Chap 7 Flow of a Real Fluid viscosity, turbulence

Chap 7 Flow of a Real Fluid

viscosity, turbulence

7.1 Laminar and Turbulent Flow

• Laminar flow (層流):

- 완만하고 규칙적인 흐름

-

흐름의 층 사이에 대규모 혼합이 없음

• Turbulent flow (亂流):

- 불규칙하고 무질서한 흐름

-

흐름 내에 대규모 혼합이 존재함

• Reynolds experiment (Read text) For the same fluid (same viscosity),

V ↑ : laminar→turbulent (upper critical velocity) V ↓ : turbulent→laminar (lower critical velocity) lower critical velocity < upper critical velocity more engineering importance

Reynolds number,

force viscous

force inertia

=

=

= μ ν

ρ Vd

R

Vd

where d =characteristic length scale

(pipe flow: diameter, open channel: depth) Rc

R

>

: turbulent flow, _R

_<

_Rc : laminar flow where _Rc = critical Reynolds number

7.2 Turbulent Flow and Eddy Viscosity

{ { 123

directions and

in

ns fluctuatio turbulent

velocity mean time

velocity ous instantane

'

y x

y

x

v

v v

v

= + +

−

root-mean square (rms) of

_v

_x or

_v

_y

=

( ) v

x^{2 1/2} or

( ) v

²y ^1/2 = turbulence intensity

Laminar flow

dy μ dv

τ =

; shear stress due to viscosity b/w molecules

fluid viscosity (property of fluid)

Turbulent flow

dy ε dv

τ =

; shear stress due to momentum transfer

eddy viscosity (property of flow)

Time-averaged (or mean) shear stress due to momentum transfer:

Note: Positive shear stress retards the fluid motion.

Prandtl (1926):

dy v dv

v

_x

,

_y

∝ l

where l=mixing length (unknown function of

^y

)

2

2

⎟

⎠

⎜ ⎞

⎝

= ⎛

−

=

∴ dy

v dv

v

_{x y}

ρ l ρ

τ

dy

2

dv ρ l ε =

∴

Near a wall,

_v

_x

_{, v}

_y↓ → l↓

At the surface of a wall,

_v

_x

_{, v}

_y

_{= 0}

→ l = 0

Assume l =

κ y

(linear variation with

_y

), which is in good agreement with experimental data, where

κ

=von Karman constant ( ≈ 0.4 ) and

y

=distance from the wall.

2 2

2

⎟

⎠

⎜ ⎞

⎝

= ⎛

∴ dy

y dv

ρκ

τ

7.3 Fluid Flow Past Solid Boundaries

• For a real fluid flow,

v

= 0 at solid boundary (no-slip condition)

• Effect of surface roughness:

- Laminar flow: Viscosity is dominant.

Roughness has no effect on the flow.

- Turbulent flow: Viscous sublayer is formed near the surface with thickness

δ

_v^.

If roughness

<< δ

_v → smooth boundary

→ no effect of roughness on the flow If roughness

≥ δ

_v

_{/ 3}

→ rough boundary

→ roughness has effect on the flow Note: For the same roughness, the boundary can be either smooth or rough depending on the property of the flow (

_Q δ

_v depends on the flow characteristics). In general,

_{v ↑ ⇒}

R

_{↑ ⇒} δ

_v

_↓

.

7.4 – 7.8 External Flows

(over a wing or flat plate, etc)

• External flows are less important in civil and environmental engineering, so it is not taught in this class.

