Velocity profile in laminar boundary layer

CL Parameter in prescribed velocity profile in laminar boundary layer theory (—)... 

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... 

This transition has profound effects in all fluid dynamics, and certainly so in aerodynamics. The velocity profile in (he boundary layer becomes fuller neat the surface on account of Ihe higher average kinetic energy of the layer created by turbulent energy exchange from layer lo layer. The effective viscosity is therefore larger in turbulent than laminar flow, ihe turbulent boundary layer thickens more rapidly downstream, the skin friction increases. 

Consider the vertical flat plate shown in Fig. 7-1. When the plate is heated, a free-convection boundary layer is formed, as shown. The velocity profile in this boundary layer is quite unlike the velocity profile in a forced-convection boundary layer. At the wall the velocity is zero because of the no-slip condition it increases to some maximum value and then decreases to zero at the edge of the boundary layer since the free-stream conditions are at rest in the free-convection system. The initial boundary-layer development is laminar but at... 

Velocity profiles in turbulent boundary layer are determined experimentally since solutions to the flow equations are not available. Based on experimental data, universal velocity profile is proposed in which the boundary layer is divided into four regions—laminar sublayer, buffer zone, turbulent core, and turbulent wake. The turbulent velocity profile is given in terms of dimensionless velocity defined as... 

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 ... 

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

Obtain the momentum equation for an element of boundary layer. If the velocity profile in the laminar region may be represented approximately by a sine function, calculate the boundary-layer thickness in terms of distance from the leading edge of the surface. 

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. ... 

At velocities greater than the critical, the fluid velocity profile in the conduit is uniform across the conduit diameter except for a thin layer of fluid at the conduit wall. This boundary layer continues to move in laminar flow. In connection with flow measurement, most flowmeters have constant coefficients under turbulent flow conditions. Some flowmeters have the advantage of constant coefficients over Reynolds Number ranges encompassing both turbulent and laminar flows. See also Fluid and Fluid Flow and Reynolds Number. 

The way in which the momentum integral equation is applied will be discussed in detail in the next chapter. Basically, it involves assuming the form of the velocity profile, i.e., of the variation of u with y in the boundary layer. For example, in laminar flow a polynomial variation is often assumed. The unknown coefficients in this assumed form are obtained by applying the known condition on velocity at the inner and outer edges of the boundary layer. For example, the velocity must be zero at the wall while at the outer edge of the boundary layer it must become equal to the freestream velocity, u. Thus, two conditions that the assumed velocity profile must satisfy are ... 

A number of experimental investigations have shown that the velocity profile in a turbulent boundary layer, outside the laminar sublayer, can be described by a one-seventh-power relation... 

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. 

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... 

Structure of the flow. Velocity profile. The flow in the boundary layer on a flat plate is laminar until Rex = U[X/v 3 x 105. On a longer plate, the boundary layer becomes turbulent, that is, its thickness increases sharply and the longitudinal velocity profile alters. 

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 6.5 Velocity profiles in the laminar boundary layer of a liquid,... 

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. 

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