Ico__,.)Mr'l - NASA Technical Reports Server (NTRS)

/'//--._/_' NASA

Technical

ICOMP-94--21;

Memorandum

106721

CMOTr-94--6

A New k-e Eddy Viscosity Model for High Reynolds Number Turbulent Flows-Model Development and Validation

T.-H. Shih, W.W. Liou, A. Shabbir, Z. Yang, and J. Zhu Institute for Computational Mechanics in Propulsion and Center for Modeling of Turbulence and Transition Lewis Research Center Cleveland,

Ohio (NASA-TM-106?2I)

A NEW k-EPSILON

EODY VISCOSITY REYNOLDS NUMBER

MODEL (NASA.

MODEL FOR TURBULENT

DEVELOPMENT Lewis

AND

Research

N95-I1442

HIGH FLOWS:

VALIDATION Center)

Uncl as 32

P G3/34

August

0022322

1994

Ico__,.)Mr'l National Aeronauticsand Space Administration

A New

k-_

Eddy

Turbulent

T.-H.

Viscosity Flows-Model

Shih,

W. W.

Center and

Model

NASA

Liou,

A. Shabbir,

Reynolds and

Z. Yang

of Turbulence

for Computational

Lewis

High

Development

for Modeling

Institute

for

Research

and

Validation

and

:l. Zhu

Transition

Mechanics

Center,

Number

in Propulsion

Cleveland,

OH

44135

Abstract

A new tion

and

model

k-s eddy

a new

lation

Schwarz'

a set

are examined

include:

a mixing

layers

flows.

model

with

The

model

from

the

standard

that

the

present

model.

shear

(i) rotating

homogeneous

predictions k-E eddy

and

a pressure

round

viscosity

is a significant

model

in this

The

positivity

that

shear

flows;

(iii)

a channel

with are

improvement

and

(iv)

available

the

backward experimental

the

present

and

stresses

model

The

plate step

data.

k-s

with

flows shear

flat

facing

new

formu-

Reynolds

for comparison. standard

The

viscosity

of flows.

flow,

equa-

mean-square

(ii) boundary-free

also included over

paper.

eddy

of normal

We find

rate

of the

new

well for a variety

jets;

dissipation

equation

number.

gradient;

are compared

dynamic

stresses.

perform

model

is proposed

the

can

planar

of a new

the

Rcynolds

coefficients

without

model

on

constraints;

for turbulent

layer,

and

turbulent

consists

formulation,

is based

realizability

inequality

of unified

including

the

which

viscosity

equation

at large

on

model,

eddy

rate

fluctuation is based

and

ary

realizable

dissipation

vorticity

viscosity

that flows

bound-

separated The

results

It is shown eddy

viscosity

1.

Introduction

The eddy

main

viscosity

viscosity well

task

formulation

model,

for

viscosity

does

not

known

anomaly

to the

model

viscosity eddy

not for flows

the

about

but cases

appropriate

rate

to predict

viscosity

model

formulations

for both

significantly

improve

exact

the

be

rate

The

dissipation

rate

equation

can

standard

model

shear

rate

a round

these

purpose

of this

viscosity

be written

as,

rate

equation the

well-

jet is mainly

due

of the

in the

study

k-_

eddy

existing

k-e

is to propose

and the eddy

k-_ eddy

standard

ability

deficiencies

quite

or a massive

For example,

the

eddy

by the

dissipation

to improve

k-e

performs

is overpredicted

rate equation

of the

mean

jet versus

flows,

an appropriate

dynamics,

for turbulence.

In order

performance

a high

of a planar

turbulent

The

fluid

standard scale

removed.

model

dissipation

the

is to provide

equation.

viscosity

length

complex

rate

with

eddy

equation.

should the

the

spreading

model

in computational

In addition,

the

dissipation

model

The

flows

in these

give

dissipation

used

formulation.

always

viscosity

is widely

layer

because

a k-6 eddy

and a model

which

boundary

separation, eddy

in developing

viscosity

new

that

can

model.

2v

where

e = _,uijui,j,

t and

x_.

term

ve,li,

used

for

scales not

All

terms

are new

of turbulence,

energy terms kinetic resulting

equation. which

are

energy model

- 2L'2U_,jt, Ui,:ik

2vui,kuj,_uij and

on the

hand

side

Thus,

they

must

is extremely

difficult.

That

equation which

is, the

assumed

has

dissipation

rate

eddy

equation

(la)

for the

derivatives the

viscous

be modeled

before

this

equation

with.

which

in the

similar

one

to that also

time,

related Eq.

creates of the

k/E.

and

in the

this

following

be

small

is usually

a simple turbulent

dissipation

With

can

to the

and

to

diffusion

(la)

has generation

to the production

be written

are

literature,

Instead,

equation

can

respect

except

unknowns,

turn-over

with

(la),

Therefore,

rate

- 2L'_-_,,]-u_,kU_,k

of Eq.

a structure

to be proportional large

new

to work

dissipation

by the

( ),i stand

right

of these

equation

divided

( ),t,

Modeling

as a useful rate

-

unknowns.

applications.

dissipation

- 2v_,ku_U_,_

e' = uuijuij

the

considered

- 2v_U_,k_

model kinetic

destruction of turbulent

assumption,

the

form:

g2

+

=

-

-

e

-

Ce2-_

(lb)

Eq.(lb) used

is the in

Eq.(lb)

standard

various have

form

turbulence

also been

closure

proposed

lent

flows 1-4

and

rate

equation

based

on the

between

eddies

interactions scale

has

form

from

from

large

free

also

for the

inal

_ equation

(lb).

dissipation but

This

is achieved

mean-square

vorticity

a model

dissipation

at large

Reynolds

The

eddies

rate

and

was

his new

the

more

can

which

statistical

energy

is not only

physically

more

Once

equation

the

be readily

the

standard for the

dynamic obtained

a new

related

dynamic

the

model

to the

orig-

equation

equation

for _

by using

time

turbulent

dissipation

equation

to the

transfer

of some

of deriving

than

due

is of a different

possibility

robust

turbu-

for an inverse

prediction

of

a dissipation

transfer

the

a model

wiwi.

