Evaluation of the Equations of State for the 1-D Modelling of the Thermally Choked Ram Accelerator

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1 Evaluation of the Equations of State for the 1-D Modelling of the Thermally Choked Ram Accelerator Tarek Bengherbia 1 and Yufeng Yao Kingston University, London SW15 3DW, UK Pascal Bauer 3 Laboratoire PPRIME, CNRS, 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 unsteady one-dimensional (1-D) modelling of ram accelerator thrust performance, computational fluid dynamics solutions of Reynolds-averaged Navier-Stokes equations have been used to investigate the reacting flow field of a projectile accelerated in the sub-detonative velocity regime. 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 was used as a tool to implement the unsteady 1-D modelling with an accurate Mach number dependent heat release zone length. A significantly better agreement of the predicted thrust-mach number behaviour with experimental data was observed. Additional improvement has been conducted through an extensive study on the influence of available equations of state. 1 PhD, Faculty of Science, Engineering and Computing, Kingston University, Roehampton Vale, London SW15 3DW, UK, AIAA Member Reader, Faculty of Science, Engineering and Computing, Kingston University, Roehampton Vale, London SW15 3DW, UK, AIAA Senior Member 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 3550, Seattle WA, 98195, USA, Associate Fellow AIAA 5 Professor, Department of Aeronautics and Astronautics, Box 3550, Seattle WA, 98195, USA, Fellow AIAA 6 Consultant, Exobal Consulting, Saint-Louis la Chaussée, 68300, France 1

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 η = caloric imperfection, h/(c p T) T I. Introduction he ram accelerator [1] is a propulsion concept based on using shock-induced combustion to accelerate projectiles. In ram accelerator, a projectile travels at supersonic speeds in a launch tube pre-filled with a gaseous combustible mixture. Since this novel concept was first introduced in 1980's [1], extensive experimental studies have been carried out at laboratories around the world, notably at the University of Washington, Seattle, WA, USA, where the 38-mm-bore ram accelerator facility has been used and operated at propellant fill pressures up to 0 MPa [1-3], at the French-German Research Institute (with 30-mm-bore and 90-mm-bore), Saint Louis, France [4], at the US Army Research Laboratory, Aberdeen, MD, USA (with 10-mm-bore) [5], and the Tohoku University, Sendai, Japan (with 5-mm-bore) [6].

3 Several modes of ram accelerator propulsion have been investigated, one of which operates in the sub-detonative velocity regime (Fig. 1); i.e., below the Chapman-Jouguet (CJ) detonation speed of the propellant. In this mode the thrust is generated by the high projectile base pressure resulting from normal-shock-wave that is stabilized on the body of the projectile by thermal choking of the flow at full tube area behind the projectile [1-3]. As a projectile travels at supersonic speed in quiescent propellant, a system of oblique shock waves between the projectile and the tube walls is produced, i.e. shock train, as a result of strong shock-wave/boundary-layer interactions. At the projectile aft-body, the shock induced auto-ignition of propellant could happen, due to higher temperature after a normal shock-wave. This combustion zone at the projectile base will be stabilized by the flame holding ability of the bluff-body of a projectile. Furthermore, it generates high pressure and thus produces higher thrust, which continuously accelerates the projectile to very high speeds. The combustion itself, which happens in a subsonic region downstream of the projectile, appears to have a reaction time scale that is dependent to incoming Mach number. In addition, the choking location is also largely affected by both the large-scale reactive flow structures presented in the combustion region and the shape of the projectile (e.g. the inviscid-viscous interaction effects). Theoretical one-dimensional calculations have been successfully used to predict the thrust in the thermally choked propulsive mode [7, 8]. Figure 1 provides the key flow features associated with the thermally choked propulsive mode. In general, inviscid flow models can predict the position of the shockwave patterns and the theoretical operational bounds of the thermally choked propulsive mode, i.e. the control volume, can be determined based on experimental observations [1]. This and other studies based on quasi-steady, one-dimensional modelling [3] confirmed that the projectile acceleration and the pressure at the thermally choked point could be determined with fairly good agreement with experimental data. Extensive modeling using computational fluid dynamics has been done [9, 10] to simulate the reactive flow field in the ram accelerator. However, at high initial pressures, better agreement was observed with an unsteady modelling approach [11-13]. Figure 1. Flow over ram accelerator in the thermally choked propulsive mode. 3

