This study proposes a novel design concept that leverages the geometric effects of channels with varying diameters and bends to boost flame acceleration and detonation in microchannels. A quasi-direct numerical simulation with detailed chemical kinetics is used to evaluate the processes of flame acceleration and detonation. A comparative study on the impact of equidistant spiral and converging spiral shapes of detonation channels on flame acceleration is conducted and discussed. The results indicate that in the low-speed and high-speed stages of flame propagation, Fermat's spiral channel exhibits a more significant promoting effect on flame acceleration compared to the Archimedes spiral channel. Fermat's spiral channel has significant advantages in terms of combustion efficiency and can save about 30% of fuel during the detonation process. This study helps to further reduce the scale of the detonation system.

The pulse detonation engine (PDE) is an engine that utilizes pulsed detonation waves to generate thrust, characterized by its simple structure, high thrust-to-weight ratio, and broad operating range, thus garnering significant attention. Despite zero-dimensional analyses indicating significant thermodynamic advantages of detonation combustion, translating these advantages into corresponding improvements in propulsion efficiency still presents several challenges, including the need for rapid ignition techniques. The typical cycle of a pulse detonation engine comprises several fundamental processes, including fresh mixture filling, ignition initiation, propagation of detonation waves, and exhaust processes. Among these, the time required for filling, ignition initiation, and exhaust processes is directly proportional to the ignition distance, consuming a significant portion of the working cycle, serving as the primary bottleneck limiting the engine's operating frequency. Therefore, reducing the ignition distance not only aids in reducing the engine's size and weight but also decreases the intake and exhaust time, thereby enhancing the engine's operating frequency.

To reduce the ignition distance, researchers have conducted extensive studies and proposed various methods, primarily including the following:

  • Installing turbulence generators upstream of the detonation channel: Turbulence generators, such as spiral-shaped blunt bodies or orifice-type flow disruptors, effectively enhance disturbance, promoting the transition of laminar flames to turbulent flames, thereby increasing the flame propagation speed. Schelkin1 first proposed the addition of spiral metal wires at the head of the detonation channel, which can control the ignition distance within a relatively wide range. Studies have found that the blockage rate and the length of the metal wires collectively affect the ignition distance; although high blockage rates of spiral wires can promote DDT (deflagration to detonation transition) to occur at shorter distances, they also reduce the peak thrust.2,3 Placing discrete obstacles inside the detonation channel4–6 can also achieve similar effects. Additionally, Wang et al.7 proposed a method of using thermally active fluid obstacles to accelerate ignition, whose operating mechanism is similar to that of solid obstacles.

  • Assembling shock wave reflectors downstream of the detonation channel. The enhancement of turbulence on flame propagation speed is limited. Shy et al. found that when the turbulent flame propagation speed reaches its limit (generally 10–20 times the laminar flame propagation speed), further increase in the turbulence intensity leads to local flame extinguishment, reducing the heat release intensity and thus decreasing the flame propagation speed.8 Therefore, in the later stages of flame acceleration, it is necessary to promote the formation of detonation waves through the rational organization of the interaction between shock waves, flames, and walls. Installing shock wave reflectors at a certain distance downstream of the detonation channel helps strengthen the shock wave intensity, achieving the effect of ignition assistance. De Wit et al.9 tested two types of reflectors, namely conical and disk-shaped, and found that the leading shock wave, when reflected from the shock wave reflector, reinforced the original shock wave, significantly increasing the pressure and temperature of the mixture near the wall, prompting the mixture near the wall to generate an explosion center, which then develops into a detonation wave. It should be noted that the use of shock wave reflectors can lead to relatively significant total pressure losses, and caution must be exercised when using them.

Despite the effectiveness of the aforementioned ignition assistance measures in reducing the ignition distance, they also have some drawbacks. For instance, turbulence generators and shock wave reflectors can lead to noticeable total pressure losses, thereby decreasing the propulsion performance of the engine.10 Furthermore, the high-temperature environment and periodic shock loads easily cause erosion and damage to the ignition assistance devices. In recent years, the ignition assistance potential of curved detonation channels has been receiving increasing attention. To address the issue of excessive axial length in pulse detonation engines, the Northwestern Polytechnical University detonation research team proposed a spiral detonation channel structure.10–12 Research has indicated that the smaller the spiral curvature radius, the shorter the DDT (deflagration to detonation transition) distance. Subsequently, Kuznetsov et al.,13 our research team,14–16 as well as Gai et al.17 and Pan et al.,18,19 observed significant differences in the morphology of the accelerated flame in curved and straight channels: The curved channels resulted in an asymmetric stretching of the flame front, with the flame propagating on the inner wall side. It is precisely this unique characteristic that gives curved channels a significant advantage in promoting flame acceleration.

