Momentum boundary layer thickness

The procedure here is similar to that adopted previously. A heat balance, as opposed to a momentum balance, is taken over an element which extends beyond the limits of both the velocity and thermal boundary layers. In this way, any fluid entering or leaving the element through the face distant from the surface is at the stream velocity u and stream temperature 0S. A heat balance is made therefore on the element shown in Figure 11.10 in which the length l is greater than the velocity boundary layer thickness S and the thermal boundary layer thickness t. 

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. 

Explain why it is necessary to use concepts, such as the displacement thickness and the momentum thickness, for a boundary layer in order to obtain a boundary layer thickness which is largely independent of the approximation used for the velocity profile in the neighbourhood of the surface. 

Momentum boundary layer calculations are useful to estimate the skin friction on a number of objects, such as on a ship hull, airplane fuselage and wings, a water surface, and a terrestrial surface. Once we know the boundary layer thickness, occurring where the velocity is 99% of the free-stream velocity, skin friction coefficient and the skin friction drag on the solid surface can be calculated. Estimate the laminar boundary layer thickness of a 1-m-long, thin flat plate moving through a calm atmosphere at 20 m/s. 

Equation (E4.4.1) is a nonlinear partial differential equation, because of the velocity u that appears in front of the velocity gradient du/dx. The boundary layer thickness is generally defined as the distance from the plate where the momentum reaches 99% of the free-stream momentum. We will assign (Blasius, 1908)... 

The boundary layer equations may be obtained from the equations provided in Tables 6.1-6.3, with simplification and by an order-of-magnitude study of each term in the equations. It is assumed that the main flow is in the x direction. The terms that are too small are neglected. Consider the momentum and energy equations for the two-dimensional, steady flow of an incompressible fluid with constant properties. The dimensionless equations are given by Eqs. (6.46) to (6.48). The principal assumption made in the boundary layer is that the hydrodynamic boundary layer thickness 8 and the thermal boundaiy layer thickness 8t are small compared to a characteristic dimension L of the body. In mathematical terms,... 

Integrate x-momentum equation with respect to y over the boundary layer thickness 8(x). Eliminate velocity component v(x,y) in the equation by means of the continuity equation, resulting in the momentum integral equation. 

The momentum boundary layer thickness is represented by 8, and the thermal boundary layer thickness is represented by 8,. 

Consider next the application of the conservation of energy principle to the control volume that was used above in the derivation of the momentum integral equation. The height, , of this control volume is taken to be greater than both the velocity and temperature boundary layer thicknesses as shown in Fig. 2.21. 

The boundary layer integral equations have been derived above without recourse to the partial differential equations for boundary layer flow. They can, however, be determined directly from these equations. Consider, for example, the laminar momentum equation (2.140). Integrating this equation across the boundary layer to some distance from the wall, i being greater than the boundary layer thickness, gives because du/dy is zero outside the boundary layer and because dp/dx is independent of y ... 

To determine the turbulent-boundary-layer thickness we employ Eq. (5-17) for the integral momentum relation and evaluate the wall shear stress from the empirical relations for skin friction presented previously. According to Eq. (5-52),... 

To show how one might proceed to analyze a new problem to obtain an important functional relationship from the differential equations, consider the problem of determining the hydrodynamic-boundary-layer thickness for flow over a flat plate. This problem was solved in Chap. 5, but we now wish to make an order-of-magnitude analysis of the differential equations to obtain the functional form of the solution. The momentum equation... 

Continuous Cylindrical Surface The continuous surface shown in Fig. 6-48fe is applicable, for example, for a wire drawn through a stagnant fluid (Sakiadis, AIChE J., 7, 26-28, 221-225, 467- 72 [1961]). The critical-length Reynolds number for transition is Re, = 200,000. The laminar boundary layer thickness, total drag, and entrainment flow rate may be obtained from Fig. 6-49 the drag and entrainment rate are obtained from the momentum area 0 and displacement area A evaluated at x = L. 

