Viscous multi-phase detonation of a pulse detonation engine with plasma jet ignition

Abstract

The two-dimensional viscous space-time conservation element and solution element (CE/SE) method is used to calculate the multi-phase detonation of plasma jet ignition. The effects of viscosity on the detonation flow field are compared with the N–S equation and the Euler equation as the governing equtions, and the effects of the jet temperature, time and initial droplet radius on the deflagration-to-detonation transition are analysed. The results show that the effect of viscosity on the propagation of detonation waves is very small, but the viscosity has certain effects on the detonation parameters. It is possible to significantly shorten the (deflagration-to-detonation transition) DDT distance of the stable detonation by increasing the temperature and time of the initial jet ignition. When the plasma jet has already fully ignited the detonable mixture, increasing the jet time has little effect on shortening the DDT distance. When the droplet radius is less than 50 μm, the peak pressure of the detonation wave increases with an increase in the droplet radius, and the peak pressure decreases with an increase in the droplet radius when the droplet radius is more than 50 μm.

Keywords

pulse detonation engine plasma jet deflagration to detonation transition

1. Introduction

A pulse detonation engine (PDE) is an engine using a high-temperature and high-pressure gas generated by pulse detonation waves to obtain thrust, and it is considered to be one of the sources with development prospects [1,2]. Because liquid fuel is easy to carry, achieving the rapid initiaton and detonation of gas-liquid two-phase detonation is one of the most urgent problems to be solved in engineering applications.

Research has shown that jet flame ignition can shorten the deflagration-to-detonation transition (DDT), which uses the jet flame formed by the pre-combustion chamber to ignite the combustible mixture in the detonation tube [3,4]. Z.W. Wang [5] numerically simulated the initiation process of a transverse jet in a detonation chamber filled with a mixture of propane/air in a chemically equivalent ratio and analysed the initiation and mechanism of the transverse jet. J.L. Yu [6] showed that jet ignition could effectively increase the initial velocity of flame in the detonation tube in the DDT experiment of an acetylene/air mixture. However, jet ignition needs to be realized by means of pre-combustion, and the structure of a transverse jet is rather complicated. Plasma jet ignition has the advantages of possessing a simple structure, high ignition energy, short pulse discharge time and high energy transfer efficiency. This technology has been widely used in the fields of chemistry and spaceflight [7,8]. There are few studies on the applications of this technology to gas-liquid detonation. Hence, as the use of plasma jet ignition is still in its exploratory stage, plasma jet ignition initiation is used in this paper.

A large number of theoretical studies have been carried out for gas-liquid two-phase detonation at home and abroad. The Euler equation is used as the control equation for the simulation of a gas-liquid two-phase PDE, and thus, the effect of viscosity on the detonation process is ignored [9,10]. However, the actual process is characterized by stickiness. Therefore, the viscous multi-phase detonation model of a PDE with plasma jet ignition, which is solved by the viscous space-time conservation element and solution element (CE/SE) method, is established in this paper. The effects of viscosity on the detonation parameters are compared using both the N–S equation and the Euler equation as the governing equations. On this basis, the effects of the plasma jet temperature, time and initial droplet radius on the DDT are analysed.

2. Model and governing equations

2.1. Computational model

Figure 1 shows a schematic diagram of the PDE model with plasma jet ignition at the wall. Since the detonation process in a PDE is axisymmetric, a two-dimensional calculation model can be used. The part above EF is a plasma generator. The area surrounded by ABEF is the half-plane of the detonation tube, where AF, BE, JK and AB are the thrust wall, the wall of the PDE tube, the inlet of the plasma jet and the central axis of symmetry, respectively. AF and EK are the wall surfaces.

Fig. 1.

General configuration consists of a pulse detonation engine with plasma jet ignition. Gasoline/air mixture in the detonation tube is ignited by plasma jet from the segment JK of plasma generator.

