Numerical simulation of heat transfer in the separated and reattached flow on a blunt flat plate

INT. OOMM. HEAT M A S S T R A N S F E R 0735-1933/86 $3.00 + .00 Vol. 13, pp. 665-674, 1986 ©Pergamon Journals Ltd. Printed in the United States

NUMERICAL SIMULATION OF HEAT TRANSFER IN THE SEPARATED AND REATTACHED FLOW ON A BLUNT FLAT PLATE

M.C. Thompson, K. Hourlgan and M.C. Welsh Commonwealth S c i e n t i f i c and I n d u s t r i a l R e s e a r c h O r g a n i z a t i o n D i v i s i o n of Energy T e c h n o l o g y , I Q g h e t t , V i c t o r i a , A u s t r a l i a

(C~ll~nicated by J.P. Hartnett and W.J. Minkowycz)

ABSTRACT The tlae-dependent heat transfer process in the region of a turbulent separation bubble at the leading edge of an isothermal square leading edge plate Is modelled numerically. A dlscrete-vortex model is used to d e t e r u f l n e t h e v e l o c i t y f i e l d and a t h l r d - o r d e r up~rlnd d i f f e r e n c i n g t e c h n i q u e i s used t o c a l c u l a t e t h e t h e r m a l f l e l d . The p r e d i c t i o n of t h e mean N u s s e l t numbers i s compared w i t h e x p e r i m e n t . The model p r e d i c t s t h e i n s t a n t a n e o u s s t r e a m l i n e s , i s o t h e r m s and l o c a l N u s s e l t numbers a t t h e p l a t e s u r f a c e . The i n f l u e n c e of t h e l a r g e - s c a l e v o r t e x s t r u c t u r e s on t h e l o c a l h e a t t r a n s f e r i s d e t e r m i n e d .

Introduction

Heat

transfer

in separated,

reattached and redeveloped flow regions

is

important in many engineering sltuatlons. There has been a large number of

investigations These

include

facing

steps

carried

out on such f l o w s f o r a wide v a r i e t y

clreular (e.g.

cyllnders

Gooray e t

e x p a n s i o n s or c o n t r a c t i o n s years,

(e.g. [2]),

in tubes

(Ota

and

MacCormlek et al. tlme-mean heat reattach.

Kon

[I]),

forward

r o u g h n ess

[ 4 ] , Zelenka

and

bodies. backward

e l e m e n t s and a b r u p t

Sparrow and O ' B r l e n [ 3 ] ) .

In r e c e n t

have been u n d e r t a k e n t o d e t e r m i n e t h e

of such f lo ws around b l u n t and

[7], Motwanl et al.

Loehrke

flat

plates

[ 5 ] , Cooper

et

which a r e al.

[6],

[8]). These measurements show that the

transfer is augmented when the flow is made to separate and

The heat

significantly;

Morgan surface

(e.g.

a number of e x p e r i m e n t a l s t u d i e s

mean t h e r m a l c h a r a c t e r i s t i c s heated

al.

of b l u f f

a

transfer

local

characteristics

minimum

in

along

the Nusselt

the plate

number

separation bubble and a local maximum near reattachment. 665

is

surface

found

inside

vary the

666

M.C. Thompson, K. Hourigan and M.C. Welsh Recently,

proposed

mechan~sm~

to explain (e.g.

reattachment

large-sca]e

Jnvolvlng

the enhanced K~ya

and

heat

Sasakl

Vol. 13, No. 6

vortex

structures

have

been

transfer near the polnt of t~me--mean

[9]). A determination of the mechanJsms

leading to the var~atlon of the heat transfer ~n different regions of the flow requires evaluatlon of the instantaneous numerical wthod the

turbulent

presented.

for Invastlgatlng

structures

formed

A dlscrete-vortex

Peclet

numbers.

convection transfer

in

separated

high

transfer

Reynolds

r a t e of

number f l o w s

solutlon as

obtained

the

valuable

flow

insights

is

of

restricted

energy

into

is

t h e h i g h Reynolds number

scheme i s used t o s o l v e t h e e n e r g y e q u a t i o n .

th e

However,

processes,

In this paper, an economical

model i s used to p r e d i c t

f l o w and a f l n l t e - d l f f e r e n c e economy of c o m p u t a t i o n ,

flow.

t h e i n f l u e n c e on t h e h e a t

is

here

still

t h e mechanisms

For

t o moderate doalnated

leading

by

to heat

a u g m e n t a t l o n a r e found t o emerge from t h i s type of a p p r o a c h .