energy

of the

widely

versions

s proposed

equation

in the

been

in near-wall

E equation

physics

we explore

developing

equation

spectral

successful

which

and

fluctuation rate

mimics

equation

by first

Lumley

has

modified

for example,

A new transport with

which

several

applications,

sizes.

study,

simpler

equation

addition,

of non-equilibrium

model

present

also

In

in conjunction

to small

rate

flows 5. Recently,

of different

This

fiows 6. In the

form

concept

of Eq.(lb).

eddies

schemes.

turbulent

suggested

dissipation

for different

in rotating

been

that

shear

of the model

of the

is modeled,

relation

E = vwiwi

number.

standard

eddy

viscosity

formulation

for incompressible

turbulence

is

2

-u uj

=

(2a)

+ T(U ,j +

c. = 0.09 It has

been

known

mean

strain

rate

can

become

strain

(e.g.,

negative

realizability, mean

for long

the rate.

this model

Schwarz' coefficient

of C_, is quite

sublayer

of a flat

0.09

in the

0.05

in a homogeneous

new

formulation

this

paper.

inertial

shear

for C,,

following

which

sections,

flow was

not

different

boundary of Sk/e suggested

stresses

layer

in each layer

by Reynolds

the

and

the

must

7 and

example,

above Shill

development

To insure

be related

= 3.3, and

to the

of large stresses

be violated.

For Sk/e

case

normal

and homogeneous

case.

in which

= 6. According

describe

in the

can

be a constant

on boundary

we will first

non-realizable because

for shear

C_, must

value

the

become

S = _),

inequality

the experiments

show

will

> 3.7 where

In fact,

also

In the

that

Sk/c

and

model

that

(2c)

to the

shear C,

flows

is about

C_ is about

considerations,

et aI. s, is adopted

of a new

model

a in

dis-

sipation The

rate

performance

rotating

of the

jets),

backward

new

a channel

facing

step

equation

The

equation

exact

and

mean

boundary-free

boundary

new

dissipation

viscosity

with

flows

of flows

(e.g.,

a mixing

and without

pressure

formulation. which

include

layer,

planar

gradients,

and

rate

equation

wiwi

"

TT. ( OJiOJi _

•

+ v -y-m

the fluctuating

vorticities

eddy

in a variety

shear

layers

new

for wlw_ is

--y-),t

Ui are

of the

flows.

for

OJiOJi

ui and

development

will be examined

flows, flow,

of the

the

model

separated

Dynamic

where

then

shear

Development

2.1

and

homogeneous

and round

2.

equation,

which

are

•wiwi.

+

-

+ wiwjuid

-

and

defined

1

-

= v(--5-

mean

+ w- U ,j

(3)

uwi,jwi,j

velocities,

and wi and

fli are

the fluctuating

by

wi = eljku_,j,

(4)

_i = eiikUk,j

and 1V. Tennekes first

two

transport

and

Lumley

9 clearly

terms

on

right

vortex

9li_i

with

the

the

fluid,

Reynolds

The

it appears

source

production

hand

same

sign,

fourth with

produced due

and

opposite

by

mean

to fluctuating

respectively. numbers,

the

sixth

and the

meaning

of each

viscous

transport

the

term

is the

vorticity.

This

it will

either

the

sign in the

vortex

and

third

represents

vortex

Tennekes

physical

represent

mean

hence,

term

the

side The

stretching

the

simultaneously. because

described

of bJ_w_, respectively.

fluctuating for

the

1

equation

have

also

shown

terms

sixth

dissipation that,

7./,3 _3/2

._ O[,--_.ut

seventh

are the \

)

is produced

by

equation and and

term

wiwi _'li_i,

represents

terms

to the

largest

The

turbulent

wiwi

fifth

due

the

_i_i

at sufficiently

in Eq.(3) ---

uwi,jwi,j

The

in Eq.(3).

in the

between

and

of order: wiwju_,j,

which

or decrease

for _i_i.

the

and

appears

exchange

The and

seventh

term

vorticity

stretching

term

increase

stretching.

Lumley

source

term

are

the

viscosity

high terms

of

turbulent and

are

All the

remaining

of order Rt

-

and

(u3/l

terms

on the

_) or (u/1)_R_.

is the

turbulent

length

scales

of turbulence,

in Eq.(3),

then

As pointed

out

Reynolds

the

(_i0)i

hand

except

analysis,

number,

TT. ( Wiogi

and

Lumley,

the

l are the of order

be described

1

term,

denotes

terms

j,_+ v_- V),_ = -_(_),j

by Tennekes

second

u and

If the

of _viw_ would

the "O"

and

respectively.

evolution

_

side,

In the above

ul/u

kept

right

are

smaller,

order

either

of magnitude,

characteristic (u s/l 3)Rt

by the

velocity

or larger

following

were

equation,

+ w,_ju_,j- .w_.j_,,j

at very

large

Reynolds

numbers,

(5)

Eq.(5)

becomes,

w_wjui,5 = vwi,_w_,j Or

equivalently,

wlwjui,j

is always

created The

by the

vortex

eddy

production

sizes.

equals

positive.

vortex

In

tends

However,

this

process

effect

of viscosity.

microscale

which

corresponds

2.2

This

can

which

verified

Modeling

of the

dynamic

Modeling

of

from

wiwjui,j.

that

to the

end

length

and

at a certain

that

the

scale

indicates

there

terminal

for the

the

a broad

of eddy eddy

size

size

derivative

term

length

of fluctuating

to create

level

that

is a new

derivative

size of eddies

We expect

be easily

relation

it indicates

the

must

to the

This

is related

to reduce

smoothing

wi,j.

addition,

stretching

stretching

dissipation.

