1 Introduction

The study of nozzle has a long history. Lin and Pu^[1] obtained an exact solution for the incompressible potential flow through a two-dimensional Laval nozzle through a careful examination of the variation of the velocity along the centerline and the contour of a Laval nozzle. Maddahian et al.^[2] investigated the development of turbulent swirling flow in the entrance region of a conical nozzle with the boundary layer integral method. The nozzle of pulse detonation rocket engine (PDRE) based on detonation and combustion is a powerful device to improve the performance of PDRE by converting the thermodynamic energy under high temperature and high pressure gas discharged from the combustion chamber into kinetic energy. However, the shape design of PDRE's nozzle becomes very complicated due to the intermittent style of the PDRE, which makes the shock waves and expansion waves in the flow of nozzle unsteadily move, evolve, and interact with each other^[3]. Therefore, in order to facilitate the nozzle into PDRE, a lot of world-widely experimental and numerical studies have been carried out on the exhaust nozzle of pulse detonation engine (PDE) in recent years.

Allgood et al.^[4] conducted an experimental study on a multi-cycle PDE by using various converging and diverging nozzles under different filling fractions with hydrogen/oxygen as the fuel and oxidant, and showed that the optimum nozzle area ratio was a function of the filling fraction. Yan et al.^[5] presented an experimental study of the effects of various injectors and nozzles on the PDRE performance, and showed that the highest thrust augmentation of 27.3% could be obtained through adjusting the contraction ratio or the expansion ratio of the nozzle. Wang et al.^[6] experimentally studied 12 different configurations of tail nozzles on the two-phase PDRE detonation characteristics and propulsion performance. They showed that the Laval nozzle could generate the highest thrust augmentation for the full filled rate and frequency of 10 Hz, the export area ratio of the Laval nozzle had a great effect on the PDE thrust augmentation, while the length ratio of convergence to expansion had little effect on the thrust augmentation. Zhang et al.^[7] studied the effects of nozzles on a valveless PDRE without the purge process with oxygen-enriched air and gasoline as the oxidant and fuel. They showed that a maximum thrust rise of 25% could be obtained when the nozzles were utilized, but the expansion ratio almost had no effect on the operation of the valveless PDRE if the expansion ratio of the diverging nozzles was less than five and the thrust augmentation of the converging or diverging nozzle was only affected by the filled rate.

Yungster and Perkins^[8] studied the effects of nozzle shape and filled condition on the PDE performance with the conservation element/solution element (CE/SE) method. Li et al.^[9] studied the effects of nozzle shape and filled condition on the PDE performance with the grid adaptation and finite-rate kinetic chemical reaction model. Qin et al.^[10] simulated different nozzle structures of PDE, and showed that the diverging nozzle could produce the maximum instantaneous thrust peak and specific impulse.

Therefore, the optimum nozzle geometries are different under different initial conditions. It is time-consuming and laborious to establish the optimal nozzle geometry experimentally or numerically. Therefore, it is necessary to develop a feasible and simple theoretical design model. In 2002, Talley and Coy^[11] proposed a constant volume cycle (CVC) analytical model of PDRE, in which the constant volume combustion process was used to replace the detonation combustion process, i.e., the mass of gas discharged from the chamber in the detonation process could be ignored since the propagation time of detonation wave in the PDRE was very short. Therefore, the CVC model^[12] can be used to describe the working process of PDRE. In 2014, Fan and Li^[3] built a nozzle design method with the CVC model. However, this model will lead to somewhat large error and uncertainty for the optimum contraction ratio of the nozzle since the contraction ratio depends on the experimental or empirical exhaust time.

Based on the previous studies, in this paper, we will present a new model to improve the CVC model by introducing the filling duty cycle and considering the detonation frequency, in which the dependence of nozzle configuration design on experiment and experience is eliminated theoretically. Then, various nozzle configurations are designed with the improved CVC model. Finally, the feasibility and reliability of the improved CVC model and design method are verified through the single-cycle accurate numerical simulation.

2 CVC analysis of nozzle performance 2.1 CVC model

In the CVC model, the basic PDRE cycle is assumed to consist of a constant pressure (CP) filling process, a constant volume (CV) combustion process, a CV blowdown process, and a CP blow down process (see Fig. 1).