Numerlca] Model

The f l o w c o n s i d e r e d heated flow.

flat

plate,

is

which I s

The f l o w u p s t r e a m

The f l u i d

is

that

assumed

is

past

is

taken

t o be o f u n i f o r m v e l o c i t y

t o he I n v l s c l d ,

allow representation

focussed

on t h e

The v o r t l c l t y negligible

shear

generated

effect

Incompresslble

Invlscld

of t h e

layer at

the

lo w e r

field

both

is th e

determined steady

due t o o t h e r

flow across

every-

a r e l o c a t e d . These

scale flow structures.

Attention

from t h e t o p l e a d i n g edge c o r n e r .

l e a d i n g edge c o r n e r

is

assumed t o have

model

e n t e r i n g t h e f l o w from t h e l e a d i n g edge

of t h e p l a t e i s a p p r o x i m a t e d by a d i s t r i b u t i o n vortex

and t m m p e r a t u r e .

on t h e s h e a r l a y e r a t the top of t h e p l a t e .

The c o n t i n u o u s s h e e t of v o r t i c i t y

field,

set/-Inflnlte

and i r r o t a t l o n a l

llne vortices

larger

separating

Vortlclt 7 E~uatlon: Discrete-vortex

each

two-dlmenslonal,

e l l g n e d w / t h a s q u a r e l e a d i n g edge normal t o t h e

where e x c e p t a t p o i n t s where s l a p l e vortices

a rigid,

and

line

the s o l i d

through

vortices.

plate

the

fluctuating

contributions components,

In o r d e r

boundary,

of l i n e v o r t i c e s .

to s a t i s f y

from and

the

from

The m o t i o n o f irrotattonel the

vorticity

the condtcton of zero

the following S c h w a r z - C h r i s t o f f e l

f o r m a l t r a n s f o r m a t i o n i s used t o l o c a t e image v o r t i c e s :

z = ().(~2 - 1) 1/2 - a r c o s h (A)) 21t/s + iH •

con-

VOl. 13, No. 6 This ~nto

SEP~A~D~FI/3WGNAPLATE

transformation

the

flow

Naps th e

bounded

by t h e

placed st conjugate p o s i t i o n s to

satisfy

vortices

the

flow around

rea3

axis

(~-p]ane).

of

st

the

An i m p o r t a n t

the

annlhilatlon

of v o r t l c l t y

the tangential

model,

adjusted turn

is

into

the

fluld

~tth

scheme used

(e.g.

here

is

u p s t r e a m of t l m e - ~ e a n r e a t t a c h m e n t fraction fractlonal

18

determlned value

untll

by the

running net

vortlclty.

is

th e

is

of t h e

responslble

for

vort-

(Morton

leads to a

[12]).

In t h e

the

surface,

code

is

which i n

For l o n g p l a t e s ,

found t o be a p p r o x l a a t e l y [13])

by a f i x e d

to

the

a t t h e c e n t e r of t h e l e a d l n g

strengths

vortlclty

fraction and

unity,

and by t h e p r e s e n t model. of v o r t i c e s

Iteratlvely

generation

reclrculatlng

each t l m e - s t e p ;

rate

is

w i t h i n one p e r c e n t of t h e tlme-mean p r e s s u r e drop a l o n g t h e p l a t e

Energy E q u a t i o n : F i u l t e - d l f f e r e n c e

strength

recognition

and of t h a t

same r a t e

and C h e r r y

reduce

the

drop a l o n g i t s

introduced

Itilller

to

of its

shed I n t o t h e f l o w a l o n g t h e p l a t e

downstream of r e a t t a c h a e n t

both experlnentally

The i n t r o d u c t i o n

i n a homogeneous f l u l d ,

in the nonaallzed pressure coefficient

edge and t h a t

[10].

