One-Dimensional Modeling of Thermally Choked Ram Accelerator Based on CFD Simulations

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1 50th AIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition January 2012, Nashville, Tennessee AIAA One-Dimensional Modeling of Thermally Choked Ram Accelerator Based on CFD Simulations Tarek Bengherbia 1 and Yufeng Yao 2 Kingston University, London SW15 3DW, UK Pascal Bauer 3 Laboratoire de Combustion et de Détonique(LCD), UPR9028CNRS, ENSMA Poitiers 86961, France Carl Knowlen 4 and Adam Bruckner 5 University of Washington, Seattle, WA 98195, USA and Marc Giraud 6 Exobal Consulting, Saint-Louis la Chaussée, 68300, France In order to improve one-dimensional unsteady modeling of ram accelerator thrust performance, computational fluid dynamics solutions of Reynoldsaveraged Navier-Stokes equations have been used to investigate the reacting flow field of a projectile accelerated in the sub-detonative velocity regime. Both shearstress transport turbulence and eddy dissipation combustion models were used, including a detailed chemical kinetic mechanism with six species and five-step reactions. Simulations for a series of incoming Mach numbers were performed to estimate the length within which the combustion reactions were completed. This in-depth calculation of the flow field allowed implementing the unsteady 1- D modeling with an accurate Mach number dependent heat release zone length. A significantly better agreement of the predicted thrust-mach number behavior with experimental data was observed. 1 PhD, Faculty of Science, Engineering and Computing, Kingston University, Roehampton Vale, London SW15 3DW, UK, Member AIAA 2 Reader, Faculty of Science, Engineering and Computing, Kingston University, Roehampton Vale, London SW15 3DW, UK, Senior Member AIAA 3 Professor, Laboratoire de Combustion et de Détonique, PPRIME, ENSMA Poitiers 86961, France, Associate Fellow AIAA 4 Research Scientist, Department of Aeronautics and Astronautics, Box , Seattle WA, 98195, USA, Associate Fellow AIAA 5 Professor, Department of Aeronautics and Astronautics, Box , Seattle WA, 98195, USA, Fellow AIAA 6 Consultant, Exobal Consulting, Saint-Louis la Chaussée, 68300, France 1 Copyright 2012 by the, Inc. All rights reserved.

2 Nomenclature a p = constant, acceleration of projectile CV = control volume c p = heat capacity at constant pressure D a = Damköhler number e = total energy F = net axial force h = specific enthalpy I = non-dimensional thrust, F/(pA) k = turbulence kinematic energy L p = projectile length L cv = control volume length M = Mach number m p = projectile mass N = gas species P = pressure ratio p = static pressure Q = non-dimensional heat release parameter R = reaction, gas constant T = temperature v = molecular volume Γ = adiabatic heat capacity rate, (dh/de) s α = L cv a p ε = turbulence dissipation rate ω = specific dissipation rate φ = diameter σ = compressibility factor γ = heat capacity ratio = caloric imperfection, h/(c p T) I. Introduction The ram accelerator is a novel hypervelocity propulsion concept [1]. It uses shock-induced ignition and bluff base-stabilized combustion to accelerate the projectile, which travels at supersonic speed in a launch tube filled with premixed gaseous propellants. Detailed experimental and theoretical investigations have been carried out with a particular interest in the thermally choked propulsive mode, which operates at Mach numbers ranging from ~2.5 to ~5 in the sub-detonative velocity regime; i.e., below the Chapman-Jouguet (CJ) detonation speed of a propellant [2,3]. This particular propulsive mode has the potential to accelerate projectiles to greater than 3.5 km/s, and is the main topic of this paper. Figure 1 provides the key flow features associated with the thermally choked propulsive mode. As a projectile travels at supersonic speed in quiescent propellant, a system of oblique shock waves is produced. The strong shock-wave/boundary-layer interactions result in a shock wave train between the projectile and the tube; and around the projectile aft-body shock ignition of propellant happens due to the relatively high static temperature of the flow. The subsequent combustion zone is stabilized by the flame holding ability of the bluff projectile base and 2

