Lecture 3• Relevant concepts in compressible flow (mostly 1D)

– Concept of compressibility and simplified governing equations for compress flow

– A representation issue for Euler equation

• Choosing a “core-set” to facilitate numerical as well as analytical

manipulation of Euler equation

– Entropy and its equation

• to replace the energy equation in describing smooth flow.

– Sound and its speed (�)– Relations linking the flow states at two ends of a propagating wave

• Contact discontinuity

• Normal Shock

– Quasi-1D Nozzle flow

• Some conveniently defined quantiles

1

Reference book chapterfor lectures of week 2

• Book reference– Modern compressible flow: with historical perspective, Third editions, Johan, D.

Anderson• Chapter 2:

• Chapter 3, one-Dimensional flow

• Chapter 5, Quasi-One-Dimensional flow

– Computational fluid mechanics and heat transfer, by J.C. Tannehill, D. A. Anderson and R. H. Pletcher

• Chapter 6: Numerical methods for inviscid flow equations.

2

3

p

Volume1

Concept of compressibility from thermodynamics

• Compressibility:

• t : property of the fluid– Water: 5x10-10 m2/N @1atm

– Air: 10-5 m2/N @1atm

Volume2

Such a process can be realized through very slow compression causing negligible velocity (|u| ≈ 0) in the fluid, this “ideal” thermodynamic process will maintain a perfectly uniform distribution of all state variables over the entire fluid.# = 1& '&'( ) * ≈+….

-./012 = &3

# = − 13 '3'( ) * ≈+…. # = − 13 '3'( ) * ≈+56./012. # = − 13 '3'( ) * ≈+16./012.

Since 0 = 788 + 7:: ⇠ 'ln- = 'ln& + dln3 ⇠ ln(-) = ln(&) + ln(3) ⇠ - = &3

4

Compressible “flow” with < ≫ 0Inner object A

Outer object BBBBThermodynamic view '( B, C'& B, C'

The three-dimensional Euler equations

5

UUC V + UUD W⃗ V + UUY Z⃗ V + UU[ \ V = 0, where V, ], W⃗, Z⃗ are vectors of 5-component

UUC&&

The 1D Euler equations

6

3D Euler 1111D Euler (C, D, Y, [)UUC

&&

A representation issue

7

UUC &&

Different representations with a different core-set

8

The state of a 1D flow system can be fully represented by choosing a “core-set” of three three three three symbols. However, there can be different choices of “core-set” :Case 1: a core-set chosen as three variables directly appeared in the conservation equations: | = &, &

Emphasis on using core-set to express the Euler equations

9

The 1D Euler equation is often expressed using the primitive variables without explicitly emphasizing on which core-set was chosen. We can try using different core-set to write Euler equation, for instance let } = &,

In additional to facilitate writing a numerical code, good representation with a suitably

chosen core-set can avoid confusion and benefit theoretical manipulation.

10

One benefit of expressing Euler equation only using the core-set symbols is to prevent us from “forget” relations such as the equation of state while our prime attention being attracted to the differential equations, for instance it is often an essential step to find the Jacobian matrix: × = | =

g g gg g g for (|) =

-` g8 + n − 1 n 8 − ` 8gwhich looks like the following for 1D Euler equation =

0 1 0− ` g8g 3 − n 8 n − 1−n 8g + − 1 8 n 8 − ` g8g n 8 for | =&&

1D Euler equations

“Conservative” vs. “non-conservative” form

11

““““NoncNoncNoncNonconservative” form of 1D Euler eq.Continuity eq 2 & + &UE< = 0Momentum eq. & 2 < + UE( = 0Energy eq. & 2 �iF + *g`

+ UE (< = 0

“C“C“C“Conservative” form of 1D Euler eq.UUC & ⋅ 1& ⋅

What about entropy I ? Where is the second law of thermodynamics?

12

CCCConservative form of 1D Euler eq.UUC &&

Isentropic/ homentropic flow

13

(1)This equation holds for 3D-Euler (zero viscosity, no-heat-conduction&adiabatic) equations using (¡2 ≡ ¡2 + < ¡E + ^ ¡¢ + _ ¡£) (2)Before encounting a discontinuity, the entropy will maintain constant along the trajectory of a

material point moving with speed

Isentropic/ homentropic flow

14

C I ≡ UIUC + < UIUD = 0

15

Sound a propagating wave of small disturbances in the state of a flow medium

• Propogation of a wave front in a flow medium;

– Speed of sound �• Small disturbance across a sound wave

– Across a sound−wave, quantities describing the flow state are continuous and differentiable– '

Mass conservation: ( &< = �GHIC. across x)

16

Speed of sound(a) Derivation using steady Euler equation in conservative form

< + '

17

Speed of sound(b) Same derivation using steady Euler Eq., but appears to be more mathematical

' &< = 0' &

18

Speed of sound(c) Derivation from the eigenvalues of the Jacobian matrix using unsteady Euler eq.

