Quantification of the Infection & its Effect on Mean Fow....

P M V SubbaraoProfessor

Mechanical Engineering DepartmentI I T Delhi

Analysis of Turbulent (Infected by Disturbance) Flows

Averaging Navier Stokes equations for Study of Turbulent Flows

uUu

p Pp

v'Vv

w'Ww

Substitute into Steady incompressible Navier Stokes equations

Continuity equation:0

zw

yv

xu

Instantaneous velocity

Averagevelocity

fluctuationaround averagevelocity

time

0

zw')(W

yv')(V

xu')(U

0

zw'

yv'

xu'

zW

yV

xU

Averaging of x-momentum Equation

uμ xpvu

τuρ

uμ xpvu

τuρ

2

2

2

2

2

2

zu

yu

xuμ

xp

zuw

yuv

xuu

τuρ

Write x-momentum equations in a short format:

uμ xpvu

τuρ

vVuUvu�

vuVuvUVV

vuVuvUVU

vuVU

kwujvuiuuvu ˆˆˆ

zwu

yvu

xuu

zwu

yvu

xuuVU

uμ xpvu

τuρ

uUu

uU

U

Reynolds Averaged Steady Turbulent Momentum Equations

uμ xpvu

τuρ

UxP

zwu

yvu

xuuVU

zwu

yvu

xuuU

xPVU

Reynolds averaged x-momentum equation for steady incompressible turbulent flow

The Reynolds View of Cross Correlation

Reynolds averaged y-momentum equation for steady incompressible turbulent flow

Reynolds averaged z-momentum equation for steady incompressible turbulent flow

zww

yvw

xuwW

zPVW

zwv

yvv

xuvV

yPVV

Reynolds Averaged Navier Stokes equations

0

zW

yV

xU

zwu

yvu

xuuU

xPVU

zww

yvw

xuwW

zPVW

zwv

yvv

xuvV

yPVV

Reynolds Stress Tensor

2

2

2

wvwuwwvvuvwuvuu

R

This is usually called the Reynolds stress tensor

2

2

2

wwvwuwvvvuwuvuu

R

Reynolds stresses : total 9 - 6 are unknown

Total 4 equations and 4 + 6 = 10 unknowns

Time averaged Infected Navier Stokes Equation

zwu

yvu

xu

zU

yU

xUv

dxdp

zUW

yUV

xUU ''''1 2'

2

2

2

2

2

2

For all the Three Momentum Equations, turbulent stress tensor:

)'()''()''(

)''()'()''(

)''()''()'(

2

2

2

infected,

wvwwu

wvvuv

wuvuu

zzzyzx

yzyyyx

xzxyxx

ij

Reynolds stresses

• Performing the Reynolds Averaging Process, new terms has arisen, namely the Reynolds-stress tensor:

''infection, jiijij uu

• This brings us at the turbulent closure problem, the fact that we have more unknowns than equations. – Three velocities + pressure + six Reynolds-stresses– Three momentum equations + the continuity equation

• To close the problem, we need additional equations to solve infected flow.

• Derivations of Reynolds-stress conservation Equations

Derivation of Conservation Equations for Reynolds Stresses

• Introduces new unknowns (22 new unknowns)

Simplified Reynolds Averaged Navier Stokes equations

0

zW

yV

xU

UxPVU t

WzPVW t

VyPVV t

4 equations 5 unknowns → We need one more ???

Modeling of Turbulent Viscosity

μ Fluid property – often called laminar viscosity

tμ Flow property – turbulent viscosity

......

-k-k-k

Re

321

Re

-k

Eq.Two

Eq.-One

TKEM

constantMVM

μon based Models

t

t

fkkll

CurvatureBuoyancyLow

LayerLayerLayer

boundedwall

Free

High

lengthmixing

MVM: Mean velocity modelsTKEM: Turbulent kinetic energy equation models

• Eddy-viscosity models• Compute the Reynolds-stresses from explicit expressions of the

mean strain rate and a eddy-viscosity, the Boussinesq eddy-viscosity approximation

MVM : Eddy-viscosity models

The k term is a normal stress and is typically treated together with the pressure term.

ijijtjiij kSuu 322

i

j

j

iij x

UxUS

21

• Prandtl was the first to present a working algebraic turbulence model that is applied to wakes, jets and boundary layer flows.