• Internal flows: flows in ducts, channels, and pipes

7.9 Flow Establishment −Boundary Layers

unestablished flow (≈ 20

d

)

↓

complicated to analyze

↓

net effect = entrance head loss (Section 9.9)

7.10

Shear Stress and Head Loss

A

= constant cross-sectional area

P

= perimeter of the pipe

P A

R

_h

= /

= hydraulic radius

Impulse-momentum equation along the pipe (l ):

{ ^{γ γ}

( ) (

^{ρ ρ}

)

^ρ

τ V dV A d V A

dl Adl dz Pdl d

A dp p

pA _o 2 2

2 )

(

force body force

shear forces

pressure

− +

+

= +

−

− +

− ⎟

⎠

⎜ ⎞

⎝

⎛

4 4 3 4

4 2 4 1

4 3 4

4 2 1

Using Q1_ρ1 =VA_ρ, Q2ρ2 = (V + dV)A(ρ + dρ),

ρ ρ

ρ Q Q

Q1 1 = 2 2 = for steady flow

⎟ ⎟

⎠

⎞

⎜ ⎜

⎝

⎟ ⎛

⎠

⎜ ⎞

⎝

⎛

_{+ = + − = =}

−

−

− ( ) 2

2

V2

Ad AVdV

V dV V

Q d Adz

Pdl

Adp _o γ ρ ρ ρ

γ τ

R dl g dz

d V dp

h

γ

o

τ γ

^⎟⎟_⎠ ^{+ = −}

⎜⎜ ⎞

⎝ + ⎛

∴ 2

2

∫

∫

^⎜⎜_⎝^{⎛ + + ⎟⎟}_⎠^{⎞ =} 2¹⁻ 1

2

2

2

dl

z R g V d p

h

γ

o

τ γ

( )

2

) 1

2 (

2 ^{2 2 1}

2 2 2

1 2 1 1

= −

Δ

=

−

⎟⎟ =

⎠

⎜⎜ ⎞

⎝

⎛ + +

⎟⎟ −

⎠

⎜⎜ ⎞

⎝

⎛ + + _L

h

o

l l EL h

z R g V z p

g V p

γ τ γ

γ

head loss due to pipe friction

Shear stress in the fluid (

τ

^):

For the streamtube of radius r ,

( )

r l

r r l R

l h l

h

L

γ

τ π

γ π τ γ

τ

2

2

2 1

2 − = =

=

l r h

_L

⎟ ⎠

⎜ ⎞

⎝

= ⎛

∴ 2

τ γ

τ

varies linearly with r from zero at the centerline to

τ

_o at the pipe wall.

7.11 The First Law of Thermodynamics and Shear Stress Effects

dE dW

dQ

_H

+ =

where

_dQ

_H =heat transferred across system boundary by temperature difference, which was neglected in derivation of work-energy equation.

dt dE dt

dW dt

dQ

_H

+ =

Examine term by term:

H H

H

m q Q q

dt

dQ = & = ρ

where

_q

_H=added heat per unit mass of fluid.

⎟⎠

⎜ ⎞

⎝

⎛ − + −

=

⎟⎠

⎜ ⎞

⎝

⎛ − + −

= _{p t}

p p E

_p

E

_t

g Q E

p E Q p

dt dW

γ ρ γ

γ

γ γ

^{1 2 1 2}

Note: no work done by shear stress because

= 0

v

at the pipe wall for real fluid

⎟⎟⎠

⎜⎜ ⎞

⎝

⎛ ⎟⎟⎠

⎜⎜ ⎞

⎝

⎛ + +

∂

= ∂

_t ∫∫∫ ^{gz V ie d V}

dt dE

CV 2

ρ

2

∫∫

∫∫ ^⎜⎜ _⎝ ^{⎛ + + ⎟⎟} _⎠ ^{⎞ ⋅ + ⎜⎜} _⎝ ^{⎛ + + ⎟⎟} _⎠ ^{⎞ ⋅}

+

in

out CS

CS

dA n v V ie

gz dA

n v V ie

gz 2 2

2

2

ρ

ρ

⎥

⎦

⎢ ⎤

⎣

⎡ ⎟⎟⎠

⎜⎜ ⎞

⎝

⎛ + +

⎟⎟ −

⎠

⎜⎜ ⎞

⎝

⎛ + +

=

1 2

2 2

2

2

V ie

gz V ie

gz Q ρ

⎟⎠

⎜ ⎞

⎝

⎛ − + −

+

∴ _H

p p E

_p

E

_t

g Q q

Q ρ ρ γ

¹

γ

²

⎟⎟ ⎠

⎜⎜ ⎞

⎝

⎛ + + − − −

=

₂ ²² _{2 1} ¹² ₁

2

2 V ie

gz V ie

gz Q ρ

(

_H

)

t

p

ie ie q

E g g z

V E p

g z V

p ⎟⎟ ⎠ + + − −

⎜⎜ ⎞

⎝

⎛ + +

=

⎟⎟ +

⎠

⎜⎜ ⎞

⎝

⎛

¹

+

¹²

+

₁ ^{2 22} ₂

1

_{2 1}

2

2 γ

γ

If no pump or turbine,

( )

