Abstract
Turbulent flows are characterised by a continuum of length and time scales, a feature that introduces some unique problems in relation to the analysis and control of errors in numerical simulations of turbulent flows. In direct numerical simulation (DNS) one attempts to fully resolve the flow field. The primary sources of error in DNS are the aliasing error, which arises due to the evaluation of the nonlinear term on a discrete grid in physical space, and, the truncation error, due to the discretization of the derivative operator. In addition there are the time-stepping errors on account of the temporal discretization. In Large Eddy Simulation (LES) only the large scale flow is computed whereas the collective effect of the small scales are modeled. In that case, in addition to the above three types of errors, there are commutation errors arising out of the averaging process used to derive the LES equations and sub-grid modeling errors that arise because in the absence of a systematic method, the subgrid stress is evaluated using an ad hoc closure model. Methods for analyzing and quantifying these errors in turbulence simulations are discussed. In some instances the analysis points to suitable methods of error control.
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References
R. Vichnevetsky and J. Bowles, Fourier analysis of numerical approximations of hyperbolic equations,SIAM, Philadelphia, (1982).
S. Lele, “Compact Finite Difference Schemes with Spectral Like Resolution,” J. Comp. Phys. 103, 16 (1992).
C. Canuto, M.Y. Hussaini, A. Quarteroni, T.A. Zang “Spectral Methods in Fluid Dynamics,” Berlin: Springer (1988).
J.W. Cooley and J.W. Tukey, “An algorithm for the machine calculation of complex Fourier Series,” Math. Comput. 19, 297 (1965).
N.A. Phillips “An example of nonlinear computational instability,” In The Atmosphere and Sea in Motion ed. B. Bolin, Rockefeller Inst. Press, New York (1959).
R. Rogallo, NASA Tech. Memo. TM81315 (1981).
G.A. Blaisdell, E.T. Spyropoulos and J.H. Qin “The effect of the formulation of nonlinear terms on aliasing errors in spectral methods” Appl. Numer. Math, 20, 1 (1996).
J.A. Domaradzki, R.W. Metcalfe, R.S. Rogallo and J.J. Riley “Analysis of subgrid-scale eddy viscosity with use of results from direct numerical simulations,” Phy. Rev. Lett. 58(6), 547 (1987).
R.M. Kerr, J.A. Domaradzki, and G. Barbier “Small-scale properties of nonlinear interactions and subgrid-scale energy transfer in isotropic turbulence,” Phys. Fluids 8, 197 (1996).
S. Cerutti and C. Meneveau “Statistics of filtered velocity in grid and wake turbulence,” Phys. Fluids 12(5), 1143 (2000).
C. Meneveau and J. Katz “Conditional subgrid force and dissipation in locally isotropic and rapidly strained turbulence,” Phys. Fluids 11(8), 2317 (1999).
A.A. Aldama “Filtering Techniques for Turbulent Flow Simulation” In Lecture Notes in Engineering, 56, Springer-Verlag, New York/Berlin (1990).
J. Kim, P. Moin and R. Moser “Turbulence statistics in fully-developed channel flow at low Reynolds number” J. Fluid Mech., 177, 133 (1987).
S. Ghosal and P. Moin, “The Basic Equations for the Large Eddy Simulation of Turbulent Flows in Complex Geometry,” J. Comp. Phys. 118, 24 (1995).
H. van der Ven, “A Family of Large Eddy Simulation (LES) filters with nonuniform filter widths,” Phys. Fluids 7(5), 1171 (1995).
O.V. Vasilyev, T.S. Lund and P. Moin “A General Class of Commutative Filters for LES in Complex Geometries”, J. Comp. Phys. 146(1), 82 (1998).
T.S. Lund “Discrete filters for LES” CTR Annu. Res. Briefs pg.83 (1997).
H. Lomax, “Finite difference methods for fluid dynamics” Lecture notes (Stanford University).
W. Gear, Numerical initial value problems in ordinary differential equations, Prentice-Hall, New Jersey, (1971).
G. Helmberg Introduction to Spectral Theory in Hilbert Spaces North Holland, Amsterdam-London (1969).
R.V. Churchill Fourier Series and Boundary Value Problems McGraw-Hill, New York (1963).
