EFFECT OF COMPRESSIBILITY CORRECTIONS TO TURBULENCE MODELS APPLIED TO A HYPERSONIC RE-ENTRY CONFIGURATION

Computation of high-speed turbulent flows using conventional turbulence models developed for incompressible flows can lead to significant errors. This is especially true for high-speed vehicles where the Mach number ranges from low subsonic at the nose to hypersonic values along the body. The flow field is further complicated by boundary layer transition on fore-body, flow separation behind the vehicle, and turbulent free shear layer recompression in the reattachment region. Several compressibility corrections to conventional turbulence models have been proposed to account for these high Mach number effects. This paper evaluates such modifications to oneand two-equation turbulence models. The test case considered is the flow around a hypersonic re-entry configuration with a large recirculation region behind the vehicle. Issues pertaining to the implementation of the models and their modifications to a practical configuration are discussed. The effect of the different models and compressibility corrections are studied in terms of the predicted boundary layer transition location, size of the recirculation region, and the resulting pressure and heat transfer rate on the vehicle. NOMENCLATURE a speed of sound, ms−1 M Mach number Mc Convective Mach number ∗Address all correspondence to this author (krish@aero.iitb.ac.in). ν kinematic molecular viscosity, m2s−1 νT kinematic eddy viscosity, m2s−1 ρ̄ mean density, kg/m3 q heat transfer rate per unit area, Watt/cm2 p pressure, Pa s body co-ordinate, m D body diameter, m U velocity, ms−1 Ω magnitude of mean vorticity vector, s−1 k turbulent kinetic energy ε turbulent kinetic energy dissipation rate ω specific dissipation INTRODUCTION Atmospheric re-entry vehicles are subjected to large aerothermal loads. During the later part of their descent the flow around the vehicle, especially in the wake region, may transition to turbulence. This can significantly increase the heat transfer to the vehicle walls. Predicting the turbulent heating rate is challenging, and the inaccuracies are generally compensated by a large safety factor in the design of heat shields [1]. Engineering prediction of turbulent flows relies mostly on Reynolds-averaged Navier Stokes (RANS) simulations that compute the time-averaged flow field. Turbulence closure is obtained using RANS turbulence models which were originally developed for incompressible flows. As noted by Paciorri in [22], fluid compressibility adds to turbulence phenomenology new specific processes and mechanisms, such as new pathways for energy ex1 Copyright c © 2006 by ASME changes, and strong coupling between momentum and energy exchanges. This can significantly alter the turbulent characteristics and the flow behavior. A detailed review of compressibility and its effects on turbulence can be found in [2]. Compressibility effects play an important role in turbulent mixing layers. Numerous studies [3], [4], [5] have highlighted that the spreading rate of a mixing layer is reduced with respect to an analogous incompressible mixing layer. Many research groups are working on methods of including these compressibility effects in conventional turbulence models to predict more realistic flow-fields for compressible conditions. Catris and Aupoix [21], Paciorri and Sabetta [22], Secundov et al [25], and Spalart [24] have shown various compressibility modifications to existing turbulence models to predict the effects of compressibility on shear layers and other aspects of a flow-field. Compressibility corrections have also been proposed by Sarkar et al [29] and Zeman [30]. Majority of these modifications aim to achieve the reduced spreading rate of free shear layer at high Mach numbers. For bluff bodies, flows are complex in nature [6] because they involve interaction of a wake, separated flow, free shear layer, and boundary layer. Brown [18] recently computed the flow around a re-entry configuration where the flow transitions on the afterbody and the sting. The compressibility corrected SST model was found most accurate. Brown also highlights the consistency and accuracy of numerous turbulence models, with and without compressibility corrections, in predicting heat transfer levels for a range of hypersonic flows. Sinha [12] computed the flow-field around Fire II re-entry configuration using standard k−ω and SA turbulence models. Large differences between the turbulence model predictions are found in the vicinity of the transition location on the forebody. Variation of separation bubble size and afterbody pressure and heat transfer as obtained by the two turbulence models are also highlighted. Transition from laminar to turbulent in high speed boundary layers is important for prediction and control of heat transfer, skin friction, etc. A lot of research is directed towards predicting correctly this transition in high speed flows. A large number of parameters influence this transition such as Mach number, transverse and streamwise curvature, pressure gradient, temperature, and cross flow [7]. A survey of flight data for boundary layer transition at hypersonic speeds is done by Schneider [8]. Lees [14] highlights the laminar-turbulent transition in hypersonic wakes and summarizes crisply the effect of Reynolds number in transition. Similar to simulations done by Sinha in [12], the current paper studies the transitional/turbulent flow along the Fire II vehicle using RANS turbulence models. Compressibility corrections are included in turbulence models to understand the effect of these corrections in predicting realistic flowfield around a re-entering vehicle. Although the temperature in the flowfield is high enough for the chemical reactions to occur, the current simulations assume perfect gas and effect of thermo-chemistry is neglected. This is because of the focus of the study is to understand the behavior of turbulence models and compressibility corrections to them at low Reynolds number and high Mach number conditions, typical of re-entry conditions. 1 SIMULATION METHODOLOGY AND TEST CASE The grid topology, freestream conditions, boundary conditions and computational approach is the same as described by Sinha in [12]. The Fire II vehicle has a spherical forebody and a conical afterbody [19]. The vehicle dimensions are shown in Fig. 1. The simulations in [12] correspond to an altitude of 35 km, where the freestream density and temperature are 0.0082kg/m3 and 237 K respectively. The vehicle velocity is 5 km/s (Mach 16) at zero angle of attack. The wall temperature is 553.3 K and the Reynolds number based on freestream flow and vehicle diameter is 1.76× 106. Note that a detailed grid refinement study is presented in Ref. [12] and the current simulations are run on the baseline grid for each case. The turbulence models and compressibility corrections are described below.