This velocity profile transforms the integral condition (3.165) for momentum into an ordinary differential equation for the still unknown boundary layer thickness 5... 

In this context the momentum boundary layer thickness y = d is conveniently defined as the point beyond which the velocity takes on its free stream value The second condition stating that the velocity gradient vanishes a,t y = S, ensures that we obtain a continuous gradient as the velocity attains its free stream value. 

It has been shown that there exists a continuous change in the physical behavior of the turbulent momentum boundary layer with the distance from the wall. The turbulent boundary layer is normally divided into several regions and sub-layers. It is noted that the most important region for heat and mass transfer is the inner region of the boundary layer, since it constitutes the major part of the resistance to the transfer rates. This inner region determines approximately 10 — 20% of the total boundary layer thickness, and the velocity distribution in this region follows simple relationships expressed in the inner variables as defined in sect 1.3.4. 

Although the dependence of the thermal boundary-layer thickness on the independent parameters Re and Pr (or Pe) remains to be determined, we may anticipate that the magnitudes of Re and Pe will determine the relative dimensions of the two boundary layers. If Pe yp Re yp 1, both the momentum and thermal layers will be thin, but it seems likely that the thermal layer will be much the thinner of the two. Likewise, if Pe Pe 1, we can guess that the momentum boundary layer will be thinner than the thermal layer. In the analysis that follows in later sections of this chapter, we consider both of the asymptotic limits Pr —> oc (Pe yy Re y> D and Pr 0 (Re yy> Pe p> 1). We shall see that the relative dimensions of the thermal and momentum layers, previously anticipated on purely heuristic grounds, will play an important and natural role in the theory. 

It may be noted that the functional dependence of the local heat flux onx is the same as the shear stress. This is a consequence of the fact that the thickness of the thermal boundary layer varies with x in the same way as the momentum boundary-layer thickness. Furthermore, the form of the correlations for large and small Prandtl numbers are also quite similar. However, this latter observation may be somewhat misleading. In the case Pr 1, the heat... 

The boundary layer thickness, 8, is defined as the distance that is required for the flow to almost reach If. We might take an arbitrary number (say 99%) to define practically what we mean by nearly, but certain other definitions are used for convenience. The displacement and the momentum thicknesses are alternative measures of the boundary layer thickness and are used in the calculation of various boundary layer assets. 

For forced convection, consider the hybrid (differential in x and integral in y) control volume shown in Fig. 5P-1. Write the conservation of mass, the balance of momentum in terms of the momentum boundary-layer thickness S and the conservation of thermal energy in terms of the thermal boundary-layer thickness S. Compare the results with Eqs. (5.44) and (5.28). 

For small particles (colloidal) the transport is entirely diffusion controlled, and the rate of diffusion decreases as the particle diameter increases. As the particle size increases so does the momentum, so that the momentum effect gradually becomes more important. Beal [1970] shows that the deposition rate reaches a minimum below which diffusion is the dominant transport mechanism and above the minimum momentum assumes the greater importance. At very large particle diameter where the stopping distance is greater than the assumed boundary layer thickness, particle size no longer has any effect, and the transport rate remains constant for a given velocity. 

In addition to the boundary-layer thickness 5, two other thicknesses occur frequently in the boundary-layer literature the displacement thickness S and the momentum thickness 6. To see the meaning of the displaicement thickness, consider the streamlines for the laminar boundary layer on a flat plate, as sketched in Fig. 11.5. 

From the 7 power distribution rule (Eq. 11.33), deduce the ratio of the momentum thickness to the boundary-layer thickness, using Eq. 11.28. Here the integration is from 0 to 5 rather than from 0 to infinity. 

The momentum boundary layer thickness is calculated in terms of the sphere radius by solving the following equation, based on (11-26) and (11-27), which is implicit in Sy/R ... 

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