of products of fluctuating quantities will not in general vanish. After averaging, the continuity and momentum equations become and Using the product rule on one of the terms of the left hand side, it is revealed that where the last term on the right hand side vanishes as a result of the averaged continuity equation. The averaged momentum equation now becomes, after a rearrangement: where the Reynolds stresses, , are collected with the viscous normal and shear stress terms, . Discussion The time evolution equation of Reynolds stress was first given by Eq.(1.6) in Zhou Peiyuan's paper. The equation in modern form is where is the kinematic viscosity, and the last term is turbulent dissipation rate. This equation is very complex. If is traced, turbulence kinetic energy is obtained. The pressure-scrambling term is so called because this term (also called the pressure-strain covariance) is traceless under the assumption of incompressibility, meaning it cannot create or destroy turbulence kinetic energy but can only mix it between the three components of velocity. Depending on the application, this equation can also include buoyant production terms (proportional to the gravitational acceleration ) and Coriolis production terms (proportional to the Earth's rotation rate); these would be present in atmospheric applications, for example. The question then is, what is the value of the Reynolds stress? This has been the subject of intense modeling and interest, for roughly the past century. The problem is recognized as a closure problem, akin to the problem of closure in the BBGKY hierarchy. A transport equation for the Reynolds stress may be found by taking the outer product of the fluid equations for the fluctuating velocity, with itself. One finds that the transport equation for the Reynolds stress includes terms with higher-order correlations (specifically, the triple correlation ) as well as correlations with pressure fluctuations (i.e. momentum carried by sound waves). A common solution is to model these terms by simple ad hoc'' prescriptions. The theory of the Reynolds stress is quite analogous to the kinetic theory of gases, and indeed the stress tensor in a fluid at a point may be seen to be the ensemble average of the stress due to the thermal velocities of molecules at a given point in a fluid. Thus, by analogy, the Reynolds stress is sometimes thought of as consisting of an isotropic pressure part, termed the turbulent pressure, and an off-diagonal part which