Subsequently, our research team conducted a systematic study of the flame acceleration mechanism in slightly curved ducts. The flame acceleration mechanism can be classified into two categories: One involves accelerating the flame by increasing the flame area, including inherent flame instability, interaction between the flame and vortices, and interaction between the flame and turbulence; the other involves increasing the mixture pressure and temperature through interaction between pressure waves and the flame. For the accelerating flame in the upstream curved duct, when the flame propagation speed is low, it can be approximated as isobaric combustion, and the primary reason for flame acceleration is the increase in flame area caused by the asymmetric stretching of the flame front. Research has shown that in ducts with large curvature, the larger the duct width, the higher the flame acceleration rate, while the opposite is true for ducts with small curvature, where the smaller the duct width, the higher the flame acceleration rate.20 For the accelerating flame in the downstream curved channel, the flame propagation speed is higher. The main reason for flame acceleration is the interaction between the flame and pressure waves. Using an incremental analysis method, we eliminated the flame stretching effect and unified the channel outlet effect, isolating the compressive effect. Research has shown that although an increase in curvature is advantageous for flame acceleration, it leads to a decrease in the maximum flame propagation speed.21 Therefore, adopting a spiral structure with gradually decreasing curvature is reasonable. It is important to note that our conclusions are aimed at accelerating flames in slightly curved channels where the turbulence effect is not significant. Based on the above conclusion, we speculate that Fermat's spiral-shaped channel may have a superior flame acceleration effect compared to the Archimedes spiral-shaped channel. The main purpose of this paper is to verify this hypothesis.

In this paper, we compare the flame acceleration and DDT processes in channels with the same volume, but with Archimedes' spiral and Fermat's spiral. The physical and mathematical models adopted for the computations are described in Sec. II. The results are discussed in Sec. III. Finally, the main findings are summarized in Sec. IV.

This paper presents two numerical cases, comprising one Archimedes' spiral channel and one Fermat's spiral channel, as illustrated in Fig. 1. The governing equation for the Archimedes spiral shown in Fig. 1(a) is as follows:
{ x = 0.02 θ cos ( θ ) y = 0.02 θ sin ( θ ) with   θ [ 0 : 8 π ] .
(1)
FIG. 1.

Schematics of the computational domains. (a) An Archimedes' spiral conduit with a constant channel width of approximately 1.26 mm. (b) Fermat's spirals with a gradually decreasing channel width from the center outward. Red dots in the figure represent ignition positions, triangles correspond to Fermat's spiral conduits with widths equal to that of the Archimedes' spiral conduit, and pentagrams indicate the onset position of detonation.

FIG. 1.

Schematics of the computational domains. (a) An Archimedes' spiral conduit with a constant channel width of approximately 1.26 mm. (b) Fermat's spirals with a gradually decreasing channel width from the center outward. Red dots in the figure represent ignition positions, triangles correspond to Fermat's spiral conduits with widths equal to that of the Archimedes' spiral conduit, and pentagrams indicate the onset position of detonation.

Close modal
The governing equation for Fermat's spiral shown in Fig. 1(b) is given by
{ x = 0.078731 θ cos ( θ + π 2 ) y = 0.078731 θ sin ( θ + π 2 ) with   θ [ 0 : 11 π ] .
(2)

In setting up the computational domains, the following considerations are taken into account:

  • The areas of all computational domains in the numerical examples are kept identical, serving as a benchmark for analyzing the influence of channel configuration on flame acceleration.

  • The circumferential distance of the channels needs to be sufficiently long to exclude the effects of the outlet on flame acceleration.

  • The case is two-dimensional, and the channel width is on the millimeter scale to suppress turbulence effects.

