Modeling of turbulent shear flows

Center for Modeling of Turbulence Research Briefs - 1991

and

Transition

N93-15797

Modeling

of

Turbulent William

1.

Motivation This

and

report

modeling resents

documents

The

energy

lier,

which

2.

status transfer

2.1

and

flows.

a new

the

a weakly

turbulent

second-order

Model

for

simulations

existence

and

3D

structures

are

model

is based

on the

is mainly

on the

energetic

affected

only

development

Firstly, turbulent

eddies. have

and

more This

process,

be generated waves These

among creation

structures.

The

objective

ear-

of these

turbulent

Turbulent

flows.

on

the

words,

energy

eddies,

has

contain when

with

the

energy

of the

large

enhances

the

vortex

already

been

enhanced,

equilibrium.

Another

spectrum that

of vorticity,

It

or the has

substantial however,

mechanism

creation

been

shown

vorticity occurs

by

Kida

is created

mainly

67

that

at the

less

they

collide

may

and

stretching

by shock

also

wave.

on

through

the the

of the

eddies

may

through shock the spectrum

contribute related

3 and

baroclinic

eddies

route

small

is strongly the

with

mechanism

usual

the

Orszag

energy,

than

pass

structures then cause

of vorticity

energetic

much

eddies

to the

in

intermittently of the

containing

as they

In addition

transfer.

is included

occur

which

small

to compressible

collision

structures

are intensified

transfer.

that

may

in

The

energy

analysis

by the

evolution

in other

flows.

spectral

that

mainly

scale

structures

mechanism

structures

shocklets

of compressibility shear

detailed

tha.t

between

compressibility a new two-scale

of its application

A more

small

effect

increased

of shocklet

vortical

the

shown

mechanism

in turbulent

the

formed

The

eddy

spectral

fluctuation. others,

shocklet

scale.

that

results

here.

are

have

transfer

are the two important are incorporated into eddies

some

Flows

turbulence

energy

directly from the passage of the large vortical processes of enhanced energy transfer may

from

non-equilibrium pressure

the

the

influence

large

stretching

to depart

rep-

developed

wave

for compressible

through

formation

the

direct The

increases

vortex

former

model

the

large

and

flows

in the Thus,

ones.

shocklet.

The

for

nonlinear

proposition

described

that

of large

efficient

smaller

briefly

turbulent

eddies

less

and

are

model

it is assumed

in compressible

other

model.

of

model

in turbulence.

compressible

and

indirectly

of the

free shear layers a NASA TM 5.

are

development

effects

Compressible

of 2D

of shocklet

The

turbulence The

transfer

and

a two-scale

compressibility large

closures

the kinetic energy and the thermal-energy effects 1,2,3,4. These compressibility effects

eddies

research include

Accomplished

"Numerical

model.

in the These

energy

to identify

refines

directly

A Two-Scale

the

Liou

progress

shear

efforts

model

is to develop

Work

current

flows

of our

models

activities

the

for turbulent

turbulent

the

W.

Flows

Objective

techniques

compressible

Shear

Lee terms.

to the to the et

al. 9, The

William Based

on the

is mainly The

picture

described

above,

the

on the energy

containing

large

eddies

large

eddies

resulting

from

respond

either

to compressibility

more

high

readily

speed

increases

through the mechanism scales, on the other hand,

of vortex are only

contained

in the

is increased low

only

energy of the

scales

as more

wavenumber

compressible

the

small

turbulent

spectrum large

large

eddy

eddy

transport

high

fluctuations.

compressible

straining

energy

of the large

transfer

wavenumber in from

spectrum.

quantities

modeled

(kp) and

the

small

eddy

to the

with

eddy

(k,)

eddies

the

these

rate

flow

eddies

due

small

scales

The smaller The energy spectrum

associated

with

Favre-averaged

two

mean

distinct

regimes

for the

kinetic

equations

and the rate

the

mean

of the energy

the large

transport

(¢p) and

part

To model

associated

the

equations

in the

The

spectral

is pumped

of the

small

in turbulence

stretching and direct generation. indirectly affected by compressibility.

we solve to the

P _t

energy

of compressibility

or the low wavenumber

to changes

the

in the

part

effect

or combustion.

effects

the

W. Liou

of energy

of energy

in the energy

transfer

dissipation

from

(¢t).

The

are

= _vv [(_ + a_,

-_v j +

#rt_

v)

_e'_

--

+

P.D.

(1)

t-p cp2p- + E.S.

(2)

_2

=

c'plg.T( ) D_

d

= P.D.

and

tively.

E.S.

