Effect of gravity on the propagation of flames in tubes

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Effect of gravity on the propagation of flames in tubes

P. Pelcé

To cite this version:

P. Pelcé. Effect of gravity on the propagation of flames in tubes. Journal de Physique, 1985, 46 (4),

pp.503-510. �10.1051/jphys:01985004604050300�. �jpa-00209990�

Effect of gravity ^{on the} propagation of flames in tubes

P. Pelcé

Laboratoire de Dynamique ^et Thermophysique ^{des Fluides,} Université de Provence, Centre St Jérôme, 13397 Marseille Cedex 13, ^France

(Reçu ^{le 26} septembre 1984, accepté le 27 novembre 1984 )

Résumé.

Il s’agit de l’étude théorique de l’influence du champ ^de pesanteur ^{sur la} propagation d’un front de flamme dans un tube. De la même façon que dans le travail de Zeldovich ^et al. [4], l’existence de zones de stagnation

dans les gaz brûlés permet d’ établir des relations intégrales ^{ne mettant en} jeu que la forme du front ^et le champ

de vitesse amont. Ces relations sont exploitées pour donner ^une estimation de la forme du front courbé ainsi que ^sa vitesse de propagation fonction de l’expansion des gaz ^et du nombre de Froude. Cette étude ne tient pas compte ^des effets thermodiffusifs apparaissant ^dans l’épaisseur ^{de flamme} puisque ^la vitesse normale de propagation ^est supposée constante. La validité de cette approximation ^est finalement discutée.

Abstract.

The influence of a gravity field upon flame propagation ^{in a} tube is studied. In the same manner as in the work of Zeldovich et al. [4], the existence of a stagnation ^zone in the burnt gas enables ^{one to} calculate integral

relations involving only ^{the front} shape ^{and the} upstream velocity field. These relations are used to estimate the curved shape of the front as well as its propagation velocity, ^{as a} function of the expansion ratio and the Froude number. This study neglects thermodiffusive effects which occur inside the flame thickness since the burning velocity

is assumed to be constant. Finally, ^the validity ^{of this} approximation is discussed.

Classification Physics ^Abstracts

47.20 - 47.70F

1. Introduction.

Many experiments have demonstrated propagation

in tubes of curved flame fronts having stationary shape ^{and constant} velocity [1-3]. ^{These works show}

that both the shape ^{and the} velocity ^are highly depen-

dent on various parameters ^{such as :} equivalence ^ratio

of the reactive mixture, ^tube _geometry, ^tube radius,

and direction of propagation. ^{Some recent} simplified

theories give qualitative ^{results, which in some cases,}

are in agreement ^{with the} observed flame shapes ^and

velocities.

In a first study, Zeldovich et al. [4] ^{assume that the}

flame front is a surface of hydrodynamic discontinuity separating ^two incompressible gases, the fresh and burnt mixture, ^with respective ^densities po and _Pb.

The flame propagates ^{with a constant normal relative} burning velocity UL and the effects of buoyancy ^and viscosity ^{are not} taken into account. With these

hypotheses, ^Darrieus (1938) [5] ^{and Landau} (1944) [6]

have shown that the plane ^flame front is unstable for small perturbations because of the deflection of the

velocity ^field through the wrinkled flame due to gas

expansion. Zeldovich et al. [4] ^have suggested that,

in a tube, ^{due to} non-linear effects, ^{the unstable} plane

flame front can restabilize as a curved front, with a

stationary shape ^convex toward the fresh mixture,

and constant propagation velocity. Using ^conserva-

tion low arguments, which will be discussed in detail

later, they evaluate the propagation velocity ^U

as U ⁼ UL f(0153), ^where f ^{is a} numerically ^determined

function taking ^values larger ^{than one, and a =} pb/P o

the _gas expansion ^ratio.

In other works, Levy [12], Bregeon et ^al. [13], ^Von

Lavante and Strehlow [3] consider the upward pro-

pagation ^{of a flame in a vertical} tube. They ^show

that the regime ^{of constant} propagation velocity ^of

such a flame _may be compared ^{to the motion of a} gas bubble rising ^{in a} vertical tube filled with water. This type of motion has been studied initially by ^Davies

and Taylor [7]. ^{When an} interface between two fluids of different densities and negligible viscosity ^{is sub-}

mitted to an acceleration perpendicular ^{to it, the sur-}

face is stable if the acceleration is directed from the

more dense to the less dense fluid and unstable if the acceleration is in the opposite direction. When the

plane ^{interface is} unstable, ^{it can} restabilize as a

curved _one, with stationary shape ^{convex towards}

the more dense fluid and rising ^{with a constant} velocity.