• The model is based on mixing length hypothesis deduced from experiments and is analogous, to some extent, to the mean free path in kinetic gas theory.

Algebraic MVM

Molecular transport Turbulent transport

dydU

lamxylam ,

mfppeclam lv21 where,

dydU

turbulentxyturbulent ,

Kinetic Theory of Gas• The Average Speed of a Gas Molecule

mkTvpec

3

Kinetic Theory of Gas Boundary Layer• Motion of gas particles in a laminar boundary layer?

Microscopic Energy Balance for A Laminar BL

Random motion of gas molecules

Solid bodies Dissipate this energy by friction

Thermal EnergyEnthalpy = f(T)

Macro Kinetic Energy

Gas Molecules Dissipate this energy by viscosity at wall

http://www.granular.org/granular_theory.html

Prandtl’s view of Viscosity

• For a gas in a state of thermodynamic equilibrium, the quantities such as mean speed, mean collision rate and mean free path of gas particles may be determined.

• Boltzmann explained through an equation how a gas medium can have small macroscopic gradients exist in either (bulk) velocity, temperature or composition.

• The solutions of Boltzman equation give the relation between the gradient and the corresponding flux in each case in terms of collision cross-sections.

• Coefficients of Viscosity, Thermal conductivity and Diffusion are thereby related to intermolecular potential.

2

21

16

5

d

mkT fmpthlam lv

21

Pradntl’s Hypothesis of Turbulent Flows

• In a laminar flow the random motion is at the molecular level only.

• Macro instruments cannot detect this randomness.• Macro Engineering devices feel it as molecular viscosity.• Turbulent flow is due to random movement of fluid

parcels/bundles.• Even Macro instruments detect this randomness.• Macro Engineering devices feel it as enhanced

viscosity….!

Prandtl Mixing Length Hypothesis

U

X

Y

y

UU

UU

00

uv

00

uv

The fluid particle A with the mass dm located at the position , y+lm and has the longitudinal velocity component U+U is fluctuating.This particle is moving downward with the lateral velocity v and the fluctuation momentum dIy=dmv. It arrives at the layer which has a lower velocity U. According to the Prandtl hypothesis, this macroscopic momentum exchange most likely gives rise to a positive fluctuation u >0.

U

0vu

Definition of Mixing Length• Particles A & B experience a velocity difference which can be

approximated as:

dydUl

yUlU mm

The distance between the two layers lm is called mixing length.Since U has the same order of magnitude as u, Prandtl arrived at

dydUlu m

By virtue of the Prandtl hypothesis, the longitudinal fluctuation component u was brought about by the impact of the lateral component v , it seems reasonable to assume that

vu dydUlCv m1

Prandtl Mixing Length Model

• Thus, the component of the Reynolds stress tensor becomes

22

2,

dydUlCvu mxyturbulent

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

• The turbulent shear stress component becomes

22

2

dydUlCvu m

dydUlmturbulent

2

Prandtl Mixing Length Model

• Thus, the component of the Reynolds stress tensor becomes

22

1,

dydUlcvu mxyturbulent

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

• The turbulent shear stress component becomes

22

1

dydUlCvu m

dydUlmturbulent

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:

xP

zU

yU

tl

2

2

2

2

With 0&0

zP

yP

xU

Turbulent Viscosity is a Flow Property

zygllf mmt ,&

xP

zU

zyU

yzU

yU

ttl

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:

dxdP

drdUr

drd

drdUr

drd

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

uUU

yuy

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

Ay

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:

0.126

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.

Steady Turbulent flow

A Segment of Reconstructed Turbulent Flame in SI Engines

Large Scales: Parents Vortices

Creation of Large Eddies an I.C. Engines

• There are two types of structural turbulence that are recognizable in an engine; tumbling and swirl.

• Both are created during the intake stroke. • Tumble is defined as the in-cylinder flow that is rotating

around an axis perpendicular with the cylinder axis.

