Boundary layer laminar-turbulent transition

Arnal, D. (1986). Three-dimensional boundary layer Laminar-turbulent transition. AGARD eport No. 7fl 1-34. 

Putting the constant equal to zero, implies that <5 = 0 when x = 0, that is that the turbulent boundary layer extends to the leading edge of the surface. An error is introduced by this assumption, but it is found to be small except where the surface is only slightly longer than the critical distance xc for the laminar-turbulent transition. 

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... 

Figure 4 Hydrodynamic boundary layer development on the semi-infinite plate of Prandtl. <5D = laminar boundary layer, <5t = turbulent boundary layer, /vs = viscous turbulent sub-layer, <5ds = diffusive sub-layer (no eddies are present solute diffusion and mass transfer are controlled by molecular diffusion—the thickness is about 1/10 of <5vs)> B = point of laminar—turbulent transition. Source From Ref. 10.

Fig. 5.11 Position of boundary layer separation and laminar/turbulent transition in the critical region and beyond. Experimental results of Achenbach (A3) and Raithby and Eckert (R3).

During recent years experimental work continued actively upon the macroscopic aspects of thermal transfer. Much work has been done with fluidized beds. Jakob (D5, J2) made some progress in an attempt to correlate the thermal transport to fluidized beds with transfer to plane surfaces. This contribution supplements work by Bartholomew (B3) and Wamsley (Wl) upon fluidized beds and by Schuler (S10) upon transport in fixed-bed reactors. The influence of thermal convection upon laminar boundary layers and their transition to turbulent boundary layers was considered by Merk and Prins (M5). Monaghan (M7) made available a useful approach to the estimation of thermal transport associated with the supersonic flow of a compressible fluid. Monaghan s approximation of Crocco s more general solution (C9) of the momentum and thermal transport in laminar compressible boundary flow permits a rather satisfactory evaluation of the transport from supersonic compressible flow without the need for a detailed iterative solution of the boundary transport for each specific situation. None of these references bears directly on the problem of turbulence in thermal transport and for that reason they have not been treated in detail. 

Leehey, P. and Shapiro, P. (1979). Leading edge effect in laminar boundary layer excitation by sound. In Laminar Turbulent Transition (eds. R. Eppler and H. Fasel), 321-331, Springer Verlag. 

Michel, R., Arnal. D., Coustols, E. and Jullien, J.C. (1984). Experimental and theoretical studies of boundary layer transition on a swept infinite wing. In Proc. of lUTAM S3unp. On Laminar-Turbulent Transition Novosibirsk, USSR, Springer Verlag. 

If the term q x, s,) is defined as the heat flux at point x in a turbulent boundary layer with instantaneous transition from laminar to turbulent flow at s and q" , (x) is defined as the heat flux at point x with a laminar boundary layer beginning at the leading edge, the heat flux caused by the intermittency of turbulence in the transition zone is then... 

The first term is the product of the laminar heat flux and the fraction of time the boundary layer is laminar at x. The second term accounts for both the fraction of time the boundary layer is turbulent and the effect of the moving transition location. The term qZ,(x, s,) is sufficiently complex mathematically that Eq. 6.240 is normally solved by numerical integration. If it is assumed that the energy thickness remains unchanged as the laminar boundary layer changes instantaneously into a turbulent boundary layer, then... 

Now, to ascertain how large an error we made by assuming that the entire boundary layer jwas turbulent, we assume that transition from laminar to turbulent flow takes place at an of 10. From the above, this corresponds to a distance of of the length of the plate so the boundary layer over the first 0.2 ft presumably is laminar. For this area the drag due to a laminar boundary layer is given byiEq. 11.19 as... 

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. 

The transition from laminar to turbulent flow on a smooth plate occurs in the Reynolds number range 2 x 10 to 3 x 10 , as shown in Fig. 3.10-1. When the boundary layer is turbulent, a thin viscous sublayer persists next to the plate. The drag caused by the viscous shear in the boundary layers is called skin friction and it is the only drag present for flow past a flat plate. 

However, the transition Reynolds number depends on free-stream turbulence and may range from 3 X 10 to 3 X lO ". The laminar boundary layer thickness 8 is a function of distance from the leading edge ... 

Continuous Flat Surface Boundaiy layers on continuous surfaces drawn through a stagnant fluid are shown in Fig. 6-48. Figure 6-48 7 shows the continuous flat surface (Saldadis, AIChE J., 7, 26—28, 221-225, 467-472 [1961]). The critical Reynolds number for transition to turbulent flow may be greater than the 500,000 value for the finite flat-plate case discussed previously (Tsou, Sparrow, and Kurtz, J. FluidMech., 26,145—161 [1966]). For a laminar boundary layer, the thickness is given by... 

When a fluid flowing at a uniform velocity enters a pipe, the layers of fluid adjacent to the walls are slowed down as they are on a plane surface and a boundary layer forms at the entrance. This builds up in thickness as the fluid passes into the pipe. At some distance downstream from the entrance, the boundary layer thickness equals the pipe radius, after which conditions remain constant and fully developed flow exists. If the flow in the boundary layers is streamline where they meet, laminar flow exists in the pipe. If the transition has already taken place before they meet, turbulent flow will persist in the... 

As seen in Fig. 11-2, the drag coefficient for the sphere exhibits a sudden drop from 0.45 to about 0.15 (almost 70%) at a Reynolds number of about 2.5 x 105. For the cylinder, the drop is from about 1.1 to about 0.35. This drop is a consequence of the transition of the boundary layer from laminar to turbulent flow and can be explained as follows. 

You want to perform an experiment that illustrates the wake behind a sphere falling in water at the point where the boundary layer undergoes transition from laminar to turbulent. (See Fig. 11-4.) If the sphere is made of steel with a density of 500 lbm/ft3, what should the diameter be ... 

The boundary layer thickness gradually increases until a critical point is reached at which there is a sudden thickening of the boundary layer this reflects the transition from a laminar boundary layer to a turbulent boundary layer. For both types, the flow outside the boundary layer is completely turbulent. In that part of the boundary layer near the leading edge of the plate the flow is laminar and consequently this is known as a... 

When the Reynolds number Rep reaches a value of about 300000, transition from a laminar to a turbulent boundary layer occurs and the point of separation moves towards the rear of the sphere as discussed above. As a result, the drag coefficient suddenly falls to a value of 0.10 and remains constant at this value at higher values of Rep. 

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. 

---