Reynolds stresses velocity-pressure-gradient term

The closed PDF transport equation given above can be employed to derive a transport equation for the Reynolds stresses. The velocity-pressure gradient and the dissipation terms in the corresponding Reynolds-stress model result from... 

The physical interpretation of the terms in the equation is not necessary obvious. The first term on the LHS denotes the rate of accumulation of the kinematic turbulent momentum flux within the control volume. The second term on the LHS denotes the advection of the kinematic turbulent momentum flux by the mean velocity. In other words, the left hand side of the equation constitutes the substantial time derivative of the Reynolds stress tensor v v. The first and second terms on the RHS denote the production of the kinematic turbulent momentum flux by the mean velocity shears. The third term on the RHS denotes the transport of the kinematic momentum flux by turbulent motions (turbulent diffusion). This latter term is unknown and constitutes the well known moment closure problem in turbulence modeling. The fourth and fifth terms on the RHS denote the turbulent transport by the velocity-pressure-gradient correlation terms (pressure diffusion). The sixth term on the RHS denotes the redistribution by the return to isotropy term. In the engineering literature this term is called the pressure-strain correlation, but is nevertheless characterized by its redistributive nature (e.g., [132]). The seventh term on the RHS denotes the molecular diffusion of the turbulent momentum flux. The eighth term on the RHS denotes the viscous dissipation term. This term is often abbreviated by the symbol... 

At high Reynolds numbers, the fluctuating pressure field term dominates the fluctuating shear stress term in (12.4.2-6). It results from interactions between the turbulence and the mean velocity gradients, the so-called rapid-pressure contribution, and from the turbulence proper, the so-called slow-pressure contribution. To simulate the rapid-pressure related process. Pope [1985] proposed a linear deterministic model ... 

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