Flat plate boundary-layer solution

Blasius steady-flow, laminar, flat-plate, boundary-layer solution is a numerical solution of his simplification of Prandtl s boundary-layer equations, which are a simplified, one-dimensional momentum balance and a mass balance. This type of solution is known in the boundary-layer literature as an exact solution. Exact solutions can be found for only a very limited number of cases. Therefore, approximate methods are available for making reasonable estimates of the behavior of laminar boundary layers (Prob. 11.8). 

In this case, the diffusion boundary layer is embedded in the viscous boundary layer and the velocity it sees is that close to the wall. The Blasius solution for flat plate boundary layer in the series form is... 

Problem 10-9. Translating Flat Plate. Consider the high-Reynolds-number laminar boundary-layer flow over a semi-infinite flat plate that is moving parallel to its surface at a constant speed (7 in an otherwise quiescent fluid. Obtain the boundary-layer equations and the similarity transformation for f (r ). Is the solution the same as for uniform flow past a semi-infinite stationary plate Why or why not Obtain the solution for f (this must be done numerically). If the plate were truly semi-infinite, would there be a steady solution at any finite time (Hint. If you go far downstream from the leading edge of the flat plate, the problem looks like the Rayleigh problem from Chap. 3). For an arbitrarily chosen time T, what is the regime of validity of the boundary-layer solution ... 

From Blasi us boundary-layer solution we can calculate the drag on any part Of a flat plate. Because the solution is based on laminar flow of a... 

In many electrolysis cells it is the solution rather than the electrode which moves, and as an example of such systems, we shall consider briefly the flow of solution over a flat plate. As the solution flows across the plate, two forces act upon it the first is the cause of the flow and is known as the inertial force (i.e. that generated by the pump or solution head), while the second opposes flow and results from viscous forces between the plate and the solution. Hence as the solution flows over the plate, the layer adjacent to the surface will continuously be slowed down, and the boundary layer, where the rate of flow is less than that in the bulk, will expand into the solution. This is illustrated in Fig. 4.6. The shape of the flow contours and the thickness of the boundary layer will depend on the relative importance of the forces leading to solution flow and those leading to the retardation of flow at the plate/solution interface. Because of the importance of this ratio of inertial/viscous forces, it is given a name, the Reynolds number. Re, This is a dimensionless parameter defined by... 

I. Turbulent, local flat plate, natural convection, vertical plate Turbulent, average, flat plate, natural convection, vertical plate Nsk. = — = 0.0299Wg=Ws = D x(l + 0.494W ) )- = 0.0249Wg=W2f X (1 + 0.494WE )- [S] Low solute concentration and low transfer rates. Use arithmetic concentration difference. Ncr > 10 " Assumes laminar boundary layer is small fraction of total. D [151] p. 225... 

Fig. 4. Migration contribution to the limiting current in acidified CuS04 solutions, expressed as the ratio of limiting current (iL) to limiting diffusion current (i ) r = h,so4/(( h,so, + cCuS(>4). "Sulfate refers to complete dissociation of HS04 ions. "bisulfate" to undissociated HS04 ions. Forced convection" refers to steady-state laminar boundary layers, as at a rotating disk or flat plate free convection refers to laminar free convection at a vertical electrode penetration to unsteady-state diffusion in a stagnant solution. [F rom Selman (S8).]...

Similarity Variables The physical meaning of the term similarity relates to internal similitude, or self-similitude. Thus, similar solutions in boundary-layer flow over a horizontal flat plate are those for which the horizontal component of velocity u has the property that two velocity profiles located at different coordinates x differ only by a scale factor. The mathematical interpretation of the term similarity is a transformation of variables carried out so that a reduction in the number of independent variables is achieved. There are essentially two methods for finding similarity variables, "separation of variables (not the classical concept) and the use of "continuous transformation groups. The basic theory is available in Ames (1965). 

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. 

These relationships have been used by Spalding in the dimensionless presentation both of theoretical values obtained in his approximate solution of the boundary layer equations (58) and of the experimental data (51, 55, 60). Emmons (3), who has solved the problem of forced convection past a burning liquid plane surface in a more rigorous fashion, shows graphically the rather close correspondence between values obtained from his exact solution and that of Spalding, and between the calculated values for flat plates and the experimental values for spheres. 

With the initial values for ug, Eq. (10.30) may be solved for Uj+ij explicitly, usually by starting from the flat plate and working outward until Ujj+i/uj+i, = 1- e = 0.995 or some other predetermined value of e. Because of the asymptotic nature of the boundary layer condition, the location of the outer boundary is found as the solution proceeds. The values of Vj+ij can be computed from Eq. (10.31), starting at the point next to the lower boundary and computing upwards in the positive y direction. The stability criteria for this method are... 

