Wall bounded flows 2/2

TURBULENCE: THEORY AND MODELING

LECTURE 6

Main categories

• Channel flow

• Pipe flow

• Flat plate boundary layer

• 2D

• Simplified equations can be derived

• Average velocity profile

• Reynolds stresses

• Is there any ’universal rule’ ?

• Models

Nondimensionalized governing equations (2D)

0*

*

*

*=

∂∂

+∂∂

yv

xu

*

**

*

**

2*

*2

2*

*2

*

*

*

**

*

**

*

*

Re1

Re1

yvu

xuu

y

u

x

uxp

yuv

xuu

tu

∂′′∂

−∂

′′∂−

∂

∂+

∂

∂+

∂∂

−=∂∂

+∂∂

+∂∂

*

**

*

**

2*

*2

2*

*2

*

*

*

**

*

**

*

*

Re1

Re1

yvv

xvu

y

v

x

vyp

yvv

xvu

tv

∂′′∂

−∂

′′∂−

∂

∂+

∂

∂+

∂∂

−=∂∂

+∂∂

+∂∂

Simplify • Assumptions

– Statistically stationary

– Fully developed, i.e. For velocities:

– Also:

– No * in the followings x

y

U V

xy ∂∂

>>∂∂

0=∂∂t

vu >>

δ>>L

Simplify y-momentum

yvv

xvu

yv

xv

yp

yvv

xvu

tv

∂′′∂

−∂

′′∂−

∂∂

+∂∂

+∂∂

−=∂∂

+∂∂

+∂∂

2

2

2

2

Re1

Re1

0=∂∂t xy ∂

∂>>

∂∂

( ) ( ) ( )( ) ( ) ( )δδ

δδδ kOkOO

OypOO −−++

∂∂

−=+Re

1Re

uL

vyv

xu δ

∝⇒=∂∂

+∂∂ 0

O(1)

Y-momentum

• Integrate

• Differentiate on x

P0, v’=0

0=∂

′′∂+

∂∂

yvv

yp

0ppy →⇒∞→

vvpp ′′−= 0

xvv

dxdp

xp

∂′′∂

+=∂∂ 0

2D Boundary Layer Equations

0=∂∂

+∂∂

yv

xu

yvu

yu

dxdp

yuv

xuu

∂′′∂

−∂∂

+−=∂∂

+∂∂

2

20

Re1

Using Bernoulli:

dxdUU

dxdp 0

00 =−

Special case: Fully developed channel flow

0)()( ==− hvhv

0=∂∂

yv

yvu

yu

dxdp

∂′′∂

−∂∂

= 2

20

Re1

0=v

Rewrite ( ) ( )

yydxdp xyxy

∂∂

−∂

∂=

turbvisc0 ττ i.e. the total stress is

independent of x and varies linearly with y

• Continuity:

• BCs: • X-momentum:

U

yvu

yu

dxdp

yuv

xuu

∂′′∂

−∂∂

+−=∂∂

+∂∂

2

20

Re10=

∂∂

xu

Wall shear stress Skin friction

( ) ( )

yyydxdp xyxyxy

∂

∂=

∂

∂−

∂

∂=

τττ turbvisc0

• wall shear stress antisymm.

• Normalize: friction coefficient

U

0=∂∂

xu

2δ

τw

−τw

δτ w

dxdp

=− 0

τw

−=

δττ yy w 1)(

= 2

021/ uc wf ρτ

Total shear stress near the wall

Viscous scales • Close to the wall viscosity dominates

• Wall shear stress, τw, important

• Friction velocity:

• Viscous lengthscale:

• Friction Reynolds number:

• Wall units:

ρτρτ ττ

ww uu =⇒= 2

τν

νδu

=

ν

ττ δ

δνδ

==uRe

νδτ

ν

yuyy ==+

0

100%

Viscous stresses

Reynolds stresses

Y+ 10 20 50

Re

Wall regions and layers Inner layer y/δ<0.1 u independent of U0 and δ Viscous wall layer y+<50 significant viscous contribution to shear stress Viscous sublayer y+<5 viscous stress dominates Outer layer y+>50 viscous effects on u negligible Overlap layer y+>50, y/δ<0.1 overlap between inner and outer layer log-layer y+>30, y/δ<0.3 region where the log-law holds Buffer layer 5<y+<30 region between viscous sublayer and log-layer