The rising velocity of the bubble was evaluated as

U ⁼ k gR, where k is a numerical constant, g ^the acceleration of gravity, and R the tube radius.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01985004604050300

504

In flame propagation experiments, both deflection of the velocity ^field through the tilted front and

buoyancy ^{effects occur} together. Therefore the observed flame shape ^and propagation velocity ^must depend ^{on the relative} intensity ^{of these two} influences, determined by the Froude number F ⁼ gRl7ru’ ^and

on the direction of propagation of the combustion front. In the following ^{we consider} only ^the propaga- tion of a flame in a vertical tube. Upward propagation

will be represented conventionally by ^a negative

Froude number and downward propagation by ^a positive number. The two theories mentioned above describe the cases F ⁼ 0 and F ^{= -} oo. The _purpose of this work is to investigate theoretically ^the possible

flame shape ^{and the} corresponding propagation velocity ^{for an} arbitrary Froude number by imple- menting the method developed by Zeldovich et al. [4].

2. The model.

The curved flame propagates ^{in a} vertical channel with rectangular cross-section, ^{where one of the}

dimensions can be considered as infinite and the other of half length ^{R. Let x the} longitudinal ^coordi-

nate (the direction of increasing ^{x is} opposite ^{to the}

direction of propagation ^{of the} front), and y ^the

transversal coordinate. With these coordinates the

gravity ^{field can} be written as g ^{= -} gx, the case g > 0 corresponds ^{to the downward} propagation

of the flame (Fig. 1).

Both fresh and burnt mixtures are assumed to be

incompressible gases with negligible viscosity. ^The

flame is taken to be an infinitely thin interface between the fresh and burnt mixture which propagates ^with

constant relative burning velocity UL. A ^discussion concerning the influences of the flame thickness on

observed shapes and velocities is carried out later.

For the moment, ^the infinitely thin flame model

means that spatial variations of the flame shape ^are

of the scale of the tube radius which is an order of

magnitude larger than the flame thickness.

Fig. ^1.

Convex flame propagating ^{in a tube.}

2. .1 FLOW _EQUATIONS AND BOUNDARY CONDITIONS AT THE FLAME.

With the assumptions ^mentioned above, ^the velocity ^{field w, in a frame fixed to the flame}

with stationary shape ^{and constant} velocity, ^satisfies

the incompressible ^Euler equations.

The equation ^{of mass} conservation is :

The equation ^{of momentum} conservation is :

where S ⁼ _p + pgx is the effective _pressure of the burnt and fresh mixture.

The density ^{p is} equal ^to po in the fresh mixture and to pb in ^{the burnt} gas.

On the stationary ^front, given by the function

x ⁼ ç(y), ^{the four} following boundary ^conditions

must be satisfied :

1) ^{normal mass flow} conservation,

2) tangential velocity conservation,

3) conservation of normal momentum component,

4) the normal relative burning velocity is constant,

where n is the unit vector normal to the flame front, directed toward the fresh mixture, ^{t the} tangential

unit vector, and the subscript - ^f resp. (+ f) ^means

that the values of the velocity ^{field are taken on the} front, in the fresh side (in ^{the burnt} side, respectively).

3. Integral ^relations.

3.1 THE FLOW STRUCTURE.

Since there is no vorti-

city ^at infinity in the fresh mixture (the gas is at rest in the laboratory frame) ^{and as} vorticity ^{is conserved} along ^the streamlines, ^the upstream ^{flow is} potential.

The flow at infinity ^is parallel ^and uniform and equal

to U, ^the propagation velocity of the flame. On the front the density jump ^creates vorticity ^{which is}

carried along the downstream lines, which leads to a

rotational flow. So the downstream flow is rotational.

At infinity ⁱⁿ the burnt mixture the flow is parallel ^but

not uniform. Furthermore, ^as observed in experi-

ments [1], ^{the flame front is convex} towards the fresh

mixture and meets the wall of the tube at a finite angle.

This leads to a deflection of velocity ^at the front due to the jump ^of density towards the centre of the tube.

In this case a stagnation region, where the flow is at rest with respect ^{to the flame must occur} in the burnt gas (Fig. 2). Therefore the flow structure is extremely complex and it appears unlikely ^{that exact} analytical

solutions of flames propagating ^with stationary shape

and constant velocity ^can be found. Nevertheless we can obtain useful information about the flow field from exact integral ^relations involving ^the upstream velocity field and the flame shape.

