Numerical investigation of the acceleration performance of a thermally choked ram accelerator

Abstract

The flow fields of the thermally choked mode of ram accelerators were studied numerically. The goal of this study was to investigate the effects of the gas reaction rate, the velocity and the projectile shapes on the performances of ram accelerators. The thermally choked mode was demonstrated to occur only when the variations in the reaction rate and projectile velocity were within certain limits. The use of a boat-shaped projectile can extend this limit. The flame front can stand at both its shoulder and the base, and the thrust is the greatest at its shoulder.

Introduction

Conceived by Hertzberg et al. ¹ in 1983, the ram accelerator is a novel projectile launcher that uses chemical energy to accelerate projectiles to hypersonic speeds. The ram accelerator is similar to a supersonic air-breathing ramjet in the tube and consists of a stationary tube and a projectile. The projectile is shaped conically, both in the forebody and in the afterbody, which is analogous to the centre region of the ramjet, and travels in the tube, which functions in a manner similar to the outer cowling of a ramjet. Unlike the traditional launch,^2,3 the launch tube of a ram accelerator is filled with premixed combustible gas that is ignited by the projectile; the combustion generates a high-pressure area at the base of the projectile, thereby accelerating the projectile to a high speed.

There are two different primary modes of the ram accelerator propulsive cycles, which are distinguished by the velocity regime: the thermally choked mode and the superdetonative mode; the first mode operates at subdetonative velocities and its propulsive cycle acts as if the flow was thermally choked behind the projectile, while the superdetonative propulsive mode operates above the propellant Chapman–Jouguet (CJ) detonation speed.^4–6

The thermally choked mode is the first stage of the projectile during propulsion; many experiments have been performed to study this mode. Higgins et al. ⁷ investigated the operational limits of the thermally choked ram accelerator and determined that a certain minimum heat release is required to maintain the combustion stay on the projectile; at the same time, if the heat release is too high, the projectile will undergo the process of unstart. Hertzberg et al. ² performed ram accelerator experiments and determined the thrust variation with the velocity of the projectile. Their experimental results were determined to be in good agreement with the theoretical model for the thermally choked propulsive cycle that the velocity regime below 85% of the CJ detonation speed; however, as the velocity approached the CJ detonation speed, the acceleration was found to be well above the theoretical prediction. The experimental results obtained by Kull et al. ⁸ demonstrated that all of the propellant mixture used in the thermally choked propulsive mode allowed the projectiles to approach the detonation velocity of the mixture; in fact, in several experiments, the projectiles actually passed through the detonation velocity and continued to accelerate. Sasoh et al. ⁹ experimentally investigated the effect of the obturator on the subdetonative ram accelerator at the start and found that the perforated obturators generate weaker shock systems and that the chances for propellant ignition are lower; however, solid obturators were found to induce stronger shock systems and increase the chances for propellant ignition, thereby increasing the probability of inducing the wave unstart.

Numerical simulations have also been performed to predict the characteristics of the ram accelerator. Higgins ¹⁰ reported that one-dimensional models of performance are adequate to predict the acceleration of the ram accelerator. Based on the modified one-dimensional model to account for real gas effects, Bauer et al. ¹¹ obtained the variation of thrust with the projectile velocity of a ram accelerator, which agrees well with the experimental results: the velocity remains below approximately 90%–95% of CJ detonation speed. Nusca ¹² numerically simulated the reacting in-bore flow field for the ram accelerator projectile propulsion system and found that sufficiently high projectile velocities can induce an unstart as the combustion wave precedes the projectile. With the use of Reynolds averaged Navier–Stokes (RANS) equations, Bengherbia et al. ¹³ numerically investigated the reacting flow field around the ram accelerator in the thermally choked combustion regime.

