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Notes on 1.63 J/2.21J Fluid Dynamics Instructor: C. C. Mei, [email protected], 1 617 253 2994
February 25, 2007
3-7unsteadyBL.tex
3.7 Unsteady boundary layers
Let us begin from the full momentum equation
~qt + ~q · ∇~q = −
ρ
∇p + ν∇^2 ~q (3.7.1)
Let the veloicty and times scales be Uo and T , the tangential length scale be L and the transverse length scale be δ ∼
νT. Hence the suitable normalization is
x′^ = x/L, y′^ = y/
νT , t′^ = t/T,
u′^ = u/Uo, v′^ =
vL Uoδ
v Uo
√ L^2 νT
p =
pT ρUoL
, U ′^ = U/Uo.
If primes are omitted for brevity, the dimensionless equations are,
ux + vy = 0, (3.7.3)
ut +
UoT L
(uux + vuy) = −px +
νT L^2
uxx + uyy (3.7.4)
νT L^2
[ vt +
UoT L
(uvx + vvy)
] = −py +
νT L^2
[ (^) νT
L^2
vxx + vyy
] (3.7.5)
Outside the viscous boundary layer,
Ut + (
UoT L
)U Ux = −
ρ
px (3.7.6)
Two parameters control the motion: UoT /L (inertia) and νT /L^2 (viscosity). Several scenarios are possible:
- Low amplitude and slow motion: UoT /L ≪ 1 , νT /L^2 = O(1). The tangential and transverse scales are comparable. To the leading order, the approximate equations in physical coordinates are ux + vy = 0, (3.7.7)
~qt = −
ρ
∇p + ν∇^2 ~q (3.7.8)
This is just the Oseen’s approximation.
- Finite amplitude, fast motion, UoT /L = O(1), νT /L^2 ≪ O(1). The boundary layer is thin. To the leading order, nonlinearity is important in the boundary layer.
ux + vy = 0, (3.7.9)
ut +
UoT L
(uux + vuy) = −px + uyy = Ut +
UoT L
U Ux + uyy (3.7.10)
or, in physical coordinates,
ut + (uux + vuy) = Ut + U Ux + νuyy (3.7.11)
- Small-amplitude and fast motion. νT /L^2 ≪ UoT /L ≪ 1. This is a limit of the preceding case; linearization is possible. Examples are : the initial stage of transient motion starting from rest, oscillating flow around a vibrating body, or wave motion (sound or sea waves) past a body (or a droplet, a bubble), etc.
We now give examples of transient boundary layers of small-ampliutude motion.