Specific impulse is an important parameter to evaluate engine performance. It reflects the extent of engine energy conversion, and is defined as the thrust produced by unit weight flow propellant. In this paper, the combustor transient pressure, temperature, and density are, respectively, represented by P, T, and ρ, and the subscript "0" denotes the initial condition after the CV combustion but before CV blowdown, and the subscript "e" denotes the plane at the nozzle exit. The dimensionless density in the combustor (r) and the density ratio for the nozzle exit to the combustion chamber (r_e) are defined as follows:

(1)

The combustor density ratio at the end of CV blowdown is

(2)

where P_f and ρ_f are the filling pressure and filling density, respectively, and the constant parameter γ is the specific heat ratio.

The nozzle area expansion ratio (the nozzle exit area A_e to the nozzle throat area A^*) has the following relation:

(3)

where

(4)

The specific impulse in CP blowdown (I_{sp, cv}) can be derived as follows:

(5)

where ι is the dimensionless impulse of the CV blowdown process, and can be written as

(6)

in which

c₀ is the local sound speed, and g is the local gravitational acceleration.

Then, the CV blowdown time and the CP blowdown time can be, respectively, expressed as

(7)

(8)

Considering the volume CV and CP blowdown processes, we can obtain the total specific impulse of the PDRE cycle operation with fixed nozzle design as follows:

(9)

The average thrust is

(10)

where

Finally, the contraction ratio of nozzle can be obtained by taking a transformation of Eq. (7) as follows:

(11)

where L is the length of the detonation tube.

A nozzle design method is built in Ref. [1] with the contraction ratio ε' given by Eq. (11). However, since the exhaust time in Eq. (11) is determined from experience, the prediction ability of the model will be seriously affected. Therefore, we want to develop an improved model to overcome the above defect.

2.2 Improved CVC model

Because the actual CP blowdown time should include the filling time, in our improved CVC model, the time of CV combustion in the PDRE cycle can be negligible. We let the actual CP blowdown time be α times the CP blowdown time. Then, the duty ratio (S) of CP blowdown can be written as

(12)

(13)

where f is the detonation frequency.

Introducing Eqs. (12) and (13) into Eq. (11), we can obtain ε' as follows:

(14)

It follows that

(15)

Substituting it into Eq. (9) yields

(16)

From Eq. (10), we have

(17)

where

Equations (16) and (17) show that the total specific impulse (I_sp) and the average thrust (F_avg) are functions of the frequency (f), and they both increase with the frequency. These results are in accord with the actual experiment, and are better than the conclusions obtained by Wintenberger et al.^[13], where it was shown that the specific impulse of detonable mixture and fuel specific impulse would not change with the operating frequency and the volume of detonation tube.

The above results confirm that the improved CVC model relates the impulse and thrust with the frequency by introducing the duty ratio and the time of the filling process, and thus can overcome the defect of the empirical time in the original CVC model and is more suitable for the design of nozzle configuration.

2.3 Relationship between propulsion and frequency

The expansion ratio of the nozzle always matches with the pressure ratio and the Mach number Ma becomes 1 when the nozzle expansion ratio is reduced to 1. To avoid the appearance of shock waves and expansion waves in the tube, the range of r_cv should be

(18)

Therefore, the range of the corresponding frequency can be deduced through Eq. (15) as follows:

(19)

Therefore, when the initial pressure, the ambient pressure, the filling pressure, and the working frequency of the combustion chamber are fixed, there will be an optimal nozzle expansion ratio corresponding to the optimum specific impulse which can be obtained by numerical calculations.

For example, we assume that P₀ =1.013× 10⁶ Pa, T₀ =2 800 K, ε =2, L=1 m, and α =4. The variations of the optimal nozzle expansion ratio, the average thrust coefficient, and the specific impulse of the PDRE with the operating frequency can be obtained by the numerical iteration algorithm (see Figs. 2-4).

From Fig. 2, we can see that, when the ratio ϕ₀ of the initial pressure to the ambient pressure is constant, the optimal nozzle expansion ratio increases with the increase in the working frequency. When the frequency f is constant, the optimal nozzle expansion ratio decreases with the increase in the ambient pressure. When φ₀ decreases, f increases, which means that, in order to prevent the shock from appearing in the nozzle, the operating frequency of the fixed nozzle should be high.

