5.1 Euler’s Equation (1-D steady flow)

Chap.5 Flow of an Incompressible Ideal Fluid

• Ideal fluid (or inviscid fluid) → viscosity=0

→ no friction b/w fluid particles or b/w fluid and boundary walls

• Incompressible fluid → ρ ^=const, d ρ / dt = 0

5.1 Euler’s Equation (1-D steady flow)

- pressure force: pdA − ( p + dp ) dA = − dpdA

- body force: gdAdz

ds

gdsdA dz ρ

ρ ⎟ = −

⎠

⎜ ⎞

⎝

− ⎛ - acceleration:

ds V dV s

V V t

V dt

a dV =

∂ + ∂

∂

= ∂

=

Newton’s 2nd law: F = ma ds dAdsV dV gdAdz

dpdA − ρ = ρ

−

= 0 +

+ gdz VdV dp

ρ ← 1-D Euler ’ s equation

= 0 +

+ dV

g dz V dp

γ

2 0

2

⎟⎟ ⎠ =

⎜⎜ ⎞

⎝

⎛ + + g z V

d p

γ ^→ 2 ^const

2

= +

+ g

z V p γ

5.2 Bernoulli ’ s Equation with Energy and Hydraulic Grade Lines

For 1-D steady flow of incompressible and uniform density fluid, integration of the Euler equation gives the Bernoulli equation as follows:

{ { { { const (along a streamline ) 2

head total head

elevation head

velocity 2

head pressure

=

= +

+ z H

g V p

γ

정지유체 ( V = 0 ) : p + z = const

γ (2.6)

Assumption: The stream tube is infinitesimally thin, so that it can be assumed to be a streamline.

g H V z + p + =

= 2

EL

2

γ γ z + p

= HGL

g V HGL 2 EL

=

2

−

5.3 1-D Assumption for Streamtubes of Finite Cross Section

Assumptions:

1. Streamlines are straight and parallel.

2. V = constant across a cross section (ideal fluid → no friction)

Straight and parallel streamlines

→ No cross-sectional velocity

→ No cross-sectional acceleration

Newton ’ s 2nd law in x-sectional direction:

0 force body

force

pressure + =

c

=

F

(Q no x-sectional acceleration) Read text

¹ ₁

p

²

z

₂

p + z = +

γ γ

or p + z = constant

γ across the x-section

Ex) Ideal fluid flowing in a pipe (IP 5.1)

g z V

p g z V

p

_B

B B

A A

A

2 2

2

2

= + +

+

+ γ

γ ( _Q same streamline)

g z V

p g z V

p

_C

C C

B B

B

2 2

2

2

= + +

+

+ γ

γ ( Q V

_B

= V

_C

)

Therefore, the Bernoulli’s equation can be

applied not only between A and B (on the same

streamline) but also between A and C (on

different streamlines) in the pipe shown below.

5.4 Applications of Bernoulli ’ s Equation (1) If z ≈ const ^, const

2

2

≈ + g

V p γ

∴ high velocity → low pressure (2) Torricelli ’ s equation

Bernoulli equation b/w 1 and 2:

₂

2 2 2

1 2

1 1

2

2 z

g V z p

g V

p + + = + +

γ γ

Using z

₁

= , z

₂

p

₁

= γ h , p

₂

= 0 , V

₁

≈ 0 ,

g

h V 2

2

=

2

→ V

₂

= 2 gh ← Torricelli equation

For application of Torricelli’s equation, see IP

5.2 in text.

Note:

In the reservoir ( V = 0 ^),

Water surface = EL ( z

g V p + +

= 2

2

γ ⁾

= HGL ( = p + z γ ⁾

= const. in static fluid (reservoir) = z |

_surface

( Q p = 0 at surface )

(3) Cavitation

Cavitation occurs if p

_abs

= p

_gage

+ p

_atm

≤ p

_v

or p

_gage

≤ − ( p

_atm

− p

_v

) At initiation of cavitation,

) (

_atm

gage

p p

v

p = − −

p

c

(critical gage pressure in text p. 134)

= 0

(4) Pitot tube (for measuring velocity)

) EL 2 (

2

2

2

+ = + + = =

+ z H

g V z p

g V p

S S

S O

O

O

γ

γ

⎟ ⎠

⎜ ⎞

⎝

⎛ −

= γ

^S

γ

^O

O

p g p

V 2

can be measured

∴ V

_O

can be known.

(5) Overflow structure (e.g. spillway of a dam)

Pressure and velocity on the spillway surface at 2?