(6)

scale

vorticity. spectrum

because

is the

of

of the

Kolmogorov

of fluctuating

vorticity

Eq.(6).

equation

for

We first

define

wiwi

a fluctuating

anisotropic

tensor

bit using

wiwj wiwj

1_ .

b_ = _,:,.,,_ -_,_

(7)

,_iwju_,_ = b'5w_wkui,j

(8)

then

We

expect

that

that

the

anisotropy

the

anisotropy

the

b_

vortex

stretching

b_ is mainly may

be assumed

tends

due to the

to align anisotropy

to be proportional

vortex of the to the

lines

strain

sij

where

s_j = (u_,j + u_,_)/2 5

the

fluctuating

b,3 _ --, 8 s = (2s_js_ff/:,

with

strain

strain rate

sij.

rate rate;

That

and

hence, is,

(91

This leads to wiwyui,j

o¢ WkWk iiSi Ui"---'---_ 0¢ w_w_

S

(10)

8

If we further

assume

that

w_wk

and

(2sijso)

1/2 are well

correlated,

we may

write

(II)

w_w_ui,_ 0¢wkwk ¢_ Noting

that

wiwi

= 2_

at large

Reynolds

wiwjul,j

Eqs.(ll)

and

should

(12)

both

numbers,

o¢ wkwk

indicate

that

_-_Wi

the

we may

also

write

-- -WkWk WiWi

model

for wiwjui,j

(12)

is of order

(u3/t3)R_/2

as it

be.

Modeling must

be

terms

in

of

of order

wiwjui,j

-

(u_/£3)Rt,

Eq.(5).

because

Therefore,

the

iv_wk

wiwi/_)

in such

wkwt,

w_w_/_)

by an order

be related

to the

following

_'wi,jwi,j. that

model

a way

two

Eq.(5) is the of

that

their

of p_/2.

This

indicates

order

of the

-uwidwi,j difference suggests

that

magnitude

must

cancel

is smaller that

wiwjui,j

the

than

sum

for

_'wi,jwid the

wi, wk

¢_sij

(or

w_wk

_sij

(or

of these

two terms

wkw} -wiwi

k(_

both

the

ratio

of s to

u 2) denotes

the

turbulent

As a result,

the

dynamic

can

terms:

w_w_ S, since

other

S and

the

equation

V

ratio

kinetic

of k/u

energy

and

for fluctuating

wiwi. U / wiwi _ --5-- )'_+ j'--T ''j

(13)

_ + to

_

S is the vorticity

mean can

1 = --2(_)'_ WkWk

are

of order

strain

R_/2.

rate

be modeled

Here,

(_). as

+ Cl_-_S WiWi

(14)

V

Note

that

the

denominator

number

turbulence

it there

in case

This

also

reflects

since k vanishes the

fact

of the the

term

last _

somewhere that

the

term

in Eq.(14)

is negligible in the

parent

term

flow field of the

should

be k/u

compared

to k/u._However,

to prevent model,

for large

unnecessary

Eq.(12),

shows

Reynolds we keep singularity.

no singularity

anywhere

in the

in Eq.(14) model

2.3

flow

models

for either

It should

the last

two terms

individual

Modeling

of the

Noting readily

field.

that

obtain

also

be pointed

in Eq.(5)

out

that

as a whole

the

sum

of last

and should

not

two

terms

be viewed

as a

term.

dissipation

at large

a modeled

rate

equation

Reynolds

number

dissipation

rate

_ = _'wiwi

and

multiplying

Eq.(14)

by _, we

equation, g2

e,, + uj e,j = -(uje'),j The

model

as the

coefficients,

Reynolds

body

number

rotation

ing,

wiwjuij,

the

other

the

calculation

the

evolution

example,

rotating

of e through,

in a decaying must

shear

both

the

flow, the

"source"

two

The

term

types

model

stresses

do not

will

more

robust

used

in conjunction

with

than

the

posed that

by

Reynolds

the

present

stretching

and

than

s which form

present

is based

of the model

dissipation

terms

on the

the model

rate

This

rate

appropriately.

right

rate

especially

of spectral

compared

case

rate

energy describes

For

side of Eq.(15) of homogeneous with

time

be positive.

C2.

Eq.(15),

term.

and the

The

dissipation equation

Reynolds rate when

behaves with

is similar transfer.

so

In fact,

C1 and

the

in

determined.

S normally

term

affected

will also affect

increase

for cases

to

k, as shown

hand

equation,

model

since

equation

be easily

"source"

dissipation

"production"

concept

7

4.1.

coefficients

present

schemes,

of the

dissipation

the

is the

calculations,

form

say

C1 must

dissipation

closure

more

hence

Eq.(lb),

standard

field,

For the

and its dissipation

model

in numerical the

be positive.

Consequently, the

on the

for determining

equation,

second-order

stresses

In addition, Lumley

rate

term

be positive,

present

in Eq.(15).

tion

conditions.

be

dissipation

energy

be used

the

last

C2 can

weak

stretch-

be substantially

flows in section

C2 must

must

turbulent

by solid

vortex

is rather

will first

number

be affected

of fluctuation

effect

of C1 and

only the

kinetic

will

between

appear

shear

hence

in Eq.(15)

of flows l°'n

difference

standard

turbulent

this

of the Reynolds C2 may

reduction

of the

signs

turbulence,

C1 and

stresses

change

homogeneous

be negative,

the

Reynolds

say, k. The

grid

that

however,

in a substantial

of the

and

We note

by BardinaS;

(15)

k+

to be independent

through

For example,

result

is non-zero

these

shown

mechanisms. and

large.

on turbulence

as was

E - c2

C2, are expected

becomes

imposed

by rotation

that

C1 and

+

poor to that

equait is better initial pro-

We believe

turbulent

vortex

Eq.(15) the

turbulent

turbulence

can

be applied

transport closure.

be described

term

Here,

in the

in conjunction

next

(_-_ui),i

we apply section,

with

needs

any level

of turbulence

to be modeled

Eq.(15)

differently

to a realizable

and where

eddy

(6-_ui),i is modeled

closure;

however,

at different

viscosity

model

levels which

will

as

(16)

= The

3.

model

coefficients

Realizable

Shih form

C1, C2 and

eddy

et al. s proposed

represents

ae will be determined

viscosity

later.

model

a realizable

an isotropic

eddy

Reynolds

viscosity

stress

algebraic

equation

model.

Its

(17.1)

= _,T(U_,_+ U_,_)- -_k&_ k2

UT

the

inertial other

coefficient sublayer

hand,

experiment

C_ is not of a channel

a constant.

and

shear Corrsin

uau_2

n.

Shih

et al. s proposed

experimental

layer

flow

as well

suggest

C_ = _____/_k ov /e _y

Based

on the

(a

-- 1,2,3)

(a

= 1,2,3;8-

that

which

realizability

as DNS C_

data

on the

=

0.09.