In all cases, the channel wall is subject to non-slip, non-catalytic, adiabatic wall conditions. Some wall segments are positioned within the flow domain and set as zero-thickness walls. At the channel exit, nonreflecting boundary conditions are applied. At the initiation of the computation, a stoichiometric mixture of H2/O2 uniformly premixes across the entire computational domain. The initial pressure and temperature are set to 1 atm and 300 K, respectively. The premixed gas is ignited by a hot spot with a diameter of 0.4 mm located at (0.3, 0.1) mm. The hot spot region uniformly injects energy at a rate of 40 MW/m2 over a duration of 6 μs.

The propagation process of the flame in the channel is simulated using the quasi-direct numerical simulation solver combustionFoam.22 The simulation neglects the effects of Soret and Dufour, radiation, and assumes the gas to be an ideal gas with zero bulk viscosity. The corresponding governing equations are
ρ t + · ( ρ U ) = 0 ,
(3)
( ρ U ) t + · ( ρ U U ) · τ ¯ ¯ = p ,
(4)
ρ Y i t + · ( ρ Y i U ) + · ( ρ Y i V i ) = w ̇ i , i = 1 , , N 1 ,
(5)
( ρ h s ) t + · ( ρ h s U ) + ρ K t + · ( ρ K U ) = · q + p t + τ ¯ ¯ : U + Q ̇ r ,
(6)
p = i = 1 N ρ i R 0 M i T ,
(7)
τ ¯ ¯ = 2 3 μ ( · U ) I ¯ ¯ + μ [ U + ( U ) T ] ,
(8)
q = λ T + ρ i = 1 N h s , i Y i V i ,
(9)
where t is the time, ρ is the density, p is the pressure, U is the vector velocity, τ ¯ ¯ is the viscous stress tensor, hs is sensible enthalpy, T is the temperature, K is the kinematic energy, q is the heat flux, Q ̇ r is the net heat production rate, v v q is the heat flux, N is the number of mixture species, μ is the dynamic viscosity of the mixture, I ¯ ¯ is the unit tensor, and λ is the thermal conductivity of the mixture. Additionally, V i , w ̇ i, Yi, h s , i represent the diffusion velocity, net reaction rate, mass fraction, and sensible enthalpy of the ith component, respectively.
The diffusion velocities of species are modeled using the mixture-averaged approach:
V i = D i X i X i .
(10)
Here, Di and Xi represent the mixture-averaged diffusion coefficient and mole fraction of the ith species, respectively. To ensure mass conservation, the mass fraction and diffusion velocity of the “inert species” are explicitly calculated.

The computation employs a simplified chemical reaction mechanism suitable for high-pressure conditions, developed by Burke et al.23 This mechanism comprises 13 species and 27 reactions. The thermodynamic parameters and transport coefficients for individual species are calculated using the NASA 7-coefficient polynomial parameterization model and the Chapman–Enskog expression, respectively. The dynamic viscosity and thermal conductivity of the mixture are computed using the Wilke mixture rule and Mathur mixture rule, respectively. The chemical reaction rates are determined through the Arrhenius equation.

In the computational domain, a quadrilateral grid is employed for discretization, with control over its maximum grid size to ensure uniformity as much as possible. Numerical resolution tests were conducted by varying the value of the maximum grid size, as shown in our previous study.14 Four different maximum grid sizes were used in the tests: d x max = 10, 7.5, and 5 μm. The results indicate that a value of d x min = 7.5μm is sufficient to capture the main features of flame acceleration and DDT. The presented computational results are obtained with d x max = 5 μm, corresponding to a grid resolution of 48 grid cells per flame thickness at the initial condition. This resolution is higher than those reported in recent related studies, such as Xiao and Oran6 and Zhang et al.,24 which used 10 grid cells, and close to the 50 grid cells employed by Han et al.25 

Figure 2 (Multimedia view) illustrates a time series of selected contour plots depicting the propagation and transition to detonation of the flame in Fermat's spiral. From these plots, we can observe the acceleration of the flame, the formation of shock waves, and the timing and location of the transition to detonation. Initially, a weak ignition generates a spherical laminar flame. Due to the thermal expansion of combustion products, the flame begins to accelerate and expand outward. Influenced by the curved wall surface, the flame gradually elongates on the inner wall side, reaching an extent of nearly one and a half revolutions at its maximum (32 μs). This flame length significantly exceeds that reported in relevant experiments in the literature.13,17–19 This is primarily attributed to the large curvature of the inner ring segment in our configuration. Previous studies have indicated that higher channel curvature favors the asymmetric expansion of the flame surface.20 For the initial slow flame development, where pressure wave interaction with the flame is not significant, flame acceleration primarily depends on the growth of the flame surface area. Therefore, the spiral structure we employed, resembling a logarithmic spiral, is more conducive to flame acceleration compared to a ring structure.

