Navier-Stokes equations time-averaged

Theoretical approaches to the horizontal-plane hydraulic problem are often based upon momentum equations derived from the Navier—Stokes equations being averaged over the flow depth. This approach was developed by Rodi, Emtsev, Sherenkov, Beffa, and other authors [48, 171, 540], In the case where the vegetation is present, the resulting two-dimensional shallow-water equations for a time-dependent flow read as follows [540] ... 

Eq. (25.3) and the subsequent time averaging yield the Navier-Stokes equations for averaged flow variables [Reynolds-averaged Navier-Stokes (RANS) equations] ... 

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 ... 

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 ... 

TURBULENCE ON TIME-AVERAGED NAVIER-STOKES 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. 

The incompressible, time-averaged continuity and the Navier-Stokes equations can be written as... 

On the continuum level of gas flow, the Navier-Stokes equation forms the basic mathematical model, in which dependent variables are macroscopic properties such as the velocity, density, pressure, and temperature in spatial and time spaces instead of nf in the multi-dimensional phase space formed by the combination of physical space and velocity space in the microscopic model. As long as there are a sufficient number of gas molecules within the smallest significant volume of a flow, the macroscopic properties are equivalent to the average values of the appropriate molecular quantities at any location in a flow, and the Navier-Stokes equation is valid. However, when gradients of the macroscopic properties become so steep that their scale length is of the same order as the mean free path of gas molecules,, the Navier-Stokes model fails because conservation equations do not form a closed set in such situations. 

Since MPC dynamics yields the hydrodynamic equations on long distance and time scales, it provides a mesoscopic simulation algorithm for investigation of fluid flow that complements other mesoscopic methods. Since it is a particle-based scheme it incorporates fluctuations, which are essential in many applications. For macroscopic fluid flow averaging is required to obtain the deterministic flow fields. In spite of the additional averaging that is required the method has the advantage that it is numerically stable, does not suffer from lattice artifacts in the structure of the Navier-Stokes equations, and boundary conditions are easily implemented. 

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

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... 

Consider, next, the x-wise Navier-Stokes equation, i.e., Eq. (2.88). Substituting the values of the variables as given by Eq. (2.92) into this equation and taking the time average of each of the terms gives, after multiplying the terms out ... 

Compared to the x-wise Navier-Stokes equation for two-dimensional laminar flow, the equivalent equation for turbulent flow has, when the time averaged values of the... 

Consider a single-phase homogeneous stirred-tank reactor with a time-invariant velocity field U(x, y, z ) a single reaction of the form /) —> B. (This approach can be extended to the case of time-dependent velocity fields. If the flow in the tank is turbulent, then the velocity field is the solution of the Reynolds averaged Navier-Stokes equations). The tank is divided into a three-dimensional network of n spatially fixed volumetric elements, or n-interacting... 

The equation of motion for the turbulent flow of an incompressible fluid is obtained from the Navier-Stokes equations by replacing the instantaneous values of each point quantity by the sum of the average and its fluctuating component, and time averaging. This results in the Reynolds equations for incompressible turbulent motion in which there are more unknowns than available equations. Therefore additional relations are needed to solve the equations. 

Turbulence is the most complicated kind of fluid motion. There have been several different attempts to understand turbulence and different approaches taken to develop predictive models for turbulent flows. In this chapter, a brief description of some of the concepts relevant to understand turbulence, and a brief overview of different modeling approaches to simulating turbulent flow processes is given. Turbulence models based on time-averaged Navier-Stokes equations, which are the most relevant for chemical reactor engineers, at least for the foreseeable future, are then discussed in detail. The scope of discussion is restricted to single-phase turbulent flows (of Newtonian fluids) without chemical reactions. Modeling of turbulent multiphase flows and turbulent reactive flows are discussed in Chapters 4 and 5 respectively. 

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