Modelling of turbulent heat transport: A state-of-the-art

von Karman Institute for Fluid Dynamics

.

...

.

Chaussée de Waterloo, 72

B-1640 Rhode Saint Genèse - Belghun

Technical Memorandum 47

MODELLING OF TURBULENT HEAT TRANSPOltT

A STATE-OF-THE-ART

c.

Benocci

TABLE OF CONTENTS

Abstract ... 1

1. Introduction ... 1

2. Turbulence equations ... 3

3. Turbulence modelling. Where do we stand? ... 5

4. Eddy viscosity models ... '.' ... 6

Zero equation model ... 7

One equation model ... 8

Two equation model ... 10

5. Near wall treatment for eddy viscosity turbulence models ... 13

Wall function modelling of turbulence close to a wall ... 13

Low-Reynolds-number modeis ... 17

6. The k - € model for heat transfer calculations ... 23

7. The Reynolds stress model of turbulence ... 26

Aigebraic Reynolds stress model ... 26

Full Reynolds stress model ... 28

Wall boundary condition for full Reynolds stress model ... 32

Low-Reynolds-number modeIs ... 33

8. Conclusions ... 34

References ... 35

Annex 1 : The turbulent kinetic energy equation ... 39

ABSTRACT

An overview is given of the current state of single-point turbulence modelling for flows including heat transfer. The different levels of eddy viscosity closure are covered, together with algebraic and fuU Reynolds stress modeis. Emphasis is given to the different approaches possible for the modelling of wall effects on turbulence. The specific problems related to the modelling of heat transfer are reviewed and the performances of the different models discussed on the basis of selected examples taken from the most recent literature. The conclusion is reached that the fuU Reynolds stress model is the only closure offering reliable results for turbulent heat transfer in complex flow problem.

1. INTRODUCTION·

Turbulence is a

fl~id

flow

phen~menori

of extreme

co~ple:xity,

so complex indeed as to appear purely random and unpre~ictable in its·,detai,~s to the observer. Unpredictability is the feature of turbulerite most stressed by thos~.,who ha,ve a~tempted to provide ageneral definition, like, for example, Hinze [1], who propos es to define.the turbulent fluid motion as "an irregular condition of flow in which the various quantities show a random variation with time and space coordinates, so that statistically distinct average values can be discerned".

In principle, however, turbulence is a fully determinist ic phenomenon, namely one particular case of time dependent and tridimensional motion of a viscous fluid, which is fully described by the N avier-Stokes equations and should, therefore, be predictabie by mean of any of the many numeri cal techniques (see for example Hirsch [2]) which have been devised for the solution of the latter. The reason why this approach turns out to be unfeasible for most flow problems is entirely practical and directly related to the phenomenology of turbulent motion. As it can be observed looking to a temporal record of any flow property (Figure 1) turbulence is composed by structures (" eddies") exhibiting a very wide range of scales (in terms of either temporal or spatial separation) : the size of the largest turbulent scales is limited by the dimensions of the flowfield itself while for the smallest is set by viscosity. All si zes between these extremes are present in any turbulent flow.

The scales controlled by viscosity will obviously be the ones whose Reynolds number RTt = ~ is of order 1, where v is the velo city scale, Tl the leng th scale and v the kinematic viscosity. For air at ambient conditions, a v of 0.01 ms-l _{would give Tl ~ 0.0015m which}

is extremely small in comparison with the size of any flowfield of practical interest.

More precisely, the ratio between the largest and the smallest scale can be shown (Reynolds [3]) to be proportional to the Reynolds number of the turbulence to the power 3/4. Therefore the number of mesh points necessary to resolve numerically all the scales of turbulence in 3 dim en si ons increases with the power 9/4 of the Reynolds number and, for all but the simplest cases, exceeds by far the capabilities of any current or expected computer. Furthermore, to be consistent, such a calculation would need the initial and boundary conditions to be known with an resolution much better than the one of the numerical solution for the inner field (Frisch and Orszag [4]), requirement which, by the same token, cannot be satisfied by either numerical or experiment al means. Therefore, direct numerical simulation of turbulence will remain for the foreseeable fut ure (Moin and

Rogallo [5]) all instrument to perform, nrumerical e~p6rimen~ aim~d to achieve a better understanding of tl1e physlCs of turbulence and not ~ tooI for pr~ctical predictions.

From all engineeri~g point of view, the actua.l prediction of turbulent flows must start from the accept at ion of the randQm appearance turbulence presents to most observers and attempt to determine its relevant statistica.l quantitjes.

2 .. TURBULENCE EQUATIONS

Each variabie

F

In à

ra~dom

field can ~e

,

repr~sented ~the

,sum of its ;;tverage

F

and its fluctuation

f :

(1) where the average of. the' fluctuaÜou'

r

is 'équallo O. '

! . ;" ~. "

For flows with st rong variations of the fluid density P it is convenient to introduce the

density weighted average :

where: - I

F=F+!

- pF

F=-P

(2) (3)

and the density weighted average of the fluctuation

PI'

is equal to O. This formulation removes from the resulting equations for the mean flow all extra products of density fluc-tuations with other fluctuating quantities, which would be present if averaging procedure (1) was used.

The appropriate averaging procedure will depend on the nature of the flow : for a statistically steady flow this will be a time averaging, for a statistically homogeneous one a volume averaging, for all other cases an ensemble averaging [3].

For reasons of simplicity and time this review will be confined to incompressible flows and averaging operation (1) will be applied. The resulting transport equations for the

mean flow are : Continuity equation :

Momenttim equation :

Scalar transport equation :

de

dt

-

0

(4) (5)

~

[r

ae]_ au

j()

+

S ax} ' ax} ' 8x' } (6) 3

where Ui and Fi are, respectively,

the

veloçity and th~ body (buoy8.Ac:y or Coriolis) force component along the coordinate directio~ i (l;j:li

1,

2, 3), P )~ the presaure, S a S01,1rce

term

and

r

the molec"lar difl'usivity

pf

the transporte<l sçal~ 9.

