Analysis of swirling jet turbulent mixing and combustion

ANALYSIS OF

SWIRLING JET TURBULENT

MIXING AND COMBUSnOll

by LUCHTVM .•

"'-Klu ... · f\."] i - ~l:LFT

,.

ANALYSIS OF

SWIRLING JET TURBULENT MIXING AND COMBUSTION

Submitted February, 1981

Marc.h, 1981

by

J. P. Sis1ian

UTIAS Report No. 249

Acknowledgements

I appreciate the discussions with Prof. I. I. Glass during the preparation of this report.

I wish to thank Mrs. Winifred Dillon for typing the manuscript. This research was financially supported by the Natural Sciences and Engineering Research Council of Canada under strategie Grant G0105. This support is gratefully acknowledged.

'

.

Sunnnary

The present +eport is concerned with the analytical investigation of flows generated by swirling j et turbulent mixing and combustion. The full Navier-Stokes equations for momentum and energy conservation, as weIl as for global continuity and species continuity are derived for the mean mass-averaged variablep of a campressible, turbulent, nonswirling or swirling plane or axisymmetric flow in orthogonal curvilinear body-oriented system of coordinates, which contain, in general, the six components of the

Reynolds (turbulent) stress tensor. The Saff'man-Wilcox modelling approach is used in deriving the Reynolds stress model equations in the coordinate system considered. A simplified cambustion model is assumed, represented by a one-step rate process between just two species, fuel and oxidant. No attempt is made to include fluctuations of concentrations and temperature in the mean reaction rate. In order to test the validity and reliability of the derived model equations in the simpIer case of turbulent weakly swirling and combusting flows, a complete set of second-order boundary-layer (parabolic) equations is derived. It is believed that the numerical

solution of the presented equations will provide more insight into the mechanism of swirling jet turbulent mixing and combustion than has been

achieved so faro More over , through this work direct checks on the validity of a variety of turbulence medels may be made via extensive camparison with experimental data and an objective viewpoint expressed.

Contents

Acknowledgements _ii

Sumnary _iii

Notation

1. INTRODUCTION

2. MEAN FLOW EQUATIONS

3. DERIVATION OF THE REYNOLDS STRESS MODEL EQUATIONS

4.

TURBULENT KINETIC ENERGY AND DISSIPATION RATE EQUATIONS 5 • M:>DELLING OF THE TURBULENl' HEAT AND MASS FLUXES

6.

TWO-EQUATION MJDEL OF TURBULENCE

7.

THE COMBUSTION MODEL

8.

BOUNDARY-LAYER TYPE EQUATIONS OF MOTION

9.

BOUNDARY CONDITIONS

10. C ONC LUS IONS REFERENCES FIGURES

APPENDIX A: CALCULATION OF THE DERIVATIVES OF A SECOND-ORDER TENSOR IN BODY -ORIENTED ORTHOGONAL CURVILINEAR COORDINATES APPENDIX B: DERIVATION OF THE BOUNDARY-LAYER TYPE (PARABOLIC)

EQUATIONS FOR SWIRLING JET TURBULENT MIXING AND C OMBUST ION v 1 3 10

18

19

21 21

23

29

29

31

a cp Cl' C2 , C₃, C4 e H h h i -;7 J k L Le t M Pr p ~ q R Re r Sc T t u v Notation

speed of sound; denotes type of flow: a

=

0 plane flow, a

=

1 axisymmetric flow

specific heat at constant pressure closure coefficients

mass-averaged turbulent kinetic energy total (stagnation) enthalpy

specific static enthalpy of the mixture, h

_{= 'a.h.}

L, ). ).

specific static enthalpy of species i i diffUsional flux factor

heat conductivity

characteristic length of flow in x-direction Lewis number

dissipation length scale molecular weight

Prandtl number pressure

heat conductidn vector

universal gas constant; radius of curvature of the body surface Reynolds number distance to x-axis Schmidt number temperature time x-component of velocity y-component of velocity

w

V

Wi x Y

cx, cx*

t3, t3*

I 5 t € K I-l V L v_t ~*, _{~l' ~2} p CT, cr* # T cp W

V

Sub scripts t i Super script s t ,....

circumferential velocity component

total velocity vector

pet mass rate of production of chemical species

longitudinal coordinate along surface of the body

coordinate normal to the body

species mass fraction; angle of body with x-axis; turbulence closure coefficients

turbulence closure coefficients

specific heat ratio

characteristic length of flow in y-direction

mass-averaged turbulent dissipation rate ;

curvature of the body

coefficient of viscosity

kinematic viscosity

eddy diffUsivity

turbulence closure coefficients

density

turbulence closure coefficients

viscous stress tensor

cylindrical or azimuthal coordinate

dissipation rate quantity

divergence operator

denotes turbulent quantity

d ene es t ~ .th spec~es .

denotes turbulent quantity

mass-averaged value

*

11 nondimensional quantity time-averaged value time-averaged fluctuations mass-averaged fluctuations

1. INTRODUCTION

The shortage of energy, the achievement of a clean environment, and health and safety issues, have resulted in very serious research and develop-ment activities in the areas of combustion and combustion-system design. Increased concern exists for predicting and restricting combustion-generated pollutant emissions while simultaneously maintaining and improving thermo-dynamic efficiencies and fuel economies of fossil-fuel combustion devices. Indoubtedly, this will require new, perhaps radically different, cambustor designs and, possibly, the use of alternative fuels.

All combustion processes depend on conversion of fuel and oxidant ~o

combustion products by chemical kinetic mechanisms as weIl as by the usually turbulent transport of mass, momentum and energy resulting from the fluid motion. An adequate description of kinetically-influenced phenomena such as ignition, propagation and extinction of flames, the role of short lived reaction intermediates, and the reaction paths to pollutant formation requires hundreds of chemical reactions and chemical species. The rate constants for many of these reactions are uncertain within a factor of two to ten, and for some complex fuels complete mechanisms and rate constants are simply not available. The problem is further compounded by the turbu-lent motion of gases in the combustor. Turbulence is highly desirabIe in cOnIDustion processes at it pramotes rapid fluid transport of species and energy for optimum combustion efficiency. However, reliable and generally applicable turbulence models have not yet been developed even for non-reacting flows.

All combustor problems entail the solution of many simultaneous non-linear and stiff equations. The stiffness is due to characteristic-time or length scales which are extremely disparate. The fluid-time scales are very lID.lch slower than chemical-time scales which themselves are disparate by many factors. Problems are classed according to degree of realism and refinement, as represented by their dimensionality and type (parabolic or elliptic). The governing system of equations is usually so complex that only computational methods yield practical solutions. However, modelling of both the detailed chemical kinetics and turbulent fluid mechanic pheno-mena would tax even advanced computers beyond their capabilities.

