Energy equation boundary layer, turbulent

The same line of reasoning can be applied to the energy equation and if it is assumed that the turbulence terms (kT /M wt - T ) and (v T )/ui(Twr - T ) in the resultant equation have the same order of magnitude as the turbulence terms in the momentum equation, i.e., (8/L), then the energy equation for turbulent boundary layer flow becomes... 

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 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 ... 

Nodal points used in obtaining the finite difference forms of the momentum and energy equations for turbulent boundary layer flow. 

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). 

Bodin, S., 1979. A predictive numerical model of the atmospheric boundary layer based on the turbulent energy equation. Report Meteorology and Climatology 13, Noirkoping, Sweden, Swedish Meteorological and Hydrological Institute, SE-60176. 

Nonuniform Surface Temperature. Nonuniform surface temperatures affect the convective heat transfer in a turbulent boundary layer similarly as in a laminar case except that the turbulent boundary layer responds in shorter downstream distances The heat transfer to surfaces with arbitrary temperature variations is obtained by superposition of solutions for convective heating to a uniform-temperature surface preceded by a surface at the recovery temperature of the fluid (Eq. 6.65). For the superposition to be valid, it is necessary that the energy equation be linear in T or i, which imposes restrictions on the types of fluid property variations that are permitted. In the turbulent boundary layer, it is generally required that the fluid properties remain constant however, under the assumption that boundary layer velocity distributions are expressible in terms of the local stream function rather than y for ideal gases, the energy equation is also linear in T [%]. 

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