Momentum integral equation

Let / b be the radius of curvature of the pipe axis and R4 be the radius of the circular cross section of the pipe. Define U as the axial velocity component and (=/ d — r) as the distance normal to the wall. Denote 0 as the angle in the transverse plane with respect to the outward direction of the symmetry line and 4> as the angle measured in the plane of the curved pipe axis, as shown in Figs. 11.9(a) and (c). Assume that the changes of the flow pattern along the axis of the bend can be neglected. Thus, the momentum integral equations... 

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. 

Substitute u(x,y) into the momentum integral equation derived in Step 1, and integrate with respect to y. The ordinary differential equation for 8(x) is obtained solve for 8(x). 

This is termed the boundary layer momentum integral equation. As previously mentioned, it is equally applicable to laminar and turbulent flow. In laminar flow, u is the actual steady velocity while in turbulent flow it is the time averaged value. 

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 ... 

Other boundary conditions for laminar flow are discussed in the next chapter. In this way, the velocity profile is expressed in terms of u and 5. If the wall shearing stress rw is then also related to these quantities, the momentum integral equation (2.173) will allow the variation of S with x to be found for any specified variation of the freestream velocity, u. ... 

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. 

In writing this equation, it has been noted that since be lies in the freestream where the temperature is constant, there can be no heat transfer into the control volume through it. Longitudinal conduction effects have also been ignored because the boundary layer is assumed to be thin. This is consistent with the neglect of the effects of longitudinal viscous forces in the derivation of the momentum integral equation. 

Therefore, using the expression for the mass flow rate through be that was derived when dealing with the momentum integral equation gives ... 

Using these and other boundary conditions, some of which for laminar flow will be discussed in the next chapter, the temperature distribution is expressed as a function of St. If qw is then related to the wall thermal conditions and St, the energy integral can be solved, using the solution to the momentum integral equation, to give... 

The variation of velocity boundary >ayer thickness, 6, with the distance along the plate from the leading edge, x, is first determined by solving the momentum integral equation. In order to obtain this solution, it is assumed that the velocity profile can be represented by a third-order polynomial, i.e., by... 

Now for the case of flow over a flat plate for which u is constant, the momentum integral equation (2.173) can be written as... 

Having obtained the solution to the momentum integral equation, attention must now be turned to the energy integral equation. To solve this, the form of the temperature profile must be assumed. As with the velocity, a third-order polynomial will be used for this purpose, i.e., it will be assumed that ... 

Comparing this with the momentum integral equation result given in Eq. (3.152) shows that A is again a constant and is given by... 

Since the variation of 8 with x has been derived by solving the momentum integral equation, Eqs. (3.153) and (3.156) together constitute the solution of the energy equation. The variation of A with Pr that they together give is shown in Fig. 3.14. 

The analogy solutions discussed in the previous section use the value of the wall shear stress to predict the wall heat trans er rate. In the case of flow over a flat plate, this wall shear stress is given by a relatively simple expression. However, ir, general, the wall shear stress will depend on the pressure gradient and its variation has to >e computed for each individual case. One approximate way of determining the shear stress distribution is based on the use of the momentum integral equation that was discussed in Chapter 2 [1],[2],[3],[5]. As shown in Chapter 2 (see Eq. 2.172), this equation has the form ... 

Using this, equation, derive the boundary layer momentum integral equation for this type of flow. 

On an impervious flat plate with uniform pressure, the momentum integral equation is... 

Re, and Te. (If the viscosity is expressed as a power-law function of temperature, the dependence on Te can be eliminated.) From the von Karman momentum integral equation for a flat plate,... 

He also assumed the logarithmic velocity distribution law in the shedding layer for small D/D, simplified the momentum integral equation for the shedding layer and derived the following expression... 

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