Numerical study of thermodiffusive flame structures interacting with adiabatic walls using an adaptive multiresolution scheme Olivier Roussel ?† and Kai Schneider †‡§ ? Institut f¨ ur Technische Chemie und Polymerchemie, Universit¨ at Karlsruhe (TH), Kaiserstr. 12, 76128 Karlsruhe, Germany. † Laboratoire de Mod´elisation et Simulation Num´erique en M´ecanique, CNRS et Universit´es d’Aix-Marseille, 38 rue Fr´ed´eric Joliot-Curie, 13451 Marseille cedex 20, France. ‡ Centre de Math´ematiques et d’Informatique, Universit´e d’Aix-Marseille I, 39 rue Fr´ed´eric Joliot-Curie, 13453 Marseille cedex 13, France. Abstract. We report on numerical simulations of planar, circular, and spherical thermodiffusive flames in one, two and three space dimensions, respectively. Attention is focused on the interaction of spherical flames with adiabatic walls or equivalently on their interaction with their mirror image. The numerical scheme is based on an adaptive multiresolution discretization which allows self-adaptive grid refinement in regions of the thin reaction zone. We show that the Lewis number determines the behaviour of the flame-wall interaction. When the flame is approaching the wall, we observe for Lewis numbers smaller than unity that the reaction rate is decreased, for unitary Lewis number the reaction rate neither increases nor decreases, and for Lewis numbers larger than unity the reaction rate increases. Due to tangential diffusion the flame front curvature is also modified, i.e. for small Lewis numbers the spherical flame contracts, for large Lewis numbers it spreads out, while for a unitary Lewis number the flame front remains perpendicular to the wall. The observed phenomena present similarities with capillarity effects in fluid mechanics when a droplet hits a wall. Keywords. flame ball, adaptivity, multiresolution, wavelet, finite volume

PACS numbers: 47.70.Fw, 82.40.Ck, 02.70.Fj

Submitted to: Combust. Theory Modelling April 5th, 2004, revised August 30th, 2005, accepted September 20th, 2005.

§ To whom correspondence should be addressed ([email protected])

Thermodiffusive flame structures interacting with adiabatic walls

2

1. Introduction The numerical, theoretical and experimental investigations of spherical premixed flame structures have drawn some attention on the combustion community. Already 60 years ago Yakov Zeld’ovich [28] theoretically predicted spherical flames which have then been discovered accidentally in drop tower experiments by Paul Ronney [19]. Since then they have been studied extensively in drop-towers, aircraft experiments and in the space shuttle [20]. A microgravity environment plays hereby a crucial role to obtain spherical symmetry and also to avoid buoyancy-induced extinction of the flames. Typically, flame balls are generated by point ignition of a premixed gas which leads to the generation of multiple balls which then may interact or drift due to some g-jitter effects. In experiments it was also observed that walls tend to repel a flame ball [2]. One dimensional simulations of spherical flames imposing spherical symmetry have been presented in [27], including a study on the effects of the far-field boundary radius on flame ball properties. Direct numerical simulations of stable 3D flame balls have been studied in [6], but far from the walls. Furthermore, propagating flames interacting with walls have been widely studied (see e.g. [26, 17]). The objective of the paper is to assess the interaction of spherical flame structures with adiabatic walls and to investigate the role of the Lewis number, i.e. the ratio of the thermal diffusivity of the gas mixture to the mass diffusivity of the chemical reactant. The interaction of spherical flames with adiabatic walls can also be considered as the interaction of a flame with its mirror image. The adiabatic wall approximation can be justified for combustion chambers made with ceramics, foam, or any strong isolating material. Typical applications are encountered in combustion engines and furnaces. The goal is to assess the amount of remaining unburnt gas close to the wall, which reduces the efficiency and leads to pollutant diffusion [17]. Adaptive numerical methods are particularly attractive for simulating reactive flows due to the large number of spatial and temporal scales involved. Different approaches have been used so far, see e.g. [13, 5, 17, 8]. In [22, 23, 24] we developed an adaptive multiresolution method which automatically refines the grid in regions of strong gradients, e.g. in the thin chemical reaction zone. The principle of this method is to represent a set of data given on a fine grid as values on a coarse grid plus a series of prediction errors at different levels of nested dyadic grids. These prediction errors, the so-called details, contain the information of the solution when going from a coarse to a finer grid. In particular, they are small in regions where the solution is smooth. Hence they can be removed from memory to compress the data, which therefore allows to reduce CPU time. The original idea was introduced one decade ago by Harten [10, 11, 12] for 1D hyperbolic conservation laws, to trigger locally the flux computation. Later the multiresolution concept has been used to perform local grid adaption for 2D hyperbolic problems [14]. The fully adaptive concept has also been analytically investigated [4], and its algorithm has been extended to three dimensions [22, 24]. The paper is organised as follows: The physical problem together with the

