Novel WKB Solutions to the Non-Isentropic Helmholtz Equation in a Non-Uniform Duct with Mean Temperature Gradient and Mean Flow

We derive the generalized Helmholtz equation (GHE) governing non-isentropic acoustic fluctuations in a quasi 1-D duct with non-uniform cross-section, mean temperature gradient and non-uniform mean flow. Non-isentropic effects are included via heat conduction terms in the mean and fluctuating energy equations. To derive the Helmholtz equation exclusively in terms of the fluctuating pressure field, a relationship between density and pressure fluctuations is needed, which is shown to be a second-order differential equation for nonisentropic motions. Novel analytical solutions that are accurate for both low/high frequencies and small/large mean gradients are presented for the GHE based on the Wentzel–Kramers–Brillouin (WKB) method. WKB solutions are developed using the ansatz that the pressure fluctuation field has a travelling wave form, ˆp(x) = exp [∫0x (a + ib) dx], where x is the axial coordinate. Substituting this form into the Helmholtz equation yields coupled, nonlinear ordinary differential equations (ODEs) for a and b. Analytical solutions to the ODEs are obtained using the approximations of high frequency and slowly varying mean properties. This simplification allows us to obtain the lower order solutions b0 and a0. We then enhance solution accuracy by using a0 to solve for b1 without any approximations. Finally, b1 is employed to get a1, giving us the higher order solution. The ˆp calculated from (a1, b1) is in good to excellent agreement with numerical solution of the GHE for both low and high frequencies and for a range of mean Mach numbers, including M ≥ 1.