Realizing that complete models of real combustors would not fit on present computers, investigations have pursued the avenue of partial

mOdelling. When complex chemical IOOchanisms are incorporated, the primary goal of the model is the prediction of a "fundament al combustion process"

(for example, pollutant formation); in this case, due to the large nurnber of species conservation equations needed and their stiffness, only relatively simple flow situations are considered. When complicated fluid dynamic

phenomena is considered the models are regarded as "system simulations ", and account for rather complicated COnIDustor flow description but approxi-mate the chemical kinetics by relatively simple rate processes. Thus, the two types of models must be used together to gain maximum information from the modelling effprt.

The present investigation is part of a research programme re.cently initiated at the Institute for Aerospace Studies, University of Toronto, on turbulent swirling and combusting flows. The emphasis i s on utilizing new modelling and diagnostic techniques of turbulent fuel/air mixing processes,

and, therefore, may be characterized as system simulation modelling rather than fundament al processes modelling. It is believed that for the type of combusting flows considered, fundament al processes modelling (i.e., the inclusion of complex chemical mechanisms) would not yield realistic results if the system simulation mOdelling is not adequate.

Usually all practical industrial flames have the form of turbulent jets issuing from round orifices, the fuel-gas being introduced through a central jet and the air through an annulus surrounding it. However, in order to enhance the fuel/air mixing and combustion processes, the primary or the secondary jet or both are given a certain degree of swirl. Schwartz's experiment (Ref. 1) demonstrated that rotating a propane-air mixture in an annular combustion chamber results in appreciable shortening of the flame length, increased flame divergence, improved stability characteristics and delayed blow-off. Swirl-can combustors (Ref. 2) have also utilized fluid rotation as well as special geometrie configurations to achieve greater combustion stability. There are two important features that distinguish the turbulent mixing of swirling flows from the mixing of nonswirling flows. Firstly, the shorter mixing distance achieved in the swirling flow may be thought of as resulting fram a change in the system's turbulent viscosity, and, secondly, swirling jets with a sufficient flux of angular momentum

may experience adverse axial pressure gradients severe enough to cause

recirculating flow.

Swirling flows with or without combustion in round and annular tubes have been studied both experimentally and theoretically (see, for example, Refs.

3-29).

Almost all analytical studies have utilized various

exten-sions of Prandtl'-s mixing length concept to flows with swirl. More recent work on calculating turbulent swirling flows (see, for example, Ref. 22) has postulated that the turbulence may be adequately described by two

quantities: the kinetic energy of turbulence and some other quantity which may be the length scale of turbulence, the dissipation function (Ref. 30) or the dissipation rate of turbulence (Ref. 31). The turbulent viscosity is expressed in functions of these two variables (eddy viscosity hypothesis) and two extra partial differential equations are required to determine them. These models exhibit greater universality than the mixing-length model

for flows without swirl, transport effects on turbulent viscosity can be accounted for, and the length scale, being determined by a differential equation, does not have to be given an empirical distribution. These "two-equation mo<Îels" of turbulence determine a unique value of the turbu-lent viscosity independent of the direction of the shear plane (isotropic turbulence). The turbulent stresses are obtained by multiplying this vis-cosity by the corresponding components of the second invariant of the mean flow rate of deformation tensor. However, Lilley and Chigier (Refs. 15, 16) have shown that swirling flows exhibit a significant anisotropy of the turbulent viscosity, the effects of which cannot be satisfactorily allowed for by any self-evident arbitrary extension of turbulence models which

serve to predict nonswirling flows. For accurate prediction of turbulent swirling and combusting flows, attention should turn to the modelling of the transport equations for the Reynolds (turbulent) stresses themselves

(i.e., the eddy-viscosity hypothesis should be dropped).

_,

'

It is difficult to assess the degree of validity or the limitations of calculation procedures based on such advanced models of turbulence because

of the lack of adequate time-averaged flowfield measurements, which could provide reliable test cases. Furthermore, there is a lack of

turbulent-structure information, which could be used to assess the validity of

present turbulence models or to illustrate their shortcamings and guide the formulation of improvements. Turbulent swirling and combusting flows have been extensively studied experimentally in the past and this has been reviewed

in Ref s.

3

and

19,

Flowfield measurement s have been made wi th directi on sensitive water-cooled pressure probes, hot-wire anemometers and cooled turbulence probes • However, experimental investigation of complex turbulent and cambusting flowfields, such as occurring in swirl cambustors, is difficult to perform with any degree of reliability using conventional instrumentation. For instance, streamline curvature and the associated static-pressure varia-tions, characteristic of swirling flows, make conventional mean flow instru-mentation techniques unreliable. Problems arising in performing

turbulent-structure measurements are even more severe, because linearized hot-wire data interpretations are not accurate in these highly-turbulent flows. Besides, hot-wires are difficult to use in the hostile, high-temperature environment present in flames. Fortunately, with the advent of the laser-Doppler veloci-metry, linear, nonperturbing, fluid mëchanical measurements of complex three-dimensional flowfields are now feasible. Attempts at using laser-Doppler velocimetry as applied to turbulent swirling and combusting flows have already been reported in the literature (see, for example, Refs. 24, 25).

The present report is concerned with the analytical investigation of

swirling jet turbulent mixing and combustion problem. The full

Navier-Stokes equations for momentum and energy conservation, as well as for global continuity and species continuity are derived for the mean mass-averaged variables of a compressible, turbulent, swirling or nonswirling axisymmetric

or plane flow in curvilinear orthogonal body-oriented system of coordinates, which contain, in general, the six components of the Reynolds (turbulent)

stress tensor. These stresses are determined from a turbulence model based on the compressible Reynolds stress equations. Saffman-Wilcox modelling approach is used (Refs.

31-33)

in deriving these equations in the coordinate system considered. A simplified combustion model is assumed, represented by one-step rate process between just two species, fuel and oxidant (Ref.

34).

No attempt was made to take into account fluctuations of concentrations and temperature in the mean turbulent reaction rate. It is reasonable to test the validity and reliability of the derived model equations in the simpler

case of weakly swirling, turbulent combusting flows. To this end, a complete

set of second-order boundary-layer (parabolic) type equations is derived. The results of numerical calculations and their comparison to experiment al

results for the mean flow velocity components of a weakly swirling jet and the second-order velocity correlations (Reynolds stresses) obtained by laser-Doppler velocimetry, will be presented in a subsequent report.

2. MEAN FLOW EQUATIONS

The general equations describing flows of a reacting mixture of perfect gases in vector form are given by:

Species Continuity: Momentum.: Energy:

de

*~-p)

+

ft· [ pV

H +

ë{

+

I

h_i

~

-

(~

. '"

J

= 0 i Equation of State:

I

0:. P

=

RTp ..2:. M. • l. l. (2) (4 )

where

'f/

is the divergence operator; p, ~ pare the density, temperature and pressure of the mixture, respectively; V the velocity vector;O:i the mass

fraction of the i th species; ~ the diffusional mass-flux vector,

Ji

=

p(Vi-~; Wi the net mass rate of production of species i per unit volume by chemical

reaction (grams of i per cm

3

per sec);

~

the viscous stress tensor; H =

~

o:.h.+

l. l. l.