4 4 3 4

4 2 1

2 1

1 2

2 2 2 2

1 2 1

1 1

2 2

= −

−

−

⎟⎟ =

⎠

⎜⎜ ⎞

⎝

⎛ + +

⎟⎟−

⎠

⎜⎜ ⎞

⎝

⎛ + +

hL

qH

ie g ie

g z V z p

g V p

γ γ

( ) (

_H

)

h o

L

ie ie q

g R

l

h

₋ =

l

² − ¹ = 1 ₂ − ₁ −

2

1

γ

τ

Note:

1. head loss = energy converted to heat (

q

_H) and internal energy (

_ie

).

2. In hydraulic engineering, it is not necessary to evaluate

_q

_H and

_ie

separately, and it is more convenient to use the concept of head loss.

3.

ie

₂

− ie

₁

= c ( T

₂

− T

₁

)

where

_c

= specific heat (=4180 J/kg⋅K for water)

7.12 Velocity Distribution and Its Significance

Total flowrate ⁼

∫∫

A

vdA Q

Mean velocity ⁼

∫∫

⁼

A

A

vdA Q V A

1

Total flux of K.E. =

_⎟⎟

⎠

⎜⎜ ⎞

⎝

= ⎛

∫∫ = ¹ _{2 2}

2

2 2

3

V

Q V

Q dA

v

A

α ρ ρ

ρ α

Total momentum flux =

v dA Q V

A

ρ β ρ ∫∫

² = where

α

,

β

= correction factors:

∫∫ ∫∫

∫∫ =

= vdA

dA v V V

Q

dA

v

³

2 2 3

1 2

2 ρ ρ α

∫∫ ∫∫

∫∫

₌

=

vdA

dA v V V

Q

dA

v

² 1 ²

ρ

β ρ

• EL’s and HGL are parallel but not horizontal because of head loss.

• Straight and parallel streamlines

→ hydrostatic pressure distribution

→

p

/

γ

+ z = const. over cross-section

• EL is different for different locations over the cross-section (Q is different)

v

• mean EL =

g z V

p

2 α

2

γ ^{+ +}

Flow through a contraction:

Exact

_⎟⎟

⎠

⎜⎜ ⎞

⎝

⎛ + +

⎟⎟ −

⎠

⎜⎜ ⎞

⎝

⎛ + +

−

=

2

2 2

2 2 1

1 2

1

1

2 2

2

1

p z

g z V

p g h

_L

V

α γ α γ

Conventional

_⎟⎟

⎠

⎜⎜ ⎞

⎝

⎛ + +

⎟⎟ −

⎠

⎜⎜ ⎞

⎝

⎛ + +

−

=

2

2 2

2 1

1 2

1

2 2

2

1

p z

g z V

p g h

_L

V

γ γ

Exact – Conventional =

( ) ( )

g V g

V

1 2 1 2

2 2 2

2 1

1

− − α −

α

( ) ( )

g V g

h V

h

_{L L}

1 2 1 2

Exact al

Convention

2 1 1

2 2

2 2

1 2

1

= + − − −

∴

_{− −}

α α

loss head n

dissipatio energy

local Flow

Secondary 7.14

Separation

7.13 → →

⎭⎬

⎫

7.15 Derivation of Navier-Stokes Equations

Euler equation + viscosity 2-D, unsteady, incompressible flow

z x

σ

σ ,

= normal stresses (positive outward normal to the surface)

zx xz

τ

τ ,

= shear stresses

(1st subscript: plane, 2nd subscript: direction) Sign convention: Stresses are positive if positive direction at positive plane or negative direction at negative plane.

