Flow over a flat plate

Similar correlations are available with regard to other physical situations such as fluid flow over a flat plate, a sphere or a cylinder. 

For flow parallel to an electrode, a maximum in the value of the mass-transfer rate occurs at the leading edge of the electrode. This is not only the case in flow over a flat plate, but also in pipes, annuli, and channels. In all these cases, the parallel velocity component in the mass-transfer boundary layer is practically a linear function of the distance to the electrode. Even though the parallel velocity profile over the hydrodynamic boundary layer (of thickness h) or over the duct diameter (with equivalent diameter de) is parabolic or more complicated, a linear profile within the diffusion layer (of thickness 8d) may be assumed. This is justified by the extreme thinness of the diffusion layer in liquids of high Schmidt number ... 

An illustration of the three methods is given for the problem of incompressible flow over a flat plate. The flow is steady and two-dimensional, as shown in Figure 12.1, with the approaching constant flow speed designated as u00. 

For flow over a flat plate of length L the flow is laminar if Re/, <10, and... 

Mass transfer can produce films of nonuniform thickness because the deposition rate can depend on the velocity field u over the sohd. Regions with high whl have the highest deposition rates in a mass-transfer-hmited process. For flow over a flat plate of length L the average Sherwood number for laminar flow is given by the expression... 

If a fluid such as air flows over a flat plate placed with its surface parallel to the stream, particles in the vicinity of the surface are slowed down by viscous forces. Fluid particles adjacent to the surface stick to it and have zero velocity relative to the boundary. Other fluid particles are retarded as a result of sliding over the immobilised particles. The effects of viscous forces originating at the boundary extend for a certain distance (5, the boundary layer thickness). The effects of viscous forces originating at the boundary are not extensive and the velocity soon approaches free stream velocity. 

The boundary layer equations were derived in a previous chapter, or may be deduced from the general convection equations in the early part of this chapter. For two-dimensional, steady flow over a flat plate of an incompressible, constant-property fluid, the continuity, x-momentum and the energy equations are as follows ... 

Consider flow over a flat plate. The computation is started by assuming that uy = u,y, at the leading edge and v, = 0. The value of vy is needed in the explicit algorithm to move on to the i+1 level. It is not required to specify the initial values of vy in the formal mathematical formulation of the partial differential equation. A suitable initial distribution for vy can be obtained by using the continuity equation to eliminate du/5x from the x-momentum equation. For a laminar, incompressible flow, this means that... 

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. 

Similarity solutions for a few cases of flow over a flat plate where the plate temperature varies with x in a prescribed manna can also be obtained. In an such cases the solution for the velocity profile is, of course, not affected by the boundary condition... 

For any prescribed values of Pr and n, 0 It o will have a specific value. It therefore follows that qw will be proportional to (Tw - T )/xo s. Hence, the case where the heat flux at the surface of the plate is uniform corresponds to the case where n - 0.5, i.e., a similarity solution exists for flow over a flat plate with a uniform surface heat flux. 

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]. 

Now the form of the similarity variable, 17, for flow over a flat plate was arrived at by noting that if the velocity profiles were similar then ... 

It may be noted that the case flow over a flat plate while the case of equal to 7r, i.e., m equal to one, corresponds to flow over a plate set normal to the direction of the undisturbed freestream as shown in Fig. 3.11. The latter flow closely represents flow near the stagnation point on a bluff body. 

The profile for m equal to zero corresponds to flow over a flat plate and is identical to the solution given in the previous section for this type of flow. In making a comparison with this solution it should be noted that for m = 0, r/ is equal to y/u /2x v, i.e., tj = 17/ v 2.17 being the similarity variable used in deriving the flat plate solution. 

The boundary conditions on 0 are the same as those for flow over a flat plate, i.e.,... 

The values for m 0 are, of course, the same as the values of a given in the previous section for flow over a flat plate. 

Now for the case of flow over a flat plate for which u is constant, the momentum integral equation (2.173) can be written as... 

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. 

When Pr is equal to 1, A is, of course, equal to 1 because the form of the assumed velocity and temperature profiles are identical and when Pr is equal to one the momentum and energy integral equations have the same form for flow over a flat plate. 

The problem to which the integral equation method was applied in the above discussion, i.e., flow over an isothermal plate, is, of course, one for which a similarity solution can be found. The usefulness of the integral equation method, however, arises mainly from the fact that it can be applied to problems for which similarity solutions cannot easily be found. In order to illustrate this ability, consider flow over a flat plate which has an unheated section adjacent to the leading edge as shown in Fig. 3.15. 

Some consideration must be given to the conditions existing along the initial i = 1 line which were assumed to be known in the above discussion. The actual conditions will depend on the nature of the problem. For flow over a flat plate, because the boundary layer equations are parabolic in form, the use of these equations requires that the plate have no effect on the flow upstream of the plate. Hence, in this case, the variables will have their freestream values at all the nodal points (except at the point which lies on the surface) on the initial line which is coincident with the leading edge. At the nodal point on the surface, the known conditions at the surface must apply. This is illustrated in Fig. 3.24. 

Solution. Because flow over a flat plate with an isothermal surface is being considered, the freestream velocity and surface temperature are constant. Therefore, the inputs to the program are ... 

VISCOUS DISSIPATION EFFECTS ON LAMINAR BOUNDARY LAYER FLOW OVER A FLAT PLATE... 

The equations governing the flow were discussed in Chapter 2. Because two-dimensional flow over a flat plate is being considered, these equations are ... 

In order to find the best value to use for the constant Ci in this equation, numerical solutions in which the effects of property variations are accounted for can be obtained for some simple situations such as for flow over a flat plate and these numerical results can then be used to deduce the value of Ci that leads to the best agreement between the results obtained accounting for property variations and results obtained assuming constant fluid properties evaluated at Tprop- This procedure indicates that Ci should be taken as 0.22 [17],[18],[19],[20],[21],[22], i.e., that ... 

Consider laminar forced convective flow over a flat plate at whose surface the heat transfer rate per unit area, qw is constant. Assuming a Prandtl number of 1, use the integral equation method to derive an expression for the variation of surface temperature. Assume two-dimensional flow. 

Air at 5°C and 70 kPa flows over a flat plate at 6 m/s. A heater strip 2.5 cm long is placed on the plate at a distance of 5 cm from the leading edge. Calculate the heat lost from the strip per unit depth of plate for a heater surface temperature of 65°C. Use the appropriate integral equation result... 

Air at a Mach number of 3 and a temperature of -30°C flows over a flat plate that is aligned with the flow. The plate is kept at a temperature of 25°C. The flow in the boundary layer is laminar. Is lmat transferred to or from die plate surface ... 

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 ... 

Transition in boundary layer flow over a flat plate. 

Consider transition in the boundary layer flow over a flat plate. Using the expression for the thickness of a laminar boundary layer on a flat plate given in Chapter 3, find the value of the Reynolds number based on the boundary layer thickness at which transition begins. 

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. 

If the turbulent Prandtl number, Prj. is taken as 1 and if flow over a flat plate is considered, Cf then being assutned to be given by Eq. (6.17), i.e., by ... 

The analogy solutions discussed in the previous section use the value of the wall shear stress to predict the wall heat trans er rate. In the case of flow over a flat plate, this wall shear stress is given by a relatively simple expression. However, ir, general, the wall shear stress will depend on the pressure gradient and its variation has to >e computed for each individual case. One approximate way of determining the shear stress distribution is based on the use of the momentum integral equation that was discussed in Chapter 2 [1],[2],[3],[5]. As shown in Chapter 2 (see Eq. 2.172), this equation has the form ... 

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