ISTP-16, 2005, PRAGUE 16TH INTERNATIONAL SYMPOSIUM ON TRANSPORT PHENOMENA

1

Abstract

The incompressible Navier-Stokes equations with heat transfer are solved by an implicit pressure correction method on unstructured Cartesian meshes. The Volume of Body (VOB) function is used to identify the solid body in the flow field. The domain inside the immersed body is viewed as being occupied by the same fluid as outside with a prescribed velocity and temperature field. With this view the pressure inside the immersed body satisfies the same pressure Poisson equation as outside. In this paper we show some test results for the forced convection over a circular cylinder and the natural convection over a heated cylinder inside a square cavity.

1 Introduction

Recently numerical methods for solving incompressible flows on non-body-fitted Cartesian grids are gaining popularity for their relative ease in treating complex immersed bodies [1-6]. The differences among these methods lie in the different ways the boundary conditions for the immersed bodies are enforced. The immersed boundary method [1-3] simulates the no-slip boundary condition by adding appropriate momentum forcing terms to appropriate grid cells. In Cartesian cut-cell methods [4], the flow variables for the cut cells are solved based on the actual shape of the cut cells. The fictitious domain method [5] is a finite-element method that enforces the no-slip boundary condition using Lagrange multiplier. In Xiao [6], the solid body is viewed as a

material with distinct properties different from the surround fluids, and its motion is tracked by a color function.

In this work, we first developed an implicit fractional step pressure correction method to solve the incompressible Navier-Stokes equations on unstructured Cartesian meshes. Multigrid methods are developed to solve the difference equations for pressure, velocity and temperature field. Then a simple and effective computational model for the immersed bodies is implemented. The domain inside the solid body is viewed as being occupied by the same fluid as outside with a prescribed divergence-free velocity field and temperature field. In this view a fluid-body interface is similar to a fluid-fluid interface encountered in the Volume of Fluid (VOF) method for the two-fluid flow problems. Since the velocity field inside the body is assumed divergence-free, the pressure inside obeys the same pressure Poisson equation as outside. For the grid cells containing the fluid-body interface, the velocity is obtained by the volume average of fluid velocity and body velocity. A similar volume average is used to obtain the temperature of interface cells. In this paper we show the test results for the forced convection over a circular cylinder and the natural convection around a heated cylinder in a cavity to validate the current method.

2 Implicit Pressure Correction Method

The incompressible Navier-Stokes equations are

AN IMMERSED BOUNDARY METHOD ON UNSTRUCTURED CARTESIAN MESHES FOR

INCOMPRESSIBLE HEAT CONVECTION PROBLEMS

Dartzi Pan Department of Aeronautics and Astronautics

National Cheng Kung University, Tainan, Taiwan, ROC dpan@mail.ncku.edu.tw, Tel: +886-6-2757575 ext 63658

Keywords: Immersed Boundary Method, Unstructured Cartesian Mesh, Heat Convection

2

0v =⋅∇ vv

0SPd

dVRe

rGSdv

Re1

SdvvdVvt

CS

CV2

CSCSCV

=+

+⋅∇−⋅+∂∂

∫

∫∫∫∫v

vvvvvvvv θ

(1)

0SdPrRe

1SdvdV

t CSCSCV

=⋅∇−⋅+∂∂

∫∫∫vvvv θθθ

where vv , P and θ are Cartesian velocity vector, pressure and temperature, respectively; Re is the Reynolds number; Pr is the Prandtl number; rG

v

is the vector of Grashof number representing the thermal buoyancy; CV is the control volume considered and CS is the boundary surface of CV. Applying the Divergence theorem, adapting the backward time differencing scheme and keeping the pressure fixed at the current time level n, the momentum and temperature equations can be discretized as:

0VRH

RRt

QcQcQc

nP

visinv

1n3

n21

=++

−++− −

∆∆

)

(

*

***

(2)

where the superscript “*” represents the intermediate state, “n” the current time level;