3

Thermodiffusive flame structures interacting with adiabatic walls

governing equations are presented in section 2. The numerical scheme using an adaptive multiresolution discretization is provided in section 3. Section 4 presents several numerical simulations of flame-ball wall interactions in one, two and three dimensions for different Lewis numbers. Finally, some conclusions are drawn and perspectives for further work are given. 2. Physical problem and governing equations A flame ball is a stationary or slowly propagating spherical flame structure in a premixed gaseous mixture. Such flames have been experimentally observed for low Lewis numbers under micro-gravity conditions, to avoid that buoyant convection destroys the structure [15, 16, 19, 20]. The study of such configurations for lean mixtures is of great interest, because it allows the determination of flammability limits in an “absolute” way, i.e. in function of the reaction mechanism only, independently of the setup [25]. In Figure 1, a flame ball is represented schematically together with the adiabatic wall and the physical mechanisms involved. The chemical reaction occurs in the circular zone delimiting the ball. Inside this ball, the gas is burnt. The rest of the domain is filled by fresh premixed gas. The pre-heat zone is not represented in this figure.

COLD PREMIXED GAS

radiation

heat conduction ADIABATIC

HOT BURNT GAS reactant diffusion

WALL

Figure 1. Schematic view of the flame ball-wall interaction and the involved mechanisms.

The thermodiffusive approximation is well adapted for the computation of flame balls, because the flame velocity is very small [1]. Considering Stefan-Boltzmann blackbody radiation model, constant density and one-step chemical kinetics approximations, the system of equations modelling such a flame structure is [1] ∂T = ∇2 T + ω + s ∂t 1 2 ∂Y = ∇ Y −ω (1) ∂t Le

Thermodiffusive flame structures interacting with adiabatic walls   Ze2 Ze(1 − T ) ω(T, Y ) = Y exp − 2 Le 1 − α(1 − T ) h
4
4 i s(T ) = γ T + α−1 − 1 − α−1 − 1

4

with apropriate initial and boundary conditions, further expressed in section 4. In this system, T denotes the dimensionless temperature, Y the partial mass of the limiting reactant, ω the dimensionless reaction rate, s the dimensionless heat loss due to radiation, Le the Lewis number, Ze the dimensionless activation energy, so-called Zeldovich number, α the burnt-unburnt temperature ratio, and γ the dimensionless radiation coefficient. We also introduce the global reaction rate Z R(t) = ω(x, t) dx (2) Ω

and we define

¯ = R(t) R(t) R(0)

(3)

where Ω denotes the computational domain, which corresponds here to the combustion chamber. In 1D, since the flame velocity is Z vf (t) = ω(x, t) dx Ω

¯ we have vf (t) = vf (0) R(t).

3. Adaptive multiresolution scheme In the following we summarize the principle of the adaptive multiresolution scheme used in the subsequent computations. For more details on the numerical method and on its implementation using dynamical data structures, we refer to [24]. The basic idea of this approach is to accelerate a given finite volume scheme on a uniform grid without loosing accuracy. Applying a multiresolution transform followed by a thesholding of small coefficients a locally refined adaptive grid is defined. The threshold is chosen in such a way to guarantee that the discretization error of the reference scheme is balanced with the accumulated thresholding error which is introduced in each time step. This allows to reduce memory and CPU requirements without loosing the precision of the computations. The modelling equations (1) are a system of diffusion-reaction equations which can be written in the form ∂t U = Λ∇2 U + S(U )

(4)

with initial condition U (x, t = 0) = U o (x), where U = (T, Y )t ∈ R2 , Λ is a 2×2 diagonal matrix, x ∈ Ω, Ω ⊂ Rd , d being the space dimension, and S(U ) is a non-linear function of U .

5

Thermodiffusive flame structures interacting with adiabatic walls 3.1. Multiresolution analysis and its data representation

The principle of the multiresolution analysis is to represent the data on a set of nested dyadic grids. The data given on a fine grid is decomposed into values on a coarser grid plus a series of differences at different levels of dyadic grids. These differences contain the information of the solution when going from a coarse to a finer grid. In particular, these coefficients are small in regions where the solution is smooth [9, 11] and yield hence high compression rates for functions with inhomogeneous regularity.

