ON F R O N T A L C O M B U S T I O N IN A S L I G H T L Y

I N T E N S E L A R G E S C A L E T U R B U L E N T S T R E A M *

V. N. V i l y u n o v a n d I. G. D i k UDC 536.46

In a f i rs t approximation, the increase in react ion zone motion for turbulent pulsation scales L some- what g rea te r than the laminar flame thickness 6 can be considered due to the increment in the combustion surface. In the limit case L / 6 ~ ~, the picture of the phenomenon should not depend on the heat conduction. However, if the heat conduction coefficient a--* 0, then the heat l iberation function ~ simultaneously be- comes infinite in a na r row zone outside whose l imits it is negligible in magnitude. Such an approach affords the possibi l i ty of writing the kinematic equations of combust ion front propagation. However, the kinematics is not capable of taking into account the p roces s of rebuilding the heat fields occur r ing in a s tat is t ical ly de- formable flame. The effect of such nonstat ionary p r o c e s s e s will be more essent ial the smal le r the rat io between the cha rac te r i s t i c t ime of a hydrodynamic mole ~-~ = L / u ' and the thermal relaxation t ime in the flame T o = a / u 2, where u' and u n are the s t r eam veloci ty pulsation and the normal flame velocity. On the average, nonsta t ionary p r o c e s s e s increase the combustion veloci ty and should be taken into account in a second approximation.

1o If the turbulent pulsations are a low-frequency and l a rge - sca l e par t of the spec t rum T 1 > To, L > 6, then the formulation of the problem of the veloci ty of turbulent combustion, due to Ya. B. ZeUdovich [2], r e - duces to the following. The field turbulence veloci t ies V (x, y, z, t) is given, where there exists a t rue (not averaged) flame surface F (x, y, z, t) = 0 which separa tes the two phases : f resh (a) and burning q3). At each point the surface F =0 is displaced relat ive to the burning gas at the normal veloci ty u n. It is a s sumed that for anyini t ia l shape, the surface takes some shape as t -*~o (stat ionary on the average), and this shape is propagated at the veloci ty u t which is to be determined.

It is essent ia l in such a formulation that the normal veloci ty be u n r 0. If it is assumed that u n =0,

then just diffusion between (~) and (fl) occurs ; the domain where there is (~) and (fl) expands as Ct on both sides, i.e., there is no motion of the averaged surface as t -~r Un, d i rec ted f rom (fl) to (a), will produce an a s y m m e t r y between them whereupon the averaged surface will also move f rom (/3) to (a). Hence, it is impossible to calculate ut for any turbulent veloci ty field V (x, y, z, t) by neglecting the normal velocity; the phenomenon vanishes for u n =0,

For simplici ty, let us consider the two-dimensional case. Let the equation of the flame surface sat isfy the equation

F(x, y. t)=0. ( 1 . 1 )

The kinematic equation of propagation of the surface of discontinuity in a coordinate sys tem coupled to the moving gas is

OF / [ OF ) ' [ oF ]' (1.2) o-T=u, y ~,a,,j +lo~j"

The ratio

OF : OF . . . . u (1 .3)

Ot ,' Ox

* This paper is an elucidation of the repor t to the Third Scientific Conference on Mathematics and Mechanics of Tomsk State Universi ty [1] and to the All-Union School-Seminar on Combustion Theory.

Tomsk. Transla ted f rom Fizika Goreniya i Vzryva, Vol. 11, No. 2, pp. 223-229, March-Apri l , 1975. Original ar t ic le submitted March 26, 1974.

019 76 Plenum Publishing Corporation, 22 7 West 1 7th Street, New York, N. Y. 10011. No part o f this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission o.f the publisher. A copy o f this article is available from the publisher for $]5.00.

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has the meaning of the instantaneous ve loc i ty of sur face advancement in the x direct ion. with r e spec t to the t ime will yield the veloci ty of turbulent gas combust ion

t

�9 1 ~ ~gF : OF U t :: hm-7- . ] - ' - - dl'.

t ~.,~ Or" : d x o

Averaging (1.3)

(1.4)

If it is a s s u m e d in a f i r s t approximat ion that u n =const , then the p rob l em b e c o m e s pure ly kinematic . It should be noted that the approach taken is equivalent to applying the H u e y - M i c h e l s o n p r i n c i p l e if the t ime ave raged u t se lec ted here for a given sect ion of the front equals the ave raged value of u t along the y axis pe rpend icu la r to the d i rec t ion of f lame motion for some t ime. Indeed, the complex ~fl + (a F/@y)2/(BF/sx) 2 is a re la t ive inc rease in the f lame sur face . If u n =0 then u t = 0 a n d t h e su r face moves p a s s i v e l y toge ther with the gas.

