Hyperbolicity of pressure–velocity equations for computational hydro acoustics

A set of model equations is proposed to simulate waves generated by unsteady, low-speed, nearly incompressible air and water flows. The equations include the continuity and momentum equations with pressure and velocity as the unknowns. Compressibility effect associated with waves motion is directly tracked by time-accurate calculation of pressure fluctuations. The corresponding density changes are modeled by using the bulk modulus of the medium. The three-dimensional equations are shown to be hyperbolic by analyzing eigenvalues and eigenvectors of the composite Jacobian matrix of the equations. Specifically, the matrix is shown to be diagonalizable and have a real spectrum. Moreover, an analytical form of the Riemann invariants of the one-dimensional equations is derived. To validate the model equations, the space–time Conservation Element and Solution Element (CESE) method and the SOLVCON code are employed to solve the two-dimensional equations. Aeolian tones generated by air and water flows passing a cylinder and over an open cavity are simulated. Numerical results compare well with previously reported data.