The mode of a steady thermally choked combustion is correlated with many factors, such as the reaction rate, the incoming velocity, the projectile shape and the gas component. With the increase or decrease of the reaction rate, the flame front may not be stable at the base of the projectile; in this case, thrust cannot be generated. To investigate the effects of the reaction rate and the projectile shape on the thermally choked mode, in this article, based on the two-dimensional (2D) Navier–Stokes (N-S) equations and the hybrid Roe/Harten-Lax-Van Leer (HLL) scheme, the immersed boundary method (IBM) and an adaptive mesh refinement (AMR) technique were combined to simulate the reactive flow field of a cubic ram accelerator for the thermally choked mode. The influences of the reaction rate, the inflow velocities and the shape of the projectile on the acceleration performance of a thermally choked ram accelerator were simulated and discussed.

Numerical methods

The governing equations used for describing the flow fields of the ram accelerator are the compressible 2D N-S equations with the chemical source terms. The reaction rate is calculated using the Arrhenius equation.

An explicit second-order Godunov-type numerical scheme incorporating a hybrid Roe/HLL method is used to discretise the convection term. ¹⁴ To decouple the stiffness between the hydrodynamic transport and chemical reaction, the time-operator splitting method is adopted. However, the shock wave simulations require a fine mesh to describe its structure, and an AMR technique is utilised to ensure the required resolution locally on the basis of the hydrodynamic refinement criteria. ¹⁵

Because the rectangular Cartesian grids are required for higher order shock-capturing schemes, they can only be used for simple geometrical domain. To extend the rectangular Cartesian grids to cases of complex geometry, the IBM is used. ¹⁶

The computational model is chosen to be a 2D rectangular ram accelerator. ¹⁷ Our computational domain area is taken as a 20 × 125 grid. The flow is considered as the premixed stoichiometric combustible gas that enters the domain in a direction parallel to the x-axis from the left side. The right side is the outflow, and the non-reflective characteristic boundary condition is applied. The surface of the projectile and the top and bottom sides of the domain are considered to be adiabatic. For the velocity of the flow below the CJ detonation speed of the premixed combustible gas, the initial temperature is T0 = 1, the ratio of specific heats is γ = 1.2 and the heat release is q = 26.72.

Figure 1 presents (a) the mesh distribution around the boat-shaped projectile and (b) the magnified view of the partial grids around the projectile wall at t = 100. The length of the ramp of the forebody is 33 and that of the afterbody is chosen to be 20. The angle between the horizontal symmetrical line and the surface of both the forebody and afterbody is 11.1°. The shoulder refers to the horizontal line between the forebody and the afterbody, and its length is taken to be 0 (Case 1) and 10 (Case 2).

Figure 1.

The mesh distribution around the projectile and the magnified view of the partial grids at t = 100. (a) The mesh distribution around the projectile at t = 100. (b) Magnified view of the partial grids around the wall.

Results and discussion

The effect of the reaction rates and the inflow velocities on the thrust

Figure 2 shows the pressure (left) and temperature (right) contours of the thermally choked propulsive mode of Case 1 with an activation energy of Ea = 9.0 and the initial incoming velocity of u0 = 4.4. The flow field is observed to be almost symmetric. At the tip of the projectile, oblique shock waves are formed; these shock waves reflect twice between the tube and the afterbody of projectile walls, which causes increase in the pressure and the temperature behind the shock wave fronts. However, because of the expansion fan caused by the diffuser behind the projectile shoulder, the pressure and temperature decrease rapidly, and a low-pressure area forms. Obviously, an expansion fan exists at the base of the projectile (t = 20, Figure 2).

Figure 2.

Pressure (left) and temperature (right) contours of the flow field in the tube with Ea = 9.0: (a) t = 20, (b) t = 40, (c) t = 60, (d) t = 80 and (e) t = 400.

