Turbulence modeling approach
The Reynolds-averaged Navier–Stokes equations (RANS equations) are time-averaged
equations of motion for fluid flow. The idea behind the equations is Reynolds decomposition, whereby an instantaneous quantity is decomposed into its time-averaged and fluctuating quantities, an idea first proposed by Osborne Reynolds. The RANS equations are primarily used to describe turbulent flows. These equations can be used with approximations based on knowledge of the properties of flow turbulence to give approximate time-averaged solutions to the Navier–Stokes equations.
For a stationary flow of an incompressible Newtonian fluid, these equations can be written in Einstein notation in Cartesian coordinates as:
formula_1
The left hand side of this equation represents the change in mean momentum of a fluid element owing to the unsteadiness in the mean flow and the convection by the mean flow. This change is balanced by the mean body force, the isotropic stress owing to the mean pressure field, the viscous stresses, and apparent stress formula_2 owing to the fluctuating velocity field, generally referred to as the Reynolds stress. This nonlinear Reynolds stress term requires additional modeling to close the RANS equation for solving, and has led to the creation of many different turbulence models. The time-average operator formula_3 is a Reynolds operator.
Derivation of RANS equations.
The basic tool required for the derivation of the RANS equations from the instantaneous Navier–Stokes equations is the Reynolds decomposition. Reynolds decomposition refers to separation of the flow variable (like velocity formula_4) into the mean (time-averaged) component (formula_5) and the fluctuating component (formula_6). Because the mean operator is a Reynolds operator, it has a set of properties. One of these properties is that the mean of the fluctuating quantity is equal to zero formula_7. Thus,
formula_8 where formula_9 is the position vector. Some authors prefer using formula_10 instead of formula_11 for the mean term (since an overbar is sometimes used to represent a vector). In this case, the fluctuating term formula_12 is represented instead by formula_4. This is possible because the two terms do not appear simultaneously in the same equation. To avoid confusion, the notation formula_14, formula_15, and formula_16 will be used to represent the instantaneous, mean, and fluctuating terms, respectively.
The properties of Reynolds operators are useful in the derivation of the RANS equations. Using these properties, the Navier–Stokes equations of motion, expressed in tensor notation, are (for an incompressible Newtonian fluid):
formula_17
formula_18
where formula_19 is a vector representing external forces.
Next, each instantaneous quantity can be split into time-averaged and fluctuating components, and the resulting equation time-averaged,
 = 0 ;/math ;
formula_20
Note that the nonlinear terms (like formula_21) can be simplified to
formula_22
to yield:
formula_23
formula_24
The momentum equation can also be written as,
formula_25
On further manipulations this yields,
formula_26
where, formula_27 is the mean rate of strain tensor.
Finally, since integration in time removes the time dependence of the resultant terms, the time derivative must be eliminated, leaving:
formula_28
Equations of Reynolds stress.
The time evolution equation of Reynolds stress is given by:
formula_29
This equation is very complicated. If formula_30 is traced, turbulence kinetic energy is obtained.
The last term formula_31 is turbulent dissipation rate. All RANS models are based on the above equation.
References.
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