The averaged eq,uatiQns conta.in the Reyn91c:4; streM u'~'il j lUlrd tp.e turbulent sca.lar

fiWf;

u/J as additional unknowns whose vNues have to be determined for the clO$ure of system (4)-(6) to be possible. D~tennination of theui~J aAdUi9 requir~s the introduction of a

3. TURBULENCE MODELLING. WH ERE DO WE STAND?

Models öf turbulence allowing the closure of the Reynolds averaged transport equa-. tions for thiri 'attached shear layers are available at least since 1925, when Prandtl first

introduced the mixing length model (see below). In the 70's the pioneering work at the Imperial College of London has produced the first models suited for the computation of complex flows, including separation and recirculation regions (Launder and Spalding [6]). Turbulence models have since been extensively applied to the study of a variety of flows of engineering interest . .

Unluckily for everybody concerned, however, extensively does not necessarily mean

accurately, especially when referred to heat transfer problems. A striking example if offered

by the result of an IAHR workshop, pr~s~nted in a recent review article by Launder [7] : different predictions of the N usselt number downstream of an abrupt pipe enlargement display a scatter of one order of magnitude at least, as shown in Figure 2. It can safely be concluded that much work has still to be done before the day fully reliable predictions of complex turbulent flows will be feasible.

A wareness of the shortcomings of many established models has led to a dramatic increase in the effort de~oted to their improvement and to the development of better ones, and, in the last years, ~n unprecedented ~timber of review and position papers has appeared, attempting to make the point on the current state-of-the-art and on the trends of research for the next future. A very partiallist of the most significative contributions should include the ones from Ferzinger [8], Jon~s

[9]

Launder [10], Leschziner [11], Nallasamy [12] and Speziale [13].

The single most relevant conclusion to emerge from the literature is that Reynolds stress modeis' will remain, for the foreseeable future, the only practical mean for the prediction of high Re flows of engineering interest. It is, therefore, justifiabie to confine the present discussion to this cl~s of model; readers interested to more advanced approaches to the prediction of turbulent flows. should refer to [4], [5] or to the very recent review by Schumann [14].

4. EDDY VISCQSlTY l\fODELS

Most of the general flow solvel's currently available fOf engineering use are still based upon the eddy visc.osity approJ;l.ch to the modelling of turb\.llence, firstly introduced by

Boussinesq at the end of past century.

The eddy viscosity model is based up on the observation that the main effect of turbulence is to increaae transport of the conserved properties and dissipation effects with respect to the laminar state. As, for laminar Hows, these processes are controlled by the viscosity of the Huid, it becomes natural to represent the turbulence in tel1lls of an increased viscosity, associated to the seemingly chaotic motion on the turbulent length scale (hence the use of the term "eddy").

The Reynolds stresses ean be rewdtten function of the turbulent viscosity IIt as :

(7) where

(8) is the kinetic energy associated with thc tUl'bul~nc~ divided by the fluid density and bij is the Kronecker operator.

Analogy with physical, visçosjty also leads to expres~ 111, in dimensional terms, as the

product of a velocity acale u and a length scale 1

IIt ~ uI (9)

where u and 1 must be representative of the energy carryitlg structures of turbulence. Obviously an eddy diffusivity

r

t must be defiped for every transported scalar :

(8e)

u·f}

=

rt

~

1 8x₁ ' (10)

Turbulent diffusivities of momentum and energy (or m8$s) can be related using a turbulent Prandtl (or Schmidt) number :

Most turbulent models assume Ut to be constant, at least for fluids whose molecular

. Prandtl number is in the order of unitYi commonly used values are 0.7 for free shear layers and 0.9 for wall bounded flows. A detailed discussion on the subject can be found in [15], while a complete review of methods to determine Ut for different fluids has been made by Reynolds [16].

lntroducing (7) and (10) the equations for the average transport of momentum and energy (or mass) become

dUj

dt

(5')

(6')

where the isotropic part of the Reynolds stress teiisor ï6jjk has been lumped together with the averagepressure in the operator

P.

(5') and (6') together with a relationship for Vt form a closed system of equations which can be solved numerically.

Different algebraic and differential models are available to formulate Vt function of mean flow properties, k and k-re~ated overall properties. of turbulence.

Zero equation model

As their name implies "zero equation" models do not include any partial differential equation to model the transport ·of turbulent properties, but solve a relationship of the type:

Vt = cul (12)

where c is a coefficient, specifying u and I function of the mean flowfield and empirical data.

The oldest and most widely applied of these modeis, is the already quoted mixing

. length, which assumes the length scale to correspond to the typical travel distance of a turbulent eddy Im and specifies u on the basis of the dimensional requirement that the velocity gradient for the energy carrying eddies must be of the same order as the one for

the mean flow :

= (13)

u

U

I L

where L and U are the length 8J;ld velocity scale for the mean flow. Using (12) and (13) the general form of Vt becomes :

I

2

[BUi (BUi

8Uj)]

V t = m - -

+

-BXj -BXj

Bx,

(14)

Eddy diffusivities of heat or mass can then be found applying (11).

Equation (14) has been s~ccessfully applied to the prediction of momentum transfer in many 2D thin shear layers, like attached boundary layers, pipe flows, jets and wakes, . for which it is relatively easy to prescribe lm with simple empirical formulae (see Rodi [17] for a review of the appropriate values for different flo~s). On the other side it is of very limited usefulness for more complex flows, apd especially for 3D ones, where it becomes extremely difficult tQ specify Im in a satisfactory way.

FUrthermore, the mixing length approach suffers of a eonceptual limitation which makes it ill suited for industrial heat transfer calculations : as it is immediately evident from (14) Vt is computed using loeal properties. only and ignoring any influence of the history of the flow. In other words the hypothesis is made that turbulence is in eq1f.i1ibrium, controlled by a balance of local production and düsipation; eonvective and diffusive transport of turbulence are neglected. A strikingly consequence of this simplification is that (14) predicts Vt = 0 on a symmetry line, like for e;,çample the centerline of a pipe flow; such an error is of relatively m~nor importance for the prediction of momentum transport because the turbulent shear stres:;;es are zero on the symmetry line, but may bring to a catastrophic error in the prediction of heat transfer, which, in the core region of pipe flow, is dominated by turbulent transport. Therefore reliable heat transfer predictions with the mixing length model are practically confined to attached boundary layers of aeronautical interest. One equation model

The failure of zero equation models to take into account turbulent transport phenom-ena leads to the use of transported quantities of turbulence to determine Vt. The most straightforward approach is to use kO.s as velocity sc ale leading to :

where k is found solving the transport equation for the turbulent kinetic energy, which can be written in a simple descriptive form as :,

(16)

where the right hand terms represent respectively Production, Tran3port, ViSCOU3

Diffu-sion, Dissipation (to be henceforth called f. to conform to commonly accepted terminology)

and Production/ Destruction by body forces. The exact form of the different terms can be found in Annex 1.

