### 2.1.1. Statistical Turbulence Models and the Closure Problem

When looking at time scales much larger than the time scales of turbulent fluctuations, turbulent flow could be said to exhibit average characteristics, with an additional time-varying, fluctuating component. For example, a velocity component may be divided into an average component, and a time varying component.

In general, turbulence models seek to modify the original unsteady Navier-Stokes equations by the introduction of averaged and fluctuating quantities to produce the Reynolds Averaged Navier-Stokes (RANS) equations. These equations represent the mean flow quantities only, while modeling turbulence effects without a need for the resolution of the turbulent fluctuations. All scales of the turbulence field are being modeled. Turbulence models based on the RANS equations are known as Statistical Turbulence Models due to the statistical averaging procedure employed to obtain the equations.

Simulation of the RANS equations greatly reduces the computational effort compared to a Direct Numerical Simulation and is generally adopted for practical engineering calculations. However, the averaging procedure introduces additional unknown terms containing products of the fluctuating quantities, which act like additional stresses in the fluid. These terms, called ‘turbulent’ or ‘Reynolds’ stresses, are difficult to determine directly and so become further unknowns.

The Reynolds (turbulent) stresses need to be modeled by additional equations of known quantities in order to achieve “closure.” Closure implies that there is a sufficient number of equations for all the unknowns, including the Reynolds-Stress tensor resulting from the averaging procedure. The equations used to close the system define the type of turbulence model.

#### 2.1.1.1. Reynolds Averaged Navier-Stokes (RANS) Equations

As described above, turbulence models seek to solve a modified set of transport equations by introducing averaged and fluctuating components. For example, a velocity may be divided into an average component, , and a time varying component, .

| (2–1)

The averaged component is given by:

| (2–2)

where is a time scale that is large relative to the turbulent fluctuations, but small relative to the time scale to which the equations are solved. For compressible flows, the averaging is actually weighted by density (Favre-averaging), but for simplicity, the following presentation assumes that density fluctuations are negligible.

For transient flows, the equations are ensemble-averaged. This allows the averaged equations to be solved for transient simulations as well. The resulting equations are sometimes called URANS (Unsteady Reynolds Averaged Navier-Stokes equations).

Substituting the averaged quantities into the original transport equations results in the Reynolds averaged equations given below. For details, see Transport Equations. In the following equations, the bar is dropped for averaged quantities, except for products of fluctuating quantities.

| (2–3)

| (2–4)

where is the molecular stress tensor (including both normal and shear components of the stress).

The continuity equation has not been altered but the momentum and scalar transport equations contain turbulent flux terms additional to the molecular diffusive fluxes. These are the Reynolds stresses, . These terms arise from the nonlinear convective term in the un-averaged equations. They reflect the fact that convective transport due to turbulent velocity fluctuations will act to enhance mixing over and above that caused by thermal fluctuations at the molecular level. At high Reynolds numbers, turbulent velocity fluctuations occur over a length scale much larger than the mean free path of thermal fluctuations, so that the turbulent fluxes are much larger than the molecular fluxes.

The Reynolds averaged energy equation is:

| (2–5)

This equation contains an additional turbulence flux term, compared with the instantaneous equation. For details on this, see Equation 1–86. The term in the equation is the viscous work term that can be included by enabling `Viscous Work` in CFX-Pre.

The mean Total Enthalpy is given by:

| (2–6)

Note that the Total Enthalpy contains a contribution from the turbulent kinetic energy, k, given by:

| (2–7)

Similarly, the Additional Variable may be divided into an average component, , and a time varying component, . After dropping the bar for averaged quantities, except for products of fluctuating quantities, the Additional Variable equation becomes

| (2–8)

where is the Reynolds flux.

Turbulence models close the Reynolds averaged equations by providing models for the computation of the Reynolds stresses and Reynolds fluxes. CFX models can be broadly divided into two classes: eddy viscosity models and Reynolds stress models.

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