Identification of Macro Mean Free Path....

P M V SubbaraoProfessor

Mechanical Engineering Department

I I T Delhi

Estimation of Prandtls Mixing Length

Prandtl Mixing Length Model

• Thus, the component of the Reynolds stress tensor becomes

2

21,

dy

dUlcvu mxyturbulent

• This is the Prandtl mixing length hypothesis. •Prandtl deduced that the eddy viscosity can be expressed as

• The turbulent shear stress component becomes

2

21

dy

dUlCvu m

dy

dUlmturbulent

2

Fully Developed Duct Flow

• For x > Le, the velocity becomes purely axial and varies only with the lateral coordinates.

• V= W = 0 and U = U(y,z).

• The flow is then called fully developed flow.

For fully developed flow, the Reynolds Averaged continuity and momentum equations for incompressible flow are simplified as:

x

P

z

U

y

Utl

2

2

2

2

With 0&0

z

P

y

P

x

U

Turbulent Viscosity is a Flow Property

zygllf mmt ,&

x

P

z

U

zy

U

yz

U

y

Uttl

2

2

2

2

The true Reynolds Averaged momentum equations for incompressible fully developed flow is:

Fully Developed Turbulent flow in a Circular Pipe: Modified Hagen-Poiseuille Flow

• The single variable is r.

• The equation reduces to an ODE:

dx

dP

dr

dUr

dr

d

dr

dUr

dr

d

r t

The solution of above Equation is: ?????

•Engineering Conditions: •The velocity cannot be infinite at the centerline.•Is this condition useful???

Estimation of Mixing Length

• To find an algebraic expression for the mixing length lm, several empirical correlations were suggested in literature.

• The mixing length lm does not have a universally valid character and changes from case to case.

• Therefore it is not appropriate for three-dimensional flow applications.

• However, it is successfully applied to boundary layer flow, fully developed duct flow and particularly to free turbulent flows.

• Prandtl and many others started with analysis of the two-dimensional boundary layer infected by disturbance.

• For wall flows, the main source of infection is wall.

• The wall roughness contains many cavities and troughs, which infect the flow and introduce disturbances.

Quantification of Infection by seeing the Effect

• Develop simple experimental test rigs.

• Measure wall shear stress.

• Define wall friction velocity using the wall shear stress by the relation

u

UU

yu

y

Define non-dimensional boundary layer coordinates.

wallu

U

y

Approximation of velocity distribution for a fully turbulent 2D Boundary Layer

yU

CyU ln1

Cufy ,,

U

y

Approximation of velocity distribution for a fully turbulent 2D Boundary Layer

yU

CyU ln1

Cufy ,,

For a fully developed turbulent flow, the constants are experimentally found to be =0.41 and C=5.0.

Measures for Mixing Length

• Outside the viscous sublayer marked as the logarithmic layer, the mixing length is approximated by a simple linear function.

kylm •Accounting for viscous damping, the mixing length for the viscous sublayer is modeled by introducing a damping function D. •As a result, the mixing length in viscous sublayer:

kDylm

A

y

D exp1

The damping function D proposed by van Driest

with the constant A+ = 26 for a boundary layer at zero-pressure gradient.

• Based on experimental evaluation of a large number of velocity profiles, Kays and Moffat developed an empirical correlation for that accounts for different pressure gradients and boundary layer suction/blowing.

• For zero suction/blowing this correlation reduces to:

0.1

26

abPA

With

0.925.4 0

a

bPfor

0.929.2 0

a

bPfor

5.12

1

w

dxdP

P

Van Driest damping function

Distribution of Mixing length in near-wall region

Mixing length in lateral wall-direction

Conclusions on Algebraic Models

• Few other algebraic models are:

• Cebeci-Smith Model

• Baldwin-Lomax Algebraic Model

• Mahendra R. Doshl And William N. Gill (2004)

• Gives good results for simple flows, flat plate, jets and simple shear layers

• Typically the algebraic models are fast and robust

• Needs to be calibrated for each flow type, they are not very general

• They are not well suited for computing flow separation

• Typically they need information about boundary layer properties, and are difficult to incorporate in modern flow solvers.