×

zbMATH — the first resource for mathematics

Performance of very-high-order upwind schemes for DNS of compressible wall-turbulence. (English) Zbl 1407.76044
Summary: The purpose of the present paper is to evaluate very-high-order upwind schemes for the direct numerical simulation (DNS) of compressible wall-turbulence. We study upwind-biased (UW) and weighted essentially nonoscillatory (WENO) schemes of increasingly higher order-of-accuracy [D. S. Balsara and C.-W. Shu, J. Comput. Phys. 160, No. 2, 405–452 (2000; Zbl 0961.65078)], extended up to WENO17 [the authors, Very-high-order WENO schemes. AIAA Paper 2009-1612, 47. Aerospace Sciences Meeting, Orlando, FL, U.S.A., 5–8 January 2009]. Analysis of the advection-diffusion equation, both as \(\Delta x \longrightarrow 0\) (consistency), and for fixed finite cell-Reynolds-number \(Re_{\Delta x}\) (grid-resolution), indicates that the very-high-order upwind schemes have satisfactory resolution in terms of points-per-wavelength (PPW). Computational results for compressible channel flow (\(Re_{\tau_w}\in [180,230];\bar M _{\text{CL}}\in [0.35,1.5]\)) are examined to assess the influence of the spatial order of accuracy and the computational grid-resolution on predicted turbulence statistics, by comparison with existing compressible and incompressible DNS databases. Despite the use of baseline \(O(Delta t^2)\) time-integration and \(O(\Delta x^{2})\) discretization of the viscous terms, comparative studies of various orders-of-accuracy for the convective terms demonstrate that very-high-order upwind schemes can reproduce all the DNS details obtained by pseudospectral schemes, on computational grids of only slightly higher density.

MSC:
76F65 Direct numerical and large eddy simulation of turbulence
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Coleman, A numerical study of turbulent supersonic isothermal-wall channel flow, Journal of Fluid Mechanics 305 pp 159– (1995) · Zbl 0960.76517
[2] Huang, Compressible turbulent channel flows: DNS results and modelling, Journal of Fluid Mechanics 305 pp 185– (1995) · Zbl 0857.76036
[3] Guarini, Direct numerical simulation of a supersonic turbulent boundary-layer at Mach 2.5, Journal of Fluid Mechanics 414 pp 1– (2000) · Zbl 0983.76039
[4] Maeder, Direct simulation of turbulent supersonic boundary-layers by an extended temporal approach, Journal of Fluid Mechanics 429 pp 187– (2001) · Zbl 1007.76031
[5] Lechner, Turbulent supersonic channel flow, Journal of Turbulence 2 pp 001.1– (2001)
[6] Morinishi, Direct numerical simulation of compressible turbulent channel flow between adiabatic and isothermal walls, Journal of Fluid Mechanics 502 pp 273– (2004) · Zbl 1134.76363
[7] Pirozzoli, Direct numerical simulation of impinging shock-wave/turbulent-boundary-layer interaction at M=2.25, Physics of Fluids 18 pp 065113.81– (2006)
[8] Friedrich, Compressible turbulent flows: aspects of prediction and analysis, Zeitschrift für Angewandte Mathematik und Mechanik 87 pp 189– (2007)
[9] Buell, A hybrid numerical method for 3-D spatially-developing free-shear flows, Journal of Computational Physics 95 pp 313– (1991) · Zbl 0725.76072
[10] Morinishi, A DNS algorithm using B-spline collocation method for compressible turbulent channel flow, Computers and Fluids 32 pp 751– (2003) · Zbl 1083.76542
[11] Adams, A high-resolution hybrid compact-ENO scheme for shock/turbulence interaction problems, Journal of Computational Physics 127 pp 27– (1996) · Zbl 0859.76041
[12] Sesterhenn, A characteristic-type formulation of the Navier-Stokes equations for high-order upwind schemes, Computers and Fluids 30 pp 37– (2001) · Zbl 0991.76056