Viscous diffusion effects are negligibly small whenever the turbulent Reynolds number

Ret ~: is higher than 100, condition which is normally satisfied, in a developed turbulent flow except in the immediate neighborough of asolid wall as will be discussed later. Therefore, in many practical applications, the term Dij

Jk

is dropped, yielding the

high-Reynolds-number model of the turbulent kinetic equation, which will be used in the remainder of the present chapter.

Closure requires the right side terms to be modelled as function of Ui, k and V"~

Production of kis function of Reynolds stress,and mean velocity gradientsj it can therefore be recast at once in terms of Vt and

~~~.

Transport of k Can be reca'st, by analogy with

J .

the eddy viscosity approximation,

as

a diffusion process function of a turbulent diffusivity of k. Dimensional arguments show that. Dissipation. is related to kinetic energy as :

k1.5

€::::::--1 (17)

If the body force is created only by buoyancy effects, which is the case for most heat and mass transfer problems, Fk can be modelled as a function of turbulent diffusivity and mean gradient of the transported scalar (heat or mass) giving birth to it.

The resulting modelled transport equation for the turbulent kinetic energy is :

(18)

where

f3

is the body force coefficient

(*

for mass transport, *~ for heat transport) for, gk

the gravitational acceleration, Uk the ratio between turbulent diffusivities of momentum

and turbulent kinetic energy and Cd an empirical coefficient.

The model contains three empirical coefficients (Tk, C~, Cd as weIl as the length scale I,

which all have to be specified to close the equation. Adjusting the coefficients to fit known data for simple turbulent shear layers, the following relationships can be obtained :

(19)

For simple shear layers the form of 1 can be found requiring that, for equilibrium flows with no transport, the solution must coincide with the one provided by the mixing length model. This yields :

( ) 0.25 1 - 1 Cd - m 3 c~ (20) .

The turbulent diffusivity for the transported scalar E> is again found by applying (11). One equation models are, historically, the first to have offered the mean to make realistic predictions of non equilibrium turbulent flows and represent, in this sense, a fundamental advance with respect to zero equation ones. However, as should be evident from (20), they share with the former one fundamental weakness, namely the need to specify the length scale. As alreaçly pointed out, the task is simple for

20

thin shear layers without recirculation regions, but quite difficult for more complex problems. Rodi [17] reviews most of the formulations which have been proposed to determine algebraically 1

in genera! flows, but points out that none has ever gained widespread acceptance. Using recent results from Direct Simulations of Turbulence some progress has been made toward a formulation of 1 suita.ble for wall flows including separated regions [18], but it is much too early to assess whether bet ter physica! understanding of turbulence will breathe new life in one equations modeis.

To conclude, one equation models are basically limited to the same thin shear layers which can also be treated with the mixing length approach. They offer the advantage of providing much improved predictions in flows dominated by transport phenomena (wall boundary layers with streamwise pressure gradients, core flow in pipes), but do not make pos si bIe to attempt the prediction of complex turbulent flowsj such task requires a more flexible and general mean to determine the 'length scale.

Two equation model

Analogy with the argument used to introduce the turbulent kinetic energy equation leads to conclude that the way to overcome the limitations of one equation models would

be to introduce a transport equation for the leng th scale thereby obtaining a two equation

model.

Matter of fact, attempts to obtain directly a transport equation for the length scale have not proved satisfactorYi recently some promising models (Zeierman and Wolfshtein

[19], Speziale and al. [20)) based upon a transport equation for a time scale T = ~

have been proposed, but they are still undergoing validation and cannot, at the present time, be considered mat ure for industrial application. All the other models available (sre

Vandromme [21] for a recent review) use a transport equatio~ for some grouping ka[b. The

only model which has seen really widespread application to non aeronautical fields, is the

k - € model [6], based, as the name implies, up on a modelled transport equation for the

..

kinetic energy dissipation € (a

=

1.5 andb

=

-1).

A transport equation for € can be derived from the N avier-Stokes equations starting

from the definition :

€

BUi BUi v

-Bx_J ·Bx_J · (21)

but modelling it has proved much more difficult than for the kinetic energy equation (Launder [22)) due to the dominant rale played by smal I scales, which have no characteristic

length, but simply adjust themselves to the energy level of .the larger eddies.

Therefore modelled forms of the transport equation for € rely heavily, to quote again

Launder, on "dimensional analysis and intuition". The standard form used in the k - €

model is:

d€

dt (22)

where Gf is the body force coritribution whose formulation is discussed in details for

example in [17].

Equation (22) is obviously solved together with the kinetic energy equation :

dk

( BUi BUj) BUi B

(Vt

Bk) F

- Vt

--+-- --+-- ---- -€+

k

dt BXj BXi BXj BXk Uk BXk .

(23)

The expres sion for the turbulent viscosity then becomes :

(24)

The model contains five empirical eonstants to be adjusted: they are, to some extent, problem dependent and many slightly different sets of values can be found in the literature. For ineompressible flows the commonly aecepted values are [17] :

Cp

=

0.09 j Cl

=

1.44 j C2

=

1.92 j (jk = 1.0 j (jE

=

1.3 (25)

It must be stressed that, despite having been in use for more than 20 years the model is still undergoing revisions and modifieations to improve its reliability in complex flow problems. Unsurprisingly, in view of the heavy empirieism used in its formulation, the transport equation for € has since long revealed itself the weak point of the model and

many attempt have been made to improve it [22]. One of the main weaknesses has been identified [7] in the fact that (23) prediets too low values of € and, consequently (12) too

high values of the length seale I in pon-equilibrium regions like separated wall shear layersj Yap (as reported in [7] and [10]) has reeently proposed to add to the right side of (~3) a souree term SE :

(26)

where y is the distanee from the wall (or free surfaee) and Cl ;::; 2.44. The correction

has been shown to be quite successfull in different cases of separated flow including heat transfer [7].