On

the

is about

0.05

from

the

conditions:

(18)

< 1

_ and

The

flow,

u__>o

Reynolds

(17.2)

or boundary

for a homogeneous of Tavoularis

_ Cf * g

--

linear

model: 2

-_uj

Here

of

the

following

1,2,3)

formulation

for the

coefficient

of Cu:

1 Cu = Ao + A_U (*)k-

(19)

E

In the

formulation

of Shih

et al. s,

U(*) = _f s_i&i + _ _'lij = flij

-

2eijkwk

flij

-

eij_w_

-- flij

(20)

m

where

_ij

velocity

is the

wk.

mean

The

rotation

parameter

rate

viewed

in a rotating

A8 is determined

reference

frame

with

the

angular

by 1

A8 = V_cos ¢,

¢ = _arccos(v_W)

(21) W-

SijSj_Sk_

_3 Calibration

of the

of Eqs.(17),

(19),

(20)

a realizable

model.

Ao

such

as a homogeneous

layer

flow

is a constant,

Cu -

in hope This

For the

and

The

that

sublayer.

which

C_, = 0.09.

only

the

the

value

the

model

1 which

shows

that

to the

standard

form

can

be

closer

to the

of the the

and

form

boundary

layer

homoge,

shear

us go back

to the

value

of C_, also

the

of the

log-law in the

hence

is

for simplicity simple

we choose

= 0.09

Corrsin

flows,

a boundary

of the inertial

inertial sublayer.

11, Eq.(19),

with

A0 = 4.0, gives

of 0.05

that

of the

for

than both

produces

component

the

reasonable

flows

standard is listed

in

b12 compared

coefficients

present

-0.149 -0.274

-0.149

-0.149 -0.142

modeled

k and ¢ equations,

b12

b12

standard

b12

e,t + uje,j = (

•

one

the

b12 (_-_/2k)

1. Anisotropy

k,, + r_jk,_= (_k,j),j

and determine

by

to reproduce to C,

and

If we assume

Here,

exp.

let

is A0.

formulation

of C_. Table

Now

viscosity Eq.(18),

flow.

experimental

present

eddy

constraints

layer

corresponds

anisotropy

new

be calibrated

able

flow of Tavoularis

component

Table

realizability

or a boundary will

The

coefficient

of A0

to A0 = 4.0 which shear

A0.

undetermined

flow

is much

The

satisfies

shear that

leads

coefficient

(21)

then

homogeneous

0.06

model

-0.18

(22)

- u-_V_,j - E

(23)

e,j),j + ClS _ - 6"2k + v_

in Eq.(23).

~

Calibration bulence

at

large

of the Reynolds

model number,

coefficients the

equations

C1,

C2

and

for turbulent

a_.

In decaying kinetic

energy

grid k and

turits

dissipation

rate

6 are g2

Let

the

following

equations

can

be obtained

from

the

a=n+l,

k and e equations: n+l

C2=_

(24)

n

Experiments choose data

1° show

that

C2 -- 1.9 which of homogeneous

which

is found

strain,

the

decay

corresponds shear

exponent

n varies

to n = 1.11.

After

flow n and boundary

to be a simple

function

from

C: is chosen,

layer

of the time

1.08 to 1.30.

ratio

of the

we

we use the experimental

flow to determine

scale

In this study

the

coefficient

turbulence

to the

C1 mean

7: C1 -

max{0.43,

(25)

5 + rl }

where Sk rI = _,

S = _/2Sij

S_j

g

The

value

relations

of ae will be estimated hold

in the

inertial

using

the log-law

1

-

_r

dissipation

flow.

The

following

rate

log u__yy + C

K

---'l.KO

the

layer

sublayer: U

Analyzing

in a boundary

,_

V

2 U.r ,

equation

(26) _u---_OU

,_,

oy

in the

log-law

region,

we obtain

/¢2

ae =

where

the

von Karman

constant

c2

_ = 0.41.

cl The

model

= 1.20

coefficients

(28)

are summarized

.

Table

Uk

Ue

1.0

1.2

2. Model

coefficients

C2

C1

1.9

Eq.(25)

10

Ao

Eq.(19)

4.0

in Table

4.

Model

applications

The shown free

results

in this shear

dients,

section.

flows,

and

different initial

are made

cases

(which

conditions

value

k0, with

the

present

the

able

for

kinetic

to pick

evolution

trend

of these

and

f_/S

-

which with

4.2

the

growth

is already the

LES

the

other

rate

of turbulence

as it shows

Boundary-free

a planar profiles

and from

the

the

flows

the

present

a round model

jet.

not

the

Figures

predictions

gra-

standard

LES

not.

more

rate

the case.

between

hand

decay

the

of the

and the

and

the

11

rate

of the

of the

turbulence

present

model

is able

present than

the

model

the

_2/S

the

For

to pick

as

case

of

the

with

the

a result

agreement

of the for

up

is not

time.

for a mixing

comparisons

measurements

= -0.5.

case,

is in reasonable

performed

the

as it gives

no rotation

energy

is

kinetic

LES

on turbulence

kinetic

were

and

ske model.

of rotation

turbulence

= 0.5 and

model

model

1 (d) compare

rate

present

various

shows

present

1 (c) and

as it did for the

4 show

1 (b)

growth The

both trends

growth

ske models

2, 3 and

case.

case the

of _2/S

effect

energy

show

Figure

the

= 0.296.

this

cases

is a lot better the

that

Figures

The

the

data.

rate

The

by its initial

For

models

LES

shows

_/S=0.50).

and eo/Sko

-- 0.0.

hereafter)

to the

does

and

et al. 5 for four

normalized

of f_/S

no rotation

that

show

other

shear

using

The

kinetic

On the

the

of Bardina

energy,

case

closer

shows

it still

does

are

pressure

and

turbulence

kinetic

for two

agreement

model

data

LES

cases,

model

(ii) boundary-

without

_/S=0.25,

by ske

ske model

no rotation

ske

known.

Calculations

the

flows,

present

simulation

St for the

over

energy

the

although

as it is for the -0.5

cases over

good

same

kinetic

of the

to isotropic

= 0.25.

the

with

results

_/S=-0.50,

model

f_/S

while

and

eddy

e (denoted

is increased

trend

is decreased

time

present

case

energy

up this

first

energy

the

the

layers

of turbulence

k -

the

of turbulence

For the

this

with

large

correspond

evolution

standard

LES,

comparisons

turbulence

the

non-dimensional

and by

the

shear

turbulence

and experiments.