2.2. Governing equations of the pulse detonation engine

Based on some assumptions [11], the axisymmetric viscosity equation of gas-liquid flow in a PDE with plasma jet ignition is shown as follows: $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle {\displaystyle \frac{\partial \boldsymbol{U}}{\partial t}}+\frac{\partial \boldsymbol{F}}{\partial x}+{\displaystyle \frac{\partial \boldsymbol{G}}{\partial y}}={\displaystyle \frac{\partial \boldsymbol{F}_{v}}{\partial x}}+\frac{\partial \boldsymbol{G}_{v}}{\partial y}+\boldsymbol{R}-{\displaystyle \frac{\boldsymbol{G}-\boldsymbol{G}_{v}}{y}}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \boldsymbol{U}=\left[\begin{array}{@{}l@{}}{\phi}_{g}{\rho}_{g},{\phi}_{l}{\rho}_{l},{\phi}_{g}{\rho}_{g}u_{g},{\phi}_{g}{\rho}_{g}v_{g},{\phi}_{l}{\rho}_{l}u_{l},{\phi}_{l}{\rho}_{l}v_{l},{\phi}_{g}{\rho}_{g}\boldsymbol{Y },\\[6.0pt] {\phi}_{g}{\rho}_{g}(1-Y)E_{gg},{\phi}_{g}{\rho}_{g}\mathit{YE}_{gp},{\phi}_{l}{\rho}_{l}E_{l}\end{array}\right]^{T}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \boldsymbol{F}=\left[\begin{array}{@{}l@{}}{\phi}_{g}{\rho}_{g}u_{g},{\phi}_{l}{\rho}_{l}u_{l},{\phi}_{g}({\rho}_{g}u_{g}^{2}+p),{\phi}_{g}{\rho}_{g}u_{g}v_{g},{\phi}_{l}({\rho}_{l}u_{l}^{2}+p),{\phi}_{l}{\rho}_{l}u_{l}v_{l},{\phi}_{g}{\rho}_{g}u_{g}\boldsymbol{Y },\\[6.0pt] {\phi}_{g}u_{g}({\rho}_{g}(1-Y)E_{gg}+p_{gg}),{\phi}_{g}u_{g}({\rho}_{g}\mathit{YE}_{gp}+p_{gp}),{\phi}_{l}u_{l}({\rho}_{l}E_{l}+p)\end{array}\right]^{T}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \boldsymbol{G}=\left[\begin{array}{@{}l@{}}{\phi}_{g}{\rho}_{g}v_{g},{\phi}_{l}{\rho}_{l}v_{l},{\phi}_{g}{\rho}_{g}u_{g}v_{g},{\phi}_{g}({\rho}_{g}v_{g}^{2}+p),{\phi}_{l}{\rho}_{l}u_{l}v_{l},{\phi}_{l}({\rho}_{l}v_{l}^{2}+p),{\phi}_{g}{\rho}_{g}v_{g}\boldsymbol{Y },\\[6.0pt] {\phi}_{g}v_{g}({\rho}_{g}(1-Y)E_{gg}+p_{gg}),{\phi}_{g}v_{g}({\rho}_{g}\mathit{YE}_{gp}+p_{gp}),{\phi}_{l}v_{l}({\rho}_{l}E_{l}+p)\end{array}\right]^{T}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \boldsymbol{F}_{v}=\left[\begin{array}{@{}l@{}}0,0,{\tau}_{g,xx},{\tau}_{g,xy},{\tau}_{l,xx},{\tau}_{l,xy},0,\\[6.0pt] u_{g}{\tau}_{g,xx}+v_{g}{\tau}_{g,xy}+r_{c}(\partial T_{g}/\partial x),0,u_{l}{\tau}_{l,xx}+v_{l}{\tau}_{l,xy}+r_{c}(\partial T_{l}/\partial x)\end{array}\right]^{T}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \boldsymbol{G}_{v}=\left[\begin{array}{@{}l@{}}0,0,{\tau}_{g,yx},{\tau}_{g,yy},{\tau}_{l,yx},{\tau}_{l,yy},0,\\[6.0pt] u_{g}{\tau}_{g,yx}+v_{g}{\tau}_{g,yy}+r_{c}(\partial T_{g}/\partial y),0,u_{l}{\tau}_{l,yx}+v_{l}{\tau}_{l,yy}+r_{c}(\partial T_{l}/\partial y)\end{array}\right]^{T}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \boldsymbol{R}=\left[\begin{array}{@{}l@{}}I_{d},-I_{d},I_{d}u_{l}-F_{dx},I_{d}v_{l}-F_{dy}+{\phi}_{g}(p-{\tau}_{g,{\theta}{\theta}})/y,-I_{d}u_{l}+F_{dx},\\ -I_{d}v_{l}+F_{dy}+{\phi}_{l}(p-{\tau}_{l,{\theta}{\theta}})/y,0,-Q_{d}+Q_{c}-(F_{dx}u_{l}+F_{dy}v_{l})+I_{d}(E_{l}+p/{\rho}_{l})+Q_{p},\\[6.0pt] -Q_{p},Q_{d}+(F_{dx}u_{l}+F_{dy}v_{l})-I_{d}(E_{l}+p/{\rho}_{l})\end{array}\right]^{T}\nonumber\end{eqnarray}$

