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Steady Flow over a Rotating Disk: von Kármán Swirling Flow
Steady Flow over a Rotating Disk: von Kármán Swirling Flow
The steady flow induced by a infinite disk that rotates in its own plane at is a classical problem in fluid mechanics. It is one of the few examples of a viscous flow that involves all three components of velocity and admits an exact solution to the Navier–Stokes equations. The velocity field for the swirling flow is given by: . The Navier–Stokes equations reduce to:
z=0
v(r,z)=u(r,z)+v(r,z)+w(r,z)
e
r
e
θ
e
z
1
r
∂
∂r
∂w
∂z
u+w-=+νu++u-
∂u
∂r
∂u
∂z
2
v
r
-1
ρ
∂p
∂r
2
∂
∂
2
r
1
r
∂u
∂r
2
∂
∂
2
z
u
2
r
u+w+=νv++v-
∂v
∂r
∂v
∂z
vu
r
2
∂
∂
2
r
1
r
∂v
∂r
2
∂
∂
2
z
v
2
r
u+w=+νw++w
∂w
∂r
∂w
∂z
-1
ρ
∂p
∂z
2
∂
∂
2
r
1
r
∂w
∂r
2
∂
∂
2
z
The boundary conditions at the disk surface are no slip and impenetrability conditions:
u(r,0)=0,v(r,0)=Ω,w(r,0)=0
where is the rotational speed of the disk. Far away from the surface of the disk, viscous effects are negligible, such that
Ω
u(r,z)0,v(r,z)0asz∞
However, the viscous pumping action of the disk is balanced by a uniform axial inflow at infinity: .
w(r,z)constantasz∞
von Kármán was able to show that the above equations admit a self-similar solution defined by: , , , and , where , and the functions , , , and are determined by:
u(r,z)=ΩrF(η)
v(r,z)=ΩrG(η)
w(r,z)=
Ων
H(η)p(r,z)=ρΩP(η)
η=z
Ω
ν
F
G
H
P
H'+2F=0
F''=-++F'H
2
G
2
F
G''=2FG+HG'
and ,
P'=2FH-2F'
subject to the boundary conditions:
F(0)=H(0)=P(0)=0,G(0)=1
and .
F(∞)=G(∞)=0
It is evident from the structure of the equations that the equation for the pressure field is decoupled from the equations that define the velocity field given by the functions , , and . The boundary value problem for the velocity field can be solved by a shooting method using Mathematica. Also of interest are the fluid particle trajectories in the swirling flow; they can be computed from the velocity field by solving:
P
F
G
H
dr
dt
dz
dt
Ων
H(η)
r
dθ
dt
=ΩrG(η)
It is noteworthy that the boundary layer thickness in the above swirling flow is constant over the disk surface; this feature of the flow is exploited in electrochemistry to study mass transfer.
This Demonstration shows the velocity components of the swirling flow as a function of the angular velocity of the disk and the kinematic viscosity of the fluid. The thickness of the boundary layer is also displayed. Fluid particle trajectories are also shown for different starting positions in the flow.