Reynolds stresses production term

In turbulent flow, there is direct viscous dissipation due to the mean flow this is given by the equivalent of equation 1.98 in terms of the mean values of the shear stress and the velocity gradient. Similarly, the Reynolds stresses do work but this represents the extraction of kinetic energy from the mean flow and its conversion into turbulent kinetic energy. Consequently this is known as the rate of turbulent energy production ... 

Thus, only the normal Reynolds stresses (i = j) are directly dissipated in a high-Reynolds-number turbulent flow. The shear stresses (i / j), on the other hand, are dissipated indirectly, i.e., the pressure-rate-of-strain tensor first transfers their energy to the normal stresses, where it can be dissipated directly. Without this redistribution of energy, the shear stresses would grow unbounded in a simple shear flow due to the unbalanced production term Vu given by (2.108). This fact is just one illustration of the key role played by 7 ., -in the Reynolds stress balance equation. 

Note that if the Reynolds stresses are known, then the production term Vf defined by... 

These extra turbulent stresses are termed the Reynolds stresses. In turbulent flows, the normal stresses -pu 2, -pv 2, and -pw 2 are always non-zero because they contain squared velocity fluctuations. The shear stresses -puV, -pu w, -pv w and are associated with correlations between different velocity components. If, for instance, u and v were statistically independent fluctuations, the time average of their product uV would be zero. However, the turbulent stresses are also non-zero and are usually large compared to the viscous stresses in a turbulent flow. Equations 10-22 to 10-24 are known as the Reynolds equations. 

The averaging process creates a new set of variables, the so-called Reynolds stresses, which are dependent on the averages of products of the velocity fluctuations UjUj (which for i = j simply represent the standard deviations of the velocity components). This creates a closure problem, which is one of the fundamental issues that has to be addressed in the modeling of turbulent flows. Importantly, Equation 3.2.12 also indicates that the Reynolds stress terms, which in line with Taylor s fundamental result should be related to the dispersion parameters, are coupled to the gradients of the mean flow velocity. 

The first-order closure models are all based on the Boussinesq hypothesis [19, 20] parameterizing the Reynolds stresses. Therefore, for fully developed turbulent bulk flow, i.e., flows far away from any solid boundaries, the turbulent kinetic energy production term is modeled based on the generalized eddy viscosity hypothesis , defined by (1.380). The modeled fc-equation is... 

In Eq. (1.6), the two terms on the left side represent, respectively, the increase in Reynolds stress with respect to time and coordinate (three-dimensional) the terms on the right side denote, respectively, the molecular diffusion, the turbulent diffusion, the stress production, the pressure-stain of the flow, and the dissipation of turbulent kinetic energy. 

The second term (molecular diffusion) and the third term (production of Reynolds stress) are remained unchanged. 

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... 

All of these relations contains terms involving statistical correlations among various products of fluctuating velocity, pressure, and stress terms. This renders them considerably more complex than their laminar flow counterparts. Reynolds succeeded in partially sol ving this dilemma by the expedient of introducing the turbulent stress tensor f, defined by... 

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