Blasius solution

Forced convection, burning of a flat plate. This classical exact analysis following the well-known Blasius solution for incompressible flow was done by Emmons in 1956 [7], It includes both variable density and viscosity. Glassman [8] presents a functional fit to the Emmons solution as... 

EXAMPLE 4.4 Development of a momentum boundary layer over a solid surface (Blasius solution)... 

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. 

Similarity is perhaps best know in the context of external boundary-layer flow, such as the Blasius solution (cf., the books by Schlichting [350] or White [429]). In these cases an independent-variable transformation is found in which a single new independent variable is a special combination of the physical spatial coordinates. In this book we are generally more concerned with internal flows where the approaches to finding similarity can differ. 

According to these equations, which are valid for Pr = 1 and ideal gases, the temperature profile can be described by the velocity profile and the parameter t +. If we put in Blasius solution from section 3.7.1.1, as an approximation for the velocity profile we obtain, for example, the temperature profile in Fig. 3.55 for Mas = 2, as a function of the distance from the wall r]+. This only reproduces an approximate temperature profile because of the simplifications made. 

Consider laminar flow past a flat plate. Create a geometry as a long rectangle. Set the velocity to zero on the plate, and set it to a constant on the left. Use a neutral boundary at the top and convective flux at the right-hand side. Compare your solution with the Blasius solution in your textbook. 

C. Streaming Flow Past a Horizontal Flat Plate - The Blasius Solution... 

C. STREAMING FLOW PAST A HORIZONTAL FLAT PLATE - THE BLASIUS SOLUTION... 

But this is just the Blasius problem, and /(y) is therefore the Blasius function. Clearly we need solve only a single trial problem with initial value /"(0) = A to obtain the asymptotic value f (oo) = B, and we can transform directly to the Blasius solution, which is obtained as an initial-value problem with /"(0) = A / IF/2. 

A second property of the Blasius solution is worth mentioning because it reflects a general property of the boundary-layer equations. The fact is that the Blasius solution is actually independent of the length of the plate. This may seem strange or even incorrect at first because we have obviously used the plate length L to nondimensionalize the equations leading to (10 59), and the plate length L also appears in the Reynolds number that was used to rescale the boundary-layer equations (10-64). However, we have shown that... 

Here we consider, in detail, only the simplest case with m = 0 and /J = 0, namely, a semi-infinite flat plate with 9S = 1 for all x. In this case, the velocity field is given by the Blasius solution... 

The similarity variable rj, (11-17), is the same as for the Blasius solution. Substituting (11-18) into (11-6) leads to the similarity equation... 

The results in (11-51) and (11-52) are, of course, completely general for arbitrary k(x). Here, for illustration purposes, we evaluate them only for the special case of a flat plate (parallel to the free stream) where, according to the Blasius solution from Chap. 10,... 

The Blasius solution shows that the longitudinal velocity profiles are affinely similar to each other for all cross-sections of the boundary layer. 

This problem has also been solved numerically, and the function j(rf) is tabulated in [424]. We point out that in this case the solution differs from the corresponding Blasius solution. Thus, although physical consideration suggests that the inversion of flow is possible, the solution shows that it is impossible from the mathematical viewpoint. This is due to the fact that problems (1.7.5) and (1.7.11) are nonlinear. 

Figure 6.5 shows the velocity distributions in a boundary layer of a liquid with Pr , = 100 (e.g., sulfuric acid at room temperature). For this Prandtl number, the thermal boundary layer penetration into the liquid is much less than the flow boundary layer, and the regions where viscosity variations occur are confined close to the surface. The curve corresponding to p /pe = 1 is the Blasius solution (see Fig. 6.1). The curve labeled p ,/pe = 0.23 corresponds to a heated surface where the low viscosity near the surface requires steeper velocity gradients to maintain a continuity of shear with the outer portion of the boundary layer. The heated free-stream cases reveal the opposite effects. In general, the outer portions of the flow boundary layers act similarly to the velocity distribution of the Blasius case except for their being displaced in or out by the effects that have taken place in the thermal boundary layer.

FIGURE 6.7 Effects of viscosity changes across a laminar liquid boundary layer on surface shear and heat flux—reference shear from Blasius solution with free-stream properties, reference heat flux from Pohlhausen solution with wall properties [7]. 

Note that cfl2 in Eq. 6.42 differs in magnitude from that given by Eq. 6.16 because of the departure from the Blasius solution by the existence of Cr(T )jn Eq. 6.37. 

It is important to recall that the Blasius solution for the velocity boundary layer did not involve a velocity in the y direction at the surface. Accordingly, equation (2-40) involves the important assumption that the rate at which mass enters or leaves the boundary layer at the surface is so small that it does not alter the velocity profile predicted by the Blasius solution. 

Blasius solution for the laminar boundary layer on a flat plate, shown in Fig. 11.3, rests on a considerable string of assumptions and simplifications. However, it has been tested by numerous investigators and found to represent the experimental data very well (note that Fig. 11.3 shows the comparison between Blasius solution and Nikuradse s experimental data). Thus, these assumptions and simplifications seem to be justified. 

Blasius solution for the laminar boundary layer on a flat plate and Nikuradse s experimental tests of same. [From J. Nikuradse, Laminar Reibungsschichten an der laengsangestroemten Platte (Laminar friction layers on plates with parallel flow), Monograph Zentralefuer Wiss. Berichtwesen, Berlin (1942). 

Equation 11.14 says that, according to Blasius solution, a plot of local drag coefficients versus the Reynolds number should be given by... 

Figure 11.4 shows such a plot of experimental local drag coefficients. Those for which the flow is laminar agree very well with Blasius solution. However, if the flow is turbulent, the result is quite different. We discuss turbulent boundary layers in Secs. 11.3 and 11.5. 

At the leading edge of the plate (a = 0) the Reynolds number is zero so, according to Eq. 11.16, the drag coefficient should be infinite. This is physically unreal and leads to the conclusion that Blasius solution is not correct in the... 

Local drag coefficient for a flat plate. Experimental data are compared with Blasius solution (Eq, 11.16) and with Prandtl s equation (Eq. 11.36). [From H. W. Liepmann and S. Dahwan, Direct measurements of local skin friction in low-speed and high-speed flow, Proc. First U.S. Natl. Congr. AppL Mech, ASME, New York, 1952, p. 873. Reproduced with the permission of the publisher,] I... 

For Blasius solution we can perform the integration in Eq. 11.2 graphically on Fig. Ills by noting that the region above and to the left of the solid curve is... 

From Blasius solution (Fig. 11.3) we wish to find Vy (the y component of the velocity) at any point. Starting with the mass balance equation, Eq. 11.2, show that... 

Comparing this value with the drag coefficient based on Blasius solution (Eq. Ill 14), we find that this approximate solution gives a drag coefficient of... 

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