_/S=O.O,

cases

homogeneous

The

LES

new

flows

with are

in all these

compares

exhibited

DNS,

the proposed

boundary

flows.

shear

comparisons of _/S

and

step with

using

(i) rotating

flow

homogeneous

1 (a)

the

a channel

are compared

Figure

the

include

backward-facing

Rotating

The

flow calculations

These

(iii)

(v)

k - _ models

4.1

of turbulent

layer,

self-similar

mixing

layer,

planar

and round

Reynolds

shear

the

results

jets,

respectively.

stress

and

are shown

In these

the turbulent

kinetic

in a self-similar

that

y0.1, Y0.5, and of the

velocity agree

free

profiles well

with

predictions

ske model.

jet

are

are

of the

mixing

the

present

For the

model

round

the experimental

spreading

rates

while

of these

model

yields

round

is always the

anomaly

3. The

Case

2 shows

present

model

model,

with

The

the

is slightly

made

16 and

15.

by the

current

predicts

a much

wider

of the turbulent

level

distributions

than

and

the

overall

for the

the

model

given data.

the

measured

in Figure agree

distribution.

and 4. The

well

distribution. stress

by The

predictions

model

shear

planar

measurements

than

l:todi lr and are shown

predicted

prediction

better

experimental

the

mean

gives

predictions

lower

between

the

to

or the ske model

predictions with

velocity

that

however, stress

compared

well

centerline

Fielder

The

Hekestad

agree

are

shear

levels. are

mean

with

Significant profile

over the

The calculated

with

measurements

and are shown

in Table

predictions

than

the ske model;

especially,

well-known

than

measurements)

layer,

are compared

better

smaller

mixing

jet

the centerline

of planar

Table

model

the ske model

flows

the

local

Figure

present

peak

14, and

velocity

in the

the

For the mixing

of the

and the Reynolds

comparisons

mean

The

predictions

ske

and

the ratio

by either

for their

at the

the

of both

rate

contradicts

level

of the

spreading jet

energy

the

is also achieved in terms

present

and

jet,

data,

ske model

The

of Pate112.

of Wygnanski

distributions

improvement

data

model

velocity

as

respectively.

predicted

The

are presented.

mean

y0.5

where

layer

1_, Bradbury

energy

measurements

profile

3.

for the

Y0.9 - y0.1

and 0.9,

true

profiles

r/defined

y -

locations

0.5,

kinetic

Wyguanski

kinetic

values.

0.1,

is especially

in Figure and

turbulent

the

experimental

This

shown

of Gutmark

the

stream

of the turbulent

the

both

Yo.9 denote

the

energy

coordinate

rl =

where

figures,

and that

round

(i.e.,

the

measured

jet,

but

the

of a planar

is removed spreading

jets

the

spreading

model

prediction

completely.

rates

of turbulent

free shear

flows

measurement

ske

present

layer

0.13-0.17

0.152

0.151

planar

jet

0.105-0.11

0.109

0.105

round

jet

0.085-0.095

0.116

0.094

12

rate

3.

of a

usually

4.3 Channel flow and boundary layer flows Turbulent were the

channel

calculated present out

down

to y+

values

were

were

The shown

used

profile

in Figure

5. This

the present the

number

up

Wieghardt

skin

19.

Both

flow

formance gradient

7 shows

the

flat

and

model

Since

the integration

was

calculations.

channel

At

y+

= 80,

channel

flow and

flow

Re_-

direct

boundary

ske

model

the

give

a slightly

the

layers

with

wall

-- 395

is

simulation.

DNS

data.

with

the

Figure Reynolds

experimental

good

better

at

numerical

well with

is made

gives

flows.

flows.

plate

the

wall,

turbulent

reasonably

comparison

present

in the

turbulent

agree

model

the

by Kim is using

for the

Here,

present

the

results

favorable

to that

of the

studied pressure

better

for the

pressure

gradient

The

21 and

studied

Herring

and

gradient.

ske model.

by Bradshaw

in Figure

results

agreement

prediction

of

with

for

the

boundary

and

flow 2°, which

present

model

gives

layer

under

boundary

turbulent

by Samuel

8 and Figure

The

turbulent

the

Norbury

boundary

Joubert

9, respectively.

layer

22 were

In both

is a boundary

compariable adverse

under

the

pressure

increasingly

also calculated.

cases,

per-

The

present

adresults

model

gives

predictions.

Backward-facing

The through

calculations

other

24) with

calculations finite-volume

discretized terms

by

flows

of the present for two

(KKJ-case

conservative

other

step

performance

to benchmark

the

the

under

are shown

were

coefficient

layer

gradients

for wall bounded

from

wall,

pressure

development.

layer

the

calculated

model

for the

developed

ske model

-" 16000.

Overall,

Figure

4.4

and the

friction

to Reo

experiments.

verse

model

to the

boundary

fully

with/without

away

conditions

turbulent

flow was

flows

than

boundary

for 2D

flows

of the present

= 80, rather

for the

layer

for turbulent

as the

velocity

6 shows

layer

used

boundary

performance

is proposed

functions

Both

to test the

and

model

carried DNS

flow

for complex

backward-facing larger

step

of separated procedure.

a second-order

by the

model

standard

step

flows,

one

both

of which

flows.

The

calculations

accurate

convection and

differencing

13

terms

bounded scheme.

flows

(DS-case

height,

The

central

recirculating

have

2_) with been

were of the

is demonstrated

extensively performed

governing

differencing Sufficiently

smaller

scheme fine

and used

with

a

equations 25, and grids,

all with

201x 109 points in the DS-caseand 199x91 points in the KKJ-case, were usedto establish numerical credibility of the solutions. The computational domain had a length of 50 step heights, one fifth of which was placed upstream of the step. The experimental data were used to specify the inflow conditions, the fully-developed flow conditions were imposed at the

outflow

boundary,

viscous

sublayer

Figures

10-14

the

near

velocity

quantities

free-stream

standard

the wall.

compare

and the mean All

and the

the

Table

were

velocity

coefficients of the

A new

normalized

of the

reattachment

point

to bridge

the

the

lengths.

bottom

downstream

wall

locations.

experimental

reference

locations

ske

present 6.02

6.35

7.50

of the

size

of the

separation

that

the

overall

performance

buble,

the

skin

friction,

and is better

of the

present

model

in this

paper.

