Boundary layer thickness displacement

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. 

Obtain the boundary layer thickness and its displacement thickness as a function of the distance from the leading edge of the surface, when the velocity profile is expressed as a sine function. 

Solution to the nondimensional axisymmetric stagnation-flow problem is plotted in Fig. 6.3. Since the viscous boundary layer merges asymptotically into the inviscid potential flow, there is not a distinct edge of the boundary layer. By convention, the boundary-layer thickness is defined as the point at which the radial velocity comes to 99% of its potential-flow value. From Fig. 6.3 it is apparent that the boundary-layer thickness S is approximately z 2. In addition to the boundary-layer thickness, a displacement thickness can be defined. The displacement thickness is the distance that the potential-flow field appears to be displaced from the surface due to the viscous boundary layer. If there were no viscous boundary layer (i.e., the inviscid flow persisted right to the surface), then the axial velocity profile would have a constant slope du/dz = —2. As shown in Fig. 6.3, projecting the constant axial-velocity slope to the surface obtains an intercept of u = 0 at approximately z = 0.55. Since the inviscid flow would have to come to zero velocity at the surface, z = 0.55 is the distance that the potential flow is displaced due to the viscous boundary layer. Otherwise, the potential flow is unaltered by the boundary layer. 

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. 

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. 

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. 

Comparing Eq. 11.24 with Eq. 11.9, we see that the displacement thickness for a laminar boundary layer is 1.72/5, or about one-third of the boundary-layer thickness. ... 

The Nernst boundary layer thickness is a simple characteristic of the mass transfer but its definition is formal since no boundary layer is in fact stagnant and least of all boundary layers on gas-evolving electrodes furthermore, the Schmidt number, known to influence mass transfer, is not incorporated in the usual dimensionless form. For this reason, lines representing data from gas evolution in two different solutions can be displaced from one another because of viscosity differences. Nevertheless, the exponent in the equation = aib (32)... 

The primary cause of efficiency losses in an axial-flow turbine is the buildup of boundary layer on the blade and end walls. The losses associated with a boundary layer are viscous losses, mixing losses, and trailing edge losses. To calculate these losses, the growth of the boundary layer on a blade must be known so that the displacement thickness and momentum thickness can be computed. A typical distribution of the displacement and momentum thickness is shown in Figure 9-26. The profile loss from this type of bound-ary-layer build-up is due to a loss of stagnation pressure, which in turn is... 

This relation for the thickness of the boundary layer has been obtained on the assumption that the velocity profile can be described by a polynomial of the form of equation 11.10 and that the main stream velocity is reached at a distance 8 from the surface, whereas, in fact, the stream velocity is approached asymptotically. Although equation 11.11 gives the velocity ux accurately as a function of v, it does not provide a means of calculating accurately the distance from the surface at which ux has a particular value when ux is near us, because 3ux/dy is then small. The thickness of the boundary layer as calculated is therefore a function of the particular approximate relation which is taken to represent the velocity profile. This difficulty cat be overcome by introducing a new concept, the displacement thickness 8. ... 

When a viscous fluid flows over a surface it is retarded and the overall flowrate is therefore reduced. A non-viscous fluid, however, would not be retarded and therefore a boundary layer would not form. The displacement thickness 8 is defined as the distance the surface would have to be moved in the 7-direction in order to obtain the same rate of flow with this non-viscous fluid as would be obtained for the viscous fluid with the surface retained at x = 0. 

Calculate the thickness of the boundary layer at a distance of 150 mm from the leading edge of a surface over which oil, of viscosity 0.05 N s/m2 and density 1000 kg/m3 flows with a velocity of 0.3 m/s. What is the displacement thickness of the boundary layer ... 

Explain the concepts of momentum thickness" and displacement thickness for the boundary layer formed during flow over a plane surface. Develop a similar concept to displacement thickness in relation to heat flux across the surface for laminar flow and heat transfer by thermal conduction, for the case where the surface has a constant temperature and the thermal boundary layer is always thinner than the velocity boundary layer. Obtain an expression for this thermal thickness in terms of the thicknesses of the velocity and temperature boundary layers. 

It is found that the velocity at a distance y from the surface may be expressed as a simple power function (u oc y" for the turbulent boundary layer at a plane surface. What is the value of n if the ratio of the momentum thickness to the displacement thickness is 1.78 ... 

Displacement thickness of boundary layer 673, 677 Distillation columns, mass transfer 576 Distributors for water cooling towers 762 Ditius-Boelter equation 417 Di ttos, F. W. 417.563... 

This equation relates the gradient of the velocity in the core region to the rate of growth of the boundary layer displacement thickness. 

The integral equation analysis given in Chapter 6 solved for the boundary layer momentum thickness, 62, which is related to the displacement thickness by the form factor, H, which is defined by ... 

In Eqn. (2.6.106), U(y) and W(y) are the parallel mean flow and Re is the Reynolds number based on the displacement thickness of the boundary layer and primes indicate derivatives with respect to y. 

Mean flow is obtained using the similarity co-ordinate p, while the stability equations are solved using the independent variable, y = y /5, where y is the dimensional height over the plate and S is the displacement thickness of the boundary layer. In terms of rj, the displacement thickness is given... 

In applying the general criterion of equation (30) to detonations confined by walls, available approximations [58], [117] may be employed for relating dA/dx to the growth rate of the displacement thickness <5 of the boundary layer roughly,... 

Air Sparging Gas sparging or injection of air bubbles has been effectively used to reduce concentration polarization and enhance mass transfer. " The secondary flows around bubbles promote mixing and reduce the thickness of the concentration polarization boundary layer. When the bubble diameter exceeds that of the membrane (tubular or hollow fiber), slugs are then formed further increase in bubble diameter has no effect on flux improvement. Large slugs can displace most of the boundary layer and cause the pressure to pulsate. This results in enhancing the flux. 

The displacement thickness is defined by considering the total mass flow through the boundary layer. This mass flow is the same as if the boundary layer were completely at rest, with a thickness, 8 ... 

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