[13] Sandham, Entropy splitting for high-order numerical simulation of compressible turbulence, Journal of Computational Physics 178 pp 307– (2002) · Zbl 1139.76332
[14] Pirozzoli, Conservative hybrid compact-WENO schemes for shock-turbulence interaction, Journal of Computational Physics 178 pp 81– (2002) · Zbl 1045.76029
[15] Martín, A bandwidth-optimized WENO scheme for effective direct numerical simulation of compressible turbulence, Journal of Computational Physics 220 pp 270– (2006) · Zbl 1103.76028
[16] Taylor, Optimization of nonlinear error for weighted essentially non-oscillatory methods in direct numerical simulations of compressible turbulence, Journal of Computational Physics 223 pp 384– (2007) · Zbl 1165.76350
[17] Adams, Direct simulation of the turbulent boundary-layer along a compression ramp at M=3 and Re\(\theta\)=1685, Journal of Fluid Mechanics 420 pp 47– (2000) · Zbl 1009.76043
[18] Foysi, Compressibility effects and turbulence scalings in supersonic channel flow, Journal of Fluid Mechanics 509 pp 207– (2004) · Zbl 1066.76035
[19] Yee, Entropy splitting and numerical dissipation, Journal of Computational Physics 162 pp 33– (2000) · Zbl 0987.65086
[20] Ponziani, Development of optimized WENO schemes for multiscale compressible flows, International Journal for Numerical Methods in Fluids 42 pp 953– (2003) · Zbl 1055.76039
[21] Pirozzoli, Direct numerical simulation of isotropic compressible turbulence: influence of compressibility on dynamics and structures, Physics of Fluids 16 pp 4386– (2004) · Zbl 1187.76418
[22] Pirozzoli, Direct numerical simulation and analysis of a spatially evolving supersonic turbulent-boundary-layer at M=2.25, Physics of Fluids 16 pp 530– (2004) · Zbl 1186.76423
[23] Martín, Direct numerical simulation of hypersonic turbulent-boundary-layers-part 1-initialization and comparison with experiments, Journal of Fluid Mechanics 570 pp 347– (2007)
[24] Wu, Direct numerical simulation of supersonic turbulent-boundary-layer over a compression-ramp, AIAA Journal 45 pp 879– (2007)
[25] Wu, Analysis of shock motion in shock-wave and turbulent-boundary-layer interaction using DNS data, Journal of Fluid Mechanics 594 pp 71– (2008)
[26] Ringuette, Coherent structures in direct numerical simulation of turbulent-boundary-layers at March 3, Journal of Fluid Mechanics 594 pp 59– (2008) · Zbl 1159.76342
[27] Shu, Efficient implementation of essentially nonoscillatory shock-capturing schemes II, Journal of Computational Physics 83 pp 32– (1989) · Zbl 0674.65061
[28] Liu, Weighted essentially nonoscillatory schemes, Journal of Computational Physics 115 pp 200– (1994)
[29] Strand, Summation by parts for finite-difference approximations for d/dx, Journal of Computational Physics 110 pp 47– (1994) · Zbl 0792.65011
[30] Williamson, Low-storage Runge-Kutta schemes, Journal of Computational Physics 35 pp 48– (1980) · Zbl 0425.65038
[31] Suresh, Accurate monotonicity-preserving schemes with Runge-Kutta time stepping, Journal of Computational Physics 136 pp 83– (1997) · Zbl 0886.65099
[32] Jiang, Efficient implementation of weighted ENO schemes, Journal of Computational Physics 126 pp 202– (1996) · Zbl 0877.65065
[33] Balsara, Monotonicity preserving WENO schemes for increasingly high-order of accuracy, Journal of Computational Physics 160 pp 405– (2000) · Zbl 0961.65078
[34] Gerolymos GA, Sénéchal D, Vallet I. Very-high-order WENO schemes. AIAA Paper 2009-1612, 47. Aerospace Sciences Meeting, Orlando, FL, U.S.A., 5-8 January 2009.
