The initial stages of hydrogen–air flame propagation in tubes and the mechanism of tulip flame formation are investigated using a high-order numerical code to solve the fully compressible reactive Navier–Stokes equations for a spark or planar igniting flame at the closed end of a tube and propagating to the opposite closed or open end. It is shown that the mechanism of tulip flame formation is universal for both sparked and planar ignited flames in tubes with both ends closed. Flame front inversion results from the tulip-shaped profile of the unburned gas axial velocity near the flame front, which is the result of the superposition of the unburned gas flow generated by the accelerating flame and the reverse flow generated by the rarefaction wave during flame deceleration. In a half-open tube, this mechanism is valid for spark ignited flames. In the case of planar ignition, there is no rarefaction wave, but the growth of bulges on the sidewalls leads to the formation of a tulip flame.

The dynamics of a flame propagating in closed or half-open tubes, is important for understanding combustion processes under confinement, such as explosions and safety issues as well as for industrial and technological applications, e.g., combustion in gas turbines and internal combustion engines. The inversion of the flame front propagating from the closed end of the tube, from a convex shape with a tip in the direction of unburned gas to a concave shape with a tip in the direction of burned gas, is known as tulip flame formation and has been observed in many experiments and numerical simulations. The first photographs of the inversion of the flame front during the propagation of a premixed flame in a tube were published by Ellis1–3 almost a century ago. The concave flame front with lateral petals in the direction of the unburned gas was called the tulip-flame by Salamandra et al.4 

It should be noted that flame front inversion can be caused by various processes, such as the interaction of the flame with a shock or pressure wave, various hydrodynamic instabilities inherent to the propagating flame, such as Darrieus–Landau (DL) or Rayleigh–Taylor (RT) instabilities. This makes the concept of tulip flame formation not entirely definite, and has led to a number of different scenarios underwent in attempts to explain the mechanism of tulip flame formation. Since the first experimental observation of tulip flames,1–3 many studies have been conducted in an attempt to explain the physical process of the flame front inversion leading to tulip flames, but the actual cause of tulip flames has not yet been definitively established.

Markstein5,6 hypothesized that the collision of a convex flame front with a shock wave, leading to an inversion of the flame front, could explain the formation of tulip flames, but the formation of a tulip flame does not involve sufficiently strong shock waves. Many authors7–18 have considered the DL instability as the cause of flame front inversion, but the characteristic timescale of the DL instability is much larger than the characteristic timescale of the flame front inversion observed in experiments and numerical simulations. Clanet and Searby19 assumed that the tulip flame formation is a manifestation of the RT instability of the flame front during the flame deceleration. The interaction of vortices, apparently generated as a result of the baroclinic effect near the tube sidewalls and observed in experiments and numerical models, with the flame front has been considered by many authors8,20–29 as a seemingly plausible scenario for flame front inversion, although no evidence has been obtained that this is actually the case. For example, it was suggested in Refs. 21, 23, and 25 that the DL instability serves as a trigger for the emergence of vortices, which then participate in the inversion of the flame front. On the contrary, numerical simulations30,31 based on a non-viscous model with zero Mach number have been considered as an argument that tulip flames can form in the absence of vortices. The reader can find a dramatic history of the experimental, theoretical, and numerical studies conducted in an attempt to explain the mechanism of tulip flame formation in the comprehensive review by Dunn-Rankin.32 According to experimental studies by Ponizy et al.,33 tulip flame formation is a purely hydrodynamic phenomenon resulting from the interaction between the flame and gas motion, and the inherent instabilities of the flame front do not involve in the formation of tulip flames.

Of particular note is the work of Guénoche,34 who suggested that the key process for understanding the mechanism of tulip flame formation may be the deceleration wave generated by the flame during the phase. However, with the computational capabilities that existed in the 1960s, it was difficult to prove Guénoche's hypothesis. In the recent work of Liberman et al.35 it was shown that the formation of tulip flames is indeed a purely hydrodynamic phenomenon closely related to the rarefaction waves generated by the flame during the deceleration phases. In Ref. 35, the formation of tulip flames in channels with closed ends and different aspect ratios, as well as in a half-open channel, was investigated by simulation of the fully compressible Navier–Stokes equations combined with a detailed chemical model for a stoichiometric mixture of hydrogen/air. While in the acceleration phase the flame creates a flow of unburned fuel that moves in the same direction (x>0) as the flame, in the deceleration phase, the flame front acts like a piston moving out of the tube with negative acceleration and creates a simple rarefaction wave. In the classical case, when the gas ahead of the piston is at rest, the rarefaction wave creates a reverse flow with a maximum (negative) velocity near the piston surface, which vanishes at the head of the rarefaction wave which propagates in the direction x>0 at the speed of sound. Therefore, in the case of a flame propagating from the closed end, the flow of unburned gas mixture during the decelerating phase is a superposition of the flow generated earlier by the accelerating flame and the flow created by the rarefaction wave. This leads to the decrease in the unburned gas flow velocity in the near zone of the flame front and to an increase in the boundary layer thickness. As a result, the axial velocity profile in the unburned flow near the flame front acquires a tulip shape, and since the velocity of each point of the flame front in the thin flame model can be considered as the sum of the laminar flame velocity and the velocity with which the flame is entrained by the unburned flow, the flame acquires a tulip shape.

Usually, the premixed flame is initiated by an electrical spark, which is modeled in simulations as a small circular pocket of hot burned gas near the closed end of a tube. For the spark ignition, Clanet and Searby19 identified four stages of flame evolution during the tulip flame formation: (1) ignition followed by a hemispherical flame expansion unaffected by side walls of the tube; (2) a finger-shaped flame with exponential growth of the flame surface and, consequently, the flame speed; (3) the lateral parts of the flame skirt touch the side walls of the tube, resulting in a decrease in the flame surface and, consequently, the flame velocity; and (4) the flame front inversion and a tulip flame formation.

