Boundary layer equations turbulent

If the turbulent momentum equation is expressed in nondimensional form in the same way as was done in deriving the laminar boundary layer equations then the additional term becomes ... 

Now, the rest of the terms retained in the boundary layer equations have the order of magnitude of unity and, therefore, for the boundary layer equations to apply, the dimensionless turbulence terms (u 2lu ) and (u v /u ), which are assumed to have the same order of magnitude, will have the order of magnitude of (8/L) at most. The first term in Eq. (2.154) is, therefore, negligible compared to the rest of the terms in the boundary layer equations. Therefore, the x-wise momentum equation for turbulent boundary layer flow is ... 

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]

The program assumes the flow is turbulent from the leading edge and that 62 = 0 when x = 0. The program can easily be modified to use a laminar boundary layer equation solution procedure to provide initial conditions for the turbulent boundary layer solution which would then be started at some assumed transition point. 

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 ... 

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. 

Since the speed near the surface in a laminar boundary layer has a lower velocity than its turbulent counterpart, the laminar boundary layer is more likely to separate. When this occurs, the laminar boundary layer leaves the surface and usually undergoes a transition to a turbulent flow away from the surface. This process takes place over a certain distance that is inversely related to Re, but if it happens quickly enough, the flow may reattach as a turbulent boundary layer and continue along the surface. To compute when the separation will occur, we can solve the Navier-Stokes equations or apply one of the several separation criteria to the solutions of the boundary layer equations. 

From this equation, the dependency of the mass transfer coefficient yff,- on the diffusion coefficient D, and the boundary layer thickness d of the fluid flow, may be seen. The laminar boundary layer and turbulent bulk cannot be distinguished exactly, due to the continous transition the boundary layer thickness 3 is, therefore, a formal complementary variable. 

In a similar fashion, the integral momentum analysis method used for the turbulent hydrodynamic boundary layer in Section 3.10 can be used for the thermal boundary layer in turbulent flow. Again, the Blasius 7-power law is used for the temperature distribution. These give results that are quite similar to the experimental equations as given in Section 4.6. 

Boundary layer flows are a special class of flows in which the flow far from the surface of an object is inviscid, and the effects of viscosity are manifest only in a thin region near the surface where steep velocity gradients occur to satisfy the no-slip condition at the solid surface. The thin layer where the velocity decreases from the inviscid, potential flow velocity to zero (relative velocity) at the sohd surface is called the boundary layer The thickness of the boundary layer is indefinite because the velocity asymptotically approaches the free-stream velocity at the outer edge. The boundaiy layer thickness is conventionally t en to be the distance for which the velocity equals 0.99 times the free-stream velocity. The boundary layer may be either laminar or turbulent. Particularly in the former case, the equations of motion may be simphfied by scaling arguments. Schhchting Boundary Layer Theory, 8th ed., McGraw-HiU, New York, 1987) is the most comprehensive source for information on boundary layer flows. 

In some convection equations, such as for turbulent pipe flow, a special correction factor is used. This factor relates to the heat transfer conditions at the flow inlet, where the flow has not reached its final velocity distribution and the boundary layer is not fully developed. In this region the heat transfer rate is better than at the region of fully developed flow. 

It may be noted that no assumptions have been made concerning the nature of the flow within the boundary layer and therefore this relation is applicable to both the streamline and the turbulent regions. The relation between ux and y is derived for streamline and turbulent flow over a plane surface and the integral in equation 11.9 is evaluated. 

Equation 11.12 does not fit velocity profiles measured in a turbulent boundary layer and an alternative approach must be used. In the simplified treatment of the flow conditions within the turbulent boundary layer the existence of the buffer layer, shown in Figure 11.1, is neglected and it is assumed that the boundary layer consists of a laminar sub-layer, in which momentum transfer is by molecular motion alone, outside which there is a turbulent region in which transfer is effected entirely by eddy motion (Figure 11.7). The approach is based on the assumption that the shear stress at a plane surface can be calculated from the simple power law developed by Blasius, already referred to in Chapter 3. 

For the inlet length of a pipe in which the boundary layers are forming, the equations in the previous section will give an approximate value for the heat transfer coefficient. It should be remembered, however, that the flow in the boundary layer at the entrance to the pipe may be streamline and the point of transition to turbulent flow is not easily defined. The results therefore are, at best, approximate. 

For turbulent flow on a rotating sphere or hemisphere, Sawatzki [53] and Chin [22] have analyzed the governing equations using the Karman-Pohlhausen momentum integral method. The turbulent boundary layer was assumed to originate at the pole of rotation, and the meridional and azimuthal velocity profiles were approximated with the one-seventh power law. Their results can be summarized by the... 

In turbulent flow, the edge effect due to the shape of the support rod is quite significant as shown in Fig. 6. The data obtained with a support rod of equal radius agree with the theoretical prediction of Eq. (52). The point of transition with this geometry occurs at Re = 40000. However, the use of a larger radius support rod arbitrarily introduces an outflowing radial stream at the equator. The radial stream reduces the stability of the boundary layer, and the transition from laminar to turbulent flow occurs earlier at Re = 15000. Thus, the turbulent mass transfer data with the larger radius support rod deviate considerably from the theoretical prediction of Eq. (52) a least square fit of the data results in a 0.092 Re0 67 dependence for... 

ShaJSc113 as indicated by the thin solid line. This 0.67 power of Re agrees with the result of a turbulent heat transfer measurement on a rotating sphere [40], Since the flow induced by a rotating sphere is also characterized by an outflowing radial jet at the equator caused by the collosion of two opposing flow boundary layers on the sphere, the 0.67 power dependence on Re is clearly related to the radial flow stream away from the equator. 

Equation (6-31) applies to the laminar sublayer region in a Newtonian fluid, which has been found to correspond to 0 < y+ < 5. The intermediate region, or buffer zone, between the laminar sublayer and the turbulent boundary layer can be represented by the empirical equation... 

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