[35] Henrick, Mapped weighted-essentially-non-oscillatory schemes: achieving optimal order near critical points, Journal of Computational Physics 207 pp 542– (2005) · Zbl 1072.65114
[36] Harten, Uniformly high-order accurate essentially nonoscillatory schemes III, Journal of Computational Physics 71 pp 231– (1987) · Zbl 0627.65102
[37] Titarev, Finite-volume WENO schemes for 3-D conservation laws, Journal of Computational Physics 201 pp 238– (2004) · Zbl 1059.65078
[38] Johnsen, Implementation of WENO schemes in compressible multicomponent flow problems, Journal of Computational Physics 219 pp 715– (2006) · Zbl 1189.76351
[39] Urbin, Large-Eddy simulation of a supersonic boundary layer using an unstructured grid, AIAA Journal 39 pp 1288– (2001)
[40] Pirozzoli, On the spectral properties of shock-capturing schemes, Journal of Computational Physics 219 pp 489– (2006) · Zbl 1103.76040
[41] Črnjarić-Žic, On different flux-splittings and flux-functions in WENO schemes for balance laws, Computers and Fluids 35 pp 1074– (2006) · Zbl 1177.76218
[42] Rai, Direct numerical simulations of turbulent flow using finite-difference schemes, Journal of Computational Physics 96 pp 15– (1991) · Zbl 0726.76072
[43] Lele, Compact finite difference schemes with spectral-like resolution, Journal of Computational Physics 103 pp 16– (1992) · Zbl 0759.65006
[44] Pirozzoli, Performance analysis and optimization of finite-difference schemes for wave-propagation problems, Journal of Computational Physics 222 pp 809– (2007) · Zbl 1158.76382
[45] Martín, A parallel implicit method for the direct numerical simulation of wall-bounded compressible turbulence, Journal of Computational Physics 215 pp 153– (2006) · Zbl 1088.76021
[46] Yoon, Lower-upper symmetric-Gauss-Seidel method for the Euler and Navier-Stokes equations, AIAA Journal 26 pp 1025– (1988)
[47] Candler, Data-parallel lower-upper relaxation method for reacting flows, AIAA Journal 32 pp 2380– (1994) · Zbl 0824.76061
[48] Kim, Turbulence statistics in fully developed channel flow at low-Reynolds-number, Journal of Fluid Mechanics 177 pp 133– (1987) · Zbl 0616.76071
[49] Moser, Direct numerical simulation of turbulent channel flow up to Re\(\tau\)=590, Physics of Fluids 11 pp 943– (1999) · Zbl 1147.76463
[50] del Álamo, Spectra of the very large anisotropic scales in turbulent channels, Physics of Fluids 15 pp L41– (2003) · Zbl 1186.76136
[51] del Álamo, Scaling of the energy spectra of turbulent channels, Journal of Fluid Mechanics 500 pp 135– (2004) · Zbl 1059.76031
[52] Hoyas, Scaling of the velocity fluctuations in turbulent channels up to Re\(\tau\)=2003, Physics of Fluids 18 (2006)
[53] Hoyas, Reynolds number effects on the Reynolds-stress budgets in turbulent channels, Physics of Fluids 20 pp 101511(1-8)– (2008) · Zbl 1182.76330
[54] Chan, Stability analysis of finite difference schemes for the advection-diffusion equation, SIAM Journal on Numerical Analysis 21 pp 272– (1984) · Zbl 0553.65059
[55] Wesseling, von Neumann stability conditions for the convection-diffusion equation, IMA Journal of Numerical Analysis 16 pp 583– (1996) · Zbl 0862.65046
[56] Lomax, Fundamentals of Computational Fluid Dynamics (2001) · doi:10.1007/978-3-662-04654-8
[57] Guyker, On the uniform convergence of interpolating polynomials, Applied Mathematics and Computation (2008) · Zbl 1172.41002 · doi:10.1016/j.amc.2007.07.032
[58] Macon, Inverses of Vandermonde matrices, American Mathematical Monthly 65 pp 95– (1958) · Zbl 0081.01503