Obviously, the dynamics of flame propagation in a tube and the evolution of the flame front surface depend on how the flame was initiated. If the flame is initiated by an ignition source whose dimensions are comparable to the tube width, the flame skirt touches the tube walls from the very beginning and the flame bypasses the first two acceleration stages and the third deceleration stage. A flame initiated by planar ignition, i.e., a flat narrow strip of hot burned gas equal in size to the width of the tube, is a typical case opposite to the “classical” flame ignition by a small spark. Numerical experiments36–39 have shown that a tulip flame is also formed in the case of planar ignition, but the mechanism of tulip flame formation in this case has not been studied and remains unclear. In Ref. 39, similar to the earlier publication,29 the formation of cusps on tulip petals—the formation of a distorted tulip flame (DTF)—was attributed to “combined effects of the vortex motions and the Rayleigh–Taylor instability driven by pressure waves,” but no evidence was provided to support the correctness of this statement.

In the present paper, we consider the earlier stages of premixed flames propagating in a tube with non-slip walls, and the physical processes responsible for tulip flame formation for flames initiated by planar ignition compared to the classical ignition by a small spark. For this purpose, numerical simulations of the fully compressible reactive Navier–Stokes equations coupled with a one-step chemical model, which was calibrated to correctly describe the pressure dependence of the laminar velocity of a stoichiometric hydrogen–air flame, were performed using a high-order DNS solver.

As it was pointed above, the meaning of tulip flame formation is not entirely definite and can be caused by different physical processes. In the present paper, we consider the formation of a tulip flame, which is an inversion of the flame front from a convex to a concave shape that occurs faster than the characteristic times of the development instabilities inherent to the flame front. This means that tulip flame formation is a purely hydrodynamic process, which is consistent with the findings of experimental studies by Ponizy et al.33 We show that for a flame initiated by planar ignition in a tube with both ends closed, the flame front inversion is due to a tulip-shaped profile of the unburned gas axial velocity in the immediate vicinity of the flame front, resulting from the superposition of the forward flow of unburned gas generated by the previously accelerating flame and the reverse flow of unburned gas generated by the rarefaction wave. This scenario is universal and occurs for spark-ignited flames under all downstream conditions and for flames initiated by planar ignition in the tube with the closed ends. In the case of a spark-ignited flame, the strong reduction of the flame surface in the third phase and the resulting strong deceleration of the flame creates a strong rarefaction wave, so that flame collisions with reflected pressure waves play a minor role. On the contrary, in the case of a flame initiated by planar ignition, the surface area of the flame front increases monotonically, resulting in an increase in the heat release rate per unit frontal projected area and the higher volumetric combustion rate. Brief periods of flame deceleration in this case are due to the flame collisions with pressure waves reflected from the opposite closed end of the tube. It is shown that a chain of such short flame deceleration phases leads to the generation of relatively weak rarefaction waves, which in turn leads to the formation of a tulip-shaped profile of the unburned gas axial velocity in the immediate vicinity of the flame front, resulting in tulip-shaped flame formation. In the case of a flame initiated by planar ignition in a semi-open tube, there are no reflected pressure waves, and the increase in flame surface area is localized in a narrow band near the tube walls, roughly of the order of the boundary layer thickness. This leads to the formation of bulges at the flame front near the tube walls, which grow to form a tulip-shaped flame when they meet at the tube axis.

We consider a flame which is ignited at the left closed end of the two-dimensional (2D) rectangular channel of width D (y[D/2,D/2]) and propagates along the x-axis to the right closed or open end. The early stages of flame dynamics ignited by a small spark near the left closed end are well known,19 and the mechanism of tulip-shaped flame formation was for the first time explained in Ref. 35 for flames propagating in channels with different aspect ratios and in a half-open channel.

To investigate the dynamics of a flame initiated by planar ignition, consider the flame initiated at the left closed end of the channel by a planar strip of high-temperature combustion products of width D and thickness 1 mm. The dynamics of flame propagating in a tube depends on the boundary conditions and the tube width, which determine the flame acceleration compatible with the boundary conditions. Below, we adopt the adiabatic no-slip boundary condition on the tube walls. The dynamics of an initially plane flame ignited at the closed end of the channel differs significantly from the dynamics of the flame ignited by a small spark.

After the flame was ignited near the closed end of the tube, thermal expansion of the high-temperature combustion products pushes the unburned gas toward the opposite closed or open end of the tube, creating a plane-parallel flow of unburned fuel ahead of the flame. At the initial moment, the velocity of the unburned flow is u=(Θ1)Uf, and the flame propagates with normal laminar velocity Uf relative to the unburned fuel and with velocity UfL=ΘUf in the laboratory reference frame,40,41 where Θ=ρu/ρb is the expansion coefficient—the ratio of the unburned ρu and burned ρb gases densities, respectively. Because of wall friction, the velocity of the unburned gas flow is almost constant in the bulk, decreases in the thin boundary layer and vanishes at the channel walls. The velocity profile in the unburned gas flow depends on the channel width and gas viscosity. In the wide channel, the velocity is uniform in the bulk and drops to zero in the thin boundary layer, whereas in the thin channel a parabolic velocity profile (Poiseuille flow) develops in the unburned gas in a short time tPD2/100ν.41 Therefore, in the thin channel, the flame is accelerated all the time and no tulip-shaped flame is formed.

Consider first a flame initiated by planar ignition at the left closed end and propagating to the right open end of the tube. For theoretical analysis, we use the classical approach of an infinitely thin flame front.40,41 Each part of the flame front moves relative to the unburned mixture at a laminar speed Uf and is simultaneously entrained by the flow of unburned gas at a local flow velocity u+(x,y) immediately ahead of that part of the flame front. The shape of the flame front is determined by the relative motion of different parts of the flame front, which in turn is determined by the velocity profile in the unburned gas flow in the immediate vicinity ahead of the flame front. The local velocity of every small part of the flame front in the laboratory reference frame can be written as
UfL=Uf+u+(x,y).
(1)
It is interesting to note that, although Eq. (1) is the result of the thin flame model in which the flame front is treated as a discontinuity surface, comparison with numerical simulations shows that Eq. (1) is fulfilled with surprisingly good accuracy (see  Appendix B).