3. Simulation procedures

3.1. The CE/SE method

The CE/SE method is derived based on the conservation principle of space and time flux, which is applied to the multi-phase detonation calculation. The calculation format of the two-dimensional viscous CE/SE method is as follows, and the derivation process is shown in the literature [12,13]. $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \boldsymbol{U}_{\boldsymbol{i},\boldsymbol{j}}^{\boldsymbol{n}}-{\displaystyle \frac{{\Delta}\boldsymbol{t}}{2}}\boldsymbol{I}_{\boldsymbol{i},\boldsymbol{j}}^{\boldsymbol{n}}={\displaystyle \frac{1}{4}}\left[\left(\boldsymbol{U}+{\displaystyle \frac{{\Delta}\boldsymbol{x}}{4}}\boldsymbol{U}_{\boldsymbol{x}}\right)-{\displaystyle \frac{{\Delta}\boldsymbol{t}}{4}}\left(\boldsymbol{G}_{\boldsymbol{y}}-{\displaystyle \frac{8}{{\Delta}\boldsymbol{x}}}\boldsymbol{F}-\boldsymbol{F}_{\boldsymbol{x}}-{\displaystyle \frac{2{\Delta}\boldsymbol{t}}{{\Delta}\boldsymbol{x}}}\boldsymbol{F}_{\boldsymbol{t}}\right)\right]_{\boldsymbol{i}-1/2,\boldsymbol{j}}^{\boldsymbol{n}-1/2}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle +∼\frac{1}{4}\left[\left(\boldsymbol{U}-\frac{{\Delta}\boldsymbol{x}}{4}\boldsymbol{U}_{\boldsymbol{x}}\right)-\frac{{\Delta}\boldsymbol{t}}{4}\left(\boldsymbol{G}_{\boldsymbol{y}}+\frac{8}{{\Delta}\boldsymbol{x}}\boldsymbol{F}-\boldsymbol{F}_{\boldsymbol{x}}+\frac{2{\Delta}\boldsymbol{t}}{{\Delta}\boldsymbol{x}}\boldsymbol{F}_{\boldsymbol{t}}\right)\right]_{\boldsymbol{i}+1/2,\boldsymbol{j}}^{\boldsymbol{n}-1/2}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle +∼\frac{1}{4}\left[\left(\boldsymbol{U}+\frac{{\Delta}\boldsymbol{y}}{4}\boldsymbol{U}_{\boldsymbol{y}}\right)-\frac{{\Delta}\boldsymbol{t}}{4}\left(\boldsymbol{F}_{\boldsymbol{x}}-\frac{8}{{\Delta}\boldsymbol{y}}\boldsymbol{G}-\boldsymbol{G}_{\boldsymbol{y}}-\frac{2{\Delta}\boldsymbol{t}}{{\Delta}\boldsymbol{y}}\boldsymbol{G}_{\boldsymbol{t}}\right)\right]_{\boldsymbol{i},\boldsymbol{j}-1/2}^{\boldsymbol{n}-1/2}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle +∼\frac{1}{4}\left[\left(\boldsymbol{U}-\frac{{\Delta}\boldsymbol{y}}{4}\boldsymbol{U}_{\boldsymbol{y}}\right)-\frac{{\Delta}\boldsymbol{t}}{4}\left(\boldsymbol{F}_{\boldsymbol{x}}+\frac{8}{{\Delta}\boldsymbol{y}}\boldsymbol{G}-\boldsymbol{G}_{\boldsymbol{y}}+\frac{2{\Delta}\boldsymbol{t}}{{\Delta}\boldsymbol{y}}\boldsymbol{G}_{\boldsymbol{t}}\right)\right]_{\boldsymbol{i},\boldsymbol{j}+1/2}^{\boldsymbol{n}-1/2}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle +∼\frac{1}{4}\left[-\frac{{\Delta}\boldsymbol{t}}{{\Delta}\boldsymbol{x}}\boldsymbol{F}_{\boldsymbol{v}}\left(\boldsymbol{x}_{\boldsymbol{A}_{2}^{\prime }}+{\displaystyle \frac{{\Delta}\boldsymbol{x}}{8}},\boldsymbol{y}_{\boldsymbol{A}_{2}^{\prime }}+\frac{{\Delta}\boldsymbol{y}}{8},\boldsymbol{t}_{\boldsymbol{A}_{2}^{\prime }}+\frac{{\Delta}\boldsymbol{t}}{4}\right)+\right.\frac{{\Delta}\boldsymbol{t}}{{\Delta}\boldsymbol{y}}\boldsymbol{G}_{\boldsymbol{v}}\left(\boldsymbol{x}_{\boldsymbol{A}_{2}^{\prime }}+\frac{{\Delta}\boldsymbol{x}}{8},\boldsymbol{y}_{\boldsymbol{A}_{2}^{\prime }}+\frac{{\Delta}\boldsymbol{y}}{8},\boldsymbol{t}_{\boldsymbol{A}_{2}^{\prime }}+\frac{{\Delta}\boldsymbol{t}}{4}\right)\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle -∼{\displaystyle \frac{{\Delta}\boldsymbol{t}}{{\Delta}\boldsymbol{x}}}\boldsymbol{F}_{\boldsymbol{v}}\left(\boldsymbol{x}_{\boldsymbol{A}_{2}^{\prime }}+\frac{{\Delta}\boldsymbol{x}}{8},\boldsymbol{y}_{\boldsymbol{A}_{2}^{\prime }}-\frac{{\Delta}\boldsymbol{y}}{8},\boldsymbol{t}_{\boldsymbol{A}_{2}^{\prime }}+\frac{{\Delta}\boldsymbol{t}}{4}\right)-\left.\frac{{\Delta}\boldsymbol{t}}{{\Delta}\boldsymbol{y}}\boldsymbol{G}_{\boldsymbol{v}}\left(\boldsymbol{x}_{\boldsymbol{A}_{2}^{\prime }}+\frac{{\Delta}\boldsymbol{x}}{8},\boldsymbol{y}_{\boldsymbol{A}_{2}^{\prime }}-\frac{{\Delta}\boldsymbol{y}}{8},\boldsymbol{t}_{\boldsymbol{A}_{2}^{\prime }}+\frac{{\Delta}\boldsymbol{t}}{4}\right)\right]_{\boldsymbol{i}-1/2,\boldsymbol{j}}^{\boldsymbol{n}-1/2}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle +∼\frac{1}{4}\left[\frac{{\Delta}\boldsymbol{t}}{{\Delta}\boldsymbol{x}}\boldsymbol{F}_{\boldsymbol{v}}\left(\boldsymbol{x}_{\boldsymbol{A}_{4}^{\prime }}-\frac{{\Delta}\boldsymbol{x}}{8},\boldsymbol{y}_{\boldsymbol{A}_{4}^{\prime }}-\frac{{\Delta}\boldsymbol{y}}{8},\boldsymbol{t}_{\boldsymbol{A}_{4}^{\prime }}+\frac{{\Delta}\boldsymbol{t}}{4}\right)-\right.