FIG. 2.

Time-sequence of numerical shadow photographs showing flame acceleration and DDT in a Fermat's spiral channel. Multimedia available online.

FIG. 2.

Time-sequence of numerical shadow photographs showing flame acceleration and DDT in a Fermat's spiral channel. Multimedia available online.

Close modal

Another aspect to clarify is that the attachment of the flame surface to the inner wall, resulting in asymmetric expansion, is primarily due to the asymmetry induced by the acceleration flow field. It is essential to emphasize that the flame's asymmetric expansion is characteristic of an accelerating flame. Steady-state flames in curved channels, as reported in the literature,26 do not exhibit such asymmetric expansion of the flame surface. The accelerated flame propels unburned gas ahead, creating an accelerated flow field. Due to angular momentum conservation, the flow velocity near the inner wall is higher than that near the outer wall. As shown in Fig. 3, the flow field of the unburned gas can be divided into three regions: the boundary layer regions on both sides near the boundaries and the central inertial region. In the inertial region, the velocity near the inner wall is higher than that near the outer wall, exhibiting a linear distribution. In contrast, for a straight channel, the velocity in the inertial region shows a uniform characteristic.27 The flow field of the unburned gas serves as the background for flame propagation, and it is the non-uniformity of the flow field in the inertial region that leads to the flame propagating along the inner wall in the curved channel. For additional details, please refer to Ref. 20.

FIG. 3.

The velocity profile of unburned gases ahead of the flame, corresponding to Section A at 20 μs in Fig. 2.

FIG. 3.

The velocity profile of unburned gases ahead of the flame, corresponding to Section A at 20 μs in Fig. 2.

Close modal

As the flame continues to accelerate, a normal shock appears in front of the flame near the outer wall (32 μs). At this point, the continued acceleration of the leading edge of the flame is somewhat restrained, and the trailing edge of the flame gradually catches up. During this stage, the flame area decreases, and the flame acceleration is primarily attributed to the compressive effect, elevating the temperature, density, and pressure of the unburned gas. At 34 μs, the shape of the flame front is similar to the shapes observed in the experiments reported in the literature.17,18 At this time, the elongated funnel-shaped region between the flame front and the outer wall is in a high-temperature, high-pressure state. In the subsequent evolution, a series of localized explosions occur in this region (36 μs), continuing to sweep through the unburned gas and spreading to the rest of the unburned mixture, ultimately forming a complete detonation wave (37 μs). The literature19 reports the existence of another initiation mode in annular channels, namely initiation at the leading shock wave. However, this initiation mode was not observed in our operating conditions. Nevertheless, we speculate that increasing the initial pressure of the premixed gas may potentially lead to initiation occurring primarily at the leading shock wave.

In our previous published paper, we provided a detailed analysis and description of the dynamic process of flame propagation along Archimedes' spiral channels. We suggest that interested readers refer to Ref. 14. Compared with Archimedes' spiral channels, the difference in flame acceleration process is mainly reflected in the process of the root of the flame surface chasing the head of the flame surface. In addition, although there are differences in flame acceleration due to the influence of channel shape, the acceleration mode of flames and the initiation mode of detonation, as well as the asymmetric stretching of flame surfaces and the pressure wave flame interaction, are similar. At time 32 μs in Fig. 2, it can be observed that the middle of the flame surface will first contact the outer wall, forming a breakpoint, that is, there are two relatively independent flame surfaces in the following period of time. This phenomenon is not present in the Archimedes spiral channel. When flames propagate in an equally wide Archimedean spiral channel, their flame surface remains continuous. This process is also reflected in the evolution of flame surface area over time. Figure 4 shows the evolution process of flame surface area in two channels. It can be seen that in the Fermat spiral channel, there is a turning point (at the upper triangular symbol) during the descent stage of the flame surface, while there is no turning point in the Archimedean channel. This inflection point is located at the point where the flame segment on one side of the inner circle of the channel disappears. In addition, we can also observe that the peak of the flame surface area of Fermat's spiral is larger (at the circle symbol). The area of the flame surface after detonation (at the pentagram symbol) is related to the channel width at the detonation position, so the value of Fermat's spiral is smaller.

