REVISITING THE SHARP INTERFACE IMMERSED BOUNDARY FOR VISCOUS FLOWS
6.6 Discussions and remedial approaches
Stanton number distribution in the flow past a compression ramp, the estimates of wall heat fluxes for the flow past blunt geometries have been quite poor despite the use of locally refined grids. From a conventional viewpoint, it may be argued that Cartesian IB-FV approaches would not perform well for viscous flows owing to their inability to resolve the boundary layers accurately unlike body-fitted meshes in the FV approach. We must stress that our studies herein have shown that this is not entirely true and that solution reconstruction plays a more dominant role in determining the gradient estimates in the near-wall regions. It must also be asserted that adaptive grid refinement to acceptable number of levels does not necessarily compensate for the non-conservation errors arising from temperature reconstruction. Even in case of geometries with adiabatic walls where surface heat transfer is zero, the reconstruction of temperature can possibly cause inaccuracies in estimation of skin temperatures with the inaccuracies likely compounded when the freestream Reynolds and Mach numbers are higher. It must be reiterated that our findings are far from trivial and may be encapsulated, albeit based on the limited number of numerical experiments, in the following conjecture.
Conjecture: “It is not always possible to accurately estimate wall heat fluxes or wall temperatures (for isothermal and adiabatic walls respectively) at acceptably fi- nite near-wall grid resolutions using the present sharp-interface immersed boundary approach.”
The inability to have accurate estimates of wall heat flux while being able to make quite accurate numerical predictions of wall shear stress may appear to be a violation of Reynolds analogy. However, it must be recognised that the Reynolds analogy applies to zero pressure-gradient boundary layers and no generalisations for flows past arbi- trary geometries are available. We are also of the conviction that this conjecture would apply to all immersed boundary approaches that belong to the sharp-interface cate- gory such as those in [66,72] as well as several meshfree approaches. This is because most meshfree approaches and current state-of-art sharp-interface IB approaches do not satisfy discrete conservation principles in the near-wall regions and though there have been no studies that discuss the performance of these approaches for laminar hypersonic flows, we speculate that they would perform no better than the HCIB ap- proach discussed in this study.
The inaccurate estimation of wall heat transfer in highReflows past blunt geome- tries using the immersed boundary approach may be viewed as a significant limitation from an engineering viewpoint. This is because the stagnation heat transfer and heat
loading are critical to design of thermal protection systems of hypersonic vehicles. It is therefore imperative to discuss alternatives that does not compromise on the ease and simplicity offered by the immersed boundary approach while ensuring accurate estimations of skin friction and heat transfer in all scenarios. While mitigating the problem of under-prediction is neither a trivial task nor in the scope of this study, we do outline a few possible remedial strategies.
1. Quasi-conservative immersed boundary approach: The idea behind this strategy is to employ the conservation laws even in the near-wall regions, but ensuring that the boundary conditions are suitably accounted for. This can be realised by adopting a diffuse interface approach such as the one in [41] or the Brinkman pe- nalisation technique in [47]. In fact, [47] has employed this methodology wherein
“unified” conservation laws are solved everywhere in the domain for compress- ible flows although its utility for hypersonic viscous flows has not yet been ex- plored. We call this approach as “quasi-conservative” because the conservation equations are constructed by combining the Navier-Stokes equations with the boundary conditions (based on some scalar parameter such as permeability in [47]) thereby leading to a diffusion of the true interface.
2. Overset grid/immersed boundary approach: In this approach, a curvilinear mesh may be generated in the vicinity of the geometry that resolves the boundary layer quite accurately. The body with this “viscous padding” may then be immersed into a background Cartesian mesh with the curvilinear mesh overlapping with the Cartesian mesh. The solution to the near-wall region can then be obtained akin to a body-fitted mesh and the solution reconstruction then “shifts” from the near-wall region to away from it at the edge of the structured mesh. However, this necessitates solution interpolation between the overlapping meshes and a very similar approach has only been studied for limited number of very low- speed viscous flows recently in [142].
3. Virtual sub-grid approach: This methodology uses a virtual discretisation in the normal direction to the body along which a reduced form of the energy equation (say, steady without convection terms) may be solved using a finite volume approach so as to approximately ensure conservation. This is inspired by the work in [143] for low-speed turbulent flows where an efficient wall model is developed with the inner layer virtually resolved and a simplified form of the thin boundary layer equations solved.
The quasi-conservative approach sacrifices the sharpness of the interface while the overset grid/immersed boundary technique does not lend itself to complete automa- tion, requiring an “immersed grid” (body with a conformal structured viscous layer).
The third approach is therefore the only methodology that is in the true spirit of the sharp-interface immersed boundary approaches. While these three strategies broadly fall within the Cartesian immersed boundary framework, they remain obviously unex- plored for hypersonic flows. We do believe that these approaches would be remedial because they are all bound by the same principle - of enforcing conservation, although the specific means and the extent are quite different. Furthermore, it is also important to investigate the existing class of sharp-interface IB approaches (and these possible remedial strategies) on a wide range of canonical flows past generic configurations such as the HB2 geometry [144] and the double cone configuration [145] among others, to gain a greater insight into the strengths and limitations of different variants for com- plex hypersonic flow problems.