\frac{{\Delta}\boldsymbol{t}}{{\Delta}\boldsymbol{y}}\boldsymbol{G}_{\boldsymbol{v}}\left(\boldsymbol{x}_{\boldsymbol{A}_{4}^{\prime }}-\frac{{\Delta}\boldsymbol{x}}{8},\boldsymbol{y}_{\boldsymbol{A}_{4}^{\prime }}-\frac{{\Delta}\boldsymbol{y}}{8},\boldsymbol{t}_{\boldsymbol{A}_{4}^{\prime }}+\frac{{\Delta}\boldsymbol{t}}{4}\right)\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle +∼\frac{{\Delta}\boldsymbol{t}}{{\Delta}\boldsymbol{x}}\boldsymbol{F}_{\boldsymbol{v}}\left(\boldsymbol{x}_{\boldsymbol{A}_{4}^{\prime }}-\frac{{\Delta}\boldsymbol{x}}{8},\boldsymbol{y}_{\boldsymbol{A}_{4}^{\prime }}+\frac{{\Delta}\boldsymbol{y}}{8},\boldsymbol{t}_{\boldsymbol{A}_{4}^{\prime }}+\frac{{\Delta}\boldsymbol{t}}{4}\right)+\left.\frac{{\Delta}\boldsymbol{t}}{{\Delta}\boldsymbol{y}}\boldsymbol{G}_{\boldsymbol{v}}\left(\boldsymbol{x}_{\boldsymbol{A}_{4}^{\prime }}-\frac{{\Delta}\boldsymbol{x}}{8},\boldsymbol{y}_{\boldsymbol{A}_{4}^{\prime }}+\frac{{\Delta}\boldsymbol{y}}{8},\boldsymbol{t}_{\boldsymbol{A}_{4}^{\prime }}+\frac{{\Delta}\boldsymbol{t}}{4}\right)\right]_{\boldsymbol{i}+1/2,\boldsymbol{j}}^{\boldsymbol{n}-1/2}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle +∼\frac{1}{4}\left[-\frac{{\Delta}\boldsymbol{t}}{{\Delta}\boldsymbol{x}}\boldsymbol{F}_{\boldsymbol{v}}\left(\boldsymbol{x}_{\boldsymbol{A}_{3}^{\prime }}-\frac{{\Delta}\boldsymbol{x}}{8},\boldsymbol{y}_{\boldsymbol{A}_{3}^{\prime }}+\frac{{\Delta}\boldsymbol{y}}{8},\boldsymbol{t}_{\boldsymbol{A}_{3}^{\prime }}+\frac{{\Delta}\boldsymbol{t}}{4}\right)\right.-\frac{{\Delta}\boldsymbol{t}}{{\Delta}\boldsymbol{y}}\boldsymbol{G}_{\boldsymbol{v}}\left(\boldsymbol{x}_{\boldsymbol{A}_{3}^{\prime }}-\frac{{\Delta}\boldsymbol{x}}{8},\boldsymbol{y}_{\boldsymbol{A}_{3}^{\prime }}+\frac{{\Delta}\boldsymbol{y}}{8},\boldsymbol{t}_{\boldsymbol{A}_{3}^{\prime }}+\frac{{\Delta}\boldsymbol{t}}{4}\right)\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle +∼\frac{{\Delta}\boldsymbol{t}}{{\Delta}\boldsymbol{x}}\boldsymbol{F}_{\boldsymbol{v}}\left(\boldsymbol{x}_{\boldsymbol{A}_{3}^{\prime }}+\frac{{\Delta}\boldsymbol{x}}{8},\boldsymbol{y}_{\boldsymbol{A}_{3}^{\prime }}+\frac{{\Delta}\boldsymbol{y}}{8},\boldsymbol{t}_{\boldsymbol{A}_{3}^{\prime }}+\frac{{\Delta}\boldsymbol{t}}{4}\right)-\left.\frac{{\Delta}\boldsymbol{t}}{{\Delta}\boldsymbol{y}}\boldsymbol{G}_{\boldsymbol{v}}\left(\boldsymbol{x}_{\boldsymbol{A}_{3}^{\prime }}+\frac{{\Delta}\boldsymbol{x}}{8},\boldsymbol{y}_{\boldsymbol{A}_{3}^{\prime }}+\frac{{\Delta}\boldsymbol{y}}{8},\boldsymbol{t}_{\boldsymbol{A}_{3}^{\prime }}+\frac{{\Delta}\boldsymbol{t}}{4}\right)\right]_{\boldsymbol{i}-1/2,\boldsymbol{j}}^{\boldsymbol{n}-1/2}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle +∼\frac{1}{4}\left[\frac{{\Delta}\boldsymbol{t}}{{\Delta}\boldsymbol{x}}\boldsymbol{F}_{\boldsymbol{v}}\left(\boldsymbol{x}_{\boldsymbol{A}_{1}^{\prime }}+\frac{{\Delta}\boldsymbol{x}}{8},\boldsymbol{y}_{\boldsymbol{A}_{1}^{\prime }}-\frac{{\Delta}\boldsymbol{y}}{8},\boldsymbol{t}_{\boldsymbol{A}_{1}^{\prime }}+\frac{{\Delta}\boldsymbol{t}}{4}\right)+\right.\frac{{\Delta}\boldsymbol{t}}{{\Delta}\boldsymbol{y}}\boldsymbol{G}_{\boldsymbol{v}}\left(\boldsymbol{x}_{\boldsymbol{A}_{1}^{\prime }}+\frac{{\Delta}\boldsymbol{x}}{8},\boldsymbol{y}_{\boldsymbol{A}_{1}^{\prime }}-\frac{{\Delta}\boldsymbol{y}}{8},\boldsymbol{t}_{\boldsymbol{A}_{1}^{\prime }}+\frac{{\Delta}\boldsymbol{t}}{4}\right)\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle -∼\frac{{\Delta}\boldsymbol{t}}{{\Delta}\boldsymbol{x}}\boldsymbol{F}_{\boldsymbol{v}}\left(\boldsymbol{x}_{\boldsymbol{A}_{1}^{\prime }}-\frac{{\Delta}\boldsymbol{x}}{8},\boldsymbol{y}_{\boldsymbol{A}_{1}^{\prime }}-\frac{{\Delta}\boldsymbol{y}}{8},\boldsymbol{t}_{\boldsymbol{A}_{1}^{\prime }}+\frac{{\Delta}\boldsymbol{t}}{4}\right)+\left.\frac{{\Delta}\boldsymbol{t}}{{\Delta}\boldsymbol{y}}\boldsymbol{G}_{\boldsymbol{v}}\left(\boldsymbol{x}_{\boldsymbol{A}_{1}^{\prime }}-\frac{{\Delta}\boldsymbol{x}}{8},\boldsymbol{y}_{\boldsymbol{A}_{1}^{\prime }}-{\displaystyle \frac{{\Delta}\boldsymbol{y}}{8}},\boldsymbol{t}_{\boldsymbol{A}_{1}^{\prime }}+{\displaystyle \frac{{\Delta}\boldsymbol{t}}{4}}\right)\right]_{\boldsymbol{i}+1/2,\boldsymbol{j}}^{\boldsymbol{n}-1/2}\nonumber\end{eqnarray}$