Newton’s 2nd law:

( )

^mv

dt a d m F = =

∑

Surface forces + Body forces:

∑

∑

∑

^{F = Fxex + Fzez}

where _{e ,x ez} = unit vectors in x and z directions z dxdz

F_x x^{x zx} ⎟

⎠

⎜ ⎞

⎝

⎛

∂ + ∂

∂

= ∂

∑

^{σ τ} (7.56) dxdz

x g

F_z z^{z xz} ⎟

⎠

⎜ ⎞

⎝

⎛ −

∂ + ∂

∂

= ∂

∑

^{σ τ ρ} (7.57)

Reynolds transport theorem:

( ) _∫∫∫

_⎟⎟⁺

_∫∫

^⋅

⎠

⎜⎜ ⎞

⎝

⎛

∂

= ∂

CS CV

dA n v v V

d t v

v dt m

d ρ ρ

where

( )

_⎟⎟

⎠

⎜⎜ ⎞

⎝

⎛ +

∂

= ∂

⎟⎟⎠

⎜⎜ ⎞

⎝

⎛

∂

∂

∫∫∫ ∫∫∫

CV

z x

CV

V d e w e t u

V d

t ρv ρ

( )

^{e dxdz}

t e w

t dxdz u

e w e

t u ^{x z} ρ ^{x z} _⎟ρ

⎠

⎜ ⎞

⎝

⎛

∂ + ∂

∂

= ∂

∂ +

= ∂

∫∫ ∫∫ ∫∫

∫∫

∫∫

^{⋅ = + + + ⋅}

BC CD DA

AB CS

dA n v v dA

n v

vρ ρ

For example, for the plane AB,

z

x dz e

z w w dz e

z u u

v ⎟

⎠

⎜ ⎞

⎝

⎛

∂

−∂

⎟ +

⎠

⎜ ⎞

⎝

⎛

∂

−∂

= 2 2 , n = −ez,

2 dz z w w n

v ∂

+ ∂

−

=

⋅

∫∫

^{⋅ =⎜}_⎝^{⎛ − ∂}_∂ ^⎟_⎠^⎞⎜_⎝^{⎛− + ∂}_∂ ^⎟_⎠^⎞

AB

ex

dz dx z w w dz

z u u dA n v

vρ ρ

2 2

dz dxez

z w w dz

z

w w _⎟ρ

⎠

⎜ ⎞

⎝

⎛

∂ + ∂

⎟ −

⎠

⎜ ⎞

⎝

⎛

∂

− ∂

+ 2 2

ex

dz dx z w z u dz z w u dz z u w

uw _⎟⎟ρ

⎠

⎜⎜ ⎞

⎝

⎛

∂

∂

∂

−∂

∂ + ∂

∂ + ∂

−

= 2 2 4

2

ez

dz dx z w z w dz

z w w dz z w w

w _⎟⎟ρ

⎠

⎜⎜ ⎞

⎝

⎛

∂

∂

∂

− ∂

∂ + ∂

∂ + ∂

−

+ 2 2 4

2 2

∫∫

^{⋅ =⎜}_⎝^{⎛ ∂}_∂ ^{+ ∂}_∂ ^{+ ∂}_∂ ^⎟_⎠^⎞

∴

CS

ex

z dxdz w u

z u w x u u dA

n v

vρ 2 ρ

ez

x dxdz w u

x u w z

w w _⎟ρ

⎠

⎜ ⎞

⎝

⎛

∂ + ∂

∂ + ∂

∂ + 2 ∂

z

x dxdze

x u w z w w e

z dxdz w u

x

u u ρ _⎟ρ

⎠

⎜ ⎞

⎝

⎛

∂ + ∂

∂ + ∂

⎟⎠

⎜ ⎞

⎝

⎛

∂ + ∂

∂

= ∂

( )

