Momentum 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... 

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 ... 

The time-dependent simulations of free jets discussed here focus on the vortex dynamics and transition to turbulence downstream of the jet exit. For the sake of computational efficiency, the author concentrates on the study of jet flow initialized with laminar conditions with a thin rectangular vortex sheet having slightly rounded-off corner regions and uniform initial momentum thickness [9]. Initial conditions for the simulated jets involve top>-hat initial velocity profiles... 

An important open question relates to whether an optimal AR exists with regard to entrainment enhancement. Laboratory jet experiments with pseudo-elliptical geometries [27] suggest that an optimal AR with regard to nozzle-geometry-enhanced entrainment might be at a value AR = 3. However, the experiments are not conclusive since they involved AR up to 3.5 and nonuniform momentum-thickness distributions, which are known to also affect the entrainment process [5]. Moreover, the possible effects on jet entrainment of other more complicated interactions such as vortex-ring bifurcation still need to be established. 

The goal of this work has been to characterize the effects of the unsteady vor-ticity dynamics on jet entrainment and nonpremixed combustion. The main focus of the numerical simulations of rectangular jets has been on the vortic-ity dynamics underlying axis switching when the initial conditions at the jet exit involve laminar conditions, negligible streamwise vorticity, and negligible azimuthal nonuniformities of the momentum thickness. 

The first part of this problem is discussed in Section 11.1. If the displacement and the momentum thicknesses are 8 and 8m respectively, then ... 

In the discussion of the use of the Reynolds analogy for the prediction of the heat transfer rate from a flat plate it was assumed that when there was transition on the plate, the x-coordinate in the turbulent portion of the flow could be measured from the leading edge. Develop an alternative expression based on the assumption that the momentum thickness before and after transition is the same. This assumption allows an effective origin for the x-coordinate in die turbulent portion of the flow to be obtained. 

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 Fig. 2.10, Reynolds number based on momentum thickness at transition onset is shown plotted against the shape factor, as reported in Arnal (1984). 

Figure 2.10 Momentum thickness Reynolds number and shape factor at transition onset [Arnal, 1984].

Comparisons were limited to three integral parameters of the mean flow available from all computations the momentum thickness 6, the shape factor H, and the friction factor Ct. Mean-profile comparisons were made by many authors, and a few even made comparison with turbulence data. 

MVFN methods have been used with some success in compressible flows. Figure 9 shows a prediction of Herring and Mellor (H5) of the Mach number correction to the skin friction factor for a flat-plate boundary layer. Figure 10 shows their prediction for the boundary layer on a waisted body of revolution. We note that, while the momentum thickness is quite accurately predicted, the velocity-profile details are in considerable error. 

Figure 6.10 Momentum thickness, m, a proxy for shear layer width, and velocity difference, AU/U, where U = (U + I/2)/2. The two parameters are plotted versus distance downstream of canopy edge. Beyond 4-m the shear-layer growth has ceased.

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 the laminar boundary layers, 8 is about 1/3 of the distance to the edge of the boundary layer, 8. The momentum thickness, cf>, is defined similarly, using the momentum flux rather than the mass flux ... 

Given the momentum thickness at some location on a body with streamwise pressure gradients, the evaluation of the local skin friction coefficient from Eq. 6.157 requires knowledge of the shape factor H. Alternatively, the same input information can be used to evaluate both the skin friction coefficient and the shape factor by solving Eqs. 6.152-6.156 iteratively. 

A convenient relationship expressed explicitly in terms of the momentum-thickness Reynolds number that agrees with Eqs. 6.160 and 6.161 within a few percent in the range 5000 < Ree < 50,000 is given by... 

FIGURE 6.41 Compressible turbulent boundary layer transformation parameter for momentum-thickness Reynolds number, r(0) = 0.9, T, = 400 R (222 K). 

A later study by Hancock [142] revealed a significant dependence of the boundary layer momentum thickness on the free-stream turbulence level. Consequently, significant effects were observed when skin friction data were compared at the same momentum-thickness... 

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. 

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