AMS 691Special Topics in Applied

MathematicsLecture 2

James Glimm

Department of Applied Mathematics and Statistics,

Stony Brook University

Brookhaven National Laboratory

Review of Equations of Fluid Flow

Reference: Chorin-Marsden. Also Landau-Lifshitz is a good reference

We assume basic laws of Newtonian physics, applied to continuousrather than to discrete particle systems.

domain in space

( , , ) point in space

( , ) fluid velocity at point x

Basic conservation laws (from physics)

mass, momentum and energy are conserved

Let . Let ( ) amount of mass W

D

x x y z

u x t

W D m W d x

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in .

mass per unit volume

W

Conservation of Mass

( , ) ( , ) ( , )

; outward normal; element of surface area

= -

Last equation by divergence theorem;

previous by principle of conservation of mass.

W W

W

W

d dm W t x t d x x t d x

dt dt t

u ndA n dA

u dx

( ) 0

( ) 0

W

u d xt

ut

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Acceleration

Streamline: ( ( ), ( ), ( ), ) ( ( ), ,

( , ) ;

( ) ( ))

x y z t

t

u x t y t z t t t

d xu x t

dt

du u u u ua

dt x y z t

u Duuu vu wu u u u

t DtD

uDt

y t z tx

yx z

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Pressure = surface forceforce across per unit area = ( , )

pressure; = unit normal to surface

stress (force) on boundary

force per unit volume =

body force

force = mass acce

W

W

W

W p x t n

p n W

S pndA W

pdV

p b

b

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leration

Dup b

Dt

The flow map( ) ( , , ), ( , , ), ( , , )

determinant of Jacobean matrix of

1st column + 2nd column + 3rd column

But of posit

x x y z x y z x y z

x x x

Jy y y

z z zJ

t t t t

t

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ion coordinate or derivative = velocity or velocity derivative

sou u u u

t x x t x x x y x z x

J u v wJ J J J u

t x y z

Incompressible flow

0

0 0 0

1 0 flow is incompressible

= determinant of Jacobean matrix of flow map

ut

Du

Dt tD

u uDt

JJ

tJ

Thermodynamicsp = pressure = density; V = 1/ = specific volumeT = temperatures = entropyw = enthalpy = internal energy

Any two are independent. Any third is a function of these two.P = p(s, )

First law of thermodynamics:

2

isentropic: 0

( )

Isentropic gamma law gas:

d Tds pdV

ds

pd pdV d

p p

p A

Euler Equations for Compressible Isentropic Flow

0

;

( ) for gamma law gas

ut

uu u p b

t

p p A

If flow is not isentropic, we need one more equation for conservationof energy. For gamma law gas, Reference: Courant-Friedrichs( )A A s

Transportation Theorem

image of under flow map , ( , )

Theorem

Proof ( ( , ), ) ( ( , ), ) ( , )

( ) ( ) . But / / , so

t t

t

t

t

W W

W W

W

W W f f x t

d DffdV dV

dt Dt

d D ffdV x t t u f x t t J x t dV

dt Dt

Df u f dV D Dt t u

Dt

D

Dt

( ) 0, so

.

Likewise,

.

t t

t t

W W

W W

u ut

d Dff dV dV

dt Dt

d Dff dV dV

dt Dt

Work and Energy

2

work done by fluid per unit time = change of energy

1Energy , work per unit of time = -

2

( )

( ) ( ) 0

(( ) ) 0. Conservatio

t

t t t t

W

t

W W W W

t

t

e u pu ndA

DedV edV pu ndA pu dV

Dt

e eu pu

e e p u

n of energy

Conservation Equationsand Rankine Hugoniot Relations

0

0

( ) 0

[ ] [ ]

[ ] [ ]

[ ] [( ) ]

RH is 3 equations in 4 unknowns (1D), [ ] [ ]

solution space = curve = wave curve = states joined

by single type of wave (l or r

t

t

t

u

u uu P

e e P u

u

u uu P

e e P u

U F

shock, l or r rarefaction, contact)

Definitions: wave curve for shock = Hugoniot. For rarefaction = adiabat.

Contacts and Shocks

0 0 1 1

2 20 0 0 1 1 1

0 0 0 1 1 1

0 0 1 1

For stationary discontinuity (choice of Galilean frame), 0.

The R-H become 0,

( ) ( )

Let

Definition: If 0, discontinuity is contact or slip line.

