Turbulence and turbulent flame propagation—A critical appraisal

C O M B U S T I O N A N D F L A M E 24, 285- 304 ( 1975)

285

Turbulence and Turbulent Flame Propagation-A Critical Appraisal G. E. ANDREWS*, D. BRADLEY and S. B. LWAKABAMBA Department o f Mechanical Engineering, Leeds University, Leeds LS2 9JT, England

Current theories of turbulence that seem relevant to the structure of turbulent flames are reviewed. The compatibility of such theories with different turbulent flame models is discussed. It is suggested that the turbulent Reynolds number, R~k, of the reactants is an important controlling parameter in turbulent flame propagation. When R ~ 100, a wrinkled laminar flame structure is unlikely and the turbulent flame propagation is probably associated with small dissipative eddies. It is proposed that the ratio of turbulent burning velocity to laminar burning velocity can be correlated with R~k.

1. Introduction The diverse range of turbulent combustion, from that on a bunsen burner to that in a stirred reactor, makes any attempt at a unified theory of turbulent flame propagation a prodigious task. Most practical flames are turbulent and many explosion hazards involve turbulent flame propagation. Yet, in spite of this, the mechanism by which turbulence can drastically increase the rate of flame propagation has not yet been fully elucidated. A completely analytical theory of turbulent combustion only can be developed via extensions of the statistical theory of turbulence to include fluctuations o f temperature, species concentrations, velocities and chemical reaction rates. This type of approach was advocated several years ago by yon Karman [1] and Hawthorne [2], but as the statistical theory has not yet reached the stage where it can predict time averaged quantities, there is little immediate hope of its extension to include chemical reactions. Flame propagation is governed by the conservation equations for global mass, species and energy. In recent years, such equations, together with the appropriate chemical rate data, have predicted successfully the burning velocity, temperature, and composition profiles for some laminar flames [3, 4]. However, for turbulent conditions, if the conservation equations include terms for all the fluctuating *Present address: Lucas Aerospace Ltd., Combustion Research Laboratories, Burnley.

quantities and then are expanded, various double and triple correlations arise. It is not surprising that attempts to solve the equations have resorted to numerous arbitrary simplifying assumptions [5-7]. Apart Irom the work of Prudnikov [8-10] and Predvoditelev [11,12] which retain some aspects of the statistical nature of turbulence, the remaining analytical predictions of the turbulent burning velocity, u t, result in grossly oversimplified relationships. No realistic solution of the turbulent flame equations has yet been achieved, nor is it likely to be, in the near future. Because of this, various models have been developed that can form a basis for quantitative predictions of u t. Qualitatively, the effects of turbulence upon combustion have been known for many years. Mallard and Le Chatelier [ 13] noted in 1883 that turbulence increased both energy transfer and flame surface area and also formed new centres of inflammation. Effectively, they were anticipating three different models for turbulent combustion and also implying that no single one of them would be comprehensive. Historically, models for turbulent flames have lagged behind contemporary understanding of isothermal turbulence. The latter, in turn, has been conditioned by the degree of sophistication of the available experimental techniques for turbulence studies. Such models and their development have been the subject of several reviews [14-22]. The purpose of the present work is to examine various models in the light of current ideas on the Copyright © 1975 by the Combustion Institute Published by American Elsevier Publishing Company, Inc.

286

G.E. ANDREWS, D. BRADLEY, and S. B. LWAKABAMBA

nature of turbulence. The most relevant factors in theories of turbulence are introduced and existing models of turbulent flame propagation examined. Where necessary, new suggestions are introduced in an attempt to develop a turbulent flame model more in harmony with theories of turbulence structure. 2. The Structure of Turbulence Osborne Reynolds [23] showed that the fluctuating velocities of turbulent flow create stresses in the fluid additional to, and often much larger than, the viscous stresses created by molecular motions. Turbulent energy is fed into a flowing fluid by virtue of the straining that takes place against such Reynolds stresses. For spatially uniform conditions in the steady state, this rate of turbulent energy production is equal to the rate of energy dissipation. This is valid in an overall sense; it is not true locally. For fluctuating motion of length scale, L, and velocity scale, u', the eddy lifetime may be taken as L/u' and the rate of production of turbulent energy therefore is proportional to u'3/L [24]. The time scale for molecular dissipation if the length scale were the same would be L 2/v, where v is the kinematic viscosity, and the rate of viscous dissipation of energy would be proportional to u'2u/L 2 [24]. The ratio of the rate of production to the rate of dissipation of turbulent energy gives a Reynolds number, R L , equal to u'L/v [25]. At high values o f R L as long as the length scale is the same for production and dissipation, there can be no balance between source and sink. Balance can only be achieved by the existence of a smaller length scale for dissipation than for the production of turbulent energy. Thus smaller and smaller dissipative eddies must be created as R L is increased, and an energy transfer process must exist between the large energy containing and the small dissipative eddies. The Reynolds stresses arise principally from the larger eddies, which are thus the main agency for the extraction of energy from the mean flow. The large eddies are of a size comparable to that of the width of flow and are anisotropic. Experimental evidence [26-29] suggests the large eddies that are most effective in extracting energy

