12.520 Lecture Notes 23
Navier-Stokes Equation – dimensional form
−
∂ 2 vi
Dv
∂p
+η
+ fi = ρ i
∂xi
∂x j∂x j
Dt
Assume:
Characteristic velocity v0
Characteristic length L
Characteristic stress ηv0 / L
Characteristic time v0 / L
Choose non-dimensional variables
v ' = v / v0 x ' = x / L p ' =
v
L
p t' = 0 t
ηv0
L
or
v = v ' v0 x = x ' L p =
∂ 1 ∂
=
∂x L ∂x '
f '=
2
L
f
ηv0
ηv0
L
p' t ' =
L
t' v0
∂ v0 ∂
=
∂t L ∂t '
Substitute into Navier-Stokes equation
−
∂ 2 vi '
v Dvi '
1 ηv0 ∂p ' 1
1
+ 2 v0η
+ 2 v0η f i ' = ρ v0 0
L L ∂xi ' L
L Dt '
∂x j '∂x j ' L
−
∂ 2 vi '
ρ v L Dvi '
∂p '
+
+ fi ' = 0
η Dt '
∂xi ' ∂x j '∂x j '
or
where
ρ v0 L
= Re
η
⇒ Re gives importance of inertial terms relative to viscous terms.
Re
1
viscous forces ≈ balance
acceleration negligible Re
1
inertia dominates Note: in dimensionless form, Re is the only parameter in the Navier-Stokes equation.
⇒ for given geometry (boundary conditions) ALL equivalent
(non-dimensional) problems at same Re give same result!
Examples:
1. Low Reynolds number flow past a cylinder. Re
1
Symmetry, like in the sphere problem. Figure 23.1
Figure by MIT OCW.
2. Re = 10
∂vi
= 0 (steady)
∂t
vj
∂vi
≠0
∂x j
Asymmetry; eddies in wade.
Figure 23.2
Figure by MIT OCW.
The figure below is experimental.
102
CYLINDER
10
CD
1
10-1
10-1
1
10
102
103
104
105
106
107
Re
Figure 23.3. Variation of drag coefficient with Reynolds number for circular
cylinder.
Figure 23.3
Figure by MIT OCW.
Re < 1
CD ∼ 1 / Re
D∼V
102 < Re < 3⋅105
CD ∼ const.
D ∼ V2
Re ∼ 3 ⋅105
big drop in CD!
Recall
Earth’s mantle: Re ∼ 10-19
canoe: Re ∼ 2 ⋅105
Summary:
For low Re, inertia not important Navier-Stokes equation linear
“simple” results (analytic theory) For high Re, inertia important
Navier-Stokes equation nonlinear
time dependent
complicated – experimental approach → empirical