Mean velocity profiles • We could derive u, but du/dy

appears directly in the equations

• δν – relevant lengthscale in the viscous sublayer

• δ – relevant lengthscale in the outer layer

Φ=

∂∂

δδν

τ yyy

uyu ,

( ) ( )

yydxdp xyxy

∂∂

−∂

∂=

turbvisc0 ττ

Nondimensional, unknown function

Correct units

Law of the wall • Prandtl (1925) • Inner layer: independent of δ

and U0

Φ=

∂∂

δδν

τ yyy

uyu ,

Φ=

∂∂

ν

τ

δy

yu

yu

)(1, +++

++ Φ== y

ydydu

uuu

τ

∫+

Φ== ++y

dyyy

yfu0

''' )(1)(

Law of the wall – viscous sublayer

0)0(0 =⇒= fu 1)0(' =f

++

+++

=

+≈++=

yuyyOfyfyf )()0(')0()(

2

∫+

Φ== ++y

dyyy

yfu0

''' )(1)(

Law of the wall – the log law • For y+>30 & y/δ << 1(still inner layer)

– Viscosity small effect

– U0 – no effect yet

– von Kármán: κ=0.41 B=5.2

– Universal, err=5%

Φ=

∂∂

δδν

τ yyy

uyu ,

Byuydy

du+=⇒= ++

++

+

ln11κκ

Velocity defect law • Outer layer, y+>50

– Viscosity neglected

– Integrate between y and δ

» u0 – centerline velocity

» FD – NOT universal

– For large Re inner and outer layers overlap

Φ=

∂∂

δδν

τ yyy

uyu ,

∫ Φ=

=

− 1

/

0 ')'('

1

δτ δ yD dyy

yyF

uuu

Log(y/δ)

(u0-u)/uτ

10 ln1 ByyFu

uuD +

−=

=

−δκδτ

Velocity-defect law for small y/δ

Friction law & Re • Inner layer

• Outer layer

10 ln1 Byu

uu+

−=

−δκτ

Byuu

+

=

ντ δκln1

NOT universal, but small

Log-law not valid, but small contribution to total mass flow.

BBuu

BBuu

++

=

++

=

−

1

1

00

10

Reln1

ln1

τ

ντ

κ

δδ

κ

Friction coefficient Outer/inner parameter ratios

BBuu

uu

++

=

−

1

1

00

0 Reln1

ττ κ

Solve for u0/uτ :

Friction coefficient:

2

0

20 2

21/

=

=

uuuc wf

τρτ

Log(Re)

cf

Laminar

Turbulent

Log(Re)

Log(δ/δν)

Log(Re)

u/uτ

Re-dependence • Re ↑ → Visc. Subl.↓ • E.g. Re=1e5, δ=2cm

� δν=1e-5m – y(y+=100)=1mm – u(y+=10) =

u(y=0.1mm) = 0.3u0

Reynolds stresses

Y+

<ui’uj’>/k

50

<u’2>

<v’2>

<w’2>

Largest fluctuations In viscous layer Anisotropy

0

Production/dissipation, timescales

Production & dissipation almost in balance

No production on centerline

Largest production In viscous layer

1≈εP

Normalized shear stress 3.0≈

′′kvu

Time scale ratio

3≈′′

=εεP

vukSk

dyudS =

εk

Time scales:

Lengthscales

• Timescales approximately constant (see previous slide) • Lengthscale:

LC

Pkvu

uvu

yLε

κτ

2/3−′′′′=

ε

2/3kL = 5.2≈LC

Eddy viscosity luT ′=υ yulu m ∂

∂=′

yu

yulvu m ∂

∂∂∂

=′′− 2ρρ

Mixing length yuvu T ∂

∂=′′− ρυρ

vuu ′′=′

Pipe flows • Similar to channel flows

• Cylindrical coordinate system

• Pipe wall roughness

• Nikuradse diagram (pp.295, Fig. 7.23)