The turbulent ve loc i ty field can be taken into account by wri t ing (12) in a fixed coordinate s y s t e m

a.F (1.5) at ~" ~ grad F : u . lgrad FI.

In this ease (1.4) yields the f lame ve loc i ty re la t ive to a fixed o b s e r v e r . The x, y components vl(x , y, t) and v2(x , y, t) of the vec to r ~ , under the assumpt ion of incompress ib i l i ty , sa t i s fy the continuity equation (the influence of t h e r m a l expansion is not considered)

art § &', =0. (1.6) Ox 0~

If at l eas t one component , say vl(x, y, t), is known, then by ass igning some initial shape of the sur face , c o m - putat ions of the nonl inear equation (1.5) can be made and u t can be calculated.

It should be noted that the initial conditions for (1.5) should exe r t no influence on the f lame ve loc i ty u t because of r a n d o m n e s s of the s t r e a m . On the o ther hand, by v i r tue of (1.4) it is suff icient to know just the ra t io between cor respond ing der iva t ives . The expres s ion for u is obtained d i rec t ly f r o m (1.5)

U = u,J/1 + ~ 2 - - v l - - v ~ . (1.7)

Here, the notation r = (OF/Oy)/(OF/Ox) with the sense of the slope to the y axis has been introduced in addi- t ion to u. F o r ~ =0 (plane flame)

Let us introduce the Monge v a r i a b l e s

0F

u-----u.--Vl. (1.8)

OF bF P = - ~ " ; q= a--~-

and let us wri te down the p ro jec t ions of the in tegra l sur face (1.5) on the plane of the v a r i a b l e s f , p, q

dr dp dq

where ~i a r e t he app rop r i a t e de r iva t ives of the functions

8vl. ~o~. 0vi or,. dr, Or,

(1.9)

By v i r tue of the equal i t ies

df = U + au dq = ~ § p a~ - ~ P - ~ - ; . dp dp ,

then by using (I.9) and the re la t ionship between the notation introduced, we obtain

a , L% + , ~ , @s +*~ , ~2 (1.10)

The di f ferent ia l equation cannot be in tegra ted in genera l f o r m since the function ~i is a function of u, r

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In the s implest case of a shear flow, we have

v2~-0; cp2=~4~---cp6=~3=0

The equality cp 3 =0 resul t s f rom (1.6). Now, taking account of (1.8),

u = -~ (1.11)

Some mean value ~1/~5 is taken outside the integral sign, and we approximate it by using the turbulent pulsation p a r a m e t e r

(~1/r ~ BL/~:I ~ B u ' . (1.12)

Here and henceforth, u' = ~ , for brevity, and B is a coefficient on the o rder of one. Using (1.12), we obtain f rom (1.11) and (1.7)

Bu" /u,, (1.13) : 2 1 - - (Bu'/'Un)2"

Let us note that the approximation of the der ivat ives ~Pl and q~5 f rom (1.2) is equivalent substantial ly to averaging the hydrodynamic field. Hence, taking account of (1.13) (in the sys t em coupled to the moving gas), we obtain the formula

ut 1 + (Bu' /u.) ' (1.14) U n l -- (Bu'/Un) z"

In par t icu lar , the analog of the Tucker formula (cited in [3])

2_c~ = 1 § ( B u ' / u , f " I~R

follows f rom (1.14) to the accu racy of (u'/un) 4 t e rms . (The B in Tucke r ' s formula depends on the heat of react ion Q in such a way that B ~ o~ as Q --- 0, and is hardly physical ly explainable.)

Another (equivalent) approximation

is obtained when ~ is taken f rom (1.13)

..u-A-t = r l -J- 4B~(-~. ) 2 (1.15)

, = 2 Bu_~' (1.16) U n

The expression (1.15) is a K. L Shchelkin resu l t . Another means to obtain i t is given in [4].

Therefore, in the s imple case considered known dependences of u t on u' have been obtained which were derived f rom simple geometr ic and kinematic es t imates . The method used here may turn out to be prospect ive ly more fruitful in studying a flame in a nonuniform turbulent veloci ty field, where it is diffi- cult to const ruct hypotheses about the behavior of the flame.