) fluid ideal

2

( Q V V

V

_s

=

_b

=

Bernoulli equation at points

s

^and

b

^:

b b

b s

s

s

z

g V z p

g V

p + + = + +

2 2

2 2

γ

γ

^←

) (

_{s b}

b

z z

p = −

∴ γ

Continuity:

V

₁

y

₁

= V

₂

y

₂

Bernoulli equation at water surfaces at 1 and 2

z

s

g V y p

g V

p + + = + +

2 2

2 2 2

1 2

1

1

γ

γ

2 equations for 2 unknowns (

y

₁ ^and

V

₁⁾

→ can be solved for

y

₁ ^and

V

₁^.

See also § 5.3 const / + z =

p γ

across a x-section

5.5 Work-Energy Equation

• Addition or extraction of energy by flow machines - pump: add energy to flow system

(or a pump does work on fluid) - turbine: extract energy from flow system (or fluid does work on a turbine)

• Mechanical work-energy principle

Work done on a fluid system is equal to the change in the mechanical energy (potential + kinetic energy) of the system

dt dE dt

dE dW

dW = → =

Note: Work, energy, and heat have the same unit (J=N⋅m)

• Control volume analysis (Reynolds transport theorem)

Reynolds transport theorem for mechanical energy:

⎥ ⎦

⎢ ⎤

⎣

⎡ ⎟⎟

⎠

⎜⎜ ⎞

⎝

⎛ +

⎟⎟ −

⎠

⎜⎜ ⎞

⎝

⎛ +

=

⎟⎟ ⎠

⎜⎜ ⎞

⎝

⎛ +

⎟⎟ −

⎠

⎜⎜ ⎞

⎝

⎛ +

=

⎟⎟ ⎠

⎜⎜ ⎞

⎝

⎛ +

⎟⎟ −

⎠

⎜⎜ ⎞

⎝

⎛ +

=

⎟⎟ ⋅

⎠

⎜⎜ ⎞

⎝

⎛ + +

⎟⎟ ⋅

⎠

⎜⎜ ⎞

⎝

⎛ + +

⎟⎟ ⎠

⎜⎜ ⎞

⎝

⎛ ⎟⎟ ⎠

⎜⎜ ⎞

⎝

⎛ +

∂

= ∂

∫∫

∫∫

∫∫∫

g z V

g z V

Q

A g gV

z V A

g gV z V

A V V

gz A

V V gz

dA n V v

gz dA

n V v

gz

V V d

t gz dt

dE

in

out CS

CS CV

2 2

2 2

2 2

2 2

2

2 1 1 2

2 2

1 1 2

1 1 2

2 2

2 2

1 1 2 1 1 2

2 2 2 2

2 2

2

γ

ρ ρ

ρ ρ

ρ ρ

ρ

Rate of work done on the flow system

⎟

⎠

⎜ ⎞

⎝

⎛ = ? dt dW

1. flow work rate done by pressure force

∫∫ ^{⋅ = − = ⎜} _⎝ ^{⎛ − ⎟} _⎠ ^⎞

=

CS

p Q p

V A p V A p dA v

p

_{1 1 1 2 2 2}

γ γ

¹

γ

²

2. machine work rate done by pumps or turbines

( E

p

E

t

)

Q −

= γ

t

=

p

E

E ,

work rate per unit weight of fluid

3. Shear work rate done by shearing forces on the control surface = 0 (

Q

ideal fluid)

⎟ ⎠

⎜ ⎞

⎝

⎛ − + −

=

∴ p p E

_p

E

_t

dt Q dW

γ

γ γ

^{1 2}

Since

dt dE

dW = dt

^{, we get}

g z V

E p g E

z V p

t

p

2

2

2 2 2

2 2

1 1

1

+ + + − = + +

γ γ

Work-energy equation (per unit weight of fluid)

• Power of pump or turbine Power = rate of work done

t

=

p

E

E ,

rate of work done per unit weight of fluid

∴

Power of pump or turbine

= ( ^E

p

or ^E

t

) × ^{Q γ}

(ft⋅lb/s or J/s=W)

total weight of fluid passing the x-section per unit time

5.6 Euler ’ s Equations (2-D Steady Flow)

Newton’s 2nd law:

x

-direction:

⎟ ⎠

⎜ ⎞

⎝

⎛

∂ + ∂

∂ + ∂

∂

= ∂

⎟ ⎠

⎜ ⎞

⎝

⎛

∂ + ∂

⎟ −

⎠

⎜ ⎞

⎝

⎛

∂

− ∂

z w u x u u t dxdz u dx dz

x p p

dx dz x

p p ρ

2 2

z w u x u u x p

∂ + ∂

∂

= ∂

∂

− ∂ ρ 1

z

-direction:

⎟ ⎠

⎜ ⎞

⎝

⎛

∂ + ∂

∂ + ∂

∂

= ∂

⎟ −

⎠

⎜ ⎞

⎝

⎛

∂ + ∂

⎟ −

⎠

⎜ ⎞

⎝

⎛

∂

− ∂

z w w x u w t

dxdz w gdxdz

dz dx z p p

dz dx z

p p ρ ρ

2 2

z g w w x u w z

p +

∂ + ∂

∂

= ∂

∂

− ∂ ρ 1

Continuity:

