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)