l l l l Ω

0, 0

= = = =

... 2 1 0

= Ω

Figure 2. Graded tree data structure in 1D

The tree structure is composed of a root cell, which is the basis of the tree, the nodes which are elements of the tree, and the leaves which are the upper elements. In d dimensions, a parent cell at a level l has always 2d children cells at the level l + 1. In Figure 2, a tree structure in 1D is represented. For the incoming and outgoing flux computations, a leaf at the level l has sometimes no neighbour at the same level and needs to get information from a leaf at the level l − 1. Therefore, virtual leaves are created. They only exist for the flux computation and no time evolution is made on them. In order to be graded, the tree must verify that each leaf at a level l has always adjacent cells of level at least equal to l−1 in each direction, the diagonal being included. Here we use a finite volume discretization. Hence each node of the tree contains the cell-average value of u. To compute the average value of a cell at level l from the ones of cells at level l + 1, we use the projection (or restriction) operator Pl+1→l . It is exact and unique, given that the cell-average value of a parent cell is the weighted average value of its children cell-averages. The prediction (or prolongation) operator Pl→l+1 maps U¯l to an approximation Uˆl+1 of U¯l+1 . In contrast with the projection operator, there is an infinite number of choices for the definition of Pl→l+1 . Nevertheless, in order to be applicable in a graded tree structure, it needs to be local, i.e. based on an interpolation using the s nearest neighbours in each direction, and consistent with the projection, i.e. Pl+1→l ◦Pl→l+1 = Id. The detail is the difference between the exact and predicted values. The vector ¯ ¯ l = U¯l − Uˆl . Thanks to the consistency Dl of the details at level l therefore verifies D assumption, the sum of the details on all the children of a parent cell is equal to zero [11]. Therefore, in d dimensions, the knowledge of the 2d children cell-averages of a given parent cell is equivalent to the knowledge of the parent cell-average and 2d − 1 details, ¯l−1 , D ¯ l ). Repeating the operation on L levels, one gets the so-called i.e. U¯l ←→ (U

Thermodiffusive flame structures interacting with adiabatic walls

6

multiresolution transform [11] ¯L 7−→ (U ¯0 , D ¯ 1, . . . , D ¯ L ). ¯ :U M

(5)

The threshold operator T() consists in removing leaves where details are smaller than a prescribed tolerance , without violating the graded tree data structure. In order to account for translation of the solution and the possible generation of finer scales the index set of the wavelet coefficients is expanded by adding neighbour wavelets, which corresponds to the so-called security zone. This procedure allows to track the evolution of the flow in scale and space. Since the time integration is fully explicit, only one neighbour in each direction is added. The tolerance is chosen such that the discretization error of the reference numerical scheme is balanced with the accumulated threshold error[3]. 3.2. Time evolution At each time step tn = n∆t, a time evolution is performed on the leaves only. Time integration is performed with a third-order explicit Runge-Kutta scheme. For the space discretization, we use a second-order accurate centered scheme in each direction [24]. For the computation of the source term, we approximate the cell-average value of U with the value of U at the center of the cell, which also yields a second-order accuracy. 3.3. Adaption strategy - Algorithm When a leaf at level l has no neighbour of the same level in a given direction, the flux at the interface is computed using the cell-average value of the adjacent virtual leaf, which is computed by projection from its parent at the level l − 1 (Figure 2). In order to maintain a strict conservativity in flux computations, the ingoing flux for this parent cell (at the level l − 1) is taken as the sum of the fluxes going out of the adjacent leaves at the level l. In the following, we briefly summarize the algorithm. For more details we refer to [24, 21]. First, depending on the initial condition, an initial graded tree is created. Then, given the graded tree structure, a time evolution is made on the leaves. Finally, details are computed by multiresolution transform, in order to remesh the tree. ¯ Denoting by E(∆t) the discrete time evolution operator, the global algorithm can schematically be summarized by ¯ n+1 = M ¯ −1 · T() · M ¯ · E(∆t) ¯ ¯n U ·U

(6)

¯ is the multiresolution transform operator, and T() is the threshold operator where M with tolerance .