+

V . V/2

the total (stagnation) enthalpy; hi the specific enthalpy 0 f the

i th species;

ct

the heat-conduction vector; Mi the molecular weight of the i th species; R the univers~l gas constant. Constitutive relationships for the viscous stress tensor

'f,

for the diffusional mass flux vector ~ and for the heat conduction vector ~must be added to these equations in order to obtain a closed system. Thus, for Newtonian fluids, Le. fluids such that the vis-cous stress tensor is a linear function of the rate of strain, the visvis-cous stress is given by:

(6)

with def

V = grad V

+ (grad "V)T, the superscript T 5ienoting the transpose of a tensor, ~ being the coefficient of viscosity and

Y

the unit tensor. The diffusion of each species is assumed to depend only on the gradient of the particular species mass fraction. The assumption requires that the binary.

diffusion coefficients Dij for each species are equal and the simple Fick's law is applicable. Thus

where Le and Pr and the Lewi s and Prandtl number s, re specti vely. Furthermore, the fluid is assumed to obey Fourier's law of heat conduction for ~

-~ q

=

-k grad T c ~ - - P~ grad T ( 8)

-where cp = ~ O:.cp., cp' being the specific heat of species i.

The body-oriented orthogonal curvilinear coordinate system used is shown in Fig. 1, where x is the distance along the surface of the body

(measured from a certain point 0) and y the di stance ~ormal to this surface •

The corresponding components of the velocity vector V are u, v and w. R(x)

is the radius of curvature of the body, reckoned positive for a convex body, and a(x) its angle with and ~(x) its distance from the axis. The length element is given by

where the metric coefficients are

r

=

r_b + yCOffY-, k(x)

=

l/R(x)

₍₁₀₎

a

=

0 for plane and a

=

1 for axisynnnetric flows, ojOcp

=

0, and cp is the

lateral (z) coordinate in plane flow and the azimuthal angle in axisymmetric

flow. Then from the usual relations for vector operators in orthogonal

cur-vilinear coordinates the continuity equation, Eq. (1), is found to be

(11)

Using Eq. (7), the species continuity equation, Eq. (2), becames:

i

=

1,2, ••• , N (12)

where Sc = Fr/Le is the Schmidt number, and N the number of species considered.

The Navier-Stokes momentum equations are (Ref. 35):

x-momentum: .

o~@u)

+

~

~3

r

~

_{(h3 Bxx)} +

d

dY

(hl h3 Byx)

J

+ kB

h~y-

asin:x _r B

=

0

<pep (13)

y-momentum:

o~ev)

+ 1

r

0

(h ) _{+ dy} o _{(hl h3 Byy)}

J

_- kB

_h~-

aco~

Bcpcp

=

0

y-momentum:

where

2

8

=

pu + P - ~ .

XX

'xx'

and the relationships

2 8 _yy

=

pv P - T _yy ; 8 _qxp =pw +p--r 2 _qx:p 8

=

pvw - T ycp ycp 1

dr

.

- ""-"= = s~nx hl OX and 1

dr

h 2

dY

= eog); (16)

were used. The viseous stresses _{TXX ' ••• ,T}ycp are given by Eq.

(6),

and in the eoordinate system eonsidered are:

T

=

IJ. [

2

(~

+

kv) -

g

di

vvl .

xx

hl OX 3

J '

Tcpcp = IJ.

r

2:

(usinx + veog);) -

~ diV~l

With Eqs. (7-8) and relations

~

=

I

_{hi (T)}

:1

+

I

ai epi

~

+

~ (~2

)

where C_pi

equation,

i i

/ 2 2 2 2 /

= dhi dT, V = (u + v + w ) 2, and assuming Le = 1, the energy Eq. (4), beeomes:

d(pH-p) 1

r

d(h3 puH) d

l

1 { d

r

h3 11 d (, V2

)l

~

+ hlh3

dX

+

dY

(~h3PVH)

J

= hlh3

dX

hl

Fr

dX

\H -

'2

J

+ +

~

[

hlh3

~r ~

(H -

t)]

+

~

hh

_xxu ₊

;

l0,{V

_{+ Tx<pW) ] +}

~

hh3

h

l0,{U + + T V + T w)

l

1

yy yq>

J

r

(18)

In order to obtain the governing conservation equations for turbulent flows, i t is convenient to replace the instantaneous quantities in the above equations by their mean and their fluctuating quantities. In this treatment, the mass-weighted-averaging procedure is used. With this procedure the resulting equations for the average turbulent flow quantities will have a form very similar to the equations for laminar flow.

If f is any flow quantity, the conventional time average of this quan-tity is denoted by

f.

Then, the mass average of any quantity except density, pressure and viscous stresses (fluctuations of transport coefficients are neglected) is given by

The quantity f may then be written as

'"

f = f + f"

where f" is the departure oF fluctuation from the mass-averaged value f. It should be noted that

f"

f

0; f" = -p'f'/p. Also, it is easy to show that pfll = O. Representing the density, pressure and viscous stresses by the sum of their time average and its departure or fluctuation from the time average, namely

-p

=

p + p', p=p+p', T . .

=

T . . + T! .

1J 1J 1J

(19)

and the veloeities, the total enthalpy, statie enthalpy, temperature and mass fractions by their mass average and its departure from the mass ave rage , namely '" ..., _{= ;; + Wil =}

_H

_{+ H"} ...., U = U + Uil, _{v=v + v",} W

,

H

,

h = h + h" ( ai (J:Xi =

~i

) (20) =

T

+ Til ..., +

a:'

T

,

a.

₁ =

a.

_{1 1}

=--P P

sUbstituting these quantities into Eqs. (11)-(18), and app1ying the Reyno1ds rules of averaging (Ref. 36) yie1ds the fo11owing conservation equations for p1ane or axisymmetric turbulent f1ows:

G10bal Continuity:

Species Continuity:

i

=

1, 2, ... , N

x-momentum:

o (-'"

1

[0 (

-",2 +

0

(h h -"""')

l

+

kpw

asira -"'2 _

dt

pu) +

h1l

~ h3 pu) ~ 1 3 puv ~ - r

Pw-1 3 -J.

asi~ - - t

r (Tqxp + p T~

y-momentum:

O~pv)

_1_

r

0

(h _.""') +

0

(h h

-~)]

_

kpu

2 _ acos:x -",2

=

t + hl h3

dx

3 PIN

dY

1 3 pv: hl r Pw di) 1

{

'

o

r (-

-

t )

0

r (-

-

t

J}

= - dy + ~ h3

dx

h3 T xy + P T xy +

dY

hl h3 T yy + P T_yy) , -k(

:r

xx +

P

T

~)

h 1 acoS:X (- - t ) - - - T .... P T r qxp ~~ (21) (22) (23) ( 24)

" cp-moment um.: a [ (- - t ) (- - t )

J

+ - T + P T sira + T + P T cosCi r xcp xcp ycp ycp (25) Energy:

d {

r

j.l 1

d (",

V2 )

-

t "'( - - t ) ... ( - - t ) =~ h... --~ H -2 - pa +UT +pT +VT +PT + OX j Pr hl OX "'X xx xx xy xy + ';-(

T

_yy +

P

Tt ) +;( _yy

T

_ycp +

P

Tt ) _ycp

J }

(26)

where the mass-averaged Reynolds stresses and turbulent heat and mass fluxes are defined by t pu"u" T

= -

.:...--xx -_p t t

= _

pu"w" T~

=

TcpX _ , p pV"V" p t pw"w" Tycp

= -

_ '

p t t T_xy

=

Tyx

=

-t

<1y=

t t pv"w" T = T =-.:...-- t

=

pu"h"

<ix

ycp cpy pu''a:~t

D~

=

--~

~x

-p

-

_p

-

_p pvt'a~t D~

=

~ ~y

-p pU"V"

-

_p

-

_p (27)

Here

where e =~~--~~----~~-pu"u" + pv"v" + pw"w"

is the mass averaged specific turbulent kinetic energy.