from the mean flow are those with vortices whose principal axes are roughly aligned with the principal axis of positive rate of strain. The energy transfer is believed to be associated with a vortex stretching process [30]. Vortex stretching gives rise to an increase in the kinetic energy of rotation, as a consequence of the conservation of angular momentum. Thus, perpendicular to the axis of stretching, turbulent length scales are reduced and velocity components increased. These components increase small scale stretching along their own axes. In this way, vortex stretching produces ever smaller scales of turbulence with ever increasing vorticity and velocity gradients. Viscosity finally dissipates the energy into randomised molecular motion, but plays no essential part in the stre ching process. Through this cascade of turbulent energy, vortex stretching eradicates the original anisotropy of the large eddies and creates isotropy amongst the small scale eddies. These latter tend towards an independence of both the large eddies and the mean flow and towards a universal isotropic structure. 2.1. The Microscale Reynolds Number, R~. Taylor [31 ] has shown that the kinetic energy associated with turbulent velocity fluctuations can be analysed according to the spectral distribution of the associated frequencies. At a given fixed point in a turbulent flow with constant mean velocity, the large eddies are associated with velocity fluctuations of low frequency, whilst the small eddies are associated with high frequency fluctuations. Taylor [32] also showed that the average size of the large, low frequency, and small, high frequency, eddies can be deduced from correlations of instantaneous velocity at two measuring points. The variation of the correlation with the distance apart of the two points yields, after integration, the mean size of the large eddies, or macroscale of turbulence, l. Taylor [32] further has shown that for isotropic turbulence the rate of energy dissipation per unit mass, e (m2s-3), is given by

e = 15v (3u--) 2 . Ox

(1)

TURBULENT FLAME PROPAGATION

287

Taylor defined a length scale, X, to be called the Taylor microscale, such that ( a u ) 2 - u '2

~x __m

X~'

(2)

1/2

where u' = (u 2) with u as the fluctuating component of velocity. Hence e =

15vu "2 ____ _ X2

(3)

The rate at which large eddies supply energy to small eddies can be taken to be proportional to the kinetic energy per unit mass of the large eddies divided by the time scale of the large eddies, l/u'. Thus the rate of production of turbulent energy is proportional to u" 3ft. In the steady state, this rate is equal to the rate of viscous dissipation and thus e ~

u'3/l.

(4)

From Eqs. (3) and (4), it is seen that the Taylor microscale is related to the macroscale, l, by x2=~

l

u'

(5)

Dryden [33] has used experimental data and shown that for isotropic turbulence X2 - 48.64 u _ . l u'

(6)

small eddies. Thus RX is proportional to the ratio of time scales for large to small eddies, as well as being the ratio of inertial to viscous forces. As R x increases, the time scale of large eddies decreases, but the time scale of small eddies decreases even more. 2.2. The Mictostructure of Turbulence

As the rate of straining of a fluid increases, more energy is dissipated, R x increases, and the dissipative eddies become smaller. When R X > 100, it has been found that the spectral separation between the large eddies and the small dissipative eddies is sufficient for the vortex stretching process to ensure isotropy of the small scale eddies [35]. Most of the turbulent energy is associated with the large scale motion and most of the vorticity is associated with the small scale motion. It was shown, at the beginning of Sec. 2, that the length scale of the dissipative eddies adjusts to the rate of energy dissipation, e, and that it is also dependent upon the value of the kinematic viscosity, v. The isotropy of the dissipative eddies makes their detailed structure independent of that of the large eddies. However, the definition of the Taylor microscale, X, essentially is based upon large eddy parameters. Kolmogorov [3639] in his universal equilibrium theory analysed the isotropic regime and derived microscale values of length, rl, velocity, v.r/, and time, r_, q based upon e and v only. These Kolmogorov scales are given by

= (vale) ¼ = X 15"¼Rx "y2, Taylor introduced a turbulent Reynolds number R x, based on the microscale X, such that R X _ U'X

(7)

V

From Eqs. (6) and (7)

RX x/u

(8)

Now l/u" is the time scale of the large eddies and X/u" is the time scale of the small scale fluctuations [24, 34], a measure of the time scale of the

(9)

vrl = (eu) ¼ = u/~ = 151/*u'/Rx Y2,

(10)

rr~ = (vie) ~ = rffVr~ = x / 1 5 V2u" .

(11)

Typical numerical values may be obtained from these equations for a given macroscale, 1, and from Eqs. (6) and (7). The macroscale principally is a function of the turbulence generator and is relatively independent of R x. A value of l was taken equal to 10 mm, typical of many turbulent combustion situations, v was taken as 15.6 × 10 "6 m2sec "1, the value for air at atmospheric pressure and

288

G.E. ANDREWS, D. BRADLEY, and S. B. LWAKABAMBA

300°K, and u'ranged from 0.5 to 10 m sec °l. The variations of X, ~7, vI?, and r I? with R -a are shown in Fig. 1. The values of X, r/and Trt all decrease with increase in R x. It is of particular interest in the context of turbulent flame propagation that the value of isotropic velocity, v~, although less than u', is nevertheless appreciable, and increases with R X.

'U

10 ---

~o

(11

E E

E

.,,< "ID to U I/)

-1.1-E

C'-"

m_

JO.01 1000

0.1 100

R X ( log scale ) Fig. 1. Variation of Kohnogorov scales with R~; l = 10mm, p = 15.6 X 10-6 m2sec-1. Both the Taylor and Kolmogorov microscales are theoretical parameters and they may not indicate the precise details of the actual microstructure of turbulence. In particular, the assumption implicit in the definition of r/, that energy is dissipated uniformly and continuously throughout the fluid may be invalid. Signals from anemometers reveal that the flow may not be continuously turbulent, but that it may exhibit intermittency.