It consists

the

pressure than

that

Remarks

k-_ eddy

viscosity

model

is based

eddy

viscosity the

in various

effect

of mean

backward

standard

anomaly

step

equation

especially,

and

The

jets

is expected when

dynamic

rotating layer results

in almost

round

viscosity

it is used

for fluctuating

3 ensures stresses.

show all the

with that

The

cases

is completely to enhance in conjunction

and

The

new

model

vorticity.

The

tested.

The

numerical

stability

with

advanced

more

gradients;

performs

well-known

In addition,

is tested

boundary-free

pressure

model

contains,

model

flows;

without

present

removed. the

model

and

present

shear

the

of a new

realizability

homogeneous flows

14

formulation.

equation

on turbulence

boundary

flows.

eddy

in Section

including:

k - e model

of planar rate

rotation

and flat

facing

on the described

flows

channel

is proposed

a new realizable

formulation

benchmark

flows;

calculations,

the

4.99

equation

dissipation

h and

6.26

rate

rate

at three

7=t= 0.5

dissipation

the

height

along

DS

and

than

distribution

KKJ

equation

and

step

used

of the reattachment

profiles

measurement

rate

shear

pressure stress

by the

4. Comparison

dissipation

as well,

the

comparison

26 was

model.

5. Concluding

new

the

approach

U_f.

suggest

ske

4 shows

skin friction,

Case

comparison

function

as well as the turbulent

Table

The

wall

the

better spreading

new

in turbulent closure

model flow

schemes,

such as

second

dissipation that

rate

the

initially

order

initial

closures.

equation decay

isotropic

We have

into

the

behavior

rotating

also just

finished

LRR 27 second

of k and

order

s and

homogeneous

shear

implementing

the

closure.

effect

flows

the

present

Preliminary

of rotation

model

results

on both

show

k and

6 for

are well captured.

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1974,

5.0

5.0

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/

3.0

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/

3.0

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/

///

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2.0

2.0 ._

j,j,t

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1.0

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lo.o

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ske model

/

3.0

/

i'

/

2.0

/

/

/

/

1.0

.- ' -q r'l 13 13

evolution

/

/

/

j

8:o ' 8:o' ,o.o

°°o.o' £o ' 4:o' e:o' 8:o ' lO.O

$t

1. Time

/ / /

f

°°o.o' £o ',:o

Figure

/ I

present model

/ j

2.0

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-----

/

3.0

I

n LESdata,Q/S=-0.50

/

present model

1.0

I

/

D LES data, D/S= 0..._0 ----ske rnodel

4.0

of normalized

St

turbulence

kinetic

18

energy in various

rotating

homogeneous

shear flows.

1.2 (a} 1.0

"_::_

---

[] Patel ske model presentmodel

0.8

U/Ue

0.6

0.4

0.2

0.0 -0.5

-1.0

0.0

0.5

1.0

1.5

T_

0.012

0.04

(c) Co)

0 o

0.03

n Patel --s,4emodel _ presentmodel

n /f_o_

_

B/C\\

0.010

17Patel

0.008

k

-uv

0.02

0.004

/

0.01

o o

0.006

17

0.002

't_\\

c 17 0.00 -1.0

i

•0.5

I

,

0.0

I

0.5

.

0.000

.

1.0

1.5

-1.0

-0.5

0.0

Figure Reynolds

2.

Self-similar shear

profiles

0.5

1.0

1.5

kinetic

energy;

(c)

"q

11

for

a plane

mixing

layer.

(a)

stress.

19

mean

velocity;

(b)

turbulent

1.2 (a) 1.0

A Gutmark &Wygnanski 0 Btaclbury 0 Heskestad

0.8

U/Ue

0.6

0.4

0.2

0.0 0.00

0.05

0.10

0.15 y/x

0.12

0.20

0.25

0.30

0.04 (b)

0.10 A

0.08

A

A

_._D '_"

,,,_ rm

A O 0 -----

/ Gutmark & Wygnansk Bradt_jry Heskestad ske model

_

presentmodel

(c) /,, Gutmark & Wygnanski [] Bradbury ----ske model

0.03

present model

--at

uv 0.02

0.06

0.04 0.01

[

%,

0.02

I

0"000.00

i

0.00 0.00

-

0.10

0.20

0.30

0.;5

I

I

I

/

0.10

0.15

0.20

0.25

0.30

y/x

Figure stress.

3. Self-similar

profiles

for a plane

jet.

(a) mean

velocity;

2O

(b) turbulent

kinetic

energy;

(c) Reynolds

shear

1.2 (a) 1.0

_\\

A Wygnanski& Fielder D Rodi

0.8 ----_

\ UIUe

ske model presentmodel

0.6 \%\ 0.4

;-,.,

0.2

0.0 0.00

, 0.05

, 0.10

, 0.15

, 0.20

, 0.25

y/x

0.035

0.15

(c)

0.030 A Wygnanski& Fielder [] Rodi ----- ske model presentmodel

mmmm_.

0.10 oo

o_%

Z_Wygnanski& Fielder r-t Rodi ----ske model ,,,--..,.

0.025 !

_

/

\

%

",,

0.020

%%

presentmodel

0.015 0.05

0.010

0.005

[%',,, P

0.00

i

0.00

0.20

0.10

I_._,._

0.000

I

0.00

L30

0.()5

0.10

0.15

y/x

Figure

4.

shear

stress.

Self-similar

profiles

_.,_°

I

0.20

0.25

ylx

for

a round

jet.

(a)

mean

21

velocity;

(b)

turbulent

kinetic

energy;

(c)

Reynolds

1.30

5.0

25.0

0 EP

(a)

[]

4.0

20.0

[]

3.0

15.0 r'l i-i

*

[] o

_

Wesentmodel 1.0

13

rl

8

0.0

13

2.0

0

5.0

0

o DNS data skemodel presentmodel

----_

k+ ElDNSdata ----- skemodel

10.0

[3 13

0.0

lO_.O 2o_.o ' 3o_.o ' 4oo.o

I

0.0

lo_o

I

0.0

_o

3o_o

400.0

#t

y+

Y

0.30 (c) (d) 13 0.20 0.6

-----

13

-lzv+

E+

13DNSdata ske model presentmodel

13 rm O

0.4 0.10

13 13

0.2 Q 0.0

0

, 100.0 _ ,

0.0

. 200.0

, 300.0

0.00 400.0

'-' "' _ _"-' ,", :_ ,'-,,", -- _ ,-.. ,-.,

olo

=

I

100.0

=

I

i

200.0

y+

!

l

300.0

400.0

y+

o_

Figure

5.

