Displacement thickness

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. 

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. 

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 ... 

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. 

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... 

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. 

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 ... 

Similarly, using the definition of the displacement thickness it follows that ... 

The onset of instability is predicted at Rccr = 519 (based on displacement thickness as the length scale). It is important to realize that instability and transition are not synonymous. Actual process of transition begins with the onset of instability but the completion may depend upon multiple factors those form the basis for adjunct topics like secondary tertiary and... 

Let us therefore discuss about spatial instability of parallel flows, mainly the flow past a flat plate at zero angle of attack- a problem that enjoys a canonical status for instability analyses. For the spatial instability problem associated with two-dimensional disturbance held of two-dimensional primary flows, the disturbance quantities will have the appearance of Eqn. (2.3.28) with /3 = 0. Thus for a fixed Re, one would be looking for complex a when the shear layer is excited by a fixed frequency source of circular frequency, lvq- If we define Re in terms of the displacement thickness S as the length scale, then Re = and the results obtained will be plotted as contours of constant amplification rates Oj in Re — lvo)— plane, as shown in Fig. 2.2. 

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. 

If all the lengths are non-dimensionalized by the displacement thickness and the velocity by then the periodic vortices impose a time scale on the flow given by oiq = 2Kcja The periodicity of the vortices excites the shear layer at circular frequencies wq, 2o o, 3o o.etc. Thus, the distur-... 

The abscissa is actually the Reynolds number based on local displacement thickness. In this case, only the local solution is predominant at early times. This component also disperses and decays, as can be seen from the solution at t = 100. The observed single peak at t = 0 that is due to the local solution disperses into multiple peaks- as can be noted for all subsequent times. This dispersion of solution is due to the presence of upstream prop>-agating modes and the presence of multiple harmonics for the downstream propagating modes. The adjective upstream here is to be understood with respect to the local condition of the disturbance field. More details about this dispersion mechanism and tracking of the upstream propagating modes were first discussed in Sengupta et al. (1999) and will be discussed again in the next subsection. 

The Reynolds number based on displacement thickness of the undisturbed flow at the outflow of the computational domain is 472. Thus, the flow is fully sub-critical in the computational domain (with respect to linear stability theory criticality). 

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,... 

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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