FIG. 4.

Flame front area as a function of time for channels of different shapes. The circles represent the instances at which the flame surface reaches its maximum extent, the upward-pointing triangles indicate the moment when the flame surface experiences a breakpoint, and the pentagrams represent the moment of ignition.

FIG. 4.

Flame front area as a function of time for channels of different shapes. The circles represent the instances at which the flame surface reaches its maximum extent, the upward-pointing triangles indicate the moment when the flame surface experiences a breakpoint, and the pentagrams represent the moment of ignition.

Close modal

We address the following two questions: What are the reasons for changes in the flame's topological structure? Is such a change advantageous or detrimental to flame acceleration? First, alterations in the flame's topological structure may stem from various factors. In the case of a spiral channel, it is primarily influenced by the channel width. By examining streamline structures, it is observed that when the flame propagates laterally along the channel, its propagation speed is minimally affected by the convective field and mainly depends on the local laminar flame speed. Therefore, the distance from the flame front to the channel wall is primarily influenced by the flame propagation time. Due to the shorter propagation time of the flame head in the lateral direction compared to the flame base, a wedge-shaped region is formed between the flame front and the outer channel wall in the Archimedes spiral channel with a fixed width. The main difference between Fermat's spiral and Archimedes' spiral is that the former gradually decreases in channel width, with a trend of rapid reduction followed by a slower decrease, as shown in Fig. 5. Hence, when the rate of reduction in channel width is significant, the flame front cannot form a wedge-shaped region with a narrow base and a wider head against the outer channel wall. This, in turn, leads to the flame encountering the wall in the later stages of evolution, causing a change in the flame's topological structure. The above analysis also indicates that this phenomenon is influenced by the properties of the premixed gas (e.g., laminar flame velocity and expansion rate) and the rate of channel width contraction.

FIG. 5.

The variation of channel width with respect to the polar angle, where θ represents the angle in polar coordinates, and D denotes the channel width.

FIG. 5.

The variation of channel width with respect to the polar angle, where θ represents the angle in polar coordinates, and D denotes the channel width.

Close modal

Now, let us turn to the second question: the impact of flame topology changes on flame acceleration. Generally speaking, the acceleration of a flame is closely related to its topology. Figure 6 depicts the streamlines in Fermat's spiral channel when there are two flame fronts, with the streamlines colored by velocity magnitude. Comparing the streamlines passing through the two flame fronts, it can be observed that the rear flame front, while continuing to consume premixed gas, induces a flow field primarily along the channel width direction. Therefore, its contribution to the acceleration and area growth of the front flame is relatively small. In contrast, the primary direction of the induced flow field by the front flame front is along the channel length direction, with higher velocities near the inner side compared to the outer side. This is favorable for stretching the flame front and increasing its area, thus forming a positive feedback loop that promotes flame acceleration. In summary, the change in flame topology here is unfavorable for flame acceleration, and efforts should be made to avoid such situations as much as possible when designing channel shapes. Therefore, the Fermat spiral channel configuration studied in this article is not the optimal structure and can be further improved.

FIG. 6.

In the context of Fermat's spiral channel with two flame segments, heat release rate contour and streamline diagrams are presented, with streamlines colored by velocity magnitude.

FIG. 6.

In the context of Fermat's spiral channel with two flame segments, heat release rate contour and streamline diagrams are presented, with streamlines colored by velocity magnitude.

Close modal

In engineering, people are most concerned about the detonation distance, which is one of the important indicators for measuring the performance of detonation channels. For straight passages, the detonation distance is easily determined and representative. However, for curved channels, especially those with variable widths, the definition of detonation distance is not directly applicable. Therefore, in this article, we use a dimensionless fuel consumption ratio instead. The calculation domain area of the two examples is equivalent, and the channel distance is long enough. The outlet has no effect on flame propagation. Therefore, we set the fuel consumption rate as a function of the ratio of the area consuming hydrogen to the calculation domain area over time. Figure 7 shows the fuel consumption rate of channels with different shapes. The fuel consumption here is obtained by integrating the area swept by the flame surface. From the graph, it can be seen that the fuel consumed for detonation in Archimedes' spiral channel is about 50%. The Fermat spiral channel consumes less fuel, about 35%. Therefore, designing a reasonable Fermat's spiral-shaped channel can effectively reduce the fuel required for detonation.

