Boundary-layer flow kinetic energy

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 the actual case of a boundary-layer flow problem, the fluid is not brought to rest reversibly because the viscous action is basically an irreversible process in a thermodynamic sense. In addition, not all the free-stream kinetic energy is converted to thermal energy—part is lost as heat, and part is dissipated in the form of viscous work. To take into account the irreversibilities in the boundary-layer flow system, a recovery factor is defined by... 

The flow in the diffuser is usually assumed to be of a steady nature to obtain the overall geometric configuration of the diffuser. In a channel-type diffuser the viscous shearing forces create a boundary layer with reduced kinetic energy. If the kinetic energy is reduced below a certain limit, the flow in this layer becomes stagnant and then reverses. This flow reversal causes... 

At the inlet to the pipe the velocity across the whole section is constant. The velocity at the pipe axis will progressively increase in the direction of flow and reach a maximum value when the boundary layers join. Beyond this point the velocity profile, and the velocity at the axis, will not change. Since the fluid at the axis has been accelerated, its kinetic energy per unit mass will increase and therefore there must be a corresponding all in its pressure energy. 

Recall our short discussion in Section 18.5 where we learned that turbulence is kind of an analytical trick introduced into the theory of fluid flow to separate the large-scale motion called advection from the small-scale fluctuations called turbulence. Since the turbulent velocities are deviations from the mean, their average size is zero, but not their kinetic energy. The kinetic energy is proportional to the mean value of the squared turbulent velocities, Mt2urb, that is, of the variance of the turbulent velocity (see Box 18.2). The square root of this quantity (the standard deviation of the turbulent velocities) has the dimension of a velocity. Thus, we can express the turbulent kinetic energy content of a fluid by a quantity with the dimension of a velocity. In the boundary layer theory, which is used to describe wind-induced turbulence, this quantity is called friction velocity and denoted by u. In contrast, in river hydraulics turbulence is mainly caused by the friction at the... 

This transition has profound effects in all fluid dynamics, and certainly so in aerodynamics. The velocity profile in (he boundary layer becomes fuller neat the surface on account of Ihe higher average kinetic energy of the layer created by turbulent energy exchange from layer lo layer. The effective viscosity is therefore larger in turbulent than laminar flow, ihe turbulent boundary layer thickens more rapidly downstream, the skin friction increases. 

Jensen Webb (Ref 43) examined the data predicting the extent of afterburning in fuel-rich exhausts of metal-modified double-base proplnt rocket motors so as to determine the amt of an individual metal which is required to suppress this afterburning. The investigatory means they used consisted of a series of computer codes. First, an equilibrium chemistry code to calculate conditions at the nozzle throat then a nonequilibrium code to derive nozzle plane exit compn, temp and velocity and, finally, a plume prediction code which incorporates fully coupled turbulent kinetic energy boundary-layer and nonequilibrium chemical reaction mechanisms. Used for all the code calcns were the theoretical environment of a static 300 N (67-lb) thrust std research motor operating at a chamber press of S.SMNm 2 (500psi), with expansion thru a conical nozzle to atm press and a mass flow rate... 

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 ... 

In turbulent flow, momentum is constantly fed into the layer adjacent to the wall because of the momentum transfer between layers at different velocities. The kinetic energy of the fluid elements close to the wall does not decrease as rapidly as in laminar flow. This means that turbulent boundary layers do not become detached as quickly as laminar boundary layers. Heat and mass transfer close to the wall is not only promoted by turbulence, the fluid also flows over a larger surface area without detachment. At the same time the pressure resistance is lower because the fluid flow does not separate from the surface for a longer flow path. 

Pacanowski and Philander, 1981 Peters et al., 1988). More sophisticated methods are based on prognostic equations for the turbulent kinetic energy k and a second quantity, which is either the dissipation rate e or a length scale in the turbulent flow see Burchard (2002) for a recent review and applications of two-equation turbulence closures for onedimensional water column models. A two-equation turbulent closure has been applied by Omstedt et al. (1983) and Svensson and Omstedt (1990) for the Baltic Sea surface boundary layer under special consideration of sea ice, whereas the application in three-dimensional circulation models is described by Burchard and Bolding (2002) and Meier et al. (2003). 

Kinetic energy of the flow relative to the boundary layer enthalpy difference Forced convection (compressible flow)... 

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