Thermodiffusive flame structures interacting with adiabatic walls

7

4. Numerical results 4.1. Flame front-wall interaction in 1D : head-on quenching configuration In this part, we consider a planar flame interacting with an adiabatic wall. The goal is to assess the influence of the Lewis number in the flame-wall interaction. For a general description of the flame-wall interaction in a head-on quenching configuration, we refer to [17]. The physical problem is now one-dimensional. The computational domain is Ω = [x0 , xM ] = [0, 30]. At x = xM , we impose a Neuman boundary condition. The boundary at x = x0 has no influence on the computation. For reasons of simplicity, we decided to set the same condition on this border. Thus we have ∂T ∂T (x0 , t) = (xM , t) = 0 (7) ∂x ∂x ∂Y ∂Y (x0 , t) = (xM , t) = 0 (8) ∂x ∂x We choose the following initial condition ( 1 if x ≤ xc ; (9) T (x, 0) = exp(xc − x) if x > xc . ( 0 if x ≤ xc ; Y (x, 0) = (10) 1 − exp[Le(xc − x)] if x > xc . The position of the flame front at t = 0 is xc = 22, so that no long computation is required before the flame reaches the wall, i.e around 8 time units since vf ≈ 1. The Zeldovich number and the temperature ratio are set to Ze = 10 and α = 0.64. Heat losses due to radiation are neglegted here. The time evolution of the temperature and the reaction rate are plotted in Figure 3 for three different values of the Lewis number: 0.3, 1, and 1.4. The case Le = 0.3 corresponds to a 6.5% H2 -air mixture, the case Le = 1 to a 5% CH4 -air mixture, and the case Le = 1.4 to a 2.65% C2 H6 -air mixture [20]. For Le = 0.3, the intermediate state corresponds to tm = 4 and the final one to tf = 20. For Le = 1, we have tm = 4 and tf = 7. Finally, for Le = 1.4, we have set tm = 5 and tf = 6. In these plots, we observe the influence of the Lewis number on the flame when interacting with an adiabatic wall. For Le = 0.3, the reaction rate is gradually decreased, and the flame almost extinguishes at the contact with the wall. For Le = 1, ω is almost constant, slightly increased. For Le = 1.4, ω significantly increases. ¯ for different values of the Lewis number. In Figure 4, we plot the time evolution of R ¯ is proportional to the flame velocity. The limit between the two We recall that, in 1D, R behaviors at the boundary - reaction rate increasing or decreasing - is close to Le = 1. The phenomenon can be explained this way: for Le < 1, the speed of diffusion of the species is larger than the one of the heat. Consequently, when the flame front is close to the wall, since less and less reactant is available, the reaction rate is decreased. However,

8

Thermodiffusive flame structures interacting with adiabatic walls 3

3 T ω

2

3 T ω

2

1

1

1

0

0

0

0

10

20

30

0

10

x

20

30

0

2

1

0

0

0

30

0

10

x

20

30

0

T ω

T ω

2

1

0

0

0

x

30

0

10

20 x

T ω

2

1

20

30

3

1

10

20 x

3

0

10

x

3 2

T ω

2

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20

30

3 T ω

1

10

20 x

3 T ω

0

10

x

3 2

T ω

2

30

0

10

20

30

x

Figure 3. Interaction of a flame front with an adiabatic wall for Le = 0.3 (top), Le = 1 (middle), and Le = 1.4 (bottom). Profiles of T (plain) and ω (dashed) at t = 0 (left), t = tm (center), and t = tf (right).

the drop is gradual, because only little heat diffuses from the reaction zone. Moreover, since the maximum of the reaction rate is decreased, the flame velocity also diminishes. Therefore the decrease lasts a long time (Figure 4, left side). On the other side, when Le > 1, the heat release involves an accelerated ignition of the reactant at the interphase between preheat and reaction zone [18]. Here we observe a strong increase in the reaction rate and, therefore, also an increasing flame velocity. However, as soon as the flame hits the wall, all the reactant is burnt, and the flame instantaneously extinguishes (Figure 4, right side). To summarize, the Lewis number determines the behavior of a planar flame encountering an adiabatic wall. When Le < 1, the reaction rate is decreased, and the flame gradually extinguishes. When Le > 1, the reaction rate is increased, the

9

Thermodiffusive flame structures interacting with adiabatic walls 2

8 7

Le = 0.3 Le = 0.7 Le = 0.8 Le = 0.9

1.5

Le = 0.95 Le = 1 Le = 1.4

6

1

¯ R

¯ R

5

0.5 0

4 3 2 1

0

2

4

6 t

8

10

12

0

0

2

4

6 t

8

10

12

Figure 4. Interaction of a flame front with an adiabatic wall for Le < 0.95 (left) and ¯ for different values of the Lewis number. Le ≥ 0.95 (right). Time evolution of R

flame accelerates, then quickly extinguishes when all the reactant is burnt. 4.2. Flame ball-wall interaction in 2D In this part, we investigate the curvature effects in the interaction flame-adiabatic wall. Therefore we choose a flame front being initially circular inside a closed box with adiabatic walls. The modelling equations are the same as in the previous subsection, i.e. (1). Since the walls are adiabatic, the boundary conditions are given by ∂Y ∂T = = 0. (11) ∂n ∂n ∂Ω

∂Ω

For the initial condition, as in 1D, we apply a translation in the x-direction, so that the flame ball almost touches the wall at t = 0. Denoting by xc the initial x-coordinate of the center of the flame ball, T and Y at t = 0 are given by ( 1   if r ≤ r0 ; T (r, 0) = r exp 1 − r0 if r > r0 . Y (r, 0) =

