A Mapped Chebyshev Pseudospectral Method for Unsteady Flow Analysis

Abstract

A mapped Chebyshev pseudospectral method is developed as an accurate and yet efficient approach to solve unsteady flows. Preserving the conservation laws, the method discretizes a spatial derivative term implicitly while a time derivative term is treated explicitly using the mapped Chebyshev collocation operator. As standard Chebyshev points make the corresponding spectral derivative matrix ill-conditioned due to uneven distribution of the Chebyshev points clustered heavily towards the ends of the interval, an inverse sine mapping function is applied to the Chebyshev collocation operator so that the point distribution becomes more uniform in a time domains and mitigates numerical instabilities correspondingly. Computations of unsteady flows, including onedimensional Burgers' equation and two-dimensional airfoil flows under oscillation and plunging motions, are carried out. Numerical results of the present study are compared with those of the conventional time-marching solution method and the harmonic balance method for each application, and demonstrate high accuracy at much reduced computational cost. The method shows great potential to solve general unsteady flows involving both periodic and non-periodic motions. Nomenclature A = amplitude of oscillation a = Chebyshev coefficient b = Chebyshev coefficient for first differential D = matrix in harmonic balance equation D ch = Chebyshev collocation matrix D T = Chebyshev coefficient matrix D T = Chebyshev coefficient matrix for first differential F = Fourier transform matix G, H = fluxes g = a mapping funtion g' = derivative of a mapping function k c = reduced frequency M = matrix in frequency domain equation M ch = number of reconstruction points by Chebyshev M ∞ = freestream Mach number N hs = number of harmonics p, q = constant of polynomial mapping function Q = conservative variables R = residual T = Chebyshev polynomial T max = time interval t = time V, W = flux Jacobian matrixs in mapped Chebyshev method y = variable of a mapping function α, β = constant of inverse sine mapping function δ, ε = constant of hyperbolic sine mapping function φ = frequency ρ, σ = constant of tangent mapping function

Helicopter Rotor Flow Analysis Using Mapped Chebyshev Pseudospectral Method and Overset Mesh Topology

“…Moreover, it is reported in the reference paper [23] that unsteady flows around an airfoil plunging at constant speed with nonzero freestream velocity are equivalent to steady flows of an stationary airfoil fixed at a certain angle of attack with the same freestream velocity, which is explained graphically in Figures 6(a) and 6(b). An overset mesh topology for the plunging airfoil is constructed to facilitate the dynamic motion of the airfoil with a near-body grid of 257 × 45 and a background grid of 301 × 151 as shown in Figure 7.…”

“…* is the nondimensional time and Reynolds number based on the chord length is 5.5 × 10 6 . For the pseudo time integration, the mapped Chebyshev spectral derivative term, a DADI (diagonal alternate direction implicit), method is applied [15,23,31]. A Roe's upwind flux-difference splitting (FDS) method [32] is employed for the flux term discretization with a third order MUSCL scheme to improve solution accuracy.…”

“…Khater et al [21] applied the method to the spatial derivative term of the flow governing equations of the Burgers equations in various forms and excellent agreements were observed in comparison to the exact solutions. Im et al [23] applied a Chebyshev collocation operator to the time derivative term of the URANS equations and employed a conformal mapping function to improve the numerical stability of the method. They applied the method to the oscillating airfoil with a prespecified frequency and compared the computation results with those of the dual-time stepping method and the harmonic balance method.…”

Prediction of Dynamic Stability Using Mapped Chebyshev Pseudospectral Method

“…The computational expenses increase rapidly with the number of harmonics. Hall et al [1] suggested an alternative way to resolve such problems by casting inverse Fourier transformation to (10) as shown in…”

“…Im et al [10] solved two-dimensional airfoil flows under forced oscillation by the Euler and Navier-Stokes solvers with the mapped Chebyshev pseudospectral method. The results obtained very similar characteristics compared to those of the Fourier pseudospectral method in terms of both solution accuracy and computational efficiency.…”