FIG. 7.

Comparison of fuel consumption rates for channels of different shapes.

FIG. 7.

Comparison of fuel consumption rates for channels of different shapes.

Close modal
Figure 8 depicts the trend of the heat release rate of regional integration over time, which can be utilized to assess the rate of chemical energy conversion in detonation channels. The integral heat release rate (iHRR) over the domain area S is defined by
iHRR = S k = 1 N ω ̇ k h c , k d V ,
(11)
where ω ̇ k and h c , k are the net production rate and chemical formation enthalpy of the k th species, respectively. N is the number of species. It can be observed that Fermat's spiral channels exhibit a distinct decreasing phase in iHRR, lasting approximately 0.8 μs. This is attributed to the reduction in flame area due to quenching near the wall in the middle part of the flame front. It can also be reflected from here that changes in the topology of the flame surface are also detrimental to energy conversion. Following initiation, the magnitude of iHRR is determined by the channel width at the location of the detonation wave. Fermat's spiral exhibits a smaller iHRR after initiation, indicating that the channel width at the initiation point of the detonation wave in this structure is smaller. Therefore, to enhance the rate of chemical energy conversion in the detonation channel, the optimal structure involves transitioning the detonation channel behind the initiation point to a gradually widening configuration while ensuring self-sustenance of the detonation wave.
FIG. 8.

Comparison of heat release rate integration for channels of different shapes. The colored regions signify distinct phases in the flame surface evolution, characterized by a bifurcation followed by a subsequent convergence into a unified phase.

FIG. 8.

Comparison of heat release rate integration for channels of different shapes. The colored regions signify distinct phases in the flame surface evolution, characterized by a bifurcation followed by a subsequent convergence into a unified phase.

Close modal

In general, DDT in Fermat's spiral channels is more readily achieved, primarily due to the converging effect of curved channels. During the initial stages of flame propagation, the velocity of the flame is relatively slower, with acceleration predominantly dependent on the stretching of the flame front. Previous studies have indicated that for channels with a high degree of curvature, an increase in channel width correlates with a rise in the dimensionless acceleration rate of the flame; the opposite trend is observed for channels with lower curvature.20 The underlying mechanism involves a more pronounced stretching effect on the flame surface. Consequently, in the inner channels where curvature is greater, a larger tube diameter significantly enhances the rate of flame acceleration. Additionally, according to the law of mass conservation, tapering of the channel leads to a noticeable disparity between the propagation speeds at the leading and trailing edges of the flame, facilitating further stretching of the flame surface, thereby contributing to an increase in the area of the flame front.

Another mechanism contributing to flame acceleration is the interaction between pressure waves and the flame front. Short et al. explored the acceleration of flames in converging channels, where the channel width is on the order of the flame thickness.28 They found that the tapering of channels accelerates flame propagation by influencing the rate of pressure diffusion. Intuitively, convergence of the channel tends to elevate pressure, thereby increasing the pressure and temperature in the flame reaction zone, which promotes further acceleration of the flame. Figure 9 displays the average pressure at the flame front, illustrating that even in the segments of the channel with the most significant tapering, the difference in average pressure at the flame front is minimal. Therefore, for the two cases compared, the effect of pressure increase due to channel convergence on flame acceleration can be considered negligible. Furthermore, in the later stages of flame acceleration, where the flame propagation speed is higher and the surface area of the flame front gradually decreases, acceleration primarily relies on the interaction between the flame and pressure waves. Previous studies have noted that curved channels, by suppressing convection and diffusion processes, cause a rapid increase in the unburned gas pressure, thereby enhancing the flame acceleration rate.21 Intuitively, the centrifugal force generated by the flow leads to an increase in pressure within the reaction zone, thus increasing the reaction rate and, consequently, a greater acceleration of the flame. However, as shown in Fig. 1, the curvature difference (measured along the channel's centerline) from the points of equal width to the ignition points between the two channels is minimal, indicating that this effect can also be disregarded for the cases compared.

FIG. 9.

The average pressure on the flame front at different times. The circles represent the instances at which the flame surface reaches its maximum extent.

FIG. 9.