As shown in Figs. 3 and 4, with the increase in the operating frequency, the average thrust coefficient and specific impulse of the optimal nozzle design gradually increase. Under the vacuum condition, the extrema of the optimal thrust coefficient and the specific impulse are 2.236 9 s and 321.01 s, respectively. With high operating frequency, the average thrust coefficient of PDRE in the optimal nozzle design will be significantly improved and the specific impulse will increase when the ambient pressure is reduced.

2.4 Nozzle design method based on improved CVC model

By the improved theoretical model, if the detonation frequency, filling pressure, and ambient pressure are known, the contraction ratio (ε') of the nozzle can be obtained through Eq. (14). Then, substituting the contraction ratio (ε') and the detonation frequency (f) into Eq. (16), we can obtain the expansion ratio (ε) corresponding to the maximum specific impulse. Finally, the average thrust (F_avg) can be obtained by Eq. (17) with the related parameters. In this paper, the thermodynamic parameters after CP combustion, CV combustion, and detonation can be calculated by the chemical equilibrium and applications (CEA) code developed at the NASA Lewis Research Center^[14]. The flowchart of the nozzle design is shown in Fig. 5.

3 Numerical study of nozzle propulsion performance 3.1 Physical model

In this paper, hydrogen and air are, respectively, chosen as the fuel and oxidant, which are assumed in the stoichiometric ratio. The PDRE is modeled by a single tube with a 10 mm inner diameter and a total length of 120 mm. With the nozzle length of 10 mm (the length of the contraction and expansion section is 5 mm), we can obtain the ratio of the nozzle length to the tube length as follows:

Therefore, the effect of the nozzle length on the partial filling can be negligible^[15], and the performance change of PDRE caused by the change of the filling fraction can be neglected. When the filling pressure and ambient pressure are both at one atmospheric pressure and the filling temperature and the ambient temperature are both 300 K, the thermodynamic parameters of the products after combustion can be obtained as the values referenced in Ref. [3] (see Table 1). The nozzle contraction-expansion ratio and nozzle structure parameters are set as the values shown in Table 2.

3.2 Numerical simulation and validation

This paper intends to use the numerical simulation to verify the predicted results from the improved CVC model. Based on the finite volume method and the realizable k-ε turbulence model, the PISO algorithm is adopted to solve the two-dimensional axisymmetric field. The spatial discretization uses the second-order upwind scheme. The time steps 1× 10^-8 s and 5× 10^-8 s are adopted for the detonation combustion and exhaust process, respectively. A finite rate chemical reaction model is used, and the high energy direct ignition is used in the numerical simulation.

The detonation tube calculation field is shown in Fig. 6, where the outer region length of the flow field is 20 times the inner diameter, the width is 10 times the inner diameter, and the quadrilateral meshes with 0.1 mm are adopted. The detonation tube is filled with hydrogen/air mixed gases, and the initial pressure and temperature of the flow field are 1.013×10⁵ Pa and 300 K, respectively. The ignition radius is 1 mm, and the temperature and pressure of the gas are T=1 500 K and P=1 MPa, respectively. The detonation tube is located along the axis from the closed end of 5 mm.

The single-circle numerical simulation in the straight nozzle is firstly conducted to verify the numerical method. Figure 7 shows the variation of the pressure at different locations with time, which means that the detonation wave is formed between 15 mm and 30 mm, the peak pressure is about 1.8 MPa, and the pressure behind the detonation wave is 0.55 MPa. When the detonation wave propagates into the nozzle, it degenerates into strong shock wave without a chemical reaction, and the peak pressure decreases to 1.28 MPa. Figure 8 shows the contour map of velocity and pressure at 0.2×10^-3 s, 0.3×10^-3 s, and 0.4×10^-3 s. The upper half of Fig. 8 is a velocity contour, and the lower half is the pressure contour. When exhaust is expelled from the outlet, a spherical shock wave will be produced. The produced wave will then expand around the outlet, and move into the tube as time goes on.

