Construction of transparent boundary conditions for aeroacoustics with a non-uniform main flow

A method for calculating the operator of transparent boundary conditions (TBC) in the outlet section of a wind tunnel under the assumption that the downstream pressure is described by a linear scalar equation with the varying over the section sound speed $c(y)$ and flow velocity $u(y)$ is proposed. 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 (QTBC) for the anisotropic elasticity equations with variable coefficients. To obtain a close-to-best rational approximation of the elements of the Poincaré-Steklov matrix operator arising in the QTBC construction algorithm, a method of optimal choice of algorithm parameters is proposed, exponential convergence of the approximation with respect to the number of poles for the functions arising in the problem is shown, an example of constructing a TBC for a specific type of dependencies $c(y)$ and $u(y)$ is considered. The same TBCs can also be used for a streamtube containing a streamlined object.