(

0





1 − exp Le 1 −

r r0



if

r ≤ r0 ;

if

r > r0 .

p where r = (x − xc )2 + y 2 , r0 denotes the initial radius of the flame ball, and xc = 25. In the following computations, the domain is set to Ω = [−30, 30]2 , the initial radius is r0 = 2, and the final time is tf = 10. In Figures 5-7, the time evolution of a flame ball encountering an adiabatic wall is plotted for three different values of the Lewis number. First, we observe an effect similar to the one described in the 1D case, i.e. for Le < 1, the reaction rate is gradually decreased when the flame reaches the wall, whereas it is almost constant for Le = 1

Thermodiffusive flame structures interacting with adiabatic walls

10

Figure 5. Top: Isolines of temperature from T = 0.1 to T = 0.9 every 0.1 at t = 0 (left), t = 5 (center), and t = 10 (right) for the interaction of a flame ball with an adiabatic wall, Le = 0.3. Middle: Corresponding reaction rate ω with the same scale. Bottom: corresponding meshes.

and increased for Le > 1. This is confirmed by the cuts at y = 0 (Figure 8). Then, we also observe a modification of the front curvature, due to tangential diffusion, when the flame front reaches the wall. This phenomenon presents similarities with capillarity effects in fluid mechanics. When a fluid droplet encounters a wall, the angle between the surface of the droplet and the wall at the intersection can be smaller or bigger than π2 depending on the interfacial tension between both materials. For more details on this phenomenon, we refer e.g. to [7]. Here, we may interpret that the Lewis number characterizes this kind of capillarity. The flame ball contracts when Le = 0.3, whereas it spreads out when Le = 1.4. For Le = 1, the angle between the flame ball and the wall is close to π2 . One can explain the phenomenon this way. On one side, when Le < 1, the ingoing reactant flux is strong everywhere except in the region where the reactant is deficient.

Thermodiffusive flame structures interacting with adiabatic walls

11

Figure 6. Top: Isolines of temperature from T = 0.1 to T = 0.9 every 0.1 at t = 0 (left), t = 5 (center), and t = 10 (right) for the interaction of a flame ball with an adiabatic wall, Le = 1. Bottom Corresponding reaction rate ω with the same scale.

Therefore, close to the wall, the reaction rate is decreased and the flame front slows down whereas, everywhere else, the reaction rate remains high. This phenomenon has already been observed in the 1D case. When the flame has extinguished on the wall, the diffusion regulates the temperature, and thus the temperature is lower on the wall than inside the flame ball. On the other side, when Le > 1, the ignition is accelerated in the region close to the wall. When the flame has touched the wall, the acceleration is visible at the intersections between the wall, the fresh gas mixture and the burnt gas mixture. The reaction rate in these regions is stronger than in the rest of the flame front, which explains that the flame ball spreads out. Below the isolines of temperature and reaction rate, the corresponding meshes are given. Here the chosen tolerance is  = 10−2 , which is sufficient for L = 8 scales. We observe that the mesh follows well the flame front propagation. The CPU time and memory compressions, with respect to the refence finite volume computation on the regular finest grid, are given in Table 1. In Figure 8, the cuts at y = 0 confirm the behavior of a flame encountering an adiabatic wall in the 1D case. For Le = 0.3, the reaction rate is decreased when the flame front is close to the wall, whereas it is almost stable for Le = 1 and increased for Le = 1.4. ¯ is plotted for three different values In Figure 9 (left), the global reaction rate rate R of the Lewis number. At t ≈ 3, the flame front reaches the wall located at x = 30. This

Thermodiffusive flame structures interacting with adiabatic walls

12

Figure 7. Top: Isolines of temperature from T = 0.1 to T = 0.9 every 0.1 at t = 0 (left), t = 5 (center), and t = 10 (right) for the interaction of a flame ball with an adiabatic wall, Le = 1.4. Bottom: Corresponding reaction rate ω with the same scale.

