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

FLUID MECHANICS 3 - LECTURE 1

BASIC EQUATIONS AND THEOREM IN THE

THEORY OF IDEAL FLUID FLOWS

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

( )

D D

t t

t

0 D



Dt

      

 

 

   υ       υ υ   υ

For a stationary (steady) flow …

(  )   0

 υ      υ υ

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

 

( υ    ) υ

¹₂



²

  ω υ

 

^{1 2}

t 2

1 p

 

  υ       ω υ f

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 1

P p p

x x x

   

 

     

 x x   x

x x



P ¹ p

   

Example – adiabatic flow of the Clapeyron gas …

( )

^P

P p c T i

1

  

  



(specific enthalpy)

 The volumetric force field is potential, i.e.

f   

for some scalar field



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



¹²

^

²

^{P }

^f



     υ ω

We choose arbitrary streamline and write (

τ

- unary vector tangent to this streamline)



^{21 2 f}

  ^:

^{12 2 f}

 ^{( )}

d 1

P P 0

d    

        τ  υ υ ω    τ

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

1 2

2

   P 

f

 C

B

If the cross product

υ ω   0

then

1 2

2 f

1 2

2

 P 

f

0  P  const

 

 

    

    

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

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)

(

²

) ( )

D 1

Dt u 2 p

     υ   f υ 

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

(

²

)

D 1

Dt u 2 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!

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

(

_t

)

_Dt^D

0

       



       

f υ υ υ

Moreover, due to flow steadiness we have

t DD

0

p p p

t

p



     

υ υ

Next, from the mass conservation equation

DtD

     υ 0

we get the following expression for divergence of the velocity field

1 DDt

 

   υ

The energy equation can be now written in the following form

( / )

(

²

)

₂

DtD

D 1 p D 1 D D

Dt Dt

p

Dt Dt

u

2

_{ } p



  



   

or

(

²

)

D 1

Dt 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

(

²

)

D 1

Dt i  2     0

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

( i ¹ ₂  

²

) 0

   

υ

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

1

2

2 e

i      C  const

In particular, for the Clapeyron gas

i  c T

_p and we get

p

1

2

T 2

c      const

^,

c

_p



_^_1

R

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

If C_e 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

1 2

2 B

P      C  const

Thus, when the flow is barotropic then

e B

i P C    C  const

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

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

The equation of internal energy of an ideal fluid reads

Dt D u p

    υ

We know that

1 DDt





   υ

Thus, the equation for the internal energy

u

can be written as follows

( / )

2

D p D D D

Dt u   _ Dt    p Dt 1    p Dt 

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 / 

.

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 D

Dt 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

global 1 2

2 e

i      C

or equivalently

( i

¹₂

 

²

) 0

   

^.

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 1^st 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 C_B 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).

The Crocco Equation

Consider again the Euler equation in the Lamb-Gromeko form

(

¹₂



²

)

_¹

p 

       υ ω

Using the entropy/enthalpy form of the 1^st thermodynamic principle, we can re-write the above equation in the following form called the Crocco Equation

( ¹ ₂

²

)

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.

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 v

2

Clapeyron equation

c p c R c 1 c

dS dT d dT d dT d

T T T T

   

 

 ^

     

Using the Clapeyron equation can write

2

1 1 p 1 d p p d p d

dT d d

T T R TR p

 

   

 

 

 

 

   

    

Thus

c

^v

c

^v

c

^v

c

^p

d S d p d d p d

p p

    

   

Flow is isentropic, hence

d S  0

and

S const

d p d d p p

RT 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 p a

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.

If an external force field is absent, then the energy integral can be written as 1 2

i 

2

  const

The mass-specific enthalpy can be expressed in several forms 1 2

p 1 1 1

i c T  

_^_

RT 

_^_

p  

_

a

Mach number:

V M  a

We define:

 Stagnation parameter: the parameter’s value at such point where

  0

; e.g.

T

₀

 Critical parameter: the parameter’s value at such point where

  a M (  1 )

; e.g.

T

_

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

( ) ( )

( )

1 2

p 2 p 0

2 0

1 2

0 2

2

2 0 2

1 2

c T c T

p p

1 1

a

a 1

1 1 2 1 a





 

   

 

  

^

 

 

 

   

  

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

T  0

⁾

max max

2 2

1 1

p 2 p 0 2 p 0

c T    c T      2 c T

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

2 2

0 0 2 0

2 2

p 1

T T 1 T

1 1 1 M

2 c T T

_

a T 2 T

  



        

We have

( )

1 2 0

T 1

M 1 M

T 2



^

 

 

 

  

and

( )

1 2 2

0

a 1

M 1 M

a 2



^

 

 

 

  

If the flow is also isentropic, we have

p / 

^

 const

and

p   RT

. Then

( )

1 2 1

0

M 1 1 M

2

 





 



 

 

  

,

( )

^{2 1}

0

p 1

M 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)

Normal shock wave – summary of main formulae and results

Equations for conserved quantities:

(1) Mass _n

ds 0







 

1D case





_{1 1}

u  

₂

u

₂

(2) Linear momentum

(

_n

p ds )







 

 ^{υ n 0}

1D case





_{1 1}

u

²

 p

₁

 

_{2 2}

u

²

 p

₂

(3) Energy

( ) ( )

2 2

1 1 2 1

1 2

2 2

1 2

p p

u 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 1

1 1

2

1 1

1 1 1

p 1

p 1

 

   

 

 

   

 

 

 

 

 

 

 

 

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

p 2/p 1

Normal shock wave (Rankin-Hugoniot)

Isentropic flow (Poisson)

Rankin-Hugoniot and Poisson adiabats ( = 1.4)

Physically it means that weak shock waves are nearly isentropic and

. .

3

2 2 2

1 Hugoniot 1 isentropic 1

p p

C 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 ln

2 1 2 1 2 2

v v 2 1 1 1

s s s p p p

c c p



  

 

  

       

             

       

From the 2^nd Principle of Thermodynamics we conclude that physically admissible shock waves must be compressing shocks (the entropy cannot diminish while crossing the shock).

The most important relations concerning the normal shock wave are:

 The Prandtl’s relation

2

u u

1 2

 a

_

 Relation between Mach numbers

( )

₁²

2 2

1

2 1 M

M 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 1

1 1 1 1 2 1

1 2 2 1 2 2 1 0 is 0 is

u M a M a a

M M M M M M 1

u M M a M M a a





   



       

   

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

( )

2 2

2 2

2

u u

p 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

( )

( )

2

2 1

1 2

1 2 1

p 1 M

M 1

p 1 M M



 ^

 



We can also write

 

  ^{( )}

( )

[ ( )]

0 2

2

1

1 0 2 1

T T M

T M 1

T  T T M M 

where

( )

1 2 0

T 1

M 1 M

T 2

 

^

 

     

Entropy of the gas increases while crossing the shock wave. The formula derived earlier can be written for stagnation parameters, namely

   

ln ln

2 1

02 01 02 01

v

s s

0 p p

c ^   

  

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

Clapeyron Equati

02 02

01 02 0

0 1

on 1 0

T T T p

p

 

   

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

 

( )ln

_{02 0}

v

1

1 02 0

0 s 1 p p

c p p 1

 

    