Kinetic energy equations

Classical mechanics in volves studying motion (trajectories) on the potential surface where the classical kinetic energy equates to the temperature. The relationship is that the average kinetic energy of... 

The source terms on the right-hand sides of Eqs. (25)-(29) are defined as follows. In the momentum balance, g represents gravity and p is the modified pressure. The latter is found by forcing the mean velocity field to be solenoidal (V (U) = 0). In the turbulent-kinetic-energy equation (Eq. 26), Pk is the source term due to mean shear and the final term is dissipation. In the dissipation equation (Eq. 27), the source terms are closures developed on the basis of the form of the turbulent energy spectrum (Pope, 2000). Finally, the source terms... 

An attractor does not necessarily correspond to a confined state. By taking the limit where the external potential tends to zero, with zero -kinetic energy, equation (6) with Q replaced by has the limit of an interacting n free-electron state system. The system does not dissociate into fragments by manipulations of the PCB. The limit is one of the free electron quantum states. This is a positive... 

This equation represents the complex theoretical kinetic-energy equation of McMillen (M5) within 2.5% for all values of... 

In the many traditional methods of calculating turbulent flows, these turbulence terms are empirically defined, i.e., turbulence models that are almost entirely empirical are used. Some success has, however, been achieved by using additional differential equations to help in the description of these terms. Empiricism is not entirely eliminated, at present, by the use of these extra equations but the empiricism can be introduced in a more systematic and logical manner than is possible if the turbulence terms in the momentum equation are completely empirically described. One of the most widely used additional equations for this purpose is the turbulence kinetic energy equation and its general derivation will now be discussed. 

As was the case with the full equations, these contain beside the three mean flow variables u, v, and T (the pressure is, of course, by virtue of Eq. (2.157) again determined by the external in viscid flow) additional terms arising as a result of the turbulence. Therefore, as previously discussed, in order to solve this set of equations, there must be an additional input of information, i.e., a turbulence model must be used. Many turbulence models are based on the turbulence kinetic energy equation that was previously derived. When the boundary layer assumptions are applied to this equation, it becomes ... 

The turbulence kinetic energy equation for forced convection was derived in this chapter. Rederive this turbulence kinetic energy equation bv starting with the momentum... 

In order to utilize this equation it is necessary to use other equations to describe some of the terms in this equation and/or to model some of the terms in this equation. To illustrate how this is done, attention will be given to two-dimensional boundary layer flow. For two-dimensional boundary layer flows the turbulence kinetic energy equation, Eq. (5.S2), has the following form, some further rearrangement having been undertaken ... 

Substituting Eqs. (5.57) and (5.60) into Eq. (5.53) gives the following modeled form of the turbulent kinetic energy equation for two-dimensional boundary layer flow ... 

In order to utilize this equation to determine the turbulent shear stress it is necessary to obtain an additional equation relating, for example, the eddy viscosity to die quantities involved in the turbulent kinetic energy equation. If it is assumed that ... 

It is of interest to note that if the convection and diffusion terms are negligible in the turbulence kinetic energy equation, i.e., if the rate of production of kinetic energy is just equal to the rate of dissipation of turbulence kinetic energy, Eq. (5.62) reduces to ... 

In the case of axially symmetrical pipe flow, the turbulent kinetic energy equation has the following form when the terms are modeled in some way as was done... 

The turbulent kinetic energy equation was derived in Chapter 5 using the momentum equations and assuming buoyancy force effects were negligible. Re-derive this equation starting with momentum equations in which the buoyancy terms are retained. Assume a vertically upward flow and use the Boussinesq approximation. 

Studies on thermodynamic restrictions on turbulence modeling show that the kinetic energy equation in a turbulent flow is a direct consequence of the first law of thermodynamics, and the turbulent dissipation rate is a thermodynamic internal variable. The principle of entropy generation, expressed in terms of the Clausius-Duhem and the Clausius-Planck inequalities, imposes restrictions on turbulence modeling. On the other hand, the turbulent dissipation rate as a thermodynamic internal variable ensmes that the mean internal dissipation will be positive and the thermodynamic modeling will be meaningful. 

In the method developed by Martilli et al., 2002 [393] the contributions of every urban surface type (canyon floor, roofs and walls) on the momentum, heat and turbulent kinetic energy equation are computed separately ... 

The equation of motion (1.78) will now, with a little reformulation of the terms, be reintroduced into the kinetic energy equation... 

In the turbulent case the total kinetic energy equation (1.447) is readily generalized to the form... 

If we compare the k equation (1.401) with the mean kinetic energy equation (1.453) we see that they both contain a term describing the interaction between the mean flow and turbulence. We are of course referring to the velocity... 

Alternatively, the internal energy balance can be deduced from the averaged counterparts of the kinetic energy equation (1.124) and the total energy equation (1.113). 

The volume averaged kinetic energy equation can be derived by talking the scalar product of the averaged velocity with the momentum equation (3.188). Without introducing the normal simplifications assumed valid for dispersed flows, the result is ... 

A convenient form of the macroscopic kinetic energy equation can be expressed as ... 

In this section the application of multiphase flow theory to model the performance of fluidized bed reactors is outlined. A number of models for fluidized bed reactor flows have been established based on solving the average fundamental continuity, momentum and turbulent kinetic energy equations. The conventional granular flow theory for dense beds has been reviewed in chap 4. However, the majority of the papers published on this topic still focus on pure gas-particle flows, intending to develop closures that are able to predict the important flow phenomena observed analyzing experimental data. Very few attempts have been made to predict the performance of chemical reactive processes using this type of model. 

As a first estimate the optimum flow velocity can be calculated by means of the F factor (= loading factor = measure of the kinetic energy Equation 2.4-15) ... 

The dimension of the eddy diffusivity is a length scale times a velocity scale. These are proportional to prod-nets of eddy size and eddy velocity in the corresponding directions, and they typical vary with height, wind speed, stability, etc. Generally, the edrfy diffusivities can be modeled by using the kinetic energy equation of Eq. (3) and an appropriate choice for the length scale. As a simplification, it appears that the eddy diffusivity is often well described by a profile function such as... 

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