Adaptive CESE Method for Solving Unsteady Euler Equations

Computing & Microfluidics Lab 國立成功大學工程科學系計算與微流體晶片實驗室

Adaptive CESE Method for Solving Unsteady Euler Equations

Ruey-Jen Yang

National Cheng Kung University

Department of Engineering Science 1

To Tony Sheu

Computing & Microfluidics Lab 2

Wishing you

Good spirit, Good health,

Keep random walk with deterministic mind

Enjoy life!

Introduction

Computing & Microfluidics Lab

Shock Vortex Interaction

Flow discontinuities, Unsteady waves,

Accuracy, Easy treatment of BCs, Length Scales

NASA Glenn Research Center website: http://www.grc.nasa.gov/WWW/microbus/

3

The Inventor of the CESE Method


Dr. Sin-Chung Chang NASA Glenn Research Center

Chang SC, (1995), The method of space-time conservation element and solution element– a new approach for solving the Navier-Stokes and Euler equations, Journal of Computational Physics, 119, 295-324.

CESE Method


(1) The method is a complete explicit scheme.

(2) Non-dissipative scheme.

(3) Add numerical dissipation as desired.

(4) Enforced space-time flux conservation, local/global flux conservation.

(5) The spatial derivative is treated as unknown variable w/o

discretization.

(6) No flux reconstruction at mesh interface (Riemann problem).

ux∂∂

Our Former CESE Applications Shock Diffraction


Tseng T and Yang RJ 2005, Shock Waves, Vol. 14, 307-311

Our Former CESE Applications Supersonic Flow over a Wedge


Tseng T and Yang RJ 2006, AIAA Journal, Vol.44, 1040-1047

CV model DPL model

1-D Thermal Waves

Temperature

Heat flux vector

Diffusion

Presentation of wave, wavelike and diffusion behaviors

wave wavelike diffusion

Chou Y and Yang RJ, 2008, International Journal of Heat and Mass Transfer, Vol. 51, 3525-3534

2D Thermal Wave Wave behavior in the condition of B=0.0

Wavelike behavior in the condition of B=0.1

Chou Y and Yang RJ, 2009, International Journal of Heat and Mass Transfer, Vol.52, 239-249

Zone Electrophoresis

power supply

Samples 1+2+3+ -

+ -

buffers

(a)

(b)

anode cathode

123 cathodeanode

Introduced in the 1960s, the technique of capillary zone electrophoresis (CZE) was designed to separate species based on their size to charge ratio in the interior of a small capillary filled with an electrolyte.

Validation of CESE scheme in ZE

2 μA, ∆x=25µm, 8001 grids

Yu JW, Chou Y and Yang RJ, 2008, Electrophoresis, Vol.29, 1048-1057

IsoTachoPhoresis (ITP)

Isotachophoresis (ITP) (Greek: iso = equal, tachos = speed, phoresis = migration) is a technique in analytical chemistry used to separate charged particles.

power supply

samples 1+2+ -

(a)

anode cathode

leading electrolyte

terminating electrolyte

E

+ -

(b)leading

electrolyte


12

E

cathodeanode

+ -

(c)

leading electrolyte


12

E

cathodeanode

In isotachophoresis the sample is introduced between a fast leading electrolyte and a slow terminating electrolyte.

Validation of CESE scheme in ITP

∆x=100µm, 1201 grids

Chou Y and Yang RJ, 2009, Electrophoresis, Vol.30, 819-830

The pH at which charge reversal occurs, i.e. where the net charge is zero, is called the isoelectric point, or pI.

Isoelectric Focusing (IEF)

pH

Net Charge

+3

+2

+1

0

-3

-2

-1 4 5 6 7 8 9

A

B

pIApIB

Sample application

(a)

4 5 6 7 8 9

pH

B A

Focusing

Migration

(b)

20s 40s

65s 110s

IEF results- transition solutions

Chou Y and Yang RJ, 2010 Journal of Chromatography A, Vol. 1217, 394-404

Motivation


(1)Applying a fine mesh to solve complex fluid flow problems is

computationally expensive.

(2) Adaptive mesh provides suitable mesh as necessary to save

computational cost.

(3) The solutions in the current time-step are computed directly from the

solutions obtained in the previous time-step without the need for

extrapolation or interpolation.

Traditional CESE Method (Stationary Mesh)


0=∂∂

+∂∂

xua

tu

(a is constant speed)

Example:

),( uauh =

0)()( =∂∂

+∂∂

=⋅∇ ut

aux

h

∫ ∫∫∫∫

=⋅−⋅=−⋅

=⋅⋅=⋅⋅∇

s ss

Sv

dxudtaudxdtuau

dnhdvh

0),(),(

σ

t

x

A

B

rrdr + rd rd

dt

dx

dt

),( dxdtn −=

A surface element ds and a line segment dr

on the boundary S(V).

Two Eqs. → Solve u and 𝑢𝑢𝑥𝑥 .