The intermittency factor at a point is defined as that fraction of the time for which flow is turbulent. Experimental investigations of the spectral distribution of turbulent fluctuations show that hot wire signals reveal continuous turbulence at the low frequency, associated with the macroscale, and the intermittency factor is close to unity. However, a higher frequency component of the signal is intermittent and has a lower value of the factor. This phenomenon first was observed experimentally by Batchelor and Townsend [40] in grid turbulence and has been found to be characteristic of all turbulent flows [41-51 ]. It is not surprising that as the size of the dissipative eddies decreases with increase in RX, then the proportion of the total volume occupied by these eddies also decreases and gives rise to a decrease in the intermittency factor. This feature of the intermittency of dissipative eddies has been confirmed by Kuo and Corrsin [51] and others [46-50]. The experimental observation of intermittency led Townsend [41] to postulate a turbulence structure in which small scale eddies might be represented as a random tangle of vortex sheets (locally parallel vortex lines). Corrsin [52] suggested that the vortex sheets have a thickness of the order of the Kolmogorov microscale, 7?, and a spacing of the order of the macroscale, l. Tennekes [53] has suggested that dissipative eddies consist of vortex tubes with a diameter o f the order of the Kolmogorov microscale and a spacing of the order of the Taylor microscale, X. In both models, intermittency is explained by the spacing between dissipative regions. More recently, Kuo and Corrsin [54] have attempted to identify experimentally the geometric character of the regions of random fine structure. They tentatively concluded that these regions are more likely to be rod-like than blob-like or slab-like. They suggested that random, slightly "stringy" structures might overlap each other. The average linear dimensions of these fine structure regions is considerably larger than the turbulent fine structure within them. From this brief survey of the structure of turbulence, some factors emerge that might explain the increase in burning velocity due to turbulence. A coarse macrostructure suggests an increase in burning rate due to an increase in flame area. An

TURBULENT FLAME PROPAGATION intermittent microstructure suggests an increase in burning rate due to preferential combustion within the isotropic dissipative eddies. These allow the reacting gas to advance rapidly, leaving behind comparatively large volumes of unburnt gas, which is burnt in subsequent dissipative eddies. However, before discussing such possibilities in detail, this is an opportune point at which to review critically some of the existing models of turbulent flame structure. 3. Models of Turbulent Flame Structure Two distinct categories of turbulent flame models have been proposed, one based on molecular and the other on turbulent transport processes.

289 contain large velocity perturbations, over a wide range of frequency. For the wrinkled flame model to be viable, a laminar reaction zone must be stable to such perturbations. All the theories of Table 1 assume that the flame front propagates, relative to the unburnt gas, at the one dimensional laminar burning velocity, u l. However, the actual laminar burning velocity may deviate from u l as a consequence of flame curvature [86-88], velocity gradients [62, 89-91] and burnt gas recirculation [92, 93]. Both flame curvatures and velocity gradients are minimal in the large scale eddies and this is conducive to the occurrence of laminar combustion in the low frequency part of the turbulence spectrum.

3.1. The Wrinkled Laminar Flame Model

3.1.1. Laminar Flame Stability

This is the best known molecular transport model and was introduced by Damkohler [55]. It assumes that the significant effect of turbulence is to make a larger flame surface area available for molecular transport. Table 1 summarises the theoretical approaches that have been developed. Indeed, a majority of workers seems to adopt this model, although it has no unique interpretation. It assumes essentially that a turbulent flame consists of an array of laminar flames. Most of the theories assume that the macroscale is significantly greater than the laminar flame thickness, ~l" The assumption I > > 61 implies that small scale eddy transport, over a distance less than l has no significant effect and the sole effect of turbulence lies in an increase in flame surface area due to the action of the large scale eddies. This increase is associated with the value of u'. The turbulence microstructure described in Sec. 2.2 suggests that such a flame structure is more compatible with values o f R ~ < 100. In support of the wrinkled flame model, there is a wealth of experimental evidence purporting to identify isolated laminar reaction zones within a turbulent flame [17, 59, 79-85]. Often the model is claimed to be applicable to intensely turbulent flames [59], but it is sometimes limited to weakly turbulent flames. The widespread acceptance of this model requires further consideration of its feasibility and possible regime of validity. It was shown in Sec. 2 that turbulent flows

The high frequency part of the turbulence spectrum is associated with large velocity gradients and the wrinkled laminar flame model requires the flame to be stable to these. Not only are laminar flame stability studies important in this context, but also for an understanding of the transition from laminar to turbulent combustion. Experimental investigations of the interaction of a laminar flame with a turbulent flow could yield valuable information on the mechanism of turbulent flame propagation. The transition from one regime to the other is also important in the development of detonation, which has been attributed to laminar flame instability and turbulent acceleration of the flame [94, 95]. The potential importance of such stability and transitional studies seems evident from the voluminous literature on the stability of nonreacting laminar flows and their transition to turbulence [25, 92-102]. Both Reynolds [23] and Rayleigh [103] conceived of turbulence as arising from laminar flow instability. The mode of development from an unstable laminar state, through a sinusoidal wave motion to turbulence, was visualised experimentally by Rayleigh [103]. Heisenberg [104] showed that it is only through viscous stabilising forces that laminar motion is possible and that if the energy input to the fluid is increased sufficiently, then a transition to turbulence ensues. The increase in the spectral range of frequencies as the energy increases can explain the observed existence of a critical Reynolds number [105].

290

G. E. A N D R E W S , D. B R A D L E Y , a n d S. B. L W A K A B A M B A TABLE 1 Wrinkled Laminar Flame Model, Predictions for ut, Turbulent Flame Thickness, 6 t and Mean Combustion Time 7"

Author

Ref.

Damk~Jhler

55

Shchelkin

56

Assumptions

57

Equations for fit and 7"

Continuous wrinkled ut - l + - - - u ' laminar flame. Large- u1 ul scale turbulence l ~ / , laminar flame u t = u (u'>~Ul) thickness, ut cc Re

u'~Ull>~bl.C°n" tinuous conically

wrinkied laminar flame, u' >~ul l ~ S l. Surface split into separate islands.

FrankKamenetskii

Equations for ut

Surface split into separate eddies, burning with laminar

Conclusions

u t i n d e p e n d e n t of scale of turbulence

ut ' Ul - (1 +B(_U_)ul 2 ) %

B - const.

utindependent of scale of turbulence. 7"~ / u'

, ut ~u

t 1 u t = , ~lu- . ~ Vz

7- ~ l__

u t ~ (ul u') ½

Ul

u1

Leason

Karlovitz et al.

58

5962

Turbulent flow is composed of eddies of diameter l, with a sinusoidal velocity distribution through each eddy giving a rms vel. u'. Considers intersection of these eddies with a laminar flame.

