Testing a random phase approximation for bounded turbulent flow

Tractable implementation of a spectral closure requires that the modal representation of the energy satisfy a restricted random phase approximation (RRPA). This condition is exactly satisfied when the statistical system is homogeneous and the basis functions are Fourier modes. In this case, the ensemble average of the spectral covariance diagonalizes, i.e., =delta(k(1)+k(2)), where c(k,t) is a Fourier coefficient in a Galerkin representation of the velocity field. However, for inhomogeneous statistical systems in which the Fourier system is inappropriate, the RRPA requires validation. We use direct numerical simulations (DNSs) of the Navier-Stokes and truncated Euler equations to test the degree to which the RRPA is satisfied when applied to a recent representation due to Turner (LANL Unclassified Report No. LA-UR-96-3257) of a bounded turbulent rectangular channel flow with free slip, stress free walls. It is shown that a complete test of the RRPA for a fully inhomogeneous DNS with N3 grid points actually requires N3+1 members in the ensemble. The "randomness" of the phase can be characterized by a probability density function (PDF) of the modulus of the normalized spectral covariance. Results reveal that for both the Navier-Stokes and Euler systems the PDF does not change in time as the turbulence decays, and that the PDF for the Euler system is virtually identical to the one produced from an ensemble of random fields. This result is consistent with the equipartition of energy for the Euler system, in which the RRPA becomes an exact result rather than an approximation as the number of realizations approaches N3+1. The slight differences observed between the PDF produced from the random fields and the one from the Navier-Stokes system are thus shown to be entirely a result of the presence of a finite viscosity. It is also shown that there is great variation between statistics computed over the ensemble and those for a single realization.