Laminar Incompressible Flow
over a Rotating Disk
Numerical Analysis for Engineering
John Virtue
December 1999
Problem Description
The Navier Stokes Momentum Equations
u
  2u   u   2u 
u v 2
u
1 p
 w

  2     2 
r r
z
 r
r  r  z 
 r
u
u
  2v   v   2v 
v uv
v

 w   2     2 
r r
z
r  r  z 
 r
z  0 : u  0, v  r , w  0
z   : u  0, v  0
  2 w 1 w  2 w 
w
w
1 p
w

  2 


r
z
 z
r r z 2 
 r
Can be transformed into the following
System of Non-Linear Second Order
Differential Equations as a function of Zeta only
F 2  G 2  HF   F   0
2GF  HG   G   0
2F  H   0
  0 : F  0, G  1, H  0
   : F  0, G  0
Numerical Formulation
The Second Order Differential Equations
can be reduced to a system of First Order Equations by
Introducing the following variables
y1  F , y 2  F 
y3  G, y 4  G 
y5  H
The resulting System of Non-Linear First Order
Differential Equations are...
y1  y2
y2  y12  y32  y5 y2
y3  y4
y4  2 y3 y1  y5 y4
y5  2 y1
Runge-Kutta Fourth Order
Solution
th
For j differential equation of the system:
y j  f j ti , y j 
y ti    j
th
The i RK4 step (step size, h) approximation is given by:
w j ,i   j
k j ,1  hf j (ti , w1,i ,..wn ,i )
h
1
1


k j , 2  hf j  ti  , w1,i  k1,1 ,..., wn ,i  k n ,1 
2
2
2


h
1
1


k j ,3  hf j  ti  , w1,i  k1, 2 ,..., wn ,i  k n , 2 
2
2
2


k j ,3  hf j ti  h, w1,i  k1,3 ,..., wn ,i  k n ,3 
w j ,i 1  wi 
1
k j ,1  2k j ,2  2k j,3 , k j,4 ,..., k j,1  2kn,2  2kn,3 , kn,4 
6
Note that each of the kj,1 must be solved before kj,2 can be computed.
Figure 5.1: Flow Near a Rotating Plate
Comparison of Numerical Results (h = 0.2) to Published Results [Orstrach & Thorton]
1.2000
1.0000
0.8000
F -Published
G -Published
-H -Published
F-Numerical Results
G-Numerical Results
-H - Numerical Results
0.6000
0.4000
0.2000
0.0000
0.000
1.000
2.000
3.000
4.000
5.000
Zeta
6.000
7.000
8.000
9.000
10.000