Computational Methods for the Euler Equations

Before discussing the Euler Equations and computational methods for them, let’s look at what we’ve learned so far: Method Assumptions/Flow type 2-D panel 2-D, Incompressible, Irrotational Inviscid Vortex lattice 3-D, Incompressible, Irrotational Inviscid, Small disturbance Potential method 3-D, Subsonic compressible, Irrotational, Inviscid, Prandtl-Glauert Small disturbance Euler CFD 3-D, Compressible (no ∞M limit), Rotational, Shocks, Inviscid The only major effect missing after this week will be viscous-related effects.

2-D Euler Equations in Integral Form Consider an arbitrary area (i.e. a fixed control volume) through which flows a compressible inviscid flow: ≡nv outward pointing normal (unit length) ≡dS elemental (differential) surface length

jdxidydSnvvv −=

Note: Path around surface is taken so that interior of control volume is on left.

y

x c

dS

Cδ

nv

dSnv

dy

dx

dS

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Conservation of Mass

Cinmassof

dAdtdchangeofrate

dACinMass

Cofoutflowmassofrate

Cinmassofchangeofrate

C

C

∫∫

∫∫=⇒

=

=⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛

ρ

ρ

0

where fluidofdestiny≡ρ

Now, the rate of mass flowing out of C : Mass flow out of ∫ =•=

CudSnuC

δρ vvv velocity vector

Conservation of x-momentum Recall that: total rate of change momentum ∑= forces For x-momentum this gives:

∑=⎟⎟⎠

⎞⎜⎜⎝

⎛ −+⎟⎟

⎠

⎞⎜⎜⎝

⎛− Cofout

momflowxofrateCinmomentumx

ofchangeofrate Forces in x-direction

∫∫∫ ∑=•+CC

dSnuuudAdtd

δρρ vv Forces in x-direction

Now, looking closer at x-forces, for an inviscid compressible flow we only have pressure (ignoring gravity). Recall pressure acts normal to the surface

∫ •−=∑⇒C

dSinpxinsForceδ

vv

∫∫∫ =•+CC

dSnudAdtd

δρρ 0vv ⇒

dS

dSnpv−

Into surface

Normal to surface

Gives x-direction

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Conservation of y-momentum This follows exactly the same as the x-momentum: Conservation of Energy Recalling your thermodynamics:

⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛Cto

addedheatCinfluid

ondoneworkCinenergyof

changeofratetotal

For the Euler equations, we ignore the possibility of heat addition.

⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛Cofoutflow

energyofrateCinenergy

ofchangeofrateCinenergyof

changeofratetotal

The total energy of the fluid is:

)(21 22 vueE ++= ρρρ

Note: Tce v= where ≡vc specific heat at constant volume Static temperature So,

∫∫ ∫ •+=⎟⎟⎠

⎞⎜⎜⎝

⎛

C C

dSnuEEdAdtd

Cinenergyofchangeofratetotal

δ

ρρ vv

The work done on the fluid is through pressure forces and is equal to the pressure forces multiplied by (i.e. acting in) the velocity direction: ( ) ( )∫ •−=

C

dSunpworkδ

vv

∫∫ ∫ ∫ •−=•+C C C

dSinpdSnuuudAdtd

δ δ

ρρvvvv

∫∫ ∫ ∫ •−=•+C C C

dSjnpdSnuvvdAdtd

δ δ

ρρvvvv

Total energy

Internal energy Kinetic

energy

Pressure force

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⇒ Summary of 2-D Euler Equations

∫∫ ∫ ∫

∫∫ ∫ ∫

∫∫ ∫ ∫

∫∫ ∫

•−=•+

•−=•+

•−=•+

=•+

C C C

C C C

C C C

C C

dSnnpdSnuEEdAdtd

dSjnpdSnuvvdAdtd

dSinpdSnuuudAdtd

dSnudAdtd

δ δ

δ δ

δ δ

δ

ρρ

ρρ

ρρ

ρρ

vvvv

vvvv

vvvv

vv 0

These are often written very compactly as:

( )∫∫ ∫ =•++c sc

dsnjGiFUdAdtd 0vvv

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

≡

Evu

U

ρρρρ

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

+≡

uHuv

puu

F

ρρρ

ρ2

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

+≡

vHpv

vuv

G

ρρ

ρρ

2

ρpEenthalpytotalH +≡≡

Ideal gas: ⎥⎦⎤

⎢⎣⎡ +−−== )(

21)1( 22 vuERTp ρργρ

A Finite Volume Scheme for the 2-D Euler Eqns. Here’s the basic idea: (1) Divide up (i.e. discretize) the domain into simple geometric shapes (triangles

and quads)

