Page 151 - Bird R.B. Transport phenomena

§4.4 Flow near Solid Surfaces by Boundary-Layer Theory 135

2
1
1
This suggests that d v /dx 2 « d v /dy , so that the former may be safely neglected. In
x
x
the boundary layer it is expected that the terms on the left side of Eq. 4.4-2 should be of
the same order of magnitude as those on the right side, and therefore
'^1 or ^ = O | J ^ | = O[-|=j (4.4-7)
The second of these relations shows that the boundary-layer thickness is small compared
to the dimensions of the submerged object in high-Reynolds-number flows.
Similarly it can be shown, with the help of Eq. 4.4-7, that three of the derivatives in
Eq. 4.4-3 are of the same order of magnitude:
2
dv xl dv u d\ /Vs \ d v u
n
J o u
- - - ^ ~ - - (4.4-8)
Comparison of this result with Eq. 4.4-6 shows that d^/dy « d^/dx. This means that
the y-component of the equation of motion is not needed and that the modified pressure
can be treated as a function of x alone.
As a result of these order-of-magnitude arguments, we are left with the Prandtl
boundary layer equations: 1
dv dv y
1
(continuity) -г- + — = 0 (4.4-9)
dx dy
1 d°P d V Y
2
(motion) v -^ + v, -г± = -± ^f + v — \ (4.4-10)
dv y
dV Y
x dx XJ dy P dx dy 1
The modified pressure 2P(x) is presumed known from the solution of the corresponding
potential-flow problem or from experimental measurements.
The usual boundary conditions for these equations are the no-slip condition (v = 0
x
at у = 0), the condition of no mass transfer from the wall {v = 0 at у = 0), and the
y
statement that the velocity merges into the external (potential-flow) velocity at the
outer edge of the boundary layer (v (x, y) —> v (x)). The function v {x) is related to &(х)
x e e
according to the potential-flow equation of motion in Eq. 4.3-5. Consequently the term
-(l/p)W97dx) in Eq. 4.4-10 can be replaced by v (dv /dx) for steady flow. Thus Eq. 4.4-10
e e
may also be written as
2
dv x dv x dv e d v x
v ——h Pw^— = v -— + v—- (4.4-11)
e
x
dx У dy dx dy 2
The equation of continuity may be solved for v by using the boundary condition that
y
Vy = 0 at у = 0 (i.e., no mass transfer), and then this expression for v may be substituted
y
into Eq. 4.4-11 to give
( \ 2
dV x ГУ dV x J dV x dv c d V x
*J dy
л dx x f \J I dx x j . . \ r _ „ e dx с , _. dy 2 x (4.4-12)
0
This is a partial differential equation for the single dependent variable v .
x
1 Ludwig Prandtl (1875-1953) (pronounced "Prahn-t'D, who taught in Hannover and Gottingen and
later served as the Director of the Kaiser Wilhelm Institute for Fluid Dynamics, was one of the people
who shaped the future of his field at the beginning of the twentieth century; he made contributions to
turbulent flow and heat transfer, but his development of the boundary-layer equations was his crowning
achievement. L. Prandtl, Verhandlungen des III Internationalen Mathematiker-Kongresses (Heidelberg, 1904),
Leipzig, pp. 484-491; L. Prandtl, Gesammelte Abhandlungen, 2, Springer-Verlag, Berlin (1961), pp. 575-584.
For an introductory discussion of matched asymptotic expressions, see D. J. Acheson, Elementary Fluid
Mechanics/' Oxford University Press (1990), pp. 269-271. An exhaustive discussion of the subject may be
found in M. Van Dyke, Perturbation Methods in Fluid Dynamics, The Parabolic Press, Stanford, Cal. (1975).