# Compressible turbulence in a high-speed high Reynolds number mixing layer

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## Abstract

Compressible turbulence in a high-speed, high Reynolds number, supersonic free shear layer was studied. A two-dimensional free mixing layer was chosen to study turbulence rather than a wall bounded flow due to the experimental fact that the effects of compressibility become significant at lower Mach numbers. The mixing layer was generated by supersonic injection of air (M_{s} = 1.8, P_{ts} = 0.5 atm. T_{ts}= 295K. and Re/m = 7x10⁶) through a rearward facing tangential slot, into a supersonic free stream (M_{∞} = 4.0, P_{t∞} = 12.5 atm, T_{t∞} = 290K, and Re/m = 70x10⁶). Flow visualization was accomplished by nanosecond Shadowgraph photography. The overall flow structure was documented with the Shadowgraph and conventional mean flow probes (Pitot pressure, cone-static pressure, and thermocouple probes). The turbulent structure of the flow field was also clearly depicted in the Shadowgraphs. Image processing techniques were developed in order to determine root-mean-square index of refraction (density) fluctuation levels from the Shadowgraph plates. Multiple overheat normal and cross-wire techniques were developed and/or improved for this study. The present research concentrated on the Reynolds averaged form of the Navier-Stokes equations. where the effects of compressibility are manifested through "apparent mass" terms (i.e. p′u′_{i}). These terms appear in all of the Reynolds averaged Navier-Stokes equations (continuity, momentum, and energy). A new turbulence transformation, coupled with innovative experimental methods. allowed the full compressible Reynolds shear stress (the typical incompressible term, pu′_{i}u′_{j} as well as the apparent mass terms) to be directly measured. The full compressible heat flux and apparent mass terms were also estimated from the cross-wire results. Profiles were obtained at four downstream stations which were strategically located to map different levels of development of the shear flow. The first station was very close to the injector, about one free stream boundary layer thickness downstream (x/δ_{∞} ≈ 1), hence, it is in the initial region. The second station was located at x/δ_{∞} ≈ 28, which was near the beginning of the fully developed zone. The third station, x/δ_{∞} = 83, was just prior the shear layer and floor boundary layer merging. The last station was positioned just aft of the layer merging, x/δ_{∞} = 106. Reynolds averaging of the compressible Navier-Stokes equations implies that the compressible turbulence affects all of the governing equations. It was found, experimentally, that the effects of compressibility on turbulence were more than significant accounting for about 75% of the total level of the Reynolds shear stress formulation for the present study (i.e. the apparent mass term multiplied by the axial velocity was about 3-4 times the typical incompressible shear term). For the present mean adiabatic flow, the compressible turbulence accounted for 100% of the turbulent heat flux. The apparent mass in the continuity equation was, by definition, only due to compressibility. These results led to the development of anew Compressible Apparent Mass Mixing Length Extension (CAMMLE) model that accounts for compressible turbulence in all of the governing equations (i.e. the turbulence terms in the continuity, momentum, and energy were all consistently formulated). The CAMMLE formulation is a generalization of the Situ-Schetz compressible mixing length formulation, which was developed to account for the apparent mass terms in the momentum equation. A total of seven turbulence models were experimentally evaluated, the CAMMLE model, the Prandtl incompressible and the Situ-Schetz compressible mixing length models, the Prandtl and Bradshaw turbulent kinetic energy (TKE) formulations, and two compressible TKE extensions that are based upon a newly defined compressible TKE formulation. The measured turbulence data was used to assess the various models, where the measured mean flow profiles were used in the model formulations. The incompressible formulations were generally successful in representing the measured incompressible part of the Reynolds shear stress. However, this term only accounted for about 25% of the total shear stress level. All of the compressible extensions provided accurate estimates of the full compressible Reynolds shear stress. In addition, the newly developed CAMMLE model was also successful in representing the apparent mass terms in the continuity equation. The CAMMLE model was also the only formulation to accurately predict the measured compressible turbulent heat flux in the energy equation. The CAMMLE, Situ-Schetz, and Prandtl incompressible mixing length models were all incorporated in to a 3-D finite volume Navier-Stokes code (GASP 2.0). The numerical simulations indicated that the new compressible apparent mass mixing length extension performed very well. The CFD results also enlightened a misuse with all of the current compressible turbulence models. With the exception of the new apparent mass formulation, all existing turbulence models neglect the compressible turbulence effects on the continuity equation and treat the energy equation in an ad hoc effective eddy viscosity and thermal conductivity fashion. The numerical and theoretical studies indicated that this led to poor prediction of the mixing layer width for cases where the free stream Mach number was significantly higher than the injection Mach number.