3.2 INTEGRAL RELATIONS.

In the case of planar

geometry, ^{the mass} conservation between tube sec-

tions located at x ^{= -} oo and x ⁼ + oo is

where a (a R), is the radius of the transverse section of the tube filled with the moving fluid, ^at infinity ⁱⁿ

the burnt mixture. If one introduces the stream func-

tion 0 ^{defined as,}

Fig. ^2.

^{The flow structure.}

one obtains,

The momentum conservation integral ^is

or

We now write the quantities T - . - T , . ^and u + 00 (t/J) ^{in terms of the} upstream variables. From the Bernoulli law we have

Using ^the boundary conditions on the flame we obtain

But, in the stagnation ^{zone, filled} by the combustion products, the fluid is at rest with respect ^{to the flame} front and _so,

It follows that

In order to compute u+oo(t/!) ^{we use} the Bernoulli law along the streamline where the stream function takes the

value ql. ^So

506

Using ^the boundary conditions ^on the flame front and relation (14), ^{one can} finally ^{obtain :}

Using ^the continuity ^{of the stream function on the front, one can write the momentum integral as,}

Furthermore, using the relation (8), ^{one can} deduce the half-width, a :

or

4. Flame shape ^and propagation velocity.

The momentum integral relation and the one which determines the half-width of the section of moving

fluid at infinity in the burnt _gas, a, involve only ^the

values of the upstream velocity ^{field on the flame}

front. There is an explicit dependence ^{on the} upstream field through ^{the terms} U, u_ f(R) ^and w - f. There is an

implicit dependence ^{on the field} through ç(y). ^{As a}

matter of fact, ^{when the} upstream velocity ^{field is} known, ^the integration of the relation _{w _ f} n( ç(y), y) _

uL, with ç(O) ⁼ 0, gives ^{the flame} shape.

These integral ^{relations can be used to} evaluate the

approximate shape ^and corresponding propagation velocity ^{of a} curved flame. First take a potential velocity ^field satisfying ^the proper boundary ^conditions (uniform velocity U, ^at infinity, ^and vanishing ^trans-

verse velocity ^{at the} wall). ^{In order to be} considered a

« good » upstream flow field for a curved flame, ^this potential ^{field must also} satisfy four constraints :

1) ^The equation ^{w_ f} n(ç(y),y)=UL’ ^{with ç(O)=O,}

must have a solution.

2) ^{The flame} shape ç(y) ^{which is} computed ^must

meet the wall with an angle between 0 and n/2. ^{This is}

to prevent ^the burnt gas from being deflected towards the wall of the tube.

3) ^{The momentum} integral ^relation (18) ^{must be}

satisfied.

4) ^The half-width of moving ^{fluid at} infinity ^{in the}

burnt mixture, a, which is computed ^{from relation} (20),

must have a value between 0 and R.

The spirit of the method is the same as that used for the evaluation of the rising velocity ^{of a} gas bubble in the work of Davies-Taylor. They ^{take a} potential

flow with the _proper boundary conditions (uniform

velocity ^at infinity ahead of the bubble and vanishing

transverse velocity ^{at the} wall). The constraints on

this flow are that the stream function vanishes around the tip of the bubble and that the _pressure (which, ⁱⁿ

the exact solution, ^{would be constant on the bubble} surface) ^{is the same at two} well chosen points ^{on the}

bubble surface. It is evident that such a method cannot lead to the exact solutions of curved flame propagation

or even rising bubbles, but, ^{as shown in} [7], ^the approxi-

mate rising velocity of the bubble found in this _way agrees very well with experiments. ^We expect ^the

same success for the flame analysis.

The simplest ^{non trivial} upstream ^{w _ which satisfies} the previously ^mentioned boundary conditions is

with

and

ML-!7

Since, at the origin ^{u =} uL, A ⁼ k - ^{It follows}

that w _ is a velocity ^field depending ^{on one} parameter

e ⁼ (U - uL)/uL, and defined as

The equation w - f - n( ç(y), y) ^{= UL, with ç(O) = 0,}

is solved numerically. As the flow does not depend explicitly ^{on the} gravity intensity, g the solution of this equation çe(Y) ^{is the same as} that found in [4].

