1--Conservation Laws of Fluid Motion and Boundary Conditions

8/10/2019 1--Conservation Laws of Fluid Motion and Boundary Conditions

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Computational Fluid Dynamics (CFD)

Tel: 13683389591

Instructor: Shilong Lan

Email: [email protected]


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CFD

Theory (Science)

Applications (Tool)

Fluid flow (physics)

Model (mathematics)

Algorithms (numerics)

CFD

Physicist

Mathematician

Engineer

To understand CFD and CFD codes

The aim of course :


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CFD:1. Pre-processor

2. Solver

Geometry, domain ,grid/mesh, initial conditionsBoundary conditions

3. Post-processor

schemes

data analysis, graphic output


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Continuity

div( ) 0U t

Governing equations of the flow of a compressibleNewtonian fluid

( )div( ) div( grad ) Mx

u puU u S

t x

X-momentum

Energy( )

div( ) div( ) div( grad ) ii

iU p U k T St


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Equations of state

( , )( , )

p p T i i T

For perfect gas

v

p RT

i C T

Unknowns:

, , , , , ,u v w p T i


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General transport equations

( )div( ) div( grad ) Mx

u puU u S

t x

( )div( ) div( grad )U S t

convective term diffusive term source term

Differential form for a fluid element


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( )d div( )d div( grad )d d

CV CV CV CV V U V V S

t

Integral form for a control volume(CV)

( )div( ) div( grad )U S

t

d n ( )d n ( grad )d dCV A A CV

V U A A S Vt

Integrate with respect to time t over a small interval t

dCV

V

dt t


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Classification of physical behaviours

Equilibrium problems steady fluid flowssteady state conductive heat transfer

2

2 0T T k t x

1T T 2T T

2 2

2 2 0T T

x y

Elliptic equation

Boundary-value problems


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Irrotational and imcompressible flow (potential flow)

2 0

BCs: 0 1,r (1, ) ( ) f r

Solution: 0

1

( , ) ( cos sin )2

nn n

n

ar r a n b n

,n na b depend upon the all BCs

y

x

P

y

x

P

boundar y

Disturbance signalstravel in al l directions


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Marching problems unsteady fluid flows

Transient heat transfer

2

2

T T k

t x

0T T 0T T

Parabolic equation

Initial-boundary-value problems

0T

x

0t

t

1t Disturbance at oneinstant only influencesevents at later time


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2nd order wave equation2 22

2 2 0u uc

t x

( ,0) ( )u x f x ( ,0) ( )u x t g x ( , ) x ICs:

Another kind of marching problem described byhyperbolic equation

1 1( , ) ( ) ( ) ( )2 2

x ct

x ct u x t f x ct f x ct g r dr

c

DAlembert solution:

Appendix 1

The solution of P is the sum of half of the values at point P 1 and P 2,at the same time, it depends on the initial function between P 1 and P 2


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x ct x ct

t

x

P

P 1 P 2

( , ) x t

1 1( , ) ( ) ( ) ( )2 2

x ct

x ct u x t f x ct f x ct g r dr

c

Domain of dependence and region of influence


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Observation in fluid flow

V/c


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Classification of PDE based on characteristics

Classification of quasi-linear PDE

Cramers rule or Eigenvalue method

2 2 2

2 22u u u

A B C D x x y y

A,B,C,D are functions of , , , , x y u u x u y

Example (Cramers rule)

(1)


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Total differential2 2

2( ) u u

d u x dx dy x x y

2 2

2( ) u u

d u y dx dy x y y

(2)

(3)

(1),(2) ,(3)2 2 2

2 2, ,u u u x x y y

if are indeterminate

2

0 0

0

A B c

dx dy

dx dy


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2 0 B AC 2

0 B AC 2 0 B AC

hyperbolic

parabolicelliptic

2dy B B AC dx A

Steady, isentropic, inviscid, compressible flow over an aerofoil

2 22

2 2(1 ) 0 M x y

Example:

Steady, inviscid, compressible flow over an aerofoil

2D steady Euler equations


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2 2

2 ( )u u

dt dx d u t t t x

total differential of

(2)

,u t u x

2 2

2 ( )u u

dt dx d u x x t x

(3)