It should be noted that a s imi lar formulation of the problem was real ized in [5] to some extent, but a whole se r ies of unjustified assumptions (let us mention the decomposit ion of the root in an equation of type (1.7) real ized there, the introduction of an a p r io r i hypothesis about the nature of the surface s tat is t ics , etc.) resul ted in a doubtful resul t so that a special assumption about the relat ionships between L, T~, and u' or u n for different u ' /ua must be formulated. The final resul t in the paper

u J u ' = 1 + ~-~(u'lu,,)6/5

differs f rom that obtained above.

2. A kinematic formulation of the problem cannot naturally touch upon the question of the influence of the flame s t ructure on the combustion p roces s because the charac te r i s t i c scales 5 and L have dropped out.

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For L >> 5 the f lame sect ion toge ther with the heating zone is en t ra ined by the pulsa t ions as a whole without rebui lding the i r s t ruc tu re in p rac t ice . If th is inequali ty is not v e r y deep, then the ' n o r m a l " t e m p e r a - tu re prof i le will be de fo rmed more s t rongly (because of the nons ta t ionary convect ive flows) the g r e a t e r the ra t io ~0/Tl" Combustion becom es nons ta t ionary in a local domain. It is imposs ib le to cons ider u n=cons t .

Keeping in mind the case of equali ty of the heat and m a s s t r a n s f e r coeff ic ients a =D, let us just ex - amine the ene rg y equation

OT V grad T = /02T O~T~ - ~ -+- a/o- ~ + 0 - 7 / + (I) (T). (2.1)

The left s ides of (2.1) and (1.5) a re identical. It is natural to take some i so the rm in the f lame Tt =const as the sur face under considera t ion. Now the equation of the sur face is r e p r e s e n t e d as

F(x, g, t )=T~--T(x , y, t)---O.

Let us cons t ruc t the equation

u.[ grad TI =aV2T--{-@(T), (2.2)

f r o m (2.1) and (LS), which d i f fe r s in f o r m for a normal i so the rm veloc i ty re la t ive to the medium f r o m that which i t had obtained in [6] (apparent ly f i r s t ) . The equivalence of these r e su l t s has not yet been shown successfu l ly . Let us note that (2.2) has a s imple meaning: u n should be of such a magnitude that the con- vect ive heat flux would cancel the heat t r a n s f e r by conduction and its l ibera t ion by the chemica l source .

Let us go over to the angle function

or/0V = Og / Ox

so that

OT Igrad TI = -~xV1 + ap 2.

The Laplac ian 02T , Oxi 0 ! OT\ 02T . [ , OT /02T]O~p ) �9 = ~ {I + 4 2

is conver ted analogously. Here x 1 and Yl belong to the i so the rm, hence dx 1/dy 1 = r The ra t io

OT /OST ~ /b-~. =lo,

where l0 equals 5 in o rde r of magnitude, can c h a r a c t e r i z e the heat scale of the f lame. This is a constant for the Michelson profi le ; let us se lec t some mean value in the more genera l case .

Af te r the manipulat ions p resen ted , we will have instead of (2.2)

u . ~ ~ a + +10, 0-~+r (2.3)

If r does not v a r y downst ream, then because of the spat ia l invar iance of the equation fo r the s t a t ionary f lame, the f lame veloc i ty is invariant . Here the nonsta t ionar i ty of the f lame is a s soc ia t ed with the fact that for a fixed f lame e lement , the heat exchange conditions v a r y together with

ai =- a( l @ ~2 + lo~ ~

continuously (and chaotical ly) , which exe r t s influence on u n and i ts a s soc ia t ed quantity

u,=-u.~l+~ 2".

Provid ing a pseudo-one -d imens iona l and s t a t ionary equation resu l t ing f r o m (2.3)

aT O~T ul ~ = ai ~ + (D (T),

(2.4)

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by the ord inary conditions of no heat flux far f rom the combustion zone, and giving the initial mixture t e m - pera ture T_, which together with the heat of react ion Q and the specific heat governs the tempera ture of the combustion products T+ in the adiabatic case we formal ly obtain the c lass ica l formulation of the problem about the rate of gas combustion.