Two equation models form the basis of most general flow solvers [23] currently used to perform industrial calculations and have seen extensively application to the prediction of heat and mass transfer for a variety of flows of different complexity with somewhat mixed results, as it will be shown in Chapter 6. However, in order to complete the present discussion attention must now be drawn to the fact that most practical applications concern flow partially or totally confined by solid walls. It is therefore necessary, in order to complete the present discussion to review the probleni of modelling the behavioUT of

.

tUTbulence close to a wall in the frame of the k - € model.

. . ~ . 5. NEAR WALL TREATMENTFOR EDDY VISCOSITY

TURBULENCE 'MODELS

The reasons why turbulence close to a wall poses so big a modelling problem can be

made very clear by a quick look to Figure 3, wbich'shows the evolution of turbulence close

to the wan for an equilibrium turbulent boundary layer.

The noslip condition requires turbulence intensity to fall to 0 at the wall, and Figure 3

shows how turbulent fluctuations decrease from their maximum value to 0 over a very thin

layer, whose thickness is in the order of 30 units in the wall coordina,te y+

=

U:r

where

y is the distance from the waU and UT the surface velocity defined in function of the wall

( ) 0.5

shear stress T was UT

=

7

.

Keeping this behaviourinto account the wall region of the boundary layer is usually

. thought to be divided in a viscous sublayer where turbulent effects are much smaller than

viscous transport and the mean velocity profile is therefore linear, and an equilibrium region

of developed turbulence where the mean velocity obéys the weU known logarithmic law,

with the transition between the two regions at y+ = 12. The lin -log velocity profile and

typical experiment ar data are compared in Figure 4 ..

One main point emerges from tht'; observation of Figures 3 and 4 : flow properties change dramatically over a very thin layer, with turbulent kinetic energy increasing by

one order of magnitude for 2

<

y+

<

20. Over most of this region, Ret

<

100 and the

high-Reynolds-number models discussed in the previous chapter is no more valid; it is not

only necessary to reintroduce the neglected Viscous Transport term, but the production

and dissipation models must also be modified to properly take into account the influence of the wall upon the development of turbulence.

It is therefore easy to understand why, in the past, the option has of ten be taken to

bypass the whole problem confining the numerical solution of (23) and (24) to the high

Ret region (or, in other words locating the first mesh point away from the wall weil out of

the viscous sublayer) and introducing the effect of the wan through the mean of empirical

wall functions.

wan function modelling of turbulence close to a wall

The simplest way to model the influence ~f a wallover the development of turbulence

is to assume that, close to the wall a· region of local equilibrium always exists where flow

properties vary in accord with the Zin - log profile of Figure 4. Therefore, if the first internal mesh point is located well within the logarithmic region (say 20

<

y+

<

100), the local velo city U must match the law :

. (27) where U+ = JL, K, the von Karman constant, is equal to 0.42 and E is an empirical

UT

coefficient equal to 9.8 for smooth walls.

The temperc~ture

e

can be similarly expressed as :

(28) where

e+

is the local temperature adimensionalised with respect to wall temperature

e

w , specific heat cp and heat flux qw :

(29)

kis equal to 0.46 and

E

is

a

coefficient depending of the molecular Prandtl number as :

1 E p+ ::=

-:::-ln-::-k E (30)

(30')

From the assumptjon of local equilibrium, the transport equation for the turbulent kinetic energy reduces to :

and Vt must fits the mixing length relationship (14). Therefore :

2 k

=

.:!!:::.-C 0.5 I' (31) (32) (33)

As it ean be 'observed, under these hypotheses k is constant through the logarithmic layer.

Equations (28) to (33) make possible to cló~e' the equations for the mean flow and the turbulence model at the first intern al mesh point, avoiding entirely the viscous sublayer. In

reason of their simplicity, they have been widely' applied in industrial flow solvers (see, for

example Laurence [24]); they ~re however really valid onIy when momentum and thermal

boundary layers are ~lose to equilibriuIIi~ anel their accuracy sharPly deteriorates when

this condition is not satisfied .. The w~rst performances are encountered in 3eparated flow

regions : as T w = 0 at separation and reattachment (32) predicts k

==

0 ahd ever worse (28)

predicts

e+

= 0 while, in reality heat transfer is maximum at the reattachment position.

It is, therefore, not surprising that .a good deal of effort has been devoted over the

years, to develop wall, function formulations of more general applicability.

The most successfull proposalhas been made by Spalding [25] and is based upon the

assumption that, close to the wall a uniform-stress shear layer (T

=

T w) always exists

(Figure 6), where Vt can be modelled with (12). Using this hypothesis UT) U and k can be

related

as :

(34)

from which a modified logarithmic law can be obtained :

U* 1 . .

-

-ln(E*y~) ",,* (35) e* - -:::-ln(E*y*) 1 -",,* (36) where Yk 0.5 * v .y = c V and e* (37)

kv is the values of the turbulent kinetic energy at the upper edge Yv of the viscous sublayer

defined as the position where. (35) can be ma.tched to the linear velocity law U+ = y+,

which turns out to be Yv

=

20.4. The values of the other model coefficients can be found

requiring the results of (35) and (36) to match (28) and (29) at equilibrium; this gives

",,* = 0.23, K* = 0.255, E* =5.4 and P* = 1.826P:+.

WIP Ie (35) and (36) coincide w~th the logarithmic law in equilibrh,lm flow, they form a much better basis for extrapolation to the case of non equilibrium; for example (36) does not implies a.nymore that beat transfer is 0 at reattachment point. Their use in non equilibrium situation is hetter understood in the fr~e of a Finite Volum~ discretization as the one shown in Figure 6, where the first intemal cell includes the viscous sublayerand part of the overlying region of developed turbulence. On the basis of experiment al data the behaviour of k and f within the viscous sublayer is modelled as :

(

ak

OoS )2

f = 2v

-ay

(38)

(39)

It is a1so assumed, that, above the viscous sublayer and over the entire Finite Volume

eeU, f can be approximated with the usual formulation :

(40)

Practieally in can be assumed, with little error, tha.t k has a constant value k ::;::: kv

over the ceIl for y

>

Yv (figure 6) and, therefore, the transport equation for k can be solved over the eeU :

(41)

where the overbar indiçates Finite Volume average Qver the cello

Advection through the wall is, by def?nition, eClual to 0; diffusion at the wall is also 0 from (38); Production and Dissipation can be averaged over the cell using (35) to (40) to give:

2vk kloS Yn

f - - -

+

---ln-YvYn C'Yn Yv

where Yn is the distance from tbe wall of the upper side of the cello

