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

P M V SubbaraoProfessor

Mechanical Engineering Department

I I T Delhi

Modeling of Turbulent Flows

Simplified Reynolds Averaged Navier Stokes equations

0

z

W

y

V

x

U

Ux

PVU t

Wz

PVW t

Vy

PVV t

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

......

-k

-k

-k

Re

3

2

1Re

-k

Eq.

Two

Eq.-One

TKEM

constantMVM

μon based Models

t

t

fk

kl

l

Curvature

Buoyancy

Low

Layer

Layer

Layer

bounded

wall

Free

High

lengthmixing

LES: Large Eddy simulation models

RSM: Reynolds stress models

Additional models:

Modeling of Turbulent Viscosity

• 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 3

22

i

j

j

iij x

U

x

US

2

1

• 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

dy

dUlamxylam ,

fmpthlam lv2

1 where,

dy

dUturbulentxyturbulent ,

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

m

kTvth

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

2

1

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

0

0

u

v

0

0

u

v

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:

dy

dUl

y

UlU 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

dy

dUlu 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 dy

dUlCv m1

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