The

present

two-scale

model

for incompressible

the been

constructed similar

through to the

et al. 9. The

compressibility into

into

Fig. function

k -

second-order 1 shows

constants

the

variation

of convective

Mach

to close

the

due

corrections Viegas models, of the number.

is built for the

upon

terms

Its

that

they

and

Rubesin

Speziale vorticity

Sarkar

thickness

vorticity

Vs-Vo

68

we have

the

effects

of

model

has

has

a M_

depen-

s and

Sarkar

k -

been

implemented

w models,

Wilcox

n

n. growth

thickness,

"

model

by Zeman have

l°, into

and

(dU/dy),_,_

coefficient

The

for the

analysis,

a simple

proposed

TM.

a parallel

To model

proposed

respec-

NASA

responsible

In this

terms.

model

shocklets,

in the

equations.

reasoning.

The

eddy

to compressibility,

dissipation

e models,

and be found

turbulence

dimensional

closure

can

(4)

kt

pressure-dilatation

transfer

dilatation

k,

et al. 6. Models

needed for the

energy

successfully and

Duncan

are

7 model

spectral

dence,

(3)

of pressure-dilatation

model

for compressible

effects

Sarkax's

increased

effects

flows,

compressibility adopted

the

of the

kp

) d_]

+ --- + aT, dr'

denote

definition

_T

-

rate,

d6,,/dx,

6_, is defined

as a by

(5)

Modeling The

convective

velocity

of the

stream has

speeds

been

the

and

large

Papamoschou

free

presented

Papamoschou

flows,

structures, and

in Fig.

and

parameter

Roshko

vorticity

to the

Mach

experimental

thickness

been

(d_,/dx)i(O,

is obtained

and

rates

for

normalized

Ua/Uf,p,/pl), by using

a relation,

+ (2)'/') 1+

u

(6)

p.

constant of proportionality is obtained by the present model

formed in the limit of Mc

---* 0. Measured

calculations per-

data are denoted by open symbols in Fig.

the compressibility corrections,the current two-scale model

two-scale model

free

number data

growth

have

flows,

(1-

1. Without

convective

13,

~ The

convective

U,/Uf,p,/&f),

of (d_/dz)_

average

'to the free stream,

to correlate

The

value

of the

13. The

for incompressible

1. The

ratio

relative Roshko

(d&,/dx)(Mc,

values

Flows

as the

of compressibility.

shear

corresponding

are

scale

to be an appropriate effects

Shear

is defined

of sound,

the

compressible

number

dominant

shown

to identify by

Mach

of Turbulent

developed

by Kim

and Chen 14 (KC)

the growth rate only at very high convective Mach

and the

predict a large reduction of

numbers.

With

the inclusion of

the effectsof eddy shocklets and the pressure work, the current compressible twoscale model

predicts a smooth

the convective Mach

number

at high convective Math the convective Mach number Mach

reduction of the vorticitythickness growth rate as increases. The calculated growth rate curve levelsoff

numbers.

number

It should be noted that in the present analysis

of the shear layer is increased by increasing the Mach

of the high speed stream. number,

According

there existsa maximum

layer of the same

fluid with matched

r =. U,/U!

and

For a value of R=O.1 of air is about 2.0. Since tions

and

it is the

the

Reynolds

#t.

That

value

shear

shear

directly varies

stress

the

(

ratio

stress

of the

that

to the

appears of Me. mean

cTy

69

(r) heats

number

of the

mean

mean

flow,

Note

that

working

for a plane

in the

of the

is,

and

specific

Mach

development

as a function

is related

for a plane mixing

is,

)1/=

2

convective

the

number

l--r

=

limiting

Reynolds

influences

see how its peak

7 denotes , the

convective Mach

total temperature. That

tim M --oo where

to the definition of the convective

shear

momentum

fluid. layer

equa-

it is interesting

in the

current

analysis,

flow by a turbulent

eddy

viscosity,

to

r

William

In Fig. 2, the peak Reynolds two-scale model are compared predictions KC model inclusion

of the present are also shown of some

tested,

including

calculations model the

the

trend

with

model

independent of the number the

that

the

value

of the

is similar rate

present Elliott

ratio

of the

shear

to the

predicted of the I5.

the

peak

of the is consistent

argue

shear

that,

two

the

the

an

However,

to each

the

other.