Le 0.3 1.0 1.4

d L  −2 2 8 10 2 8 10−2 2 8 10−2

% Mem % CPU 14.1% 25.5% 11.1% 21.5% 11.8% 21.0%

Table 1. CPU and memory compression for the flame ball-wall interaction in 2D. The percentage is given compared to the reference finite volume computation on the regular finest grid.

contact is characterized by a small peak on the curve of R, but only for Le ≥ 1. For Le = 0.3, as already observed in 1D, the reaction rate is gradually decreased when the flame front reaches the wall. Consequently, no peak is observed. Then, the flame front simultaneously touches the upper and lower walls, respectively located at y = 30 and y = −30. For Le = 0.3, this happens at t ≈ 21. Even if no peak is observed, R is strongly decreased thereafter, because the flame ball can no more expand. Moreover, the flame front reaches these same walls only at t ≈ 29 for Le = 1, and t ≈ 30 for Le = 1.4. This shows that the flame velocity is larger for Le = 0.3 than for Le = 1, and larger for Le = 1 than for Le = 1.4. This dependency of vf on Le has already been observed in the previous subsection. The opposite wall, located at x = −30, is reached at t ≈ 42 for Le = 0.3, t ≈ 55 for Le = 1, and t ≈ 61 for Le = 1.4. Finally, when all the reactant is burnt, the chemical reaction stops.

13

Thermodiffusive flame structures interacting with adiabatic walls 3

3 T ω

2

3 T ω

2

1

1

1

0 -30 -15

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0 x

15 30

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2

1 15 30

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1 0 x

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15

30

Figure 8. Profiles of T (plain) and ω (dashed) for y = 0 at t = 0 (left), t = 2.5 (center), and t = 5 (right) for the interaction of a flame ball with an adiabatic wall, Le = 0.3 (top), Le = 1 (middle), Le = 1.4 (bottom).

When the flame front touches one of the walls, we observe a strong peak for Le = 1.4, a moderate peak for Le = 1 and no peak for Le = 0.3. Since the flame velocity is equal to R in 1D, we can conclude that the qualitative behavior of the 2D flame ball touching an adiabatic wall is similar to what we observed in the 1D case. Let us remark that, in the case Le = 0.3, the flame ball perturbated by the interaction with the wall does not split. This is because the flame front reaches the upper and lower walls before the splitting. 4.3. Flame ball-wall interaction in 3D In this part, we extend the previous results to three space dimensions, in order to investigate the 3D effects of the interaction. As for the 2D case, the system of modelling equations is (1) and the boundary conditions are given by (11). They correspond to a closed box with adiabatic walls. The spherical initial condition is stretched in one

Thermodiffusive flame structures interacting with adiabatic walls 10

70

Le = 0.3 Le = 1 Le = 1.4

8

60 50 ¯ R

¯ R

6

14

Le = 0.3 Le = 1 Le = 1.4

40 30

4

20 2 0

10 0

10 20 30 40 50 60 70 t

0

0 10 20 30 40 50 60 70 t

¯ in Figure 9. Interaction of a flame ball with an adiabatic wall. Time evolution of R 2D (left) and 3D (right) for different values of Le.

direction and a rotation on two axes is applied. Hence it writes ( 1   if r ≤ r0 ; T (r, 0) = r if r > r0 . exp 1 − r0 ( 0    if r ≤ r0 ; Y (r, 0) = 1 − exp Le 1 − rr0 if r > r0 .

where

r= with

r

X2 Y 2 Z2 + + a b c

   X = (x − xc ) cos θ − y sin θ Y = ((x − xc ) sin θ + y cos θ) cos ϕ − z sin ϕ   Z = ((x − x ) sin θ + y cos θ) sin ϕ + z cos ϕ c

The computational domain is Ω = [−30, 30]3 , the parameters of the ellipsoid are a = b = 1, c = 1.25, the rotation angles are θ = π3 and ϕ = π4 , the initial radius of the flame ball is r0 = 4, and the initial x-coordinate of the center of the flame ball is xc = 18, so that the flame ball almost touches the wall at t = 0. We perform numerical simulations from t = 0 to t = 15 with L = 7 scales for Le ≥ 1, 8 scales for Le < 1, and a tolerance set to  = 5 · 10−2 . The case Le < 1 corresponds to an unstable case, where the perturbations - due either to the curvature, or to a too coarse discretization - are amplified. To avoid the latter case, we have decided to perform computation with one more level when Le < 1. The numerical solution for the three different values of Le is plotted in Figure 10. We can observe the same capillarity effect as in the 2D case. However, being given the configuration, this effect is more difficult to visualize. The corresponding CPU time

Thermodiffusive flame structures interacting with adiabatic walls

15

Figure 10. Isosurfaces T = 0.5 (black) and T = 0.1 (gray) at t = 0 (left), t = 7.5 (center), and t = 15 (right) for the interaction of a flame ball with an adiabatic wall, Le = 0.3 (top), Le = 1 (middle), Le = 1.4 (bottom).