Introduction of one-dimensional adaptive CESE method

18

Mesh redistribution


Physical Coordinate:

( ) 0=ξξωx

),.....,,( 21 dxxx=x

),.....,,( 21 dξξξ=ξ

Monitor Function: ( )21 Φ∇+= βω

Computational Coordinate:

Quasi-Static Equidistribution Principle

Φ: Flow variables

β : scaling parameter

Space-time domain


The space-time is divided into non-overlapped rectangular regions as the conservation

elements and each mesh nodes (marked by filled circle) are setting as the solution elements. 20

Stationary Moving

The solution element (SE) and conservation element (CE)


Flow properties are assumed continuous within SE. For any (x, y) SE( j, n), um (x, t)

and fm (x, t) are approximated by u*m(x, t) and f *

m(x, t).

They are defined as :

∈

( ) ( ) ( ) ( ) ( ) ( )nnjmtj

njmx

njmm ttuxxuunjtxu −+−+=,;,*

( ) ( ) ( ) ( ) ( ) ( )nnjmtj

njmx

njmm ttfxxffnjtxf −+−+=,;,*

Solution element in point (j,n)

Basic conservation element CE+ and CE-

The CE combines two basic CEs, CE+ and CE-, and to enforce

flux conservation across CE surfaces we have

( ) ( ) ( ) 3,2,1 ,2/ 32121 =∆+++−= ++−− mxFFFFu njm

where the , , and denote the fluxes of cross the

line segment BE ,EC ,CF and FD, respectively.

−1F −2F +2F +1F

21


( )

( ) ( )( ) ( ) ( ) ( ) ( ) ( )[( ) ( ) ( ) ( )( )]2/2/

2/2/2/

12/12/1

21

2/12/1

22/12/11

2/12/1

2/12/1

*1

txfxu

tfxutf

BE

dF

njmx

njmx

njmt

njm

njm

mm

EB

∆∆+∆−

∆+∆−∆−=

−=

⋅=

−−−

−−−

−−

−−−

−−

→Γ∫−

ϕϕ

sh

( )

( ) ( )( ) ( ) ( ) ( ) 2/2

22/12/12

2/12/1

*2

−−−

−−−

→Γ

∆−∆−=

−=

⋅= ∫−

xuxu

EC

dF

njmx

njm

mm

CE

ϕϕ

sh

( )

( ) ( )( ) ( ) ( ) ( )[ ]2/2

22/12/12

2/12/1

*2

+−+

+−+

→Γ

∆−∆−−=

−=

⋅= ∫+

xuxu

CF

dF

njmx

njm

mm

FC

ϕϕ

sh( )

( ) ( )( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( )( )2/2/

2/2/2/

12/12/1

21

2/12/1

22/12/11

2/12/1

2/12/1

*1

txfxu

tfxutf

FD

dF

njmx

njmx

njmt

njm

njm

mm

DF

∆∆+∆−

∆+∆−∆=

−=

⋅=

+−+

+−+

−+

+−+

−+

→Γ∫+

ϕϕ

sh

(1) (2) (3) (4)

The compounded conservation elements

22


( ) ( ) ( ) ( ) ( ) ( ) ,3,2,1 ,2/2/12/11

2/12/1

2/12/12/1 =∆+∆+=′ −

±±−

±

−

±± mtuxuuu njmt

njmx

njm

njm

The is the first-order Taylor’s series approximation of at . ( )njmu 2/1±′ mu ( )nj ,2/1±

( ) ( ) ( )3,2,1 ,ˆ

2/1

2/1 =−

−′=

±

±± m

xxuu

u nj

nj

njm

njmn

jmx

( )( ) ( ) ( ) ( )

( ) ( )

ˆˆ

ˆˆˆˆˆ αα

αα

njmx

njmx

njmx

njmx

njmx

njmxn

jmx

uu

uuuuu

−+

+−−+

+

+=

The denote first-order spatial derivatives of the backward and forward differential

schemes at point (j,n). ( )n

jmxu ±ˆ

The solution gradient are evaluated by ( )njmxu

23

( ) ( )( )0ˆˆ >+ −+

njmx

njmx uu Schematic of weighting function

Adaptive-CESE Algorithm


(1) Give the initial solutions, and with a uniform mesh to the physical domain at

time .

(2) Compute and smooth the monitor function values .

(3) Move each grid point from its current location to a new location using the Gauss-

Seidel iteration method.

(4) Compute the flow variables, and using the adaptive CESE scheme.

(5) If output time, set and and then return to

step (2) for the next time step.

( )0jmu ( )0

jmxu

0=t

2/1+jωµjx 1+µ

jx

( )njmu ( )njmxu

Tt n <+ 2/1 ( ) ( ) 2/10 += njmjm uu ( ) 2/2/1

2/12/12/1

0 +−

++ += n

jnjj xxx

Shock tube problems (1)


(1)The Sod’s shock tube problem

( ) ( )( )

≥≤

=5.0 1.0 ,0 ,125.05.0 0.1 ,0 ,0.1

,,xifxif

puρ



X0 0.2 0.4 0.6 0.8 1

0.2

0.4

0.6

0.8

1

tt=0.23

26



(2) The problem of two interaction blast waves

( )( )( )( )

≥≤≤

≤=

.90 100 ,0 ,1.900.1 01.0 ,0 ,1

.10 1000 ,0 ,1 ,,

xifxif

xifpuρ



t=0.038


Introduction of two-dimensional adaptive CESE method

29

Space-time domain


The space-time grid points at three adjacent time levels. 30

Solution element


x

yt1B′

2B′

3B′

4B′

1A

2A

3A

4A

1B

2B3B

4B

Q

Q ′′

4B ′′ 1B ′′

2B ′′3B ′′

Q′

Flow properties are assumed continuities within SE. For any

and any , , , and

, respectively, will be approximated by ,

, and to be defined immediately.

