These models are based on transport equations for all components of the Reynolds stress tensor and the dissipation rate. These models do not use the eddy viscosity hypothesis, but solve an equation for the transport of Reynolds stresses in the fluid. The Reynolds stress model transport equations are solved for the individual stress components.

Algebraic Reynolds stress models solve algebraic equations for the Reynolds stresses, whereas differential Reynolds stress models solve differential transport equations individually for each Reynolds stress component.

The exact production term and the inherent modeling of stress anisotropies theoretically make Reynolds stress models more suited to complex flows; however, practice shows that they are often not superior to two-equation models.

The Reynolds averaged momentum equations for the mean velocity are:

(2–86)

where is a modified pressure, is the sum of body forces and the fluctuating Reynolds stress contribution is . Unlike eddy viscosity models, the modified pressure has no turbulence contribution and is related to the static (thermodynamic) pressure by:

(2–87)

In the differential stress model, is made to satisfy a transport equation. A separate transport equation must be solved for each of the six Reynolds stress components of . The differential equation Reynolds stress transport is:

(2–88)

where and are shear and buoyancy turbulence production terms of the Reynolds stresses respectively, is the pressure-strain tensor, and C is a constant.

Buoyancy turbulence terms also take the buoyancy contribution in the pressure strain term into account and are controlled in the same way as for the and model. See the discussion below Equation 2–27.