Turbulent

shear

stress;

(d)

channel

dissipation

flow

at Re,

=

395.

(a)

mean

rate.

22

velocity;

(b)

turbulent

kinetic

energy;

(c)

turbulent

0.005

1.20

1.00

0 0.004

-----

o EXP. skemodel presentmodel

O

0.60 o UIUe

C.f

0.60 o

0.003

o EXP. ----skemodel presentmodel

0.40

0.002 0.20

0.00

i

0.000

0.020

I

0.001

I

50'00

0.060

0.040

,

i

,

L

E

15000

20000

.

0.008

i

(o}

(c) ,-.

10000 Re 8

y (m)

0.010

'

0.008

0.006

0

-._

0.006

0

0.004 ,_ _,_

Eo 0.004

E

OEXP. ----- skemodel

./-J

-----

ske model

_

present model

U

0.002

0.002

_

/

present model

o =l

|

0"00%.0

1.0

310

410

510

i

I

,

I

J

°°°°.0 1'.0 2'.0 310 4.0 s.o

i

2.0

6.0

6.0

x (m)

x (m)

°_

Figure6. Zero pressuregradientturbulentboundarylayer.(a)mean velocity at /?_e=8900;(b)skinfriction coefficient;(d) displacement

thickness; (d) momentum

thickness.

23

1.20

•

•

,

-

•

i

-

-

,

-

0.005

-

(a)

(b)

1.00 0.004 0.80

0

[]

0

[]

[]

0.003 U/Ue

0.60 [] ----0.40

nEXP. ske model present model

0.002

nEXP. ske model

----1

0.20

_e_

model

0.001

0.00 0.0''

0'.3 '

' 0'.6 _ _ 019

1'.2

1.5

0"0000'.0 '

1_.0 '

2'.0 '

y (in)

0.15

310 '

4'.0 '

5'.0

6.0

x (ft)

2.0

L

(c) 1.8

0.12

-----

nEXP. ske model

1

0.09

17

0

[]

present model

1.6

0

H E 2 0.06

,5

-----

1.4

OEXP. ske model present model

3

[]

0.03

0"000.0

o 17

o

n

1.2

'

1'.0 '

2'.0 '

31.0 ' 410 ' 5'.0 '

6.0

1.0010

'

1.0' .....2'.0

x (ft)

Figure 7. Favorable pressure at x=4.0 ft.; (b) skin friction

3'.0

4'.0

5'.0

6.0

x (ft)

gradient turbulent boundary layer (tterring and Norbury coefficient; (c) momentum thickness; (d) shape factor.

24

flow).

(a) mean

velodty

0.005

1.2

(b)

(a)

D EXP.

0.004 1.0

o EXP. -----

ske model

-----

ske model

--

present model

dr'/

0.003

c/ U/Ue

0.8

presentmodel

--

/"

[] []

0.002

0.001 0.6

/0

0.4.2

i

i

-1

0

0"0001.0

2.0 I

,

30 I

*

4.0 I

,

5.0 I

,

6.0

7.0

8.0

x (ft)

log(y) (in)

1.60

0.40 (c)

1.55

(c0 OEXP.

0 EXP.

0.30

1.50

----

ske model

--

present

/.

model

////I

-----

ske

--

present

model model

1.45 [3

.o

H

E 2

1.40

O 0

0

1.35 0.20 1.30 1.25

0"101.0

....... 3.0 4.0

2.0

Figure 8. Adverse pressure (b) skin friction coefficient;

5.0 x (ft)

61 ...... 0 7.0 8.0

1.20 1.0

2.0

3.0

4

io'i'i 50 60

i'

70

8.0

x (ft)

gradient turbulent boundary layer (Bradshaw (c) momentum thickness; (d) shape factor.

25

flow).

(a) mean velocity

at x=5.5

ft.:

1.2

0.004 (a)

(b) -----

1.0

0.003

U/Ue 0.8

Cf _"

0.6

r.

_

2'

0.4 .

_,

- .5

present model

0.002

[] EXP. -----skemodel

,___

-

-- rn'_%

D_N_x 0.001

presentmodel

-1.5

D EXP. ske model

o.ooo 0.0 ....... 1.0 2.0

-1

log(y) (m)

3.0

4.0

x (m)

0.030

0.020

(d) (c) v

/f/ /

rl EXP. skemodel

0.015

/I

D EXP. ----- ske model

/I /

0.020 .2 0.010

0.010

:_ 0.005

0.000 0.0

1'.o '

2:o

3'0

0.000 0.0

4.0

x (m)

1:o

2o

3o

4.0

x (m)

t_

Figure x=1.76

9. Adverse pressure gradient turbulent boundary m.; (b) skin friction coefficient; (c) displacement

layer (Samuel and Joubert flow). (a) mean thickness; (d) momentum thickness.

26

velocity

at

3

II|llllllilillllll|il|illllll|llll||lll

(a) DS-case

2 0 (:D (:D (:D I"

o

D _ []-- E_--_

oo 2"_

1

u EXP

,_ 0

ske model present model

• _[][]

-1

[]

-2

I

I

I

I

I

I

I

I

I

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I

!

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I

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i

I

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10

0

I

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t

i

,

i

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i

#

20

|

i

i

i

i

i

!

,

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i

30

40

x/h

3

Illllllllllllllllllllllllilll||llllllll

(b) KKJ-case

2 rO C:) (:D O

1 //

0

,',, //

-1 -2

!

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Figure

,

!

I

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!

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f

I

I

1

I

I

I

f

I

I

I

10

I

10. Friction

I

I

I

'

'

!

