A study of supersonic flow past a plate using a two-fluid turbulence model

This paper presents a numerical study of supersonic flow past a sharp flat plate at zero incidence using a two-fluid turbulence model written in von Mises variables and solved with an implicit marching finite-difference scheme. Velocity, skin-friction, temperature, and turbulent Prandtl number distributions were obtained for M = 2 and M = 5 with adiabatic (Tw/Tr=1) and cooled (Tw/Tr=0.5 and 0.2) walls, Tr being the recovery temperature, in the range Rex≤3⋅107 (Reθ up to about 2 × 104). Laminar-to-turbulent transition was captured by varying the amplitude of the initial disturbance from 0.05% to 5%. The skin-friction coefficient was compared with the van Driest II correlation, with direct numerical simulation (DNS) data at matched Mach number and wall-temperature ratio, and with the recent skin-friction scaling law of Zhao and Fu. For adiabatic walls, the model agrees with van Driest II within 9% (mean deviation 4%–5%) for 4000≤Reθ≤12 000. For cooled walls, the model predicts a skin friction 11%–21% lower than van Driest II. This reproduces the tendency, found in DNS, of van Driest II to over-predict the skin friction on cold walls but overestimates its magnitude: the model under-predicts the DNS skin friction by 3%–26% (14% on average) for 0.15≤Tw/Tr≤0.5. Besides the skin friction, the two-fluid model describes phenomena that are not described by empirical correlations: the dependence of the transition onset on the intensity of the incoming disturbances and the anisotropy of turbulent heat transfer in a compressible boundary layer, which cannot be represented by models based on the Boussinesq hypothesis. The computed turbulent Prandtl number varies strongly in the transition region and tends to a constant value of 0.855 in the turbulent zone.

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