For a wide channel the velocity of unburned flow ahead of the flame u+(x,y) is nearly uniform in the bulk and drops to zero within a thin boundary layer of thickness δl5X/ReD, where Re=u+(x,0)D/ν is Reynolds number, ν is the kinematic viscosity. The shape of the flame front “replicates” the “inverted” shape of the unburned flow velocity profile, remaining nearly flat in the bulk, with the edges of the flame skirt stretched backward within the boundary layer. A stretched flame consumes fresh fuel over a larger area, resulting in a higher rate of heat release per unit of a frontal projected area and a higher volumetric burning rate. It should be emphasized that, unlike a flame ignited by a small spark, the increase in the surface area of the flame initiated by a planar ignition is localized in the narrow strip near the side walls of the tube, approximately the thickness of the boundary layer, resulting in the formation of bulges on the flame front near the tube walls, while the central part of the flame front remains flat [see Fig. 8(b)]. As the flame velocity is directed along the normal to the flame surface, the bulges in the flame front near the tube walls begin to grow. Since the velocity of the unburned flow ahead of the flame u+(x,y) is maximum at yδl, the tip of the bulge expands faster in the x-axis direction. The expansion of the bulges toward the tube axis “eats up” the nearly flat central part of the flame front until the bulges growing from either side of the tube axis “eat up” the entire central part of the flame front and converge on the tube axis to form a tulip-shaped flame.

In the case of a tube with both ends closed, the evolution of the flame front shape initiated by planar ignition is mainly determined by the flame collisions with pressure waves reflected from the right closed end of the tube. Each such collision reduces the flame velocity, and the decelerating flame creates a weak rarefaction wave that alters the velocity profile in the unburned flow, thereby favoring the formation of a tulip-shaped velocity profile in the unburned gas in the immediate vicinity ahead of the flame, which in turn causes a tulip-shaped profile of the flame front. Later, the flame collisions with the pressure waves lead to the distortion of the tulip petals and the formation of a distorted tulip flame (DTF), which does not occur in a half-open channel.

The two-dimensional computational domains modeled using high-resolution DNS are D=1cm wide rectangular channels with an aspect ratio of L/D=6, with both ends closed, and a 1 cm wide rectangular half-open channel with the right end open. The simulations solve the 2D time-dependent, reactive compressible Navier–Stokes equations including molecular diffusion, thermal conduction, viscosity with a single-step Arrhenius model for chemical reaction of a stoichiometric hydrogen/air mixture. The governing equations are
ρt+(ρui)xi=0,
(2)
(ρui)t+(Pδij+ρui2)xj=τijxj,
(3)
(ρE)t+[(ρE+P)ui]xi=(τijui)xiqixi,
(4)
ρYkt+ρuiYkxi=xi(ρVikYk)+ω̇k,
(5)
P=ρRBT(i=1NsYiWi),
(6)
σxx=2μuxx23μ(uxx+uyy),
(7)
σyy=2μuyy23μ(uxx+uyy),
(8)
σxy=σyx=μ(uxy+uyx).
(9)
The gas mixture is assumed to be an ideal gas, P=ρRBT(i=1NsYiWi), with a constant ratio of specific heats γ=cP/cV, and the total energy is
E=Pρ(γ1)+12(ux2+uy2).
(10)
Here, ρ, ui, T, P, E, τij, qi are density, velocity components, temperature, pressure, total energy, components of viscosity stress tensors, heat flux, RB is the universal gas constant. Yi,Wi,Vi,j are the mass fraction, molar mass, and diffusion velocity of species i. The viscosity coefficients μk are calculated using the standard method.42,43 For species k,
μk[kgms]=2.67×106WkT/1000σk2Ω(2,2)(T),
(11)
where σk is the collision diameter. The collision integral42,
Ω(2,2)(T)=1.0313(T)0.1193+(T+0.43628)1.6041
(12)
is a function of reduced nondimensional temperature T=εT/kB. Here, ε is the maximum attraction energy between a pair of molecules, and kB is the Boltzmann constant. The viscosity coefficient of mixture is obtained using the semiempirical formula42,43
μ=12{k=1NsXkμk+(k=1NsXkμk)1},
(13)
where Xk is the molar fraction of species k.
The coefficient of heat conduction of species k was obtained using Cantera,44 
λk=(i=04ai,kln(T)i)T
(14)
with parameters ai,k presented in Table I.
TABLE I.

Parameters ai,k used in Eq. (14).

H2 O2 H2O N2
a0,k  −2.34  2.58 × 10−1  −9.83 × 10−1  6.33 × 10−3 
a1,k  1.39  −1.54 × 10−1  6.12 × 10−1  3.86 × 10−3 
a2,k  −3.04 × 10−1  3.44 × 10−2  −1.42 × 10−1  −2.38 × 10−3 
a3,k  2.93 × 10−2  −3.36 × 10−3  1.44 × 10−2  4.00 × 10−4 
a4,k  −1.04 × 10−3  1.23 × 10−4  −5.40 × 10−4  −2.01 × 10−5 
H2 O2 H2O N2
a0,k  −2.34  2.58 × 10−1  −9.83 × 10−1  6.33 × 10−3 
a1,k  1.39  −1.54 × 10−1  6.12 × 10−1  3.86 × 10−3 
a2,k  −3.04 × 10−1  3.44 × 10−2  −1.42 × 10−1  −2.38 × 10−3 
a3,k  2.93 × 10−2  −3.36 × 10−3  1.44 × 10−2  4.00 × 10−4 
a4,k  −1.04 × 10−3  1.23 × 10−4  −5.40 × 10−4  −2.01 × 10−5 
The coefficient of mixture heat conduction is calculated using the semi-empirical formula similar to Eq. (13). The species diffusion coefficient is determined by assuming that the Lewis number is unity for all species,
Dk=λmixtureρcp.
(15)

Our recent high-resolution numerical simulations with a detailed chemical scheme for hydrogen–air flames35 have shown that the formation of tulip flame is a purely hydrodynamic phenomenon. A similar conclusion was reached earlier by Ponizy et al.33 as a result of an experimental study of tulip shape formation in a stoichiometric propane–air flame ignited at the closed end. Therefore, a simplified one-step chemical model was used. This model gives a correct pressure dependence of the laminar flame velocity.

