Modeling Status of Normal Field Gradient Effects and Impact on ALIST - - PowerPoint PPT Presentation

Modeling Status of Normal Field Gradient Effects and Impact on ALIST Free Surface Options - Part I TASK I

Presented by Sergey Smolentsev APEX-15 Project Meeting Los Angeles April 24-25, 2001

SEVERAL MODELS HAVE BEEN DEVELOPED UNDER TASK II TO ANALYZE WALL-NORMAL B-FIELD EFFECTS

FLOW

(x) a y B r

X Y Z

Sketch of the induced electric currents in an open channel MHD flow under a space varying (in x- direction) wall-normal magnetic field. There are two current loops.

Changing the magnetic field over the flow length gives rice to an axial potential difference, which drives currents in the axial direction. These currents, while interacting with the magnetic field, produce electromagnetic forces driving the liquid from the core to the side walls. This also results in a higher MHD drag than that in a uniform magnetic field. Unlike a uniform magnetic field, the Lorenz force does not directly affect the flow structure. The interaction occurs indirectly through redistribution of the pressure field. Such flows demonstrate 3-D features and they are very difficult to analyze in a full 3-D formulation. Present modification of FLOW-3D incorporates basic MHD features but computations are time consuming. One PC calculation using relatively coarse meshes takes about 1 week. Here, we are developing a simpler 1.5-D model, which allows us to estimate the streamwise flow thickness variation under a space varying magnetic

• field. This model is based on our calculations of fully

developed and developing flows using different approaches developed under TASK II.

OUR 2-D FULLY DEVELOPED FLOW CALCULATIONS EXPLAIN BASIC PHYSICAL FLOW MECHANISMS

• 1
• 0.8 -0.6 -0.4 -0.2

0.2 0.4 0.6 0.8 1

z / b

• 1
• 0.8
• 0.6
• 0.4
• 0.2

0.2 0.4 0.6 0.8 1

y / a

• 0.14
• 0.107
• 0.074
• 0.041
• 0.008

0.025 0.058 0.091 0.124

• 1. Changing the

magnetic field over the flow length gives rice to an axial potential difference, which drives currents in the axial direction. Contour lines

• f the induced

electric current

• 2. The induced current

interacts with the gradient magnetic

• field. This creates

non-uniformly distributed pressure gradient, dP/dx,

• pposing the flow.
• 3. Under the influence of

the opposing pressure gradient, the velocity profile is deformed and the mean velocity is reduced.

FLOW3D-M CALCULATIONS GIVE ADDITIONAL DATA ON THE FLOW STRUCTURE IN DEVELOPING REGIME

The flow area can be subdivided into two sub-areas: the core (about 90%) and the near-wall regions (about 10%) Flow in the core is about 2-D: No variations in Z-direction Moderate thickening in X-direction (20%) Moderate velocity reduction (20%) About parabolic velocity distribution in Y-direction Flow in the near-wall region is essentially 3-D: Local flow thickness increase Local velocity increase

FLOW IN THE CROSS-SECTIONAL AREA Initial thickness = 2 mm Initial velocity = 10 m/s Chute inclination angle = 90° Wall-normal magnetic field gradient = 0.2 T/m

A x=0.1 m B x=0.15 m

Z Y DOWNSTREAM VARIATIONS OF THE FLOW THICKNESS AND THE VELOCITY AT THE CHUTE MIDPLANE Y

BASED ON THE OBSERVATIONS WE DEVELOPED A 2-D MODEL DESCRIBING DOWNSTREAM FLOW VARIATIONS

In this model, 3-D flow equations are integrated (averaged) analytically in the Z-direction. It results in 2-D formulation, which takes into account flow variations in X and Y directions. The information on the distributions of the flow quantities in Z-direction is included through the 1-D analytical solution.

; ~ ~ sin 5 . | ~ Re ~ 5 . ) cos ~ ~ (sin 1 ~ ~ Re 1 ~ ~ ~ ~ ~ ~ ~ ~

1 1 ’ 1 1 ~ 1 ~ 2 2 2

• MHDII

y y MHDI z z

z d B Fr Ha z d df U x h Fr y U y U V x U U t U

∫

− = − =

+ > < + ∂ ∂ − + ∂ > < ∂ = ∂ > < ∂ > < + ∂ > < ∂ > < + ∂ > < ∂ α β α α ; ~ ~ ~ ~ = ∂ > < ∂ + ∂ > < ∂ y V x U . ~ ~ ~ ~ ~ > =< ∂ > < ∂ > < + ∂ > < ∂

s s

V x h U t h

MHD I stands for the viscous friction at the side walls; MHD II stands for the opposing pressure gradient;

z d z U z U z f

fd fd

~ ) ~ ( ~ 5 . ) ~ ( ~ ) ~ (

1 1

∫

−

=

describes z-distribution of the velocity

) ~ ( ~ ), (

’ z

B z f

y

are taken from the 1-D solution

∫

−

>= <

b b

dz z y x U b U ) , , ( 2 1 . 2 / ; ) / ( / ; / ; / Re

5 . 1 2 1 2

b h dx dB b Ha gh U Fr h U

y y

= = = = β νρ σ ν

h0, U0 are the flow thickness and the mean velocity at x=0; 2b is the chute width; α is the chute inclination angle

