wall free shear flow turbulent flows free of solid boundaries jet two-dimensional image of an...

WALL FREE SHEAR FLOW

Turbulent flows free of solid boundaries

JET

Two-dimensional image of an axisymmetric water jet, obtained by the laser-induced fluorescence technique. (From R. R. Prasad and K. R. Sreenivasan, Measurement and interpretation of fractal dimension of the scalar interface in turbulent flows, Phys. Fluids A, 2:792–807, 1990)

x

y

Irrotational

Turbulent

WAKE

http://www.ifh.uni-karlsruhe.de/science/envflu/

SHEAR LAYER

Laser-Induced Fluorescence (LIF) VISUALIZATION OF AN AXISYMMETRIC TURBULENT JET, C. Fukushima and J. Westerweel, Technical University of Delft, The Netherlands

x

x

Laser-Induced Fluorescence (LIF) VISUALIZATION OF AN AXISYMMETRIC TURBULENT JET, C. Fukushima and J. Westerweel, Technical University of Delft, The Netherlands

x

Turbulent Kinetic Energy (q2) Balance in a Jet

2

222

2wvu

q

x

y

q2

<v2 > <u2 >

<w2 >

-<uv >

y

m2 /

s2

2

222

2wvu

q

Laser-Induced Fluorescence (LIF) VISUALIZATION OF AN AXISYMMETRIC TURBULENT JET, C. Fukushima and J. Westerweel, Technical University of Delft, The Netherlands

x

Turbulent Kinetic Energy (q2) Balance in a Jet

02

22

0 vpvqyy

uuv

y

qv

x

qu

No local accelerationsNo viscous transportPart of the shear production = 0No buoyancy production

wg

x

uuueuuqup

xdt

dq

oj

i

jiijijjoj

21 2

2

x

y

02

22

0 vpvqyy

uuv

y

qv

x

qu

ym2 /

s3

Gai

nL

oss

x

qu

2

y

qv

2

y

uuv

02 vpvq

y

http://www.symscape.com/node/447

U0

WALL-BOUNDED SHEAR FLOW

Nominal limit of boundary layer

0.99U0

Viscous sublayer

For fully developed, bounded turbulent flow (not changing in x):

zx

p

0 uwz

u

zx

p

0Function of x only Function of z only

CONSTANTS!

uwz

u

z

centerline or surface

zz

uw

x

uu

0

1in boundary layer:

uwz

u

z

z

edge of boundary layer

stress is a function of x and z

Near the wall – Different Layers

zuu ,,, 0

http://furtech.typepad.com/

z

ū (x)

u(x,z)

0,Only involve mass dimension

Should appear together in nondimensional groups

0

* u Friction Velocity

zuuu ,,*

zuuu ,,*

This relates 4 variables involving the dimensions of length and time

According to the PI THEOREM, this relationship has 4 variables and 2 dimensions

Then, only two (4 – 2) non-dimensional groups can result:

zf

zuf

u

u

*

*

Law of the Wall

Inner part of the wall layer, right next to the wall, is called the viscous sublayer – dominated by viscous effects

*u

u

*u

u

z z

z (m

) =

z+ν/

u*

viscous sublayer

z

u

0

zu 0

zu

u

*

buffer layer

logarithmic layer

*u

u

z

viscous sublayer

buffer layer

logarithmic layer

zf

zuf

u

u

*

*

outer layer

F

zF

u

Uu

*

Velocity defect law

Law of the wall

dz

dfu

z

u

2*

d

dFu

z

u *

z

fz

d

dF

1

Karman constant = 0.41

Equating and multiplying times z/u*

z

fz

d

dF

1

Karman constant = 0.41

Integrating: Azzf ln1

BF

ln

1

From experiments: 5ln1 *

*

zu

u

u1ln

1

*

z

u

Uu

*u

u

z

Velocity distributions for theOverlap layer,Inertial sublayer,Logarithmic layer

Logarithmic velocity distribution near a boundary can also be derived from dimensional analysis

z

u

can only depend on z, and the only relevant velocity scale is u*

z

u

z

u *1

Czu

u ln*

*zu

*u

u

*

5

u

*

30

u

*

300

u

0@0 zzu 0

* lnz

zuu

0

* lnz

zuu

m005.0

sm04.0

0

*

z

u

0

* u

Pa20

Data from Ponce de Leon Inlet

FloridaIntracoastal Waterway

Florida