1/36 Gridless Method for Solving Moving Boundary Problems Wang Hong Department of Mathematical Information Technology University of Jyväskyklä 28.05.2009

1/36

Gridless Method for Solving Moving Boundary Problems

Wang Hong

Department of Mathematical Information Technology

University of Jyväskyklä

28.05.2009

2/36

Content

Introduction

Principle of Gridless Method

Steady Simulation of Euler Equations and Applications

Unsteady Simulation of Euler Equations: Validation on Referenced Airfoils

Conclusion and Future Research

3/36

At present, CFD (Computational Fluid Dynamics) community has

many methods in solving moving boundary problems, for instance,

dynamic mesh method and fictitious domain method and so on.

The dynamic mesh method contains different techniques, for

example, mesh reconstruction methods and mesh deformation

methods.

Introduction

4/36

Mesh reconstruction methods have been proposed based on the

regeneration of mesh according to the moving boundaries. For

example, when we use unstructured Cartesian mesh to solve

moving boundary problems, we should fasten the mesh; the

moving boundaries cut the mesh elements. These kinds of methods

need to consider the cutting elements in every time step, and refine

and coarsen the grids to satisfy the distribution requirements.

Introduction

5/36

Introduction

Mesh deformation methods are basically relied on the control models.

Spring approximation model was firstly introduced by Batina to solv

e the vibrating airfoil flows. The basic idea is that treat every edge of

mesh elements as a spring, the coefficients of the spring are related t

o the length of every edge of mesh elements; after the movement of t

he boundaries, the new positions of the mesh points are defined by s

olving the force equilibrium of the spring system.

6/36

A fast dynamic cloud method based on Delaunay graph mapping

strategy is proposed in this presentation. A dynamic cloud method

makes use of algebraic mapping principles and therefore points can

be accurately redistributed in the flow field without any iteration. In

this way, the structure of the gridless clouds is not necessary

changed so that the clouds regeneration can be avoided

successfully.

Introduction

7/36

Gridless node definition

Spatial discretisation


8/36

Global and close-up views of typical structure for gridless clouds

Gridless Node Definition

9/36

k k ih x x

k k il y y 1

i

fa

x

2

i

fa

y

If we keep the first order of , then we have the approximation

1 2k i k kf f a h a l Define the total error in the cloud of points as

2

1

M

k kk

G f f

in order to minimize the total error , let

1 2

0G G

a a

Ax＝ b

1 ik ik ia f f 2 ik ik ia f f

Spatial Discretisation – Least Square Method

2 2

1 2 ( , )k i k k k kf f a a O h lh l

kf

10/36

Steady Simulation of Euler Equations and Applications

Governing Equations

Boundary Conditions

Spatial Discretisation

Time Discretisation

Steady Flow Simulation Results

NACA0012

RAE2822

11/36

0t x y

W E F

2

2

[ , , , ]

[ , , , ( ) ]

[ , , , ( ) ]

u v e

u u p uv e p u

v vu v p e p v

T

T

T

W

E

F

Whereρis the density ， u and v are the velocity components ， p is the pressure, e is the total energy per unit volume, for an idea gas, it can be written as

2 21( )

1 2

pe u v

Governing Equations

12/36

V

V

pn

pw

Vn

p*

V*

p

pw

V

h

h V

Boundary Conditions – Solid Wall

(1) Direct method (2) Mirror method

13/36

Boundary Conditions – Far Field

M<1

inner field

u+c u-c

t

u

outer field

x

u

M<1

outer field

u+c u-c

t

u

inner field

x

u

M>1

inner field

u+c u-c

t

u

outer field

x

u

M>1

outer field

u+c u-c

t

u

inner field

x

u

(1) Subsonic inflow (2) Subsonic outflow

(3) Supersonic inflow (4) Supersonic outflow

14/36

0i i

t x y

W E Fi

ix y

E FQ

i ik ik i ik ik i

ik ik ik ik ik i ik i

ik i

Q E E F F

E F E F

G G

U u v

G E F( )

