Reynolds Method to Diagnosize Symptoms of Infected Flows.... P M V Subbarao Professor Mechanical...

Reynolds Method to Diagnosize Symptoms of Infected Flows....

P M V SubbaraoProfessor

Mechanical Engineering Department

I I T Delhi

Reynolds Averaged NS Equations

Averaging of x-momentum Equation

uμ x

pvu

τ

uρ

uμ x

pvu

τ

uρ

uUuU

τ

uUρρ

'

uU

uμ x

pvu

τ

uρ

vVuUvu

vuVuvUVU

vuVuvUVU

vuVU

kwujvuiuuvu ˆˆˆ

z

wu

y

vu

x

uu

z

wu

y

vu

x

uu

z

wu

y

vu

x

uuVU

uμ x

pvu

τ

uρ

uUu

uU

U

Reynolds Averaged Steady Turbulent Momentum Equations

uμ x

pvu

τ

uρ

Ux

P

z

wu

y

vu

x

uuVU

z

wu

y

vu

x

uuU

x

PVU

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

z

ww

y

vw

x

uwW

z

PVW

z

wv

y

vv

x

uvV

y

PVV

Reynolds Averaged Navier Stokes equations

0

z

W

y

V

x

U

z

wu

y

vu

x

uuU

x

PVU

z

ww

y

vw

x

uwW

z

PVW

z

wv

y

vv

x

uvV

y

PVV

Reynolds Stress Tensor

2

2

2

wvwuw

wvvuv

wuvuu

R

This is usually called the Reynolds stress tensor

2

2

2

wwvwu

wvvvu

wuvuu

R

Reynolds stresses : total 9 - 6 are unknown

Total 4 equations and 4 + 6 = 10 unknowns

Time averaged Infected Navier Stokes Equation

z

wu

y

vu

x

u

z

U

y

U

x

Uv

dx

dp

z

UW

y

UV

x

UU

''''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

z

W

y

V

x

U

Ux

PVU t

Wz

PVW t

Vy

PVV 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

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

MVM: Mean velocity modelsTKEM: Turbulent kinetic energy equation models