We consider the irreversible global reaction,
H2+0.5O2H2O.
(16)
The reaction rate is taken in the form a one-step Arrhenius-type chemical kinetics. For pressures P<2bar,
ω̇=dYH2/dt=Aexp(EaRBT)(ρYH2WH2)(ρYO2WO2),
(17)
while for pressures P>2bar,
ω̇=dYH2/dt=Aexp(EaRBT)(ρYH2WH2)0.9(ρYO2WO2)0.9.
(18)
The reaction order is n=2 for P<2bar, and for P>2bar the reaction order is n=1.8. A comparison of the laminar flame velocity vs pressure obtained using a one-step model [Eqs. (17) and (18)] with the results obtained using a detailed chemical model50 is given in  Appendix A.

Two-dimensional (2D) direct numerical simulation (DNS) is used to solve the governing Eqs. (2)–(9) using the DNS solver, which is a weighted essentially non-oscillatory (WENO) fifth order finite difference scheme for solving the convective terms of the governing equations.45 The advantage of the WENO finite difference method is the capability to achieve arbitrary high order accuracy in smooth regions while capturing sharp discontinuity. To ensure the conservation of the numerical solutions, the fourth order conservative central difference scheme is used to discretize the non-linear diffusion terms.46 The time integration is third order strong stability preserving Runge–Kutta method.47 

The initial conditions are P0=1atm, T0=298K. In simulations, we used adiabatic no-slip reflecting boundary conditions at the tube walls
u=0,T/n=Yk/n=0,
(19)
where n is the normal to the wall (y-axis). To model a half-open tube, the nonreflecting outflow boundary condition48 is used to calculate the subsonic outflow.

Reliable modeling of reactive flows requires proper resolution of the internal structure of the flame. Since the thickness of the laminar flame decreases with increasing pressure, higher resolution is required to determine the flame structure at maximum pressure. The maximum pressure during the tulip flame formation is about 2 bar in the case of a short tube with both ends closed, resulting in the decrease in the thickness of the flame front from Lf=350 to Lf130μm. A uniform mesh with a resolution of Δx12.5μm, which corresponds to 28 grid points across the flame width at the beginning of the process and 14 grid points at maximum pressure for a tube closed at both ends, was used in the simulations. Thorough resolution and convergence (a grid independence) tests were performed in previous publications35,49 by varying the value of Δx to ensure that the resolution is adequate to capture details of the problem in question and to avoid computational artifacts. The parameters used in simulations are shown in Table II.

TABLE II.

Model parameters for simulating a stoichiometric hydrogen–air flame.

Initial pressure  P0  1.0 atm 
Initial temperature  T0  298 K 
Initial density  ρ0  8.5×104g/cm3 
Pre-exponential factor P<2bar  A  2.95×1013cm3/mol3s 
Pre-exponential factor P>2bar  A  2.1×1012(cm2.4/mol2.4s) 
Activation energy  Ea  27RBT0 
Laminar flame velocity  Uf  2.43 m/s 
Laminar flame thickness  Lf  0.0325 cm 
Adiabatic flame temperature  Tb  2503 K 
Expansion coefficient (ρu/ρb Θ  8.34 
Specific heat ratio  γ=CP/CV  1.399 
Sound speed  as  408.77m/s 
Initial pressure  P0  1.0 atm 
Initial temperature  T0  298 K 
Initial density  ρ0  8.5×104g/cm3 
Pre-exponential factor P<2bar  A  2.95×1013cm3/mol3s 
Pre-exponential factor P>2bar  A  2.1×1012(cm2.4/mol2.4s) 
Activation energy  Ea  27RBT0 
Laminar flame velocity  Uf  2.43 m/s 
Laminar flame thickness  Lf  0.0325 cm 
Adiabatic flame temperature  Tb  2503 K 
Expansion coefficient (ρu/ρb Θ  8.34 
Specific heat ratio  γ=CP/CV  1.399 
Sound speed  as  408.77m/s 

We consider the tulip flame formation in the two-dimensional rectangular channel of width D=1cm, aspect ratio L/D=6 with both ends closed. Figures 1(a) and 1(b) shows the time evolution of the local velocities of the flame front along the centerline UfL(y=0) and near the sidewall for the case of a flame ignited by a small spark at the tube axis near the left closed end [Fig. 1(a)] and for the flame ignited by a planar strip of high temperature combustion products at the left closed end [Fig. 1(b)].

FIG. 1.

(a) Time evolution of the flame surface area Ff (dashed-dotted), combustion wave speed Sf, local velocities of the flame front at the tube axis y=0 and at near the wall y=0.42cm for a flame initiated by a small spark at the axis at the left end of the tube; and (b) the same for the flame initiated by planar ignition.

FIG. 1.

(a) Time evolution of the flame surface area Ff (dashed-dotted), combustion wave speed Sf, local velocities of the flame front at the tube axis y=0 and at near the wall y=0.42cm for a flame initiated by a small spark at the axis at the left end of the tube; and (b) the same for the flame initiated by planar ignition.