(42)

Equations (40) and (41) have be.en extensively used in the calculation of turbulent flows, including separated regions, even if"(35) is not a realistic model of the velocity profile near a separation or reattachment point; Improved formulations can be obtained assuming a linear varÎation of k and T across the turbulent layer (Figure 7) or explicitly

modelling the buffer layer between viscous sublàyer and full turbulent region [12]. None of these improvements however can solve two basic limitations of the model, namely the assumption that the value of 20.4 for Yv'" is· universal arid the use of (41), which assumes

€ to be function of the distance from th~ wall only.

The first drawbaék éould be overcome by formulations Üke' [7] :

... 20

Yv = I'

+

3 1 h ok

. • k" oy

(44)

where the partial derivative of k is computed at Yv with a non centered form using only values from the turbulent region.

The second, on the contrary, is ~uch more fundament al bècause is connected to the already discussed weakness of the current models for € in separ~ted regions. Therefore,' the only way to drastically improve the modelling of wall regions in complex turbulent flows is to dispense with wall functions and' solve the flow equations down to the wall through the viscous sublayer. This approach requires to adapt the turbulence model to the region of low values of·the tlJ.rbulent Reynolds numl?er Ret close to the wall, hence the name of

low-Reyn.olds-mt,mber ,models.

Low-Reynolds-number rnodels

Close to a wan the level of turbulence fans to 0 as shown in Figures 3 and 5 with a

I

steep gradient that none of the models des cri bed up to now can match. Therefore the need to introduce damping functions to decrease the production of turbulence in the viscous sublayer has emerged quite early. The oldest, but quite successfull, modification is the one proposed by Van Driest for the mixing length model:

(45)

where A has been calibrated from boundary layer data:

A

=

26 (46) (1

+

11.81p

'!u!:;;)

dz UT 0.5

and X is the direction parallel to the wall.

Obviously (45) reduces the value of 1, and consequently the one of VI, in the viscous

sublayer with respect to the one given by the standard mixing length model. It has been extensively [7] used as viscous sublayer model for attached duet fiows together with a k-E

model for the region of developed turbulence. The values of k and E at the fust such point

can be obtained from Vt using (42) and (43).

Prediction of heat transfer within the viscous sublayer can be improved adopting a turbulent Prandtl number functi<;>n of wall distance as [7] :

Ut

=

1.43 - 0.17y

+0.211

(47)

The mixing length approach to sublayer modelling is simple and economical, but cannot be used for separated fiow~, where qlQre sophisticated roodels are needed; one obvious generalisation of (41) would be to solve the kinetic e,nergy equation a.cross the viscous sublayer to take into account the deviation

of

k from its equilibrium behaviour (38).

Different versions of th~ kmodel for th,e viscous sublayer can be found in the literature and impressive re$ults have ben presented [26], [27] for separated wall fiows. Extensive calculations in problems including heat transfer have been perfQrmed with the model proposed by Yap (see [7]) :

Vt = c_liko.5

[c,y

(1 -

exp( - AIiRe_{t )) ]} (48)

k1.5

€ - (49)

qy(l - exp( -AERet))

where the different coefficients are :

The role of the two damping terms (1 - eÁ )

'is

to dèerease the production of k and inerease its dissipation within the viseous sublayer' to reproduce the deeay of turbulenee

. level with distanee from the wal!.

Ret is a modified turbulent Reynolds number where f ~s repla,eed by the quantity :

(

8k

O.S)2

€

=

f2v

-. 8y (52)

whose signifieation will diseussed below.

Obviously the wan boundary condition for (50) is k

=

O.

The one equation sublayer model has given good results for the predietion of heat transfer in pipes, but, for the already quoted case of flow in a sudden exp~nsion, whieh will be diseussed in more details in the next chapter, the results are even WOTse than for the wall function approach. The conclusion to be drawn is that, in separated flows, f cannot . be eomputed using the equilibrium length seale assumed by (49); instead the transport

'equation for dissipation must be solved across the viseous sublayer .

. To use the k - ~ model right to the wan would seem the "naturaI" way to d~al with wall bounded turbulent flows, but involves overcoming serious modelling and numerical

.problems related to the f transport equation.

The main difficulty lies in the. fact that, as implied by (39), f is not equal to zero at

t • , . • .

the wan but takes a fiilite and unknown value. Different formulations for fw ean be found

in the literature. From (39) fw ean obviously be defined as :

fw _ 2v (8k O • S ) 2 8 y w

Alternatively the linearized version of (53) :

as wen as :

have also been proposed.

fw -

2V(

k₂) . y w

.

(8k

2 ) fw

=

V 8 2 Y w

19

(53) (54) (55)

In view of the obvious difficulty to perform accurate, roeasurements very close to the wall, the first real insight into the behaviour of f for y+ in the order of 1 WijS obtained only

when it became pos si bie to perform very refined Direct Simulation of simple wall flows. On the basis of their numerical results Chapman and Kuhn [28] found (55) to have the correct limiting behaviour and also proposed two simpier ways to extrapolate fw from values of E

and k in the viscous sublayer :

4k

fw

=

V - - f

y2

which is a 1% accurate fit of the numerical results for y+

<

0.6 and :

fw

=

v

(6k _

~

8k)

y2 y 8y

which is 1% accurate for y+

<

1.3.

(56)

(57)

Either (56) or (57) can be used to obtain fw from the computed value fl at the first internal grid point YI provided Yl is close enough to the wall (y+

<

1).

Besides the numerical difficulties introduced by the need to accurately compute

gra.-dient~ of k for y+ ~ 1 the fact that the wall value of f is different from 0 leads to a more

conceptual difficulty, namely that the relationship between dissipation and length scale, expressed as :

(58) does not apply in the viscous sublayer. Matter of fact

k:

·

a

goes to 0 as y3 while I goes to

o

as y. It is therefore necessary to replace f in (58) with a modified dissipation :

i = f-D (59)

which goes to 0 as y2 and approaches f far from the wall. i must replace f any time the

latter is used to define the length scale, as, for example, in the definition of turbulent viscosity Vt

=

cJl k

E 2

• The form of D depends of the one chosen for fw. It is now obvious

that (52) is the formulation of € consistent with (53).

Apart from prescribing € in such a way as to have a correct behaviour of the length scale in the viscous sublayer, a low-Reynolds-number k - f model, like the previously

described k one, requires the introduction of damping functions, reducing the production of Vt and increasing the product ion of f, in order to reproduce the decrease of turbulence