= 0

∂ + ∂

∂

∂

z w x

u

3 equations for 3 unknowns

( p , u , w )

5.7 Bernoulli ’ s Equation

Euler’s equations:

z dx w u x u u x

p ⎟ ×

⎠

⎜ ⎞

⎝

⎛

∂ + ∂

∂

= ∂

∂

− ∂ ρ 1

dz z g

w w x u w z

p ⎟ ×

⎠

⎜ ⎞

⎝

⎛ +

∂ + ∂

∂

= ∂

∂

− ∂ ρ

1

+

( ) ^gdz

z u x

wdx w udz

z dz w w z dz

u u x dx

w w x dx

u u

gdz z dz

w w x dz

u w z dx

w u x dx

u u z dz

dx p x p

⎟ +

⎠

⎜ ⎞

⎝

⎛

∂

− ∂

∂

− ∂ +

⎟ ⎠

⎜ ⎞

⎝

⎛

∂ + ∂

∂ + ∂

⎟ ⎠

⎜ ⎞

⎝

⎛

∂ + ∂

∂

= ∂

∂ + + ∂

∂ + ∂

∂ + ∂

∂

= ∂

⎟ ⎠

⎜ ⎞

⎝

⎛

∂ + ∂

∂

− ∂ ρ 1

Using

z dz dx p

x dp p

∂ + ∂

∂

= ∂

z dz w w z dz

u u x dx

w w x dx

u u

z dz w dx u

x w w u

u d

∂ + ∂

∂ + ∂

∂ + ∂

∂

= ∂

∂ + + ∂

∂ +

= ∂ +

2 2

2 2

) (

) ) (

(

2 2

2 2

2 2

we get

ρ ^{( )} ξ

1 2

) (

^{2 2}

udz g wdx

g dz w u

d g

dp + + + = −

Let

X

₁

= ( x

₁

, z

₁

)

^and

X

₂

= ( x

₂

, z

₂

)

^. Integrating from

X

₁ ^to

X

₂^,

∫ ⁻

⎟⎟ =

⎠

⎜⎜ ⎞

⎝

⎛ + + +

²

1 2

1

) 1 (

2

2

2 X

X X

X

udz g wdx

g z w u

p ξ

γ

If the path between

X

₁ ^and

X

₂ is a streamline (

u = dx / dt

^,

w = dz / dt

), the RHS vanishes so that

2 0

2

1

2

2

⎟⎟ ⎠ =

⎜⎜ ⎞

⎝

⎛ + + +

X

X

g z w u

p γ

Therefore,

) ( const 2

2 2

C g z

w u

p + + + =

γ

along a streamline.

But, in general, the constant(

C

) may be different for different streamlines.

On the other hand, for an irrotational flow,

ξ = 0

^{, thus}

2 0

2

1

2

2

⎟⎟ ⎠ =

⎜⎜ ⎞

⎝

⎛ + + +

X

X

g z w u

p γ

for any two points in the flow. Thus

)

const 2 (

2

H g z

V

p + + = γ

over the whole flow field for irrotational flow.

5.8 Applications of Bernoulli ’ s Equation

For irrotational flow of ideal incompressible fluid, the Bernoulli’s equation applies over the whole flow field with a single energy line.

g z V H = p + +

2

2

γ

We are interested in

p

^and

V

at a point.

Exact velocity field → Exact pressure Difficult to solve

Semi-quantitative (approximate) approach

(1) Tornado or bathtub vortex: Higher pressure and lower velocity outwards from the center (Read text)

(2) Flow in a curved section in a vertical plane (Fig.

5.12)

- gravity effect >> centrifugal force (low velocity) - outer wall: sparse streamlines → lower velocity

→ higher pressure - inner wall: dense streamlines → higher velocity

→ lower pressure

(3) Flow in a convergent-divergent section

At section 2:

Center: sparse streamlines → lower velocity

→ higher pressure Wall: dense streamlines → higher velocity

→ lower pressure

g z V

p g z V

p

_L

L L

U U

U

2 2

2

2

= + +

+

+ γ

γ

L

U

V

V =

→ ^U _U

p

^L

z

_L

p + z = +

γ γ

U

L

z

z <

→

p

_U

< p

_L

If cavitation occurs, it will occur first at the upper wall.