The average pressure on the flame front at different times. The circles represent the instances at which the flame surface reaches its maximum extent.

Close modal

Consequently, in the design of the initial segments of detonation channels, it is advisable to maximize the channel diameter to facilitate the early stages of flame acceleration, particularly when the flow speed is below 0.3 Mach.

Detonation velocity deficit is a key parameter in evaluating the performance of detonation channels. Figure 10 shows the average flame propagation speeds for two channels, calculated by the ratio of the volume of fuel consumed per unit time to the area of the flame front. In both channels, after the injection of ignition energy is complete, there is an initial decline in the average flame propagation speed. Subsequently, at roughly the same moment, the speed rapidly increases and surpasses the Chapman–Jouguet (CJ) velocity. It is important to note that the average flame speed described here is different from the propagation speed at the flame tip. It reflects the average velocity across all positions on the flame front. Therefore, although the graph shows that the times of speed increase in the two channels are almost identical, this does not imply that their detonation run-up times are nearly the same. When the overdriven detonation waves gradually settle into a steady-state detonation, the variation in speed across the flame front becomes minimal, and the calculated speed at this state represents the propagation speed of the detonation wave. The results indicate that the detonation propagation speeds for both channels are close and slightly below the CJ speed. The deficit in speed is mainly caused by two factors: The first is heat loss. Given that our model uses adiabatic walls, this factor does not apply to our case. Furthermore, the coiled morphology of the spiral channel's walls naturally offers a thermal advantage by enveloping burned gases and using peripheral heat to warm unburned gases, as opposed to straight channels. The second factor, wall friction, is the main source of speed deficit in these two channels. Figure 10 indicates that the differences in speed deficit between the two channels are negligible.

FIG. 10.

Average flame propagation speed in the two channels. The estimation of Chapman–Jouguet (CJ) velocity is calculated by the Shock and Detonation Toolbox.29 

FIG. 10.

Average flame propagation speed in the two channels. The estimation of Chapman–Jouguet (CJ) velocity is calculated by the Shock and Detonation Toolbox.29 

Close modal

In this study, we compared the propagation and detonation characteristics of flames in Archimedes' spiral and Fermat's spiral channels using numerical simulation methods. We observed the acceleration of flames and the formation of shock waves, as well as the specific time and location of detonation transition, through numerical shadow plots of time series. The initial laminar flame begins to accelerate and expand due to thermal expansion. Unlike steady-state flames in curved channels, the asymmetric expansion of the flame surface is caused by the asymmetry of the accelerating flow field, while the curved structure of Fermat's spiral channel further promotes the asymmetric expansion and acceleration of flame surface. In addition, unlike Archimedes' spiral channels where the flame topology changes continuously, if the width of Fermat's spiral channel changes at a large rate, it can lead to breakpoints during flame propagation, which is not conducive to flame acceleration.

In addition, we found that compared to Archimedes' spiral channels, Fermat's spiral channels perform better in reducing the fuel required for detonation, which is of great significance for designing efficient and energy-saving detonation channels. We discussed three key factors that affect flame propagation and acceleration: (1) the influence of channel curvature and width on flame acceleration, (2) the promoting effect of channel width reduction on flame propagation, and (3) the influence of curvature effect on flame acceleration in curved channels. Therefore, we propose that when designing detonation channels, the shape and size of the channels should be considered to optimize the flame propagation process, reduce the required detonation fuel, and thus improve the overall efficiency of the system.

This study shows that there is still room for further improvement and optimization in Fermat's spiral channels. Future research can explore different channel geometries and conditions to find better flame acceleration and detonation initiation effects. In addition, the study comparing Fermat's spiral and Archimedes' spiral channels not only reveals the commonalities and differences in flame acceleration processes but also provides theoretical basis and reference for designing efficient combustion systems. This discovery can be used to shorten the detonation distance required for the development of detonation and applied to efficient combustion technology. It may be used for the development of pulse detonation engines, internal combustion engines, and ignition devices.

This work was supported by the National Natural Science Foundation of China (Grant Nos. 52206184 and 52176139).

The authors have no conflicts to disclose.

Tao Li: Conceptualization (equal); Data curation (equal); Investigation (equal); Methodology (equal); Project administration (equal); Writing – original draft (equal). Xing Li: Supervision (equal). Baopeng Xu: Supervision (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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