U

uU p

vU p

e p U

G


15/36

i ik i Q G G

,ik L RG G W W

i k

ik

L R

Center node i and satellite node k. (“+” denotes the right wave,

“ －” denotes the left wave) Roe Scheme


16/36

1n ni i

it

W W

R

(0)

(1) (0) (0)1

(2) (0) (1)2

(3) (0) (2)3

(4) (0) (3)4

1 (4)

ni i

i i i i

i i i i

i i i i

i i i ini i

t

t

t

t

W W

W W R

W W R

W W R

W W R

W W

1 2max , , ,iM

CFLt

A A A

( 1,2, , 4)k k

represents the stage coefficients, and

1

2

3

4

0.0833

0.2069

0.4265

1

2 2( ) u v c A

Time Discretisation

17/36

Global and close-up views of the computational domain for the NACA0012 airfoil.

Steady Flow Simulation Results – NACA0012

x

y

-10 -5 0 5 10-10

-5

0

5

10

x

y

-1 -0.5 0 0.5 1-1

-0.5

0

0.5

1

337 nodes on the airfoil and 5557 nodes in the flow field.

18/36

0.8, 0.0M

x/c

Cp

0 0.2 0.4 0.6 0.8 1

-1.5

-1

-0.5

0

0.5

1

1.5

M = 0.8 = 0.0o

X

Y

-1 0 1 2-2

-1

0

1

2

M = 0.8 = 0.0o

Flow field pressure coefficients and mach number distributions for NACA 0012 airfoil.


19/36

0.8, 1.25M

x/c

Cp

0 0.2 0.4 0.6 0.8 1

-1.5

-1

-0.5

0

0.5

1

1.5

M = 0.8 = 1.25o

XY

-1 0 1 2-2

-1

0

1

2

M = 0.8 = 1.25o



20/36

0.85, 1.0M

x/c

Cp

0 0.2 0.4 0.6 0.8 1

-1.5

-1

-0.5

0

0.5

1

1.5

M = 0.85 = 1.0o

X

Y-1 0 1 2

-2

-1

0

1

2

M = 0.85 = 1.0o



21/36

1.2, 7.0M

x/c

Cp

0 0.2 0.4 0.6 0.8 1

-1.5

-1

-0.5

0

0.5

1

1.5

M = 1.2 = 7.0o

X

Y

-1 0 1 2-2

-1

0

1

2

M = 1.2 = 7.0o



22/36

Steady Flow Simulation Results – RAE2822

x

y

-10 -5 0 5 10-10

-5

0

5

10

x

y

-1 -0.5 0 0.5 1-1

-0.5

0

0.5

1

Global and close-up views of the computational domain for the RAE2822 airfoil.


23/36

0.725, 2.55M

x/c

Cp

0 0.2 0.4 0.6 0.8 1

-1.5

-1

-0.5

0

0.5

1

1.5

M = 0.725 = 2.55o

X

Y

-1 0 1 2-2

-1

0

1

2

M = 0.725 = 2.55o


Flow field pressure coefficients and mach number distributions for RAE2822 airfoil.

24/36

0.75, 3.0M

x/c

Cp

0 0.2 0.4 0.6 0.8 1

-1.5

-1

-0.5

0

0.5

1

1.5

M = 0.75 = 3.0o

X

Y-1 0 1 2

-2

-1

0

1

2

M = 0.75 = 3.0o


Flow field pressure coefficients and mach number distributions for RAE2822 airfoil.

25/36

Unsteady Simulation of Euler Equations and Validation

A Fast Dynamic Cloud Method

Unsteady Flow Simulation Results

NACA0012

NACA64A010

26/36

(a) Global view (b) Close-up view

Back ground mesh for NACA0012 airfoil based on Delaunay triangulationBack ground mesh for NACA0012 airfoil based on Delaunay triangulation


27/36

XY

0.39 0.4 0.41-0.385

-0.38

-0.375

-0.37

-0.365

-0.36

-0.355

(a) Spring analogy strategy (b) Delaunay graph mapping strategy

Moved gridless clouds of 30°pitching airfoil


28/36

Close-up views of computational domain for the NACA0012 airfoil for 2.51° pitch.

