Boundary layer equations laminar

To describe the velocity profile in laminar flow, let us consider a hemisphere of radius a, which is mounted on a cylindrical support as shown in Fig. 2 and is rotating in an otherwise undisturbed fluid about its symmetric axis. The fluid domain around the hemisphere may be specified by a set of spherical polar coordinates, r, 8, , where r is the radial distance from the center of the hemisphere, 0 is the meridional angle measured from the axis of rotation, and (j> is the azimuthal angle. The velocity components along the r, 8, and (j> directions, are designated by Vr, V9, and V. It is assumed that the fluid is incompressible with constant properties and the Reynolds number is sufficiently high to permit the application of boundary layer approximation [54], Under these conditions, the laminar boundary layer equations describing the steady-state axisymmetric fluid motion near the spherical surface may be written as ... 

C17. Curie, N., The Laminar Boundary Layer Equations. Oxford Univ. Press (Clarendon), London and New York, 1962. 

After the Burgers equation, the numerical analysis of the incompressible boundary layer equations for convection heat transfer are discussed. A few important numerical schemes are discussed. The classic solution for flow in a laminar boundary layer is then presented in the example. 

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

L Howarth. On the Solution of the Laminar boundary Layer Equations. Proc R Soc (London), A164 546, 1938. 

PL Donoughe and N B Livingood. Exact Solutions of Laminar Boundary-layer Equations with Constant Property Values for Porous Wall with Variable Temperature. NASA Technical Report 1229,1958. 

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 ... 

NUMERICAL SOLUTION OF THE LAMINAR BOUNDARY LAYER EQUATIONS... 

By numerically solving the two-dimensional laminar boundary layer equations, determine how the local heat transfer rate in W/m2 varies along the plate. 

Some of the commonly used methods for obtaining solutions to problems involving laminar external flows have been discussed in this chapter. Many such problems can be treated with adequate accuracy using the boundary layer equations and similar ity integral and numerical methods of solving these equations have been discussed. A brief discussion of the solution of the full governing equations has also been presented. 

An implicit finite-difference procedure for solving the laminar boundary layer equations was discussed in this chapter. Discuss how these boundary layer equations could be solved using an explicit procedure. In such a procedure, the terms duldy and d1uldyl are evaluated on the (i - 1) line. The continuity equation is trea ed in the same way in both procedures. Write a computer program based on this procedure and show by numerical experimentation with this program that instability develops if ... 

J0. Show how die numerical method for solving die laminar boundary layer equations discussed in this chapter can be modified to allow for viscous dissipation. Use a computer program based on this modified procedure to estimate the importance of this dissipation on the heat transfer rate along an isothermal flat plate in low speed flow. 

The program assumes the flow is turbulent from the leading edge and that 62 = 0 when x = 0. The program can easily be modified to use a laminar boundary layer equation solution procedure to provide initial conditions for the turbulent boundary layer solution which would then be started at some assumed transition point. 

Solutions to the boundary layer equations are, today, generally obtained numerically [6],[7],[8],[9],[10],[11],[12]. In order to illustrate how this can be done, a discussion of how the simple numerical solution procedure for solving laminar boundary layer problems that was outlined in Chapter 5 can be modified to apply to turbulent boundary layer flows. For turbulent boundary layer flows, the equations given earlier in the present chapter can, because the fluid properties are assumed constant, be written as ... 

The boundary layer equations for free convective flow will be deduced using essentially the same approach as was adopted in forced convective flow. Attention will, as discussed above, be restricted to the case of two-dimensional laminar boundary layer flow. Attention will initially be focused on a plane surface that is at an angle, 4>, the vertical as shown in Fig. 8.4. The x-axis is chosen to be parallel to this surface as shown in Fig. 8.4. 

A numerical solution to the laminar boundary layer equations for natural convection can be obtained using basically the same method as applied to forced convection in Chapter 3. Because the details are similar to those given in Chapter 3, they will not be repeated here. 

Consider laminar free-conveqtive flow over a vertical flat plate at whose surface the heat transfer rate per unit area, qw, is constant. Show that a similarity solution to the two-dimensional laminar boundary layer equations can be derived for this case. 

Before turning to a discussion of other methods of solving the laminar boundary layer equations for combined convection, a series-type solution aimed at determining the effects of small forced velocities on a free convective flow will be considered. In the analysis given above to determine the effect of weak buoyancy forces on a forced flow, the similarity variables for forced convection were applied to the equations for combined convection. Here, the similarity variables that were previously used in obtaining a solution for free convection will be applied to these equations for combined convection. Therefore, the following similarity variable is introduced ... 

Tbe numerical procedure for solving the laminar boundary layer equations for forced convection that was described in Chapter 3 is easily extended to deal with combined convection. The details of the procedure are basically the same as those for forced convection and the details will not be repeated here [16]. A computer program, LAMBMIX, based on the procedure is available in the way discussed in the Preface. This program can actually allow the wall temperature or wall heat dux to vary with X but as available, the program is set for the case of a uniform wall temperature or a uniform wall heat flux. 

The numerical procedure described above for solving the laminar boundary layer equations is easily extended to deal with situations in which the free-stream velocity is varying with x, i.e., to deal with situations involving flow over bodies of arbitrary shape. 

Gemmerich J, Hasse L (1992) Small-scale surface streaming under natural conditions as effective in air-sea gas exchange. Tellus 44B 150-159 Howarth L (1937) On the solution of the laminar boundary layer equations. Proc Royal Soc London A 164 547-579... 

The original papers describing the behavior of solutions of the boundary-layer equations at a separation point are S. Goldstein, On laminar boundary-layer flow near a position of separation, Q. J. Mech. Appl. Math. 1, 43-69 (1948) K. Stewartson, On Goldstein s theory of laminar separation, Q. J. Mech. Appl. Math. 11, 399-410 (1958). 

Problem 10-9. Translating Flat Plate. Consider the high-Reynolds-number laminar boundary-layer flow over a semi-infinite flat plate that is moving parallel to its surface at a constant speed (7 in an otherwise quiescent fluid. Obtain the boundary-layer equations and the similarity transformation for f (r ). Is the solution the same as for uniform flow past a semi-infinite stationary plate Why or why not Obtain the solution for f (this must be done numerically). If the plate were truly semi-infinite, would there be a steady solution at any finite time (Hint. If you go far downstream from the leading edge of the flat plate, the problem looks like the Rayleigh problem from Chap. 3). For an arbitrarily chosen time T, what is the regime of validity of the boundary-layer solution ... 

Since the speed near the surface in a laminar boundary layer has a lower velocity than its turbulent counterpart, the laminar boundary layer is more likely to separate. When this occurs, the laminar boundary layer leaves the surface and usually undergoes a transition to a turbulent flow away from the surface. This process takes place over a certain distance that is inversely related to Re, but if it happens quickly enough, the flow may reattach as a turbulent boundary layer and continue along the surface. To compute when the separation will occur, we can solve the Navier-Stokes equations or apply one of the several separation criteria to the solutions of the boundary layer equations. 

Governing Differential Equations. For a fluid as general as a gas in chemical equilibrium, the boundary layer equations for laminar flow over a flat plate are ... 

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