V∆ is the volume of the considered cell; t∆ is the time increment. The vector of conserved variables is defined as [ ]TvuQ θ= . The surface integral of inviscid fluxes invR , the pressure flux PR , the viscous fluxes visR and the source term H will be defined later. The constants are 1c =1.5, 2c =2 and 3c =0.5 for the second-order accurate backward differencing scheme, and 1c =1, 2c =1 and 3c =0 for the first order Euler implicit scheme. The intermediate velocity ∗vv generally does not satisfy the divergence-free condition. The velocity and the pressure are corrected as

nn1n

n1n

PP

tvv

φ

φ∆

+=

∇−=+

∗+vvv

(3)

where φ is the pressure correction. By requiring 1nv +v be divergence-free, we obtain the Poisson equation for φ :

0tvn2 =

⋅∇−∇

∆φ

*vv (4)

Equations (2)~(4) constitute the implicit pressure correction method used in this work.

3 Finite-Volume Method

A cell-centered finite volume method on unstructured Cartesian grid is developed in this work. The flow variables are stored at the center of the cell volume. The flow states at a cell vertex are obtained by averaging the surrounding center values with the weighting constant proportional to the inverse of the distance to the particular center. The variable gradients at the cell center are obtained by differencing the vertex values for the particular cell. The left and right variable states at a cell-face center are linearly reconstructed from the neighboring center values using the estimated gradient information. The mean variable states at the cell-face centers are obtained by a simple average:

)(. Rf

Lf

Mf QQ50Q += (5)

where the subscript f represents the face index 21j /± or 21k /± ; the superscripts M, L and

R represent the mean, left and right states of face f. The inviscid flux integral InvR is defined as

Snvvu

V1

R Mf

RL

CSinv ∆

θ∆

ˆ

/

⋅

= ∑ v (6)

where S∆ is the face area; n is the outward unit surface normal; and the summation operator is done over all surfaces of the cell. The subscript L/R represents the velocity upwinding:

( ) ( )( )

≤⋅•>⋅•

=•0nvif0nvif

Mf

R

Mf

L

RL ˆˆ

// vv

(7)

The pressure integral is computed as

∑

⋅⋅

=CS

MfP SP

0jnin

V1

R ∆∆

ˆˆ

ˆˆ

(8)

3

where i and j are the Cartesian unit vectors. The viscous fluxes visR are

S

nPr1

nvnu

VRe1

RCS

vis ∆

θ∆ ∑

⋅∇

⋅∇⋅∇

=ˆ

ˆˆ

v

vv

(9)

The variable gradients at the cell-face centers are obtained by differencing the neighboring cell-center values. The source term representing thermal buoyancy is

=

0GrGr

Re1

H y

x

2θθ

(10)

where xGr and yGr are the Grashof number in x and y direction, respectively. On a regular Cartesian grid, formulations (6) to (10) are second-order accurate schemes for inviscid and viscous flux terms in Eq. (1).

4 Pressure-Velocity Coupling

To compute the divergence of velocity, a face-normal velocity fU is defined independently for each cell face, which is different from the cell center velocity. Specifically, for a cell face f with surface unit normal n , the intermediate state of the face-normal velocity is defined as:

nePPt

cdispnvU Lf

Rf

Mff ˆˆ)(ˆ* ⋅−−⋅=

lv

∆∆

(11)

where l∆ is the distance between the right and the left neighboring cell centers; e is the Cartesian unit vector normal to the face; cdisp is an input constant for the pressure dissipation. The divergence of the intermediate velocity for a particular cell is computed as

∑ ∑

∑

⋅−

−⋅=

=⋅∇

CS CS

Lf

RfM

f

CSf

SnePP

Vtcdisp

SnvV1

SUV1

v

∆∆∆

∆∆

∆

∆∆

ˆˆˆ

)( **

lv

vv

(12) The last term on the right hand side of Eq. (12) is a background dissipation term based on the

pressure field. It has similar effects as the wildly used momentum interpolation method suggested by Rhie and Chow [7]. On a regular Cartesian grid this dissipation is proportional to

)(4

42

4

42

yP

ybxP

xacdisp∂∂

+∂∂

⋅ ∆∆ where a and b

are some constants. Note that in Eq. (11) the dissipation term is written in a simple form suitable for unstructured meshes.

The intermediate face-normal velocity is corrected similarly as the cell center velocities by

ntUU fn

f1n

f ˆ)(* ⋅∇−=+ φ∆ (13)

where the gradient of pressure correction is computed at the cell-face center. By demanding

1nfU + be divergence-free, a discrete Poisson

equation for the pressure correction is constructed as

0SUVt

1Sn

V1

CSf

CSf

n =−⋅∇ ∑∑ ∆∆∆

∆φ∆

*ˆ)(v

(14)

Equation (14) is a compact discretization of Eq. (4). It is used to compute the pressure correction nφ at the cell centers, which in turn determines the cell-center pressure and velocity at the new time level n+1 in Eq. (3).

5 Treatment of Immersed Bodies

The body velocity Bvv and the body temperature Bθ of the immersed bodies are assumed known from some appropriate governing equations. In this work, we use a level set function Lφ defined on the cell vertices to track the presence of the immersed bodies. W chose Lφ to be a signed distance function whose absolute value equals the shortest distance to the body surface. It is defined

0L >φ outside the body, 0L <φ inside, and 0L =φ on the body surface. With Lφ known,

the volume fraction occupied by the immersed body in a cell, or the Volume of Body (VOB) function Bφ , can be computed easily. The volume of body function is used to identify the body cells ( Bφ =1), the fluid cells ( Bφ =0) and the interface cells ( 10 B << φ ). For

4

convenience, the body surface is represented by the contour of 50B .=φ . Since the velocity inside the body surface is known, we modify Eq. (2) to

0t

QcQcVRH

RRt

QcQcQcV1

1nB11

BnP

visinv

1n3

n21

B

=−

+++

−++−

−

+

−

)()

()(

**

***

∆∆φ

∆∆φ

(15)

where [ ]TBBBB vuQ θ= represents the

component of body velocity and the body temperature. Equation (15) recovers Eq. (2) for fluid cells, and it yields 1n

BQQ +=* for body cells. As for interface cells, the solution of Eq. (15) is a volume-averaged mixture of the body velocity (temperature) and the velocity (temperature) computed by the flux conservation. This volume averaging is a simple and effective treatment to connect the flow solution in fluid cells to the boundary conditions 1n

BQ + in body cells. However, the averaged interface variables do not satisfy Eq. (2). They are merely values interpolated between body cell values and fluid cell values using Eq. (15). Thus interface cells should not be included in the residual computation.

Note that the boundary surface of the domain occupied by all body cells with Bφ =1 is a closed stair-step like zigzag contour. The volume averaging of flow variables in interface cells essentially smoothens the stair-step representation of the body surface. The true location of the body surface is embedded in the interface cells with an uncertainty of one cell size. This implies that effective body dimension will be somewhat smaller than the true body dimension. Fortunately this error in geometry will decrease when the grid is refined. In this work the capability of local refinement on unstructured Cartesian grid is developed, which is useful to enhance the grid resolution around the body surface.

In essence, the domain inside the body surface is viewed as being occupied by the same fluid as outside with a prescribed incompressible velocity distribution. In this view there is no property jump across the body

surface, and the pressure inside the body surface obeys the same governing equation as outside. Thus, the Poisson equation for pressure correction, Eq. (14), is used for the entire computational domain without distinguishing the cell type. When the given 1n

Bv +v is divergence-free, the source term of Eq. (14) is zero automatically. The pressure inside the body has no physical meaning. The elliptical nature of the Poisson equation ensures that the pressure field inside the body adjusts itself according to the pressure field outside.

To compute the forces acting on the immersed body, the surface integral of the pressure and viscous stresses over the body surface is transformed into a volume integral over the body volume. Similarly, the conduction heat over the body surfaces is also transformed into a volume integral. Then, the total force acting on the immersed body and the total heat transfer over the body surface are obtained by

)(

)(

θ∆φ

∆φ

2

cellBbody

2

cellBBody

PrRe1

VH

vRe1

PVf

∇=

∇+∇−=

∑

∑ vvv

(16)

6 Implicit Relaxation Method

Applying Newton’s method to Eq. (15) and writing it in delta-law form, the implicit time integration equation can be written as

s

1nB1

s1

BnP

s

svis

sinv

1n3

n2

s1

B

s1ssvis

sinv

B1

Rest

QcQcVRH

RRt

QcQcQcV1

QQQ

RRV1

tVc

=

−+++

−++−

−−=

−

∂

−∂−+

+

−

+

)()

()(

)()(

)(

∆∆φ

∆∆φ

∆φ∆∆

(17) where the superscript s is the index for sub-iteration, and sRes is the right hand residual. Note that on the implicit side of Eq. (17) the Jacobian of source term H is neglected. Equation (17) can be represented by

5

[ ] { }s

ss1s

Res

SourceQRHSQQLHS

=

−−=−+ )()( (18)

where )(RHS− is the right hand side operator involving only the current variable sQ ; Source represents all terms involving known states of

nP , nQ , 1nQ − and 1nBQ + in the residual; LHS is

the implicit operator. When the sub-iteration in s converges, the solution is *QQ 1s =+ , satisfying the time-accurate Eq. (15). Note that term Source is constant during the sub-iteration. In operator LHS, simplifications are obtained by using the first order upwind scheme for the convective fluxes and the 3-point compact difference operator for the viscous fluxes. With these simplifications, the stencil of LHS extends only to the neighboring cell centers while the right hand side operators are kept second-order accurate.

Splitting LHS into the sum of a diagonal part D, a lower triangular part L and an upper triangular part U, a two-step implicit relaxation method is used to invert Eq. (18) as

ss1s

sss

ss1s

QQQ

Res2QUDQL

Res2QLDQU

∆

ωω∆ω∆ω

ωω∆ω∆ω

+=

−=++

−=++

+

−

)()(

)()(*

*

(19)

where the superscript *s indicates the intermediate stage in sub-iteration s; ω is a relaxation parameter, and here we take

190 ≤≤ ω. . The initial conditions are ns QQ = and 0Q 1s =−∆ for s=1. For further

simplifications, in D, L and U the inviscid flux Jacobians are replaced by their eigenvalue matrices such that the solution process of Eq. (19) requires only scalar operations.

It is our experience that the convergence rate of Eq. (19) will deteriorate for highly refined grids. Thus to accelerate the convergence, a V-cycle multigrid method is developed on unstructured Cartesian grid for Eqs. (18) and (19). The tree data structure generated in the grid generation provides a natural sequence of grid coarsening from fine to coarse grids.

7 Relaxation Method for Poisson Equation

Applying the Newton’s method to Eq. (14), the implicit relaxation method can be written as

s

CS CSff

s

s1s2

Res

SUt

1Sn

V

φ

∆∆

∆φ

∆φφ

=

−⋅∇−=

−∇

∑ ∑

+

*ˆ)(

)(v

(20)

where s is the index for sub-iteration. When the iteration in s converges, the solution is

n1s φφ =+ , which satisfies the Poisson equation, Eq. (14). The left hand side operator is a compact Laplacian operator whose stencil extends only to the neighboring cell centers. Equation (22) can be further represented by [ ] { }

s

ss1s

Res

Source-)RHSLHS

φ

φφ φφφ

=

−=−+ ()( (21)

where the term φSource is related to the divergence of velocity field. Note that this equation is in the same form as Eq. (18). Hence, similar to the method described above, an implicit relaxation method and a V-cycle multigrid method are developed to solve Eq. (21). The initial conditions are 0s =φ and

021s =− /φ∆ for s=1.

8 Forced Convection Over A Cylinder

The forced convection over a circular cylinder of unit diameter are computed on unstructured Cartesian grids. The grid is refined around the cylinder surface such that there are about 384 interface cells. The cylinder volume computed by summing up VB∆φ is 0.78536, which is about 0.01% less than the true value of

π250. . The outer boundaries are 30 diameters away from the origin. The total number of grid cells is 30,304. The cylinder temperature is one and the far stream temperature is zero. The Prandtl number is 0.71. The uniform flow condition is set to the inflow boundary and the two boundaries in y. The downstream boundary follows the upwind differenced equation of

0xQUtQ n =∂∂+∂∂ )/()/( , where nU is the computed normal outflow velocity at the boundary.

For the steady case of forced convection at

6

Re=40, the Euler implicit method is used with the maximum CFL number around 25. The infinity norm of the steady state residual for fluid cells, or Eq. (15) with Bφ =0 and without the term involving t∆ , has dropped 4 orders of magnitude in 500 steps. The computed pressure coefficient contours temperature contours and the streamlines are plotted in Fig. 1. For convenience, the contour of level set function 0L =φ is displaced as the cylinder wall. The flow is steady with a separation bubble behind the cylinder. Note that the pressure inside the cylinder adjusts itself automatically to the pressure field outside. The pressure contours intersect the cylinder wall in a nearly orthogonal manner. The streamlines around the cylinder are smooth, indicating the effects of the interface cells in smoothening the stair-step representation of the cylinder surface.

To examine the surface pressure distribution, the pressure coefficient at the intersection of the cylinder surface and the grid lines is interpolated using the surrounding center values and plotted in Fig. 2. The data from Fornberg [10] are also included for comparison. The two results follow the same general trend, but the current computation predicted a slightly lower surface pressure distribution on the lee side. This is not unexpected. In the current approach the cylinder surface has an uncertainty of one grid cell, hence the effective cylinder radius is somewhat smaller than the true cylinder radius.

Table I lists the wake length (Lw) normalized by the diameter (d), the computed lift coefficient (Cl), drag coefficient (Cd) and the average Nusselt number (Nu) using Eq. (16). The comparison with the work of others is generally satisfactory. The curve fit of experimental data for average Nu in Holman [11] is used as reference data. The difference between the computed Nu and the experimental curve fit is less than 5%.

9 Heated Cylinder In A Square Cavity

The next validation examines the natural convection heat transfer from a heated cylinder placed in a square cavity with unit length. This

problem has been studied by Moukalled and Darwish [12] and Sadat and Couturier [13]. The radius of the cylinder is 0.1. The center of the cylinder is at (0.35, 0.35) from the left lower corner of the cavity. The temperature of the cylinder is one and the wall temperature is zero. The base grid is 6464 × covering the cavity with a four level refinement around the cylinder surface. The finest cell is 1/1024 in length such that there are about 577 interface cells around the cylinder. The total cell number is 13036. This grid is shown in Fig. 3.

The cases of a Rayleigh number Ra = 410 , 510 and 610 are computed in this study.

Figures 4 and 5 show the computed streamlines and isotherms, respectively. The streamlines and isotherms are similar to those presented in [12, 13]. The formation of a thermal plume at Ra = 610 is clearly observed. In Table 2 the computed average Nusselt number over the cylinder is compared with the results in [12, 13]. One can see that the agreement is very good.

10 Conclusions

The incompressible Navier-Stokes equations with heat transfer are solved by an implicit pressure correction method and an immersed boundary method. The immersed body is identified by the Volume of Body function. The domain inside the body is viewed as being occupied by the same fluid as outside with a prescribed velocity and temperature distribution. In this view the pressure field inside the body obeys the same Poisson equation used for the fluid flow outside. The flow variables at the interface cells are obtained as a volume-averaged mixture of the body variables and the variables estimated by the conservation equations. Test examples of the forced convection over a circular cylinder and the natural convection over a heated cylinder inside a square cavity show good comparisons with published data.

Acknowledgement

This work is funded by National Science Council under the grants NSC91-2212-E006-

7

104 and NSC92-2212-E006-105. The support is highly appreciated.

References

[1] Peskin C.S., The Immersed Boundary Method, Acta Numerica, pp. 1-39, 2002.

[2] Fadlun E.A., Verzicco R. , Orlandi P.. and Mohd-Yusof J. , Combined Immersed-Boundary Finite-Difference Methods for Three-Dimensional Complex Flow Simulations, J. Comput. Phys, Vol. 161, pp. 35-60, 2000.

[3] Kim J., Kim D. and Choi H., An Immersed-Boundary Finite-Volume Method for Simulations of Flow in Complex Geometries, J. Comput. Phys., Vol. 171, pp. 132-150, 2001.

[4] Ye T., Mittal R., Udaykumar H.S.. and Shyy W., An Accurate Cartesian Grid Method for Viscous Incompressible Flows with Complex Immersed Boundaries, J. Comput. Phys., Vol. 156, pp. 209-240, 1999.