Cases

ratios and the can be made. To further

free

In this case, the

a wider

the

layer

plane mean

of the

computed

model

under-predicts

outer

and

the

Reynolds

shear

2.2

A New

Energy

the the

shear

application

more

detailed

The

of the

measured peak stress,

Model weakly of the

fluctuation

into

a component

with

respect

representing

=

to the the

F, long

shear stress

agree for

very

shear

layer

the

Mach growth

nearly

equal

as the

speed

statement

it is applied

to the

Me = 0.51

and r = 0.36

is

reasonably possible

well with

causes

4 shows

the

The

present 12%.

with

the

Free

The

profile

of

Flows

developed and some

briefly

two-scale

measurement.

Shear

models model

for the

comparison

described

by Liou results of here.

A

TM is.

into three +

are such

by about

are

analysis,

16.

stress.

nonlinear wave energy transfer

made

and Elliott

Fig.

well

of

1 in Samimy

many

Turbulent

/,

+

components, /_

time-average

large-scale

motion,

70 i

are

thickness

any conclusive

3 agrees

zone.

shear

in a NASA

are split ],

The

free

is included

flow properties

there

however,

Case

in Fig.

mixing

0.2,

model,

of air with

Reynolds

Reynolds

to an incompressible

random

layer shown

previously,

region

Transfer

analysis

to the

shear

profile

The model is built upon the and Morris 17. The development its

two-scale

Mach

convective thickness

conditions, before

variation

observation

integral

0.1 and

of operating

to be examined

corresponding

expanded

in the

range

here,

compressible

As described

difference

considered

need

present

calculated

measurement.

small

ratios

fluids,

shear

a fully The

speed

with

working validate

compressible examined.

two

to be

characteristic

decreasing trend of the level of the Reynolds shear stress, as the number is increased, is due mainly to the decrease of momentum rate.

appears

vorticity

with

has

streams.

convective

This

on

stress

predicted

of the

number.

based

stress by the

free

stress

normalized

model

shear

shear

of the

In fact,

Mach

They

Reynolds

However,

up consistently

as a function

variation

in the two-scale

stress.

Reynolds

Reynolds

stress

models

satisfactorily

peak

difference

convective

two-scale

Samimy

that

two-scale

shear

is picked

of the free streams.

as a function and

Note velocity

Reynolds

compressible

number

and of the without the

compressible

Reynolds

the level of the

of the value

present

of the peak

Mach

of the

performs

The

model.

square

corrections show that,

none

KC model,

layers.

that

convective

of the velocity

peak

growth

experiment

the

also shows

and the value

two-scale by

corrections,

shear

absolute

in the

normalized

The

free

increasing

compressible

been

model

by the present compressible Elliott and Samimy 15. The

compressibility The results

of compressibility

the

observed

current

model, without for comparison.

current

under-predicts

"

shear stresses predicted with measured data,

of compressible

decreases

by

forms

W. Liou

(10) component, ]i,

and

Fi, is separated one

representing

Modeling the

residual

fluctuations,

is denoted

of Turbulent

]_.

The

the

large-scale

fluctuation,

{u, The

bold

face

physical

v,

time-average

A(x)

-i

of tl/e

instantaneous

value

form

a complex

large-scale

The

can

(11)

of solution exp

solution

structures,

frequency.

structures

/0 TI Adt

T,

[fi(y),0(y),_(y)]

denote

w the

large

F_ =

a separable

=

of the

and

of the

p}

quantities

properties

wavenumber tions

Flows

by an overbar,

1,= =

For

long

Shear

is assumed,

[i(ax-

whose

real

a (= a_ + iai)

governing

be reduced

(12)

wt)].

equations

part

describes

denotes

for the

to the

Rayleigh

_

o___y 2 } 0

local

equation

the

a complex distribu-

in terms

of

0, d2

{( aU

w ) (

_

d2U

dY 2

(_2)

--

0

(13)

The amplitude, A(x), appears as a parameter in the local calculationfor the _, 0,;_ and is determined separately from the large scale turbulent kineticenergy equation,

ok

__OUi

U_ox I

-

o

ui%ox#

+

e-

(_ < ulu)>)

+

viscous

0 Ox--'_ (ul

where ture.

k = ½_. Note that

scale

structure

represents large-scale val much

sum

of the

spectrum.



smaller

background the

third

numerical spectral

tuations tude

and

needs

The

energy

scales.

fluctuation.

and

detailed

side

For the

regarding analysis the

Crighton

(14),

scales

weakly

can be found

from

the interactions nonlinear

in the

carefully.