S. Ghosal “An analysis of numerical errors in large eddy simulations of turbulence,” J. Comp. Phys. 125, 187 (1996).
W. Jones and N. March, Theoretical solid state physics, Vol.1: Perfect lattices in equilibrium, Wiley-Interscience, London, (1973).
M. Lesieur, Turbulence in fluids, Kluwer Academic Publishers, Dordrecht, The Netherlands, (1987).
A. Monin and A. Yaglom, Statistical Fluid Mechanics, Vol. 2 The MIT Press, Cambridge, (1979).
G.K. Batchelor, “Pressure fluctuations in isotropic turbulence”, Proc. Camb. Phil. Soc. 47, 359 (1951).
G. Batchelor, The theory of homogeneous turbulence, Cambridge Univ. Press, Cambridge, England, (1953).
M. Germano, U, Piomelli, P. Moin and W.H. Cabot “A dynamic subgrid-scale eddy viscosity model,” Phys.Fluids A 3, 1760 (1991).
S. Ghosal, T.S. Lund, P. Moin and K. Akselvoll “A dynamic localization model for the large eddy simulation of turbulent flow,” J. Fluid Mech. 286, 229 (1995).
C. Meneveau, T.S. Lund and W.H. Cabot “A lagrangian dynamic subgrid-scale model of turbulence,” J. Fluid Mech. 319, 353 (1996).
O.V. Vasilyev, T.S. Lund and P. Moin “A general class of commutative filters for LES in complex geometries,” J. Comp. Phys. 146(1), 82 (1998).
M. Lesieur and O. Metais “New trends in large-eddy simulations of turbulence” Annu. Rev. Fluid Mech. 28, 45 (1996).
U. Piomelli “Large-eddy simulation: achievements and challenges” Progress in Aerospace Sciences 35, 335 (1999).
B. Vreman, B. Geurts and H. Kuerten “Discretization error dominance over subgrid terms in large eddy simulation of compressible shear layers in 2D” Comm. in Num. methods Eng. 10, 785 (1994).
B. Vreman, B. Geurts and H. Kuerten “A priori tests of large eddy simulation of the compressible plane mixing layer” J. Eng. Math. 29, 299 (1995).
B. Vreman, B. Geurts and H. Kuerten “Comparison of numerical schemes in large eddy simulation of the temporal mixing layer” Int. J. Num. methods in Fluids 29, 299 (1996).
A.G. Kravchenko and P. Moin “On the effect of numerical errors in large eddy simulation of turbulent flows” J. Comp. Phys. 131, 310 (1997).
B. Kosović D.I. Pullin and R. Samtaney “Subgrid-scale modeling for Large-eddy simulations of compressible turbulence” (preprint) (2001).
D.I. Pullin (private communication).
T.J.R. Hughes, L. Mazzei and A.A. Oberai “The multiscale formulation of large eddy simulation: Decay of homogeneous isotropic turbulence” Phys. Fluids 13(2), 505 (2001).
T.J.R. Hughes, A.A. Oberai and L. Mazzei “Large eddy simulation of turbulent channel flows by the variational multiscale method” Phys. Fluids 13(6), 1784 (2001).
D. Drikakis, O.P. Iliev and D.P. Vassileva “A Nonlinear Multigrid Method for the Three-Dimensional Incompressible Navier-Stokes Equations” J. Comp. Phys. 146, 301 (1998).
J.A. Langford “Toward ideal large eddy simulation” Thesis, Univ. Illinois at Urbana Champaign (2000).
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Ghosal, S. (2002). Analysis and Control of Errors in the Numerical Simulation of Turbulence. In: Drikakis, D., Geurts, B. (eds) Turbulent Flow Computation. Fluid Mechanics and Its Applications, vol 66. Springer, Dordrecht. https://doi.org/10.1007/0-306-48421-8_4
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DOI: https://doi.org/10.1007/0-306-48421-8_4
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