Close modal

It can be seen in Fig. 1(a) that the flame propagation speed in the laboratory reference frame is closely related to the variation of the flame surface area Ff. From the beginning and up to 0.5 ms the flame surface area Ff, the combustion wave velocity Sf as well as the local velocities of the flame front UfL(y=0) and UfL(y=0.42cm) increase. After the lateral parts of the flame skirt touch the side walls of the tube, the flame surface area Ff, combustion wave velocity Sf and flame tip velocity UfL(y=0) decrease, but the local flame front velocity near the wall at y=0.42cm continues to increase and after 0.65 ms it exceeds the flame tip velocity, UfL(y=0).

The dynamics of the flame shown in Fig. 1(b), which is initiated by planar ignition, is quite different. The flame surface area Ff increases with small oscillations, while the combustion wave velocity Sf, the flame tip velocity UfL(y=0), and the local flame front velocity near the wall, UfL(y=0.42cm) strongly oscillate as the result of the flame collisions with pressure waves reflected from the right closed end of the tube, similar to oscillations in Fig. 1(a) after 0.7 ms. Nevertheless, it can be seen in Fig. 1(b), that after each oscillation the local flame front velocity near the wall, UfL(y=0.42cm) increases, while the flame tip velocity, UfL(y=0) oscillates, remains constant in the averaging, and later decreases. Finally, after several collisions after 1.1 ms, the velocity UfL(y=0.42cm) significantly exceeds the velocity of the flame tip UfL(y=0).

Figures 2(a) and 2(b) shows the evolution of the axial flow velocities of the unburned fuel at 0.5 mm ahead of the flame front at the tube axis y=0 and near the sidewall of the tube, at y=0.42cm for the case of a spark ignition [Fig. 2(a)] and for the case of a planar ignition [Fig. 2(b)]. In both Figs. 2(a) and 2(b) also shown the difference between the local velocity of the unburned gas near the sidewall of the tube and at the tube axis. Figure 2(a) shows that after 0.6 ms, Δu+=u+(y=0.42)u+(y=0) becomes positive, and from that moment on, the parts of the flame front located closer to the tube walls are entrained by the flow of unreacted mixture at a higher velocity, and, accordingly, the parts of the flame front located closer to the tube walls propagate faster than the parts of the flame front located closer to the tube axis.

FIG. 2.

(a) Velocities of the unreacted flow at 0.5 mm ahead of the flame front at y=0, y=0.42cm, and the difference Δu+=u+(y=0.42)u+(y=0) for spark ignited flame; and (b) the same for planar ignition.

FIG. 2.

(a) Velocities of the unreacted flow at 0.5 mm ahead of the flame front at y=0, y=0.42cm, and the difference Δu+=u+(y=0.42)u+(y=0) for spark ignited flame; and (b) the same for planar ignition.

Close modal

In Ref. 35, it was shown that the difference Δu+=u+(y=0.42)u+(y=0) between the axial velocities in the unburned flow is mainly due to the first strongest rarefaction wave. As the surface area of the flame decreases, the velocity of the combustion wave also decreases. The decelerating flame acts like a piston moving out of a tube with negative acceleration, creating a simple rarefaction wave in the unburned gas mixture. In Fig. 1(a), one can see oscillations of the flame velocity caused by collisions of the flame front with the pressure waves reflected from the right end of the tube. Similar oscillations in the velocity of the unburned gas near the flame front, caused by the collision of the flame with pressure waves reflected from the right end of the tube, can be seen in Figs. 2(a) and 2(b). As a result of these collisions, the velocity of the flame is reduced, causing the flame to decelerate, creating a weak rarefaction wave in the unburned gas. One of the consequences of the rarefaction waves in the unburned gas is the adverse pressure gradient in the unburned flow near the flame front. The pressure waves reflected from the right end of the tube create conditions similar to those created by the first rarefaction wave described above as a result of the reduction in flame surface area [Fig. 1(a)]. The effect of the rarefaction wave generated by each collision of the flame with the pressure wave is weaker compared to the first strong rarefaction wave [Figs. 1 and 2(a)], but repeated collisions strengthen the overall effect so that the value of Δu+=u+(y=0.42)u+(y=0) oscillates but remains positive in both cases shown in Figs. 2(a) and 2(b) already after the formation of the first rarefaction wave. This leads to the formation of a tulip-shaped axial velocity profile in the unburned gas close to the flame front, which in turn leads to the formation of a tulip-shaped flame.

It should be emphasized that in the case of a flame initiated by planar ignition, there is no stage of strong reduction of the flame surface [see Fig. 1(b)] and correspondingly strong flame deceleration, but several collisions of the flame front with pressure waves reflected from the right closed end of the tube are sufficient to create conditions in the unburned gas velocity near the tube wall u+(y=0.42) to increase above the unburned gas velocity at the tube axis u+(y=0). Figure 2(b) shows that the value of Δu+=u+(y=0.42)u+(y=0) becomes positive already after the first collision. It fluctuates, but grows and becomes positive after 0.85 ms. After 1.2 ms, the value of Δu+ became large enough, and the axial velocity profile in the unburned gas in the close vicinity ahead of the flame front becomes tulip-shaped, and, consequently, the flame front also acquired a tulip shape.

Figures 3(a) and 3(b) show the calculated Schlieren images for selected time instants and the stream lines during the formation of a tulip flame for the flame propagating in the tube D=1cm, L=6cm with both ends closed for spark ignition [Fig. 3(a)] and for planar ignition [Fig. 3(b)].

FIG. 3.

(a) Time sequence of computed Schlieren images and streamlines for the spark ignited flame propagating in the tube L/D=6 with both ends closed. (b) Time sequence of computed Schlieren images and streamlines for the planar ignition of the flame propagating in the tube L=6cm with both ends closed.

FIG. 3.

(a) Time sequence of computed Schlieren images and streamlines for the spark ignited flame propagating in the tube L/D=6 with both ends closed. (b) Time sequence of computed Schlieren images and streamlines for the planar ignition of the flame propagating in the tube L=6cm with both ends closed.