The characteristic parameters of detonation wave, such as the detonation velocity (U_C-J), the detonation temperature (T₁), and the pressure near the thrust wall (P₂), are calculated by the numerical simulation. The obtained results are compared with those calculated by the STANJAN software^[16] under the same condition (see Table 3). It indicates that the relative deviations between the numerical results and the theoretical values are below 8%. The numerical results accord with the conclusion that the detonation velocity obtained in the experiment ranges from 1 520 m/s to 2 290 m/s^[17], which shows that the numerical simulation is reliable.

3.3 Verification of the improved CVC model with numerical simulation

In the paper, the performance of the nozzle can be evaluated from thrust and specific impulse. The instantaneous thrust generated by PDRE during the exhaust is defined as follows:

(Ⅰ)

where , ν_e, A_e, and P_e are the mass flow rate, the flow velocity, the area, and the flow pressure at the nozzle exit, respectively, and P_∞ is the ambient pressure (1.013×10⁵ Pa).

The specific impulse is

(Ⅱ)

where ρ₀, V, and g are the initial density of the combustible mixture, the tube volume, and the gravity acceleration, respectively, and t is the duration from the detonation wave into the nozzle with the thrust wall pressure down to one atmospheric pressure (gauge pressure to zero).

Tables 4-6 show the comparisons between the predicted values of the improved CVC model and the numerical simulation results. From the tables, we can see that the relative deviations of the average thrust values are below 15% (see Table 4), the relative deviations of the specific impulse values are below 6% (see Table 5), and the relative deviations of the exhaust time are below 9% (see Table 6). From the results, it can be seen that the variation trend of the PDRE performance parameters is consistent with the variation of the nozzle contraction/expansion ratio. Although the values obtained from the numerical simulation are different from the theoretical data, they are acceptable to estimate the PDRE performance parameters by the improved CVC analysis theory. Therefore, the results verify the reliability of the improved CVC model and the nozzle design method.

3.4 Performance analysis of PDRE

Figures 9-11 show the profiles of the thrust values, the specific impulse values, and the exhaust time with different contraction/expansion ratios.

Figure 9 shows that the expansion nozzle has better thrust performance. The larger the contraction ratio is, the smaller the thrust value is. Moreover, the maximum thrust is produced when the contraction ratio is 1.02 and the expansion ratio is 1.2, which are consistent with the theoretical values obtained by the nozzle design of the improved CVC model.

From the theoretical analysis of the nozzle design with the improved CVC model, we have that the nozzle contraction ratio has no effect on the specific impulse value when the filling pressure is fixed (see Fig. 10(a)), the expansion ratio has a great effect on the specific impulse value, and there is an optimal value for the specific impulse (see Fig. 10(b)). This has been validated by the results obtained by the CFD numerical simulation.

Figure 11 shows the relationship between the CV blowdown time and the ratio of contraction to expansion. The time, in which the pressure of the thrust wall falls down to the ambient pressure, is independent of the expansion ratio. However, the larger the contraction ratio is, the longer the exhaust time is needed, and the smaller the corresponding detonation frequency is. It also shows that there is an optimum nozzle contraction ratio for the fixed operating frequency, which makes the pressure of the detonation tube decrease properly to avoid blowdown gas over-expansion or under-expansion.

4 Conclusions

In this paper, an improved CVC model and a theoretical design method are developed for PDRE nozzle. The conclusions can be summarized as follows:

(ⅰ) In this model, the relationship between the nozzle contraction ratio and the operation frequency is obtained by introducing the duty ratio, which can eliminate the defect of the empirical data in the theoretical design.

(ⅱ) The improved CVC model can be used for the preliminary design of the nozzle configuration of PDRE. The higher the operating frequency of PDRE is, the shorter the exhaust time is. Therefore, smaller nozzle contraction ratio should be designed, which can make high pressure gas exhaust from the detonation tube as soon as possible. The lower the ambient pressure is, the larger the expansion ratio of the nozzle should be. When the ambient pressure is 1.013×10⁵ Pa, the optimal expansion ratio will be less than 2.26.

(ⅲ) The improved model can be used to analyze the performance of PDRE conveniently and quickly. The contraction ratio of nozzle has a significant effect on the thrust performance of PDRE, and has little effect on the specific impulse. However, the expansion ratio has an effect on both the parameters. When the ambient pressure is reduced to vacuum, the extremum of the optimal thrust coefficient is 2.236 9, and the extremum of the specific impulse is 321.01 s.