Abstract
A highly accurate algorithm for the direct numerical simulation (DNS) of spatially evolving high-speed boundary-layer flows is described in detail and is carefully validated. To represent the evolution of instability waves faithfully, the fully explicit scheme relies on non-dissipative high-order compact-difference and spectral collocation methods. Several physical, mathematical, and practical issues relevant to the simulation of high-speed transitional flows are discussed. In particular, careful attention is paid to the implementation of inflow, outflow, and far-field boundary conditions. Four validation cases are presented, in which comparisons are made between DNS results and results obtained from either compressible linear stability theory or from the parabolized stability equation (PSE) method, the latter of which is valid for nonparallel flows and moderately nonlinear disturbance amplitudes. The first three test cases consider the propagation of two-dimensional second-mode disturbances in Mach 4.5 flat-plate boundary-layer flows. The final test case considers the evolution of a pair of oblique second-mode disturbances in a Mach 6.8 flow along a sharp cone. The agreement between the fundamentally different PSE and DNS approaches is remarkable for the test cases presented.
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References
N.A. Adams and L. Kleiser. Numerical simulation of transition in a compressible flat-plate boundary layer. In Transitional and Turbulent Compressible Flows—1993 (L.D. Kral and T.A. Zang, eds.). FED, Vol. 151. ASME, New York, 1993, pp. 101–110 (presented at the Fluids Engineering Conference, Washington, DC, June 20–24, 1993).
A. Bayliss, L. Maestrello, P. Parikh, and E. Turkel. Wave phenomena in a high Reynolds number compressible boundary layer. In Stability for Time Dependent and Spatially Varying Flows (D.L. Dwoyer and M.Y. Hussaini, eds.). Springer-Verlag, New York, 1985, pp. 188–205.
C. Canuto, A. Quateroni, M.Y. Hussaini, and T.A. Zang. Spectral Methods in Fluid Dynamics. Springer-Verlag, Berlin, 1988.
M.H. Carpenter, D. Gottlieb, and S. Abarbanel. The stability of numerical boundary treatments for compact high-order finite-difference schemes. ICASE Report No. 91-71, 1991.
C.-L. Chang. Unpublished personal communication, 1993.
C.-L. Chang, M.R. Malik, G. Erlebacher, and M.Y. Hussaini. Compressible stability of growing boundary layers using parabolized stability equations. AIAA Paper 91-1636, 1991.
D.F. DeSanto and H.B. Keller. Numerical studies of transition from laminar to turbulent flow over a flat plate. J. Soc. Indust. Appl. Math., Vol. 10, No. 4, 1962, pp. 569–595.
S.P.G. Dinavahi and C.D. Pruett. Analysis of direct numerical simulation data of a Mach 4.5 transitional boundary-layer flow. In Transitional and Turbulent Compressible Flows—1993 (L.D. Kral and T.A. Zang, eds.). FED, Vol. 151. ASME, New York, 1993, pp. 147–153 (presented at the Fluids Engineering Conference, Washington, DC, June 20–24, 1993).
W. Eissler and H. Bestek. Spatial numerical simulations of nonlinear transition phenomena in supersonic boundary layers. In Transitional and Turbulent Compressible Flows—1993 (L.D. Kral and T.A. Zang, eds.). FED, Vol. 151. ASME, New York, 1993, pp. 69–76 (presented at the Fluids Engineering Conference, Washington, DC, June 20–24, 1993).
N.M. El-Hady, T.A. Zang, and U. Piomelli. Dynamic subgrid-scale modeling for high-speed transitional boundary layers. In Engineering Applications of Large Eddy Simulations—1993 (S.A. Ragab and U. Piomelli, eds.). FED, Vol. 162. ASME, New York, 1993 (presented at the Fluids Engineering Conference, Washington, DC, June 20–24, 1993).
G. Erlebacher and M.Y. Hussaini. Numerical experiments in supersonic boundary-layer stability. Phys. Fluids A, Vol. 2, No. 1, 1990, pp. 94–104.
H. Fasel, A. Thumm, and H. Bestek. Direct numerical simulation of transition in supersonic boundary layers: oblique breakdown. In Transitional and Turbulent Compressible Flows—1993 (L.D. Kral and T.A. Zang, eds.). FED, Vol. 151. ASME, New York, 1993, pp. 77–92 (presented at the Fluids Engineering Conference, Washington, DC, June 20–24, 1993).
N. Gilbert and L. Kleiser. Near-wall phenomena in transition to turbulence. In Near-Wall Turbulence: 1988 Zoran Zaric Memorial Conference (S.J. Kline and N.H. Afgan, eds.). Hemisphere, Washington, DC, 1990, pp. 17–27.
B. Gustafsson. The convergence rate for difference approximations to mixed initial boundary-value problems. Math. Comp., Vol. 29, No. 130, 1975, pp. 396–406.
L. Kleiser and T.A. Zang. Numerical simulation of transition in wall-bounded shear flows. Annu. Rev. Fluid Mech., Vol. 23, 1991, pp. 495–537.
L.D. Kral. Numerical Investigation of Transition Control of a Flat Plate Boundary Layer. Ph.D. thesis, University of Arizona, 1988.