Very

the stresses, seeks normal fluctuations

71

wave

little

are

an

described

entire

average

inter-

scale

of the

interaction the

energy

between

models

and Morris turbulent

turbulence

wave

experimental large-scale linearized

17. fluc-

models,

of the

by the

transfer

wave

in Liou

ofthe

terms,

is eventually

nonlinear

information,

- < u_u_ >. mode solution

report

in the

describe

determination

strucof large

of this

time

where

here

weakly

with x°. The

of equation of the

part modes

characteristic

used

importance

to be considered

the

small

analysis

results

is of crucial

Locally,

Strange to the

first

of all the

time-average

than

hand

(14)

of the large-scale kinetic energy

in the

energy

larger

procedure

transfer

theoretical, is available The weakly nonlinear bulent

right

kp defined a short

much

terms

kinetic energy the turbulent

kinetic

presumably,

solution

of different transfer

The

represents

on the

by viscosity.

and the

mode.

fluctuation,

energy,

>)

turbulent k denotes

turbulent

T1 but

expression

dissipated

energy

than

small-scale

of large-scale

The

k denotes the in this analysis of a single

the

< u;u_

the

amplior turEuler

William equations. scale

On

the

structures

Therefore, scales

could

the

hand, be

spatial

of each

with

analysis.

That

of the

determined

is to estimate

associated

nonlinear

extent

as being

here

wavelength

weakly

the

regarded

proposition

as the

by the

other

W. Liou

the

the

modes

by the

size

structures,

large-

wavenumber,,

characteristic

large

of the

at.

of the

which

are

large

predicted

is,

I lw

=

(15) (2r

where

l_, denotes

fer can

the wavelength.

be modeled

Through

dimensional

reasoning,

the enregy

trans-

by

k] This

is the

proposed

This

estimate

scale. that

dissipation

the large is already model for

large

model

structure

unstable local

mode and

characteristic Fig. with

model to

wave

turbulence

turbulent

free

of the

motions.

Therefore,

distance.

l_ is determined

predicted

_ is a similarity

except by Liou

representation

of the

surements in this flow direction. The in Fig.

streamwise 6. After

at the

entire

region

of the

energy

equations

nonlinear

This

wave

self-contained

be important

in the

mixing

mean

average,

layer.

flow 17, the overall

in the

the

most"

most

amplifying

interactions

between

present

streamwise

Since

formulation,

most mean

the

unstable

modes.

velocity

profiles

y -

yi/2 --

(17)

Z0

where the local mean velocity is one half of the free self-similar profiles agree well with that compiled edge

17. They large

resulting

evolution a region

of

coordinate,

low speed

and Morris

the

of the

X

where Y112 denotes the location stream velocity. The predicted

behavior

to the

weakly

only by the locally

evolution

_/=•

observed

plane

modeling

scale

with

theory

the wavenumber fluctuation, this size

may

small

flows.

an incompressible

scale

the

inertia

a closure the

models

shear

to the

of turbulence

be self-contained.

in the

large

scale

characteristic

render

is used the

large

inviscid

provides

allowing

strongly

length

by Pate121

the

most

5 shows axial

by the

interacts

instability mean

nonlinear

againit

the

assumption

dictated

simple

large-scale

is tested

from

classic

in estimating

rather

to other

the

a rate

thereby

weakly

transfer

Lumley 20. Computationally, since of the equations for the large-scale

structure,

applications

The

This

energy with

at

efforts

scales.

of the

of the

future

the

no extra

large-scale

description nature

proceeds

Tennekes and of the solution

involves

the

for the

is in accord

"..

eddies", a part

containin_g

model

of the

of the

scale

spectrum

from

the

this

the

Similar

differences

difference

to the

and the

local

amplitude

of establishment,

layer.

attributed

large

of the

changes large-scale

amplitude

72

uncertainties

reaches

in the

were

also

single

mode

in the

mea-

instantaneous

structures a saturated

is shown value.

Modeling In this from

region, the

to the

the

mean

small

equation

rate of the

of Turbulent production

flow is balanced

scales.

Note

of the

by the

that,

Shear

rate

for the

Flows

large-scale

of energy

present

tfirbulent

transfer

energy

kinetic

from

transfer

the

model,

energy

large

the

scales

amplitude

becomes, dA 2 d-"_" =

G3 and

G4 denote

the

As

A3

G3(x)

normalized

G4(x)

-

positive

definite

(18)

integrals

of the

production

terms

and interaction terms across the layer, respectively. The critical points of the ear equation (18), where dA2/dx = 0, are Aa = 0 and G4(x_)A2 = G3(x_). analyses

by applying

critical

point.

the

Any

Liapunov

small

function

disturbances

method

to A2,

say

nonlinSimple

22 show

that

A1 is an unstable

would

grow

exponentially.

A_

In

fact,

ea'("') " As,

on the

other

A_, would

hand,

decay

is asymptotically

stable.

