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The formulation of the RANS equations for supersonic and hypersonic turbulent flows

Published online by Cambridge University Press:  12 October 2020

H. Zhang Open the ORCID record for H. Zhang [Opens in a new window] ,
T.J. Craft Open the ORCID record for T.J. Craft [Opens in a new window]  and
H. Iacovides Open the ORCID record for H. Iacovides [Opens in a new window]
H. Zhang*
Affiliation:
State Key Laboratory of Aerodynamics, Mianyang, China Computational Aerodynamics Institute, China Aerodynamic Research and Development Center, Mianyang, China Thermo-Fluids Group, University of Manchester, Manchester, UK
T.J. Craft
Affiliation:
Thermo-Fluids Group, University of Manchester, Manchester, UK
H. Iacovides
Affiliation:
Thermo-Fluids Group, University of Manchester, Manchester, UK
*
haoyuan.zhang@cardc.cn
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Abstract

Accurate prediction of supersonic and hypersonic turbulent flows is essential to the design of high-speed aerospace vehicles. Such flows are mainly predicted using the Reynolds-Averaged Navier–Stokes (RANS) approach in general, and in particular turbulence models using the effective viscosity approximation. Several terms involving the turbulent kinetic energy (k) appear explicitly in the RANS equations through the modelling of the Reynolds stresses in such approach, and similar terms appear in the mean total energy equation. Some of these terms are often ignored in low, or even supersonic, speed simulations with zero-equation models, as well as some one- or two-equation models. The omission of these terms may not be appropriate under hypersonic conditions. Nevertheless, this is a widespread practice, even for very high-speed turbulent flow simulations, because many software packages still make such approximations. To quantify the impact of ignoring these terms in the RANS equations, two linear two-equation models and one non-linear two-equation model are applied to the computation of five supersonic and hypersonic benchmark cases, one 2D zero-pressure gradient hypersonic flat plate case and four shock wave boundary layer interaction (SWBLI) cases. The surface friction coefficients and velocity profiles predicted with different combinations of the turbulent kinetic energy terms present in the transport equations show little sensitivity to the presence of these terms in the zero-pressure gradient case. In the SWBLI cases, however, comparisons show that inclusion of k in the mean flow equations can have a strong effect on the prediction of flow separation. Therefore, it is highly recommended to include all the turbulent kinetic energy terms in the mean flow equations when dealing with simulations of supersonic and hypersonic turbulent flows, especially for flows with SWBLIs. As a further consequence, since k may not be obtained explicitly in zero-equation, or certain one-equation, models, it is debatable whether these models are suitable for simulations of supersonic and hypersonic turbulent flows with SWBLIs.

Keywords

Hypersonic flowsSupersonic flowsSWBLIsTurbulent kinetic energyTurbulence Modelling
Type
Research Article
Information
The Aeronautical Journal , Volume 125 , Issue 1285 , March 2021 , pp. 525 - 555
Copyright
© The Author(s), 2020. Published by Cambridge University Press on behalf of Royal Aeronautical Society

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