Numerical simulation of particle concentration in a turbulent pipe flow
Hans KuertenMaurice VeenmanJoost de Hoogh
Contents• DNS - model• Concentration equations• Space and time discretisation• Early results• Finite volume method• Future planning
DNS model• Direct Numerical Simulation by M. Veenman• Axial and tangential direction
– Spectral solver
• Radial direction– Chebychev polynomial expansion
• Implementation of the concentration equations as a passive scalar
Concentration equation
Dcu c c
t
/ 2 1 2/ 2 1
( )
,/ 2 1 / 2 1
( , , , ) ( )z
z
z z
M zMi k k
Lz
k M k M
c r z t c r e
0r
c
Boundary conditions
Axial and tangential direction: periodic conditions
Radial direction:
Adams-Bashforth time integration
13 1 11
22 2 ( ) ( )n n n n nc c c t N c N c
( ) (c)cN c L
t
131 12 2
3 3 ( )nn nc c t D L c
Simple testing
1.
2.
3.
4.
0 0( , ) ( ) tc r t J r e
0
2( , , ) sin( )
zc r z
L
0 ( , , ) 0c r z
2 21 12 22 2
( 2)( ) ( )
0
1( , , )
2r z
r z
r
c r z e e
0.05
0.5r
z
First results
Movie clip!
Negative concentration
0 50 100 150 200-1
0
1
2
3
4
5
6
7
z position in nodal points
conc
entr
atio
nnegative values after 500 timesteps
Finite volume method
• Roe’s first order upwind scheme
• Muscl method (by van Leer)Second/third order depending on parameters
cdV u c dA D c dAt
r
φ
z
Results Muscl method
Movie clip!
Different grid
• New uniform grid
– Radial direction: uniform grid
– Tangential direction (MUSCL): half the points
• Velocity Interpolation
Mean radius
2
2
c r dVr
c dV
c r dVr
c dV
Muscl
Fourier
1st order upwind
Things still to do
• Implement diffusion
• Compare the model with Brethouwer’s results
• Look into forces acting on the concentration (e.g. gravity)