SUBCAPITULO I. DERECHOS FUNDAMENTALES INVOLUCRADOS EN LA
4.1.1. Derecho a la Identidad
A study of three-dimensional boundary layers on a flat plate is greatly simplified by the absence of geometrical complication. Under these circumstances the solutions of the boundary-layer equations can be found by employing some forms of similarity law (e.g. Blasius-type solutions). A study of three-dimensional flow over a stationary flat plate was discussed in great detail first by Loos (1955)
and applied in the case of a moving plate parallel to its leading edge by Kitchens et al (1975). In this test case, we compare our calculations with those reported by Kitchens et al (1975).
Specification data:
For X, y , z co-ordinate (see Fig. 5.1), a flat plate surface is defined on the X - and y -axises, and the distance normal to the surface is the z -axis. This plate is moving parallel to its leading edge (i.e. y -axis) with a velocity of Vw«u. The boundary conditions of the boundary layer equations are given in equation (5.1-5.2).
z = 0 u = w = 0
v = vwi 5.1
Z U = U e
V = Ve 5.2
where u , v and w are the velocity components along the x -, y - and z -direction respectively. The subscript "wall" means the value at the wall and "e" means the value at the boundary layer edge.
In this co-ordinate system, the velocity components at boundary-layer edge over the plate are given by equations 5.3 and 5.4.
Ue = constant 5.3
Ve = a - b X 5.4
The laminar boundary layer solution for this system can be solved by the method described in chapter 3 or by the Blasius similarity method. In the latter method, the three-dimensional boundary layers can be reduced to two ordinary
— I U e
differential equations by using the Blasius similarity variable, r\ = z J ^ = (for
V DX
detail see Yohner and Hansen (1958) or Kitchens et al (1975)). The ordinary differential equations areff" + 2 r = 0 5.5
2 h " ' + f h " - 2 f ' h ' + 2 = 0 5.6
where the f' and h are the first derivatives with respect to T). The boundary conditions can be written as
at Ti=0 f(0) = f^(0) = h(0) = h (0) = 0 5.7a
at T| —>00 f^(«») = h («») = 1 5.7b
The velocities (i.e. u, v and w ) are related to two functions f and h through;
U = U e f ' 5.8
v = U e [ a f '- b x h '- l - V w d i( l - f ') ] 5.9
w = i f - S ^ l [ T if'-f] 5.10
The solutions of equation (5.5) and (5.6) can be obtained by a numerical solution (e.g. Runge-Kutta method) or from the table in Yohner and Hansen (1958). For the solutions using Runge-Kutta method see Appendix C.
Although the system of boundary layer equations is three-dimensional, the a
flows are identical along the y -plane (e.g. —= = 0). In order to avoid the identical dy
flows in the y -axis and to get a more general test case. Kitchens et al (1975) introduced the x-, y- and z-co-ordinate in which the origin of the x-, y- and z-co- ordinate is moved from the origin of the x-, y - and z -co-ordinate by "c", and the y-axis is rotated by an angle 0 to the y -axis (see Fig. 5.1). The relations between the barred and unbarred co-ordinate and velocities are given by
X = X C O S 0 4- (y+c) sin 0 5.11
y = - X sin 0 + (y+c) cos 0 5.12
z = z 5.13
u = u cos 0 + V sin 0 5.14
V = - u sin 0 + V cos 0 5.15
w = w 5.16
Therefore, the system of boundary layer equations in the x-, y- and z-co- ordinate is a general three-dimensional flow. In our calculations, the x-, y- and z- co-ordinate is employed, and the solutions are compared with the analytical solutions (presented in Appendix C) of the x -, y - and z -co-ordinate through the equations (5.11-5.16).
In this test case "a" is set equal -1.25, "b" is set equal -1.0, "c" is set equal sin 0 (1-sin 0) and 0 = tan'* (1/6). The plate is moved paralleled to its leading edge with v ^ = -0.2.
Discussion of Results
A 101x101x51 grid point of x, y and z-directions was used in the calculation. The three-dimensional velocity profiles obtained from the exact solutions were used as an initial data (only on the first row along the y-direction). Please note that the moving wall (vwiu ) has velocity components in both the x and y-directions due to the co-ordinate transformation.
The velocity profiles in all three co-ordinate directions are illustrated in Fig 5.2. We can see that they are excellent agreements in u and v of the present method and the exact solutions. In w, the present method shows a little bit downward shifts.
In order to obtain stable solutions for the three-dimensional problem, it is necessary to obey the zone of dependence condition. This concept is presented in chapter 3. Kitchens et al (1975) show that the zone of dependence for this case can be stated as
Ax < coAyMin ^ 5.17
where the Min ^ is evaluated over the whole plane of the data, co is a safety
factor. In this calculation we set co=l. Therefore, the choice of Ax and Ay depends on the magnitude of the local streamwise and crossflow velocities. To study the zone of dependence concept, we will use a similar approach as that reported by Kitchens et al (1975). In this study Ax is unchanged and Ay is changed. The value of AXp is defined as the largest permissible step size that is calculated from equation (5.17).
The calculated values of zone of dependence are given in table 5.1. The results of this table are subjected to constant step size Ax. The step size of Ay was changed until the results are no longer acceptable. As the mesh size in the y direction is decreased, the error in the calculation increases even though the truncation error decreases. This is caused by the violation of the zone of
Ax Ax
dependence principle. The highest value of —— is about 12 with — = 2. In
general, the results agree well with the exact solution even though the zone of dependence concept is violated.
Table 5.1 Summary of the zone of dependence (Ax=0.01 and co=l)
Ax
AXp umw crmr V rm* error wTÎTIK error
5.96 5.8x10-5 1.1x10^ 5.3x10-3
8.52 5.8x10-5 1.2x10-4 5.1x10-3
11.92 7.3x10-5 1.2x10-4 1.0x10-3
Note: u = Z (uexact aolutian u)2/ (total number of grid points)
5.3 THREE-DIM ENSIONAL FLO W OVER A FLAT PLA TE W ITH