Let us consider the veloci ty of i so therm motion in a slightly curved flame front to cor respond to the regular i ty u 1 ~ ~ . Here t 1 is the cha rac te r i s t i c t ime of the chemical reaction, which depends only on the t empera tu re cor responding to the given isotherm. Then

u..~i ~ 1 / a l

Let us note that if a line where the maximum heat l iberation is obse rved is taken as the flame line, for example, then this line in a nonstat ionary flame will be related to different i so therms, and then it is cer ta in ly impossible to consider t 1 constant. Let us assume both approaches to be s ta t is t ical ly equivalent. The express ion for ul will be

ul u, V I § ,24 t0, (2.5)

The passage f rom u 1 to u t is accomplished by substituting its mean value instead of r into (2.5). For example, consider ing (1.13) and (1.14) as a f i rs t approximation, it is possible to take r ~ u ' /u n and ~r u ' /Lun, respect ively . Then the formula for the turbulent gas combust ion veloci ty becomes

It is not difficult to extract the K o v a s z n a y - K I i m o v pa rame te r [5, 6] in the last member under the radical (since l0 ~ 6)

r A ~0I~1 - l0/L__ - - u , / u " (2.7)

which indicates the degree o f n o n s t a t i o n a r i t y o f t h e p r o c e s s e s in the flame, cor responding to [7, 8].

The result (2.6) obtained in isotropic turbulence is eas i ly extended to the three-d imens ional case. The fac tors ahead of the member ~ (u'/un) 2 are hence doubled. Indeed, for the three-d imens ional case it is n e c e s s a r y to introduce the additional function 0 = (~F/3z) / (aF /0x) , which is s ta t is t ical ly equal to a i = a [1 +2r 2 +2l 0 r (0 r andul= u n 4~ +2r 2.

A formal compar ison between (2.4) and the s ta t ionary flame equation is understandably not r igorous. For example, according to [6] there is a definite corre la t ion between a 1 and a2T/0~ 2 . Hence, the resul t (2.6) should be considered approximate

The difference between the method of taking account of the nonconstancy of the normal combustion veloci ty and the phenomenological Markstein approach which a s sumes a dependence of u n on the curvature of the front should be noted. If the curvature is retained, then according to Markstein the combustion veloc- ity is invariant . Nonstat ionary equations were used in this paper; the combustion velocity var ies because of the change in slope in the flame to the direction of motion. Such a rotation of a section of the flame oc- cu rs in a turbulent flow under nonsta t ionary conditions, i.e., the heat exchange conditions va ry continuously. If the line of grea tes t heat l iberation is taken as the flame front, then the maximum reaction rate fluctuates all the t ime, so that the Markstein cor rec t ion ref lec ts either fluctuations in the react ion rate or f luctua- t ions in the rate of heat exchange. The fo rmer of the mentioned c i rcumstances was taken as the basis of the cor rec t ion in [9], and the lat ter , which yields a s ta t is t ical ly identical result , was taken as the basis of the cor rec t ion in the present paper.

Questions of the r eve r s e effect of the f lame on the s t r eam were not considered.

In conclusion, the authors a re grateful to Academician Ya. B. Zel 'dovich for formulat ing the problem.

i.

LITERATURE CITED

V. N. Vilyunov and L G. Dik, Abstracts of Reports to the Third Scientific Conference on Mathematics and Mechanics [in Russian], Izd. Tomsk. Gos. Univ., Tomsk (1973).

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2.

3. 4.

5.

6. 7. 8. 9.

Ya. B. Zel'dovich and D. A. Frank-Kamenetskii, Course in the Theory of Combustion, Detonation, and Explosion [in Russian], Pt. II, MMI, Moscow (1947). F. A. Vii'yams, Combustion Theory [in Russian], Nauka, Moscow (1971). K. I. Shchelkin and K. Ya. Troshin, Gasdynamics of Combustion [in Russian], Izd. AN SSSR, Moscow (1963). J. M. Richardson, Proc. Aerothermochemistry Gasdynamics Symposium, Northwestern Univ., Evan- ston (1956). A. M. I(2imov, Combustion and Explosion [in Russian], Nauka, Moscow (1972). L. S. Kovasznay, Jet Prop., 26, No. 6 (1956). A. M. Klimov, Zh. Prikl . Mekh. Tekhn. Fiz., No. 3 (1963). G. I. Barenblatt, Ya. B. Zel'dovich, and A. G. Istratov, Zh. Prikl . Mekh. Tekhn. Fiz., No. 4 (1962).

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