FLUID MECHANICS 3 LECTURE 1 BASIC EQUATIONS AND … filei.e., the Bernoulli constant is global (the...

24
FLUID MECHANICS 3 - LECTURE 1 BASIC EQUATIONS AND THEOREM IN THE THEORY OF IDEAL FLUID FLOWS

Transcript of FLUID MECHANICS 3 LECTURE 1 BASIC EQUATIONS AND … filei.e., the Bernoulli constant is global (the...

Page 1: FLUID MECHANICS 3 LECTURE 1 BASIC EQUATIONS AND … filei.e., the Bernoulli constant is global (the same for all streamlines) Equation of energy conservation We begin with the differential

FLUID MECHANICS 3 - LECTURE 1

BASIC EQUATIONS AND THEOREM IN THE

THEORY OF IDEAL FLUID FLOWS

Page 2: FLUID MECHANICS 3 LECTURE 1 BASIC EQUATIONS AND … filei.e., the Bernoulli constant is global (the same for all streamlines) Equation of energy conservation We begin with the differential

In this lecture, we recapitulate main equations and theorems of Fluid Mechanics, we have learnt

in the course of Fluid Mechanics I.

Differential equation of the mass conservation

Basic (conservative) form

( )t

0

υ

Other (equivalent) forms

( )

DD

t t

t

D0

Dt

υ υ υ υ

For a stationary (steady) flow …

( ) 0 υ υ υ

Page 3: FLUID MECHANICS 3 LECTURE 1 BASIC EQUATIONS AND … filei.e., the Bernoulli constant is global (the same for all streamlines) Equation of energy conservation We begin with the differential

Equation of motion of and ideal fluid (Euler Equation)

Basic form

( )t p υ υ υ f

Conservative form

( ) ( )t p υ υ υ f

The Lamb-Gromeko form

( ) 212 υ υ ω υ

21t 2

1p

υ ω υ f

Page 4: FLUID MECHANICS 3 LECTURE 1 BASIC EQUATIONS AND … filei.e., the Bernoulli constant is global (the same for all streamlines) Equation of energy conservation We begin with the differential

Bernoulli integral of the Euler Equation

Assumptions:

Flow is stationary

Fluid is in the barotropic state, hence the pressure potential can be defined

( ) : ( )1

P p d

.

[ ( )] [ ( )] [ ( )]( ) ( )i i i

1 1P p p

x x x

x x x

x x

1P p

Example – adiabatic flow of the Clapeyron gas …

( ) P

pP c T i

1

(specific enthalpy)

The volumetric force field is potential, i.e. f for some scalar field

Page 5: FLUID MECHANICS 3 LECTURE 1 BASIC EQUATIONS AND … filei.e., the Bernoulli constant is global (the same for all streamlines) Equation of energy conservation We begin with the differential

With the above assumption the Euler Equation can be written in the following form

212 fP ωυ

We choose arbitrary streamline and write (τ - unary vector tangent to this streamline)

: ( )2 21 12 f 2 f

d 1P P 0

d

τ υ υ ω

τ

Hence, the function under the gradient operator is constant along the streamline:

21

B2 fP C

If the cross product 0 υ ω then

212 f

212 f

P constP 0

i.e., the Bernoulli constant is global (the same for all streamlines)

Page 6: FLUID MECHANICS 3 LECTURE 1 BASIC EQUATIONS AND … filei.e., the Bernoulli constant is global (the same for all streamlines) Equation of energy conservation We begin with the differential

Equation of energy conservation

We begin with the differential energy equation, which in the case of an ideal fluid reduces to

(u - mass-specific internal energy)

( ) ( )2D 1Dt 2

u p υ f υ

By expanding the pressure term, this equations can be re-written equivalently as

( )2D 1Dt 2

u p p υ υ f υ

Assume now:

Flow steadiness

Potentiality of the volumetric force field

We do not assume that the flow is barotropic!

