Boundary layer equations momentum

For the case in which the Schmidt number is equal to 1, it can be shown [7] that the conservation equations [in terms of Cl see Eq. (6.17)] can be transposed into the form used for the momentum equation for the boundary layer. Indeed, the transformations are of the same form as the incompressible boundary layer equations developed and solved by Blasius [30], The important difference... 

Given the particular circumstances of the flow in a long, narrow channel, explain the reduction of the governing equations to a boundary-layer form that accommodates the momentum and species development length. Discuss the essential characteristics of the boundary-layer equations, including implications for computational solution. 

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,... 

Several approximate methods exist for solving the boundary layer equations. The momentum-integral method of analysis is an important method. The principal steps of the method are listed below. 

The boundary layer equations were derived in a previous chapter, or may be deduced from the general convection equations in the early part of this chapter. For two-dimensional, steady flow over a flat plate of an incompressible, constant-property fluid, the continuity, x-momentum and the energy equations are as follows ... 

To have a better control on the stability of the explicit method by monitoring a single criterion, the second term in the x-momentum boundary layer equation can be depicted as... 

For a mesh with a constant rectangular grid, the incompressible laminar boundary layer equations include the momentum equation as... 

If the turbulent momentum equation is expressed in nondimensional form in the same way as was done in deriving the laminar boundary layer equations then the additional term becomes ... 

Now, the rest of the terms retained in the boundary layer equations have the order of magnitude of unity and, therefore, for the boundary layer equations to apply, the dimensionless turbulence terms (u 2lu ) and (u v /u ), which are assumed to have the same order of magnitude, will have the order of magnitude of (8/L) at most. The first term in Eq. (2.154) is, therefore, negligible compared to the rest of the terms in the boundary layer equations. Therefore, the x-wise momentum equation for turbulent boundary layer flow is ... 

It is next noted that because the velocities are very low in the inner, i.e., nearwall, region, the convective terms in the boundary layer equations can be neglected in this region, i.e., in this region the momentum and energy equations can be assumed to have the form ... 

The above equations can be solved using numerical methods, i.e., using the same basic procedures as used with forced convection. There is, however, one major difference between the procedures used in forced convection and in mixed convection. In forced convection, the velocity field is independent of the temperature field because fluid properties are here being assumed constant. Thus, in forced convection it is possible to first solve for the momentum and continuity equations and then, once this solution is obtained, to solve for the temperature distribution in tike flow. However, in combined convection, because of the presence of the temperature-dependent buoyancy force term in the momentum equation, all of the equations must be solved simultaneously. Studies of flows for which the boundary layer equations are not applicable are described in [24] to [43]. 

Theoretical approaches to the prediction of H x,y,t) would involve the solution of the boundary layer equations for coupled energy and momentum transport or, more simply, the solution of the energy equations in conjunction with a constructed wind field. The application of such approaches to the prediction of inversion height has not yet been reported. Now, empirical models offer the only available means to estimate H. For those areas where it is necessary only to account for temporal variations in H, interpolation and extrapolation of measured mixing heights may be sufficient. When it is important to estimate // as a function of x,y, and t, a detailed knowledge of local meteorology is essential. 

The conclusion from the previous paragraph is that similarity solutions of the momentum boundary-layer equations should not generally be expected. An interesting question is whether similarity solutions can be obtained in any case other than the flat plate problem in the previous section. To answer this question, we start with the boundary-layer equations in their most general form ... 

Uniform Surface Temperature, Foreign Gas as Coolant. The effectiveness of air injection in reducing convective heat flux stimulated investigations into the use of other coolants With the introduction of a foreign species into the boundary layer, the boundary layer equations reduce to the continuity equation (Eq. 6.6), the momentum equation (Eq. 6.7), the energy equation... 

The simpler analyses reduce the boundary layer equations to ordinary differential equations, with the distance normal to the surface as the independent variable. This results from the assumption of Couette flow, where changes of the dependent variables in the streamwise direction may be neglected. The continuity and momentum equations then become... 

For the case of constant momentum diffusivity, v, Blasius (1908) obtained a solution for the velocity boundary-layer equations (2-21) and (2-22). He introduced the concept of a stream function, V(x, y), such that... 

Blasius steady-flow, laminar, flat-plate, boundary-layer solution is a numerical solution of his simplification of Prandtl s boundary-layer equations, which are a simplified, one-dimensional momentum balance and a mass balance. This type of solution is known in the boundary-layer literature as an exact solution. Exact solutions can be found for only a very limited number of cases. Therefore, approximate methods are available for making reasonable estimates of the behavior of laminar boundary layers (Prob. 11.8). 

Prandtl started with the Navier-Stokes equations and discarded enough terms to make his boundary-layer equations, which are the working form of the momentum and continuity equations for boundary-layer problems. 

Here, Ig is the characteristic length of the inertial sublayer, u is the characteristic velocity of the eddy, and Re is the Reynolds number (uls/v). Equations 3 and 4 indicate that with an increase in Reynolds number, the length and time scales of eddies decrease. To spatially resolve the small eddies, sensors that are of same size as the Kolmogorov length scale for that particular flow are needed. Hence, smaller sensors are required as the Reynolds number is increased [9]. Eor a flat plate boundary layer with momentum thickness, Reynolds number equal to 4,000, the... 

The momentum boundary-layer equation is very similar. 

The problems experienced in drying process calculations can be divided into two categories the boundary layer factors outside the material and humidity conditions, and the heat transfer problem inside the material. The latter are more difficult to solve mathematically, due mostly to the moving liquid by capillary flow. Capillary flow tends to balance the moisture differences inside the material during the drying process. The mathematical discussion of capillary flow requires consideration of the linear momentum equation for water and requires knowledge of the water pressure, its dependency on moisture content and temperature, and the flow resistance force between water and the material. Due to the complex nature of this, it is not considered here. 

Equation 11.12 does not fit velocity profiles measured in a turbulent boundary layer and an alternative approach must be used. In the simplified treatment of the flow conditions within the turbulent boundary layer the existence of the buffer layer, shown in Figure 11.1, is neglected and it is assumed that the boundary layer consists of a laminar sub-layer, in which momentum transfer is by molecular motion alone, outside which there is a turbulent region in which transfer is effected entirely by eddy motion (Figure 11.7). The approach is based on the assumption that the shear stress at a plane surface can be calculated from the simple power law developed by Blasius, already referred to in Chapter 3. 

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. 

Derive the momentum equation for the flow of a viscous fluid over a small plane surface. Show that the velocity profile in the neighbourhood of the surface may be expressed as a sine function which satisfies the boundary conditions at the surface and at the outer edge of the boundary layer. 

For turbulent flow on a rotating sphere or hemisphere, Sawatzki [53] and Chin [22] have analyzed the governing equations using the Karman-Pohlhausen momentum integral method. The turbulent boundary layer was assumed to originate at the pole of rotation, and the meridional and azimuthal velocity profiles were approximated with the one-seventh power law. Their results can be summarized by the... 

---