The p r e s e n c e of v i s c o s i t y the

the pressure

I n f l u e n c e d by p r e v l o u s l y

~ - p l an e

The a d v e c t i o n of t h e

r a t e of g e n e r a t i o n i s p r o p o r t l o n a l

at

t h e amount of v o r t l c l t y

t o be c o n s i s t e n t

difference

The

of v o r t l c l t y

surface pressure gradient.

f l u x of v o r t l c l t y

scheme i s

i n t h e f l o w . That i s ,

i s g e n e r a t e d a t a boundary and I t s

present

flow.

are

[II].

present

mechanism l e a d l n g t o t h e g e n e r a t i o n

(z-plane)

Jmnge v o r t i c e s

l e a d l n g edge c o r n e r end d e t e r n l n a t l o n

innovation in

plate

i n t h e upper h a l f of t h e

zero norual

bas ed on t h e scheme used by Nagano e t e l .

net

The

f o l l o w s c l o s e l y t h e scheme used by l ~ y a e t 81.

of vortlclty

Iclty

a semJ-tnftntte

to the v o r t i c e s

boundary c o n d i t i o n

667

adjusting consistent

this the to

surface.

approxlmatlon

The e n e r g y e q u a t i o n t o be s o l v e d i s g i v e n by

~T/~T + (u_.grad) T = Pe

The v e l o c i t y u 18 d e t e r a l n e d using a discrete

at

-I

d l v ( g r a d T) .

each s t e p by t h e d l s c r e t e - v o r t e x

approximation to the B i o t - S a v a r t I n t e g r a l .

scheme,

668

M.C. THOMPSON, K. Hourlgan and M.C. Welsh

Vol. 13, No.

The energy equation is solved using QUICKEST (Leonard scheme

uses

Jnc]udes manner

third

order

an estimation

to Lelth's

upstream

for the

method

differencing

truncated

for

the

[]4]). Th~s explicit advectlon

time difference

[15]. The method approaches

both time and space as the Peclet number

term~

and

terms in a slmJ]ar

thlrd order accuracy lu

approaches Jnflnlty

(for a constant

flow velocity).

GRID AND BOUNDARY CONDITIONS

51x31

GRIDPOINTS

T=T0

7--

4H T=T 0

dT ~xx=o

T:T1 t

FIG. I c o m p r e s s e d mesh u s e d f o r f l n l t e - d l f f e r e n c e wlth boundary condltlon8 marked

Quadratlcally

To a c c o m o d a t e quadratic

the higher

transformation

factor

of

Figure

1. F o r t h e p r e s e n t

for

both

(2H/V). Cyber

ten

the

is

used

for here.

205 c o m p u t e r .

temperature the vertical The

mesh

calculations

discrete-vortex

The P e c l e t

CPU t i m e p e r

--I

20 H

method

number used

is

The ( v e c t o r i z e d )

1000 t i m e s t e p s .

gradients

near

coordinate

is adopted;

and

boundary

and t h e Pe = 4 0 .

the plate

conditions

t h e mesh u s e d i s

code

scheme,

a

a compression are

shown i n

51 x 31. The t i m e s t e p

finite-difference The n u m e r i c a l

takes

surface,

scheme i s

0.05

code was r u n on a

approximately

350 s e c o n d s

of

Vol. 13, NO. 6

SEPARATI~AND R E A ~ F I E W O N

A PLATE

669

/~~,~

1.0

i|/

,°~/

--o-- EXPERIMENTAL[6]

"r

I

i

0

1.0

2-0 X/x R

FIG. 2 and observed (Cooper et al. [6]) time-mean local Nusselt numbers along the plate surface

Predicted

Results

The t l a e - ~ e a n The

Nusselt

attachment. and

a

Nusselt values number), the

9.3H.

downstream

number is

to

at

the

Nusselt

reattachment.

the

For

the

Nusselt

the normalized

experimental

results.

lower

are

shown i n F i g u r e occurring

near

minimum i n t h e s e p a r a t i o n predicted

number decreases

n u m b e r s m e a s u r e d by C o o p e r e t a l . of

surface

maximum v a l u e

a local

reattachment;

The l o c a l of

the plate

normalized

maximum n e a r

xR being

direction

numbers at

are

The N u s s e l t

local

length

llusselt

numbers

and Discussion

comparison,

tlme-mean

bubble

reattachment

monotonically the

2. re-

normalized

in the local

[6] a r e s h o w n . A l t h o u g h t h e a b s o l u t e

numbers

are

Nusselt

n u m b e r s a r e f o u n d t o be i n good a g r e e m e n t w i t h

(due

to

the

use

of

a lower

Peclet

670

M.C. Thompson, K. HouriganandM.C. Welsh A typical

time t r a c e

of the p o i n t

F i g u r e 3. I t i s seen t h a t a

new r e a t t s c h m e n t

shortened. and

result

[9];