and memory compressions are given in Table 2. We observe a better compression rate for the case Le = 0.3, but this is only due to the fact that we used 8 scales, given that the compression rate usually increases with the number of scales [24]. ¯ is different than in the 2D The time evolution of the global reaction rate rate R case, since no peak is visible when the flame front reaches the wall for Le = 1 and Le = 1.4 (Figure 9, right side). This is due to the fact the the flame extends in the two other directions. Thus, even if the chemical reaction stops close to the wall, the flame ball goes on growing in the rest of thedomain. We also remark that the peak was already much sharper in 1D than in 2D, given that, in the 1D case, the reaction stops as soon as the planar flame front reaches the wall. In the 2D case, the integral of the reaction rate is significantly modified by the interaction with the wall. In the 3D case, the extinction of the part of the flame front reaching the wall has little influence on the

Thermodiffusive flame structures interacting with adiabatic walls

16

global reaction rate. Let us finally remark that we focused on the early times. When t → ∞, the flame ball extinguishes as soon as all the fresh mixture is burnt. Le d 0.3 3 1.0 3 1.4 3

L 8 7 7

 5 · 10 5 · 10−2 5 · 10−2 −2

% Mem % CPU 4.5% 9.5% 11.3% 28.6% 10.3% 26.1%

Table 2. CPU and memory compression for the flame ball-wall interaction in 3D. The percentage is given compared to the reference finite volume computation on the regular finest grid.

Now we study the influence of the radiation on the flame ball-wall interaction. Following the experiments [19], the radiative heat loss can be modified by adding chemical products like e.g. CF3 Br to the mixture. Such products do not modify the main chemical reaction, but increase the radiative heat loss due to soot production. In this computation, the dimensionless radiation coefficient is set to γ = 0.05 (cf. [6]). The other control parameters are the same as in the case Le = 0.3. In Figure 11, we observe in the beginning of the simulation the splitting of the initially stretched flame ball into two parts (t = 10). From t = 10 to t = 20 both parts are growing and are drifting away from each other, while approaching the wall. At t = 40 we observe another splitting into smaller structures which are finally (at t = 45) interacting with the wall. 5. Conclusion We presented several numerical simulations of spherical flame structures with adiabatic walls in one, two and three space dimensions. The thermo-diffusive model was used as the flame velocity is small. The numerical scheme is based on an adaptive multiresoltution discretization which allows efficient large scale computations in three dimensions thanks to a dynamical data structure. The presented numerical simulations of the flame-wall interactions showed that the control parameter is the Lewis number which determines the different behaviours when the flame is approaching the wall. From them we can conclude that, for Lewis numbers smaller than unity, the reaction rate is decreased, while for a unitary Lewis number the reaction rate is neither increased nor decreased. On the other hand, we observed for Lewis numbers larger than unity that the reaction rate increased, which leads to an acceleration of the flame speed when the flame is approaching the wall. The two- and three-dimensional simulations have shown that the flame front curvature was modified when the spherical flame is approaching the wall due to tangential diffusion. For small Lewis numbers the spherical flame is contracted, for large Lewis numbers it spreaded out, while for unitary Lewis number the flame front remained perpendicular to the wall. The observed phenomena present similarities with capillarity effects in fluid mechanics

Thermodiffusive flame structures interacting with adiabatic walls

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Figure 11. Isosurfaces T = 0.5 (black) and T = 0.1 (gray) at t = 0 (top left), t = 10 (top center), and t = 20 (top right), t = 30 (bottom left), t = 40 (bottom center), and t = 45 (bottom right)for the interaction of a flame ball with an adiabatic wall when taking into account radiation, Le = 0.3, γ = 0.05.

70

γ = 0.05 γ=0

60 50 ¯ R

40 30 20 10 0

0

5

10

15

20 t

25

30

35

40

¯ in Figure 12. Interaction of a flame ball with an adiabatic wall. Time evolution of R 3D with and without radiation, Le = 0.3.

when a droplet hits a wall. For the flame ball-wall interaction, the Lewis number plays a similar role as the surface tension for capillarity effects.

