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DIRECT NUMERICAL SIMULATION OF A SPATIALLY EVOLVING SUPERSONIC TRANSITIONAL/TURBULENT BOUNDARY LAYER

Hiroshi Maekawa
Department of Mechanical Engineering and Intelligent Systems, the University of Electro-Communications 182-8585 Tokyo; Graduate School of Engineering, Hiroshima University 1-4-1 Kagamiyama, Higashi-Hiroshima-shi, 739-8527 Hiroshima, Japan

Daisuke Watanabe
Department of Mechanical and Control Engineering, the University of Electro-Communications 182-8585 Tokyo; Graduate School of Engineering, Hiroshima University 1-4-1 Kagamiyama, Higashi-Hiroshima-shi, 739-8527 Hiroshima, Japan; Graduate School of Engineering Toyama University 3190 Gofuku, Toyama-shi, Toyama 930-8555, Japan

Kougen Ozaki
Graduate School of Engineering, Hiroshima University 1-4-1 Kagamiyama, Higashi-Hiroshima-shi, 739-8527 Hiroshima, Japan

Hajime Takami
Environmental Engineering Division, Railway Technical Research Institute 2-8-38 Hikari-cho, Kokubunji-shi, 185-8540 Tokyo, Japan

Abstract

Transition mechanism in supersonic flat plate boundary layers with isothermal walls up to M = 3 is studied using spatially developing DNS. The compressible Navier-Stokes equations are numerically solved using high-order upwind-biased compact schemes for spatial derivatives (see Deng, Maekawa, Shen,1996). Navier-Stokes characteristic boundary conditions are used in the streamwise and vertical directions, and periodic boundary conditions in the spanwise direction. Random disturbances of isotropic homogeneous turbulence with the intensity of 4% relative to the free-stream velocity are introduced at the inlet of a computational box for M = 2:5. The transition scenario consists of the following four stages: first, generation of staggered vortical structures upstream giving rise to streaks, which is rather oblique and sometimes bifurcated, secondly, elongation of streaks in the streamwise direction, thirdly, oscillation of streaks with a couple of oblique streamwise vortices and an incipient spot, and finally breakdown of the flow due to growth of the spot. The three latter phases in this scenario have been observed in the low-speed boundary layer experiments subjected to high levels of free-stream turbulence by Mastubara and Alfredsson (2001). The longitudinal growth of the streaks is closely related to transient growth theory. The distribution of the streamwise velocity fluctuation shows to be similar to the experimental result of the incompressible transitional boundary layer, when it is scaled by the displacement thickness and normalized with their respective maximum. At higher amplitude case of 5% for M = 3, staggered structures are also observed but spanwise large-scale spots with rather complicated vortical structures are generated downstream soon. The shape factor starts around 2.6 and decreases to a value of 1.42 in the turbulent region. Reynolds number based on the momentum thickness reaches 1400 in the turbulent region. The streamwise velocity fluctuations in the turbulent regime show a good agreement with the experimental result by Konrad (1993) for M = 2:9 turbulent boundary layers. Large-scale motions observed in the turbulent regions exhibit similarities to those that have been found in incompressible boundary layers. Turbulent statistics show that the spanwise scale of the large-scale structure is about 440 wall units, which is consistent with recent PIV measurements for the M = 2 turbulent boundary layer by Ganapathisubramani et al. (2006). As Adrian (2000) and others hypothesized that the large-scale coherence in the incompressible turbulent boundary layers is a result of individual hairpin vortices convecting as groups, the numerical results show that a few hairpin vortices generated on the lifted-up low-speed streak convect in the supersonic turbulent boundary layer. As stated for the turbulent spots in the transitional region, several hairpin vortices are generated all at once when the large spots grow and merge each other in the transitional boundary layer. This fact is consistent with the experimental observations of the large-scale coherence in the supersonic/incompressible turbulent boundary layers. In the simulation the skin friction coefficient for M = 3 is found to show a good agreement with experimental results for 2:2 < M < 2:8 compiled by Coles (1954). The simulation results for M = 2:2, 2:5 and 3 indicate that transition is delayed as Mach number increases due to the near wall density gradient.