Stability of MHD Buoyancy Driven Flows

Presented by Naveen Vetcha (UCLA)

With contribution from:

Sergey Smolentsev (UCLA)

Rene Moreau (Prof., Lab. EPM, ENSHM de Grenoble, France)

Mohamed A. Abdou (Prof., UCLA)

FNST MEETING

August 18-20, 2009

UCLA

Outline

Importance of the study

Problem formulation

Linear stability analysis

Results and discussions

Conclusions

Future work

Calculations by Smolentsev.

Buoyancy flows are dominant in the blanket

• Nusselt number increase by 20-50% (Authie et. al., European Journ. of Mech. B/Fluids, 22, 2003, 203-220) can lead to higher heat loss to He which is undesirable.

• Recirculation flows may lead to “hot spots” at the interface.

• Temperature fluctuations caused due to the instability may lead to fatigue of the wall material.

• Recirculation flows may lead to local Tritium accumulation making its extraction more complicated.

• Will affect temperature distribution in the FCI.

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Linear stability analysis is performed in order to predict conditions under which flow becomes turbulent, but quantitative knowledge of the turbulent flow requires full numerical modeling or experiment.

Due to buoyancy the characteristic velocity is higher (~25-30 cm/s) compared to that in the forced flow (10 cm/s).

What type of flow regime prevails in the poloidal ducts? Is it laminar or

turbulent ?

Two particular cases: upward and downward flows

Sketch illustrating the forced flow direction with respect to the gravity vector and the magnetic field (Smolentsev et.al.).

Parameter ITER DEMO OB

Ha 6500 12,000

Re 30,000 60,000

Gr 7.0x109 3.5x1012

Basic dimension less flow parameters. Poloidal flow. ITER, DEMO OB (outboard).

In the downward flows, recirculation flows can be predicted for both ITER and DEMO. This suggests that instability mechanisms in upward and downward flows are different.

In DEMO, Gr is sufficiently higher ~ 1012. This makes the problem really complicated.

Problem formulation

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y

u

x

u

x

p

y

uv

x

uu

t

u nn

02

2

2

2

111

v

y

v

x

v

y

p

y

vv

x

vu

t

v

2

2

2

21

qcy

T

x

T

c

k

y

Tv

x

Tu

t

T

pp

1

2

2

2

2

Hartmann effect

Buoyancy term

Volumetric heat flux

laml

ayExpqq /,0

'''

0

y

v

x

u b

n = 1: buoyancy assisted (upward) flow; n = 2: buoyancy opposed (downward) flow.

•The flow is quasi-two-dimensional (Q2D)

•The FCI acts as ideal thermal/electric insulator

•Inflectional instability is dominating over “thermal-buoyant” instability

•Inlet/outlet effects are not considered

0;0 vu at 1y

Linear Stability analysis

• All the flow variables are split up into

mean and fluctuating parts.

• We look for solution txieyq ~

tyxqyQq ,,~

Modified Orr-Sommerfeld equation

2

242''''2 2

Re b

aHa

iUU

0i Stable, 0i Unstable; 0i Neutral.

0;0 at 1y

We look at the sign of by solving an Eigenvalue problem.

-wave number, ir i where i is amplification factor.

i

Results. Upward flow: flow patterns

• Upward flow Re = 10,000; Gr = 4e+07 and Ha = 100.

Base VelocityVorticity

Boundary layer

Internal shear layer

• Two types of instabilities have been observed.

•Primary instability. Inflectional instability associated with the internal shear layer.

• Secondary instability. The vortices generated at the inflection points make the side layer unstable.

Different types of Instabilities

Re = 5e+04

Re = 1e+05 Re = 2e+06

Unstable

Unstable

Unstable

Ha = 10

•At Re~103-104, Gr~106-109 the flow is stable if Ha>250-800

•In the range of flow parameters relevant to DCLL, the mixed convection upward flow seems to be stable

Results. Upward flow: neutral stability curves

Downward flow Re = 30,000; Gr = 1e+08 and Ha = 1500.

Base Velocity

Results. Downward flow: flow patterns

• As the velocity profile shows a shear layer a shear layer instability is expected and two rows of vortices appeared.

• Flow is observed to be unstable at very high Hartmann numbers also.

• This instability at high Hartmann number shows that downward flow can be unstable under the blanket conditions.

• need to extend the analysis to solve the complete set of governing equations.

Internal shear layer

Boundary layer

Vorticity

Results. Downward flow: neutral stability curves

• The critical Hartmann number is comparable to the Ha relevant to the blanket conditions.

•In the range of flow parameters relevant to DCLL, the mixed convection downward flow can be unstable.

Conclusions

Upward flow:

• The primary instability is Inflectional instability due to the internal shear layer

• Primary vortices destabilized the side layer.

• The critical Ha is very small compared to the fusion relevant values, so the upward flow is stable (considering the effects of only buoyancy).

Downward flow:

• Instability mechanism appears to be same as that of upward flow but inflectional instability is stronger due to the recirculation.

• The critical Ha is comparable to the fusion relevant values so this flow can be unstable.

• Need for the full study of the downward flow.

Future work

• The same analysis should be repeated by including the effect of thermal-buoyant instability.

• Small conductivity of walls leads to M shaped profiles in the channel which will increase the chances of flow becoming unstable, this has to be studied.

• It was observed that slip flow increase the chances of instability (Smolentsev. S, MHD duct flows under hydrodynamic slip condition, Theor. Comp. Fluid. Dyn., 2009).

• Extend the analysis to three dimensional disturbances.

• Full numerical modeling of the flow.

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