3 terminates at a point somewhat downstream where the flow has thermally choked. Furthermore, this combustion process results in high pressure being applied to the projectile base and produces substantial thrust, which smoothly accelerates the projectile to very high speeds. In general, quasi-steady 1-D modeling of the thermally choked ram accelerator propulsive mode predicts the projectile thrust remarkably well, particularly when considering that the details of flow field are ignored much in the manner used in 1-D modeling to predict the CJ detonation wave speed [1,3]. At high fill pressure (>2 MPa) the thrust predictions of the 1-D performance model is improved by including real gas equation of state [4,5]. In situations where the ratio of propellant-toprojectile density is relatively large, resulting in high acceleration, further improvements in thrust prediction are realized by incorporating the effects of unsteady projectile motion [6-8]. Nevertheless, it was apparent that the length of the combustion zone was a major factor in the unsteady 1-D modeling, thus a means to account for how it varies as Mach number increases is necessary to increase the accuracy of performance predictions carried out in this manner. The actual 3-D flow field of an accelerating ram accelerator projectile is much more complicated than implied in Fig. 1. The combustion process is turbulent and the heat release primarily occurs in a subsonic region downstream of the projectile. The chemical reaction rate time scales are highly dependent on incoming Mach number due to the influence of flow total temperature. Furthermore, the thermal choking location is affected by the large-scale reactive flow structures presented in the combustion region and the shape of the projectile (due to shock-viscous interaction effects). In order to better understand this combustion process, detailed modeling of ram accelerator operation in the sub-detonative velocity regime has been carried out using 3-D CFD simulations of axisymmetric projectiles which included a five-step kinetic reaction mechanism [9-11]. The CFD predicted thrust agreed quite well with experiment in the subdetonative velocity regime, thus the variation of length of the heat release region determined in the simulations as the projectile Mach number increases is assumed to be appropriate for unsteady 1-D modeling. Utilizing the CFD predicted control volume length, defined as the distance from projectile nose tip to the plane where the area-averaged exit Mach number is equal to one, the thrust-mach number profile is then calculated by means of an unsteady 1-D in-house computer code, TARAM [12,13], which was developed at Laboratoire de Combustion et de Détonique, PPRIME, ENSMA. The calculated non-dimensional thrust from the 1-D model as a function of Mach number is then compared with experimental data. Figure 1. Flow field schematic of thermally choked ram accelerator propulsive mode. 3

4 II. CFD Calculation Procedure A quasi-steady CFD simulation of axis-symmetric projectiles using multi-step kinetics mechanisms has been carried out for the conditions of a ram accelerator experiment in a 38-mmdiameter tube [11]. The governing equations for the chemically reacting viscous flows are the compressible Navier-Stokes equations with chemical source terms for a mixture composed of N gas species. In addition to these equations, a turbulence model and a combustion model are required. The shear-stress transport (SST) turbulence model [11] is used for all simulations, which has the advantages of both the Wilcox and the standard model; i.e., in the inner region of a boundary layer the standard model is used, while in the outer region of a boundary layer, a high-reynolds-number version of the model is applied. A blended function is used to make a smooth change at the interface between two models. The reaction rate is calculated from the Arrhenius law, and for turbulent flow, the eddy dissipation combustion model is utilized. The chemical reaction source term derived from an eddy dissipation model developed by Magnussen [14] was based on assumption that chemical reactions occur in the smallest turbulent eddies. In other words, depending upon the Damköhler number D a, which is a ratio of characteristics fluid time over characteristic chemical reaction/induction time, either the Arrhenius approach or the eddy dissipation model will be utilized. For D a << 1, i.e., when the chemical induction time is very short, then the Arrhenius law is used. However, if the reaction rate is predominately influenced by the turbulent flow, then the eddy dissipation model is selected. Bengherbia et al. [9-11] investigated more thoroughly the combustion process in the ram accelerator using the following types of reaction mechanisms: global one step, two steps, three steps, and five steps. Among these options tested, the five-step reaction mechanism shown below yields the best agreement with experimental data: CH O 2 CO+2H 2 (R 1 ) H O 2 H 2 O (R 2 ) CO+0.5O 2 CO 2 (R 3 ) CO+H 2 O CO 2 +H 2 (R 4 ) CO 2 +H 2 CO+H 2 O (R 5 ) The reaction rate is defined using the law of mass fraction and a modified Arrhenius expression [11]. The Reynolds-averaged Navier-Stokes equations are solved together with Menter s SST turbulence model [9] for viscous flow prediction. Both steady and unsteady simulations were considered for different configurations (i.e., projectile with and without guiding fins) and at different incoming Mach numbers ranging from 3.5 to 5. Furthermore, the numerical solutions of reactive flow were performed for the projectile operating in the sub-detonative velocity regime, with a premixed combustible gas mixture composition of 2.95CH 4 +2O N 2 at a fill pressure and temperature of 5.15 MPa and 300 K, respectively. This propellant has been routinely used in 4