[59] Eisinberg, On the inversion of the Vandermonde matrix, Applied Mathematics and Computation 174 pp 1384– (2006) · Zbl 1124.65028
[60] Shannon, Communications in the presence of noise, IRE Proceedings 37 pp 10– (1949)
[61] Briggs, The DFT: An Owner’s Manual for the Discrete Fourier Transform (1995) · Zbl 0827.65147 · doi:10.1137/1.9781611971514
[62] Sengupta, Analysis of central and upwind compact schemes, Journal of Computational Physics 192 pp 677– (2003) · Zbl 1038.65082
[63] Sengupta, A comparative study of time advancement methods for solving Navier-Stokes equations, Journal on Scientific Computing 21 pp 225– (2004) · Zbl 1060.76084
[64] Liepmann, Elements of Gasdynamics (1957)
[65] White, Viscous Fluid Flow (1974) · Zbl 0356.76003
[66] Graves, Bulk viscosity: past to present, Journal of Thermophysics and Heat Transfer 13 pp 337– (1999)
[67] Emanuel, Bulk viscosity in the Navier-Stokes equations, International Journal of Engineering Science 36 pp 1313– (1998)
[68] Zuckerwar, Variational approach to the volume viscosity of fluids, Physics of Fluids 18 (2006) · Zbl 1185.76867
[69] Schlichting, Boundary-layer Theory pp 327– (1979)
[70] Eckert, Analysis of Heat and Mass Transfer pp 64– (1972)
[71] Gerolymos, Implicit multiple-grid solution of the compressible Navier-Stokes equations using k-\(\epsilon\) turbulence closure, AIAA Journal 28 pp 1707– (1990)
[72] Chassaing, Efficient and robust Reynolds-stress model computation of 3-D compressible flows, AIAA Journal 41 pp 763– (2003)
[73] Vinokur, An analysis of finite-difference and finite-volume formulations of conservation laws, Journal of Computational Physics 81 pp 1– (1989) · Zbl 0662.76039
[74] Carpenter, A stable and conservative interface treatment of arbitrary spatial accuracy, Journal of Computational Physics 148 pp 341– (1999) · Zbl 0921.65059
[75] Van Leer, Upwind and high-resolution methods for compressible flow: from donor cell to residual distribution schemes, Communications in Computational Physics 1 pp 192– (2006) · Zbl 1114.76049
[76] Toro, Restoration of the contact surface in the HLL-Riemann solver, Shock Waves 4 pp 25– (1994) · Zbl 0811.76053
[77] Toro, Riemann Solvers and Numerical Methods for Fluid Dynamics (1997) · Zbl 0888.76001 · doi:10.1007/978-3-662-03490-3
[78] Batten, On the choice of wave speeds for the HLLC Riemann solver, SIAM Journal on Scientific Computing 18 pp 1553– (1997) · Zbl 0992.65088
[79] Batten, Average-state Jacobians and implicit methods for compressible viscous and turbulent flows, Journal of Computational Physics 137 pp 38– (1997) · Zbl 0901.76043
[80] Gerolymos GA, Sénéchal D, Vallet I. Analysis of dual-time-stepping with explicit subiterations. AIAA Paper 2009-1608, 47. Aerospace Sciences Meeting, Orlando, FL, U.S.A., 5-8 January 2009.
[81] Zhao, Computation of complex turbulent flow using matrix-free implicit dual-time-stepping scheme and LRN turbulence model on unstructured grids, Computers and Fluids 33 pp 119– (2004) · Zbl 1165.76353
[82] Chassaing, Reynolds-stress model dual-time-stepping computation of unsteady 3-D flows, AIAA Journal 41 pp 1882– (2003)
[83] Gerolymos, Oblique-shock-wave/boundary-layer interaction using near-wall Reynolds-stress models, AIAA Journal 42 pp 1089– (2004)
[84] Gerolymos, Influence of inflow-turbulence in shock-wave/turbulent-boundary-layer interaction computations, AIAA Journal 42 pp 1101– (2004)
[85] Pope, Turbulent Flows (2000) · Zbl 0966.76002 · doi:10.1017/CBO9780511840531
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.