Continuously wrinkled laminar flame with additional turbulence generated by the flame (u ").

lu' ~> 1 considering only increase 2ui in area Ul = ( 1 + (

)2)V2

(1) Turbulence increases the flame area. This is independent of scale. (2) Turbulence increasesrate of diffusion of active species or heat. This effect depends on I and r t./.

l > ~ l without flame gen. turb.

(i) u"~ul

ut= ut + u' (ii) u'>~u l u t = ul + (2UlU')Vz For strong turb. u' is increased

by u". ,,

Ul

u =-~-

(Pu Pb

- 1)

u l remains a more important factor than turbulence, P~ even for u >~ul .

Flame generated turbulence is much greater than u' in strong turbulence flow.

TURBULENT FLAME PROPAGATION

Author

Ref.

Scurlock and Grover

63

Wohl et al.

64, 65

291

Continuously wrinkled laminar flame. Additional turbulence generated by the flame.

Continuously wrinkled laminar flame

Equations for 8 t and 1"

Equations for u t

Assumptions

Conclusions

u t dependent on

t > > 8 I.

turbulence scale I. For high velocity confined flames, flame generated turbulence far outweighs the initial turbulence.

ut-

+ c(Y)2)V2 ~-t - (1 -C - const. Y - rms displacement of flame element dependent on eddy diffusion, flame propagation and flame generated turbulence

t independent of

Possible equations are:

ut

2u')2)½

(i) -all =(1 + ( - ~ b _ (ii) u ~ - I

_2~' + Ul

ut = (½[1 + (I (iii) ~-t , 4u'.2.½1X½

t~J Ut

J jj

r

(iv) ~ / : 1 + ( 2 [ u ~ -

1 +exp(-u'/ul)]) Ut_

(v) W -

-

fi

,

+ exp(-./.,)3) Talantov

6669

½

bl F

1 ~ 61

ut _ rb +

Surface split into separate islands

ut

½

2u' l Ul [in(1 + u~ ) 1 ½

In (1 +ulu' ) Utl.independent of

8t cc ~ u

/4 t

= / - u- r In (1 + u~- )

Sokolik

70

Model of small eddies gradually burning up in laminar flames

ut = average eddy vel. =½u(1 + ~ - ) ;

u = eddy vel.

ut 5t=81+lul T=--

l

ul Povinelli and Fuhs

71

Large scale turbulence wrinkles the flame surface, but does not affect u 1. Small scale turbulence influences the transport properties.

ut

u'

n°

=(l+h--/-/) f

F(n)dn

O

F (n) dn

+(1 + ~ - ) t/o

where no is frequency separating large and small scale fluctuations F(n) - spectral function

~_ turbulent flame gickness a n d T dependent on I. u l important even at high turbulence intensities.

292 Author

G . E . ANDREWS, D. BRADLEY, and S. B. LWAKABAMBA Ref.

Kozachenko 7276

Assumptions

Continuous wrinkled laminar flame with additional turbulence generated by the flame (u"). Only flame seats projected towards the fresh mixture take part.

Fquations for u t

t

Equations for ft and "r

Conclusions

tp

ut = Ul + u + u ut

u'

Pu/Pb -1 I

Ul = 1 + ui +

" / - I1 v3 I - ( h-7 ) UuS-l 2 [0.5 l

t

.u. > 1 ul Ul

Tucker

Ul +

N~~

-+ 1

77 Analysis of the interut _ u' 2 action of a plane -Ul -1 +S( ui ) laminar flame with isotropic turbulence.

s = f ( rb ) Tu

Richardson

78

Continuous wrinkled laminar flame

u' ~ U l ut - l + Ul

u' ~ u l 1 X/~

u' Ul

ft

= 2 l/4 /-u' Q/~lx/~ l

One dimension',d turbulence

t~

u' >~ u l ut - 1 + ul

u >>u l l 2~

This sudden, or 'catastrophic' [106] transition to turbulence occurs in tubes, ducts and boundary layers. Experiments on transition in boundary layers, using a vibrating ribbon have improved understanding [ 107-109] of the phenomenon. Unfortunately, experimental studies of the effect of flow perturbations upon laminar flames are scarce. Most work has been concerned with vibrating the flame holder [ 1 1 0 - 1 1 2 ] , rather than imposing oscillations on the unburnt gas [113]. Such investigations have been confined to low frequencies, but De Soete [114] has shown that a laminar flame becomes unstable as the frequency increases. Theoretical prediction of turbulent transition phenomena in nonreacting gas flows has not yet

u' 4/5 (u l )

l

fit = ~ -

( 2 u ' ) 2/5 uI

been achieved [96, 9 7 , 1 0 2 , 1 1 5 - 1 1 8 ] . However, careful observations of the catastrophic transition processes show that the process starts as a series o f turbulent spots that grow until the whole flow becomes turbulent [25, 119-122]. Within these spots, transition is abrupt and fully developed turbulence exists. There is less understanding of laminar flame stability, about which there are conflicting theoretical predidtions [123135]. The absence of experimental or theoretical proof of the stability of a laminar flame towards the spectrum of turbulent perturbations makes an assessment o f the validity o f the wrinkled laminar flame model difficult. It is difficult to see how such a flame could be stable for all values of Reynolds number. If it were, it would imply a transition to turbulent combustion without signif-

TURBULENT FLAME PROPAGATION icant change in the laminar combustion process. This in turn would suggest the unlikely existence of a stabilising force arising from chemical reaction, such that turbulent flow might coexist with a laminar reaction zone. At low values of Rx, (< 100), the rate of turbulent energy dissipation is low and the dissipative eddies are relatively large. Thus, both the velocity gradients and the enhancement of molecular motion in dissipative eddies is small. Perhaps a laminar reaction zone can exist within the large eddies until R X is increased to a value at which the small scale eddy motion renders it unstable. At this point another regime of turbulent combustion would arise, for the understanding of which, a more detailed knowledge of the structure of turbulence is required. 3.2. T u r b u l e n t T r a n s p o r t M o d e l s