(19)

A disturbance

about

the

A2,

exponentially,

2 • The

saturated

It indicates simple state of the

analyses

be damped may

A Two-Scale the

for the

for

two-scale

value.

large-scale

structures.

away

this

Consequently,

the

of the

the

of the

equilibrium

any deviation

from

saturation

self-similarity

The

equilibrium of the

of the

wave

flow in terms

structures.

A New Apply

Compressible

model

compressibility

Turbulent

to wall-bounded of second-order

effects

identified

Flows

flows. closure

during

the

models

that

account

development

of the

explictwo-scale

model. Energy the

Transfer

weakly

investigate the effects scale structures. 4.

exponentially.

the development

e.ddy-viscosity

(1)

is an asymptotically state

that

large-scale

Model

(2) Continue

3.2

equilibrium

an indication

of the

As,

(20)

Plans

(1) Extend itly

amplitude,

also show

out

provide

development

Future

3.1

of the

an asymptotically

amplitude

3.

value

instability would

say

Model

nonlinear

for

wave

of compressibility

Turbulent

model

Free

Shear

to compressible

on the characteristics

Flows

mixing

layers

of the coherent

to

large-

References

1 Passot, flows 2 Lee, shock

T. and in the

Pouquet,

turbulent

S., Lele, turbulence

A.,

"Numerical

regime,"

S. K. and

Moin,

interaction,"

simulation

J. Fluid P.,

Mech.,

"Direct

AIAA

73

paper

181,

numerical 91-0523.

of compressible 441-466 simulation

homogeneous

(1987). and

analysis

of

William 3 Kida,

S. and

Orszag,

turbulence," 4 Kida,

J. Sci.

S. and

lence,"

W.

flows,"

Comp.

(1991)

and

Shih,

NASA

TM

(to

B. S., Liou,

7 Sarkar,

S.,

Erlebazher,

modeling

dynamics

in forced

compressible

in decaying

model

for

compressible

turbu-

compressible

turbulent

(1992).

W. and

Shih,

NASA

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75

Shear with

Flows

Applications

and

Historical

Notes,

1.2

•

I

'

I1

0.2

I

"

I

'

Ci

[] []

0.0

,

I

0.0

Figure

1. Variation Present:

I

0.5

of relative without

corrections, Samimy

,

Ikawa and Kubota

I

1.0

growth

compressibility

cn

rate

with

1.5

convective

corrections;

,Papamoschou

,

_

Mach

2.0

number,

, Present:

with

-

0.1.

n

__,

compressibility

,

,Sullinset al;_- ,Messersmith etal;"X ,Chinzei et al;_

,

et al. 76

and Roshko; _

r

, Goebel et al; _

, KC;

and EUiott;

,

0.014

I

'

I

'

I

0.012

0.010

0.008 ..= O'J

"0

0.006

0.004

0.002

0.000

,

, 0.5

0.0

Figure

2.

Variation Present: with

of the r

--

peak 0.1,

corrections;--4---, r = 0.1;

•

Reynolds

without

compressibility

I 1.0

shear

,

stress

compressibility

corrections;---_--, Present:

, EUiott

,

r -

with

and Samimy. 77

convective

corrections; Present:

0.2,

, 1.6

with

r --

compressibility

_ 0.2,

, 2.0

Mach

number.'+,

, Present: .without

r

=

0.I,

compressibility

corrections;-_,

KC:

1.2

•

I

,

It

'

I

"

I

'

!

,

1.0

!

0.8

t

oD-i

¢.)

0.6 c_ G)

0.4

4)

0

2:

0.2

0.0

--0.2

-1.0

",

-0.5

!

i

0.0

0.5

1.0

Z/- Po.s

6_

Figure

3.

Comparison Samimy

of mean

(1991).--,

velocity Present;

profiles *

78

for

the

shear

, experiment.

layer

in Case

1 of EUiott

and

0.010

•

I

'

I

"

I

•

0.008

0.006

0.004

0.000

-0.002 -i.0

Figure 4.

'

Comparison Samimy

l -0.5

of Reynolds

(1991).--,

,

, 0.0

shear stress Present;

•

,

for the shear , experiment.

?g

, 0.5

layer

, 1.0

in Case

1 of Elliott

and

i

-0.20

-0.10

0.00

I

,

0.10

0.20

r/

Figure 5. Mean

velocity profiles.---, z = 3.82;-m

8O

4.43;-=--,

5.99;

•

, Patel 9.

0.15

H

I

I

0.10

0.05

0.00

f

0.0

,

I

100.0

I

,

200.0

f

,,

4-00.0

300.0

_/oo }

Figure

6.

Variation

of the

wave

amplitude

81

with

streamwise

distance.

F