these

reattachment

i n g e n e r a l t h e bubble s t e a d i l y

point

This

SasakJ

of s t r e a m ] i n e

Vol. 13, No. 6

ls

forms

upstream

consistent

showed the

and

vlth

the

the

continual

bubble

Js

mxperlnentsl

formation

Js shown ~n

grows Jn l e n g t h u n t i l discontinuously findlngs

of new p o J n t s

v m l o c l t y u p s t r m s s o f t h e p o i n t o f mean r e s t t a c h m s m t as l a r s m - s c a l e

of Y~lya of z e r o

structures

were r e l e a s e d from t h e s e p a r a t i o n b u b b l e . 20 -

X

15-

°'[ |

0 0.0

I TIME

7

FIG. 3 Predicted tism t r a c e of the r e a t t a c l m e n t

A t l s m - e v e r a s e d measure o f t h e i n t e n s i t y the

vertlcal

direction

eddy d l f f u s l v l t y fluctuations position

l

,l 2S.0

due t o v o r t i c e s

Dr. M~Ich i n d i c a t e s

and t u s p e r a t u r e

of t r a n s p o r t

In t h e

flow is

th e c o r r e l a t i o n

fluctuations.

o f muctmm d £ f f n s t v i t y

length for the flow nodel

Thls

Is

of thermal energy In the

vertlcal

between v e r t l c a l plotted

in the chordsdse d i r e c t i o n

thermal velocity

I n F i g u r e 4. aloq

The

t h e p l a t e 18

found In t h e n e l s h b o r h o o d of t h e position of a n s l s m a N u s n s l t number downstream o f t h e l e a d i n g edl~e. This r e s u l t fer results cool fluld

sugsests that

t h e a u g m e n t a t l o n of h e a t t r a n s -

from v o r t e x a c t i o n I n moving warm f l u i d towards t h e h e a t e d p l a t e .

away from t h e s u r f a c e and

VOl. 13, No. 6

0

S ~ > ~

4

R~D ~

FIEW ON A PIATE

8

12

16

671

20

x/N FIG. 4 PredLcted contours of the vertlcal thermal eddy dlffusivlty D v

(b)

_z~ 0

I

O

I

x H

10

,

I

20

F'rG. 5 P r e d i c t e d s n a p s h o t s of (a) I n s t a n t a n e o u s v o r t e x p o s l t l o n s and i s o t h e r m s and (b) I n s t a n t a n e o u s l o c a I N u s s e l t number p r o f i l e

672

Vol. 13, No. t,

M.C. Thc~pson, K. HOurlgan and M.C. Welsh

More ~mportantly, from the point of v~ew of understanding the mechanisms leading to the augmented heat transfer when the flow Is made to separate and reattsch,

are

the

predJctlons

of

tlme-dependent

quantities.

Character~stle

instantaneous isotherms together with the vortex positions and Nusselt numbers are shown in Figure 5. The influence of the large-scale vortex structures on

tha is

local heat

transfer

found from t h i s

large-scale

structure.

in

cooler

drawing

vortex positions.

rate

This i s

fluld

plate

and

peak i n t h e N u s s e l t number

the

as

the a c t i o n

surface

of t h e s e

immediately

b e i n g drawn c l o s e

to

structures

downstream

the i n s t a n t a n e o u s

of

the

point

of

c o o l , h a v i n g flowed o v e r the s e p a r a t i o n bubble away

surface.

this

A local

interpreted

towards

reattachment is relatlvely

the

apparent.

and o t h e r s n o t shown, t o be a s s o c i a t e d w l t h each

The f l u i d

from t h e h e a t e d

is

snapshot,

The r e l a t l v e l y

cooler

fluld

large

accounts

observed at the reattachment p o i n t .

temperature

for

difference

the higher heat

between

transfer

Inside the s e p a r a t i o n bubble,

rates

the r e c l r c u -

latlng flow is generally less turbulent and of lower velocity; this leads to a generally higher thermal resistance, as reflected by the lower values of both

t h e tlme-mean and i n s t a n t a n e o u s

It Istlcs

is,

perhaps,

along

Invlscld,

the

N u s s e l t numbers i n t h i s

significant

plate

that

surface

are

t w o - d l m e n s l o n a l model.

prlmarily

responslble

for

th e

the

large-scale

which may t h e r e f o r e

heat

quite

transfer

mell

character-

by an e s s e n t l a l l y

agreement i s not s u r p r i s i n g

:

[16] indicate that the spanwlse vortices

region

n e i g h b o r h o o d of the mean r e a t t a c h m e n t transport,

relative

However, t h i s

the findings of Rothe and Johnston

are

the

predicted

region.

of

unsteady

point;

structures,

are

reversing

flow in

t h e dominant v e h l c l e s little

influenced

the

of e n e r g y

by v i s c o s i t y ,

be n e g l e c t e d .