5 the University of Washington ram accelerator facility. The projectile body has a bi-conical shape with the nose cone having a half-angle of 10 and a length of 82 mm, whereas the aft-body is represented as a truncated cone having a convergence angle of 4.5 and a length of 71 mm (Fig. 2). The maximum projectile diameter is 29 mm and is situated at the joint of the two cones. The overall length of the projectile is 153 mm. The computational domain begins 10 mm ahead of the projectile nose tip. For the present axisymmetrical modeling, the computational domain length is 563 mm; i.e., approximately 3.7 times the projectile length, L p. Due to the axi-symmetrical nature of the projectile, only one quarter of the complete domain is considered for a finless projectile. A sequence of four meshes, from coarse to fine with the same grid topology, has been generated in order to identify a baseline grid with the appropriate grid resolution [11]. This procedure provides a good resolution without consuming excessive computational time. Hence, this baseline mesh is used for the simulations. No-slip and adiabatic thermal boundary conditions are applied on the projectile surface and tube wall. Moreover, the tube wall is assumed to be moving at the same velocity as the incoming flow. The symmetric boundaries are used in the circumferential direction of the computational domain. The results from the simulations were compared with data from a representative experiment at the University of Washington 38-mm-bore facility. In particular, the ram accelerator data used for comparison with theory come from an experiment with a titanium alloy projectile having mass of 109 g, travelling through a 16-m-long test section. In this experiment, the projectile entered the test section at 1060 m/s and accelerated throughout its length to exit at 2050 m/s. 563mm Figure 2. Sketch of CFD computational domain with projectile geometry (Note results presented hereby are from a finless model). III. Summary of Previous One-Dimensional Modeling A one-dimensional, quasi-steady, analytical model of ram accelerator propulsion was originally developed at the University of Washington [1, 3]. In this model, steady flow is assumed to enter 5

6 a control volume at supersonic velocity (denoted as state 1) and to exit the control volume at sonic velocity (denoted as state 2) as shown in Fig. 3. Inside the control volume, it is assumed that the reacting flow attains chemical equilibrium while conserving mass, momentum, and energy, resulting in a thermally choked flow at the exit. The thrust prediction from this model compares quite well with the experimental data, when the rate acceleration of a projectile and the tube fill pressure are below 10 5 m/s 2 and 2 MPa, respectively [3], i.e. the projectile velocity is less than 80% of the Chapman-Jouguet detonation speed of the propellant mixture. Projectile Figure 3. Control volume for one-dimensional ram accelerator thrust performance model. This 1-D quasi-steady model was later modified to include projectile acceleration effects [6,7] and further extended to include real-gas equations of state (EoS) for the combustion products [8,12,13,15]. The influence of projectile acceleration on the net thrust is determined as a global process between the state of the propellant entering the control volume and the state of the thermally choked flow at its exit, as shown in Fig. 3. For a control volume domain of length L CV, i.e., the distance between sections 1 and 2, the mass, momentum, and energy conservation equations were applied. The external heat addition and the rate of change of axial momentum are expressed through the non-dimensional chemical heat-release parameter Q and the net axial force F of the projectile acting on the control volume, respectively. This yields a set of conservation equations for the determination of projectile thrust which are presented in Bundy et al. [6]. After some algebraic manipulations, while specifying the end state to be thermally choked; i.e., M 2 2 = Γ 2 R 2 T 2 = 1 (where M 2 is Mach number, Γ 2 = (dh/de) s, R 2 is gas constant and T 2 is temperature at the exit plane), and introducing a real-gas EoS [7,8,13]; namely, pv/rt = σ(v,t) (where p is pressure, σ is compressibility factor), the following expressions are derived: 6