Unsteady Flow Simulation Results – NACA0012


29/36

(o)

CL

-3 -2 -1 0 1 2 3

-0.4

-0.2

0

0.2

0.4

ExperimentKirshmanComputation

(o)

Cm

-3 -2 -1 0 1 2 3-0.03

-0.02

-0.01

0

0.01

0.02

0.03

ExperimentKirshmanComputation

Comparisons of computed lift and moment coefficients with the experimental and Kirshman’s data for prescribed oscillation of NACA0012 airfoil.

Unsteady Flow Simulation Results – NACA0012

0( ) sin( ) 0.755 0.016 2.51 0.0814m 0 mt + t Ma k

30/36

Close-up views of computational domain for the NACA64A010 airfoil for 1.01° pitch.

Unsteady Flow Simulation Results – NACA64A010


x

y

-0.5 0 0.5 1

-0.5

0

0.5

x

y

-0.5 0 0.5 1

-0.5

0

0.5

x

y

-0.5 0 0.5 1

-0.5

0

0.5

31/36

x/c

Re(

Cp 1)

0 0.2 0.4 0.6 0.8 1-30

-20

-10

0

10

20

30ExperimentWanggangComputation

x/cIm

(Cp 1)

0 0.2 0.4 0.6 0.8 1-20

-10

0

10

20ExperimentWanggangComputation

Comparisons of the first Fourier mode component of surface pressure coefficients with the experimental data for oscillating NACA64A010 airfoil.

Unsteady Flow Simulation Results – NACA64A010

0( ) cos( ) 0.796 0.0 1.01 0.202m 0 mt + t Ma k

32/36

0( ) sin( ) 0.755 0.016 2.51 0.0814m 0 mt + t Ma k

Gridless Method with Dynamic Clouds of Points for Solving Unsteady CFD Problems in Aerodynamics

– accepted by International Journal for Numerical Methods in Fluids

0( ) cos( ) 0.796 0.0 1.01 0.202m 0 mt + t Ma k

Mach number distribution for pitching airfoil (NACA0012 and NACA64A010)

33/36

Conclusion and Future

Introduction


Steady Simulation of Euler Equations

Unsteady Simulation of Euler Equations

Future Research

Inverse problem with one NACA 0012 airfoil using gridless method

Inverse problem with dual NACA 0012 airfoils using gridless method

Mesh/gridless hybridized algorithms to solve other boundary moving problems

Our method is both flexible and efficient, therefore it is quite suitable to solve optimization problems, such as:

Multi element airfoil lift optimization with Navier-Stokes flows in Aerodynamics

Antennas optimal position in Telecommunications

34/36

Inverse problem of NACA 0012

x/c

Cp

0 0.2 0.4 0.6 0.8 1

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0.0(target)0.0006

generation

(o)

1 1.5 2 2.5 3 3.5 4

0.2

0.4

0.6

0.8

1

generation

f()

1 1.5 2 2.5 3 3.5 4

10-7

10-6

10-5

10-4

10-3

10-2

0.5; 0.0Ma 2*

1

min ( ) ( ) ( )M

p p ii

f C C

Genetic Algorithms Using a Gridless Euler Solver for 2-D Unsteady Inverse Problems in Aerodynamic Design

– will be submitted to EUROGEN 2009

35/36

x

y

-1 -0.5 0 0.5 1 1.5 2

-1

-0.5

0

0.5

1

1.5Mach

0.700.670.640.610.580.550.520.490.460.430.400.370.340.310.280.250.220.190.160.130.100.070.04

0.8; 1.25Ma

x

y

-1 -0.5 0 0.5 1 1.5 2

-1

-0.5

0

0.5

1

1.5Mach

1.501.441.371.311.251.181.121.050.990.930.860.800.740.670.610.550.480.420.350.290.230.160.10

0.5; 0.0Ma

Subsonic and Transonic Simulations of dual airfoils

Mach number distribution for dual aifoils

36/36

Documents

1/36 Gridless Method for Solving Moving Boundary Problems Wang Hong Department of Mathematical Information Technology University of Jyväskyklä 28.05.2009