ABSTRACT
As can be seen from Eq. (11.86), the approximate form of the Navier-Stokes equation is also a nonlinear partial differential equation [dependent variables vx, vy and their derivatives ilvxfilx, av,.jay appear as products on the left-hand side of Eq. (1!.86)]. The boundary conditions for the approximated boundary layer equations are
at for all x (11.89)
1'.- 1' .\ - ;x,oo at y = oc (11.90)
The boundary layer equations can be solved by introducing dimensionless parameter 17, as it was introduced to solve the formation of the transient boundary layer problem in the flow a flat plate [Eq. (1 1 .27)]. In this case, however, timet will be replaced xfvx,oo' which is the time required for a fluid element to reach out the distance x when it is moving at characteristic flow velocity vx,oo [the numerical coefficient 2 in the denominator of Eq. (11.27) is neglected]:
(11.92)
or
y (11.93)
Also, a stream function 1/J could be assigned for the solution of the boundary layer flow problem, the derivatives of which in the x and v directions would give the components of the velocity vector (i.e., for a two-dimensional problem). Because the velocity vector would satisfy the equation of continuity [i.e., incompressible flow, Eq. 11.85)], the stream function 1/J has to satisfy the relations:
J ax ( 11.94)
The stream function 1/r can be expressed as a function of the dimensionless parameter rJ so that the ratio would be some function of rJ. To this condition, stream function 1/r has to take the form [because drJfdy =
(11.95) from which, using Eq. (11.94), the velocity components can be expressed in terms of f(rJ):
a1fi 1 1'x = ay = v,,ocf (ry) (11.96)
a1jl 1 ;yu,,CXJ[ I J Vv =- (Jx = 2 ~ rJf (rJ) -f(rJ) (11.97)
The approximate boundary layer equation [Eq. (11.86)] can be expressed in terms of the stream function 1/J:
(11.98)
or in terms of f(rJ), f(rJ)f"(rJ) + 2/ 111 (17) = 0 (11.99)
which is a nonlinear ordinary differential equation. By considering the relations stated by (11.96) and (l the boundary conditions Eqs. (11.89) and (11.90)
become
(11.100)
(11.101) Numerical integration of Eq. (11.99), satisfying Eqs. (11.100) and (11.101) (Fig. 11.7) shows good agreement with the experimental data (a numerical solution was flrst found
Blasius in 1908). Using the numerical solution for the approximate layer equation, the fric-
tion force exerted on unit area of a plate at distance x from the leading edge is calculated as
= V ~/"(0) = V2 (vx,ooX)~l/2/"(0) j.J., x,ooy~ P x,oo Y (11.102)
or
(1 .103)
where Rex is the Reynolds number evaluated at distance x. The local drag coefficient Cn,x is defined as
C _ Tw 2f"(O) 2f"(0) D,x-(lj2)pv~ oo )vx,ooxfy ~ (11.104)
Figure 11 The numerical solution of the two-dimensional boundary layer approximate equations [integration ofEq. (11.99), satisfying Eqs. (11.100) and (11.101)]. (From Brodkey, 1969.)
or
(11.1 05)
(11.106)
The drag coefficient CD can also be defined using Eq. (11.1 06) for the total drag on one side of the plate:
F Cn = 2 (1 /2)pvx,ooAc
- F =4/"(0)-1-= (l/2)p·u~,00 WL ~ (11.107)
Transport Phenomena
r5
or 8 -=4.99-- x ~ (11.