The various theoretical models based on turbulent, as distinct from molecular, transport processes are summarised in Table 2. There are two main categories of the model, in which emphasis is on large and small scale eddy transport, respectively. 3.2.1. Large Scale Eddy M o d e l s If an entire large eddy is always at uniform temperature, then spontaneous reaction will occur throughout the eddy when the spontaneous ignition temperature has been reached. This presupposes perfect mixing within the eddy. If separate large eddies move as entities, the reaction front could be propagated by their motion counter to that of the unburnt gas. This may involve the movement of a reacting eddy over comparatively large distances and may give rise to either spontaneous ignition after turbulent mixing of a newly formed eddy or to a temperature rise of the eddy, which after further turbulent mixing reaches the spontaneous ignition temperature. Alternately, the process may involve the mixing of adjacent eddies. Thus the structure embodies fluctuating combusting eddies, each of uniform composition and temperature, and an extended reaction zone. The application of spontaneous ignition data to turbulent combustion was introduced first by Lloyd [145] and Mullins [146,147] and

293 Penner and Mullins [ 148]. They suggested that, in gas turbine combustion chambers, spontaneous ignition could be rate determining. Possibly spontaneous ignition data would give a more relevant parameter than burning velocities. On the other hand, Basevich [149] could find no correlation between turbulent propagation rates of flames and their respective spontaneous ignition characteristics. Sokolik, Karpov and Semenov [150-159] have postulated a turbulent P.ame structure in which propagation is due to turbulent mixing of burnt and unburnt gas, followed by spontaneous ignition. The precise details of the model in terms of the structure of turbulence are not altogether clear. These workers express the rate of flame propagation as the ratio of the macro Lagrangean scale to the chemical reaction time. This suggests that large eddy motion with accompanying reaction is involved. A model based on turbulent transport first was introduced by Damkohler [55] and Shchelkin [56]. They assumed that molecular transport in laminar flames would only be influenced by turbulence when l < 6 l. However, in very few practical flows is l less than 1 mm, a typical laminar flame thickness. Summerfield et al. [137-139] have developed a similar model with no restriction on l and applicable at high turbulence intensities. It is tempting to quantify this type of model using the same theoretical approach as for laminar flames, but with molecular transport coefficients replaced by equivalent turbulent ones. If thermal conductivity, k, is replaced by epCp, where e is eddy thermal diffusivity, p density, C specific heat at constant pressure, and if the t~arbulent Lewis number is unity, then the approach of Refs. [160, 161] gives _ ( epC ~ )

ut

v2

(12)

uI

If e is assumed equal to u'l and Pr is the Prandtl number, then Ut ul

_

( u'l Pr) P

Y2

(13)

G. E. ANDREWS, D. BRADLEY, and S. B. LWAKABAMBA

294

TABLE 2 Turbulent Flame Model, Predictions for u t and 5 t Author

Ref.

Assumptions k

Damk6hler

55

ut

1/2

UlCX ( C ~ o )

°:~v/~ ut

uI

~t Xt (__)1/2, where e = C ~ = lu'

_ ( e)1/2=

P-

1 <51 ( ~ )1/2

Shchelkin

56

UlCr ZX

TX= reaction time c r - E / R T

ut

Xt

--

=(1+

ut

V2 )

~-

l <5 l

Delbourg

X T =k+X t

136

cl = a c o n s t a n t

Thermalenergy T i = ignition temp.

c l Tuk (T b - T i) uI -

, l<61

;ca(r

_

ut-

CruXr(rb-r? p C p ( T i - T u)

c2

sents the nontemperature dependent terms in the turbulent reaction rate. For tube: -u- t

ut _ e (1+ ~ t ) uI

c = c t + c e repre-

u1

k

= 0.0128 Re ° A Summerfield

137-139

Thermal theory. Model applies irrespective of the scale /.

Bhaduriand Ippolitov

140-142

Zeldovitch-Frank-Kamenetskii laminar flame approach ff . . . ~k . t _ lu ~

ut ul

_ e p

ut _ Ul ul U

5l 5t

utSt e

k ~t

function of air/fuel ratio only.

U = flow velocity. '

Model applies irrespective of the scale L Predvoditelev and Tsukhanova

5t ec me , Expression for 5 t only.

143,144

ut l <5 l

Equation (5) gives, for a constant value of Pr

ut

cx R X .

(14)

uI

Although the simplicity of this approach is attractive, its basis in physical reality is slender. The theory of turbulent transport employed is

essentially a mixing length theory, based on analogy with molecular transport theory. Whilst the latter is valid for laminar flames, the former is less valid for turbulent ones. The molecular mean free path is small compared with the associated flame thickness, but the length scale of turbulence is not. Furthermore, there is little valid basis in the spectral theory of turbulence for assuming energy

TURBULENT FLAME PROPAGATION flux is proportional to temperature gradient. There are several incompatibilities between the flame models suggested in this subsection and the structure of turbulence discussed in Sec. 2. There it was shown that the large eddies have a long life time and low convective velocity. Usually large eddies are convected with a velocity similar to the mean flow velocity and are not capable of any significant motion against the flow [31,162-166]. Thus the transport of a large reacting eddy into the unburnt gas is unlikely. This, however, may be possible with small eddy motion, for which there is some evidence of high velocity motion relative to the mean flow [162,164, 165]. A further difficulty arises in models that presuppose perfect mixing within a large eddy. This is hardly compatible with the small scale structure of turbulence. This discussion, together with that on laminar flame stability, suggests the importance of the small scale eddies in turbulent flame propagation. 3.2.2. Small Scale Eddy Models