^dxdzex

z w u x u u t v u

dt m

d _⎟ρ

⎠

⎜ ⎞

⎝

⎛

∂ + ∂

∂ + ∂

∂

= ∂

∴

dxdzez

z w w x u w t

w _⎟ρ

⎠

⎜ ⎞

⎝

⎛

∂ + ∂

∂ + ∂

∂ + ∂

x : z x z w u

x u u t

u _{x zx}

∂ + ∂

∂

= ∂

⎟⎠

⎜ ⎞

⎝

⎛

∂ + ∂

∂ + ∂

∂

∂ σ τ

ρ

z: _g

x z

z w w x u w t

w σ_z τ_xz ρ

ρ −

∂ + ∂

∂

= ∂

⎟⎠

⎜ ⎞

⎝

⎛

∂ + ∂

∂ + ∂

∂

∂

Stokes hypothesis: stresses = f₍p_,u_, w_, μ₎

2 2 2

z x

z x

p z

p w

x

p u σ σ

μ σ

μ

σ ₊

−

=

⎪→

⎭

⎪⎬

⎫

∂ + ∂

−

=

∂ + ∂

−

=

⎟⎠

⎜ ⎞

⎝

⎛

∂ + ∂

∂

= ∂

= x

w z

u

zx

xz τ μ

τ

⎟⎟⎠

⎜⎜ ⎞

⎝

⎛

∂ + ∂

∂ + ∂

∂

− ∂

∂ = + ∂

∂ + ∂

∂

= ∂ 1 ²₂ ²₂

z u x

u x

p z

w u x u u t u dt

du ν

ρ

z g w x

w z

p z

w w x u w t w dt

dw ⎟⎟⎠−

⎜⎜ ⎞

⎝

⎛

∂ + ∂

∂ + ∂

∂

− ∂

∂ = + ∂

∂ + ∂

∂

= ∂ 1 ν ²₂ ²₂

ρ

These are Navier-Stokes equations for 2-D flow.

For steady flow of inviscid fluid:

_{Eulerequations}

1 1

⎪⎪

⎭

⎪⎪⎬

⎫

∂ −

− ∂

∂ = + ∂

∂

∂ ∂

− ∂

∂ = + ∂

∂

∂

z g p z

w w x u w

x p z

w u x u u

ρ ρ

3-D Navier-Stokes equations:

⎟⎟⎠

⎜⎜ ⎞

⎝

⎛

∂ + ∂

∂ + ∂

∂ + ∂

∂

− ∂

∂ = + ∂

∂ + ∂

∂ + ∂

∂

∂

2 2 2 2 2

1 2

z u y

u x

u x

p z

w u y v u x u u t

u ν

ρ

⎟⎟⎠

⎜⎜ ⎞

⎝

⎛

∂ + ∂

∂ + ∂

∂ + ∂

∂

− ∂

∂ = + ∂

∂ + ∂

∂ + ∂

∂

∂

2 2 2 2 2

1 2

z v y

v x

v y

p z

w v y v v x u v t

v ν

ρ

z g w y

w x

w z

p z

w w y v w x u w t

w ⎟⎟⎠−

⎜⎜ ⎞

⎝

⎛

∂ + ∂

∂ + ∂

∂ + ∂

∂

− ∂

∂ = + ∂

∂ + ∂

∂ + ∂

∂

∂

2 2 2 2 2

1 ν 2

ρ

Continuity: = ₀

∂ + ∂

∂ + ∂

∂

∂

z w y

v x u

4 equations for 4 unknowns (p,u,v,w)

Navier-Stokes equations in cylindrical coordinates

for axi-symmetric flows (r, z ) → Eq. (7.64) in text (gravitational force is neglected)

Continuity: = 0

∂ + ∂

∂ +

∂

r u r

u z

u_{z r r}

7.16 Applications of Navier-Stokes Equations

(1) Couette flow

⎟⎟⎠

⎜⎜ ⎞

⎝

⎛

∂ + ∂

∂ + ∂

∂

− ∂

∂ = + ∂

∂ + ∂

∂

∂

2 2 2

1 2

: z

u x

u x

p z

w u x u u t

x u ν

ρ

2

1 2

0 z

u x

p

∂ + ∂

∂

− ∂

=

∴ ν

ρ

z g w x

w z

p z

w w x u w t

z w ⎟⎟⎠−

⎜⎜ ⎞

⎝

⎛

∂ + ∂

∂ + ∂

∂

− ∂

∂ = + ∂

∂ + ∂

∂

∂

2 2 2

1 2

: ν

ρ

) ( )