Page 7: FLUID MECHANICS 3 LECTURE 1 BASIC EQUATIONS AND … filei.e., the Bernoulli constant is global (the same for all streamlines) Equation of energy conservation We begin with the differential

Since the volume force is potential, the corresponding term in the right-hand side can be

transformed as follows

( ) Dt Dt

0

f υ υ υ

Moreover, due to flow steadiness we have

D

t D

0

tp p p p

υ υ

Next, from the mass conservation equation

DDt

0 υ

we get the following expression for divergence of the velocity field

1 D

Dt υ

Page 8: FLUID MECHANICS 3 LECTURE 1 BASIC EQUATIONS AND … filei.e., the Bernoulli constant is global (the same for all streamlines) Equation of energy conservation We begin with the differential

The energy equation can be now written in the following form

( / )

( )22

DDt

pD 1 D 1 D DDt Dt

p

Dt Dt2u p

or

( )2D 1Dt 2

i

u p 0

where i u p denotes the mass-specific enthalpy of the fluid.

Thus, the energy equation can be written as

( )2D 1Dt 2

i 0

Since the flow is stationary, the above equation is equivalent to

( )212

i 0 υ

Page 9: FLUID MECHANICS 3 LECTURE 1 BASIC EQUATIONS AND … filei.e., the Bernoulli constant is global (the same for all streamlines) Equation of energy conservation We begin with the differential

Using the same arguments as in the case of the Bernoulli Eq., we conclude that along each

individual streamline

21

e2i C const

In particular, for the Clapeyron gas pi c T and we get

2p

12

Tc const , p 1c R

In general the energy constant Ce can be different for each streamline.

If Ce is the same for all streamlines then the flow is called homoenergetic.

Let us recall that if the flow is barotropic then along each streamline we have

21B2

P C const

Thus, when the flow is barotropic then

e Bi P C C const

i.e., the enthalpy i and the pressure potential P differ only by an additive constant.

Page 10: FLUID MECHANICS 3 LECTURE 1 BASIC EQUATIONS AND … filei.e., the Bernoulli constant is global (the same for all streamlines) Equation of energy conservation We begin with the differential

Internal energy equation. Entropy of a smooth flow of ideal fluid

The equation of internal energy of an ideal fluid reads

DDt

u p υ

We know that 1 D

Dt υ

Thus, the equation for the internal energy u can be written as follows

( / )2

pD D D DDt Dt Dt Dt

u p 1 p

Let us remind that the 1st Principle of Thermodynamics can be expressed in terms of complete

differentials of three parameters of thermodynamic state: entropy s , internal energy u and

specific volume /1 .

Page 11: FLUID MECHANICS 3 LECTURE 1 BASIC EQUATIONS AND … filei.e., the Bernoulli constant is global (the same for all streamlines) Equation of energy conservation We begin with the differential

The corresponding form of this principle reads

Tds du pd

For the thermodynamic process inside individual fluid element one can write

D D D D DDt Dt Dt Dt Dt

T s u p p p 0

Conclusion: If the flow is smooth (i.e., all kinematic and thermodynamic fields are sufficiently

regular) then the entropy of the fluid is conserved along trajectories of fluid elements.

We have already introduced the concept of homoenergetic flows. In such flows we have

global21e2

i C

or equivalently

( )212

i 0 .

Page 12: FLUID MECHANICS 3 LECTURE 1 BASIC EQUATIONS AND … filei.e., the Bernoulli constant is global (the same for all streamlines) Equation of energy conservation We begin with the differential

Similarly, we call the flow homoentropic if s 0 . Thus, when the flow is homoentropic

then the entropy is uniformly distributed in the flow domain.

Since the 1st Principle of Thermodynamics can be written in the following form

( / )T ds di 1 d p

then for any stationary flow one has

( / )T s i 1 p

In the case of a homoentropic flow we get

( / )i 1 p P .

Hence, any homoenergetic and homoentropic flow is automatically barotropic and the

Bernoulli constant CB is global. Note that in the case of 2D flows, it implies that the velocity

field is potential (its vorticity vanishes identically in the whole flow domain).

