Laminar boundary layer temperature distribution

Pohlhausen, 1911 [1] solved these equations first, whereas Schmidt and Beckmann, 1930 [2] solved them for Pr = 0.733 in 1930. Ostrach, 1953 [3], solved the same equations for the range 0.01 to 1000. For free convection laminar boundary layer on a heated vertical plate in that range of Pr, the velocity and the temperature distributions are shown in Figs. 9.2 and 9.3, respectively. 

Consider two-dimensional laminar boundary layer flow over a flat isothermal surface. Very close to the surface, the velocity components are very small. If the pressure changes are assumed to be negligible in the flow being considered, derive an expression for the temperature distribution near the wall. Viscous dissipation effects should be included in the analysis. 

Temperature distributions in a laminar boundary layer with and without viscous dissipation. 

Using the linear-velocity profile in Prob. 5-2 and a cubic-parabola temperature distribution [Eq. (5-30)], obtain an expression for heat-transfer coefficient as a function of the Reynolds number for a laminar boundary layer on a flat plate. 

FIGURE 6.2 Temperature distributions in the laminar boundary layer on a flat plate at uniform temperature—constant property, low-speed flow. 

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 [%]. 

D. R. Chapman and M. W. Rubesin, Temperature and Velocity Profiles in the Compressible Laminar Boundary Layer With Arbitrary Distribution of Surface Temperature, J. Aeronaut. Sci. (16) 547-565,1949. 

A. N. Tifford and S. T. Chu, Heat Transfer in Laminar Boundary Layers Subject to Surface Pressure and Temperature Distributions, Proc. 2d Midwestern Conf. Fluid Mech., p. 363,1952. 

FIGURE 4.24 Laminar and turbulent boundary layers and temperature distribution inside the boundary layer. 

Such expressions can be extended to permit the evaluation of the distribution of concentration throughout laminar flows. Variations in concentration at constant temperature often result in significant variation in viscosity as a function of position in the stream. Thus it is necessary to solve the basic expressions for viscous flow (LI) and to determine the velocity as a function of the spatial coordinates of the system. In the case of small variation in concentration throughout the system it is often convenient and satisfactory to neglect the effect of material transport upon the molecular properties of the phase. Under these circumstances the analysis of boundary layer as reviewed by Schlichting (S4) can be used to evaluate the velocity as a function of position in nonuniform boundary flows. Such analyses permit the determination of material transport from spheres, cylinders, and other objects where the local flow is nonuniform. In such situations it is not practical at the present state of knowledge to take into account the influence of variation in the level of turbulence in the main stream. 

Consider convection with incompressible, laminar flow of a constant-temperature fluid over a flat plate maintained at a constant temperature. With the velocity distributions found in either Prob. 10.1 or Prob. 10.2, compute the dimensionless temperature distribution within the thermal boundary layer for the Peclet number equal to 0.1,1.0,10.0,100.0. Use the ADI method. 

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