Interdisciplinary Applied Mathematics

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Mo    To,


Sutherland’s law can be approximated at high temperature values from the equation above and ш ^ 0.5, whereas for low temperature values ш ^ 1.


Interface and Boundary Conditions


Following the time-splitting algorithm of equation (14.1), we first consider the inviscid (Euler) equations in order to present a proper treatment of interface conditions. Specifically, the Euler equations are solved by a spectral (Gauss-Lobatto-Legendre) collocation formulation (Karniadakis and Sherwin, 1999). The flow domain is divided into elements, where collocation discretization is applied in each element locally. This procedure brings up the issue of elemental interface treatment. Since the Euler equations are hyperbolic, simple averaging at elemental interfaces is inappropriate. Instead, a characteristic treatment is necessary, as we explain next.


Specifically, the interface problem is solved in three main steps:


   linearization of the Euler equations,


   characteristic decomposition, and


•    characteristic treatment.


Let us consider the following one-dimensional system of nonlinear hyperbolic partial differential equations in conservative form:


dW dF(W) dt    dx


We first define the Jacobian A by


d F dW







Here A(W0) is a constant N x N matrix (N is the dimension of the vector W). We then transform the linearized equations into the characteristic form




(14.8)

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