Shchetinkov [20, 167-171] noted that the properties of the small scale eddies were such that laminar flame propagation was unlikely within them. He proposed that in highly turbulent flows the motion of burning small scale eddies was the major mechanism involved in turbulent flame propagation; he also emphasised the importance of the relative magnitudes of the chemical induction time within the eddy and the eddy lifetime. Shchetinkov recognised the importance of an improved knowledge of the microstructure of turbulence for the development of flame theories. At the present time, neither the size of the eddies moving in a turbulent stream, their velocity, nor their lifetime, is known accurately. Neither do we know the minimum eddy size beyond which intra eddy processes are governed by the laws of local isotropic turbulence, nor whether a correlation exists between the size of an eddy, its path length, and its velocity of motion [170]. Flame propagation can be described in terms of the small scale structure of turbulence. A small eddy may be formed from both hot and previously unreacted gas. If the lifetime of'the eddy is greater than the chemical induction time for the conditions, reaction proceeds within the eddy until its

295 decay. After the decay of this and similar eddies in partially reacted or unreacted gas, another generation of eddies is born and reaction is induced in them, initiated by the heat release, temperature rise and species formation in the original eddies. .The lifetime of a small eddy may be less than the chemical induction time, but if the gas in such an eddy becomes a component of a succeeding, but hotter eddy, then flame propagation is possible. On the other hand, under conditions of ignition, before a propagating flame kernel becomes established, the gas from the original eddy may mix with cold gas and the intended igniting hot kernel may be quenched. More recently, Chomiak [172-174] used the model of intermittency proposed by Tennekes, referred to in Sec. 2.2, to suggest an intermittent fine flame structure at high values o f R X. His interpretation of experiments suggests a luminous zone of reaction with a size approximately corre. sponding to that of the Kolmogorov microscale. He suggested that strong vortex tubes play an essential role in turbulent flame propagation and that since chemical processes are molecular in nature, they will be more effective in the smaller dissipative regions, as compared with the nondissipative background. At low values of R~, the dissipative regions are weak and flame propagation should be independent of them. At high values of R X, the flame front would be "caught" by the dissipative regions and the mechanism of flame propagation would be purely hydrodynamic. At higher values of R X, any continuous flame sheet would be disrupted. Chomiak's suggestion that propagation requires the thickness of the dissipative regions to exceed the quenching distance seems, however, to be more open to question. It seems to be probable that for high values of Rx, the reaction zone is located in the dissipative eddies. The present authors, however, would not agree that the flame propagation would be purely hydrodynamic. It is also possible that a reaction zone may propagate within the dissipative regions with a velocity relative to the unburnt gas, which is much greater than the laminar burning velocity, as a result of the considerably enhanced transport processes within these regions. In order for a reaction zone to propagate through

296

G. E. ANDREWS, D. BRADLEY, and S. B. LWAKABAMBA TABLE 3 Turbulent Flame Proposed Propagation Parameters Criterion

Author

Ref.

Damk6hler

55

Flow Reynolds number in approach tube

Richardson

78

l 1 7" (U,Ul)VZ'

Kovasznay

175

Typicalvelocity gradient in_approaching cold flow _ u' Typical velocity gradient in laminar flame uI

Wohl

176

Mean chemical reaction time Meanlifetime of a large eddy

Klim ov

177 178

S

Talantov

179

Chemicalreactiontimeinlaminartlame = (61) Residence time in turbulent combustion zone ul

Bhaduri

180183

-Ud'

Sanematsu

184

u~'_ uI

6l

~

uI

UI

, u uI

where 7"= turbulence time. -/- ~ u' 7

8l l

6l Ul

, (_

)

T Tu

61 )t

where T-

_ T b -- T 2

, where S is the relative rate of change of the area of a surface element of the flame u" (~)

exp[_(l+u,/ul) ]

where U = flow velocity; d = duct diameter.

. ~n, b = screen mesh diameter; ~ = equivalence ratio; n = constant.

l b

the dissipative eddies, the lifetime o f the eddy, probably of the order of r~, must be greater than the time for the reaction to occur. This suggests that an increase in R x might quench the flame. Experimental data on this point are inconclusive, due to the difficulties of isolating the effects o f R x on a freely propagating flame, from those of ignition and flame holding. Although there have been important advances in the understanding o f turbulence structure, in recent years, as outlined in Sec. 2, progress in prediction or measurement of the properties o f small scale dissipative eddies has shown little advance b e y o n d that reported in Ref. [ 170]. However, lack o f knowledge of the properties o f such eddies does not preclude the probability that they are determining factors in the mode o f flame propagation in highly turbulent flows. The precise details of the propagation will.be uncertain in the absence of a better understanding and measurements of small eddy size, form, lifetime, intermittency factor and convective velocity.

4. Correlation of Turbulent Burning Velocity Data tn a situation in which there is no comprehensive

theory of turbulent burning, perhaps the most satisfactory approach is to correlate turbulent burning velocity data in terms o f nondimensional groupings that take account of the determining physical and chemical phenomena. Table 3 shows some o f the dimensionless criteria that have been proposed, against which the dimensionless ratio, u t / u l , might be plotted. The dimensionless parameters of Ippolitov [142], Kovalyev et al. [141], Bhaduri [181-183] and Sanematsu [184] are not completely general, as they contain apparatusdependent terms [185]. Apart from that o f Damkohler [55], all the remaining proposed groupings contain mixed chemical and turbulence parameters. The present authors believe that there is advantage in separating these two categories o f parameter and that the cold flow turbulent Reynolds number, R x, is an appropriate practical parameter that makes allowance for the various turbulence phenomena, discussed in the previous sections. The dimensionless group, u t / u l may be retained and simple laminar flame theory shows that the effects of the determining chemistry will to some extent be incorporated in t h e magnitude o f u l . Thus it is proposed that the primary