7 , (1), (2), (3) where c p is specific heat capacity at constant pressure, η = h/(c p T), γ is the ratio of specific heats of non-reacting mixture,, a p is the projectile acceleration, σ 1 and σ 2 are the compressibility factors in the fresh gas mixture and the burned gases, respectively. P = p 2 /p 1 is the pressure ratio between the inlet and outlet planes and I = F/p 1 A is the non-dimensional thrust with F being the net thrust acting on the projectile and A is the cross-sectional area of the tube. Details of the steps required to justify the approximation of the integral terms of the unsteady conservation equations with algebraic expressions that lead to the above equation set are presented in Bundy et al. [6]. For an accelerating projectile in a quiescent propellant, however, the projectile mass is coupled with the acceleration via I = m p a p /p 1 A where m p and a p are the projectile mass and acceleration rate, respectively. An iterative approach is thus required to determine the unique I and a p which satisfy the conservation equations for a given projectile mass and inflow Mach number. The aforementioned equations showed that the non-dimensional thrust, I, is a direct function of both the control volume length and the acceleration of the projectile. An iterative procedure is used to solve for the value of α in Eq. (3) for an arbitrarily chosen value of L CV. The initial estimate of a p is typically determined from a steady-state calculation. After the α value is converged, it can be used in Eq. (1) and Eq. (2) to evaluate T 2 /T 1 and P, respectively. Later these values are used to compute a new σ 2 and the iteration process is repeated until all the α, σ 2, T 2 /T 1, and P terms have converged. Previous studies have shown that the predicted thrust-mach number profiles using this model are in better agreement with experimental results having high acceleration rather than those determined using the quasi-steady flow model with a real gas EoS [7]. Following previous 1-D modeling studies, a value of L CV =2L P is chosen as a default value. This choice derives from experimental observations of the luminosity of the flow which indicates that the combustion is achieved within one projectile length behind the projectile base during 7

8 operation at in-tube Mach numbers of ~3.4. The corresponding values of the non-dimensional thrust are compared to those in the quasi-steady case (Fig. 4). Calculations used a fill pressure p 1 =5.15 MPa with the ideal gas equation of state (EoS) at station 1, and the Boltzmann EoS at station 2 to evaluate the properties of the combustion products. Note that the length within which the combustion is completed is expected to decrease as the Mach number increases due to the rapid increase in Arrhenius reaction rates with increasing flow temperature. Therefore, the influence of variable control volume length on the non-dimensional thrust has been studied; namely, a fixed longer length of 3L p, and a linear variation of L CV from 4L p to 1L p and 6L p to 1L p over the Mach number range of M = 3 to M CJ (about M = 5.05), respectively [7,8]. Note that the quasi-steady and unsteady 1-D models predict zero thrust at the CJ detonation Mach number of the propellant, which is expected since the projectile acceleration approaches zero at this point and unsteady effects are thus negligible. The influence of control volume length on the thrust predicted by the unsteady 1-D model was compared with the experimental data and those of the quasi-steady model in Fig. 4. It is apparent that using either a fixed or a linear variation of the control volume length L CV does not significantly improve the agreement with experimental data over the whole range of Mach numbers. A further refinement of the unsteady 1-D modeling is thus required. The possibility of using a more accurate dependency of the variation of the control volume length as a function of the incoming Mach number to improve the accuracy of the thrust calculations was a major motivation for exploring the flow field of the ram accelerator combustion process in more detail with CFD simulations. Figure 4. Non-dimensional thrust vs Mach number from 1-D modeling using TARAM code with a fixed and/or a variable L CV. 8