S.K. Lele. Compact finite difference schemes with spectral-like resolution. Center for Turbulence Research Manuscript 107, Stanford University, April 1990.
L.M. Mack. Boundary-layer linear stability theory. In Special Course on Stability and Transition of Laminar Flow (R. Michel, ed.). AGARD Report No. 709, 1984, pp. 3.1–3.81.
L. Maestrello, A. Bayliss, and R. Krishnan. On the interaction between first and second-mode waves in a supersonic boundary layer. Phys. Fluids A, Vol. 3, No. 12, 1991, pp. 3014–3020.
L.L. Ng and G. Erlebacher. Secondary instabilities in compressible boundary layels. Phys. Fluids A, Vol. 4, No. 4, 1992, pp. 710–726.
L.L. Ng and T.A. Zang. Secondary instability mechanisms in compressible, axisymmetric boundary layers. AIAA J., Vol. 31, No. 9, Sept. 1993, pp. 1605–1610.
J. Nordstrom. The influence of open boundary conditions on the convergence to steady state for the Navier-Stokes equations. J. Comput. Phys., Vol. 85, 1989, pp. 210–244.
X. Normand and M. Lesieur. Direct and large-eddy simulations of transition in the compressible boundary layer. Theoret. Comput. Fluid Dynamics, Vol. 3, 1992, pp. 231–252.
T.J. Poinsot and S.K. Lele. Boundary conditions for direct simulations of compressible viscous flows. J. Comput. Phys., Vol. 101, 1992, pp. 104–129.
C.D. Pruett. On the accurate prediction of the wall-normal velocity in compressible boundary-layer flow. Internat. J. Numer. Methods Fluids, Vol. 16, 1993, pp. 133–152.
C.D. Pruett and C.-L. Chang. A comparison of PSE and DNS for high-speed boundary-layer flows. In Transitional and Turbulent Compressible Flows—1993 (L.D. Kral and T.A. Zang, eds.). FED, Vol. 151. ASME, New York, 1993, pp. 57–67 (presented at the Fluids Engineering Conference, Washington, DC, June 20–24, 1993).
C.D. Pruett and C.L. Streett. A spectral collocation method for compressible, non-similar boundary layers. Internat. J. Numer. Methods Fluids, Vol. 13, No. 6, 1991, pp. 713–737.
C.D. Pruett and T.A. Zang. Direct numerical simulation of laminar breakdown in high-speed, axisymmetric boundary layers. Theoret. Comput. Fluid Dynamics, Vol. 3, No. 6, 1992, pp. 345–367.
M. Rai and P. Moin. Direct numerical simulation of transition and turbulence in a spatially evolving boundary layer. AIAA Paper No. 91-1607, 1991.
K.F. Stetson and R.L. Kimmel. On the breakdown of a hypersonic laminar boundary layer. AIAA Paper 93-0896, 1993.
K.F. Stetson, E.R. Thompson, J.C. Donaldson, and L.G. Siler. Laminar boundary-layer stability experiments on a cone at Mach 8. Part 1: Sharp cone. AIAA Paper 83-1761, 1983.
C.L. Streett and M.G. Macaraeg. Spectral multi-domain for large-scale fluid dynamic simulations. Appl. Numer. Math., Vol. 6, 1989/90, pp. 123–139.
K.W. Thompson. Time dependent boundary conditions for hyperholic systems. J. Comput. Phys., Vol. 68, 1987, pp. 1–24.
A. Thumm. Numerische Untersuchungen zum laminar-turbulenten Stroemungsumschlag in transsonischen Grenzschictstroemunge. Ph.D. thesis, Universitaet Stuttgart, 1991.
A. Thumm, W. Wolz, and H. Fasel. Numerical simulation of spatially growing three-dimensional disturbance waves in compressible boundary layers. In Laminar-Turbulent Transition. IUTAM Symposium, Toulouse, France, 1989 (D. Arnal and R. Michel, eds.). Springer-Verlag, Berlin, 1990, pp. 303–308.
L.N. Trefethen. Group velocity in finite difference schemes. SIAM Rev., Vol. 24, No. 2, 1982, pp. 113–136.
R. Vichnevetsky. Invariance theorems concerning reflection at numerical boundaries. J. Comput. Phys., Vol. 63, 1986, pp. 268–282.
F.M. White. Viscous Fluid Flow. McGraw-Hill, New York, 1974, pp. 347–350.
J.H. Williamson. Low-storage Runge-Kutta schemes. J. Comput. Phys., Vol. 35, 1980, pp. 48–56.
T.A. Zang, C.-L. Chang, and L.L. Ng. The transition prediction toolkit: LST, SIT, PSE, DNS, and LES. Fifth Symposium on Numerical and Physical Aspects of Aerodynamic Flows, January 1992.
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Communicated by M.Y. Hussaini
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Pruett, C.D., Zang, T.A., Chang, CL. et al. Spatial direct numerical simulation of high-speed boundary-layer flows part I: Algorithmic considerations and validation. Theoret. Comput. Fluid Dynamics 7, 49–76 (1995). https://doi.org/10.1007/BF00312399
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DOI: https://doi.org/10.1007/BF00312399