Construction of Transparent Boundary Conditions for Aeroacoustics with a Nonuniform Main Flow

A method is proposed for calculating the operator of transparent boundary conditions (TBCs) in the outlet section of a wind tunnel under the assumption that the downstream pressure is described by a linear scalar equation with the sound speed c(y) and flow velocity u(y) varying over the section. The operator generalizes the known boundary conditions for constant values of c and u by using the technology of constructing the previously proposed quasi-analytical transparent boundary conditions (QTBCs) for the anisotropic elasticity equations with variable coefficients. To obtain a close-to-best rational approximation of the elements of the Poincaré–Steklov (P–S) matrix operator arising in the QTBC construction algorithm, a method is proposed for optimally choosing the parameters of the Frobenius–Padé method, and exponential convergence of the approximation with respect to the number of poles for the functions arising in the problem is shown on an example of constructing a TBC for a specific type of dependence of c(y) and u(y). The same TBCs can also be used for a streamtube containing a streamlined object.