Laminar boundary layer similarity solution

N. B. Cohen, Boundary-Layer Similar Solutions and Correlation Equations for Laminar Heat Transfer Distribution in Equilibrium Air at Velocities up to 41,000 Feet per Second, NASA Tech. Rep. R-118, 1961. 

Dhir and Lienhard [118] studied laminar film condensation on two-dimensional isothermal surfaces for which boundary layer similarity solutions exist and found that a similarity solution exists for body shapes that give g(x) = x". Nakayama and Koyama [119] extended the analysis of arbitrarily shaped bodies to include turbulent film condensation. 

Numerically determine the heat transfer rate variation with two-dimensional laminar boundary layer air flow over a fiat plate with a uniform heat flux at the surface. Compare the numerical results with those given by the similarity solution. 

Cohen, C.B. and Reshotko, E., Similar Solutions for die Compressible Laminar Boundary Layer with Heat Transfer and Pressure Gradient , NASA Report 1293, Houston, TX, 1956. 

SIMILARITY SOLUTIONS FOR FREE CONVECTIVE LAMINAR BOUNDARY LAYER FLOWS... 

A numerical solution to the laminar boundary layer equations for natural convection can be obtained using basically the same method as applied to forced convection in Chapter 3. Because the details are similar to those given in Chapter 3, they will not be repeated here. 

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. 

Before turning to a discussion of other methods of solving the laminar boundary layer equations for combined convection, a series-type solution aimed at determining the effects of small forced velocities on a free convective flow will be considered. In the analysis given above to determine the effect of weak buoyancy forces on a forced flow, the similarity variables for forced convection were applied to the equations for combined convection. Here, the similarity variables that were previously used in obtaining a solution for free convection will be applied to these equations for combined convection. Therefore, the following similarity variable is introduced ... 

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 ... 

Similar solutions for Prandtl numbers other than unity may be obtained from Eqs. 6.117 and 6.118 or their equivalent. A major simplification is the independence of the momentum equation (Eq. 6.117), from the energy equation (Eq. 6.118), which makes/independent of /. Also, the linear form of the energy equation in / permits handling arbitrary surface temperature distributions as in the case of the flat plate. (See the section on the two-dimensional laminar boundary layer.)... 

Nonuniform Surface Temperature. Nonuniform surface temperatures affect the convective heat transfer in a turbulent boundary layer similarly as in a laminar case except that the turbulent boundary layer responds in shorter downstream distances The heat transfer to surfaces with arbitrary temperature variations is obtained by superposition of solutions for convective heating to a uniform-temperature surface preceded by a surface at the recovery temperature of the fluid (Eq. 6.65). For the superposition to be valid, it is necessary that the energy equation be linear in T or i, which imposes restrictions on the types of fluid property variations that are permitted. In the turbulent boundary layer, it is generally required that the fluid properties remain constant however, under the assumption that boundary layer velocity distributions are expressible in terms of the local stream function rather than y for ideal gases, the energy equation is also linear in T [%]. 

C. B. Cohen and E. Reshotko, Similar Solutions for the Compressible Laminar Boundary Layer With Heat Transfer and Pressure Gradient, NACA Rep. 1293, 1956. 

Laminar boundary layer flow and heat transfer from a moving plate to a quiescent fluid under uniform wall temperature (UWT) and uniform heat flux (UHF) boundary conditions have been studied. Similarity solutions of the incompressible boundary layer equations with... 

The continuity equations for mass, x-direction momentum, chemical species and energy in the plane, stationary, laminar boundary layer flow have already been given as Eqs. (7.1) to (7.4). The stream function ij/, by means of which the mass continuity equation is automatically satisfied, is defined by Eqs. (7.5). Following the approaches of Lees (1956), Fay and Riddell (1958), and Chung (1965), self-similar solutions in the stagnation region are obtained via transformations from (x, y) co-ordinates to the two new variables... 

Some of the commonly used methods for obtaining solutions to problems involving laminar external flows have been discussed in this chapter. Many such problems can be treated with adequate accuracy using the boundary layer equations and similar ity integral and numerical methods of solving these equations have been discussed. A brief discussion of the solution of the full governing equations has also been presented. 

The physical situation where the fluid stream parallel to the moving plate is in an opposite direction to the motion of the plate (Fig. 18.11c) is also encountered in materials processing. Using a similarity method, Klemp and Acrivos [75] found that a critical value of the moving surface to the free stream velocity ratio (A/ = USIUJ) was 0.3541. The inability to obtain similarity solutions of the boundary layer equations for laminar flow was attributed to the boundary layer separation from the moving plate. Similarity and integral solutions for fluid friction... 

Similar to the momentum boundary layer entrance length in a closed channel, there is a thermal boundary layer development region. For laminar fully developed flow (Re < 3000), an exact solution can be found for different boundary conditions. For fully developed and laminar heat transfer in a rectangular channel with equal depth and width, the heat transfer coefficient is constant and can be determined as follows... 

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