Nonlocal form of the rapid pressure-strain correlation in turbulent flows

A new fundamentally based formulation of nonlocal effects in the rapid pressure-strain correlation in turbulent flows has been obtained. The resulting explicit form for the rapid pressure-strain correlation accounts for nonlocal effects produced by spatial variations in the mean-flow velocity gradients and is derived through Taylor expansion of the mean velocity gradients appearing in the exact integral relation for the rapid pressure-strain correlation. The integrals in the resulting series expansion are solved for high- and low-Reynolds number forms of the longitudinal correlation function $f\left(r\right)$, and the resulting nonlocal rapid pressure-strain correlation is expressed as an infinite series in terms of Laplacians of the mean strain rate tensor. This formulation is used to obtain a nonlocal transport equation for the turbulence anisotropy that is expected to provide improved predictions of the anisotropy in strongly inhomogeneous flows.

Figure 1

(Color online) (a) Comparison of exponential $f\left(r\right)$ in Eq. (39) with experimental data from axisymmetric turbulent jet [14] and (b) planar turbulent mixing layer [15] and with (c) direct numerical simulation (DNS) data from turbulent channel flow at ${\text{Re}}_{\tau }=650$ [12].

Figure 2

(Color online) Comparison of inertial-range and exponential forms for $f\left(r/\Lambda \right)$ in Eqs. (38, 39), respectively. Note that $C_{\lambda }=0.23$ in Eq. (40) gives reasonable agreement between the two forms in the inertial-range ${\text{Re}}_{\Lambda }^{-3/4}⪡\left(r/\Lambda \right)⪡1$.

Figure 3

(Color online) Comparison of rapid pressure-strain coefficients $C_2^{\left(n\right)}$ from Eq. (57) for the ${\text{Re}}_{\Lambda }⪢1$ exponential $f\left(r\right)$ in Eq. (39) and from Eq. (58) for the ${\text{Re}}_{\Lambda }\to 0$ Gaussian $f\left(r\right)$ in Eq. (41).