(28)

Equations (21)-(26) and (5) form a set of equations describing plane or axÏ"f;YDJmetric swirling or nonswirling mean turbulent reacting flows containing i distinct species. To close the system addi tional equations should be

provided to determine the Reynolds (turbulent) stresses and turbulent heat and mass fluxes. In the following paragraph, based on the Saffman-Wilcox turbulence modelling approach (Refs.

31-33),

a model is formulated for the Reynolds stress equations. Model equations for the turbulent heat and mass fluxes will be presented in a subsequent report.

3.

DERIVATION OF THE REYNOLDS STRESS MODEL EQUATIONS

The derivation of the equations for the six Reynolds stresses and the equations themseLves are made compact by the use of orthogonal tensor

notation. In this notation, the time-dependent continuity and Navier-Stokes equations for a compressible fluid with zero external body forces are:

P t + (pu.) .

= 0

, J ,J

and

(pu.) t + (pu.u. + pB .. - T .. ) .

= 0

~ , ~ J ].J ~J, J (30)

where ui (i = 1, 2 or 3) are the components of velocity in the orthogonal curvilinear coordinate system, p, p and Tij the pressure, density and the viscous stress tensor, respectively; the comma represents partial differen-tiation and repeated sUbscripts summation. Equations for the Reynolds

stresses are found by manipulation of Eqs. (29) and (30) into a moment of momentum equations in the follawing way:

u. (pu.) t + u. (pu.u. + pB .. - T .. ) .

= 0

K . ~ , K ~ J ].J ~J, J

or

u.u~P,t _{K ...} + (pu.u.) t - u.(pu_k) t + u.u. (pu.) . + (pu.u.u.) .

-~ K , ]. , ]. K J ,J ]. K J ,J - u. (pu. u.) . + u. P . - U. T.. .

=

0 ~ K J , J r ,]. K ~J, J and +U.U·-U.T =0 r,~ K ij , j

FinaJ.1y, using again Eqs. (29) and (30) we get

Expanding this equation according to Eqs. (19) and (20) and averaging yie1ds .

the sum of the mean flow terms

(pU.~) t + (pu.~.~)

.

=

~ K , J ~ K ,J

...

--u.p k _~

_,

~

p .

+;:;:

(T . . - pu:'u'!) .

+

r ,~ K ~J ~ J ,J

and the equations for the Reyno1ds stresses:

(pU~Iu!') + (u.pu~'u!j) .

=

-(pu~'u'!);:;: . _ (pu!ju'.')~ . . +

~ K ,t J ~ K ,J ~ J K,J K J ~,J

+ (U'.'T .) . + (u!'T .. ) . - (u"pu:'u!') . - (u~'p') - (tL''P')

.-~ kj ,J K ~J , J Y ~ K ,J ~,k r , ~

T

' \"

u"

+ '(U " +

u'.' ) _ ::-lT~

p- • _ ':":lTU

p-- _~J ., _{,J . - Tk · .. J ~,J} P k ' _,~~, k _,~~ . _, k or, using the definitions, Eqs. (27),

(PT~k)

t +

(U.PT~k)

. =

-(U~jTkj)

. - (u!'T .. ) . -

(u'!pu~'u!')

.

-~, J ~ ,J ~ ,J K ~J ,J J ~ K ,J

Convection MOlecular Diffusion

- t ... - t ...

- (-Ul.lp') -

(-Ukf>" ') • -

p ~ p u

k T. . . - Tir; • •

~, ,~~J ,J ... ~ ~,J

Pressure Diffusion Production

Turbulent Diffusion

+ Ti'~'· " + Tk·U . . -"

J K,J J ~,J

Dissipation

- p' (u!' . + u~' ) +

ü.'"

p .

+

u:r

P

k K,~ ~,k K . ,~ ~,

Pressure-strain

Rate Corre1ation Compressibi1ity

( 32)

(33)

(34)

The above re1ationship represents a set of exact partia1 differentiaJ. equations governing the transport of the Reyno1ds stresses. To convert Eq.

(34),

the mean-motion and continuity equations, Eqs. (21)-(26), into a

closed set of equatibns for mean flow variables and Reynolds stresses, the turbulence quantities on the right-hand si de of Eq. (34) must be represented as modelled functions of the mean flow variables and Reynolds stresses and their derivatives. The modelled terms should be dimensionally consistent and may contain e~irical constants that are expected to be insensitive to the character of individual flow fields, their values being fixed by arguments based on widely observed properties of turbulent flows.

In the present investigation, Eq. (34) for the Reynolds stresses is modelled in the following way in terms of mass-averaged dependent variables:

=

r

(~

+

o*p .:: )

T

~k

.

l

~

w

~

,J . ,j +

1____ Convection ~ L Molecular and Turbulent..J

Diffusion L Production ..J

L Dissipation -.J

~--- Pressure-Strain Rate Correlation --______ ~

-~ pe divV • Bik

L Co~ressibili ty --.J

where w is the dissipation rate of turbulent eddies (a characteristic decay time of turbulence), e the mass-averaged kinetic energy of the turbulent motion, e

=

-T~i/2,

0*,

~*,

S,

_{Cl' C2, C}

3

and C

4

are the closure coefficients,

and the tensors Pik' Dik and Sik are defined as

t t v t t v

P

=

T U. + T u.

ik im k,m km ~,m

while the quantity P is half the trace of Pik (and of Dik), i.e.,

1 1

P =

"2

P ii

="2

Dii

=

Tt _mn S _nm

(36)

Equation (35) has been arrived at by using the Saffman-Wilcox-RUbesin (Refs. 31-33, 38) and Launder-Reece-Rodi (Ref. 39) turbulence modelling approaches. The first term on the right-hand side of Eq. (35) is the model-led laminar and turbulent diffusion. It is assumed that turbulent diffusion is a simple gradient diffusion process, where the quantity a*pe/w has the dimension of a "coefficient of viscosity". The second term is the exact unmodelled production term of Reynolds stresses derived in Eq. (34). The dissipation (decay) terms in Eq. (34) are modelled [the third term in Eq.