Let

( ) ( )*,, QSEtyx ∈ 4,3,2,1=m ( )tyxum ,, ( )tyxfm ,, ( )tyxgm ,,

( )tyxm ,,h ( )** ;,, Qtyxum

( )** ;,, Qtyxf m ( )** ;,, Qtyxgm ( )** ;,, Qtyxmh

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )nQmtQQmyQQmxQmm ttuyyuxxuuQtyxu −+−+−+= ******

** ;,,

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )nQmtQQmyQQmxQmm ttfyyfxxffQtyxf −+−+−+= ******

** ;,,

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )nQmtQQmyQQmxQmm ttgyygxxggQtyxg −+−+−+= ******

** ;,,Solution element

31

Basic conservation element


x

yt

1A′

1B′4B′Q′

1A1B

4B

Q

2/1−= ntt

ntt =

Δt/2

Q′

Q

x

yt 2A′

1B′

2B′

2A1B

2B

x

ytQ′

Q

3A′2B′3B′

3A

2B3B

Q′

Q

x

yt 4A′3B′

4B′

4A

3B

4B

(a) (b)

(c) (d)

(a) The basic conservation element .

(b) The basic conservation element .

(c) The basic conservation element .

(d) The basic conservation element .

)()1( QCE

)()2( QCE

)()3( QCE

)()4( QCE

32

Two-dimensional Riemann problems


x

y

(1)(2)

(3) (4)

The spatial computation domain is defined by and the initial conditions include

four constant states.

The four interfaces number are defined as

The forward and backward shock wave is denoted and .

The forward and backward rarefaction wave is denoted and .

The forward and backward slip line is denoted and .

[ ] [ ]1,01,0 ×

( ) ( ) ( ) ( ) ( ){ }1 ,4 ,4 ,3 ,3 ,2 ,2 ,1 , =k

+kS

−kS

+kR

−kR

+kJ

−kJ

Numerical results

(a) adaptive mesh distribution

(b) adaptive CESE solution with 150x150 grid points

(c) the CESE solution with 150x150 grid points.

(d) the CESE solution with 400x400 grid points.

Initial conditions for the two-dimensional Riemann problems.

33

Riemann problems (1) Pure shock waves interaction


Sub-case : −+−+41342312 SSSS

( )( )( )( )( )

=

(4) -- 0.35 ,0.8939 ,0 ,0.5065(3) -- 1.1 ,0.8939 ,0.8939 ,1.1(2) --0.35 ,0 ,0.8939 ,0.5065(1) ------------ 1.1 ,0 ,0 ,1.1

,,, pvuρ

Riemann problems (1) Pure shock waves interaction


0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

(a) (b)

(c) (d) 35

Riemann problems (2) Pure slip line interaction


Sub-case : −−−−41342312 JJJJ

( )( )( )( )( )

=

(4) --1 ,0.5- ,0.75- ,3(3) ---1 ,0.5 ,0.75- ,1

-(2)---1 ,0.5 ,0.75 ,2(1) ---1 ,0.5- ,0.75 ,1

,,, pvuρ

Riemann problems (2) Pure slip line interaction


0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

(a) (b)

(c) (d) 37

Riemann problems (3) Interaction of multiple waves


Sub-case : ++−−41342312 SJJS

( )( )( )( )( )

=

-(4)- -- 1 ,0.7276 ,0 ,1(3) -- ----- 1 ,0 ,0 ,0.8(2)--- - 1 ,0 ,0.7276 ,1(1) -- 0.4 ,0 ,0 ,0.5313

,,, pvuρ

Riemann problems (3) Interaction of multiple waves


0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

(a) (b)

(c) (d) 39

Conclusion


(1) The CESE scheme can cover vast range of scales.

(2) The proposed adaptive CESE scheme in the current time-step is computed

directly from the solutions obtained in the previous time-step without the need

for extrapolation or interpolation.

(3) The adaptive CESE scheme captures the flow features with a significantly higher

resolution than the original CESE solver.

(5) The adaptive CESE scheme (implemented on a coarse mesh) achieves the same

(or better) resolution as the original stationary CESE solver on a fine mesh, but

with a significantly lower computational cost.

Dr. Chang, Sin-Chung My students

Dr. Tseng, Tzu-I Dr. Chou, Yin

Mr. Chen, Wun-Ching

Progress is impossible without change, and those who cannot change their minds cannot change anything.

-- George Bernard Shaw, Irish playwright

Acknowledgement

Adaptive CESE Method for Solving Unsteady Euler Equations

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