∫∫ ∫ ∫ •−=•+C C C

dSunpdSnuEEdAdtd

δ δ

ρρ vvvv

Conservative state vector

Flux vector for x-direction

Flux vector for y-direction

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Looking at this small region: Cell 0 is surrounded by cells 1, 2, & 3. i.e. cell 0 has 3 neighbors: cell 1, 2, & 3. Nearest neighbors

9

0

10 8

2 17

6 5

4 3

1112

13

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(2) Decide how to place the unknowns in the grid. (a) Cell-centered: cell-average values of the conservative state vector are stored for each cell. (b) Node-based: point values of the conservative state vector are stored at each node. The debate still rages about which of these options is best. We will look at cell- centered schemes because these are easiest (although not necessarily the best). Also, they are very widely used in the aerospace industry. (3) Approximate the 2-D integral Euler equation on the grid to determine the chosen unknowns.

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Let’s look in detail at step (3): Cell-average unknowns:

( )( )( ) ⎥

⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=

0

0

0

0

0

Evu

U

ρρρρ

( )( )( ) ⎥

⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=

1

1

1

1

1

Evu

U

ρρρρ

.............2 =U ...........3 =U

Specifically, we define 0U as:

∫∫≡00

01

CUdA

AU where

⎩⎨⎧

≡≡

00

0

0

cellofareaAcellC

Now, we apply conservation eqns:

( )∫∫ ∫ =•++0 0

0C C

dSnjGiFUdAdtd

δ

vvv

The time-derivative term can be simplified a little:

∫∫ =0

00C dt

dUAUdA

dtd

The surface flux integral can also be simplified a little: ( ) ( )

( )

( )∫∫

∫∫

•++

•++

•+=•+

a

c

c

b

b

aC

dSnjGiF

dSnjGiF

dSnjGiFdsnjGiF

vvv

vvv

vvvvvv

0δ

f b e

a c

d

12

0

3

Cells: 0,1, 2, 3 Nodes: a, b, c, d, e, f

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Combining these expressions: Now, we make some approximations. Let’s look at the surface integral from

ba → :

( )∫ •+b

adsnjGiF vvv

The normal can easily be calculated since the face is a straight line between nodes a & b. Recall, the unknowns are stored at all centers. So, what would be a logical approximation for : ( )∫ =•+

b

a abdSnjGiF ???vvv

Option #1= Option #2= Note: Option #1 ≠ option #2 in general. There is very little difference in practice between these options. Let’s stick with:

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( ) caca

a

c caca

bcbc

c

b bcbc

ababab

b

aab

snjGGiFFdSnjGiF

SnjGGiFFdSnjGiF

SnjGGiFFdSnjGiF

∆•⎥⎦⎤

⎢⎣⎡ +++=•+≡ℑ

∆•⎥⎦⎤

⎢⎣⎡ +++=•+≡ℑ

∆•⎥⎦⎤

⎢⎣⎡ +++=•+≡ℑ

∫

∫

∫

vvvvv

vvvvvv

vvvvvv

3030

2020

1010

21

21

21

21

21

21

Where

( )( )( )( )33

22

11

00

UFFUFFUFFUFF

≡≡≡≡

( )( )( )( )33

22

11

00

UGGUGGUGGUGG

≡≡≡≡

( ) ( )

( )∫

∫ ∫=•++

•++•++

a

c

b

a

c

b

dSnjGiF

dSnjGiFdSnjGiFdt

dUA

0

00

vvv

vvvvvv

b

ac2 1 0 3

abnv

No approximations so far!

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Finally, we have to approximate dt

dUA 00 somehow. The simplest approach is

forward Euler:

000 =ℑ+ℑ+ℑ+ cabcabdt

dUA

And n

abℑ etc. are defined as:

( ) ( )( )( )nn

nn

ababnnnnn

ab

UFF

etcUFF

SnjGGiFF

11

00

1010

.21

21

≡

≡

∆•⎥⎦⎤

⎢⎣⎡ +++≡ℑ vvv

For steady solution, basic procedure is to make a guess of U at 0=t and then iterate until the solution no longer changes. This is called time marching. Question What assumptions have we made in developing our 2-D Euler Equation Finite Volume Method?

001

00 =ℑ+ℑ+ℑ+

∆−+

nca

nbc

nab

nn

tUU

A

Where ( )no

no tUU ≡ and iterationntntn ≡∆≡ ,