Spectral Vanishing Viscosity Method for Large-Eddy Simulation of Turbulent Flows

The spectral vanishing viscosity (SVV) method was introduced in the late 80’s [24] in order to solve, with the spectral Fourier method, 1D non-linear conservations laws (typically, the inviscid Burgers equation). The non-periodic case was investigated later [16], using a Legendre polynomial approximation. In both cases the basic idea is to introduce a SVV term VN , only active in the high frequency range, in order to stabilize the calculations while preserving the exponential rate of convergence of the discrete solution towards the fully converged spectral approximation of the exact one. Let us follow [16] and denote Λ= (−1,1), PN(Λ) the space of polynomials of maximum degree N defined on Λ, uN(x)∈PN(Λ) the polynomial approximation of the exact solution u(x) and {Lk}k 0 the set of Legendre polynomials. Then the SVV term is written as: VN = N∂x(QN(∂xuN)), where N is a O(N−1) coefficient and where QN is a “spectral viscosity operator” such that: ∀φ, φ=∑∞k=0 φ̂kLk, QNφ≡ ∑N k=0 Q̂kφ̂kLk, where