Statistical properties of the wall pressure spectrum: correlations and wavenumber-frequency spectra
According to the ‘weak coupling approximation’ introduced above, in the present approach we consider a boundary layer developing on an infinitely extended rigid flat plate in a low Mach number flow without mean pressure gradients. In this framework, taking into account that the boundary layer thickness increases slowly in the streamwise direction, it is possible to consider the pressure field statistically homogeneous on the plane of the plate and statistically stationary in time. The homogeneous plane is described by the Cartesian axes that, for the sake of clarity, are defined as x, x2, being x aligned with the free stream velocity. The frame of reference adopted is depicted in Figure 4.
Figure 4. Frame of reference adopted to describe the statistics of pressure fluctuations.
Considering the fluctuating component of the pressure field p(x1,x2,t), the space time correlation can be written as:
Rpp(€i, b ,т) = -1 E[p(xi, x2,t)p(xi + £bx2 + 6 ,t + t )] (40)
where ap is the pressure variance and the symbol E[•] denotes the expected value. When the ergodic hypothesis holds, time averages can be used. This is an important hypothesis when pointwise pressure measurements are performed. In this case the pressure is a function of time only and the crosscorrelation is given by a much simpler expression:
Rpp(t) = -1 <p(t)p(t + t) >t (41)
where the symbol < • >t now denotes the time average. Taking the Fourier transform of Eqs. 40 and 41 one obtain the wavenumber-frequency spectrum Фр(к1,к2,ш) and the frequency spectrum Фp(ш). In this notation ш is the radian frequency and k, k2 are the components of a two dimensional wavevector. By taking the frequency Fourier transform of Eq. 40 it is possible to obtain the cross-spectrum rp(£i,£2,w) that is defined in the space-frequency domain. The experimental determination not being very difficult, Гр represents a key ingredient for the theoretical models that are presented below.
In the framework of the statistical modeling, a relevant role is played by the phase velocity w/k, being k the magnitude of the wavevector, whose magnitude spans from the order of the flow speed to sonic or supersonic values.