Therefore, the general form of the equations for a low-Reynolds-number k - E model

would be :

(60)

where

ft,

12,

f~ are damping functions, which respectively decrease the production and enhance the dissipation of turbulent kinetic energyj

rr

is the pressure transport term (see Annex 1) which is sometimes modelled separately from turbulent transport and E is an additional term introduced to force (62) to exhibit the correct behaviour in the viscous sublayer.

An impressive amount of effort has been devoted to the development of low-Reynolds

k - E models and a complete review can be found in [29]. As an example we will be giving

below the complete formulation for the very last one, proposed by Shih and Mansour [30] on the basis of results from Direct Simulation of turbulent cluinnel flow :

(63) with al

=

6 x 10-3, a2

=

4 x 10-4, a3

=

-2.5 x 10-6, a4

=

4 x 10-9

ft

= 1

(64)

f2

_-

1 - 0.22exp(

36

-Ret

)

(65)

E

(8

2

U)

2 = VVt 8y2

(66)

rr

_f~(l 0.05

- - -

Vt 82k

(67)

- exp( -y+)) Uk 8X k2

Prom the above, it is clear that one practical drawback of the low-Reynolds-number model lies in the high resolution required in the near-wall regiou. As already shown, the

first mesh point must Heat a distanee from the wall smaller than 1 in wall coordinates, and a typical mesh will therefore have 4-5 points for y+

< 4 and 15-30 overall for

y+

<

30, instead of the ONE point for the wall function formulation.

In spite of their cost, however, low-Reynolds k - € models have gained widespread acceptance in view of the sounder physical hypothesis they are based upon and the improvemeIit of results they offer for a variety of applications. Some relevant results for heat transfer problems and some conclusions about their range of validity win be presented in the next chapter.

6. THE k - € MODEL FOR HEAT TRANSFER CALCULATIONS "

As already said the k - € model is the basis of most general flow solvers available on

the open market and has been extensively used in the calculation of complex flows ([21], [24]); recently, a review of typical applications has been made by Rodi [31].

Specifically, in the domain of heat transfer, the low-Reynolds-number vers ion of the model has provided quite satisfactory predictions in a number of different cases.

Cotton and J ackson [32] have studied mixed convection regime in vertical tubes. The predicted evolution of Nusselt number along the length of the tube is in quite good agreement with experimental data, as 'shown in Figure 8 for different values of Reynolds and Grashof numbers. Adimensional velocity and temperature profiles also agree very weil with experience (Figure 9).

To and Humprey [33] have presented results for free convection along a vertical heated flat plate. Heat transfer predictions (Figure 10) are in"good agreement with most experiment al data. The same can be said for predicted velocity, mean and rms temperature profiles, which are compared with measurements in Figure lla to llc respectively.

Humprey and To also studied [34] free and mixed convection in a heated cavity (Figure 12) with equally satisfactory results for global heat transfer (Figure 13) and temperature profile (Figure 14).

Natural convection in a rectangular enclosure (Figure 15) has also been studied by Inee and Launder [35], who found good agreement for the heat transfer coefficient along both cold and hot wall (Figure 16) and for the temperature distribution across the enclosure (Figure 17). The critical importance of a correct modelling of the transport equation for

€ is put in evidence by the significative changes brought by the use of the source term SE

of (26).

In spite of these impressive results, the limitations of the k - € model quickly become

evident as soon as more complex geometries are considered. A striking example is the already quoted problem of an abrupt pipe expansion (Figure 18) : predictions of longitu-dinal evolution of N usselt number downstream of the expansion are shown for 'two different values of Raleigh number in Figures 19a and 19b respectively. The experiment al data are compared with three different calculations, using respectively wall functions (35) and (36),

k equation wall model (48) to (50), and a.1ow-Reynolds-number k-€ model: all predictions

show the strong increase of heat transfer downstream of the expansion, but they all are significatively different from measurements in an important wayj the most striking point is that all models show a far too small sensitivity to Raleigh numberj results are acceptable for the lower Raleigh but are far off the mark for the higher one. Another important point .

is that the kmodel actually performs worse than the wall functions, pointing again to the limitations of a model depending upon a near equilibrium model of the length scale when applied to separated flows.

There is a widespread agreement ([7], [9], [10], [15]) that eddy viscosity models are fundamentally inadequate for the prediction of turbulent complex flows because they cannot, by their nature, take into account some fundament al physical features. The key to this failure lies at the very heart of the model, namely with the hypothesis of isotropy

of the turbulent stress implicit in (7), which lead to a quite erroneous prediction of the normal stress UiUi. For the simple case of flow at equilibrium, (7) reduces to :

-

~k

3 (68)

and a single glance to Figure 3 will show how far this assumption is from the reality : for simple equilibrium flows like boundary layers and duct the longitudinal stress is twice as big as the two transversalones. In the case of flow at equilibrium, the error on UiUi

does not have adverse consequences upon the prediction of the mean momentum transfer, because, a;~;i

=

0, but it is an important source of errors in non equilibrium problems.

Matter of fact the most significative non equilibrium effects, such as arise from curva-ture, separation, swirl and buoyancy interact in a different way with different normal and shear stresses thereby promoting anisotropy. The reaction of individual stress components to anisotropy promoting forces applied by the mean flowfield cannot be correctly modelled if a single isotropie parameter Vt is used to relate the tWOj therefore the model fails when

any of these effects become significative.

Another problems, specific of the low-Reynolds-number k - € modeis, lies in the

treatment of the near-wall region. Experiments show that the velocity component normal to the wall decreases much more quickly than the ot her two, and that 'a significative redistribution of energy between the different components takes place. On the contrary the modelled damping terms act upon the kinetic energy and so decrease all the three fluctuating components in an isotropie manner. Therefore the model fails to reproduce a fundamental aspect of the phenomenology of turbulence near a wal!.

The limitations of the . kE . model are even morè serious for heat transfer calculations