Th

F

u u

u P u P

e P u e P u

M u u

M

0 1

0 1

en 0, so (in general, not stationary frame),

fluid moves with discontinuity. If 0, discontinuity is a shock.

Then 0 . Side of shock with gas particles entering shock is called the front sid

u u

M

u u

e

other side is the back side. We take front to have index 0, back index 1.

Hugoniot Curves

0 0 1 1

2 20 0 0 1 1 1

0 0 0 1 1 1

0 10 1 1

0 1

2 0 1

0 1

20 0 1 1

0 10 1

0 1

Recall

( ) ( )

so

Substitute /

But

( )( ), so

o

i i i

u u M

u P u P

e P u e P u

P PMu P Mu P M

u u

u M MV

P PM

V V

M MM u u

P Pu u

Hugoniot Curves, Continued

0 0 1 1

2 20 0 0 1 1 1

0 0 0 1 1 1

0 0 0 0 0 0 1 1 1 1 0 1

0 0 1 1 1 1 0 0

Recall:

( ) ( )

Rewrite, with 1/ :

( ) ( )

Cancel identical factor , get

i i

i i

u u

u P u P

e P u e P u

V

e V PV u eV PV u

u

e V eV PV PV

Hugoniot Curves, continued

2 20 0 1 1 0 0 0 0 0 1 1 1 1 1

0 1 0 1 0 1

0 10 1 0 1

0 1 1 1 0 0 1

0 1 0 1

0 10 1 0 1

1 1

2 2

1( )( )

2

( )2

From - / = -

and

( ) ( )2

Thus Recal

o

i i

e V eV u V u V

u u u u

P PMV MV

M

Mu P Mu P P P M u u

M u MV MV u u

P PV V

0 0 1 1 1 1 0 0

0 11 0 1 0

00 0 0

0 0

l

( ) 02

Define

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

0 defines the Hugoniot curve through , .

e V eV PV PV

P PV V

P PH V P V P V P V V

H V P

Contact and Adiabat CurvesContact curve defined by and [ ] 0.

Defines a curve in the (2D) space of thermodynamic variables.

Rarefaction wave = adiabat, given by solution of ODE in

thermodynamic space. For adiabat, 0,

u P

ds

de Td

, so s const

des PdV P

dV

Wave curves and the Riemann Problem

We have defined three types of wave curves: forward (right moving) shock or rarefaction wavescontactsbackward (left moving) shock or rarefaction waves

In 1D, the number of variables (dimension of state space) is 3:Conserved density, momentum, energy. Also called the primitive variables

Thus there is one type of wave curve for each dimension.If we compose the three wave curves, we sweep out a 3D region in state space. To realize this construction, we pick the three waves ina definite order, starting from the right, with the fastest first.This is the order listed above. Thus any sequence of waves (taken along a wave curve from a given starting point) defines a solutionof conservation laws, joining starting state to final state. It is a Riemannsolution, of the Riemann problem with right state = start, left state = finish.

Riemann problems

In the wave curve picture, we see that a rarefaction is a kind of negativeshock wave, in that it moves in the opposite direction. It is also close butnot identical to the Hugoniot curve followed in the “wrong” direction.Similarly, the Hugoniot curve is close but not identical to the rarefactioncurve (adiabat) defining a rarefaction in the “wrong” direction.

For most equations of state (EOS), this wave curve map, from start tofinal state covers all of the state space.

In other words, for most EOS, and any pair of left and right states, there is a (unique) solution of the Riemann problem with this data, formed ofthree waves, that is defined by the three wave curves.

These ideas are at the basis of a number of numerical schemes for conservation laws. See the book by Levesque.

Solving Riemann ProblemsThe waves at the right/left are shocks/rarefactions; the one in thecenter is a contact. Across the contact, pressure and (normal) velocityare continuous. Let u*, P* denote this common value on the two sidesof the contact.

Let , , 1/ denote left state values, and similar for

right state values. These quantities are known as data for the

Riemann problem. A left shock/rarefaction curve joins , to *, *

and simil

l l l l

l l

u P V

u P u P

arly a right shock/rarefaction joins , to *, *.

Lemma: and vary monotonically along shock/rarefaction curves (for most EOS).

Adjust strength of waves so that they are compatible, in the sense

r ru P u P

u P

that * computed

from the left = * computed from the right. Now the two left and right waves have

a linked strength, but still variable. Adjust this strength until the * from the left equals the

* fro

u

u

P

P m the right.

For a gamma law gas, and for a general EOS subject to certain assumptions, the above argument

is possible and yields a unique solution to the Riemann problem.