Similarity solutions flat plate

Similarity Variables The physical meaning of the term similarity relates to internal similitude, or self-similitude. Thus, similar solutions in boundaiy-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 theoiy is available in Ames (see the references). 

The diffusivities thus obtained are necessarily effective diffusivities since (1) they reflect a migration contribution that is not always negligible and (2) they contain the effect of variable properties in the diffusion layer that are neglected in the well-known solutions to constant-property equations. It has been shown, however, that the limiting current at a rotating disk in the laminar range is still proportional to the square root of the rotation rate if the variation of physical properties in the diffusion layer is accounted for (D3e, H8). Similar invariant relationships hold for the laminar diffusion layer at a flat plate in forced convection (D4), in which case the mass-transfer rate is proportional to the square root of velocity, and in free convection at a vertical plate (Dl), where it is proportional to the three-fourths power of plate height. 

When heat and mass are transferred simultaneously, the two processes interact through the Gr and Gq terms in Eq. (10-12) and the energy and diffusion equations. Although solutions to the governing equations are not available for spheres, results should be qualitatively similar to those for flat plates (T4), where for aiding flows (Gr /Gq > 0) the transfer rate and surface shear stress are increased, and for opposing flows (Gr Gq < 0) the surface shear stress is predicted to drop to zero yielding an unstable flow. 

The torque of rotating cylinders is also reduced in polymer solutions under turbulent flow conditions. Analytical studies of the flow of polymer solutions over flat plates have all been based upon the similarity between flow in pipes and the flat plate flow. 

An interesting class of exact self-similar solutions (H2) can be deduced for the case where the newly formed phase density is a function of temperature only. The method involves a transformation to Lagrangian coordinates, based upon the principle of conservation of mass within the new phase. A similarity variable akin to that employed by Zener (Z2) is then introduced which immobilizes the moving boundary in the transformed space. A particular case which has been studied in detail is that of a column of liquid, initially at the saturation temperature T , in contact with a flat, horizontal plate whose temperature is suddenly increased to a large value, Tw T . Suppose that the density of nucleation sites is so great that individual bubbles coalesce immediately upon formation into a continuous vapor film of uniform thickness, which increases with time. Eventually the liquid-vapor interface becomes severely distorted, in part due to Taylor instability but the vapor film growth, before such effects become important, can be treated as a one-dimensional problem. This problem is closely related to reactor safety problems associated with fast power transients. The assumptions made are ... 

SIMILARITY SOLUTIONS FOR FLOW OVER FLAT PLATES WITH OTHER THERMAL BOUNDARY CONDITIONS... 

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]. 

The basic procedure for arriving at the similarity solution is, of course, the same as that for flat plate flow. The continuity equation is first used to express v in terms of the similarity function /. For this purpose it is written as ... 

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. 

Having established that similarity solutions for the velocity profile can be found for certain flows involving a varying ffeestream velocity, attention must now be turned to the solutions of the energy equation corresponding to these velocity solutions. The temperature is expressed in terms of the same nondimensional variable that was used in obtaining the flat plate solution, i.e., in terms of 8 = (Tw - T)f(Tw -Tt) and it is assumed that 0 is also a function of ij alone. Attention is restricted to flow over isothermal surfaces, i.e., with Tw a constant, and T, of course, is also constant. 

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. 

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. 

For a Prandtl number jpf 0.7, die similarity solution for flow over an isothermal flat plate gives ... 

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. 

In order to illustrate how similarity solutions are obtained for free convective flows, see [7] to [23], consideration will initially be given to two-dimensional flow over a vertical flat plate with a uniform surface temperature. The situation being considered is thus, as shown in Fig. 8.6. 

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. 

Show that a similarity-type solution can be obtained for the case of two-dimensional flow over a flat plate in a porous medium, the plate being aligned with a forced flow... 

Similarity Solutions for Flow over Flat Plates with... 

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... 

You saw how the equations governing energy transfer, mass transfer, and fluid flow were similar, and examples were given for one-drmensional problems. Examples included heat conduction, both steady and transient, reaction and diffusion in a catalyst pellet, flow in pipes and between flat plates of Newtonian or non-Newtonian fluids. The last two examples illustrated an adsorption column, in one case with a linear isotherm and slow mass transfer and in the other case with a nonlinear isotherm and fast mass transfer. Specific techniques you demonstrated included parametric solutions when the solution was desired for several values of one parameter, and the use of artificial diffusion to smooth time-dependent solutions which had steep fronts and large gradients. 

The conclusion from the previous paragraph is that similarity solutions of the momentum boundary-layer equations should not generally be expected. An interesting question is whether similarity solutions can be obtained in any case other than the flat plate problem in the previous section. To answer this question, we start with the boundary-layer equations in their most general form ... 

The only distinction between bodies of different geometry is the function ue(x). Thus the question we wish to answer is whether similarity solutions exist for functions ue(x) other than ue(x) = const (the flat plate problem). 

In general, (10-89) requires an initial profile atx = 0 (corresponding to the most upstream point in the boundary layer). Evidently, if a similarity solution does exist, the boundary condition (10-97) must represent this initial condition, as well as the boundary condition for Y oo. This implies that g(0) = 0 that is, the boundary-layer thickness must go to zero as x 0. This can occur only for a body that has a pointed leading edge (such as the flat plate). 

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 ... 

Figure 11-7. A comparison of the asymptotic forms for the local Nusselt number correlation for Pr

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