Time-averaged equations

When the continmty equation and the Navier-Stokes equations for incompressible flow are time averaged, equations for the time-averaged velocities and pressures are obtained which appear identical to the original equations (6-18 through 6-28), except for the appearance of additional terms in the Navier-Stokes equations. Called Reynolds stress terms, they result from the nonlinear effects of momentum transport by the velocity fluctuations. In each i-component (i = X, y, z) Navier-Stokes equation, the following additional terms appear on the right-hand side ... 

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-... 

When the continuity equation and the Navier-Stokes equations for incompressible flow are time averaged, equations for the time-averaged... 

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 volume-time-averaged equations can be summarized as follows ... 

For convenience, we remove the () signs and time-averaging bars from the preceding equations. Thus, the volume-time-averaged equations can be simplified to the form... 

The time averaging equations are straightforward and will not be shown here. 

The use of CFD models for gas-liquid bubble columns has also raised considerable interest only Euler-Euler and Euler-Lagrange frameworks have been employed for the description of the gas and liquid phase states [3.38-3.42]. Bubble trays, considered as particular kinds of bubble columns, have lately presented enormous interest for the flow description by CFD. The flow patterns on a sieve tray have been analyzed in the liquid phase, solving the time-averaged equations of continuity and momentum [3.43]. 

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. 

In order to carry out computations with the time averaged equations on the form (3.227), we need to relate the average of products to products of averages and derive constitutive equations for the interfacial coupling terms. 

As in the volume averaged equations discussed in sect 3.4.1, three undetermined terms can be identified in the time averaged equations (3.259). The first term, —V (/3fc(pfcv V )rfc) denotes the covariance or correlation terms. The second term, Tkikj, accounts for the effects of interfacial stress. 

To save space the governing time averaged equations for each of the primary variables are not listed here as their mathematical form coincides with the volume averaged model formulation given in sect 3.4.1. Nevertheless, it is important to note that the physical interpretations of the mean quantities and the temporal covariance terms differ from their spatial counterparts. Furthermore, the conventional constitutive equations for the unknown terms are discussed in chap 5, and the same modeling closures are normally adopted for any model formulation even though their physical interpretation differ. 

Delhaye JM (1977) Local time-averaged equations. In Kakac S, Mayinger F Two-Phase Flows and Heat Transfer. Vol. 1, pp. 91-100, Hemisphere, Washington, DC... 

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. 

To correct laboratory P-wave velocity measurements to in situ conditions a modified time-average equation (Wyllie et al. 1956) can be used (Shipboard Scientific Party 1995)... 

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. 

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