Swirl is defined as the charge that rotates concentrically about the axis of the cylinder.

Instantaneous Energy Cascade in Turbulent Boundary Layer.

A state of universal equilibrium is reached when the rate of energy received from larger eddies is nearly equal to the rate of

energy of when the smallest eddies dissipate into heat.

One-Equation Model by Prandtl• A one-equation model is an enhanced version of the algebraic models. • This model utilizes one turbulent transport equation originally

developed by Prandtl. • Based on purely dimensional arguments, Prandtl proposed a

relationship between the dissipation and the kinetic energy that reads

t

D

lkC 2

3

•where the turbulence length scale lt is set proportional to the mixing length, lm, the boundary layer thickness or a wake or a jet width. •The velocity scale is set proportional to the turbulent kinetic energy as suggested independently. •Thus, the expression for the turbulent viscosity becomes:

5.0klC mturbulent

with the constant C to be determined from the experiment.

222

21

21 wvuuuk ii

Transport equation for turbulent kinetic Energy

uUμ x

pPvVuUτ

uUρ

x-momentum equation for incompressible steady turbulent flow:

Reynolds averaged x-momentum equation for incompressible steady turbulent flow:

subtract the second equation from the second equation to get

uμ xpVuvU

τuρ

UxP

zwu

yvu

xuuVU

Multiply above equation with u and take Reynolds averaging

uμ

xpuVuvU

τuuρ '

Similarly:

vμ

ypvVvvV

τvvρ

wμ

zpwVwvW

τwwρ

222

21

21 wvuuuk ii

Define turbulent kinetic energy as:

Turbulent Kinetic Energy Conservation Equation

The Cartesian index notation is:

Boundary conditions:

0 :at Wall 0 yk

Taylor Hypothesis• Taylor proposed an hypothesis that the energy transport contribution

of small size eddies that are carried by a large scale eddy Taylor proposed an hypothesis:

• The energy transport contribution of small size eddies that are carried by a large scale eddy, compared with the one produced by a larger eddy, is negligibly small.

• In such a situation, the transport of a turbulence field past a fixed point is due to the larger energy containing eddies.

• It states that “in certain circumstances, turbulence can be considered as “frozen” as it passes by a sensor”.

Spectral Representation of Turbulent Flows

Kolmogorov Hypotheses• Kolmogorov established his universal equilibrium theory based

on two similarity hypotheses for turbulent flows. • The first hypothesis states that for a high Reynolds number

turbulent flow, the small-scale turbulent motions are isotropic and independent of the detailed structure of large scale eddies.

• Furthermore, there is a range of high wavenumbers where the turbulence is statistically in equilibrium and uniquely determined by the energy dissipation and the kinematic viscosity .

• With this hypothesis and in conjunction with dimensional reasoning, Kolmogorov arrived at length (), time () and the velocity (v) scales. Considering the Kolmogorov’s length and velocity scales, the corresponding Kolmogorov’s equilibrium Reynolds number is

v

kol Re

Structure of Equilibrium Turbulence

35

32

kCKE K

4/13

Structure of Equilibrium Turbulence

One and Two Equation Turbulence Models• The derivation is again based on the Boussinesq approximation

• The mixing velocity is determined by the turbulent turbulent kinetic energy

• The length scale is determined from another transport equation ex.

Second equation

Dissipation of turbulent kinetic energy• The equation is derived by the following operation on the

Navier-Stokes equation

The resulting equation have the following form

The k-ε model• Eddy viscosity

• Transport equation for turbulent kinetic energy

• Transport equation for dissipation of turbulent kinetic energy

• Constants for the model

Dealing with Infected flows

• The RANS equations are derived by an averaging or filtering process from the Navier-Stokes equations.

• The ’averaging’ process results in more unknown that equations, the turbulent closure problem

• Additional equations are derived by performing operation on the Navier-Stokes equations

• Non of the model are complete, all model needs some kind of modeling.

• Special care may be need when integrating the model all the way to the wall, low-Reynolds number models and wall damping terms.