Viscous diffusion of a rectilineal vortex
Two-dimensional Flow Governing equation
Initial condition: Boundary condition: Solution:
Circulation of a Vortex
vR 1 v vz vR 0
R R z R
vR t
vR
vR R
v R
vR
vz
vR z
v2 R
1
p R
v
2vR R 2
1 R2
2vR
2
2vR z 2
1 R
vR R
2 R2
v
vR R2
v t
vR
v R
v R
v
vz
v z
vRv R
1 p
R
v
2v R 2
1 R2
2v
2
Poisson equation for temperature: Boundary conditions:
Viscous flow in pipe. Let
Non-dimensional equation:
Boundary condition:
Laplacian operator in the cylindrical coordinates
2v y 2
2v z 2
w t
u
w x
v
w y
w
w z
1
p z
2w x2
2w y 2
2w z 2
3.CV
T t
u
T x
v
T y
w
T z
k
2T x2
2T y 2
2T z 2
2
u x
2
v y
2
w z
2
1 2
u y
v x
2
1
u
w
2
1
v
w
2
2 z x 2 z x
Analyze:
(1) Linear , Steady Flow 1. Couette Flow (Couette , 1890)
直角坐标下的N-S方程
1. u v w 0 x y z
2. u t
u
u x
v
u y
w
u z
1
p x
2u x2
2u y 2
2u z 2
v t
u
v x
v
v y
w
v z
1
p y
2v x2
Boundary Condition:
Suppose: Boundary Condition:
Transformation:
1
(x, y) xf (y) f (y) (k) 2 F()
2. Von Karman Vortex Pump (Karman ,1921)
Boundary Conditions: Assumption:
Let From the boundary condition
Velocity profile
5. Starting Flow in a Tube
As we know, when
Choose the cylindrical polar coordinate.The N-S equations are as following:
We obtain
At last , we obtain the solution
u(r,t)
G
4
(a2
r2)
2Ga2
n1
J
3n
0
n
r a
J1 n
r a
exp
2n
a2
t
(3). The Nonlinear, Steady Flow
1, Stagnation-point flow (Himentz Flow, 1911)
Methodology:
1. Choose a coordinates; 2. Write down the governing equations; 3. Analyze the flow; 4. Simplification; 5. Boundary conditions, Initial value conditions; 6. Solution; 7. Analysis.
r r1
/ /
r0 ) r0 )
ln( r / r0 ) ln( r1 / r0 )
B
r02
2 0
k(T1 T0 )
3. Full-developed Flow in Tubes
U
dp c
dx
Full-developed
Poisson equation for velocity: Boundary conditions:
Inviscid Flow:
Two-dimensional Flow. Stream function :
Vorticity:
Dynamics Equation of Vorticity
Governing equation: (Vorticity equation) This is a Nonlinear Equation.
When
, should be regular,
Mass flux:
Temperature: Hagen-Poiseuille Flow ( 1838 , 1840 )
Hagen-Poiseuille Flow ( 1838 , 1840 )
(2) Linear , Unsteady Flows
Chapter 2
Viscous Fluid Motion
§2.2
The exact solutions of Navier-Stokes equation
Solution
Exact Analytical solutions …
Approximate … Numerical solutions
1. Linear , Steady Flow; 2. Linear , Unsteady Flow; 3. Nonlinear , Steady Flow;
2v z 2
1 R
v R
2 R2
vR
v R2
vz t
vR
vz R
v R
vz
vz
vz z
1
p z
v
2vz R2
1 R2
2vz
2
2vz z 2
1 R
vz R
Governing equation:
G dp dx const.
Initial condition: Boundary condition:
From Continuity Equation:
( Infinite long plane ) Constant
From the Non-Slip Boundary Condition , at We have ,
const
Equations: Boundary conditions:
Non-dimensionalization:
v R2
vz t
vR
vz R
v R
vz
vz
vz z
1
p z
v
2vz R2
1 R2
2vz
2
2vz z 2
1 R
vz R
Boundary conditions:
Solution:
T T0 T1 T0
B
r14 (1 1 / 0 )2 r14 r04
1
r02 r2
1
ln( ln(
Example:
Error function
2. Viscous Diffusion of Vorticity
Viscous diffusion of a rectilineal vortex
Navier-Stokes eqns. Dynamics Equation of Vorticity
t
0,
0
,
A
0
4
0
4 t
exp
r2
4t
v
0
2r
1
exp
r2
4t
t1
t2
t3
t4
r
Vorticity profile
v
Velocity profile
3. The First Stokes Problem
Governing equation:
Initial condition: Boundary condition:
Continuity Equation:
Z-direction N-S eqn.
Boundary Condition:
vr r
F ( z1 ),
v r
G(z1),
vz
H (z1)
vz vr
0.886
v
r
The End
1. Uniformization process with viscous dissipation
Parallel Shear Flow
Navier-Stokes equations:
Heat Conduction Equation
Initial value: heat source at
Initial value:
Using the boundary condition: