# Boundary Element Method in Viscous Fluids

Mechanics

Oct 24, 2013 (4 years and 8 months ago)

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Boundary Element Method in

Viscous Fluids

Existing computational methods in

viscous fluid mechanics

Finite Difference Method (FDM)

:

-

simple to use

-

fast for computation

-

requires regular mesh (structure mesh)

Finite Volume Method (FVM):

Finite Element Method (FEM):

Boundary Element Method (BEM)

-

uses integral form of basic equations of fluid mechanics

-

uses unstructure mesh

-

flux and gradient are not accurate

-

convenient for mesh generation

-

difficult to determine penalty parameters in penalty formulation

-

easy to set up computational model

-

small disturbance potential problems

-

incompressible fluid flows

Features of current boundary
-
domain

integral equation method

Solving full Navies
-
Stokes equations

Valid for incompressible and compressible fluids

for incompressible fluids, pressure term can be eliminated

from the system equations

Easy to be developed as meshless BEM formulations using

the Radial Integration Method (RIM) and, consequently, all

advantages of BEM can be remained

Uses primitive variables in basic equations

Velocity gradients can be accurately determined

(the accuracy is as high as velocity itself)

Automatically satisfies infinite boundary condition,

so advantageous for solving aerodynamic problems

Governing equations in viscous fluid flows

Continuity equation:

Momentum equation:

: fluid density

: velocity

: body force per unit mass

: stress tensor

: shear stress tensor

: heat conductivity

: temperature

: energy

:

: pressure

where

Energy equations:

or

k

T

E

J

p

Equation of state:

(gas),

(water)

C
onstitutive relationship

based on

Stokes’ hypothesis

: viscosity

: internal energy

: specific heat

: the Kronecker delta

: outward normal to boundary

: traction

where

e

Stress
-
pressure relationship:

Stress
-
velocity relationship:

Traction
-
stress relationship:

Energy relationship:

Weighted residual equation for
conservation of momentum

Weighted residual equation

Choose weight function

to satisfy:

is the Dirac delta function:

where

Boundary
-
domain integral equation for
conservation of momentum

Fundamental solutions for

momentum integral equation

for 2D

for 3D

where

and

Velocity divergence integral equation

Pressure integral equations for internal points

Based on continuity equation:

Pressure equations for boundary points

Based on continuity equation:

and the first invariant of strain rate:

Pressure for boundary points can be expressed as

It can be seen that in general pressure is not equal to normal traction

Weighted residual equation for
conservation of energy

Weighted residual equation for energy

Choose weight function

to satisfy:

is the Dirac delta function:

where

where

Boundary
-
domain integral equation for
conservation of energy

where

with being the enthalpy

Fundamental solutions for

energy integral equation

where

for 2D and

(i.e.,

for 3D)

and

Numerical implementation
of
incompressible flows

Condition for
:

Discretization of boundary and domain:

where is shape functions and is nodal values of .

Algebraic matrix equations for
incompressible flows

For boundary nodes:

For internal nodes:

For pressure

where

{
X
}: containing boundary unknown velocities and tractions

Numerical example: Couette flow

Velocity profile on vertical lines

Driven flow in an unitary square cavity

Re=100

Traction distribution along top wall

Velocity vectors

Vortex center: (0.6153, 0.7354) by current method

(0.6172, 0.7344) by Ghia et al in1982

Three
-
dimensional curved pipe flow

=1

and

=1

=50

Discretization of half:

672 boundary elements

2880 linear cells

719 boundary nodes

2784 internal nodes

3503 nodes (total)

Boundary conditions:

Upper end:

Lower end:

velocity vector plot for different sections
over vertical central plane

Section

x

z

u
x

u
z

p

1

10.000

-
10.250

1.9593

-
0.6504

2.0644

-
1.6262

3

8.8179

-
9.8488

1.5734

-
0.3890

1.6208

9.1021

5

7.7094

-
9.6240

1.7870

-
0.5257

1.8627

16.0389

7

6.6745

-
9.3255

1.8425

-
0.7386

1.9851

22.1204

9

5.7133

-
8.9534

1.8423

-
0.9610

2.0778

27.2860

11

4.8257

-
8.5076

1.8029

-
1.1930

2.1619

31.5646

13

4.0118

-
7.9882

1.7290

-
1.4325

2.2453

35.0271

15

3.2715

-
7.3952

1.6234

-
1.6750

2.3326

37.7730

17

2.6048

-
6.7285

1.4890

-
1.9151

2.4259

39.9218

19

2.0118

-
5.9882

1.3305

-
2.1477

2.5265

41.6027

21

1.4924

-
5.1743

1.1538

-
2.3712

2.6370

42.9407

23

1.0466

-
4.2867

0.9659

-
2.5867

2.7611

44.0433

25

0.6745

-
3.3255

0.7752

-
2.8086

2.9137

44.9647

27

0.3760

-
2.2906

0.5868

-
3.0871

3.1424

45.6331

29

0.1512

-
1.1821

0.3579

-
3.6088

3.6265

45.5448

31

0.1250

0.0000

0.0289

-
2.3579

2.3581

53.1565

Results of each section over central plane

Contour plot of pressure over vertical
central plane

Iteration history for pipe flow

Computational time in different
stages (Minutes)

Conclusion

compressible and incompressible flows.

No velocity gradients appear in the system of equations.

Pressure can be eliminated from the system of equations.

Velocity gradient can be explicitly derived from the basic integral

equation and, therefore, has high computational accuracy.