Turbulent boundary layer example

Conditions in the fully turbulent outer part of the turbulent boundary layer are quite different. In a turbulent fluid, the shear stress f is given by equation 1.95. As illustrated in Example 1.10, outside the viscous sublayer and buffer zone the eddy kinematic viscosity e is much greater than the molecular kinematic viscosity v. Consequently equation 1.95 can be written as... 

The transition to a turbulent boundary layer for a flat plate has been experimentally determined to occur at an Rcx value of between 3 x 10 and 6 x 10. For this example, the transition would occur between 15 and 30 cm after the start of the plate. Thus, the computations for a laminar boundary layer at 0.6 and 1 m are not realistic. However, the Blasius solution helps in the analysis of experimental data for a turbulent boundary layer, because it can tell us which parameters are likely to be important for this analysis, although the equations may take a different form. 

It is interesting to compare equations (6.32) and (6.33) with those for a fully developed laminar flow, equations (6.29) and (6.30). In Example 5.1, we showed that eddy diffusion coefficient in a turbulent boundary layer was linearly dependent on distance from the wall and on the wall shear velocity. If we replace the diffusion coefficient in equation (6.30) with an eddy diffusion coefficient that is proportional to hu, we get... 

The conditions under which transition occurs depend on the geometrical situation being considered, on the Reynolds number, and on the level of unsteadiness in the flow well away from the surface over which the flow is occurring [2], [30]. For example, in the case of flow over a flat plate as shown in Figure 5.6, if the level of unsteadiness in the freestream flow ahead of the plate is very low, transition from laminar to turbulent boundary layer flow occurs approximately when ... 

Calculate the turbulent-boundary-layer thickness at the end of the plate for Example 5-7, assuming that it develops (a) from the leading edge of the plate and (b) from the transition point at Recri, = 5 x 105. 

Ideal Gases at High Temperature. Three fundamentally different approaches have been applied to the treatment of the turbulent boundary layer with variable fluid properties all are restricted to air behaving as an ideal, calorically perfect gas. First, the Couette flow solutions have been extended to permit variations in viscosity and density. Second, mathematical transformations, analogous to Eq. 6.36 for a laminar boundary layer, have been used to transform the variable-property turbulent boundary layer differential equations into constant-property equations in order to provide a direct link between the low-speed boundary layer and its high-speed counterpart. Third, empirical correlations have been found that directly relate the variable-property results to incompressible skin friction and Stanton number relationships. Examples of the latter are reference temperature or enthalpy methods analogous to those used for the laminar boundary layer, and the method of Spalding and Chi [104]. 

With turbulent channel flow the shear rate near the wall is even higher than with laminar flow. Thus, for example, (du/dy) ju = 0.0395 Re u/D is vaHd for turbulent pipe flow with a hydraulically smooth wall. The conditions in this case are even less favourable for uniform stress on particles, as the layer flowing near the wall (boundary layer thickness 6), in which a substantial change in velocity occurs, decreases with increasing Reynolds number according to 6/D = 25 Re", and is very small. Considering that the channel has to be large in comparison with the particles D >dp,so that there is no interference with flow, e.g. at Re = 2300 and D = 10 dp the related boundary layer thickness becomes only approx. 29% of the particle diameter. It shows that even at Re = 2300 no defined stress can be exerted and therefore channels are not suitable model reactors. 

The approach described above is by no means complete or exclusive. For example, Lamb et al. (1975) have proposed an alternative route to assess the adequacy of the atmospheric diffusion equation. Their approach is based on the Lagrangian description of the statistical properties of nonreacting particles released in a turbulent atmosphere. By employing the boundary layer model of Deardorff (1970), the transition probability density p x, y, z, t x, y, z, t ) is determined from the statistics of particles released into the computed flow field. Once p has been obtained, Eq. (3.1) can then be used to derive an estimate of the mean concentration field. Finally, the validity of the atmospheric diffusion equation is assessed by determining the profile of vertical dififiisivity that produced the best fit of the predicted mean concentration field. 

In Example 8.5, we saw how the diffusive boundary layer could grow. The boundary never achieves 1 m in thickness or even 5 cm in thickness, because of the interaction of turbulence and the boundary layer thickness. The diffusive boundary layer is continually trying to grow, just as the boundary layer of Example 8.5. However, turbulent eddies periodically sweep down and mix a portion of the diffusive boundary layer with the remainder of the fluid. It is this unsteady character of the turbulence... 

Material transport is usually associated with thermal transport except in situations involving homogeneous phases which can be treated as ideal solutions (L4). For this reason it is necessary to consider the behavior of combined thermal and material transport in turbulent flow. The evaporation of liquids under macroscopic adiabatic conditions is a typical example of such a phenomenon. Under such circumstances the behavior in the boundary layer is similar to that found in the field of aerodynamics in a blowing boundary layer (S4). However, it is not... 

The structure of turbulence in the transition zone from a fully turbulent fluid to a nonfluid medium (often called the Prandtl layer) has been studied intensively (see, for instance, Williams and Elder, 1989). Well-known examples are the structure of the turbulent wind field above the land surface (known as the planetary boundary layer) or the mixing regime above the sediments of lakes and oceans (benthic boundary layer). The vertical variation of D(x) is schematically shown in Fig. 19.8b. Yet, in most cases it is sufficient to treat the boundary as if D(x) had the shape shown in Fig. 19.8a. 

As shown in Illustrative Examples 19.3 and 19.4, often it is not immediately known whether an exchange process is controlled by transport across a boundary layer or by transport in the bulk phase. In Illustrative Example 19.3 we look at the case of resuspension of particles from the polluted sediments of Boston Harbor. We are interested in the question of what fraction of the pollutants sorbed to the particles (such as polychlorinated biphenyls) can diffuse into the open water column while the particles are resuspended due to turbulence produced by tidal currents in the bay. To answer this question we need to assess the possible role of the boundary layer around the particles. 

Constant-Stress Layer in Flowing Fluids. In the boundary layer of a fluid flowing over a solid wall. Ihe shear stress varies with distance from Ihe wall bul ii may be considered nearly constant within a small fraction of the layer thickness. The concept is of particular importance in turbulent flow where it leads lo a theoretical derivation of the law of ihe wall," the logarithmic distribution of mean velocity. The constant stress layer is ihe best-known example of the equilibrium flow s near a wall. 

Consider, for example, assisting turbulent mixed convection over a vertical flat plate. This situation is schematically shown in Fig. 9.20. If it is assumed that the boundary layer assumptions apply, the governing equations for the mean velocity components and temperature are ... 

The zones where these gradients occur are often called boundary layers. For example, the aerodynamic boundary layer is the region near a surface where viscous forces predominate. Boundary layers exist with both laminar and turbulent flow and may be either solely laminar or turbulent with a laminar sublayer themselves (Landau and Lifshitz, 1959). 

Linear stability theory results match quite well with controlled laboratory experiment for thermal and centrifugal instabilities. But, instabilities dictated by shear force do not match so well, e.g. linear stability theory applied to plane Poiseuille flow gives a critical Reynolds number of 5772, while experimentally such flows have been observed to become turbulent even at Re = 1000- as shown in Davies and White (1928). Couette and pipe flows are also found to be linearly stable for all Reynolds numbers, the former was found to suffer transition in a computational exercise at Re = 350 (Lundbladh Johansson, 1991) and the latter found to be unstable in experiments for Re > 1950. Interestingly, according to Trefethen et al. (1993) the other example for which linear analysis fails include to a lesser degree, Blasius boundary layer flow. This is the flow which many cite as the success story of linear stability theory. 

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