, = < ± U( 8,1U& ¬ )1 = < ± �` = <

Unsteady Euler Eq. in conservative form UUC &&

19

Limit of compressibility?

x

u

xu

x

u

x

u

Can be written as:

Sound: '( = �`'&

3.0MThe limit is often set to:

< U&UD ≪ & U

Sound wave vs. shock wave(1D)

20

Difference:(1) Across a sound wave the flow state has a small and continuous variation. On the other hand, shock is associated with a large, dis-continuous jump in flow states.Similarity:(1) Both waves propagate with certain speed.(2)(2)(2)(2) SoundSoundSoundSound can be viewed as micromicromicromicro----shockshockshockshock!(3) Both are governed by the same, 3 conservation laws.Observations(1) A sufficiently-large control-volume can be placed to cover the internal structure of a sound wave and a shock save, both the two ends of the control volume will have a constant-valued flow sate.

Relations linking the two ends of a 1D propagating wave

The “normal-shock” relations

21

&

A mathematical detourHow to find the solution for two algebraic equations with two unknowns (D, Y)?

22

We may find multiplemultiplemultiplemultiple solutions as the intersection points of the two nonlinearnonlinearnonlinearnonlinear curves.

y

x

¶−22 = 3D` + DY + 4Y`1 = 5Y + 4DIt is also possible two curves do not intersect, therefore there may be no solutionno solutionno solutionno solution at all!

¸−22 = 3D` + DY + 4Y`26 = 5Yº̀ + 4DºY

DY[

Why there exists “additional” non-trivial “shock” solution? Three algebraic equations (at least a nonlinear one) for three unknowns (D, Y, [) can yield multiple solutions

23

»−2 = 3D` + DY + 4[`16 = 2D[ + 2Y[ + 3Y`21 = D + 15Y − [

&

How to find the nontrivial solution from the normal-shock relations

24

The trivial solution of (&`, (`)=(&, ()corresponds to point (1,1) plotted in a normalizednormalizednormalizednormalized coordinate of ((# = cgc , = 8g8 )

&

Normal shock relationsRankine–Hugoniot jump conditions:

25n + 1n − 1− n − 1n + 1

(#

1/1

1

Also passing piont (1,1), the “normalized” (shock) Hugoniot

curve is a unique curve which does

not depend on input values of

(&, (,

Intersection of Rayleigh line with Hugoniot curve can give physical/non-physcalsolutions relating the states on two ends of a discontinuous/continuous interface.

26

Hugoniot curve: (# = ÁÃÁÄ ¾# ÁÃÁÄ ¾# Rayleigh line :

c#¾# = −À = − *g 5 < 0A quiz: focus on the Hugoniot curve (plot in solid-black), which of following is possible? (# = (`( = ∞? = &`& = ∞?'I >0

(#

1/− n + 1n − 1 n + 1n − 1

1

1

If the slope of Rayleigh line (−À = − *g 5 ) at (1,1) matches the slope of Hugoniot curve (−n), the second solution overlaps with first trivial solution (1,1). This means two sides of across an interface share same flow-states therefore being “continuous”continuous”continuous”continuous” across, corresponding to a physical, steadily propagating sound wave. The sound speed � is equal to

When playing with straight-line and curve, we naturally wondering about the “tangency” scenario! Such a simple investigation tells, the flow at the upstream of shock must be supersonic!

27

(#

1/− n + 1n − 1

n + 1n − 1

1

1

Slope of the Rayleigh line representing a sound

wave on upsteam of a “shock” : − *g |Ê 5|Ê“Larger” (neg.) slope in case

of a shock solution : − *g|Ë 5|Ë

control volume AAAA placed relatively stationary to a true “shock”Note: Note: Note: Note: BBBB is not stationary relative to is not stationary relative to is not stationary relative to is not stationary relative to AAAA

Supersonic incoming flow � )Ì =====

(|Ì < (`|Ì&|Ì < &`|Ì�|Ì

The normal shock relationsexpressed analytically using the zÎ�ℎ number (z≡

After learning a lot theories about the non-trivial “shock”,

do not forget the simpler, trivial “contact-discontinuity”

29

&

(Quasi) 1D Nozzle flows

30

Assumption:

Thin boundary layer,

(Quasi) 1D flowsIsentropic flow with (nozzle) area changes

31

Governing Eqs. in smooth flow'(& 0'< > 0'( < 0

'< < 0'( > 0?

&u = �IC. Ð0ln + ln& + ln< = ln�IC. 7(⋅) ' + '&& + '

32

Nozzle flow(convergent)

Further decreasing (Ñ will not change the mass flow (”choked”)

since zÒÓÔ = 1.

'< > 0'( < 0''D < 0z < 1 (+

(Õ

(+ > (Õ

33

Nozzle flows(Convergent-divergent)

The original Quasi 1D Euler Eq can be used

to deal with discontinuous (shock) problem

UUC && 1z = 1

'(&

Some conveniently defined flow parameters

34

• Static pressure ( and temperature F for an fluid element in a actual flow domain (, F, u ⇒ [& = c5 , c = rmF¬ , Ma = Ö. , h = C×T, e = �iF, s = ln c8Á .q , … . ] • Imagine take this fluid element and adiabatically slow it down (if Ma>1) or speed it up (Ma

Relations between the conveniently defined flow parameters

35

'(&