction. MGG reconstruction, therefore, may be viewed as a strategy that may devoid of iterations and results in a fast, robust and accurate approach for compressible flows.
The focal point of study in this thesis is the development of an in-house Immersed Boundary Finite Volume (IB-FV) framework for compressible flows. We have devised a sharp interface immersed boundary approach that employs a non-conservative re- construction in the vicinity of the body. Two techniques, one employing an employing Inverse Distance Weighting (IDW) based reconstruction for inviscid flows and the other employing one-dimensional reconstruction for viscous flows have been proposed in this work. The IB-FV framework has been employed to solve a large range of flow problems involving hypersonic inviscid and laminar flows, including those with moving bodies. The salient findings from the numerical investigations are summarised below.
1. The IDW reconstruction approach accounts for both Dirichlet and Neumann boundary conditions and directly enforces the boundary conditions on the surface of the body, thereby preserving the sharp interface.
2. Numerical studies show that the mass conservation errors remain non-zero on finite mesh resolutions. However, these errors tend to decrease with grid refine- ment, decaying at a rate close to unity.
3. The use IDW reconstruction approach preserves the nominal second-order accu- racy of the finite volume flow solver, despite a lack of conservation in the near vicinity of the body. This clearly indicates that the reconstruction approach does not degrade the accuracy of the solution in the computational domain.
4. Despite the use of a non-conservative approach, the wall pressure and skin- friction distribution were computed with reasonable accuracy. While the pres- sure distributions could be computed accurately on relatively coarser meshes, accurate estimates of skin-friction necessitated local grid refinement near the vicinity of the body.
5. The computations of wall heat fluxes and stagnation point heat transfer are dependent on the geometry and flow conditions. While accurate results were
obtained for flow past a compression ramp, studies on aerodynamically blunt geometries yielded under-predicted heat fluxes. The problem was more severe on higher Mach and Reynolds numbers and also persist in adiabatic flows where the skin temperatures were incorrectly estimated.
6. An in-depth investigation into the underprediction of heat flux/skin tempera- ture indicated that the errors are largely due to the non-conservative nature of the one-dimensional reconstruction approach employed. The inability of the IB approach to discretely conserve the energy in the near wall cells manifests as an under prediction in stagnation point heat flux for such cases. The problem was found to increase as Reynolds number increases for a given Mach number and as Mach number increases for a given Reynolds number, indicating a failure of the methodology to accurately predict the heat loads in hypersonic laminar regimes.
The use of non-polynomial and non-linear interpolation as opposed to linear in- terpolation previously employed, was found to be equally non-conservative and inaccurate for heat-flux prediction.
The finite volume/immersed boundary framework has also employed in conjunc- tion with variable fidelity framework for design and optimisation problems that in- cluded aerodynamic shape optimisation of axisymmetric nose cone, design of planar scramjet intakes and optimal axisymmetric nozzle configurations. The conclusions from these findings are enumerated below.
1. The use of a multi-fidelity framework comprising of a computationally inexpen- sive low-fidelity framework and computationally accurate high-fidelity framework leads to a significant reduction in turn-around time from initial guess to final optimal solution.
2. The maximum ballistic coefficient bodies are found to lead to lower stagnation point heat transfer while also having a low drag value than minimum drag ge- ometries.
3. The quasi one-dimensional low-fidelity frameworks based on gas dynamic princi- ples are found to be quite useful for the design of nozzle and scramjet intake. In case of scramjet intakes, this approach leads to a design with high total pressure recovery (TPR) while leading to low non-uniformity in the isolator. Similarly, these approaches have also been used successfully to design optimal contoured nozzles that minimise the radial velocity at the exit.
4. The IB-FV solver have been employed to verify the results from the quasi one- dimensional design approach. Studies show that the two-dimensional computa- tions agree well the one-dimensional estimates.
The numerical investigations in this thesis have attempted to address some of the important issues concerning IB-FV solvers such as discrete conservation and its accuracy for skin-friction and heat-transfer estimates in hypersonic flows. The IB- FV approach is also a promising alternative that can be used in conjunction with high-fidelity frameworks for design and optimisation. It must however be noted that while the framework is quite accurate for inviscid compressible flows, there are issues in employing this solver in viscous compressible regime, that needs to be looked into more deeply. The present work should be seen as the first step in a thorough investigation towards the development of a robust and accurate industry level flow solver and some of the directions for future research are discussed below.