Turbulence time-averaged equations

Although direct numerical simulations under limited circumstances have been carried out to determine (unaveraged) fluctuating velocity fields, in general the solution of the equations of motion for turbulent flow is based on the time-averaged equations. This requires semi-... 

The most widely adopted method for the turbulent flow analysis is based on time-averaged equations using the Reynolds decomposition concept. In the following, we discuss the Reynolds decomposition and time-averaging method. There are other methods such as direct numerical simulation (DNS), large-eddy simulation (LES), and discrete-vortex simulation (DVS) that are being developed and are not included here. 

The key aspect, then, in numerical simulation of the atmoLj.)heric boundary layer is the evaluation of the turbulent momentum fluxes in the time-averaged equations of motion (24). Considering this, we review briefly some of the more promising techniques that have been used to determine these fluxes. Our objective is not to give a full review, but rather to introduce the types of approaches which in the future may permit the solution of (23) and (24) and thus the prediction of urban wind fields. 

Sha WT, Slattery JC (1980) Local Volume-Time Averaged Equations of Motion for Dispersed, Turbulent, Multiphase Flows. NUREG/CR-1491, ANL-80-51... 

The Reynolds averaging approach has thus been found to represent a tradeoff between accuracy and computational costs. Using the Reynolds averaging concept turbulence is interpreted as a waveform and described by the time averaged equations of motion and a turbulence closure. Two-equation turbulence models like the standard k-e and k-cu models are common but full Reynolds stress models have also been applied in rare cases. 

Although less well defined than turbulence of the continuous phase, the dispersed, particle phase also experiences turbulence. Turbulence in the solids is usually either treated with a simple RANS model, such as k-e, or ignored. There are other considerations, such as the influence turbulence in one phase has on turbulence in the other phase. These effects are not captured in the momentum transfer terms contained in the time averaged equations and must be separately included. 

Let us consider the steady, two-dimensional turbulent flow of air in the surface layer parallel to the ground at = 0. We assume that i = i U3) and 2 = 0. Our object is to determine U ix2) when the vertical temperature profile is adiabatic. Since in this case 9 = 0 we need consider only the x component of the time-averaged equation of motion. 

The time averaged equations of mass, morrrentum, energy, and species conservation can be written in dimensionless form for a fluid in turbulent flow past a surface. If (1) radiant energy and chemical reaction are not present, (2) viscous dissipation is negligible, (3) physical properties are independent of temperature and composition, (4) the effect of mass transfer on velocity profiles is neglected, and (5) the boundary conditions are compatible, then dimensionless local heat and mass transfer coefficients can be shown to be described by equations of the form ... 

As of this writing, the only practical approach to solving turbulent flow problems is to use statistically averaged equations governing mean flow quantities. These equations, which are usually referred to as the Reynolds equations of motion, are derived by Reynold s decomposition of the Navier-Stokes equations (18). The randomly changing variables are represented by a time mean and a fluctuating part ... 

TURBULENCE ON TIME-AVERAGED NAVIER-STOKES EQUATIONS ... 

These extra turbulent stresses are termed the Reynolds stresses. In turbulent flows, the normal stresses -pu, -pv, and -pw are always non-zero beeause they eontain squared veloeity fluetuations. The shear stresses -pu v, -pu w, -pv w and are assoeiated with eorrelations between different veloeity eomponents. If, for instanee, u and v were statistieally independent fluetuations, the time average of their produet u v would be zero. However, the turbulent stresses are also non-zero and are usually large eompared to the viseous stresses in a turbulent flow. Equations 10-22 to 10-24 are known as the Reynolds equations. 

For most applications, the engineer must instead resort to turbulence models along with time-averaged Navier-Stokes equations. Unformnately, most available turbulence models obscure physical phenomena that are present, such as eddies and high-vorticity regions. In some cases, this deficiency may partially offset the inherent attractiveness of CFD noted earlier. 

---