DNS and LES of Supersonic Turbulent Internal Flows with and without Shock Train

Abstract: Large-eddy simulations of supersonic, turbulent flow in asymmetric and symmet-ric planar nozzles are carried out using high-order compact finite difference schemesand an LES-approach based on explicit filtering. The inflow to the nozzles are fromsupersonic, fully developed turbulent channel flows at Reτ = 180 and 360 with centerline Mach number 1.2. The combined effects of expansion and acceleration of theflow in this geometry result in reductions of mean pressure, density and temperatureas well as a reduction of Reynolds stresses. The asymmetry of the nozzle leads tonotable differences in the above mentioned effects near the straight and the curvedwalls. It is shown that the decay in Reynolds stresses is more near the curved wallthan near the straight wall of asymmetric nozzle and it is nearly the same near bothwalls of the symmetric nozzle. Effects of longitudinal streamline curvature are foundto be localised near the curved wall of the asymmetric nozzle and are quantified.The above mentioned effects of acceleration, expansion and streamline curvature arefound to be similar in the flow cases with two different Reynolds numbers, althoughwith increasing Reynolds number subtle differences in these effects are found. Directnumerical simulations of supersonic, turbulent flow in asymmtric and symmetric pla-nar nozzles with incoming Mach number around 1.45 and friction Reynolds number245 are carried out using high order compact finite difference schemes. We compareasymmetric and symmetric planar nozzles with the same area ratio, and symmetricplanar nozzles with different area ratios. Also, a comparison of the symmetric planarnozzle with the axisymmetric (circular cross-section) nozzle results existing in the lit-erature with the same area ratio for incoming Mach number around 1.5 and frictionReynolds number 245 are carried out.
              

High-order bandwidth-optimized Weighted Essentially Non-Oscillatory (WENO)schemes with limiters are applied in this study to a wide variety of problems including DNS of a shock train in a channel flow with isothermal walls. A comparisonof the performance of these schemes with 3-point and 4-point stencils is shown forone-dimensional and two-dimensional test problems. For the one-dimensional andtwo-dimensional test cases, these schemes show their advantage over the standard5th-order and the 7th-order WENO schemes proposed in the literature in capturingthe small-scale structures appearing in these flows. Effects of the limiter thresh-old values are discussed for the bandwidth-optimized WENO schemes. Further, thebandwidth-optimized WENO schemes with limiters are compared with other variantsof the WENO scheme found in the literature. Direct numerical simulation (DNS)of supersonic channel flow is also performed with the bandwidth-optimized WENOscheme (with 3-point stencil) with limiter, and the results are found to be in goodagreement with the literature. This scheme is then used for performing DNS of shocktrain in a turbulent channel flow. LES results using 6th order compact central schemealong with explicit filtering based ADM approach is compared with the results of DNSusing WENO3P-LIM-5 scheme for shock-turbulence interaction in channel flow withisothermal walls.

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