because anisotropic effects are important even for cases of flow at equilibrium. It is weil known [15] that in pipe flow turbulent heat fluxes in the flow direction are two or three times larger that in the radial directionj the ratio ofaxial versus radial turbulent heat diffusivity, shown in Figure 20, clearly indicates how far the isotropy hyp~thesis is far from reality ! The discrepancy is further aggravated by the effect of buoyancy : experiment al data for vertical heated pipe flow [36] show a reversal of the direction of the axial turbulent heat flux (Figure 21), which cannot be taken into account by eddy viscosity modeis.

These limitations could be partially overcome by the introduction of an anisotropic turbulent diffusivity, like the generalized gradient diffusion hypothesis (see [7]) :

where Co = 0.3.

-

koe

Ui(J =

-COUkUi---f OXk (69)

It is, however, clear from (69), that the quality of the heat transfer prediction will dep end on the accuracy with which the Reynolds stresses are known. There is therefore, by now, a widespread agreement ([7], [10], [11], [13]) that the only way towards correct prediction of turbulent transport phenomena in complex flows lies in the solution of the transport equations for the individual Reynolds stresses or, to use the common definition,

. . ' .

in second moment models of turbulence.

7. THE REYNOLDS STRESS MODEL OF TURBULENCE

Transport equations for UiUj and u/} can be obtained from the corresponding ones for

Uj and 8, applying the Reynolds decomposition, taldng t~e fluctuating part, multiplying by Ui and averaging. The resulting equations, are, in descriptive form :

dUiO . dt

düTûj d . .

dt

=

Pro ij

+

TrSPij

+

TIij

+

DzJ Jij

+

D'I,SSij

+

Fij

- Prodi9,l

+

Prodi9,2

+

TrsPi9

+

TI i9

+

+DiJ Ji9

+

Dissi9

+

Fi,

(70)

(71)

In a similar way, an equation for the rms fluctuation of the transported scalar (}2 can also be obtained :

d:

t

2

=

Prod9

+

Trsp9

+

DiJJ9

+

Diss 9 (72) with the same nomenclature as for (16); TI is the pressure-strain term which contains the correlation between pressure fluctuations and turbulent strain rate, while _{PiS,l and Pi9},2

are respectively production by mean sca.lar gradient and by mean strain rate. The exact mathematical form of each term can be found in Annex 2.

As discussed in Chapter 51 the viscous transport term can be dropped for Ret

>

100

leading to the high-Reynolds-number model which will now be discussed in some detail.

It is easy to realize the difficulties related to the use of a Reynolds stress model for practical calculations : from the theoretical point of view it requires to model high order terms like triple correla#ons of velocity and correlation between pressure and fluctuating strain rates for which very few information were available before the advent of Direct Numerical Simulation; from the numerical point of view (70) to (72) form a set of 10 (I)

transport equations to which, as it will be seen later, the transport equation for f must be

added.

It is not, therefore, surprising, that, in the recent past, a considerable effort has gone in devising simpier relationships to replace (70) to (72).

Aigebraic Reynolds Stress Model

The most radical way to simplify the Reynolds stress model is to pass from partial differential to algebraic equations expressing the transport terms in (70) to (72) as function

of the production and dissipation of turbulent kinetic energy and of the rms f} fluctuations [7] : (73) dUif} - - -Trsp'e -_{dt .} I Uif} ( ) uif} ( .

2k_o.5 Prodk - €

+

2f}2 Prode - D'tsse) (74)

and modelling PrOdij , IIij ecc ...

A detailed review of the algebraic stress models (ASM) has been made by Lakshmi-narayana [37]; here we will show, as an example, the very simple "WET" (WEALT H ex

EARN INGS X T IM E) proposed by Launder ([7],[22]), which, more mathematically,

reads :

value(uiuj) ex Generation rate(uiUj) x Turbulent time scale (75)

leading to the expressions :

2 k 1

:;: -h·_{3' IJ} ·k - c _S -[Prod_{€ IJ} "

+

F, _IJ .. - -h _{3 IJ} .. (Prodk

+

Fk)] (76) (77) (78) with Cs

=

0.26, Ce

=

0.3 and R = 0.8.

Equations (76) to (78), or the corresponding ones for any ASM moçlel must be solved together with the transport equations for k and €, with the production term rewritten in

functibn of Ui u j instead of Vt.

(79)

(80)

Equations (76) to (78) being algebraic, boundary conditions are required only for (79) and (80); therefore any of the approaches discussed in Chapter 5 can be adopted.

ASMs make it possible to take into account the anisotropy of turbulence with a computational effort which is still moderate (roughly twice the one required for a standard

k - f closure), and, for many problems, provide a dramatic improvement in the quality of

the results. One striking example is the al ready discussed case of heat transfer downstream of a sudden expaI)sion, for which ASM predict, for the first time, the correct dependency of heat transfer on Raleigh number, as shown in Figure 22 a and b. Improvements were also found for the case of natural convection from a heated vertical plate [33] as shown in Figures 23 and 24.

There is, however, some agreement [9] that ASM usefulness is limited to relatively simple problems where the main flow is essentially uni-dimensional and the simplified transport model embodied ip. (73) and (74) is not too far from reality. The only wayopen to a truly general treatment of complex turbulent flows lies in the jull Reynolds stress closure.

Full Reynolds Stress Model

The development of a satisfactory model for the Reynolds stress (RMS) has shown it'self to be an exceedingly difficult tasks. Although modelled forms of (70) have been available since the pioneering work by Rotta in the early 50s, they did show themselves, in the beginning, quite unsatisfactory, yielding results scarcely better than the ones obtainable with eddy viscosity models. However, over the last ten years, improved numeri cal tech-niques and better physical understanding of turbulence have brought tremendous progress and definitively shown the RMS tc! be the way of the fut ure (or, at least the way of the "next" future) for the predictjon of complex turbulent flow. While further improvemèri.t is still needed (see [9] Çlnd [13] for a detailed discussion), Figures 25 b, c, d clearly show the dramatic improvement that Reynolds stress models already offer with respect to standard

k - f ones for the classical problem of boundary layer d,evelopment in a faired diffuser

(Figure 25 a).

The results of Figure 25 have been obtained using the simplest and most widely used version of the RSM [10], where the different 'tenns are modelled as :

Production :

Prod_IJ ·· (81)

Dissipation :

Turbulent Transpór~. :

(83) Buoyancy Force :

(84)

The greatest difficulty lies in modelling the pressure-strain correlation ITij, whose physical role is to redistribute energy from thestreamwise to the transversal velocity components, and provide a sink for the shear stresses thereby promoting theisotropisation of turbulence. Therefore _{ITij must be modelled to be 0 in isotropic turbulence and} proportional to the flow anisotropy otherwise.

