Laminar boundary layer profiles

Equation 11.12 corresponds closely to experimentally determined velocity profiles in a laminar boundary layer. 

To describe the velocity profile in laminar flow, let us consider a hemisphere of radius a, which is mounted on a cylindrical support as shown in Fig. 2 and is rotating in an otherwise undisturbed fluid about its symmetric axis. The fluid domain around the hemisphere may be specified by a set of spherical polar coordinates, r, 8, , where r is the radial distance from the center of the hemisphere, 0 is the meridional angle measured from the axis of rotation, and (j> is the azimuthal angle. The velocity components along the r, 8, and (j> directions, are designated by Vr, V9, and V. It is assumed that the fluid is incompressible with constant properties and the Reynolds number is sufficiently high to permit the application of boundary layer approximation [54], Under these conditions, the laminar boundary layer equations describing the steady-state axisymmetric fluid motion near the spherical surface may be written as ... 

FIGURE7-1 Schematic of flow in daughter brandies of bifurcation model for steady inspiratory flow with flat profile in parent bran<. Velocity profiles in plane of bifurcation (—) and in normal plane (—) are indicated in right branch. Orientation of secondary flows and position of laminar boundary layer are shown in left branch. Redrawn wifo permisnon from Bdl. ... 

The burner in the test facility, shown in Fig. 18.1, is an axisymmetric nozzle, which is concentrically placed into the circular suction collar. To achieve a top-hat velocity profile with laminar boundary layer at the nozzle exit, a fourth order polynomial with a large contraction ratio of 31.6 1 and an exit diameter of D = 10.16 mm is used in the design. The suction collar assembly is connected to a vacuum pump through a series of solenoid valves so that a counterflow, which is in the opposite direction of the fuel-air mixture flow, can be established... 

Figure 5.4 Schematic drawing illustrating the concentration profiles of a salt in the laminar boundary layer on both sides of a cation-exchange membrane and the flux of ions in the solutions and the membrane.

The concentration polarization occurring in electrodialysis, that is, the concentration profiles at the membrane surface can be calculated by a mass balance taking into account all fluxes in the boundary layer and the hydrodynamic conditions in the flow channel between the membranes. To a first approximation the salt concentration at the membrane surface can be calculated and related to the current density by applying the so-called Nernst film model, which assumes that the bulk solution between the laminar boundary layers has a uniform concentration, whereas the concentration in the boundary layers changes over the thickness of the boundary layer. However, the concentration at the membrane surface and the boundary layer thickness are constant along the flow channel from the cell entrance to the exit. In a practical electrodialysis stack there will be entrance and exit effects and concentration... 

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. 

Fig. B-2 Temperature profiles in laminar boundary layer with isothermal wall.

For laminar airflow in a tube, when 8 approaches the tube radius, Poiseuille flow or a parabolic flow profile is fully developed. This is accomplished by the acceleration of the central portion of the flow. However, when Re exceeds a value lying somewhere between 104 and 106, the laminar boundary layer becomes so thick that it is no longer stable, and a turbulent boundary layer develops. 

It is noted that the integral method gives only approximate values for the mass transfer coefficient as the model derivation is based on several simplifying assumptions regarding the concentration and velocity profiles. Nevertheless, the given relation has been confirmed by experiments for laminar boundary layer flows over a flat plate (e.g., [134], p 80 and p 201 [27], p 345). 

B model parameter in logarithmic velocity profile (—) b constant of integration in laminar boundary layer theory (—)... 

When a gas enters a smooth pipe from a large reservoir through a well-faired entry, a laminar boundary layer forms along the walls. The velocity profile in the main body of the How remains flat. The velocity boundary layer thickens with distance downstream from the entry until it eventually fills the pipe. If the Reynolds number based on pipe diameter is less than 2100, the pipe boundary layer remains laminar. The flow is said to be fully... 

Many numerical and series solutions for the laminar boundary layer model of mass transfer are available for situations such as flow in coeduits under conditions of fully developed or developing concentration or velocity profiles. Skellaed31 provides a particularly good summary of these results. The laminar boundary layer model has been extended to predict tha effects of high mass transfer flux on the mass transfer coefficient from a flat plate. The results of this work ate shown in Fig. 2.4-2 and. in com rest to the other theories, iedicate a Schmith number dependence of Ihe correction factor. 

FIGURE 6.1 Similar velocity profile in the laminar boundary layer on a flat plate—constant fluid properties. 

FIGURE 6.3 Temperature profiles in the laminar boundary layer on an insulated plate—constant-property, high-speed flow. 

FIGURE 6.8 Laminar boundary layer enthalpy profile function on a uniform-temperature flat plate, C, = 1 [5]. 

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 boundary layers on curved surfaces, the pressure will change with distance. This greatly complicates the solution of the boundary-layer equations compared with that on a flat plate (in which dPIdx was zero), and so very few exact solutions are known for such boundary layers. Some estimate of the behavior of such boundary layers is given by several methods. To illustrate, we apply them to the laminar boundary layer on a flat plate, where we can compare the results with Blasius exact solution. These methods begin by assuming a velocity profile of the form V tV where S is the boundary-layer thickness. 

Here a strongly flattened flow profile (Eigure 2.4-2) is evident, which drops rapidly to zero only near the wall of the pipe. The thickness S of the laminar boundary layer can be estimated by using Equation (2.4-14) ... 

If a fluid flows past a plate at bulk velocity w, then close to the surface a velocity, temperature and concentration profile develops. These profiles are approximated by laminar boundary layers, of characteristic thickness. In these boundary layers the resistances to momentum, heat and mass transfer are located. As long as the Reynolds number Re = w L p/ rj, calculated with plate length L, is smaller than 10 , the flow is laminar. 

Following assumptions are made for the analysis (1) Flow is laminar and two dimensicHial with uniform jet velocity. (2) Upon impingement, laminar boundary layer develops. (3) A quadratic velocity profile is assumed for the flow inside the... 

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