9 IV. One-Dimensional Modeling Based on CFD Data The CFD simulation has been used to determine the control volume length variation at various incoming velocities [15]. At a velocity of about 1240 m/s (M = 3.4), CFD predicted that the control volume length (L cv ) in which the combustion will be completed is about 2.8L p, i.e., the area-averaged axial Mach number equals unity at a plane located 1.8L p behind the base of the projectile. As incoming velocity increases, this length increases initially to a maximum of about 3.6L p, then decreases to about 1.1L p at a velocity of 1733 m/s (M = 4.75). Beyond this velocity, it remains nearly constant, as shown in Figure 5, which provides the CFD predicted control volume length and L cv /L p ratio as a function of incoming Mach number. The decrease in predicted control volume length at velocities near the CJ detonation speed is anticipated since the quasi-steady 1-D model predicts that the normal shock wave will mutate into a CJ detonation wave at the zero thrust operating velocity. The increase in control volume length with increasing Mach number in the lower velocity range of the sub-detonative velocity regime, however, is counter-intuitive; i.e., Arrhenius reaction rates imply that the combustion induction time will dramatically decrease with increasing of the total temperature of the flow. The interpretation of the CFD results is being more closely examined, but, for the purposes of this paper, the CFD-predicted control volume lengths shown here are used in the unsteady 1-D ram accelerator performance model as described below. Figure 5. CFD-predicted control volume length as a function of Mach number. As evident in Fig. 6, using the CFD-predicted control volume length for the unsteady 1-D modeling results in the projectile thrust calculations being in significantly better agreement with 9

10 experimental data than when using just a constant control volume length. Nevertheless, at Mach numbers of 4.6 and above, a deviation between the prediction and experiment thrust curves remains. This is accounted for in part by experiments with finned projectiles that were allowed to accelerate up to near the CJ speed, which showed that acceleration at and above the CJ speed is feasible under conditions in which there is a cessation of thermal choking of the flow behind the projectile [2]. This operational characteristic, however, was not adequately modeled in these CFD simulations, most likely because they were constrained to an axi-symmetric geometry. Indeed, other researchers have shown with non-reactive 3-D CFD simulations that the shock waves coming off the fins have a significant impact on the flow field of the ram accelerator projectile [16,17]. Simulating the chemically reactive 3-D ram accelerator flow field with the inclusion of fins on the projectile is currently in progress using the five-step kinetic rate model described here. The results of the 3-D simulations will be reported upon the completion of the study. Figure 6. Non-dimensional thrust vs Mach number for a 2.95CH 4 +2O N 2 propellant. V. Conclusions The control volume lengths determined from CFD simulations of the thermally choked ram accelerator were used to improve the thrust-mach number predictions of the unsteady 1-D model. CFD simulations in the sub-detonative velocity regime were carried out by solving the steady Reynolds-Averaged Navier-Stokes equations to determine the combustion zone length 10

