Laminar boundary layer velocity distribution

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 [%]. 

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. 

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. 

An aerosol with particles in the micron size range flows around a smooth solid sphere a few millimeters in diameter. At sufficiently high Reynolds numbers, a laminar boundary layer develop.s over the sphere from the stagnation point up to an angle of about 110 at which. separation takes place. The removal of particles by direct interception can be calculated from the velocity distribution over the forward. surface of the sphere, up to 90 from the forward stagnation point (Fig. 4.P4). 

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. 

For the laminar boundary layer flows of incompressible Newtonian fluids over a wide plate, Schhchting (Boundary Layer Theory, 6th edn.. Me Graw Hill, New York, 1965) showed that the following two equations for the velocity distributions give values of the shear stress and friction factor which are comparable with those obtained using equation (7.10) ... 

In order to see how the boundary layer affects particle detachment, let us turn to Fig. X. 1. Depending on the flow velocity, the boundary layer may be either laminar (Fig. X. 1. a) or turbulent (Fig. X. 1. b). The laminar boundary layer is characterized by a linear velocity distribution in the layer. The adherent particles may be completely submerged in this layer if the particle diameter is smaller than the boundary layer thickness (see Position I in Fig. X.l.a). Position II shows the case in which the diameter of the adherent particle is greater than the boundary layer thickness. 

Detachment of Particles Situated in a Laminar Boundary Layer. The action of an air stream on adherent particles under the conditions of a laminar boundary layer (see Fig. X.l. a) or a laminar boundary sublayer (see Fig. X.l. b) have certain features in common but others that differ. The common feature is that there is a linear velocity distribution across the thickness of the laminar layer or sublayer (line c). The difference is that the laminar boundary layer is in direct contact with an air flow having a velocity of Uoo. In the case of a turbulent boundary layer, there is no direct contact between the laminar sublayer and the free stream, which are separated by a buffer layer 3 and a turbulent core 4 (see Fig. X.l.b). These features do affect the drag. 

With a linear velocity distribution in a laminar boundary layer, the velocity at the level of the particle center can be expressed by the equation... 

Introduction and derivation of integral expression. In the solution for the laminar boundary layer on a fiat plate, the Blasius solution is quite restrictive, since it is for laminar flow over a flat plate. Other more complex systems cannot be solved by this method. An approximate method developed by von Karman can be used when the configuration is more complicated or the flow is turbulent. This is an approximate momentum integral analysis of the boundary layer using an empirical or assumed velocity distribution. 

Integral momentum analysis for turbulent boundary layer. The procedures used for the integral momentum analysis for laminar boundary layer can be applied to the turbulent boundary layer on a flat plate. A simple empirical velocity distribution for pipe flow which is valid up to a Reynolds number of 10 can be adapted for the boundary... 

In this development the turbulent boundary layer was assumed to extend to x = 0. Actually, a certain length at the front has a laminar boundary layer. Experimental data check Eq. (3.10-61) reasonably well from a Reynolds number of 5 x 10 to 10. More accurate results at higher Reynolds numbers can be obtained by using a logarithmic velocity distribution, Eqs. (3.l0-45)-(3.10-47). 

The frontal pressure (F ) depends on the thickness of the boundary layer. For a linear distribution of velocity and a laminar boundary layer [284], the force of interaction between the flow and the particles is given by Stokes law ... 

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 laminar flow of a fluid over a flat plate. Use the folly implicit method of finite differencing to compute the two dimensionless velocity-component distributions within the boundary layer. 

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. 

The cause of large drag in the case of a body like a circular cylinder is the asymmetry in the velocity and pressure distributions at the cylinder surface that results from separation. All bodies in laminar streaming flow at large Reynolds number are subjected to viscous stresses that boundary-layer analysis shows must be... 

The flow in the wake behind a body moving in an unbounded liquid possesses all the properties of free jet flows and can be calculated by methods of the boundary layer theory [427]. Note that the wake behind a moving body is almost invariably turbulent even if the boundary layer on the body surface is laminar. This is due to the fact that there are points of inflection on all velocity profiles of the wake such velocity distributions are known to be particularly unstable. 

One approach to the solution of this equation is to assume that the pressure does not vary with v, and so it is specified by the external velocity distribution, v is computed from the continuity equation, leaving a differential equation in u to be integrated. However, this holds only for the laminar flow since turbulent boundary layers are inherently unsteady. The subsequent sections deal with the solutions of these equations in more detail. 

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. 

For infinite diluted solutions, Sc 1000, therefore Sd O.ldu. Consequently, Sd Su, and the distribution of velocities in the diffusion boundary layer may be determined independently of the appropriate hydrodynamic problem. As an example, consider stationary laminar flow of viscous incompressible liquid in a flat channel. It is known that at some distance from the channel entrance, the velocity profile changes to a parabolic Poiseuille profile (Fig. 6.2) [5]. 

Concepts have now been worked out [273] for the three-dimensional structure of the turbulent boundary layer (see Fig. X. 1. b). Between the turbulent core 4 and the laminar sublayer 2 there is a buffer layer 3. In the turbulent boundary layer there is a laminar sublayer in which the velocity distribution is linear. 

Fig. X.1. Structure of laminar (a) and turbulent (b) boundary layers and velocity distribution in these layers (c) (1) laminar layer (2) laminar sublayer (3) buffer layer (4) turbulent core.

Adler measured the velocity distribution in a coiled pipe for laminar flow [86]. He found that the velocity profile differed considerably from the parabolic one which was due to the existence of secondary flow field. Using the boundary layer theory he derived the following relationship for relatively high flow rate but laminar range. 

The character of the fully developed region is determined by the character of the flow in the boundary layer at the cusp (Fig. 2.4). If it is laminar, the fully developed flow will be laminar (Fig. 2.4a). Or the flow may be transitional (Fig. 2.4b) or completely turbulent (Fig. 2.4c). The latter, in a pipe, channel, or annulus, has three zones of flow behavior which reflect those of the turbulent boundary layer. The velocity distribution depicted in Fig. 2.1 corresponds to the latter regime. 

The flow of liquid caused by electro-osmosis displays a pluglike profile because the driving force is uniformly distributed along the capillary tube. Consequently, a uniform flow velocity vector occurs across the capillary. The flow velocity approaches zero only in the region of the double layer very close to the capillary surface. Therefore, no peak broadening is caused by sample transport carried out by the electro-osmotic flow. This is in contrast to the laminar or parabolic flow profile generated in a pressure-driven system, where there is a strong pressure drop across the capillary caused by frictional forces at the liquid-solid boundary. A schematic representation of the flow profile due... 

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