However, s, which is given by ^the integral ^relation (21)

is now a function of two parameters : ^the expansion

ratio a, and Froude number, ^{F. A} numerical compu- tation of B(F), ^{with a = 2 is} given ⁱⁿ figure ^{3. The} corresponding half-width, a, for this case is shown in

figure ^{4. This} particular ^{choice of a} has been taken because it corresponds ^{to the case of flame} propagation experiments. ^We first notice that c(F) ^is globally ^a decreasing function of F (F is considered an algebraic number). ^In fact relation (21) has another branch of solutions besides the one shown, ^{in the} negative

Froude number domain, ^but this solution corresponds

to a half-width a larger ^{than R. Hence, this} branch does not satisfy ^the physical constraints and is rejected.

This fact will be emphasized ^further. The fact that s decreases when the Froude number increases can be

explained ^{in the} following way. A curved flame is a

kind of equilibrium ^between competitive stabilizing

and destabilizing influences. The latter, which is the Darrieus-Landau instability (and ^the Rayleigh-Taylor instability ^{in the case of} upward propagating flames),

tend to curve the flame. The former resulting ^{from the} stretching ^{of wave} lengths ^mentioned by ^Zeldovich

et al. [4] and from the Rayleigh-Taylor stability (in ^the

case of downward propagating ^{flames) tend to flatten}

the flame. When the Froude number is positive ^the stabilizing ^{effect are more} important ^{and so the flame} amplitude is smaller. On the other hand, negative ^a

Froude number leads to an increase of the flame ampli-

tude and consequently ^{the flame} velocity.

There is a critical positive ^{Froude number,} Fc,

above which the only solution is the flat flame. In other words, ^at least with this simple model, ^{there is}

a maximum size Rc ⁼ u2 .71 9 L ^{Fc of a} curved flame _propa- g

gating downward in a tube as soon as uL and g ^are given. So, when the flame propagates ^{in a tube of} size larger ^than R, ^it may break down into a cellular

flame, each cell being one convex flame described

above, ^{with a} size smaller than the critical ^one.

For Froude number smaller than F,, ^{there is a}

domain where 8 is a double valued function. This situation is not general; ^for larger ^{values of} a, ^s becomes single ^valued.

There is another, ^somewhat larger, negative ^Froude number (F - 40), above which the only solution is also the flat flame. The end of the branch corresponds

to the fact that for the considered velocity field, w-E, with s above Bmax ç(y) exists but relation (20) ^{cannot be}

satisfied for _any value of the Froude number. This

probably corresponds ^to the fact that for a sufficiently large negative ^Froude number, ^the amplitude ^{of the}

flame is relatively large ^{and so the} upstream flow of the flame cannot be described by ^the simple ^unimodal

flow _{w _ .} So by using ^this simple ^{flow we cannot}

Fig. ^3.

^Flame velocity ^versus Froude number for an

expansion ^ratio equal ^to 0.2. Planar geometry.

Fig. ^4.

Half-width a versus Froude number for an expan- sion ratio equal ^to 0.2. Planar geometry.

describe the probably continuous transition between the upward propagating ^{flame at} Froude number of order unity ^{and the} rising ^{bubble of} gas which corres-

ponds ^{to infinite} negative ^{Froude number.}

An analytical solution of the flame shape ^{and its} corresponding velocity ^{can be} found when the ampli-

tude of the flame is _very weak. When s is small the

integration ^of equation (6) ^{leads to :}

Limiting the solutions to flames of convex shape ^we obtain :

An expansion ^{in c of the momentum} integral ^rela-

tion (19) and of the half-width relation (20) ^{leads res-}

508

pectively ^{to the} relations :

and

Equation (28) ^{has non-zero} solution of small amplitude

if F is near + F * ^{c where} F * ^{c =} 1 . So ^{2 a relation (27)}

leads to two branches of solution beginning ^{for s small}

and positive ^{and F near ±} F *. ^{As one can see from (28)}

the branch corresponding ^{to the} negative ^{values of F}

is not admissible because it leads ^{to a} half-width larger

than R. Therefore the only branch retained is :

and

The corresponding radius, a, ^{is :}

aC corresponds ^to the limit between single ^{and double}

valued solutions.

5. Effect of tube geometry.

It is known from experiments that the flame shape

and velocity depend ^{on the tube} geometry. If the tube is a cylinder ^{of radius R one can} apply ^{the same method}

as in the planar ^case. Only the formulation of the

integral ^relations changes. ^{The mass} conservation is

and

where § ^{is the stream} function which is defined by

The momentum integral ^is

Using ^the expression of-t - OOT-t +OO ^{and u+} 00 ( t/J) given ^{above, one obtains}

and

The upstream potential ^{flow is now}

with and

where Ai is the first zero of the Bessel function J1.

