Incompressible turbulence and minimum Fisher information: Correlations between velocity and pressure

Abstract

The principle of minimum Fisher information (MFI) and the theory of random Gaussian fields are used to work out the joint distribution function of the density and velocity in homogeneous, isotropic, stationary, nearly incompressible turbulence, in the case where the velocity and pressure are correlated. The appropriate Fisher variables seem to be the mass flux, the density, and the generalized heat function (enthalpy) or pressure head. It is shown that simple constraints on the minimization may be chosen to give a good fit to the pressure distribution function found in recent direct numerical simulations and experiments, where the PDF is exponential for negative p and roughly exp[−(p/p0)3/2]p−1/2 for positive p. In this case, the fit is an improvement on a past MFI calculation, in which the correlations between p and u were not accounted for. In addition, the form of the conditional average 〈p|u〉 as found from direct numerical simulations is taken into consideration. The theory of random Gaussian velocity fields predicts 〈p|u〉=〈p|0〉−βu2, where u2≡u⋅u and β⩽1/8 is a constant. In conjunction with this theory, MFI predicts a specific dependence of the conditional average 〈ux2|p〉 on p, where ux is a typical velocity component. The conditional PDF P(ux|p) is slightly non-Gaussian, but P(ux) is Gaussian. The relation 2〈u2δp〉2=〈u2〉[〈u2(δp)2〉−〈u2〉〈(δp)2〉] is predicted.