Close modal

It can be seen in Fig. 3(a) that after 0.5 ms, as the flame surface area begins to decrease and the flame velocity decreases, the flow velocities in the combustion products are reversed from positive to negative. The reverse flow of combustion products is required by the boundary conditions at the left end for the reduced flame velocity. The reverse flow also leads to the formation of a pair of vortices in the combustion products.

Figures 3(a) and 3(b) also shows that after tulip flame formation, continued collisions of the flame with pressure waves lead to deformation of the tulip petals and formation of distorted tulip flames.29,39

Figures 3(a) and 3(b) also shows a pair of vortices created in the combustion products near the tube walls behind the flame front by strongly curved sections of the flame front near the side walls, apparently due to the baroclinic effect. However, while the vortices in Fig. 3(a) appear during tulip flame formation and persist for up to 1.2 ms, the vortices in Fig. 3(b) appear when the reverse flow of combustion products is already present and the tulip shape of the flame is almost formed, and they disappear during the final phases of tulip flame formation. It has been widely believed12–14,20,21,25,26,29,33 that vortices play an important role in the formation of a tulip flame. However, the present study convincingly shows that vortices are not related to the mechanism of tulip flame formation. In the case shown in Fig. 3(a), the recirculation of combustion products occurs due to the reduction in flame velocity and the reverse flow of combustion products. These large vortices then drift in the combustion products toward the flame front, forming a focus of streamlines near the tube axis that appear to extend in both directions from the flame front. This is a purely hydrodynamic process that does not involve flame front instabilities. A more detailed discussion of this problem was given in Ref. 35.

Figures 4(a) and 4(b) shows the time evolution of the axial velocity profiles in the unburned gas in the immediate vicinity ahead of the flame front during the tulip flame formation.

FIG. 4.

(a) Profiles of the axial velocities in the unburned gas mixture ahead of the flame front during the tulip flame formation for the flame initiated by a small spark; and (b) the same for the flame initiated by planar ignition.

FIG. 4.

(a) Profiles of the axial velocities in the unburned gas mixture ahead of the flame front during the tulip flame formation for the flame initiated by a small spark; and (b) the same for the flame initiated by planar ignition.

Close modal

In Ref. 39, flames initiated by planar ignition in 2D channels of different sizes were studied. It was found that the flame dynamics and the normalized characteristics of the flame evolution are essentially the same for the same aspect ratio. The effect of the aspect ratio for the flame initiated by the spark ignition has been studied in Ref. 35. It has been shown that the flame dynamics and the flame evolution are essentially different for channels of different aspect ratios. Figure 5 shows that the smaller the aspect ratio, the more flame collisions with pressure waves reflected from the right end.

FIG. 5.

The velocity of the flame tip along the centerline y=0, computed for tubes with different aspect ratios L/D=6,12,18 and for the half-open tube of the same width D = 10 mm.

FIG. 5.

The velocity of the flame tip along the centerline y=0, computed for tubes with different aspect ratios L/D=6,12,18 and for the half-open tube of the same width D = 10 mm.

Close modal

In the case of a half-open tube, the flame ignited at the left closed end propagates toward the right open end, so there are no reflected pressure waves and the pressure ahead of the flame remains constant. Figures 6(a) and 6(b) shows the time evolution of the flame surface area Ff, the combustion wave velocity Sf, and the local velocities of the flame front at the centerline y=0 and near the sidewall y=0.40cm for spark ignition [Fig. 6(a)], and at the centerline y=0 and near the sidewall y=0.42cm for planar ignition [Fig. 6(b)].

FIG. 6.

(a) Calculated time evolution of combustion wave velocity Sf, the flame surface area Ff (dashed–dotted) and local velocities of the flame front at y=0 and near the sidewall at y=0.4cm in the half-open tube for the flame initiated by a spark; the same as in (a) for the planar ignition.

FIG. 6.

(a) Calculated time evolution of combustion wave velocity Sf, the flame surface area Ff (dashed–dotted) and local velocities of the flame front at y=0 and near the sidewall at y=0.4cm in the half-open tube for the flame initiated by a spark; the same as in (a) for the planar ignition.

Close modal

It can be seen from Figs. 6(a) and Fig. 5 that the acceleration stage of the spark-ignited flame in a half-open channel is almost the same as for the spark-initiated flame in a channel with both ends closed in Fig. 1(a). However, the deceleration rate in a half-open channel is significantly slower than in a tube with both ends closed; the flame deceleration phase in a half-open channel lasts almost twice as long as in a tube with both ends closed, where it is enhanced by the flame collisions with reflected pressure waves. This means that the intensity of the rarefaction wave generated by the flame in the half-open channel during the deceleration phase is weaker. Thus, in a half-open channel only the first rarefaction wave affects the dynamics of the unburned gas flow and, consequently, the formation of the tulip flame.

In the case of the flame initiated by planar ignition at the closed end and propagating toward the open end [Fig. 6(b)], there is no deceleration stage, the flame surface area increases monotonically until the tulip flame is formed. Therefore, the mechanism of tulip flame formation for planar ignition in a semi-open channel differs from the mechanism of tulip flame formation for spark ignited and planar ignited flames in a closed tube. In the case of a flame initiated by planar ignition, the sides of the flame skirt expand backward along the boundary layer, while the central part of the flame front remains nearly flat [see Fig. 8(b)]. Due to the stretching of the flame surface in the strip Δyδl near the wall, the surface area of the flame increases, the flame consumes fresh fuel over a larger area, resulting in an increase in the rate of heat release per unit area of the frontal projection. An increase in the heat release rate results in an increase in the volumetric combustion rate and the formation of bulges on the flame front near the tube walls. Since the tip of the bulge propagates faster along the x-axis than the flat central part of the flame, the velocity of the unburned gas ahead of the flame is maximum at y±δl. The expansion of the two bulges (in the y direction to the tube axis) “eats” the central part of the flame front until the bulges converge on the tube axis, forming a tulip-shaped flame.