3.2. The processing of the source term

The source term is rigid since the characteristic time of the chemical reaction is much less than the characteristic time of convection. The fourth-order Runge–Kutta method is applied to deal with the source term. First, the CE/SE method is used to solve $(\boldsymbol{U})_{j}^{n}$ without considering the impact of the source term. Second, $(\boldsymbol{U})_{j}^{n}$ is used to solve the ordinary differential equation d U ∕dt = R .

3.3. The calculation of the plasma jet

The plasma jet is solved by the coupled fluid mechanics equation and the Maxwell equation [14,15], and a two-dimensional CE/SE method is used to calculate the magnetohydrodynamic equations containing the convection term and the diffusion term in this paper.

4. Initial conditions and boundary conditions

4.1. Initial conditions

The plasma generator is filled with air at atmospheric temperature and pressure, and the air is ionized to form the plasma, which is injected from section JK into the detonation tube to ignite after electricity is applied. The gasoline/air mixture is uniformly injected into the detonation tube at a certain chemically equivalent ratio of 1.

4.2. Boundary conditions

An axisymmetric boundary condition is used in section AB, and section BE uses the non-reflecting free boundary condition of the CE/SE method. Wall reflection boundary conditions are adopted in sections AF, FJ and EK. Segment JK uses an outflow boundary condition during plasma jet ignition and a solid wall boundary condition when the plasma jet stops.