TURBULENT FLAME PROPAGATION

297 Karpov, Semenov and Sokotik [150], Semenov [186], Sokolik, Karpov and Semenov [159] and Vinckier and Van Tiggelen [187]. The experiments of Karpov, Semenov and Sokolik were carried out in a nearly spherical vessel equipped with four stirrers driven by electric motors. This achieved a central region of uniform intensity of isotropic turbulence and higher values o f R X than can be aChieved from grid generated turbulence. The values o f / a r e given in the paper of Semenov. These workers summarised their results in the form of plots o f u t against u'. Experiments were carried out with a variety of fuels and diluents and the final results indicated a mixture dependence. Vinckier and Van Tiggelen used an opposed jet burner at atmospheric pressure. They investigated stoichiometric CH4-02 and C2 H2-O2 flames, both with different amounts of diluent N2, and results were correlated by plotting ut/u/A against u'. All the experimental points given by both ihe Soviet and the Belgian groups have been replotted

correlation of experimental data should be sought in plots of ut/u l against R X. In the course of such investigations, other influences affecting u t might be discerned. The simplified theory leading to Eq. (14) also supports this proposed correlation, even though the derivation of the equation grossly simplifies the relationship between the transport of energy by turbulence and the turbulent velocity with its associated scale of distance. As far as the present authors are aware this correlation has not been carried out previously. Experimentalists have recognised the importance of measuring u" in the cold flow, but very often the turbulent scale of distance has not been measured and the importance of the kinematic viscosity has been unrecognised, despite the fact that its value is a determining factor for small eddy size. Few workers have presented data that are complete enough for the proposed correlation. However, measurements of u" and l have been made using hot wire anemometers and have been reported, along with measurements o f u t , by

1

I

I

KEY REFER TO TABLE /. FOR ~ R N T MIXTURE SYMBOL MIXTURENo.SYMBOL MIXTURENg.

20

e • o [3 m @ .B-

1 2 3 & 5 6 7

[] •

8 9

'~ • 4, • ~' ~

10 11 12 13 14 15

0 m

~o

m

0 v

10

°

"

I

200 TURBULENT REYNOLDS NUMBER RA

I

300

Fig. 2. Influence o f t u r b u l e n t Reynolds n u m b e r on turbulent burning velocities. (See Table 4 for Mixtures.)

298

G. E. ANDREWS, D. BRADLEY, and S. B. LWAKABAMBA TABLE 4 Mixture Data for Figure 2 Mixture No. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Mixture CH4 + 202 +4.93 N2 CH4 + 602 + 9.07 He 2H2 +02 + 17 Ar C3H8 + 502 + 28.5 Ar CH4 + 202 + 4.9 Ar 2H2 +02 + 17 He C3H8 + 502 + 28.5 He C3H8 + 502 + 18 Ar C3H8 + 502 + 18 He CH4 + 202 + 4.50 N2 2C2H2 +502 + 27.15 N2 2C2H2 +502 + 18.48 N2 CH4 + 202 + 2.79 N2 CH4 + 202 + 1.96 N2 2C2H2 + 502 + 14.88 N2

in Fig. 2, in accordance with the proposed correlation. Values of ;k were obtained from the macroscale using Eq. (6). The viscosity of mixtures was derived using the expressions of Wilke [188]. The key to the different experimental points that are plotted is given by Table 4. The values of u I are those used by the original workers. It is seen that a fair correlation is obtained by the presentation of the rather varied data in this way. There is no consistent influence of the mixture upon the ut/u l vs R ;k relationship. It will be observed that for values of R?, above 300, mixtures numbers 3 and 4 seem to give rather low values of ut/u I and even suggest a decrease in this ratio with increase of R~. These mixtures have the lowest laminar burning velocities and, because of this, they will have some of the largest laminar flame thicknesses. The technique of Sokolik, Karpov and Semenov is dependent upon flame photography and accurate pressure measurement. It requires the determination of the ratio of u n b u r n t to burnt gas density and the expression adopted for this neglects the finite thickness of the laminar flame [189]. If this is appreciable, there is a significant underestimation of the turbulent burning velocity. This could explain the low values of utlu l for these mixtures. Clearly there is a need of accurate data for values of R?~ in excess of 300. The scatter of points of the graph is not unsatis-

uI

p X 106 (m 2 sec-1 )

(msec-l )

15.90 34.00 15.63 13.78 14.91 95.35 54.31 13.63 42.77 15.91 15.22 15.09 16.00 16.07 15.01

0.7 0.7 0.29 0.45 0.7 0.7 0.74 0.8 1.4 0.89 0.90 1.55 1.55 1.93 1.98

Reference 150 150 159 159 159 159 159 159 159 187 187 187 187 187 187

factory when due regard is paid to the possible large errors that can arise in the measurement of burning velocities. The possible errors in the measurement of laminar burning velocities have been discussed by Andrews and Bradley [189]. The accurate measurement of turbulent burning velocities is even more difficult. It is not possible from Fig. 2 to detect any secondary influences, such as equivalence ratio, on the value of ut/u l . For this to be done perhaps more data and of greater accuracy are required. If there are different regimes of turbulent combustion, these are not clearly differentiated in Fig. 2. If a wrinkled laminar flame can exist within large scale eddies only up to a value of R~ at which small scale motion renders them unstable, then the upper limit for this regime is probably in the region of R k equal to 100, where the dissipative eddies first start to acquire a size significantly less than that of the large eddies

(35). The lifetime of a large eddy is approximately

l/u' and its laminar burning time would be l/u 1. I f u ' 2> ul, then the eddy will decay before it has been consumed by laminar combustion and burning may either continue in a newly formed large eddy or cease. It is of interest to note that for hydrocarbon-air atmospheric combustion and with a value o f / e q u a l to 10 mm, the limit condition, u ' = u l, is obtained at a value o f R k

TURBULENT FLAME PROPAGATION

299

of approximately 100. The wrinkled flame theories o f Table 1 predict values of ut/u I between approximately unity [56, 57, 64, 65] and five [66-69, 72-76] for u" = u I. For higher values of Rx, beyond a possible wrinkled flame regime, the role o f dissipative eddies becomes important, but Fig. 2 gives no conclusive indication o f a regime of burning that is purely hydrodynamic, as was suggested by Chomiak [172, 173].

and is relatively unchanged by changes in the value o f u ' . The Table suggests that the practical range o f R x for turbulent combustion lies between 100 and 1000. In general, grid turbulence creates values o f R x < 100 [ 1 6 6 , 1 9 2 - 1 9 7 ] and only one example of grid turbulence with a higher R x could be found [198].