The value of s(F) ^is plotted ^on figure 5; ^{the corres-} ponding half-radius a on figure ^{6. As} before, ^the expansion ^{ratio is} equal ^{to 0.2. The} qualitative ^beha-

viour of g as a function of the Froude number is the

same as in the planar ^case; quantitatively, ^{s is} nearly

twice as large.

6. Effect of flame thickness.

The study ^of stationary ^curved flames, neglecting

thermodiffusive effects (diffusion of heat and mass occur on a scale of the order of the flame thickness

only) ^{seems to be} justified by experimental ^observa-

tions. It is an experimental observation that stationary

convex flames can propagate in tubes which are larger by ^{an order of} magnitude than the flame thickness.

Fig. ^5.

^Flame velocity ^versus Froude number for an

expansion ^ratio equal ^{to 0.2.} Cylindrical geometry.

Fig. ^6.

Half-width a versus Froude number for an expan- sion ratio equal ^{to 0.2.} Cylindrical geometry.

For instance, Von Lavante and Strehlow mention the existence of stationary ^curved CH4-Air ^flames propa-

gating ^{in a 100 mm x 100 mm tube,} when the per- centage of methane is between 5.34 % ^{and 7} % (by mass). ^{In this case, the} burning velocity ^{is about}

10 cm/s. ^The corresponding propagation velocity ^is

about 30 cm/s. ^{In order to} evaluate the relative influence of thermo-diffusive effects one can use the Clavin-Joulin [8] formula. This formula gives ^{the first}

correction of the burning velocity ^{in an} expansion ^for

small Peclet number d/3l, ^{where 3l} is the radius of curvature of the flame and d is the flame thickness.

One can write this formula

where 6UL’S the variation of the burning velocity ^and UL is the burning velocity ^{of a} plane ^{flame. C} is the Mark- stein length ^and u_ f is upstream velocity ^{field on the}

front. A is on the order of the tube radius, ^{n -} v - n

UL is of order U/R and C is of order d. It follows that the

relative ^{influence of the} thermodiffusive effect can be evaluated by :

This evaluation shows that neglecting ^thermodif-

fusive effects in order to evaluate the approximate

flame shape ^{and the} corresponding propagation ^velo- city may be justified. ^{However, when the} burning velocity ^becomes sufficiently small, ^{in the case of} very dilute mixtures for instance, the flame thickness becomes larger ^{and so} the thermodiffusive effects are more important. ^This explains, ^{as shown} by ^{Pelce and}

Clavin [9], that the flat flame propagating ^downwards

is stable at low velocity.

The condition of small burning velocity ^{is not} enough ^{to ensure} large thermodiffusive effects. As a

matter of fact, ^{one sees from} [37] ^{that if} U/uL ^{is rela-} tively large, thermodiffusion can be important ^for

any value of the Peclet number. This can be the case

for upward ^flame propagation ^at large Froude number

where, ^as suggested by ^{Lavante and Strehlow} [3],

U - ffl ^{and so U/UL -} IIF ’> ^1. This could be

partial explanation ^{of the} anomalously large ^effect

of stretch on upward propagating ^{methane flames,}

near the flammability ^{limit, as observed} by ^these

authors.

In the extreme case where the flame propagates

near the extinction limit, the thermodiffusive effects

can even dominate. As a matter of fact, ^as suggested by Joulin and Linan [10] ^and by Peters and Smooke [ 11 ], ^the competition ^between branching ^chain

reactions and recombination reactions of radicals which can occur near the extinction limit lead to

vanishing burning ^flame velocity. ^This effect, leading ^to

510

a very large ^flame thickness, ^{could be another} explana-

tion of the anomalously large influence of stretch on

upward propagating flames mentioned above. In all these _cases, the theory developed ^{above is not}

valid and thermodiffusive effects must be taken into account.

7. Conclusion.

The simplified theory presented in this work leads to good qualitative results relative to the effects of

gravity ^on vertical flame propagation ^{in the case of} moderately large ^Froude number. Nevertheless some

progress ^must be made for the description ^{of the}

transition between the upward propagating ^flame

and the rising bubble of burnt _gas which seems to occur experimentally ^at large ^and negative ^Froude

numbers. This improvement ^{must be made} by ^consi- dering upstream flows which are more realistic than the unimodal flow used in this study.

Acknowledgments.

The author is greatly ^{indebted to P.} Clavin and A.

Linan for fruitful discussions.

This work was supported by ^{the CNRS and the} AFME (AIP 5007).

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