粘性流体的边界元算法
高效伟
教授
东南大学
工程力学系
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
steady
incompressible flows
Condition for
steady incompressible flows
:
Discretization of boundary and domain:
where is shape functions and is nodal values of .
Algebraic matrix equations for
steady
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
•
Presented formulations are general, applicable to steady, unsteady,
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.
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