2 2 2

2 2

2

1 0 0

0 ( )

0 ( )

c u t

dt dx u x d u t

dx dt u x t d u x

(1),(2),(3)

2 2 2 2 2, or , oru t u x u x t indeterminate

Example:2 2

22 2 0u u

ct x

(1)


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Cramers rule

21 0

0 0

0

c

dt dx

dx dt

dx cdt

the di rect ion o r s lopeof the character is t ics

x ct x ct

t

x

P

P 1 P 2

r ight- r unning c har a c teris tic

lef t - r unning c har a c ter is tic ( , ) x t

One point two c haracter is t ics , par t of inform at ion prop agat ion

Propagat ion speeds are c


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dom ain o f d ependenc e

r e gion o f inf luenc e

x ct x ct

t

x

P

P 1 P 2

right- r unning c harac teris tic

lef t - r unning c harac teris tic

( , ) x t

Domain of dependence and region of influence


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Numerical domain must include the domain of dependence.

1c t x

1c t

x


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Algorithms Characteristics

Marching problem or Equi l ib r ium prob lem

Remarks:

(Transient or Transient-like problem ) (Boundary value problem )

CFL cond i t ion 1c t

x

Numerical domain must include the domain of dependence.

x c t

Courant-Fredir ich-Levis


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Eigenvalue method (for system of equations)

1 1 1 1

2 2 2 2

0

0

u u v va b c d x y x y

u u v va b c d

x y x y

uW

v

1 1 1 1

2 2 2 2

0

0

a c b d W W a c b d x y

0

0W W

K M x y


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10

0W W

K M x y

00W W N x y

eigenvaluesi of [ N ] i

dydx

i

hyperbolic

parabolic

elliptic

all real at least two different values

one same real value

all complex

i Propagation speed

finite value

infinity

null

Assignment


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The eigenvalues may be a mix of both real and complex valuesa mixed hyperbolic-elliptic nature

PDEs are transformed into ( , , )

Remarks:

space

the type of PDEs do not alter

Eigenvalue method can be derived from Cramers rule

Appendix 4


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Example : 1D unsteady Euler equations

0U U

At x

U u

E

2

3 2

0 1 0

( 3) / 2 (3 ) 1

( 1) 1.5( 1)

A u u

u uE u E u

0 A I 1 2 3, ,u c u u c hyperbolic

On left boundary Descriptions Numbers of BCsSupersonic entrance 3

Subsonic entrance 2

Supersonic exit null

Subsonic exit 1

0,u u c

0,u u c

0,u u c

0,u u c

Left boundary


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Well-posed problem : if the solution to a PDE exists and unique,and if the solution depends continuously upon the IBCs.

IBCs must be properly specified to ensure a well-posed problem

IBCs Characteristics

Remarks:

Number of BCs Number of characteristics enter the domain


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Example: 2D steady Euler equations

2 2 21,2 3,4 2 2/ ,

uv c u v cv uu c

2 2 2

0u v c

Supersonic flow hyperbolic2 2 2 0u v c Subsonic flow elliptic2 2 2 0u v c Sonic flow parabolic

Example : 2D unsteady Euler equations

1,2 3,4,u u c


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2D steady Euler equations Mixed type


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Appendix 1 DAlembert solution


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Appendix 2

),(),(),( y xc yu

y xb xu

y xa

Consider a line in ( x,y ) plane: )();( s y y s x x

du u dx u dyds x ds y ds

along :

bds

dy

adsdx

if is hold along ( , )du

c x yds

ODE

is characteristic, along 1st PDE become an ODE

Characteristics of 1st order PDE


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Method of characteristics

2 2

22 2 0u uct x

Along characteristicdx

cdt

0u uct x

Along characteristicdx

cdt

u uc u

t x

Method of characteristics

Appendix 3

(1)

(2)

(3)

(1) (2), (3)

Only for hyperbolic PDE

Note, along characteristics, 2nd PDE is replaced by two 1st PDE

Classic technique for inviscid supersonic flows


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Appendix 42 2 2

2 22u u u

A B C D x x y y

2 2 2

2 22u u u

A B C D

1--Conservation Laws of Fluid Motion and Boundary Conditions

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