5. Results and analysis

5.1. Effects of viscosity on detonation parameters

Figure 2 shows the distribution of the plasma mass fraction in the PDE tube with plasma jet ignition. The plasma mass fraction near the ignition section is 1 during the ignition process (as shown in Fig. 2(a)). The gasoline/air mixtures in the detonation tube near the ignition section are ignited and constantly spread around by the plasma jet due to its high energy (as shown in Fig. 2(b) and (c)). The ignition time is so short that the plasma injected into the detonation tube is limited. When the plasma jet is stopped, the plasma mass fraction decreases to 0.4 at the time t = 150 μs. The plasma propagates along the outlet of the detonation tube with the gas phase. At the time t = 190 μs, the plasma mass fraction at the position of the ignition wall rapidly decreases to 0.2 (as shown in Fig. 2(d) and (e)). The plasma is transmitted to the gas phase in the form of radiation. The plasma is mixed with the gas phase when the temperature of the plasma is reduced to the gas temperature, and the plasma mass fraction is close to 0 (as shown in Fig. 2(f)).

Fig. 2.

Results show the distribution of the plasma mass fraction in the PDE tube with plasma jet ignition.

Figure 3 shows the Mach number distributions in the detonation tube at different moments. The gasoline/air mixtures are ignited and driven by the plasma jet. A stable high-temperature and high-pressure region is formed at the wall of the detonation tube. The gasoline droplet is vaporized and mixed with the air and ignited rapidly by the high-temperature gas after the plasma jet is stopped. A circular compression wave is formed and propagates to the unburned region at a higher speed. At the time t = 27.1 μs, the Mach number of the wave surface reaches 0.65 (as shown in Fig. 3(a)). The compression wave propagates to the right under the constraint and extrusion of the left-end thrust wall. At the time t = 67.3 μs, collision occurs when the compression wave propagates to the central axis, and a local high-pressure point is formed. Meanwhile, the pressure and temperature rise sharply, and the velocity reaches the supersonic speed of Ma = 1.49 (as shown in Fig. 3(b)). A transverse wave is formed by the reflection of the compression wave and spreads to the mouth of the tube, after which it collides to form a reflection wave when it spreads to the wall position at the time t = 130 μs. The velocity of the wave surface becomes supersonic (as shown in Fig. 3(c)). The shock wave constantly collides and reflects during propagation. The unburned chemical reaction rate is accelerated by the upward and downward movements of the reflected wave, and the leading shock intensity increases continuously such that a detonation wave is formed eventually at the time t = 271 μs, and the velocity reaches a supersonic speed of Ma = 1.2 (as shown in Fig. 3(d)).

Fig. 3.

Results show the Mach number distributions in the detonation tube at different moments.

Figure 4 shows the spatial distributions of the parameters after the formation of a steady detonation wave near the tube wall under the conditions of two kinds of control equations. Figure 4(a) ∼ Fig. 4(c) show the variations in the axial velocity, radial velocity and pressure along the x axis. The detonation wave detonates successfully and propagates steadily under the condition of plasma jet ignition. The flow fields of the two governing equations, namely, the Euler equation and the N–S equation, are compared. It can be seen that the axial velocity and the pressure variations occur in the same direction based on Fig. 4(a) and (c). The detonation wave structures of the two control equations are almost the same. The pressure and axial velocity of the Euler equation are slightly higher than those under the viscous calculation condition (N–S equation). As seen from Fig. 4(b), the radial velocity is very small, and the radial effect is not obvious at the location near the tube wall under the two calculation conditions, and the radial velocity decreases under the action of viscosity.

Fig. 4.

Results show the spatial distributions of the parameters after the formation of a steady detonation wave near the tube wall under the conditions of two kinds of control equations.

Figure 5 shows the spatial distributions of the parameters after the formation of a steady detonation wave near the tube wall at 0.02 m under the conditions of two kinds of control equations. It can be seen that the variation trends of the axial velocity and pressure at a distance of 0.02 m from the wall are almost the same as those near the wall, while the radial velocity is constantly changing. The compression wave constantly reflects and moves up and down under the action of the wall before a steady detonation wave is formed, and the trend of the radial velocity is similar.

Fig. 5.

Results show the spatial distributions of the parameters after the formation of a steady detonation wave near the tube wall at 0.02 m under the conditions of two kinds of control equations.

The propagation process of the detonation wave is basically the same with or without considering the viscous effect, and the change in the pressure is in accordance with the change law of the gas velocity. The resistance increases and energy is lost during the propagation of the detonation wave due to the effect of viscosity such that the velocity decreases and the intensity of the detonation wave decreases. The viscous effect does not affect the propagation trend of the detonation wave; rather, it affects only the propagation intensity and velocity. The model based on viscosity with a plasma jet can reflect the characteristics of the deflagration-to-detonation transition more realistically.