Page 13: FLUID MECHANICS 3 LECTURE 1 BASIC EQUATIONS AND … filei.e., the Bernoulli constant is global (the same for all streamlines) Equation of energy conservation We begin with the differential

The Crocco Equation

Consider again the Euler equation in the Lamb-Gromeko form

( )21 12

p υ ω

Using the entropy/enthalpy form of the 1st thermodynamic principle, we can re-write the above

equation in the following form called the Crocco Equation

( )212

T s i υ ω

According to the Crocco Equation, any inhomogeneity in the spatial distribution of

entropy in the homoenergetic flow immediately leads to vorticity generation.

Page 14: FLUID MECHANICS 3 LECTURE 1 BASIC EQUATIONS AND … filei.e., the Bernoulli constant is global (the same for all streamlines) Equation of energy conservation We begin with the differential

Mechanics and thermodynamics of small disturbances

Consider again the First Principle of Thermodynamics …

int. . . .

( )

dU energ

v

d spec vol

TdS c dT p d 1

The differential of (mass-specific) entropy can be expressed as follows

( )v v v v2

Clapeyronequation

c c c 1 cp RdS dT d dT d dT d

T T TT

Using the Clapeyron equation can write

2

1 1 p 1 d p p d p ddT d d

T T R TR p

Thus pv v v

cc c cdS d p d d p d

p p

Page 15: FLUID MECHANICS 3 LECTURE 1 BASIC EQUATIONS AND … filei.e., the Bernoulli constant is global (the same for all streamlines) Equation of energy conservation We begin with the differential

Flow is isentropic, hence dS 0 and

S const

d p d p pdRT 0

p d

Thus, the flow is barotropic and the derivative of the pressure as the function of density is

always nonnegative function.

We can introduce the quantity a defined as a RT .

Then 2

S const

d pa

d

.

The physical unit of a is [m/s]. In has been demonstrated in the course of FM 1, that this

quantity is the velocity of small (acoustic) disturbances measured with respect to the gas.

Page 16: FLUID MECHANICS 3 LECTURE 1 BASIC EQUATIONS AND … filei.e., the Bernoulli constant is global (the same for all streamlines) Equation of energy conservation We begin with the differential

If an external force field is absent, then the energy integral can be written as

21

2i const

The mass-specific enthalpy can be expressed in several forms

21

p 1 1 1i c T RT p a

Mach number: V

Ma

We define:

Stagnation parameter: the parameter’s value at such point where 0 ; e.g. 0T

Critical parameter: the parameter’s value at such point where ( )a M 1 ; e.g. T

Page 17: FLUID MECHANICS 3 LECTURE 1 BASIC EQUATIONS AND … filei.e., the Bernoulli constant is global (the same for all streamlines) Equation of energy conservation We begin with the differential

In gas dynamics we often use three equivalent forms of the energy equation

( ) ( )

( )

21p p 02

2 012

0

222 201

2

c T c T

pp

1 1

aa 1a

1 1 2 1

Maximal velocity which can be achieved in any stationary flows is (T 0 )

max max

2 21 1p p 0 p 02 2

c T c T 2c T

If the flow is adiabatic, the temperature of the gas is a simple function of a local Mach number

2 2

20 0 0

22p 1

T T T11 1 1 M

2c T T a T 2 T

We have ( )

1

2

0

T 1M 1 M

T 2

and ( )

1

22

0

a 1M 1 M

a 2

Page 18: FLUID MECHANICS 3 LECTURE 1 BASIC EQUATIONS AND … filei.e., the Bernoulli constant is global (the same for all streamlines) Equation of energy conservation We begin with the differential

If the flow is also isentropic, we have /p const and p RT . Then

( )

1

12

0

1M 1 M

2

, ( )

12

0

p 1M 1 M

p 2

0 0.5 1 1.5 2 2.5 3 3.5 4

Mach number M

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

a/a0

T/T0

p/p0

Isentropic relations (1.4)

Page 19: FLUID MECHANICS 3 LECTURE 1 BASIC EQUATIONS AND … filei.e., the Bernoulli constant is global (the same for all streamlines) Equation of energy conservation We begin with the differential