THE MODEL IS CLOSED USING 1-D SOLUTION FOR FULLY DEVELOPED FLOW

; 2 ) 2 sin( } 2 exp{ 2 } 2 2 exp{ 1 ) 2 sin( } 2 exp{ 2 } 2 2 exp{ 1 2 ) 1 ( ~

1 1 1 1 1 1 1 1 1

1

y Ha at y y y y y y y

Ha Ha Ha Ha Ha Ha Ha Ha z z d df

y

− ±    →  + − − − − ± = ± =

>> 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ’

2 )} 2 cos( } 2 exp{ 2 } 2 2 exp{ 1 /{ } / ) 2 sin( } 2 exp{ 2 ) 2 cos( } 2 exp{ 2 2 / } 2 2 exp{ 2 / 1 } 2 2 exp{ 1 { 2 ~ ~

1

y Ha at y y y y y y y y y y y y y y

Ha Ha Ha Ha Ha Ha Ha Ha Ha Ha Ha Ha Ha Ha z d B

y

−    →  + + − + − + + − =

>> −

∫

If Hay

1>>1: Fr/sinα+MHD II=0 and MHD I =

Re / ~ 5 .

1 2 y

Ha U > < β

.

> < − ∂ ∂ + ∂ > < ∂ = ∂ > < ∂ > < + ∂ > < ∂ > < + ∂ > < ∂ U Ha x h Fr y U y U V x U U t U

y

~ Re 5 . cos ~ ~ 1 ~ ~ Re 1 ~ ~ ~ ~ ~ ~ ~ ~

1 2 2 2

β α

The liquid flows due to inertia with deceleration caused by viscous losses at the side walls and the bottom and other MHD opposing forces (if any). No moving force !

THE RESULTS CALCULATED DO NOT SHOW ANY SIGNIFICANT FLOW THICKENING DUE TO THE NORMAL FIELD GRADIENT

Parameter Midplane

Length, m 0.9 Width, m 0.3 Initial thickness, m 0.004 Initial velocity, m/s 2.0, 5.0, 10.0 Inclination angle, degree 90.0 Liquid Li Wall-normal field gradient, T/m 0.2

In calculations, only the effect of the wall-normal field gradient was included to demonstrate its impact on the flow thickness. However, our previous calculations of the NSTX chute module show that the flow thickness increase due to other MHD effects can be kept within about 50%

• f the initial thickness by isolating the

chute walls or by making them reasonably thin.

Effect of the wall-normal field gradient on the average flow thickness

0.2 0.4 0.6 0.8 1

DISTANCE, M

0.004 0.0041 0.0042 0.0043 0.0044

THICKNESS, M

U=2 m/s U=5 m/s U=10 m/s

NSTX CHUTE MODULE

CONCLUSIONS

In the open channel MHD flow under a gradient wall-normal magnetic field, the flow area can be subdivided into two sub-areas: the core and two near- side wall regions. Under NSTX conditions (midplane, with the field gradient of 0.2 T/m), the core area is about 90% of the whole cross-section. In the core, the flow demonstrates quasi 2- D character. The gravity force is balanced by the opposing MHD pressure gradient, and the liquid flows by inertia with deceleration due to viscous effects at the backplate and the side walls as well as other MHD opposing factors, such as the Hartmann drag and the Lorenz force. As calculations show, the wall-normal field gradient itself does not cause any significant changes of the flow thickness in the

• core. At the same time, the flow thickness increase due to other MHD effects can be

kept within about 50% of the initial thickness by isolating the chute or by making the walls reasonably thin. In the two regions near the side walls, the flow is essentially 3-D. The velocity is much higher and the layer is much thicker than that in the core. The model with

averaging presented does not allow calculations near the side walls. However, one may expect some “negative” flow patterns here, such as splashing and spilling the liquid off the chute side walls. Some negative effect may also arise due to interaction of the induced streamwise current with other magnetic field components. However, such situations can be hardly analyzed qualitatively or using “simple” models because of a large number of parameters. More effective computer tools at the level of commercial software are needed. In the second part of the presentation, Huang Hulin will tell us about his FLOW–3D modifications and present his preliminary calculations for the NSTX chute module performed at a more detailed level of the flow description.