(35)J by assuming the dissipative motions to be ~sotropic [the corresponding Reynolds stress is replaced by 2e/3 5ik, for, when contracted, the term in Eq. (35) yields the dissipation of the kinetic energy of turbulence e]. The

correlation of turbulent fluctuations in the pressure and rate of strain, namely,

(38)

has been modelled by Launder, Reece and Rodi (Ref. 39) as

(39) The first term on the right-hand side of this equation is Rotta's "tendency

toward isotropy" term (Ref. 40), characterizing the anisotropy of normal

Reynolds stresses. The remaining terms represent the interaction between

the Reynolds stresses and the mean flow. The terms are structured to vanish when the trace of nik is taken, as would be implied by the continuity equation

for an incompressible fluid. As shown by Launder, Reece and Rodi, synnnetry properties of the exact pressure-rate of strain correlation term imp1y that C2, C3 and C4 can be expressed in terms of a single unknown coefficient C as follows:

1

C2

=

11 (C + 8), C 3

=

11 1 (8c - 2), C4

= -

4

(15C - 1)

55 (40)

Launder et al find that a value C

=

0.4 appears most appropriate, which corre-sponds to

C

3

=

0.11 (41)

As to the coefficient Cl, the study of the decay of anisotropic turbulence and its asymptotic return to isotropy implies a value (see Ref. 39) for Cl

The modelling of the last "campressibi1ity" term on the right-hand side of Eq. (34) is given in Ref. 41.

In the coordinate system considered the modelled equations for the com-ponents of the Reyno1ds stress tensor are then, from Eq. (35):

t t t t I

eh

-'" OT

OT

{~

[h (

) (

OT

)]

- xx pu xx -'" xx 1 0 3 - e xx t

P

""""diC

+ hl """"dx'""'" + pv

""dy

= hl h3

dx

hl J..1 + 0* P

W

~ + 2k T xy +

- t

(o~

bi: )

4 a -'" t 2 - - ( t 2 )

2PTxy Oy + hl +

r

_{sinx PWTXcp} +

'3

~*pwe - C1~*PW T_XX +

'3

e +

2 [ - t (1

oli

kY) -

t (

o1i

1

?N

kÜ ) - t (1

C;;;

+ j _C2 2PTXX hl dx + ~ + PTxy 2

7JY -

hl

dX

+ ~ - PTXcp hl

di

+

2 [ - t ( 1

OU

k!- )

-

t (1

C;;

kÜ

1

~)

- t (1

C;;;

+ - C 2PT - ""\:: + - + 2PT - ~ - - - - ~ + 2pT - ~ +

3 3 xx hl OX hl xy hl OK ~ 2 u~ xcp hl oX

+ 1 asinx ....,) - t?iV - t

(r;.;

acoS:X '" ) -PTt a(li,sinx + ';cos.:x) ] +

'2

r W - PT yy dy - PT ycp dy - r W - qxp r

[ -(1

OU

1&)

1 - ",] 2 -

'"

+ _{C4 e} P hl

di

+ hl -

'3

P di vV +

'3

spe di vV (43)

2a ( -

e) ["

t 2 t t ] - t ( 1

C;;

2k~

)

- ~ . J.l + o*p - SJ.IO: co~ T + cos a( T - T"') - 2 pT - ~ - -

-re: w xy yy qxp' x:y hl OX ~

- t

C;;;

4aco~

-'" t 2 · - - ( t . 2 ) 2

- 2PT ~ + (JWT + -3 t3*pwe - C

t3*pw

T + - e

+ -

C

YYoy r ycp 1 yy 3 3 2

[ -t

2pT-, xy

.

(l

-hl ~

c;:;

ox

-~~

1.

ou)

+ 2- t

?ii -

t

(àW

+

2aco~

"",) -PTt ( _1

O~

+

k;) _

-

~

- '2

dy . PTyy dy - PTycp

dY

r W - xx hl

Ox

hl

- t

(1.

àW

asinx '" ) - t a( usiIO: +

vco~)

] + 2 C [ -

t

( 2

~

1

c;:;

+

pT xcp hl

dX -

r W - PT qxp r

'3

3

,

.

PTxy

oy -

~

ex

kU )

-

t

dV

-

t (

(i;;

aco~

tv) -

t

(1

?il

k;' )

+~ + 2PT_yy dy + PT_ycp 2 dy + r W - PT_XX hl

OX

+ hl

-- t

(1.

"Ow

asinx '" ) - t a(usiIO: +

vco~)

] + C ( -

C;;

1 - d" V", ·) +

- pT xcp ~

di -

.

r W - PT qxp - r -

4

e P

7JY -

'3

P J. V +

3.

~pe

di';; 3 t

"0 [

(

-

e)

OT

qxp ] 2a ( - e ) [ "2 t t +

dy

~~ ~ + ~P

W

--cF.ï

+ ~ J.l + o*p

W

.

_{sJ.n a(Txx -} 'T~ + (44)

2 ( t t ) . t ] - t

(1

cM

asi~ ",)

+ cos ex T_{yy -} Tqxp + 2s~~ coS:X Tx:y - 2 PT_xcp

Ei'

dX

+ r W

-2 [2-Tt

(1:....?iW

+

asi~

; ) + 2-Tt

(OW

+

~

W""') + +

"3

C2 P xcp

~

di

2r P ycp

7SY

2r

+ 2-Tt a(usinx _P + :;';cos:x) _{- - pT} - t

(1

_{'EL ~}

ou

_{+ -}k; ) _- - t _pT

(1

_{- ~}

êi/

_{+ ~}

ou ,,;; )

_{- ~}

-qxp r xx 1 oX hl xy hl oX oy hl

- t

oV ]

2

r

2-Tt a(

usi~

+ vcos:xl _ -Tt

(~OW

+ 2asinx W....,) _

- PT_yy ~y +

-3

C

₃

P P ""'\'::

oy qxp r xcp hl oX r

- t

(?iW

+ 2acos:x ...., ) - t

(1

ei{

+

k; )

-

t

(1

C;;

+

~

k,7 )

- pT ycp Öy r W - pT xx hl

dX

hl - pT xy hl

öX

dY -

hl -- t

?N ]

[ -

a( usinx + vcos:x) 1 - ...., ] 2 t -pe divV"'" - pT yy dy + C

₄

e P r - j pdi vV + j S (45)

o

t

o [

(

-

e ) Txy

J} ( -

e ) k

[0

( t t

4

t ]

+

<ET

~h3 ~ + o*p

w

dY

+ ~ + o*p

W

~2

di

_{Tyy Txx) kTxy}

-- t

(ou

ku ) -

t

(1

OU

&;

b )

2apW ( . t t - PTyy dy + ~ - pT x:y hl

öX

+ Öy + ~ + - r - s~~ T_ycp + coS:X TX~

-(46)

t t t t

èT -""

èT

èT

{~

[h (

) (

èT )]

- X pu x~ -"" x~ 1 e 3 - e x~ t

p

T

+

h:L

"'"diC

+ pv

7fY

=

h1h3

ox

hl 1.1. + a*p

w

--0-

+ kTycp +

(

- e)

ar.

t

4 .

