Numerical solutions turbulent boundary layer

In the numerical solution for boundary layer flow given in this chapter it was assumed that transition occurred at a point i.e., the eddy viscosity was set equal to zero up to the transition point and then the full value given by the turbulence model was used. Show how this numerical method and the program based on it can be modified to allow for a transition zone in which the eddy viscosity increases linearly from zero at the beginning of the zone to the full value given by the turbulence model at the end of the zone. 

There are a number of schemes for numerically approximating the boundary layer equations and many different solution procedures based on these various schemes have been developed. In the present section, one of the simpler finite-difference schemes will be described. The solution procedure based on this scheme should give quite acceptable results for most problems. The scheme is easily extended to deal with turbulent ftnws as will he Hi rncce/t i AH i[Pg.123]

NUMERICAL SOLUTION OF THE TURBULENT BOUNDARY LAYER EQUATIONS... 

Solutions to the boundary layer equations are, today, generally obtained numerically [6],[7],[8],[9],[10],[11],[12]. In order to illustrate how this can be done, a discussion of how the simple numerical solution procedure for solving laminar boundary layer problems that was outlined in Chapter 5 can be modified to apply to turbulent boundary layer flows. For turbulent boundary layer flows, the equations given earlier in the present chapter can, because the fluid properties are assumed constant, be written as ... 

In order to handle the rapid variations near a w all, one must either use a fine computational mesh in this region or else employ a special treatment. The variation in shear stress is, to a first approximation, small across this region, and the law of the wall is known to be followed by the mean-velocity profile very near the wall for most turbulent boundary layers. One simple approach is therefore to patch the numerical solution at the first computation point away from the wall to the empirical wall law,... 

When flow occurs about a sphere the solution to tins forced convection mass transfer problem is quite complex because of the complexity of the flow field. At low flow rates (creqiiiig flow) a laminar boundary layer exists about the sphere which separates from die surface at an at ular porition and moves lowani the forward stagnation point as the flow rate increases. Wake fimnation occurs st the tear of the sphere. At still higher flow rates transition to a turbulent boundary layer occurs. Solutions to the problem of mass transfer during creeping flow about a sphere (Re < 1) have been developed by a nombw of authors with the numerical solutions of Brian and Hales being perhaps the most extensive. Their result is... 

Fig. 3. Numerical values of A and a for the solution of turbulent flow boundary layer on a rotating hemisphere. The value of meridional angle, 9, is given in degrees. From [22].

This chapter has mainly been devoted to the solution of the boundary layer form of the governing equations. While these boundary layer equations do adequately describe a number of problems of great practical importance, there are many other problems that can only be adequately modeled by using the full governing equations. In such cases, it is necessary to obtain the solution numerically and also almost always necessary to use a more advanced type of turbulence model [6],[12],[28],[29]. Such numerical solutions are most frequently obtained using the commercially available software based on the finite volume or the finite element method. 

Air at a temperature of 0°C and standard atmospheric pressure flows at a velocity of 50 m/s over a wide flat plate with a total length of 2 m. A uniform surface heat flux is applied over the first 0.7 m of the plate and the rest of the surface of the plate is adiabatic. Assuming that the boundary layer is turbulent from the leading edge, use the numerical solution to derive an expression for the plate temperature at the trailing edge of the plate in terms of the applied heat flux. What heat flux is required to ensure... 

Bradshaw et al. (B3) use Eqs. (40) to derive a differential equation for the turbulent shear stress t. The transport velocity Qa is taken as (Tmei/p), where Tm x is the maximum value of riy) in the boundary layer. G and I are prescribed as functions of the position across the boundary layer, and o is essentially taken as constant. Together with Eqs. (10a,b), Eq. (36) gives a closed set of equations for U, V, and t this system is of hyperbolic type, with three real characteristic lines. Bradshaw et al. construct a numerical solution using the method of characteristics it can also be done using small streamwise steps with an explicit difference scheme (Nl A. J. Wheeler and J. P. Johnston, private communications). There is a great physical appeal to the characteristics, especially since it is found that the solutions along the outward-going characteristic dominates the total solution. This... 

The key aspect, then, in numerical simulation of the atmoLj.)heric boundary layer is the evaluation of the turbulent momentum fluxes in the time-averaged equations of motion (24). Considering this, we review briefly some of the more promising techniques that have been used to determine these fluxes. Our objective is not to give a full review, but rather to introduce the types of approaches which in the future may permit the solution of (23) and (24) and thus the prediction of urban wind fields. 

A turbulence model for small polder additives In a boundary layer on a flat plate is proposed with an account being taken of optimum polymer concentration. Results of a numerical investigation of the solution flow on a flat semi-infinite plate are presented. These results are in a satisfactory 6ig-reement with experimental data. 

The solutions to Navier-Stokes equations are typically very difficult to arrive at. This fact is attested to by the extraordinary development of numerical computation in fluid mechanics. Only a few exact analytical solutions are known for Navier-Stokes equations. We present in this chapter some laminar flow solutions whose interpretation per se is essential in this regard. We then introduce the boundary layer concept. We conclude the chapter with a discussion on the uniqueness of solutions to Navier-Stokes equations, with special reference to the phenomenon of turbulence. 

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