5. Combustion Systems and the Turbulent Reynolds Number If turbulent burning velocities can be correlated satisfactorily with RX, then there is a need to measure unburnt gas turbulent parameters in practical combustors, in order to evaluate R x. Few measurements are reported o f all parameters, but estimates can be made o f R x using Eqs. (6) and (7), and the assumption o f isotropic turbulence. Such estimates are given in Table 5 for a variety of combustors. The macroscale, l, is determined by the width o f the turbulence generator

At the present time attempts to quantify the rate o f turbulent flame propagation would seem to rest upon measurements of turbulence in the unburnt gas. Such measurements have been carried out in m o t o r e d reciprocating engines by Semenov and co-workers [199-201 ], Ivanov [202], Molchanov [203] and Horvatin and Hussman [204]. The use of Rx, or R l, as a parameter o f turbulence requires that the length scale should be measured, in addition to u'. Values of v should also be derived and measurements of the intermittency factor might prove useful. Values of u" are governed by the mean velocity of flow, whilst the macroscale, l, and hence X, are related to the size of the generator o f turbulence in the flow system of the combustor. In

TABLE 5 Values of RaA for Different Combustors Combustor Bunsen Burner (190)

l (mm)

u' (m sec-1) (approx)

10

2

p (m 2 sec-1 ) 15.6 X 10"6

R~k (approx) 250

P = 1 atm T = 293°K Internal Combustion Reciprocating Engine

20 Typical inlet valve diameter

1 Semenov (201) measurements at T.D.C.

6 X 10-6

Large Pulverised Coal Burner

300 Distance between swirl vanes

2 5% mean velocity (40 m/sec)

39 X 10-6

P= 1 atm T = 550°K

850

Gas Turbine Combustion Chamber

20 Typical primary hole diameter

3.5 5% mean annulus velocity ( ~ 70 m/sec)

Ground Idle 12 X 10-6 Max Power 4.5 X 10-6

P = 2 atm T = 375°K

550

StiJred Reactor (191)

8 (jet)

3 (recirculation zone)

400 P = 8 atm

T= 580°K

25

15.6 X 10 -6

10

15.6 X 10 -6

P = 16 atm T = 700°K

900 800 (jet) 300 (recircula tion zone)

300

G.E. ANDREWS, D. BRADLEY, and S. B. LWAKABAMBA

some instances, these distances are not design variables. Nevertheless, for a given mass flow rate the Reynolds number is inversely proportional to the size and, under these conditions, a decrease in a diameter can increase the turbulent Reynolds number. Where the number of burners or air entry ports is a design variable, a reduction in this number with an associated increase in their size, also would result in an increase in R~,. The influence of unburnt gas kinematic viscosity upon R~. in practice is principally via the influence of pressure. This is important in engine combustion, which often occurs over a range of pressures. An increase in pressure results in an increase inR~, and this, in turn, gives rise to an increase in ut/u r Measurements that have been made of the influence of pressure upon turbulent burning show that it increases u t and Ut/Ul, even in those cases where an increase in pressure gives rise to a decrease in ul ['152, 205-208] . It is to be expected that the effect of an increase in unburnt gas temperature would occur principally via an associated increase in u r It has been shown that when combustion occurs in intense turbulent flows it is most likely to be associated with the small scale eddies. The properties of these and the turbulent burning rate, are a function of R~. Hence the distribution of R~

in the unburnt gas in a combustor will be of prime importance in the performance of the combustor. Combustion will tend to be located preferentially in those regions where the rate of strain and Rk are highest, and no flame propagation is likely in those where R;~ is lowest. Similarly, if the approach flow turbulence exhibits large scale intermittency, then there will be a similar intermittency in the reaction zone. Similar considerations apply to turbulent wakes and jets, enclosed by laminar flows. The much higher rates of flame propagation in the turbulent region, due to the presence of small scale eddies, makes it impossible for propagation to occur outside this region. Thus turbulent flame boundaries tend to follow those of the equivalent turbulent wake or jet, irrespective of the value of u 1. This has been confirmed for wake turbulence by Wright and Zukoski [209]. The flame is also affected by factors, such as the rate of strain of the flow, which influence the spreading of the wake [210]. On the other hand, a flame can be stabilised in a flow that is completely turbulent, with no surrounding laminar flow, with a flame holder wake turbulence that is much less than that of the surrounding flow [211 ]. Under these conditions, a normal turbulent flame front can be generated that is quite different from that of wake turbulence.

6. Conclusions (1) A review of the structure of turbulence suggests the use of the microscale Reynolds number, RX, as an important parameter in turbulent combustion. (2) A review of existing models for turbulent flame propagation suggests that in the wrinkled laminar flame model the laminar flame becomes unstable at an upper limit of R?c For higher values of R?c the turbulent small scale structure becomes important. (3) Available experimental data support a correlation of ut/u l with R?c which is independent of mixture composition. Further investigations are necessary to ascertain secondary influences. (4) Data should be obtained on the unburnt gas turbulence parameters in practical combustors. A knowledge of values of R?~ in a combustor is valuable in ascertaining combustion performance.

The authors thank the British Gas Corporation and Leeds University for Studentships to G.E.A. and S.B.L., respectively. They also thank Mr. F. R. Mobbs o f Leeds University Mechanieal Engineering Department for informative and stimulating discussions on the nature o f turbulence.

TURBULENT FLAME PROPAGATION

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