Figures 7(a) and 7(b) shows the flow velocities along the tube axis y=0 and near the wall at y=0.40cm for the flame initiated by a spark [Fig. 7(a)] and for the planar ignited flame y=0.42cm [Fig. 7(b)] in the unburned gas at 0.5 mm ahead of the flame front.

FIG. 7.

(a) Velocities of the flow at 0.5 mm ahead of the flame front at y=0, y=0.40cm, and the difference Δu+=u+(y=0.40)u+(y=0) for the flame initiated by a spark in the half-open tube; (b) the same as in (a), but for the planar ignition y=0.42cm Δu+=u+(y=0.42)u+(y=0).

FIG. 7.

(a) Velocities of the flow at 0.5 mm ahead of the flame front at y=0, y=0.40cm, and the difference Δu+=u+(y=0.40)u+(y=0) for the flame initiated by a spark in the half-open tube; (b) the same as in (a), but for the planar ignition y=0.42cm Δu+=u+(y=0.42)u+(y=0).

Close modal

Figures 8(a) and 8(b) shows the sequences of the calculated Schlieren images for selected time instants and the streamlines during the formation of a tulip flame for the flame propagating in the half-open tubes D=1cm for the spark ignited flame [Fig. 8(a)] and for planar ignition [Fig. 8(b)]. A significant difference in flame dynamics can be seen between the tulip flame formation in a half-open channel for the spark-initiated flame and the flame initiated by planar ignition. For example, the vortical flow in the combustion products is more pronounced in Fig. 8(a) than in Fig. 8(b). The vortices in Fig. 8(b), seen at 0.9 ms, are caused by the baroclinic effect and are not related to the tulip-shaped flame formation.

FIG. 8.

(a) Sequences of computed Schlieren images and the streamlines for the premixed hydrogen–air flame propagating in a half-open tube for the flame initiated by the spark ignition; and (b) the same for the flame initiated by the planar ignition.

FIG. 8.

(a) Sequences of computed Schlieren images and the streamlines for the premixed hydrogen–air flame propagating in a half-open tube for the flame initiated by the spark ignition; and (b) the same for the flame initiated by the planar ignition.

Close modal

Figures 9(a) and 9(b) show the evolution of the axial velocity profiles in the flow of unburned gas at 0.5 mm from the flame front during the tulip flame formation for the flame initiated by spark ignition [Fig. 9(a)] and by planar ignition [Fig. 9(b)]. Figure 7(a) shows that after 0.7 ms, the flow velocity in the unburned gas closer to the side walls exceeds the flow velocity at the centerline, and after 0.9 ms, the axial velocity profile of the unburned gas acquires a tulip shape [Fig. 9(a)]. A similar scenario occurs after 0.5 ms for the case of a flame initiated by planar ignition. In both cases, the further away from the centerline, the greater the difference Δu+. This trend continues to the “inner” edge of the boundary layer, where the velocity reaches its maximum. From this point the velocity decreases within the boundary layer and vanishes at the sidewall. Accordingly, the local velocities of the flame front Ufl=Uf+u+ become minimum at the centerline, gradually increase toward the boundary layer, where they reach the maximum value, and then decrease within the boundary layer, so that the flame front acquires a tulip shape. This tendency determines the specific shape of the tulip flame: the position of the tip of the tulip petal relative to the sidewall of the tube and the height of the petal.

FIG. 9.

(a) Axial velocity profiles at 0.5 mm ahead the flame front in the unburned gas for spark ignited flame in a semi-open tube; and (b) the same as in (a) but for the flame initiated by planar ignition.

FIG. 9.

(a) Axial velocity profiles at 0.5 mm ahead the flame front in the unburned gas for spark ignited flame in a semi-open tube; and (b) the same as in (a) but for the flame initiated by planar ignition.

Close modal

This paper presents results of numerical simulations of the early stages of the dynamics of a hydrogen–air flame and explains the physical processes leading to the formation of a tulip-shaped flame during flame propagation in closed and half-open tubes. In the calculated Schlieren images Figs. 3(a) and 3(b) one can see the cusps formed on the tulip petals at later times, the phenomena called distorted tulip flame (DTF).26,29 It should be noted that no DTF occurs in the half-open channel shown in Figs. 8(a) and 8(b). This means that DTF is formed due to the flame collisions with pressure waves reflected from the opposite closed end of the tube. In Refs. 29 and 39, the formation of cusps on the tulip petals was explained by “the joint action of vortex motions and Rayleigh–Taylor (RT) instability due to the pressure wave.” Clearly, collisions of the flame front with pressure waves are the likely cause of distorted tulip flames, but the role of RT instability and vorticity is not clear. Figures 10(a) and 10(b) shows the flame acceleration in the tube with both ends closed for a spark ignited flame [Fig. 10(a)] and for the planar ignited flame [Fig. 10(b)].

FIG. 10.

(a) Spark ignited flame acceleration during a distorted tulip flame formation in the tube with both ends closed; and (b) the same for a planar ignited flame.

FIG. 10.

(a) Spark ignited flame acceleration during a distorted tulip flame formation in the tube with both ends closed; and (b) the same for a planar ignited flame.

Close modal

The increment of the initial linear stage of RT instability is σ=Agk, where A=(ρuρb)(ρu+ρb)=(Θ1)(Θ+1)0.77 is the Atwood number, g is the flame acceleration, k=2π/λ is the wave number.41 Taking a typical wavelength as the minimum size of the cusp in Figs. 3(a) and 3(b), λ1mm, we obtain σ3.8×105s1 for spark ignited flame [Fig. 10(a)], and σ1.55×104s1 for planar ignited flame [Fig. 10(b)]. In both cases, the acceleration that can lead to the development of RT instability lasts about Δt(0.10,2)ms, which means that the amplitude of RT instability does not grow too much in either case, exp(σΔt)1. Probably the acceleration can be larger and lasts for a longer time for a larger tube size and for a 3D flame.

