# Turbulence Modeling for High-Speed Flows

In Chapter 7 we have discussed attached high-speed turbulent flow by means of a very simple description, the ^-power law with the reference-temperature extension. Our understanding was and is that such a description can be used only for the establishment of general insights, for trend considerations on typical configuration elements, and for the approximate quantification of attached viscous flow effects, namely boundary layer integral properties, the thermal state of the surface, and the skin friction.

For the “exact” quantification of viscous flow effects the methods of numerical aerothermodynamics are required. These methods employ statistical turbulence models, which permit a description of high-speed turbulent flows with acceptable to good accuracy, as long as the flow is attached. Turbulent flow in the presence of strong interaction phenomena still poses large problems for statistical turbulence models, which hopefully can be diminished in the future.

In this section we give only a short overview of issues of turbulence modeling for attached viscous flows on high-speed vehicles. Like in the case of semiempirical and empirical transition prediction methods and criteria it holds that it is necessary for a given practical application to make first a assessment of the flow field under consideration, see Section 1.2. The assumptions

and the modeling in the turbulence model to be used must correspond to the considered flow field.

On RV’s the boundary layer due to the large angle of attack on a large part of the lower trajectory, Section 1.2, is a subsonic, transonic and low supersonic boundary layer with large boundary-layer edge temperatures and steep temperature gradients normal to the surface due to the surface radiation cooling, and hence steep density gradients with opposite sign. On CAV’s we have true hypersonic boundary layers at relatively small boundary-layer edge temperatures but also steep gradients of temperature and density, respectively, also due to the surface radiation cooling.[149] In the literature transonic, supersonic, and hypersonic boundary-layer flows are all together called compressible boundary-layer flows. For detailed introductions to the topic of turbulent flow see, e. g., [9, 10, 23], [144]-[146] as well as [147]-[149].

A compressible boundary-layer flow is a flow in which non-negligible density changes occur. These changes can occur even if stream-wise pressure changes are small, and also in low-speed flows, if large temperature gradients normal to the surface are present. In addition to these density changes density fluctuations p exist, in particular in compressible turbulent boundary layers.

In turbulence modeling these fluctuations can be neglected, if they are small compared to the mean-flow density: p ^ pmean. Morkovin’s hypothesis states that this holds for boundary-layer edge-flow Mach numbers Me P 5 in attached viscous flow [150].

Morkovin’s hypothesis does not hold for flows with large heat transfer, free shear flows, and turbulent combustion, see, e. g., [151, 152]. It also does not hold if the turbulent flow crosses a shock wave, for instance at the boundary-layer edge in the case of an incident shock wave, see, e. g., [153], and in strong interaction situations (shock/boundary-layer interaction, Section 9.2), see, e. g., [154]. Hence in flows with edge Mach numbers Me P 5, with shock/boundary-layer interactions, etc., explicit compressibility corrections must be applied.

For compressible flows besides the continuity and the momentum equations also the energy equation must be regarded. Hence in turbulence modeling not only velocity and pressure fluctuations must be taken into account but besides the density fluctuations also temperature fluctuations. This brings us from the Reynolds-averaging to the Favre-averaging, the latter being a mass-averaging process, which also necessitates further closure assumptions [10, 149].

Special issues appear due to turbulent heat conduction and turbulent mass diffusion, the latter in the case of non-equilibrium flow. Analogously to the Prandtl and Schmidt numbers in laminar flow, turbulent Prandtland

Schmidt numbers are introduced. These are usually taken as constant.[150] For the Prandtl number it is known since long that it is not constant in attached high speed turbulent flows [155]. With measured turbulent Prandtl numbers in attached flow 0.8 ^ Prturb ~ 1 usually a mean constant Prandtl number Prturb = 0.9 is employed in turbulence models. It is advisable to check with parametric variations whether the solution for a given flow class reacts sensitively to the choice of the (constant) Prandtl number. The same holds for the Schmidt number.

The turbulence models used are in general transport-equation models, see, e. g., [149]. They allow to compute with good accuracy attached turbulent twodimensional and three-dimensional high-speed boundary layers with pressure gradients, surface heat transfer, surface radiation cooling, surface roughness, and high-temperature real-gas effects. Very important is that the transition location is known or the flow is insensitive to the transition location. For a general introduction to three-dimensional attached viscous flow see [28].

General instructive computational results are given in [156]. There, several two-equation models with compressibility corrections were applied to flat – plate boundary layers with cooled and adiabatic wall in the Mach number regime 1.2 Si Мо 2l 10, to a hypersonic compression corner with cooled surface (two-dimensional control surface) with an onset flow Mach number M = 9.22, and to the flow past the X-38 configuration with extended body flap at Mо = 6 and a = 40° in the wind-tunnel situation.

Once the flow separates, the usual prediction problems, present already in the low-speed regime, see, e. g., [105], arise. In particular in the case of shock-induced separation (shock/boundary-layer interaction at a compression ramp) it is observed, see also [10], that the upstream influence is wrongly computed, the primary separation location is not met, and the computed pressure in the interaction zone is different from the measured one. In the interaction zone typically the heat transfer is predicted too high [157]. (See in this regard the example in Sub-Section 9.2.1.)

Downstream of the interaction zone the relaxation of pressure, skin friction and heat transfer often is different from the measured one. This is of large concern for practical applications, because the asymptotic behavior of, for instance the pressure on the ramp, is important. It governs the efficiency of an aerodynamic trim or control surface, like that of an inlet ramp [1].

Too high estimated thermal loads are not a problem per se, as too low predicted ones would be. They lead, however, to generally unwanted structural mass increments.[151] The reason for the misprediction possibly lies in the large wall-normal mean-flow density gradients present in the interaction zone.

These would warrant the employment of Reynolds-stress turbulence models [158, 159]. Such models have not yet found their place in aerothermodynamic computation methods.

Other reasons could be intrinsic three-dimensionalities in the interaction zone. This does not only concern Gortler vortex phenomena, but also embedded unsteady three-dimensional flow phenomena, as were observed in numerical simulations of laminar compression-ramp flow [160].

Massive flow separation, which as a rule is unsteady, now becomes treatable with zonal methods. These methods couple RANS and LES methods, see, e. g., [161]. The computational effort still is high. Nevertheless, these methods have an high application potential. Results of an investigation of high-speed base flow are discussed in Section 9.1.

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