An Approximate Turbulent Pressure Fluctuation Frequency Spectra for a Finite Supersonic Plate

[Abstract] Here we derive approximate physics-based models for wall pressure fluctuation frequency spectra for compressible boundary layer flows. The formulation is based upon approximate analytical solutions to Lighthill’s acoustic equation where very simple representations of the mean-turbulent tensor source term have been applied. The solutions to Lighthill’s equation provide an estimate for the wall pressure fluctuation whereby a Fourier transform for the pressure fluctuation is obtained and used as a surrogate for the actual spectral density. The finite nature of the domain is reflected by our choice of sine or cosine transform. The resulting pressure fluctuation spectrum is explicitly a function of Mach number in terms of both magnitude and spectral curve shape. The resulting expressions exhibit reasonable agreement with Goody’s recommended 1 semi-empirical model in terms of spectral peak and high frequency behavior. The low frequency behavior follows a linear behavior as opposed to the “classical” quadratic rate but passes through the zero intercept. Since compressible low frequency pressure fluctuation behavior is not necessarily correctly described by zero power at zero frequency, we examine modifications of the simplified model which yield finite power at zero frequency. We derive this modification by ignoring the very near wall behavior of the source i.e. the no-slip condition where the Mach number must be low. The consequences of ignoring the very near wall (small scale) behavior will be to provide a poor representation of high frequency behavior, but since our concern in this development is focused on compressible flow and the outer region boundary layer region we will consider this approximation acceptable. A composite model is then derived that provides reasonable estimates for frequency spectrum for supersonic boundary layer flow and shows good agreement with relevant experiments, especially for low frequency. This type of approximate analytical formulation may be of value where more complete simulation based procedures are not appropriate.