Normal shock wave – summary of main formulae and results

Equations for conserved quantities:

(1) Mass n ds 0

1D case

1 1 2 2u u

(2) Linear momentum ( )n p ds

υ n 0 1D case

2 2

1 1 1 2 2 2u p u p

(3) Energy ( ) ( )

2 21 21 11 22 2

1 2

p pu u

1 1

After a lengthy algebra, we have shown in the Fluid Mechanics I course that the nontrivial

relation between density and pressure ratios reads

1 2

2 1

1 2

2 1

1 11 12

1 11 1 1

1p

p 1

Page 20: FLUID MECHANICS 3 LECTURE 1 BASIC EQUATIONS AND … filei.e., the Bernoulli constant is global (the same for all streamlines) Equation of energy conservation We begin with the differential

This formula describes the Rankin-Hugoniot adiabat which is different from the (isentropic)

Poisson adiabat, see figure below.

Yet, for the density ration only slightly larger than unity, the difference is really small (as RH

and P lines are strictly tangent for the argument equal 1)

1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6

0

5

10

15

20

25

30

p2/p

1

Normal shock wave (Rankin-Hugoniot)

Isentropic flow (Poisson)

Rankin-Hugoniot and Poisson adiabats ( = 1.4)

Page 21: FLUID MECHANICS 3 LECTURE 1 BASIC EQUATIONS AND … filei.e., the Bernoulli constant is global (the same for all streamlines) Equation of energy conservation We begin with the differential

Physically it means that weak shock waves are nearly isentropic and

. .

3

2 2 2

1 1 1Hugoniot isentropic

p pC 1 h o t

p p

Since

ln ln ln( )

v

v p v

c

s c p c const c p const

the change of entropy between two thermodynamic states can be expressed as follows

ln ln ln ln2 1 2 1 2 2

v v 2 1 1 1

s s s p p p

c c p

From the 2nd

Principle of Thermodynamics we conclude that physically admissible shock

waves must be compressing shocks (the entropy cannot diminish while crossing the shock).

Page 22: FLUID MECHANICS 3 LECTURE 1 BASIC EQUATIONS AND … filei.e., the Bernoulli constant is global (the same for all streamlines) Equation of energy conservation We begin with the differential

The most important relations concerning the normal shock wave are:

The Prandtl’s relation 2

1 2u u a

Relation between Mach numbers

( ) 21

2 21

2 1 MM 1

2 M 1

Other important relations can be derived. We usually use either plot or tabularized values.

For instance

( ) ( ) ( ) ( ) [ ( )]( ) ( )

1

2 1 1 1 11 1 1 1 2 1

1 2 2 1 2 2 1 0 0is is

u M a M a aM M M M M M 1

u M M a M M a a

Page 23: FLUID MECHANICS 3 LECTURE 1 BASIC EQUATIONS AND … filei.e., the Bernoulli constant is global (the same for all streamlines) Equation of energy conservation We begin with the differential

To evaluate the pressure ratio (as the function of M1) we rewrite the momentum equation in the

following way

( )2 2

2 22

u up u p 1 p 1 p 1 M const

p a

Since the above expression has the same value at both sides of the shock wave, we get

( )( )

22 1

1 21 2 1

p 1 MM 1

p 1 M M

We can also write

( )( )

[ ( )]

0 221

1 0 2 1

T T MTM 1

T T T M M

where ( )

1

2

0

T 1M 1 M

T 2

Page 24: FLUID MECHANICS 3 LECTURE 1 BASIC EQUATIONS AND … filei.e., the Bernoulli constant is global (the same for all streamlines) Equation of energy conservation We begin with the differential

Entropy of the gas increases while crossing the shock wave. The formula derived earlier can be

written for stagnation parameters, namely

ln ln2 102 01 02 01

v

s s0 p p

c

The energy is conserved, hence the total temperature at both sides is the same and

ClapeyronEquati

02 0201 02 0

0 1on 1 0

pT T T

p

We conclude that the stagnation pressure drops while crossing the SW …

( )ln 02 0v

11 02 0

s0 1 p p

cp p 1