The simulation of hypersonic flows is computationally demanding due to the large gradients of the flow variables at hand, caused both by strong shock waves and thick boundary or shear layers. The resolution of those gradients imposes the use of extremely small cells in the respective regions. Taking turbulence into account intensifies the variation in scales even more. Furthermore, hypersonic flows have been shown to be extremely grid sensitive. For the simulation of fully three-dimensional configurations of engineering applications, this results in a huge amount of cells and, as a consequence, prohibitive computational time. Therefore, modern adaptive techniques can provide a gain with respect to both computational costs and accuracy, allowing the generation of locally highly resolved flow regions where they are needed and retaining an otherwise smooth distribution. In this paper, an h-adaptive technique based on wavelets is employed for the solution of hypersonic flows. The compressible Reynolds-averaged Navier-Stokes equations are solved using a differential Reynolds stress turbulence model, well suited to predict shock-wave/ boundary-layer interactions in high-enthalpy flows. Two test cases are considered: a compression corner at 15 deg and a scramjet intake. The compression corner is a classical test case in hypersonic flow investigations because it poses a shock-wave/turbulent-boundary-layer interaction problem. The adaptive procedure is applied to a twodimensional configuration as validation. The scramjet intake is first computed in two dimensions. Subsequently, a three-dimensional geometry is considered. Both test cases are validated with experimental data and compared to nonadaptive computations. The results show that the use of an adaptive technique for hypersonic turbulent flows at high-enthalpy conditions can strongly improve the performance in terms of memory and CPU time while at the same time maintaining the required accuracy of the results. Nomenclature b ij= anisotropy tensor c p = specific heat at constant pressure, pressure coefficient D ij = diffusion tensor for the Reynolds stresses, m 2 ∕s 3 δ ij = Kronecker delta d = spatial dimension E = specific total energy, m 2 ∕s 2 H = total specific enthalpy, m 2 ∕s 2 II = second invariant of the anisotropy tensor k = turbulent kinetic energy, m 2 ∕s 2 L = maximum refinement level l = local refinement level M = Mach number M ij = turbulent mass flux tensor for the Reynolds stresses, m 2 ∕s 3 μ = molecular viscosity, kg∕m · s P ij = production tensor for the Reynolds stresses, m 2 ∕s 3 p = pressure, Pa p t = total pressure, Pa q i = component of heat flux vector, W∕m 2 q t k = turbulent heat flux, W∕m 2 R ij = Reynolds stress tensor, m 2 ∕s 2 Re = Reynolds number, 1∕m Res drop = averaged density residual at which the adaptations are performed ρ = density, kg∕m 3 S ij = strain-rate tensor, 1∕s St = Stanton number T = temperature, K T w = wall temperature, K T 0 = total temperature, K t = time, s U = local velocity, m∕s u i = velocity component, m∕s W ij...
In contrast to external flow aerodynamics, where one-dimensional Riemann boundary conditions can be applied far up- and downstream, the handling of non-reflecting boundary conditions for turbomachinery applications poses a greater challenge due to small axial gaps normally encountered. For boundaries exposed to non-uniform flow in the vicinity of blade rows, the quality of the simulation is greatly influenced by the underlying non-reflecting boundary condition and its implementation. This paper deals with the adaptation of Giles’ well-known exact non-local boundary conditions for two-dimensional steady flows to a cell-centered solver specifically developed for turbomachinery applications. It is shown that directly applying the theory originally formulated for a cell-vertex scheme to a cell-centered solver may yield an ill-posed problem due to the necessity of having to reconstruct boundary face values before actually applying the exact non-reflecting theory. In order to ensure well-posedness, Giles’ original approach is adapted for cell-centered schemes with a physically motivated reconstruction of the boundary face values, while still maintaining the non-reflecting boundary conditions. The extension is formulated within the original framework of determining the circumferential distribution of one-dimensional characteristics on the boundary. It is shown that, due to approximations in the one-dimensional characteristic reconstruction of boundary face values, the new approach can only be exact in the limiting case of cells with a vanishing width in the direction normal to the boundary if a one-dimensional characteristic reconstruction of boundary face values is used. To overcome the dependency on the width of the last cell, the new boundary condition is expressed explicitly in terms of a two-dimensional modal decomposition of the flow field. In this formulation, vanishing modal amplitudes for all incoming two-dimensional modes can easily be accomplished for a converged solution. Hence we are able to ensure perfectly non-reflecting boundary conditions under the same conditions as the original approach. The improvements of the new method are demonstrated for both a subsonic turbine and a transonic compressor test case.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.