1
Lecture Notes on Fluid Dynmics
(1.63J/2.21J)
by Chiang C. Mei, MIT
3-2Hi-Re-bl.tex
3.2
Viscous Flow at High Reynolds Numbers
Let us first give a heuristic estimates of boundary layer in steady flows.
Consider a particle near the wall to be influenced by viscosity. After traveling a distance
x from the edge, it has been under viscous influence for a time of t = x/U . Let U be large.
For finite x, t is small so that vorticity is spread sideways to the width (νt) 1/2 ∼ (νx/U )1/2 .
Let us define this width to be the boundary layer, which has thickness δ = O (νx/U ) 1/2 .
Alternatively we start from Navier-Stokes equations :
∂u ∂v
+
=0
∂x ∂y
(3.2.1)
∂u
∂u
1 ∂p
u
+v
=−
+ν
∂x
∂y
ρ ∂x
∂2u ∂2u
+
∂x2 ∂y 2
!
(3.2.2)
∂v
∂v
1 ∂p
+v
=−
+ν
u
∂x
∂y
ρ ∂y
∂2v ∂2u
+
∂x2 ∂y 2
!
(3.2.3)
When viscosity is important y = O(δ), x = O(L), convective inertia is comparable to viscous
stresses.
From continuity
u
v
∼
L
δ
From x−momentum
u ux ∼ ν uyy
U2
U
∼ ν 2
L
δ
Therefore,
δ ∼ (νL/U )1/2
and
δ
ν
∼
L
UL
1/2
(3.2.4)
= Re1/2 .
(3.2.5)
Shear stress on the awall :
τ0
∂u U
=ν
=ν
∼ νU
ρ
∂y 0
δ
s
U
νL
2
Hence the drag coefficient is,
τ0
CD = 1 2 = 2
ρU
2
s
2
ν
=
.
Ux
Re
For water ν = 10−5 ft2 /sec. Let U = 1 ft/sec L = 1 ft, then Re = 105 . Hence,
δ
O
L
!
1
1
∼ 10−2
∝ √
3
Re
(δ ∼ 0.003 ft)
and
CD ∼ 0.003.
Experiments for flat plates (Schlichting, p. 133) show that: CD ∼ 0.002, but experiments
for a circular cylinder show that CD ≈ 0(1) because flow is separated for most Re .
3.2.1
Systematic Boundary-layer Approximation
Let u = O(U ), x = O(L), y = 0(δ). From continuity, v = O(U δ/L). Let u → U u, v →
Uδ
v, x → Lx, y → δy
L
U
(ux + vy ) = 0.
(3.2.6)
L
U2
P ∂p νU
νU
(uux + vuy ) = −
+ 2 uxx + 2 uyy .
(3.2.7)
L
ρL ∂x
L
δ
δ U2
P
νU δ
νU δ
(uvx + vvy ) = − py + 2 vxx + 2
vyy .
L L
ρδ
L L
δ L
(3.2.8)
ux + vy = 0.
(3.2.9)
From Eqn. (3.2.6)
From Eqn. (3.2.7)
!
P
1
uux + vuy = − 2 px +
ρU
Re
L2
uxx + 2 uyy .
δ
(3.2.10)
!
(3.2.11)
From Eqn. (3.2.8)
P L2
1
uvx + vvy = − 2 2 py +
ρδ U
Re
L2
vxx + 2 vyy .
δ
To keep the dominant viscous stress term in Eqn. (3.2.10), we must have
δ
L
!2
=
1
Re
or
δ
= Re−1/2 .
L
(3.2.12)
From Eqn. (3.2.11)
δ2
py = O
L2
!
(3.2.13)
3
and from Eqn. (3.2.10)
uux + vuy = −
P
px + uyy .
ρU 2
(3.2.14)
In physical variables, we have to leading order
ux + v y = 0
(3.2.15)
1
uux + vuy = − px + νuyy
ρ
(3.2.16)
The pressure is constant across the boundary layer and must be the same as the pressure
just outside. In the inviscid outer flow
1
U Ux + V U y = − px .
ρ
(3.2.17)
Since V = 0 on the wall, px = −ρU Ux . Hence, inside the boundary layer:
uux + vuy = U Ux + νuyy .
(3.2.18)
This is the classical boundary layer approximation for high Re flows, due to Prandtl
(1905).