It was found necessary to model separately the contribtitiön of the fluctuating strain rates:

of the main strain rates :

. IT ij,2 - -C2

(pr~dij

-

~bijProdk)

and of the buoyan~y force' :

IT"3 IJ,

(85)

(86)

(87)

A furthér term was foun'd necessary to model the teflection of pressure fluctuations from the wall and its influence upon the turb\llent behaviour in the wall region, namely the rapid decrease of the turbulent fluctuation in the direction normal to the wall and the consequent redistribution of energy to the other components :

ITij,wl (88)

( 3 3 ) k

1 .. 5

-C2 w ITk m , " m IJ 2n.·n b" -_. ' -ITL2 .. I, ' 2nLn' -.. J -ITL2 "J, ' 2nLn.. I ' - -C _L€Y (88')

(88")

where nk is the unit vector in the direct ion normal to the wall and only the terms of the

type nlnl will, obviously, be different frox;n O.

This formulation represents the simplest way in which the pressure-strain correlation ean be expressed and more sophisticated formulij.tions areava.ilable. A mOre detailed diseussion can be found in [22], while the most recent proposals are discussed in [30], [38] and [39].

The cQmmonly accepted values of the different coeffieients are [10] :

.

,

Cs

=

0.22; Cl

=

1.8 ; C2 ::;: 0.6 ; C3 ::;: 0.6 ; Clw

=

0.5 ; C2w

=

0.3; C3w

=

0; CL

=

2.5

(89) It is dear from (82) that it is still neeessary to solve the dissipation transport equation, for which the suggested form is :

(90)

The equatiQn for the I?~ar

flwc

UiB

has

received up to now, mueh less attention,

beeause of a feeling that it Was premature to devote [39) muçh effort to it before the Reynolds stress model was {uUy reliable. Therefore much work has still to be done, above all concerning the sealar dissipation J;"ate fB [9, 22, 39].

The latest modelling proposal [39], without low-Reynolds-number corrections, takes the form :

Produdion :

Dissipation :

Turbulent Transport:

ProdiB1 , ~ -UiUk"""'--'-

as

aXk -aUi - -UkB -aXk DisSi8 - 0 . TrsPi8 (91) (91') (92) (93)

Buoyancy Force :

(94)

Return to isotropy term :

IIiS,l (95)

Mean strain rate term :

(96)

Body force term :

(97)

Wall reflection term :

€ _ k1.5

. IIiS

_,

W ::;: -Cu

_,

w-kUI.:fJnkni-

_fY

(98)

where the model coefficients take the values :

Cs ::;: 0.15 ; CIS = 3.0 ; C2S ::;: 0.4 ; C3S ::;: 0.33 ; C18,w.::;: 0.75 (99)

It has to ~e remarked that a considerable uncertainty still exists on the correct model for the wall reflection te~, and that Lai and So actually conclude that it should be omitted.

Finally the transport equation for

7fi

can be modelled as : Production : Prods

-

~2Uk8-

-

.

-88

8Xk (99) Disss - ---81 f -2 Rk Dissipation : (100) Turbulent Transport : (101)

Alternatively, a transport equation for €s can be found in

[9].

As already pointed out the above described formulation is the "basic" RMS as has evolved in the 705 and early 80s. In the last couple of year5 a tremendous efforl has been

devoted to improve the modelling of the different terms and more refined formulations are presently being tested. Very promising results have been obtained for simple test problems, but none of these models has, up to now, been applied to the study of truly complex flows of practical interest. They will not, therefore, be included in the present review; interested readers are referred to 30 and to [39] to [42] for further details.

Wall Boundary Condition for Full Reynolds Stress Model

The problems related to the introduction of wall boundary conditions in the Reynolds stress model are similar to the ones discussed in Chapter 5 and most of the approaches there presented can be straightfowardly extended. Broadly, three possible methods to treat the wall layer can be identified :

k - € model

The standard k - € model can be used, together with any of the wall formulations discussed in Chapter 5 (wall functions; k low-Reynolds model, k - € low-Reynolds model), to yield values of k and € in the high-Reynolds region. The con'esponding values of UiUj, UiB, 82 can then be found using WET relationships (76) to (78) or any other ASM.

wan function

The logarithmic layer model or the constant shear layer model for the wall region (see Chapter 5) can both be extended to Reynolds stres~. ~losures. Values of the individual stress components consistent with the logarithmic law are given, for example, by Sebag and Laurence [43], but by far the most common approach is based up on an approximate solution of the transport equations for' the individuàl stress components over the first mesh cell (see, for example Leschziner [44]), using the same approach as for the turbulent kinetic energy.

Therefore, the discrete equation :

PrOdij

+

TrSPij

+

IIij - €ij

+

Fij (102)

together with the corresponding ones for ud) and B2, is solved on the control volume similar to the one sketched in Figure 6, under the assumptions that advective and diffusive transport are both 0 at the wall, uv ~~ is the only not 0 contribution to the production term, and €ij is equal to

f'f

given by equation (43).

Low-Reynolds-number models

The range of applicability of the Reynolds stress model can be extended to the viscous sublayer using modifications and damping functions similar to the ones discussed. for the k

. and f equations in Chapter 5. However, up to now, much less attention has been devoted to the subject than 'for the case of eddy viscosity model and no weU established and widely accepted model can be yet said to exist. Proposals' of low-Reynolds-number closures for the Reynolds stress equations can be found i~ [30] and [42], while the only such closure for the heat transfer equation the author is aware of is the one proposed by Lai and _, 80 [39]. This last model has been applied to the prediction of flow in a heated pipe and a very good agreement found with experiment al data, with respe<;t to both meao flow parameters (Figure 26) and turbulence (Figure 27).

8. CONCL USIONS

The present pace of progress in turbulence modelling is so fast at any review paper

risks being hopelessly out-of-date by the very time it i~ printed. The present contribution

is not and cannot hope to be an exhaustive analysis of all that has been done in the field of turbulent scalar transport modelling, but an at tempt to summarize the conclusions of recent research and the main trends for the near future.

Two main conclusions emerge : the wall function approach to near-wall turbulence is dramatically inferior to low-Reynolds-number modelling for non equilibrium flows including separated regions and RSMs have been shown to be practically feasible and definitely superior to eddy viscosity modeis, especially in so far as scalar transport is concemed.

It now seem possible that present research will soon lead to robust and accurate

low-Reynolds-number RSMs, making possible to perform heat/mass transfer predictions of engineering usefulness in complex flow regimes.

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