2

2

t ] - t

(1

?M

- 1.1. + a*p

w

7-

3S1ra coSJ:: Ty~ + ( S1n ex + cos ex) TX~ _{- PTXX} hl

dX

+

_ -Tt

( ""

èu. + ~

"')

+ 2asira -;T t _ C

A*-

Tt + -Tt

(C

~

èw _

'"

C asi:rx:x:;

.

)

+

P ~

"'fii

.

hl . r p y~ 1.... pw x~ P xx

h:L

di{

3

r

- 1 (1

(iW. a "" )

+ C4pe - - ~ - - sira w

t t t t

dT -"" dT '" dT "m 1 {-:.. [h3 ( ) ( dT t )]

- ycp pu - oT'!" 0 - e ycp

P - t - + hl

~

+ pV

dY

=

~

h3

dX

1\

IJ. + o*p

W

ex -

_kTXcp +

(

-e) a [.

t ( 2 . 2 t

l -

t

(oW

- Il + a*p

w

_{r 2 3sna coS:X TXcp} + 4cos

ex

+ Sl.n

ex)

_Tycp

J -

PT

yy .

dY

+

acoS::X .... ) 2acoS::X ""- t - t

(1

oW

asira "") - t

(1

dV

2k~

)

+ r W + r WPTcpcp - _PTxy hl

di

+ - - ; - _{W - PTXcp} hl

di -

hl

-_ -Tt [d'; + a( us ira + ;cosa)

l _

-

t - t (

ct;

acros::x w'" ) +

P ycp dy r

J

C_{l f3*PWTycp} + PT_yy C₂

7!iY -

C

3

-- t

(

ow

-:,...~

acoEa '"

)

- t

(C ""

2

OW

asira ....

)

- t

(C -:..'"

2 oV

+ pT cpcp C 3

dy -

C2 - r - - W + PT xy hl

dX -

C₃ r W + PT xcp hl

dX

+

dU kU) - t (C + C )

(d';

+

a(~sira

+ ;cos::x) ) +

+ C₃ Öy - C2 hl + pT ycp 2 3 dy r

( 48)

4 .

TURBULENl' KINETIC ENERGY AND DISSIPATION RATE EQUATIONS

Contracting Eq. (35) yields an equation for the turbulent kinetic energy, (e

=

-T~./2) as follows:

J.l.

d [

(

-

e)

de

J} -

- _ '"

where Snm is the mean strain-rate tensor, and in the coordinate system con-sidered is gi ven by 1

(C;;

acoso; ... ) S23

="2

dY -

r w S33

=

~ (~sinx

+ ;'co&:x) (50)

The derivation of the equation for the turbulent energy dissipation rate

w is analogous to that given in Ref.

41,

except that in the present

investiga-tion the product i on term for w is modelled as the product of the production

term of the kinetic energy equation, Eq.

(49),

the ratio w2

/e

and a new

modelling coefficient (see Ref. 38). Hence the equation for w is as follows:

where the length' scale of turbulence is

1/2

t

=_e_

w

The modelling clösure coefficients employed are (Refs. 32, 38,

41):

_ 3

t3 - 20'

1

0"=0*-

_{- 2'}

5 •

MODELLING OF THE TURBULENT HEAT AND MASS FLTJXES

(51)

(52)

(53)

In the present report it is assumed that the turbulent heat and mass fluxes

are simp1e gradient diffusion processes, i .e. ,

-

₁

_Oh

-

_oh

_pullhll pe _-pv"h" _ pe

=

Pr tw 4₁

di'

-;töY

_{r w} and ₍₅₄₎

-

₁

eb.

_-

ca.

_pUi'a~i pe ].. _PVf'a~f pe ]..

=

-=-r

h

öx'

₌

_sctw

_7Siï

].. Sc w 1 ].. t t

where pr and Sc are the turbulent Prandt1 and Schmidt numbers respecti ve1y, and prt = Set

=

0.89

(Let

=

1) (Ref. 32). The species continuity and energy equations used in the present investigation, Eqs. (22) and (26), become then

and + ,... -W(T _ycp + PT - t )] } _ycp N

.

+

w.

].. (55) ( 56)

Equations (21), (23)-(25), (55), (56) and (43)-(51) describe the swir1ing jet turbulent mixing and combustion processes for the flow geometry considered, and form a set of 13+N partia1 differential equations for the determination of

,... ,... ,... ,... - t ( . )" " . .

u, v, "H, H, p, e, w, T •• _J..J " six and 1'I species generated durmg conibust]..on. The _{_} mean rate of production of the N species,

W,

has yet to be ~ecifi~d. The tem-perature field is determined from the tota1 entha1py field H, and p from the equation of state, Eq. (5).

6. TWO-EQUATION M)DEL OF TURBULENCE

An analytically simpler but less accurate description of turbulent swirling flows may be arrived at by introducing Prandtl' s "turbulent vi scosi ty" concept. According to this hypothesis, the turbulent stresses, heat and mass fluxes are expressed as:

- t _{2Jl t ( S ..} 1

divV Bij) - 2 peB .. pT .. =

_{- "3}

"3

lJ lJ lJ (57) t t

_db

- t t

cE

-

Jl Jl K pqj = - Prt

dX

_j and PDj = - sct

dXj

(58) where Sij are given by Eqs. (50), Bij is the Kronecker symbol, and Jlt the

"turbulent viscosity". It should be noted that unlike Jl, the molecular vis-cosity, Jlt is not a property of the fluid. lts value varies from point to point in the flow and is determined by the structure of turbulence at the point considered. Hence, the turbulent flow field is determined if Jlt is known. By analogy to laminar viscosity, Prandtl postulates that

t -Jlo

=

pv J,

t

where Vt éi.nd l, are characte":ristic veloci ty and length scales of turbulence at a given point in the flow field. Prandtl suggested to use the square/root of the turbulence kinetic energy as the characteristic velocity, Vt

=

el 2.

Since the rate of transfer of energy fram large to small eddies, i.e., the (inviscid) dissipation rate, is proportional to Vt/l" w

=

Vt/l,

=

e l / 2/l, [see Eq. (52)], the turbulent viscosity is expressed as

-t = pe

Jl w

where e and w are the solutions of Eqs.

(49)

and (51). With this "two-equation" model of turbulence the swirling jet turbulent mixing and combustion processes

are described by Eqs. (21)-(26),

(49)-(59)

and form a set of 7+N partial differ-ential equations for the determination of

u, v,

w,

H, p,

e, w and N species generated during combustion. The mean density

p

is determined from the equation of state, Eq. (5).

7.

THE COMBUSTION M)DEL

As mentioned in the Introduction, real comb°..lstion phenomena involve hundreds of interacting chemical reactions and chemical species, and modelling of both the detailed chemical kinetics and turbulent fluid-mechanic phenomena is beyond the capabilities of even advanced computers. However, the overall effects of combustion can of ten be described simply: fuel and oxidant disappear and I

It is therefore possible and useful to introduce a model of eonbustion (Ref.