11 variation as a function of projectile velocity. After validation against available experimental measurements, the CFD-predicted combustion zone length was used in the unsteady 1-D modeling, which led to excellent agreement with the experimental data at velocities less than 90% of the CJ detonation speed. Such an improvement of the unsteady 1-D modeling makes it a readily implemented and useful tool to predict the performance of the ram accelerator in the subdetonative velocity regime, without having to resort on more computationally intensive 2-D or 3- D computational schemes. References [1] Hertzberg, A., Bruckner, A.P., and Bogdanoff, D.W., 1988, Ram Accelerator: A New Chemical Method for Accelerating Projectiles to Ultrahigh Velocities, AIAA Journal, 26, 2, pp [2] Hertzberg, A., Bruckner, A.P., and Knowlen, C., 1991, Experimental Investigation of Ram Accelerator Propulsion Modes, Shock Waves, 1, 1, pp [3] Bruckner, A.P., Knowlen, C., Hertzberg, A., and Bogdanoff, D.W., 1991, Operational Characteristics of the Thermally Choked Ram Accelerator, Journal of Propulsion and Power, 7, 5, pp [4] Bauer, P., Knowlen, C., and Bruckner, A.P., 1998, Real Gas Effects on the Prediction of Ram Accelerator Performance, Shock Waves, 8, pp [5] Bauer, P., and Knowlen, C., 2003, Compressibility Effects of Unreacted Propellant on Thermally Choked Ram Accelerator Performance, Eur. Phys. J. Appl. Phys., 21, pp [6] Bundy, C., Knowlen, C., and Bruckner, A.P., 2004, Unsteady Effects on Ram Accelerator Operation at Elevated Fill Pressures, Journal of Propulsion and Power, 20, pp [7] Bauer, P., Knowlen, C. and Bruckner, A. P., 2005, Modeling Acceleration Effects on Ram Accelerator Thrust at High Pressure, Journal of Propulsion and Power, 21(5): pp [8] Bauer, P., Knowlen, C., and Bruckner, A.P., 2005, One-Dimensional Modeling of Ram Accelerator at High Acceleration Rates in Sub-Detonative Velocity Regime, Eur. Phys. J. Appl. Phys. 29(3), pp [9] Bengherbia, T., Yao, Y.F., and Bauer, P., Knowlen, C., and Bruckner, A.P., 2007, Numerical Analysis of the Thermally Choked Ram Accelerator in Sub-detonative Regime, The 21th International Colloquium on the Dynamics of Explosions and Reactive Systems (ICDERS), ENSMA - Poitiers, France. [10] Bengherbia, T., Yao, Y.F., and Bauer, P., 2006, Computational Investigation of Transitional Viscous Flow over a Ram Accelerator Projectile in Sub-Detonative Propulsion Mode, AIAA [11] Bengherbia, T., Yao, Y.F., and Bauer, P., and Knowlen, C., 2009, Numerical Investigation of Thermally Choked Ram Accelerator in Sub-Detonative Regime, AIAA [12] Bengherbia T., Yao Y.F., Bauer, P., and Knowlen, C., 2010, One-dimensional Performance Modeling of the Ram Accelerator in Sub-detonative Regime, Aerotecnica, J. of Aerospace Sci., Tech. and Syst., 89 (1), pp [13] Bengherbia T., Yao Y.F., Bauer, P., and Knowlen, C., 2010, Equations of State for 1-D Modelling of Ram Accelerator Thrust in Thermally Choked Propulsion Mode, Int. J. Eng. Syst. Modeling and Simulation, 2(3), pp

12 [14] Magnussen, B.F., 1981, On the Structure of Turbulence and a Generalized Eddy Dissipation Concept for Chemical Reactions in Turbulent Flow, AIAA [15] Bengherbia, T., Yao, Y.F., Bauer, P., Giraud, M., and Knowlen, C., 2011, Improved onedimensional unsteady modelling of thermally choked ram accelerator in sub-detonative velocity regime, ASME Journal of Applied Mechanics, 78(5), [16] Hinkey, J.B., Burnham, E.A., and Bruckner, A.P., 1992, High Spatial Resolution Measurements in a Single Stage Ram Accelerator, 29 th JANNAF Combustion Subcommittee Meeting, NASA Langley Research Center, Hampton, VA, October [17] Henner, M., Giraud, M., Legendre, J.F. and Berner, C., 1997, CFD Computations of Steady and Non-Reactive Flow Around Fin-Guided Ram Projectiles, Ram Accelerators, Takayama K. and Sasoh A. (eds), pp