20 x/h

coeiBcient

along

27

I

I

,

I

|

_

I

I

I

30

the

bottom

,

I

,

I

i

40

wall

i

0.2

i

i

i

i

i

i

i

i

i

I

i

i

i

i

I

i

i

i

[

I

_

i

I

i

i

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i

i

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i

i

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0.1 Q.

i_

o

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0

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0

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0

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f

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'

i

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10

f

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f

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20

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i

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40

x/h

0.4

i

|

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4

.

•

4

.

|

i

i

s

_

i

i

-

|

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.

i

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a

•

s

- "_-_-n

.

.

.

•

s

|

i

i

|

•

|

.

i

f

!

I

..........

0.3 0.2 Q.

c)

0.1 ,

0.0 -0.1

f

,

.

(b) KKJ-case .

¢

0

|

.

,

i

|

.

l

|

|

|

|

!

10

i

,

|

,

.

.

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20

.

.

.

.

,

|

.

I

f

l

30

11. Static

pressure

coefficient

along

(legend

as in figure

10)

28

the bottom

:

40

x/h

Figure

l

wall

_,_,1,,.,1_

.

(a) DS-cas_ 2

t

,'t .

_,,,i,,,_1,,_11,

I

w g

t-

,4

o I

"1

.4

x/ 0 -0.5

, .

0.0

0.5

1.0

1

,! .

L==lr.

,

.

p I

,

,

,

-0.5 0.0 0.5 U/Uref

,

I

1.0

1.0

f.... .....t i-I

t-

x/h=_

"

.... _I,JP,,, .... , , _L,... ,, .... , -0.5 0.0 0.5 1.0 -0.5 0.0 0.5 1.0

0 -0.5 0.0

U/Uref

Figure

12. Streamwise (legend

mean

velocity

as in figure

29

10)

U-profiles

3

I

x/h=2

2

x/h=5

t-

/ / / /

0 -0.50.0

0.5

Figure

1.0 -0.50.00.51.0 - 100uv/U ref**2

13. Turbulent

sheax

(legend

stress

proKles

as in figure

0.51.0

-0.50.0

in the DS-case

10)

,,,i

,,,i,,,i,,,1_,,

.'1 I

''l'''l'''

I'''

I'''

.1

x/h=20

2

1

o

,I,,,I,,,I,ll

012345012345

01

2345

lO0(uu+vv)/Uref**2 Figure

14. Turbulent

normal

(legend

stress

as in flguze

3O

profdes 10)

in the

DS-case

REPORT

DOCUMENTATION

Form Approve# OMB No. 0704-0188

PAGE

Public reporting burden for this co_leoticnof informationis estimated to average 1 hour per response, includingthe time for reviewinginstructions,searching existing data sources, gathering and maintainingthe data needed, and completingand reviewing the collectionof information. Send comments regardingthisburden estimate or any other aspect of this collection of infownation, includingsuggestionsfor reducingthis burden, to WashingtonHeadquartersServices, Directoratefor Infomlation Operations and Reports, 1215 Jefferson Davis Highway, Suite 1204. Arlington, VA 22202-4302, and to the Office of Management and Budget, Paperwork Reduction Project (0704-0188), Washington,DC 20503. 1. AGENCY

USE ONLY

(Leave blank)

2. REPORT

DATE

August 4. TITLE

AND

3. REPORT

TYPE

AND

DATES

Technical

1994

SUBTITLE

COVERED

Memorandum

5. FUNDING

A New k-e Eddy Viscosity Model for High Reynolds Flows-Model Development and Validation

Number

NUMBERS

Turbulent WU-505-90-5K

6. AUTHOR(S)

T.-H. Shih, W.W. Liou, A. Shabbir, Z. Yang, and J. Zhu

7. PERFORMINGORGANIZATIONNAME(S)ANDADDRESS(ES)

8. PERFORMING ORGANIZATION REPORT NUMBER

National Aeronautics and Space Administration Lewis Research Center Cleveland,

Ohio

E-9087

44135-3191

9. SPONSORING/MONITORINGAGENCYNAME(S)ANDADDRESS{ES)

10. SPONSORING/MONITORING AGENCY REPORT NUMBER

NASA TM- 106721 ICOMP-94-21 CMOTF-94-6

NationalAeronauticsand Space Administration Washington, D.C. 20546-0001

11. SUPPLEMENTARY

NOTES

T.-H. Shih, W.W. Liou, A. Shabbir, Z. Yang, and J. Zhu, Institute for Computational Mechanics in Propulsion and Center for Modeling of Turbulence and Transition, NASA Lewis Research Center (work funded under NASA Cooperative Agreement NCC3-233). ICOMP Program Manager, Louis A. Povinelli, organization code 2600, (216) 433-5818. 1211. DISTRIBUTION/AVAILABILITY

Unclassified Subject

13. ABSTRACT

STATEMENT

12b.

DISTRIBUTION

CODE

- Unlimited

Category

(Maximum

34

200 words)

A new k-e eddy viscosity model, which consists of a new model dissipation rate equation and a new realizable eddy viscosity formulation, is proposed in this paper. The new model dissipation rate equation is based on the dynamic equation of the mean-square vorticity fluctuation at large turbulent Reynolds number. The new eddy viscosity formulation is based on the realizability constraints; the positivity of normal Reynolds stresses and Schwarz' inequality for turbulent shear stresses. We find that the present model with a set of unified model coefficients can perform well for a variety of flows. The flows that are examined include: (i) rotating homogeneous shear flows; (ii) boundary-free shear flows including a mixing layer, planar and round jets; (iii) a channel flow, and flat plate boundary layers with and without a pressure gradient; and (iv) backward facing step separated flows. The model predictions are compared with available experimental data. The results from the standard k-e eddy viscosity model are also included for comparison. It is shown that the

14.

present

model

SUBJECT

TERMS

is a significant

improvement

over the standard

k-e eddy viscosity

model.

15.

NUMBER

16.

PRICE

20.

UMITATION

OF PAGES

32 Turbulence

modeling

CODE

A03 17.

SECURITY CLASSIFICATION OF REPORT

Unclassified NSN

7540-01-280-5500

18.

SECURITY CLASSI RCATION OF THIS PAGE

Unclassified

19. SECURITY CLASSlRCATION OF ABSTRACT

OF ABSTRACT

Unclassified Standard Form 298 (Rev. 2-89) Prescribed by ANSI Std. Z39-18 298-102