(4) Stagnation points

Sharp corner of a surface →

V = 0

(stagnation point) At stagnation point, EL = HGL (

Q V = 0

⁾

(5) Sharp-crested weir

At upstream side, pressure is hydrostatic.

y p

_A

= γ

∴

_'

At point

A ''

(stagnation point),

g y V

p

_A

2

2 ''

= + γ

Pressure distribution above the weir:

Near free streamlines: dense streamlines → higher velocity

→ lower pressure

Near center: sparse streamlines → lower velocity → higher pressure Note:

p = 0

along free streamlines

(6) Flow past a circular cylinder

θ cos

1

₂

2

⎟⎟ ⎠

⎜⎜ ⎞

⎝

⎛ −

= r

U R

v

_r ^,

1

₂

sin θ

2

⎟⎟ ⎠

⎜⎜ ⎞

⎝

⎛ +

−

= r

U R v

_t

At windward stagnation point:

= 0

→

= R v

_r

r

^,

θ = π → v

_t

= 0

At leeward stagnation point:

= 0

→

= R v

_r

r

^,

θ = 0 → v

_t

= 0

At surface of the cylinder (

r = R

^):

θ sin 2

,

0 v U

v

_r

=

_t

= −

g z U

z p g U p

o o

2

) sin 2

( 2

2

2

θ

γ γ

+ − +

= + +

∴

) , , ,

( θ

γ ^{z fn p z U} p

o

=

o

+

∴

(7) Flow through an orifice

Streamlines are straight and parallel in the vena contracta.

p = 0

everywhere in the jet, but

v

^varies across the jet (Read page 131).

5.9 Stream Function and Velocity Potential

• Bernoulli equation → Relationship b/w

p

^,

V

^and

z

•

z

is known. Therefore, if

V

⁽

u

^and

v

) can be calculated, then

p

can be calculated by the Bernoulli equation.

• Introduce stream function (

ψ

) and velocity potential (

φ

^),

which are related to the velocity field (

u

^and

v

^).

• PDE for

ψ ( x , y )

^or

φ ( x , y )

+ boundary conditions → solve for

ψ

^or

φ

→

u

^and

v

→

p

(1) Stream function

•

ψ

= flowrate b/w 0 and a streamline

• Flowrate

ψ

b/w 0 and any point on streamline

A

^{is the}

same (

Q

no flow across a streamline) →

ψ

= const on streamline

A

• Flowrate b/w 0 and streamline

B = ψ + d ψ

→

) ( dq

d ψ =

= flowrate b/w streamlines

A

^and

B

physically

y dy x dx

vdx udy

d ∂

+ ∂

∂

= ∂

−

= ψ ψ

ψ

by definition

v x u y

∂

− ∂

∂ =

= ∂

∴ ψ ψ

,

Now

0

2

2

=

∂

∂

− ∂

∂

∂

= ∂

∂ + ∂

∂

∂

x y y

x y

v x

u ψ ψ

∴ ψ

(stream function) satisfies continuity equation for incompressible fluid.

For irrotational flow,

= 0

∂

− ∂

∂

= ∂

y u x

ξ v

or ²

0

2 2 2

2

= ∇ =

∂ + ∂

∂

∂ ψ ψ ψ

y

x

← Laplace equation for

ψ ( x , y ) )

, ( x y

ψ

can be solved with proper boundary conditions.

(2) Velocity potential

Define the velocity potential,

φ ( x , y )

, so as to satisfy

v y u x

∂

− ∂

∂ =

− ∂

= φ φ

,

Plug in continuity equation:

2 2 2

2

0 x y

y v x

u

∂

− ∂

∂

− ∂

=

∂ = + ∂

∂

∂ φ φ

2

0

2 2 2

2

= ∇ =

∂ + ∂

∂

∴ ∂ φ φ φ

y

x

← Laplace equation for

φ ( x , y )

Vorticity:

0

2

2

=

∂

∂ + ∂

∂

∂

− ∂

∂ =

− ∂

∂

= ∂

x y y

x y

u x

v φ φ

ξ

→ irrotational flow

∴ φ

(velocity potential) is defined only in the irrotational flow.

Note:

1. irrotational flow = potential flow

(

Q

velocity potential exists in irrotational flow) 2.

ψ

satisfies continuity for incompressible fluid → For irrotational flow,

∇

²

ψ = 0

3.

φ

satisfies irrotationality

→ For incompressible fluid,

∇

²

φ = 0

(3) Relationship between

φ

^and

ψ

⎪ ⎪

⎭

⎪⎪ ⎬

⎫

∂

= ∂

∂

→ ∂

∂

− ∂

∂ =

− ∂

=

∂

− ∂

∂ =

→ ∂

∂

= ∂

∂

− ∂

=

x y

x v y

y x

y u x

ψ φ

ψ φ

ψ φ

ψ φ

Streamlines (

ψ

=const) and equipotential lines (

φ

=const) are orthogonal (Read text p. 167-168).

Cauchy-Riemann Conditions