Reynolds decomposition

The focus of RANS simulations is on the time-averaged flow behavior of turbulent flows. Yet, all turbulent eddies do contribute to redistributing momentum within the flow domain and by doing so make up the inherently transient character of a turbulent-flow field. In RANS, these effects of the full range of eddies are made visible via the so-called Reynolds decomposition of the NS equations (see, e.g., Tennekes and Lumley, 1972, or Rodi, 1984) of the flow variables into mean and fluctuating components. To this end, a clear distinction is required between the temporal and spatial scales of the mean flow on the one hand and those associated with the turbulent fluctuations on the other hand. 

Via this Reynolds decomposition and after subsequent averaging all terms of the NS equations, the so-called turbulent or Reynolds stresses upf emerge in the transport equations, where these stresses represent the additional averaged momentum transport due to the eddies. These stresses may be resolved explicitly from separate transport equations which in suffix notation (usual in the field of turbulence) look as follows ... 

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. 

In using the Reynolds decomposition, closure of the time-averaged Navier-Stokes equations cannot readily be realized because of the unknown correlation terms such as turbulent... 

It is assumed that the instantaneous Navier-Stokes equations for turbulent flows have the exact form of those for laminar flows. From the Reynolds decomposition, any instantaneous variable, (j>, can be divided into a time-averaged quantity and a fluctuating part as... 

The decomposition of the instantaneous velocity variable into its mean and fluctuation is thus referred to as Reynolds decomposition. This procedure... 

Apply the Reynolds decomposition procedure and expand the dependent variables within the instantaneous equations into mean and fluctuating parts. 

In order to treat the equation of motion in the same way, we apply the Reynolds decomposition procedure on the instantaneous velocity and pressure variables in (1.385) and average term by term. It can be shown by use of Leibnitz theorem that the operation of time averaging commutes with the operation of differentiating with respect to time when the limits of integration are constant [154, 106, 121, 15]). 

The pressure term is normally rewritten using the conventional Reynolds decomposition concept, thus ... 

To explain the basic problem we use the Reynolds decomposition and averaging procedure, as an example. Introducing the peculiar velocity for the dispersed phase (4.103) can be re-arranged as ... 

By performing Reynolds decomposition and time averaging the given reaction rate can be expressed as ... 

It is also possible to define a fluctuating component for each quantity as the difference between the instantaneous value and the Reynolds-average value Uf = Uf - (Uf) and = np. These quantities are used for Reynolds decomposition of higher-order terms (UfWf) = turbulent flux of the NDF. Physically, this term denotes the advection of fluid elements with a given n by fluid velocity fluctuations, and thus enters the transport equation as a spatial flux. The Reynolds-average NDF can be found from Eq. (2.44) ... 

A model for axial fluctuations stems directly from the equation of motion and can be developed in the usual way by introducing Reynolds decomposition (5). Physically, we visualize a process in which interfacial fluid undergoes a renewal process. Following a burst, a relatively quiescent period of weak turbulence occurs. The time between hursts is controlled by whatever (nonlinear) processes cause the ejection of fluid from the near wall region into the core... 

After applying the Favre decomposition to the components of the velocity vector and the Reynolds decomposition to the shear stress, one obtains... 

Ca and Cb are the concentrations of species A and B. Dab, the diffusion coefficient, and kf, the reaction rate, are constant. The system is assumed incompressible and isothermal, and the scalar field has no effect on the velocity field (variations in concentration, for example, do not induce velocity gradients). Reynolds decomposition is used to separate the velocity and concentration fields into an average and fluctuating part. When these terms are substituted into eq. (2-4), the resulting equation can be averaged to give... 

Reynolds decomposition and time averaging were then applied to the instantaneous variables in the volume average model equations. However, it was assumed that none of the densities fluctuate. The terms of fluctuating quantities with order higher than two were considered small compared to those of first and second order and thus neglected. 

One should also realize that the pressure field in the above expressions is just the result of the presence of the particle. This pressure (contribution) is different from, actually is superimposed upon, the background pressure field that drives the oncoming flow and which also exerts a force on the article. This was also appreciated by Tchen (1947). It is also clearly seen in the following 3D derivation due to Rietema and Van den Akker (1983)—see also Van den Akker (1986)— which builds on the paper by Anderson andjackson (1967) and resembles the Reynolds decomposition technique known from turbulence modeling. 

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