3

4 ) whieh involves reaetion between two reaetants, the fuel and oxidant in which these combine, in fixed proportion by mass, to produce a unigue product, i.e.,

lkg fuel + rkg oxidant ~ (1 + r)kg products + H

fu (60)

The quantity r is, of course, the stoichiometrie ratio and Hfu is the heat of combustion of the fuel. Although simple, the model accords with reality in respect of the overall effects of combustion. Eguation (60) implies that the rates of creation of the chemie al species are related by

7' ..., W

=

W

Ir

fu ox'

.

W fu '"

=

-w

_pr I(l+r) or 7' W ox

=

-r~

pr I(l+r) The Arrhenius reaction is taken for the fuel specifically as

• _2 "'..., ( E )

W

=

-Cp

ex

ex

exp

-fu fu ox

RT

( 61)

(62)

where E and C are constants dependent on the particular reaetion under eon-sideration.

-The differential eguations for

ex

_{ox and afu are, from Eg.} (55),

(

-

?E

fu + p

-'"

u

?iX

fu + -'"

?iX

fu _ 1

{~

0

r

h 3 ( IJ. +

-)?iX

pe fu

J

+

P"'"'öt

~

d'X

pv

--csy- -

hl h3

dx

hl Sc Sc ;u w

dX

and

'cE

- oX p

""dt

(j;

+

0

r

h h

(L

+

pe )

fu

l

+

w

dy 1 3 Sc S t

7Si!

J

fu cfuw -+

0

r

h h

(L

+

pe )

(j;

ox

J

+

~

dy , ,1 3 Sc Sc ~ W (5'y . ox ox (63) (64)

Comparison of these two equations shows that if sc;u

=

Se~x

=

Set (it is reasonable to assume that in turbulent flows this condition is satisfied), the source term in, say Eg. (64), can be eliminated by subtracting r times Eg. (63) from Eq. (64) and taking into account Eg. (61). Carrying out this manipulation and denoting the quant i ty

a

_{ox -} rafu by

r

we get the following e guat ion:

(65)

Thus f is a conserved praperty in the sense that no source term exists. Hence

afu is determined from E~.

(63),

r,

and therefore

a

_{ox , fram E~.}

(65),

and ~r from the obvious re1ation

O:f +0: +0: =1

u ox pr

(66)

In the simp1e combustion model described above, it is assumed that the mean rate of fue],. consumption e~uals the instantaneous rate with the mean

~uantities such as temperature, pressure and species mass fractions rep1acing their instantaneous values. However, concentration measurements in turbulent f1ames with high1y active reactions show that the time-average reaction rate in such flames is 1ess than the values given by using the time-average va1ues of the species mass fraction and temperature • This effect is due to 10cal turbulent f1uctu~tions in these latter ~uantities that may give rise to situa-tions where the oxidant and the fue1 are not at the same p1ace at the same time and may cause variations in the reaction rates. The re1ated "unmixedness phenomenon" has long been recognized in experiments • At present, no al10wance for this dependence is made in the model here. More realistic combustion mode1s wi11 be considered at a later stage of the present research programme.

8. BOUNDARY-LAYER TYPE EQ,UATIONS OF MYrION

A. Reynolds stress Turbulence Model

The solution of the system of partial differential e~uations given so far (with given boundary conditions) is non-trivial, their nonlinear e11iptic character demandfng that a 1engthy numerical re1axation procedure be used. Moreover, it is reasonab1e to first test the validity of the adopted turbulence model in the casy of a simp1er c1ass of turbulent reacting swir1ing f10ws, i.e.,

swirling f10ws which have a single predominant direction and the various flux components are stgnificant only in direc-pions perpendicular to this predominant direction. Pressure variations are such that they do not al10w downstream changes to inf1uence upstream events, and. streamlines are not c10sed (there are no recirculation regions). Such f10ws are of ''boundary-1ayer type". App1ication of boundary-1ayer approximations results in the truncation of the elliptic e~uations to parabo1ic form. Fewer terms and unknown f1uxes

are left in the e~uations and a re1ative1y simp1er and ~uicker forward-marching solution procedure can be used for their numerical solution.

Boundary-1ayer type e~uations of motion can be derived by considering the re1ative orders of terms in situations when 5t/L

«

1, where 5t and L are the lateral and long~tudinal extent of the flow. Such an analysis is carried out

in detail in Appendix B. Retaining terms no smaJ.ler than (Bt/L), the tions of motion, the Reynolds (turbulent) stress model equations, the equa-tions for the kinetic energy of turbulence and its dissipation rate, Eqs. (B3), (B9), (B14)-(B17), (B19)-(B26), are in dimensional form:

Continuity:

(67) x-momentum:

pîi

d~

-'" dîi

-'''''''

asira

-.-.e

1

d:P

hl

dx

+ pv

dy

+ k puv - r pW = - hl

di

+ y-momentum:

-.-.e

-kpu aco~

p;f-

= _

oP

+ 1 d (h_h Tt ) r

dy

hlh3

dY -"].

3 yy cp-momentum:

1

d

r

h h _L

(êM

acog:x

w"')]

+ acog:x -Tt +

aco~

-L

c;;;

(70)

+

~

h3 dy 1 311 dy - r r p ycp r 11 Oy energy:

Reynolds stress model equations:

2

r -

t (

èu

,... )

,

-

t

(0;

,

acog)!

"")J

2

r -

t

(o

~

,...

)

+

3'

_C2 _ pT xy • 2

CiY

+ ku - pT ycp dy - r W ,. -

3'

C₃ '_ pT xy "dy + 2ku +

(72)

where.c and S3 denote the operators

PU

o

.

-,...

è

. .c( )

=

h

di

+ PV dy; 1 and _{-1 -pe} ~ =~ + 0 * -eff w t ) cp( t ) - t ,... 4acosa -'" t 2 -.c( T_yy

=

J.J( T_yy + 4PT _xy ku + _r pWT _ycp +

-3

l3*pwe

-C A*- ( t + 2 ) 2 C [ - t

(èu

+ 2k"'U) +

-PT~ (~

.