Log(Re)

cf

Laminar

Turbulent

Roughness

Flat plate boundary layers compared to channel flow

Boundary layers • Difficult to measure

absolute thickness (e.g. 99%u0)

• Integral quantities

– Displacement thickness

– Momentum thickness

•Continous development in the flow direction, i.e. thickness is a function of x •Wall shear stress is not known a priori •Turbulence intermittency in the outer part of the BL

( ) dyUux ∫

∞

−=

0 0

* 1δ

( ) dyUu

Uux ∫

∞

−=

0 001θ

Boundary layers

0=∂∂

+∂∂

yv

xu

yvu

yu

dxdp

yuv

xuu

∂′′∂

−∂∂

+−=∂∂

+∂∂

2

201 ν

ρ

Using Bernoulli: dx

dUUdxdp 0

001

=−ρ

dxdUU

yyuv

xuu 0

01

−∂∂

+=∂∂

+∂∂ τ

ρ

3 cases -Favourable pressure gradient -Adverse pressure gradient -0 pressure gradient (u0=const)

Boundary layers

dxdUw

θρτ 20=Wall shear stress

Similarity solution by Blasius (1908) (for laminar, zero-pressure gradient)

14.0

35.0

Re9.4

*

≈

≈

≈

δθδδ

δ

xx

υxU

x0Re =

0Ux

xυδ =

( )kk

i

ik

ik

ii xx

uxp

xuu

tuuN

∂∂∂

−∂∂

+∂∂

+∂∂

=2

ν

Derivation of the transport equations for Reynolds’ stress

Introduce the ”Navier-Stokes operator”

( ) ( ) 0=′+′ jiij uNuuNu

( ) 0=iuN

After some manipulation we get

∂

′′∂−′′′

∂∂

−Π+−=∂

′′∂+

∂

′′∂

k

jikji

kijijij

k

jik

ji

xuu

uuux

Pxuu

utuu

νε

k

ikj

k

jkiij x

uuuxu

uuP∂∂′′−

∂∂

′′−=

k

j

k

iij x

uxu

∂

′∂∂

′∂= νε 2

ikj

jkip

kijpupuT δ

ρδ

ρ′′

+′′

=)(

∂

′′∂−′′′

∂∂

−Π+−=∂

′′∂+

∂

′′∂

k

jikji

kijijij

k

jik

ji

xuu

uuux

Pxuu

utuu

νε

Production tensor

Dissipation tensor

Velocity-pressure-gradient tensor

k

pkij

ijij xT

R∂

∂+=Π

)(

∂

′∂+

∂′∂′

=i

j

j

iij x

uxupR

ρPressure-rate-of-strain tensor

Pressure transport tensor

Turbulent transport Viscous

diffusion

∂

′′∂−′′′

∂∂

−Π+−=∂

′′∂+

∂′′∂

k

iikii

kiiiiii

k

iik

ii

xuu

uuux

Px

uuu

tuu 2

1

21

21

21

212

121

νε

Take trace:

∂

′′∂−′′′

∂∂

−Π+−=∂

′′∂+

∂

′′∂

k

jikji

kijijij

k

jik

ji

xuu

uuux

Pxuu

utuu

νε

∂∂

−′′

+′′′∂∂

−−=∂∂

+∂∂

k

kkii

kkk x

kpuuuux

Pxku

tk ν

ρε 2

1

Which is the equation for turbulent kinetic energy

Note: ρpuk

ii′′

=Π21 0=iiR

k

i

k

iii x

uxu

∂′∂

∂′∂

== νεε21

j

ijiii x

uuuPP∂∂′′−==2

1

Reynolds stress → TKE

yuvu

zuwu

yuvu

xuuuP

∂∂′′−=

∂∂′′−

∂∂′′−

∂∂′′−= 222211

022222 =∂∂′′−

∂∂′′−

∂∂′′−=

zvwv

yvvv

xvvuP

033 =P

Production

Production occurs in u’u’

Production balanced by dissipation. Pressure not significant.

Pressure term IS a significant sink term!

Pressure term is a significant source term! Redistribution Important @ modeling!

ikj

jkip

kijpupuT δ

ρδ

ρ′′

+′′

=)(

∂

′∂+

∂′∂′

=i

j

j

iij x

uxupR

ρ