Fourier time spectral method for subsonic and transonic flows

The time accuracy of the exponentially accurate Fourier time spectral method (TSM) is examined and compared with a conventional 2nd-order backward difference formula (BDF) method for periodic unsteady flows. In particular, detailed error analysis based on numerical computations is performed on the accuracy of resolving the local pressure coefficient and global integrated force coefficients for smooth subsonic and non-smooth transonic flows with moving shock waves on a pitching airfoil. For smooth subsonic flows, the Fourier TSM method offers a significant accuracy advantage over the BDF method for the prediction of both the local pressure coefficient and integrated force coefficients. For transonic flows where the motion of the discontinuous shock wave contributes significant higher-order harmonic contents to the local pressure fluctuations, a sufficient number of modes must be included before the Fourier TSM provides an advantage over the BDF method. The Fourier TSM, however, still offers better accuracy than the BDF method for integrated force coefficients even for transonic flows. A problem of non-symmetric solutions for symmetric periodic flows due to the use of odd numbers of intervals is uncovered and analyzed. A frequency-searching method is proposed for problems where the frequency is not known a priori. The method is tested on the vortex shedding problem of the flow over a circular cylinder.

[1]  高超,et al.  Numerical solution of the unsteady Euler equations for airfoils using approximate boundary conditions , 2003 .

[2]  Charles H. K. Williamson,et al.  Measurements of base pressure in the wake of a cylinder at low Reynolds numbers , 1990 .

[3]  Lian-Ping Wang,et al.  Effects of subgrid-scale modeling on time correlations in large eddy simulation , 2001 .

[4]  R. Henderson.  Details of the drag curve near the onset of vortex shedding , 1995 .

[5]  Edward F. Crawley,et al.  Calculation of unsteady flows in turbomachinery using the linearized Euler equations , 1989 .

[6]  Juan J. Alonso,et al.  Application of a Non-Linear Frequency Domain Solver to the Euler and Navier-Stokes Equations , 2002 .

[7]  Lian-Ping Wang,et al.  Large-eddy simulation of turbulent collision of heavy particles in isotropic turbulence , 2010 .

[8]  David M. Schuster,et al.  Calculation of Airfoil Flutter by an Euler Method with Approximate Boundary Conditions , 2003 .

[9]  Feng Liu,et al.  Investigation of mistuning effects on cascade flutter using a coupled method , 2002 .

[10]  A. Roshko.  On the development of turbulent wakes from vortex streets , 1953 .

[11]  Shanhong Ji,et al.  Unsteady flow calculations with a multigrid Navier-Stokes method , 1996 .

[12]  A. Jameson,et al.  Numerical solution of the Euler equations by finite volume methods using Runge Kutta time stepping schemes , 1981 .

[13]  A. Jameson,et al.  Three-dimensional unsteady multi-stage turbomachinery simulations using the harmonic balance technique , 2007 .

[14]  Antony Jameson,et al.  The computational efficiency of non-linear frequency domain methods , 2006, J. Comput. Phys..

[15]  A. Jameson,et al.  Application of the Time Spectral Method to Periodic Unsteady Vortex Shedding , 2006 .

[16]  Jeffrey P. Thomas,et al.  Computation of Unsteady Nonlinear Flows in Cascades Using a Harmonic Balance Technique , 2002 .

[17]  Qing Xiao,et al.  A numerical study of transonic buffet on a supercritical airfoil , 2004 .

[18]  Meng Wang,et al.  On the Computation of Space-Time Correlations by Large-Eddy Simulation , 2004 .

[19]  Li He,et al.  Computation of unsteady flows around oscillating blades using linear and nonlinear harmonic Euler methods , 1998 .

[20]  Derek B. Ingham,et al.  Laminar boundary layer on an impulsively started rotating sphere , 1979 .

[21]  Yue Yang,et al.  Effects of subgrid-scale modeling on Lagrangian statistics in large-eddy simulation , 2008 .

[22]  Earl H. Dowell,et al.  Computationally fast harmonic balance methods for unsteady aerodynamic predictions of helicopter rotors , 2008 .

[23]  Dimitri Papamoschou,et al.  Computation of High-Speed Coaxial Jets with Fan Flow Deflection , 2010 .

[24]  C. Williamson.  Defining a Universal and Continuous Strouhal-Reynolds Number Relationship for the Laminar Vortex She , 1988 .

[25]  Min-Soo Kim,et al.  Elliptic relaxation second moment closure for the turbulent heat fluxes , 2008 .

[26]  A. Jameson.  Time dependent calculations using multigrid, with applications to unsteady flows past airfoils and wings , 1991 .

[27]  Her Mann Tsai,et al.  Calculation of Wing Flutter by a Coupled Fluid-Structure Method , 2001 .

[28]  Juan J. Alonso,et al.  Development and Validation of a Massively Parallel Flow Solver for Turbomachinery Flows , 2001 .