'"

+

2ac~sa

w",)] +

- lf-' pw T yy

'3

~

-

j 2 PT xy

7SY

"

't' u:t

f'(Ttcpcp) t - t

("oW

aco~":'

) 2 - ,. ( t 2 )

~ =~(T) - 2PT "'\:': + - W + .... l3*pwe - C_ll3*P<.u T_rrYn + -3' e ,+

cpcp ycp oy r

3

't''t'

2 [ 2- t ,(

èw

+ acog)! ",) - t

(èu '"

)l

2 [

~

t

(0;

2acosa ";'\

+

'3

_C2 _ PTycp dy 2r W - PTxy (fj - ku ,

-"3

C

3 _ PTy<p

\-oy

+ r w) +

..: t

(ou '"

)1

2 - '"

ac~~

;; ) +

c

₄

p~

(

~

-

k;)

(75) ( t ) <10..-( t ) - t

(C;;;

acoáX ",) - t

(CU '" )

-

t .L T = N T - PT ""-= + W - PTyrn ~ + ku - C1t3*PWTxrn + xcp xcp xy oy r ~ U8 ~ ( t ) ,-...,.( t ) - t

(oW

acoáX ",) - t - t (

cr;;

.L T_vm = IN T - pT "'\:: + W - C 1t3*PWT + PT C2 ~ -,,~ ycp yy oy r ycp yy 08 (77)

Fuel conservation equation:

-,..; ?iJ,

?XX

~

r ( -

L - )

di

l '"

pu fu + -'" fu _ 1 0 h h J..1 + pe fu +

W

hl

d'X

pv ~ - hlh3 dy 1 3 Sc sctw

---çs:y

_

fu (78)

where the average fuel reaction rate is gi ven b y Eq. (62).

'"

f-quantity conservation equation [see Eqs. (65), (B16)J:

(79)

Turbulent kinetic energy equation:

- t

(JU ,.... ) -

t

(C;

acosCX"'" ) - -- ,....

.L(e)

=

(e) + pT ""-= - KU + PT ~ - r W - t3*pwe - ~pedivV

---

---

---

----

---

w

I

Dissipation rate equation:

( 2) ( 2) w

2

r

-t

(èîi

rv ) - t

(C;;

acoSJ: .... )

J: w

=

~J W + 7

e

.

PT xy

7iY -

ku + PT ycp

dW -

r W

-( 81)

The fifteen partial differential equations, Eqs. (67)-(81), together with

the equation of state, Eq. (5), wi th respect to the sixteen unknown flow variables ,

rv "" ...., - ...., - G - t t t t t . . . 2 . .

u, v, ·w, p, H, p, T , T , T_{mm ,} T , T m' T cp' af ' f, e, W , descrlbe, wlthin

xx YY 't"1' xy x'j" Y u

the framework of the Reynolds stress turbulence and cambustion model introduced, the boundary-1ayer-type swirling jet turbulent mixing and combustion process' in a stagnant surrounding.

B. Two-Equation Eddy Diffusivity Model of Turbulence

With this model, the turbulent stresses, heat and mass f1uxes are represented by Eqs. (57) and (58) and taking into account Eq. (59), the Navier-stokes equations become:

x-momentum:

p~ è~

-""

è~

-""'"

asinx-"'2 hl

di

+ pv dy + kpuv - r pw

=

1

è

r

-L t

(èîi ... )

-L t

&

+ hl h3 Óy hl h3 (I-L + I-L) Óy - k u + k (I-L + I-L )

7iY

(

82 )

y-momentum:

The energy equation is

pu

dH

--- dH

:).

d

r (

~L

Il t ) (

dR

...

d~

:;

~y

)] +

hl

di

+ pv Öy

=

h:).h3 Óy hlh3 prL + prt

dY -

u

7SY -

OY

+

~h3

&: {

~h3

[

(~L

+

~

t)

(~L2

+

~/2

- k,;'2 _

ac~.a

>f' )]}

(85)

The swirling jet turbulent mixing and combustion in the boundary-layer approxima-tion is then described by ten equations~ Eqs • . (67), (82)-(85), (78)-(81) and (5),

with respect to the ten unknowns

U,

v,

w,

p,

H, p, ä

fu,

f,

e and

Jl..

C. Mixing-Length Model

Within the framework of this model, Prandtl' s mixing leng th theo:ry is general-ized to flows with swirl, and the nonisotropy of the turbulent viscosity is

allowed fol' by linking the ycp-viscosity to xy-viscosity, Le., Il':_n to !J.t , in an

t t J~ xy

empirical wa:y. Turbulent stresses T _xy and T _ycp are then represented by

- t t

(0; ;)

PT ycp

=

Ilycp

7iY - Y

(86)

Various such extensions have been proposed (see Refs. 15, 22). We adopt here the extension presented in Rei'. 22. Thus

where

t =

Î>iY0.05' À

=

0.08[1 + À s x x (S )S }, ~ t - Ilxy' ycp _ t Ier (S) x

The quantity Sx is the so-called local swirl number and is expressed as

where

00

G_x

=

J

(p~

+

:P -

p e)ydy

o

is the constant axial flux ofaxial momentum, and

28

(87)

00

J

_

...

2

Gcp = puwy dy o

is the constant axial flux of the angular momentum (see Eef. 4). The quan-tities YO.Ol and YO.05 represent~the positions where u/um

=

0.01, 0.05,

u

_m being the value of the velocity u at the axis, and Às(Sx), ~ycp(Sx) are empirically determined functions.

Hence, the problem considered is described by the seven partial differen-tial equations, Eqs. (67)-(71), (78), and (79), where pe/w

=

~iy' and the equation of state, Eq. (5). The turbulent stresses and viscosities are given by Eqs. (86)-(88).

9. BOUNDARY CONDrrIONS

The solution of the above formulated mathematical problem is.possible only when sufficient boundary conditions have been specified. Since the system of partial differential equations to be solved is of parabolic type, initial

orifice conditions at x

=

0, and edge boundary conditions at the axis of symmetry and at y

=

ot

must be given for each unknown variabIe.

A. Initial Conditions

Initial orifice conditions at x

=

0 must be assigned for each of the

unknown variables. Initial profiles might be obtained from experiment al data. If adequate experiment al data are unavailable, initial profiles might be syn-thesized by combining known results, and values approximating experiment al results. For example, the initial axial velocity might be approximated as a profile cubic in distance, or the initial swirl velocity might be given as a Rankine vortex. The accuracy of the solution will be dependent upon the appro-priateness of the initial profiles. For maximum effectiveness of the numerical solution of the partial differential equations as a predictive technique, it is desirabIe to use the minimum of experiment al data, whereas, to obtain the best accuracy possible with the numerical solution method, it is desirabIe to use available experimental data where appropriate, as for the initial profiles. B. Edge Conditions

The conditions at the outer edge (the edge adjacent to the stagnant surrounding) are obvious from physical considerations: the flow variables should tend to their values in the stagnant surrounding.

The conditions on the axis of symmetry are that the normal gradients of all variables are given zero values, except for the swirl velocity for which the condition at the axis of symmetry is

rW

=

O.

10. CONC LUS IONS

Three models of turbulence of increasing complexity and generality, and a simple combustion model have been considered to describe the turbulent mixing and combustion of a single swirling jet issuing into a stagnant

surrounding. Attention is f'ocussed on weakly swirling jets, when recirculation regions are absent alld the f'lowf'ield is of' boundary-layer type. It is believed that the numerical solution of' the presented equations will provide more insight into the mechanism of swirling j et turbulent mixing and combustion than has been achieved so faro Moreover, through this work direct checks on the validity of a variety of' turbulence models may be made via extensive camparison with experimental data alld all objective viewpoint expressed.

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