, 1 (

0 g p x z gz C x

z g p

z

p = − → = − +

∂

→ ∂

∂ −

− ∂

=

∴ ρ ρ

ρ x p z

u

∂

= ∂

∂

∂

ρ ν

1 1

2 2

1

1

1 z C

x p z

u +

∂

= ∂

∂

∂

ρ ν

2 1

2

2 1

) 1

( z C z C

x z p

u + +

∂

= ∂ ρ ν

B.C.’s: u(−h/2)= 0, u(h/2)=V

⎟⎠

⎜ ⎞

⎝⎛ +

⎥+

⎥⎦

⎤

⎢⎢

⎣

⎡ ⎟

⎠

⎜ ⎞

⎝

−⎛

∂

= ∂

∴ h

V z z h

x z p

u 2

1 2

2 ) 1 (

2 2

μ If =0

∂

∂ x

p , _⎟

⎠

⎜ ⎞

⎝⎛ +

= h

V z z

u 2

) 1

( : linear variation

If V = 0,

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡ ⎟

⎠

⎜ ⎞

⎝

−⎛

∂

= ∂

2 2

2 2

) 1

( h

x z z p

u μ : parabolic profile

(2) Hagen-Poiseuille flow

⎟⎟⎠

⎜⎜ ⎞

⎝

⎛

∂ + ∂

∂ − + ∂

∂ + ∂

∂

− ∂

∂ = + ∂

∂ + ∂

∂

∂

2 2 2 2

2 1

: 1

z u r

u r u r r

u r

p z

u u r u u t

r u^r _r ^r _z ^r ν ^{r r r r}

ρ 1 0

0 =

∂

→ ∂

∂

− ∂

=

∴ r

p r

p ρ

⎟⎟⎠

⎜⎜ ⎞

⎝

⎛

∂ + ∂

∂ + ∂

∂ + ∂

∂

− ∂

∂ = + ∂

∂ + ∂

∂

∂

2 2 2

2 1

: 1

z u r

u r r

u z

p z

u u r u u t

z u^z _r ^z _z ^z ν ^{z z z}

ρ

⎟⎟⎠

⎜⎜ ⎞

⎝

⎛ +

+

−

⎟⎟=

⎠

⎜⎜ ⎞

⎝

⎛

∂ + ∂

∂ + ∂

∂

− ∂

=

∴ dr

du r dr

u d dz

dp r

u r r

u z

p _z 1 _z 1 _z 1 _z

0 1 ₂

2 2

2 ν

ν ρ ρ

dz dp dr

du dr r

u

d _{z z}

μ 1 1

2

2 + =

dz r dp dr

du dr

u

r d ^{z z} μ 1

2

2 + =

dz r dp dr

r du dr

d _z

μ

= 1

⎟⎠

⎜ ⎞

⎝

⎛

1 2

2

1 r C

dz dp dr

r du^z = + μ

B.C.: = 0 at r =0 → C₁ =0 dr

du_z

dz r dp dr

du_z

μ 2

= 1

∴

2 2

2 2

1 r C

dz u_z = dp +

μ B.C.:

2

2 4 2

1 at 2

0 ⎟

⎠

⎜ ⎞

⎝

− ⎛

=

→

=

= d

dz C dp

r d u_z

μ

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡ ⎟

⎠

⎜ ⎞

⎝

−⎛

=

∴

2 2

2 4

1 d

dz r u_z dp

μ : parabolic profile

Time-averaged Navier-Stokes equations for turbulent flow:

pressure and

velocity (mean)

averaged -

⎪ time

⎭

⎪⎬

⎫

→

→

→ p p

w w

u u

ε ν ν

ν → _T = +