The base of projectile acts as the flame holder, in which the premixed combustible gas is ignited by the high temperature (Figure 2(a)); the burned gas travels downstream, which causes the flame to propagate downstream and fill the tube. At t = 40 (Figure 2(b)), the flow is thermally choked behind the projectile, the combustion has spread almost all of the wake area and the flame front begins to propagate upstream. The flame front is distorted due to the uneven pressure distribution. Finally, two normal shock waves form and move upstream and become stable near the projectile base (Figure 2(b)–(e)). The flame front behind the normal shock moves with the normal shock wave front and also becomes stable at the base of the projectile (Figure 2(c) and (d)). However, the flame front behind the base vibrates along with the wake flow due to the alternate shedding of the upper and lower vortices, but its front is stable at the base of the projectile (Figure 2(e) and the attached movie). It is clear that at this time, the pressure behind the projectile base is higher than that in the other regions of the tube, thereby producing thrust. The propulsive mode of this state is the typical steady thermally choked mode. Further numerical simulations indicate that with 8.6 ≤ Ea < 10.5, the thermally choked mode of the ram accelerator is steady and can accelerate the projectile.

With the variation in the activation energy, Ea, the reaction rate of the premixed combustible gas changes; for example, when Ea decreases to 8.0, the reaction rate increases, causing the thermally choked mode no longer to be stable. The corresponding pressure and temperature flow fields are presented in Figure 3. At the beginning, the changes of the flow field are similar to those in Figure 2 (Figure 3(a) and (b)); however, due to the increase in the reaction rate, the generated heat also increases, which caused the higher acceleration of the velocity of the flame front moving upstream, thereby resulting in an unstable flame front at the base. Instead, the flame front moves upstream along the afterbody of the projectile (Figure 3(c)).

Figure 3.

Pressure (left) and temperature (right) contours of the flow field in the tube, for Ea = 8.0: (a) t = 20, (b) t = 40, (c) t = 100 and (d) t = 120.

When the flame front moves to the projectile shoulder, it pauses for an interval and subsequently continues moving upstream and passes through the throat (constituted by the tube wall and the shoulder of the projectile, Figure 3(d)). At this time, the high-pressure area located behind the flame front will also pass through the throat, and the high-pressure area is located at the forebody of the projectile; therefore, the high-pressure area generates drag to the projectile rather than thrust.

Through our numerical tests, we found that for Ea < 8.6, the ram accelerator propulsive cycles are not stable for the thermally choked mode with the incoming flow at u0 = 4.4; in this case, the flame front will outstrip the projectile and result in the failure of the projectile propulsion.

However, at this time, with the increase in the inflow velocity, the flame front becomes stable again at the base of the projectile, as shown in Figure 4 (u0 = 4.8, Ea = 8.0). This result indicates that for a higher reaction rate, the incoming velocity should also be increased to ensure that the thermally choke mode remains stable; however, for a given velocity, there is a related range of the reaction rate corresponding to the stable state.

Figure 4.

Pressure (left) and temperature (right) contours of the flow field in the tube as the velocity increases at t = 170 for u0 = 4.8 and Ea = 8.0.

Figure 5 shows the pressure (left) and temperature (right) contours of the flow field in the tube at t = 200 for Ea = 10.5 and u0 = 4.4. The highest temperature is clearly located at the base of the projectile and is able to ignite the combustion; however, because the combustion rate is low, the combustion area is limited to the central region of the tube behind the projectile (Figure 5(a)), and the pressure of the projectile base is low, which results in the generation of drag. With further increase in Ea, such as Ea > 10.5, the combustion area becomes smaller; however, with a reduced incoming velocity u0, the thermally choked mode will appear, and a high-pressure area is generated and remains still at the base of the projectile to provide impulse to the projectile. From both Figures 4 and 5, we know that to preserve the thermally choke mode of a ram accelerator, the gas reaction rate and its incoming velocity should be maintained to be within a certain limit.

Figure 5.

Pressure (left) and temperature (right) contours of the flow field in the tube at t = 200 for Ea = 10.5.

Figure 6 shows the thrust histories of four different reaction rates for the incoming velocity of u0 = 4.4 (within the subdetonative velocity regime). The thrust is obtained through the x-projection of the integration of pressure along the projectile surface. On one hand, if the reaction rate is too low (Ea ≥ 10.5), then the thermally choked mode of the ram accelerator cannot be formed, and the force acting on the projectile is always negative. On the other hand, if the reaction rate is too high (Ea < 8.6), then initially the flow is in the thermally choked mode and high thrust is generated; however, its flame front is unstable at the projectile base. As a result, the flame front moves upstream along the surface of the projectile, and the thrust continues to increase until the flame front moves to the throat, and at which point, the thrust decreases rapidly and becomes negative.