The objective of the present study was to investigate the effect of ignition geometry on the mechanism of tulip flame formation. For this purpose, the tulip flame formation from the flame initiated by planar ignition near the closed end and propagating to the opposite closed or open end of the tube was studied and compared with the tulip flame formation in the case of the flame initiated by spark ignition. Our main conclusion is that the physical process that causes flame front inversion and tulip flame formation is the tulip-shaped profile of the unburned gas axial velocity created by the rarefaction wave in the vicinity of the flame front during the flame deceleration stage. This scenario is universal and works both in the case of spark ignition of a flame in a closed and semi-open tube, and in the case of a flame initiated by planar ignition in a closed tube, where the flame collisions with pressure waves reflected from the opposite closed end cause a chain of short-lived flame deceleration phases. The decelerating flame acts like a piston moving out of a tube, creating a simple rarefaction wave that generates a reverse (negative) flow of unburned gas with maximum (negative) velocity at the surface of the piston. The superposition of the already existing forward flow of unburned gas created by the accelerating flame and the reverse flow created by the rarefaction wave during flame deceleration leads to the formation of a tulip-shaped profile of the unburned gas axial velocity in the vicinity of the flame front. In the theoretical model of an infinitely thin flame, the velocity of any point on the flame front is the sum of the laminar velocity of the flame relative to the unburned gas Uf plus the velocity u+(r) at which that portion of the flame front is entrained by the flow of unburned gas. This means that if the axial velocity profile of unburned gas takes a tulip shape, the flame front also takes a tulip shape. The condition UfL=Uf+u+ is formally valid for the theoretical model of an infinitely thin flame, while the simulations are conducted for a flame front of real thickness. Nevertheless, we found that the condition UfL=Uf+u+ is satisfied with surprisingly good accuracy ( Appendix B).

In the case of a flame initiated by planar ignition at the closed end and propagating to the open end of the tube, the flame only accelerates without deceleration phases. In this case, the tulip-shaped flame is formed by the growth and convergence of two bulges near the side walls of the tube, the intersection of which creates a tulip-shaped flame front.

No specific funds were received for this particular work; research of Nordic Institute for Theoretical Physics (NORDITA) is partially supported by Nordforsk (M.L.).

The authors have no conflicts to disclose.

Chengeng Qian: Formal analysis (equal); Investigation (equal); Software (equal); Visualization (equal); Writing – review & editing (equal). Mikhail A. Liberman: Conceptualization (equal); Formal analysis (equal); Investigation (equal); Supervision (equal); Writing – original draft (equal).

The data that support the findings of this study are available within the article.

The one-step chemical model was calibrated to correctly reproduce the most important parameters of a hydrogen/air flame, such as the laminar flame velocity and thickness, the laminar flame velocity pressure dependence, the adiabatic flame temperature, and the expansion coefficient. The calibration of the pre-exponential constant A and activation energy Ea was performed by the genetic algorithm method using Cantera and is summarized in Table II. During the formation of a tulip flame, the pressure increase is negligible in the case of a half-open tube, but it increases from an initial value of 1 atm to over 2.5 atm in a short channel L=6cm. Figure 11 shows the laminar velocity of a hydrogen–air flame calculated with the present one-step model compared to the results obtained with the detailed chemical model.50 

FIG. 11.

Laminar velocity of a hydrogen air flame vs pressure calculated using a one-step model, Eqs. (17) and (18) and detailed chemical model.50 

FIG. 11.

Laminar velocity of a hydrogen air flame vs pressure calculated using a one-step model, Eqs. (17) and (18) and detailed chemical model.50 

Close modal

Figure 12 shows the flame dynamics obtained in the simulations with a detailed chemical model50 compared to the simulations with a one-step chemical model Eqs. (17) and (18).

FIG. 12.

Flame velocity calculated using detailed chemical model50 and the one-step model Eqs. (17) and (18).

FIG. 12.

Flame velocity calculated using detailed chemical model50 and the one-step model Eqs. (17) and (18).

Close modal

It can be seen from Fig. 12 that the flame dynamics, temporal evolution of flame velocity, flame surface area, etc., obtained from simulations with the one-step model are very close to those obtained from simulations with the detailed chemical model.50 

In the theoretical model of an infinitely thin flame, the velocity of any point of the flame front in the laboratory reference frame can be considered as the sum of the laminar flame velocity relative to the unburned gas and the flow velocity u+(x,y) immediately ahead of this point of the flame front, with which this part of the flame front is entrained by the flow of the unreacted mixture. Figure 13 shows the calculated flame front velocity at the axis of a half-open tube UfL(Xf,y=0) for a planar ignited flame [Fig. 6(b)], and the unburned gas velocity on the axis at 0.5 mm ahead of the flame front, u+(Xf+0.5mm,y=0). It is seen that the difference UfL(Xf,y=0)u+(x=Xf+0.5mm,0), shown by the empty squares, is very close to the laminar flame velocity Uf=2.41m/s.

FIG. 13.

The flame velocity at the tube axis UfL(y=0) and the unburned gas velocity at 0.5 mm ahead of the flame u+(y=0) for a flame initiated by planar ignition in a half-open tube. The horizontal line shows the laminar flame velocity Uf=2.41m/s, the empty squares are UfL(y=0)u+(y=0).

FIG. 13.

The flame velocity at the tube axis UfL(y=0) and the unburned gas velocity at 0.5 mm ahead of the flame u+(y=0) for a flame initiated by planar ignition in a half-open tube. The horizontal line shows the laminar flame velocity Uf=2.41m/s, the empty squares are UfL(y=0)u+(y=0).

Close modal
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