[5] Baaijens Frank P.T., A Fictitious Domain/Mortar Element Method for Fluid-Structure Interaction, Int. J. Numer. Meth. Fluids, 2001; 35, 743-761.

[6] Xiao F., A Computatioinal Model for Suspended Large Rigid Bodies in 3D Unsteady Viscous Flows, J. Compu. Phys., 1999, 155, 348-379.

[7] Rhie C.M. and Chow W.L., A Numerical Study of the Turbulent Flow Past An Airfoil With Trailing Edge Separation, AIAA J., Vol. 21, pp. 1525-1532, 1982.

[8] Armfield S.W. and. Street R, Fractional Step Methods for the Navier-Stokes Equations on Non-Staggered Grids, ANZIAM J., Vol. 42 (E), pp. C134-C156, 2000.

[9] Ghia U., Ghia K.N. and Shin C.T., High-Re Solutions for Incompressible Flow Using the Navier-Stokes Equations and a Multigrid Method, J. Comput. Phys., Vol. 48, pp. 387-411, 1982.

[10] Fornberg B., A Numerical Study of Steady viscous Flow Past a Circular Cylinder, J. fluid Mech., Vol. 98, pp. 819-855, 1980.

[11] Holman J.R., Heat Transfer, ISBN 0-07-029618-9, McGraw-Hill, pp. 243, 1981.

[12] Moukalled F. and Darwish M., New Bounded Skew Central Difference Scheme, part II: Application to Natural Convection In An Eccentric Annulus, Numerical Heat Transfer, Part B, Vol. 31, pp. 111-133, 1997.

[12] Sadat H. and Couturier S., Performance and Accuracy of a Meshless Method for Laminar Natural Convection, Numerical Heat Transfer, Part B, Vol 37, pp. 455-467, 2000.

Table I Simulation Results for Flows over a Circular Cylinder

Re Cd Lw/d Nu

Current Method 40 1.51 2.18 3.23

Fornberg [10] 40 1.50 2.24

Ye et al.[4] 40 1.52 2.27

313850 PrRe9110 /.. [11] 40 3.36

Table II Average Nusselt Number over Heated

Cylinder in Cavity

Rayleigh Number Ra

Current Method

Moukalled and Darwish [12]

Sadat and Couturier [13]

410 4.686 4.741 4.699

510 7.454 7.435 7.430

610 12.54 12.453 12.421

-1 0 1 2x

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

y

-1 -0.5 0 0.5 1 1.5 2 2.5x

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

y

Re=40 Streamlines

(a)

8

-1 0 1 2x

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

y

-1 -0.5 0 0.5 1 1.5 2 2.5x

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

y

Re=40, temperature

(b)

Fig. 1 Flows over a stationary circular cylinder, Re=40, (a) streamlines, (b) temperature contours, the contour 0L =φ is plotted to represent cylinder wall.

-1 0 1 2x

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

y

-1 -0.5 0 0.5 1 1.5 2 2.5x

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

y

Re=40, Cp

(a)

0 50 100 150Theta

-2

-1

0

1

2

Cp

Fornberg[1980]computed

Re=40, Surface Cp

(b)

Fig. 2 Flows over a stationary circular cylinder, Re=40, (a) Cp contours, (b) Surface Cp distribution, symbols: computed, line: Fornberg [1980].

0 0.25 0.5 0.75 1x

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

y

0 0.25 0.5 0.75 1x

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

y

Fig. 3. 6464 × grid with four levels of refinement around the cylinder surface.

9

0 0.25 0.5 0.75 1x

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

y

Ra=104, Streamlines

(a)

0 0.25 0.5 0.75 1x

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

y

Ra=105, Streamlines

(b)

0 0.25 0.5 0.75 1x

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

y

Ra=106, Streamlines

(c)

Fig. 4. Streamlines for (a) Ra = 410 , (b) Ra = 510 and (6) Ra= 610

0 0.25 0.5 0.75 1x

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

y

Ra=104, Isotherms

(a)

0 0.25 0.5 0.75 1x

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

y

Ra=105, Isotherms

(b)

0 0.25 0.5 0.75 1x

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

y

Ra=106, Isotherms

(c)

Fig. 5. Isotherms for (a) Ra = 410 , (b) Ra = 510 and (6) Ra= 610