Conclusions

A tlme-dependent separation vortlclty tharmal

of

the

heat

bubble has been p r e s e n t e d . fleld

fleld

difference

model

that is

is

process

The v e l o c l t y

field

p r o v i d e d by a d l s c r e t e - v o r t e x

calculated

approximation

transfer

to

slmultaneously solve

the

energy

using

near

is

a

turbulent

deduced from t h e

model of t h e f l o w . The a

equation

thlrd-order on

a

finite-

quadratically

compressed g r i d . The peak tlme-mean N u s s e l t number downstream of t h e l e a d i n g edge i s found t o be l o c a t e d i n t h e r e g i o n of t h e peak v e r t l c a l

t h e r m a l eddy d l f f u s l v l t y

and

Vol. 13, No. 6

SI~ARATI~AND ~

FIK~ONAPLATE

673

the tlme-mean point of reattachment of the separating streamline. The profile of the local tlme-mean Nusselt number compares well with those obtained experImentally

towards

at hlgher

the p l a t e

number along the

Peclet

surface plate

numbers,

leads

surface;

Vortex

action

in drawing

to i n s t a n t a n e o u s l o c a l

cooler

fluid

peaks i n the N u s s e l t

a prominent peak i s l o c a t e d a t

the p o i n t of

reattachment,

Acknowledgement s

M.C. Thompson g r a t e f u l l y

acknowledges the s u p p o r t of a CSIRO P o s t d o c t o r a l

F e l l o w s h l p Award. The computing was s u p p o r t e d by a CSIRONET Cyber 205 G r a n t .

Nomenclature

Cp

heat c a p a c i t y of f l u i d a t c o n s t a n t p r e s s u r e

Dv

vertical

H

s e m l - t h l c k n e s s of p l a t e

t h e r m a l eddy d l f f u s l v l t y

[~--r~/(DT/~y)]

la/=T k

t h e r m a l c o n d u c t i v i t y of f l u i d

Nu

S u s s e l t number [-~(T/AT)/~(y/2H)]y=O tlme-mean N u s s e l t number

mAX

maximum v a l u e of ~

occurring near reattachment

Pe

P e c l e t number CpO V 2 R / k

t

time

T

t e m p e r a t u r e of f l u i d

TO

t e m p e r a t u r e of u n h e a t e d f l u i d

T1

temperature at plate surface

V'

fluctuation

V

m a g n i t u d e of v e l o c i t y of flow a t u p s t r e a m i n f i n i t y

x

horizontal coordinate direction

of v e r t i c a l

velocity

f l u i d v e l o c i t y n o r m a l i z e d to V

xR

reattachment length

y

vertical

z-=x+ly

complex c o o r d i n a t e i n p h y s i c a l p l a n e

A

complex coordinate in transformed plane

AT

temperature difference between plate and incident fluid

coordinate direction

complex flow velocity potential O

fluid density

T

d l m e n s l o n l e s s time t V J 2 H

674

M.C. Thompson, K. Hourlgan and M.C. Welsh

Vol. 13, No. 6

References I.

V.T. Morgan, (1975).

Advances

Jn Heat

2.

A.M. Gooray, C.B. Watklns and W. Aung, J . Heat T r a n s f e r , 107p 70 ( 1 9 8 5 ) .

3.

E.M. Sparrow and J . E . (1980).

4.

T. Ota and N. Kon, J . Heat T r a n s f e r ,

5.

R.L. Z e l e n k a and R . I . (1983).

6.

P . I Cooper, J . C . 61 ( 1 9 8 6 ) .

7.

D.C. McCormick, R.C. Lessmann and F . L . ASME, 106, 276 ( 1 9 8 4 ) .

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8.

D.G. blotwani, U.N. G a i t o n d e and S . P . Sukhatme, J . Heat T r a n s f e r , ASME, 107, 307 ( 1 9 8 5 ) .

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M. Kiya and K. S a s a k l , P r o c . 5th Symp. on T u r b u l e n t Shear Flows, C o r n e l l U n i v e r s i t y , I t h a c a , New York, 5.7 ( 1 9 8 5 ) .

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ASME, 105,

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ASME, 101,