Transition to supersonic flows

in the divertor configuration

using SOLEDGE-2D

H. Bufferand – M2P2 Laboratory

G. Ciraolo, Ph. Ghendrih, L. Isoardi, E. Serre, P. Tamain

Les Houches – March, 10th 2011

Dynamics and turbulent transport in plasmas and conducting fluids

1/13

Background: the divertor configuration

Divertor configuration: Specific magnetic field topology

Confined plasma has no contact with the chamber walls

Reduction of impurities in the core plasma

High confinement: H-mode

Simple model to

evaluate energy flux on

divertor targets (Crucial

for designing divertor

materials)

Focus on estimating

Mach number on the

target plates

Divertor & limiter configuration

Objectives:

2/13

SOL

Numerical tool: SOLEDGE-2D

2D Transport code:

◦ Density

◦ Parallel particle flux ( )

◦ Te and Ti

Effective diffusion coefficients for turbulent transport

The divertor topology is simulated using a multi-domain pattern Example of SOLEDGE-2D results in the

divertor configuration

3/13

nu

SOLEDGE-2D : Fluid model

equations Density equation

Parallel momentum equation

Temperature equations

tn D n

2

i t i e i im m n T T mn

0 5/2

3 3 3

2 2 2t

coupling

nT T nT n T DT nn

T T Q

Parallel

contributions

Cross-field

diffusion

4/13

Multi-domain pattern & divertor

topology

Edge: closed field lines – SOL: open field lines

“Communications” between the domains are such that the topology is equivalent to divertor topology

┴

//

Edge

Private

SOL

SOL

EdgePrivate Private

5/13

Core plasma

Multi-domain pattern

Simulation results at high temperature (isothermal field lines)

Parallel momentum diffusion is low (conservation of total

pressure)

Edge

Private

SOL

SOL

EdgePrivate Private

5/13

1

23 3’

4 4’

Multi-domain pattern

Simulation results at high temperature (isothermal field lines)

Parallel momentum diffusion is low (conservation of total

pressure)

Edge

Private

SOL

SOL

EdgePrivate Private

5/13

1

23 3’

4 4’

Transition to supersonic along the field line

Plot of the Mach number along an open field line from stagnation point to the divertor target

Let us investigate the mechanism of the transition6/13

Expressing Mach Number dependences

One has the following definition for particle flux :

Besides, for total pressure, one has:

Hence, one can rewrite as:

Finally, one defines the reduced particle flux “A”,

function of the Mach number only:

snu nc M

2 2 21i i sp m nu m nc M

21

i

s

m M

c M

2

2 2( )

1

s

i

c MA M

m M

7/13

The reduced particle flux function

Transition to supersonic if:

Continuous transition along the fields line implies non

monotonic behavior of A

Let us check out if sign changes along the field line

/ 0dA dM

Plot of the reduced particle flux function A(M)

s A8/13

2

2 2

/ 1

s

i

c MA

m M

Study of variations of A(M) along the

field line

A-variations:

Continuity equation at steady state

Transition possible if sign changes

2

2 22

/

s s s

i i i

c dc cdA d d

ds m ds m ds m ds

2( ) ( )s rs D n s

2 ( )rn s

2

/

s

i

cA

m

Isothermal

field line

Conservation

of total

pressure

9/13

Density profile convexity change

10/13

#1

#2

The role of the boundary condition on the target

dA/ds sign changes crossing X-point

If no transition, on the target

Bohm boundary condition on the target :

Transition is necessary at X-point

1M

1M

11/13

Radial source of

particles: ↑ thus

A↑

Radial sink of

particles: ↓ thus

A↓

About neutrals

If one adds neutral source in continuity

equation,

If the source due to neutrals compensates

radial sink, the transition can be avoided.

12/13

2( ) ( ) N

s r ns D n s S

Summary

A simple 2D transport code has been developed to

study divertor SOL specificities

A transition to supersonic flow has been observed

in isothermal, low viscous simulations

The relevancy of the transition must be

investigated in more complex cases including

neutrals and parallel temperature drops.

Thank you for your attention

13/13