5.2. Effects of the ignition temperature and time on the deflagration-to-detonation transition

Figure 6 shows the velocity distribution curves at a position of 0.02 m far from the tube wall under different ignition conditions. It can be seen that the axial velocities of the four ignition conditions are basically the same. The axial velocity increases rapidly at the location of the detonation wave, and the axial velocity decreases continuously after the detonation wave propagates. The radial velocity fluctuates continuously because the compression wave moves up and down under the action of the wall in the early stage of plasma jet ignition. The fluctuation of the radial velocity slows down when the compression wave propagates towards the nozzle. The amplitude of the radial velocity decreases with decreases in the jet time and the stable distance. When the ignition temperature is T = 5000 K, the jet time is t = 2 μs, 1 μs and 0.5 μs (as shown in Fig. 6(a) ∼ Fig. 6(c)). With a decrease in the jet temperature, the radial velocity fluctuations become larger, and the maximum velocity reaches 150 m/s while the stable distance decreases. When the jet time is t = 1 μs, the jet temperatures are 4000 K and 5000 K (as shown in Fig. 7(b) and (d), respectively). The axial velocity is much larger than the radial velocity, and the radial effect is not significant at the location of the detonation wave.

Fig. 6.

Results show the velocity distribution curves at a position of 0.02 m far from the tube wall under different ignition conditions.

Table 1 shows the effects of the temperature and time of the jet on detonation. With an increase in the jet time, the DDT distance of the formation of stable detonation decreases, the rate of decrease is reduced, and the peak pressure is basically unchanged when the jet temperature is 5000 K. It can be seen that the initial plasma jet time does not affect the peak pressure. When the initial ignition plasma jet has fully ignited the explosive mixture, increasing the jet time has no obvious effect on shortening the DDT distance. The DDT distance is 0.457 m when the jet temperature is T = 4000 K, while the distance is 0.427 m when the temperature is T = 5000 K when the jet time is 1 μs. The DDT distance is shortened by 6.6%, which indicates that increasing the initial jet temperature can significantly shorten the DDT distance.

Fig. 7.

Results show the droplet radius distributions at different moments.

Table 1

Effects of the temperature and time of the jet on detonation

Temperature (K)	Time (μs)	Detonation formed	Peak pressure (MPa)	DDT distance (m)
5000	2	Yes	1.89	0.425
5000	1	Yes	1.88	0.427
5000	0.5	Yes	1.89	0.439
4000	1	Yes	1.84	0.457

5.3. Effect of the initial droplet radius on the combustion detonation

To study the effect of the initial droplet radius on the combustion process, the two-dimensional viscous CE/SE method is used to study the DDT of different initial droplet radii under a certain plasma jet ignition condition. Figure 7 shows the droplet radius distributions at different moments. The droplet radius R = 50 μm represents the initial droplet state when it has not been evaporated or stripped. Meanwhile, the droplet radius R = 0 μm means that vapor stripping is complete. The plasma is transmitted to the gas phase in the form of radiation, and a high-temperature and high-pressure area is formed at the wall end of the detonation tube after the ignition of the plasma jet. At the time t = 27.1 μs, some of the gasoline droplets around the high-temperature and high-pressure region are evaporated and ignited after mixing with the air (as shown in Fig. 7(a)). At this moment, the rate of droplet evaporation is slower. After the ignition of the plasma jet ends, a circular compression wave is formed on the wall of the detonation tube and propagates to the unreacted area. The compression wave moves up and down continuously under the wall surface, and the droplet is rapidly evaporated. The leading shock wave intensity is accelerated by the chemical reaction rate (as shown in Fig. 7(b) and (c)). At the time t = 271 μs, the droplets of gasoline at the position of the detonation wave quickly evaporate and react with oxygen in the air. The gasoline droplet radius falls rapidly from R = 50 μm to R = 0 μm (as shown in Fig. 7(d)).

The effects of different initial droplet radii on the detonation parameters are shown in Table 2. It is shown that when the initial droplet radius is less than 50 μm, the peak value of the detonation pressure increases with an increase in the droplet radius, and when the droplet radius is greater than 50 μm, the peak value of the detonation pressure decreases with an increase in the droplet radius. However, the DDT time and distance increase with an increase in the initial droplet radius. The interactions between the gas and droplets are caused mainly by the stripping effect of the droplet boundary layer and the evaporation effect caused by the difference in the temperature. The larger the droplet radius is, the longer the time required for the stripping and evaporation of fuel droplets at the same location. In a certain chemically equivalent gasoline/air mixture, the number of droplets in the unit volume decreases with an increase in the droplet radius such that the total area in contact with the oxygen in the air is smaller. The energy released is reduced due to the weak chemical reaction intensity. The energy intensity is weakened during the propagation of the detonation wave, and the time of energy accumulation required to form a stable detonation wave to stabilize increases; thus, the time for the detonation wave to stabilize increases. A stable detonation wave cannot form when the droplet radius is greater than 150 μm. This is because when the droplet radius is too large, the stripping time of the droplet and the evaporation time will be too long, which will greatly reduce the efficiency of the chemical reaction between the fuel steam and oxygen. It is not conducive to the full combustion of fuel, which causes the energy released to be too small to compensate for the energy required for the propagation of shock waves, and thus, no detonation can be formed.