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Future work deals with numerical simulations of the interaction of spherical flame structures with non-adiabatic walls and the interaction of several flame balls with each other. Acknowledgements The authors acknowledge support from the European Program TMR on “Wavelets in Numerical Simulation” (Contract No. FMRX-CT98-0184), and from the FrenchGerman Program Procope (Contract No. D/0031094). We would also like to thank Eve-Marie Duclairoir for her help in computations. References [1] H. Bockhorn, J. Fr¨ ohlich, and K. Schneider. An adaptive two-dimensional wavelet-vaguelette algorithm for the computation of flame balls. Combust. Theory Modelling, 3:1–22, 1999. [2] J. D. Buckmaster and P. D. Ronney. Flame ball drift in the presence of a total diffusive heat flux. In 27th Symposium on Combustion, pages 2603–2610, Pittsburg, 1998. [3] A. Cohen. Adaptive methods for PDE’s - Wavelets or mesh refinement ? In International Conference of Mathematics, Beijing, 2002. [4] A. Cohen, S. M. Kaber, S. M¨ uller, and M. Postel. Fully adaptive multiresolution finite volume schemes for conservation laws. Math. Comp., 72:183–225, 2003. [5] A. Dervieux, B. Larrouturou, and R. Peyret. On some adaptive numerical approaches of thin flame propagation problems. Comp. Fluids, 17(1):39–60, 1989. [6] W. Gerlinger, K. Schneider, J. Fr¨ ohlich, and H.Bockhorn. Numerical simulations on the stability of spherical flame structures. Combust. Flame, 132:247–271, 2003. [7] E. Guyon, J. P. Hulin, and L. Petit. Hydrodynamique physique. InterEditions/Editions du CNRS, 1991. [8] P. Haldenwang and D. Pignol. Dynamically adapted mesh refinement for combustion front tracking. Comp. Fluids, 31(4-7):589–606, 2002. [9] A. Harten. Discrete multi-resolution analysis and generalized wavelets. J. Appl. Num. Math., 12:153–193, 1993. [10] A. Harten. Adaptive multiresolution schemes for shock computations. J. Comput. Phys., 115:319– 338, 1994. [11] A. Harten. Multiresolution algorithms for the numerical solution of hyperbolic conservation laws. Comm. Pure Appl. Math., 48:1305–1342, 1995. [12] A. Harten. Multiresolution representation of data: a general framework. SIAM J. Numer. Anal., 33(3):1205–1256, 1996. [13] B. Larrouturou. Adaptive numerical methods for unsteady flame propagation. In Ludford, editor, Reacting flows: combustion and chemical reactors, volume 24 of Lectures in Applied Mathematics, pages 415–435, 1986. [14] S. M¨ uller. Adaptive multiscale schemes for conservation laws, volume 27 of Lecture Notes in Computational Science and Engineering. Springer-Verlag, Heidelberg, 2003. [15] H. G. Pearlman and P. D. Ronney. Near-limit behavior of high-Lewis number premixed flames in tubes at normal and low gravity. Phys. Fluids, 6(12):4009–4018, 1994. [16] H. G. Pearlman and P. D. Ronney. Self-organized spiral and circular waves in premixed gas flames. J. Chem. Phys., 101(3):2632–2633, 1994. [17] T. Poinsot and D. Veynante. Theoretical and numerical combustion. 2nd edition, R.T. Edwards, 2005.

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[18] B. Rogg. The effect of Lewis number greater than unity on an unsteady propagating flame with one-step chemistry. In N. Peters and J. Warnatz, editors, Numerical methods in laminar flame propagation, volume 6 of Notes on numerical fluid mechanics, pages 38–48. Vieweg, 1982. [19] P. D. Ronney. Near-limit flame structures at low Lewis number. Combust. Flame, 82:1–14, 1990. [20] P. D. Ronney. Premixed laminar and turbulent flames at microgravity. Space Forum, 4:49–98, 1998. [21] O. Roussel and K. Schneider. A fully adaptive multiresolution scheme for 3D reaction-diffusion equations. In R. Herbin and D. Kr¨ oner, editors, Finite Volumes for Complex Applications, volume 3, pages 833–840. Hermes Penton Science, 2002. [22] O. Roussel and K. Schneider. An adaptive multiresolution scheme for combustion problems: application to flame ball-vortex interaction. Comp. Fluids, 34(7):817–831, 2005. [23] O. Roussel and K. Schneider. Adaptive numerical simulation of pulsating planar flames for large Lewis and Zeldovich ranges. Communications in Nonlinear Sciences and Numerical Simulation, 11:463–480, 2006. [24] O. Roussel, K. Schneider, A. Tsigulin, and H. Bockhorn. A conservative fully adaptive multiresolution algorithm for parabolic PDEs. J. Comput. Phys., 188(2):493–523, 2003. [25] K. Schneider. Wavelets, turbulence and chemical reactions. PhD thesis, University of Kaiserslautern, Germany, 1996. [26] C. K. Westbrook, A. A. Adamczyk, and G. A. Lavoie. A numerical study of laminar flame wall quenching. Combust. Flame, 40:81–99, 1981. [27] M. S. Wu, P. D. Ronney, R. O. Colantonio, and D. M. Vanzandt. Detailed numerical simulation of flame ball structure and dynamics. Combust. Flame, 116:387–397, 1999. [28] Y. B. Zeldovich. Theory of combustion and detonation of gases. Academy of sciences (USSR), Moscow, 1944.