For example, the inviscid solution for flow over a flat plate is simply that the velocity is constant everywhere and equal to the velocity in the undisturbed flow ahead of the plate, say wi. In calculating the boundary layer on a flat plate, therefore, the outer boundary condition is that u must tend to u at large v. The terr large y is meant to imply outside the boundary layer , the boundary layer thickness. S, being by assumption small. 

In the preceding sections, the solution for boundary layer flow over a flat plate wav obtained by reducing the governing set of partial differential equations to a pair of ordinary differential equations. This was possible because the velocity and temperature profiles were similar in the sense that at all values of x, (u u ) and (Tw - T)f(Tw - T > were functions of a single variable, 17, alone. Now, for flow over a flat plate, the freestream velocity, u, is independent of x. The present section is concerned with a discussion of whether there are any flow situations in which the freestream velocity, u 1, varies with Jr and for which similarity solutions can still be found [1],[10]. 

It may be recalled that it was deduced from the similarity solution for flow over a flat plate that Six) = 5/jRex. The difference between the value of the coefficient in this equation, i.e., 5, and the value in Eq. (3.136), i.e., 4.64, has no real significance since, in deriving the similarity solution result, it was arbitrarily assumed that the boundary layer thickness was the distance from the wall at which u became equal to 0.99 m. 

Air flows at a velocity of 9 m/s over a wide flat plate that has a length of 6 cm in the flow direction. The air ahead of the plate has a temperature of 10°C while the surface of the plate is kept at 70°C. Using the similarity solution results given in this chapter, plot the variation of local heat transfer rate in W/m2 along the plate and the velocity and temperature profiles in the boundary layer on the plate at a distance of 4 cm from the leading edge of the plate. Also calculate the mean heat transfer rate from the plate. 

As discussed in the previous chapter, most early efforts at trying to theoretically predict heat transfer rates in turbulent flow concentrated on trying to relate the wall heat transfer rate to the wall shear stress [1],[2],[3],[41. The reason for this is that a considerable body of experimental and semi-theoretical knowledge concerning the shear stress in various flow situations is available and that the mechanism of heat transfer in turbulent flow is obviously similar to the mechanism of momentum transfer. In the present section an attempt will be made to outline some of the simpler such analogy solutions for boundary layer flows, attention mainly being restricted to flow over a flat plate. 

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... 

Solution. The following integrals arise in the approximate solution for turbulent natural convective boundary layer flow over a flat plate discussed above ... 

Using the similarity solution results, derive an expression for the maximum velocity in the natural convective boundary layer on a vertical flat plate. At what position in the boundary layer does this maximum velocity occur ... 

Consider laminar free-conveqtive flow over a vertical flat plate at whose surface the heat transfer rate per unit area, qw, is constant. Show that a similarity solution to the two-dimensional laminar boundary layer equations can be derived for this case. 

This is the momentum equation of the laminar boundary layer with constant properties. The equation may be solved exactly for many boundary conditions, and the reader is referred to the treatise by Schlichting ll] for details of the various methods employed in the solutions. In Appendix B we have included the classical method for obtaining an exact solution to Eq. (5-13) for laminar flow over a flat plate. For the development in this chapter we shall be satisfied with an approximate analysis which furnishes an easier solution without a loss in physical understanding of the processes involved. The approximate method is due to von Karman [2],... 

Derive an expression for the heat transfer in a laminar boundary layer on a flat plate under the condition = w, = constant. Assume that the temperature distribution is given by the cubic-parabola relation in Eq. (5-30). This solution approximates the condition observed in the flow of a liquid metal over a flat plate. 

To show how one might proceed to analyze a new problem to obtain an important functional relationship from the differential equations, consider the problem of determining the hydrodynamic-boundary-layer thickness for flow over a flat plate. This problem was solved in Chap. 5, but we now wish to make an order-of-magnitude analysis of the differential equations to obtain the functional form of the solution. The momentum equation... 

Temperature profiles for flow over an isothermal flat plate are similar, just like the velocity profiles, and thus we expect a similarity solution for temperature to exist. Further, the thickness of the thermal boundar y layer is proportional to /i. T/V,just like the thickness of the velocity boundary layer, and thus the similarity variable is also t), and 0 = 6(ri). Using thechain rule and substituting the It and tt e.xpres ions from Eqs. 6-46 and 6—47 into the energy equation gives... 

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