Table 2
Effect of the initial droplet radius on detonation

Initial droplet radius (μm) Detonation formed Peak pressure (MPa) DDT distance (ms) DDT distance (m)

25 Yes 1.68 0.30 0.366

50 Yes 1.90 0.43 0.435

75 Yes 1.83 >0.65 >0.6

100 Yes <1.80 >1 >1

150 No — —- —

Initial droplet radius (μm)	Detonation formed	Peak pressure (MPa)	DDT distance (ms)	DDT distance (m)
25	Yes	1.68	0.30	0.366
50	Yes	1.90	0.43	0.435
75	Yes	1.83	>0.65	>0.6
100	Yes	<1.80	>1	>1
150	No	—	—-	—

5.4. Comparison with partial calculation and experiment results

To verify the strong interruption ability of the CE/SE method to capture shock wave in a PDE flow field with plasma jet ignition, the PDE ignition process with wall ignition is numerically simulated. The calculation model is a detonation tube with a diameter of 80 mm and a length of 1 m. Since the computational model is an axisymmetric model, only half of the area is calculated. The number of grids in the calculation is 1000 × 40. Meanwhile, an experimental study of a PDE tube with the same area is carried out. The detonation tube is filled with a gasoline/air mixture, and the plasma ignition device is used for ignition in the experiment.

Figure 8 shows the pressure-time diagram of the detonation wave on the wall at the distance from the thrust wall d = 0.9 m. Figure 8(a) and (b) are the diagrams obtained from the calculation and measured by the pressure sensor in the experiment, respectively. It can be seen from these diagrams that the wave forms and trends of the two are basically the same. The calculated results are in good agreement with the experimental results. The width of the detonation wave front captured by the numerical calculation is between grid points 2 ∼ 4, and the wave front is steep, which indicates that the detonation wave is captured more effectively by the CE/SE method. Clearly, the application of the CE/SE method to calculating the multi-phase detonation of plasma jet ignition is feasible, and the result is credible.

Fig. 8.

Results show the pressure-time diagram of the detonation wave on the wall at the distance from the thrust wall d = 0.9 m. Figure (a) and (b) shows the diagrams obtained from the calculation and measured by the pressure sensor in the experiment, respectively.

6. Conclusions

The two-dimensional viscous CE/SE method is used to calculate the viscous multi-phase detonation of plasma jet ignition. The N–S equation and the Euler equation are used as the governing equations. Variations in the detonation parameters under the conditions of plasma jet ignition are considered. The effects of the plasma jet ignition temperature, ignition time and initial droplet radius on the DDT process are analysed. The conclusions are as follows:

The viscous effect does not affect the propagation trend of detonation waves; rather, it affects only the propagation intensity and velocity. The resistance during the detonation wave propagation increases, the velocity decreases, and the intensity of the detonation wave decreases due to the viscous effect of the tube wall.

It is possible to significantly shorten the DDT distance of stable detonation by increasing the temperature and time of the initial plasma jet ignition. When the plasma jet has fully ignited the explosive mixture, increasing the jet time has little effect on shortening the DDT distance.

Under the conditions with a plasma jet temperature of 5000 K and a jet time of 1 μs, when the droplet radius is less than 50 μm, the peak pressure of the detonation wave increases with an increase in the droplet radius. When the droplet radius is greater than 50 μm, the peak pressure of the detonation wave decreases with an increase in the droplet radius. When the droplet radius is more than 150 μm, the droplet evaporation and stripping times are too long, which is not conducive to the combustion of fuel. The energy released by combustion cannot provide the energy needed for shock wave propagation, and a stable detonation wave cannot be formed.

Footnotes

Acknowledgements

This study was supported by the National Natural Science Foundation of China (No. 11802039, No. 51605046) and Jiangsu Natural Science Foundation of China (No. BK20160406).

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Viscous multi-phase detonation of a pulse detonation engine with plasma jet ignition

Abstract

Keywords

1. Introduction

2. Model and governing equations

2.1. Computational model

3. Simulation procedures

3.1. The CE/SE method

3.2. The processing of the source term

3.3. The calculation of the plasma jet

4. Initial conditions and boundary conditions

4.1. Initial conditions

4.2. Boundary conditions

5. Results and analysis

5.1. Effects of viscosity on detonation parameters

Table 2 Effect of the initial droplet radius on detonation Initial droplet radius (μm) Detonation formed Peak pressure (MPa) DDT distance (ms) DDT distance (m) 25 Yes 1.68 0.30 0.366 50 Yes 1.90 0.43 0.435 75 Yes 1.83 >0.65 >0.6 100 Yes <1.80 >1 >1 150 No — —- —

Footnotes

Acknowledgements

References

Table 2
Effect of the initial droplet radius on detonation

Initial droplet radius (μm) Detonation formed Peak pressure (MPa) DDT distance (ms) DDT distance (m)

25 Yes 1.68 0.30 0.366

50 Yes 1.90 0.43 0.435

75 Yes 1.83 >0.65 >0.6

100 Yes <1.80 >1 >1

150 No — —- —