Figure 6.

Thrust curves under different reaction rates for u0 = 4.4.

Only when the reaction rate is within a proper range, 8.6 ≤ Ea < 10.5, can the thermally choked mode of the ram accelerator be stable. The total force increases slowly from negative to positive, starting from when the flow is ignited and continuing to increase until the flame moves close and sticks to the base of projectile; in this case, the thrust increases rapidly and vibrates within small range due to the unstable wake flow.

The effect of the projectile shape on the thrust

With the modification of the projectile geometry, the thrust characteristics will also change. In the following investigation, the shoulder length is increased from 0 to 10, which corresponds to a typical boat-shaped projectile. For a boat-shaped projectile, our numerical investigations indicate that the flame front can be stable both at the projectile base and just behind its shoulder; for simplicity, we only discuss the latter in this article.

Figure 7 presents the pressure (left) and temperature contours (right) of a boat-shaped projectile for Ea = 9.0 and u0 = 4.4. The process of forming the thermally choked mode of boat-shaped projectile (Case 2) is found to be almost the same as that of Case 1 shown in Figure 3 for Ea = 8.0, but the flame front in Figure 7 becomes stable just behind the shoulder rather than at the base of the projectile (as shown in Figure 3) (Case 1).

Figure 7.

Pressure (left) and temperature (right) contours of a boat-shaped projectile, for Ea = 9.0: (a) t = 40, (b) t = 80, (c) t = 120 and (d) t = 200.

The above-described difference is caused by the different reflective shock waves at the shoulder of the boat-shaped projectile. Figure 8 shows the comparison of the pressure contours for these two types of projectiles at a cold condition (without reaction). The pressure contours of these two projectiles are clearly the same at the forebody. However, on the shoulder surface, there is a reflected shock wave for the boat-shaped projectile that causes the pressure behind it to increase (Figure 8(b)). This high pressure has a blocking effect on the flame propagation; therefore, the flame front can be stable at this location within a certain reaction rate. However, there are low-pressure areas at the shoulder of the normal projectile (Figure 8(a)). Therefore, the flame front can easily move through this area (Figure 8(a)), and there is only one stable location, which is the base of the projectile.

Figure 8.

Comparison of the pressure contours of two projectiles of different shapes without reaction: (a) normal projectile and (b) boat-shaped projectile.

Figure 9 shows the thrust curves of the boat-shaped projectile under different reaction rates. Compared with the thrust curves shown in Figure 6 (Case 1) for the normal projectile, the boat-shaped projectile can clearly broaden the stable range of the thermally choked mode of the ram accelerator (8.5 < Ea < 15.0).

Figure 9.

Thrust curves with different reaction rates for the boat-shaped projectile.

When the flame front is stable behind the shoulder of the boat-shaped projectile, the generated thrust is the largest (Ea = 9.0). The largest thrust eventually becomes steady, which means that the thrust will not be influenced by the wake of the projectile. The thrust generated with the flame standing at the base is the second largest (Ea = 10.0), and the projectile vibrates under the influence of the wake.

Summary

Numerical simulations of the thermally choked mode of two types of ram accelerators were performed in this article. Our numerical results demonstrated that to generate the thermally choked mode, the reaction rate of premixed gases and inflow velocity must be within a certain limit, and the use of a boat-shaped projectile can broaden this limit. If the reaction rate is too high compared to the inflow velocity, then the flame front will propagate upstream of the projectile; however, for a low reaction rate, the flame lags behind the base of the projectile, and there is no thrust generated for both cases.

For the boat-shaped projectile, the flame front can be stable behind its shoulder, except at the base. The reason for this result is that a high-pressure area appears just behind the shoulder due to the reflection of shock wave on the shoulder, which can prevent the flame from moving upstream. When the flame front is stable behind the shoulder of the projectile, the thrust is the largest, while the projectile vibrates with the flame fixed at the projectile base.