4 II. CFD Computational Procedure Previous investigations have shown that the unsteady one-dimensional modelling could be substantially improved by utilizing the better knowledge of flow field characteristics gained through CFD simulation of the thermally choked ram accelerator. The length of the heat release region which varies as the projectile Mach number increases was evaluated by CFD modelling. A quasi-steady CFD simulation of axis-symmetric projectiles using multistep kinetics mechanisms has been carried out for a 38-mm-diameter tube experiment [14]. The governing equations for the chemically reacting viscous flows are the compressible Navier-Stokes equations with chemical reaction source terms for a mixture composed of gas species. In addition to these equations, both turbulence model and combustion model are required. The procedure is based on a five-step kinetic reaction mechanism. The reaction rate is calculated from the Arrhenius law, and for turbulent flow, the eddy dissipation combustion model is adopted. The chemical reaction source term derived from an eddy dissipation model developed by Magnussen [15] was based on assumption that chemical reactions only occur in the smallest turbulent eddies. Following the CFD predicted key parameters, such as the control volume length, the thrust was then calculated by means of an in-house unsteady 1-D computer code, TARAM [16], which was developed at Laboratoire de Combustion et de Détonique, PPRIME, ENSMA. The calculated nondimensional thrust as a function of Mach number was satisfactorily compared with experimental data. Both steady and unsteady simulations were considered for different configurations; i.e., projectile with and without guiding fins. The projectile body has a biconical shape with the nose cone having a half-angle of 10 and a length of 8 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. ). The projectile throat has a maximum diameter of 9 mm, situated at the joint of the two cones. The overall length of the projectile is 153 mm. Figure. Sketch of CFD computational domain containing a projectile geometry. Note that results presented hereby use a finless model. 4

5 III. Quasi-steady 1-D 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 (Fig. 3), steady flow is assumed to enter a control volume at supersonic velocity (denoted as state 1) and to exit the control volume at sonic velocity (denoted as state ). 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. 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 and MPa, respectively [3], i.e. the projectile velocity is less than 80% of the Chapman-Jouguet detonation speed of the propellant mixture. 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 [11, 1] and further extended to include real-gas equations of state (EoS) for the combustion products [7, 8, 13]. 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 stations 1 and, 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, which are presented in Bundy et al. [11]. While specifying the end state to be thermally choked; i.e., M = Γ R T = 1 (where M is Mach number, Γ = (dh/de) s, R is gas constant and T is temperature at the exit plane), and introducing a real-gas EoS [6]; namely, pv/rt = σ (v,t) (where p is pressure, σ is compressibility factor), this set of equations leads to the following expressions: 5

6 M 1γ 1R1 7 α η Q T c = cp1 cp T p1 1 1 T1 cp Γ R α η + 1 cp R1M 1γ 1T1, (1) R R T P= σ M γ 1 α σ Γ R R M T Rγ Γ T , () α σ γ1r α R T 1 I = + M1 1+ Γ γ 1+ M1 RT 1 1 σ1 ΓR1 RM 1 1T1 R1γ1Γ T1 σ, (3) σ 1 where c p is specific heat capacity at constant pressure, η = h/(c p T), γ is ratio of specific heat capacity, α=lcv ap and a p is a given constant, σ 1 and σ are the compressibility factors in the fresh gas mixture and the burned gases, respectively. P is pressure ratio between the inlet and the outlet planes (p /p 1 ) and I is non-dimensional thrust. For an accelerating projectile in a quiescent propellant, however, the projectile mass is coupled with the acceleration via I = m p a p /pa where m p is mass of the projectile, a p is acceleration of projectile, p is the pressure and A is the maximum cross-section area of the projectile. 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 projectile. An iterative procedure was used to solve this set of equations and get the value for L CV. Following previous 1-D modelling studies, a value of L CV = L P was 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 after the projectile base. Yet, the length within which the combustion is completed is expected to decrease as the Mach number increases due to Arrhenius reaction rates. Therefore, the influence of variable control volume length on the non-dimensional thrust was investigated in the earliest studies [16]. The deviation in predicted velocity-distance profiles demonstrated that a further refinement of the 1-D unsteady modelling was necessary. The influence on accuracy of thrust prediction due to having a more accurate value of the variation of the control volume length as a function of the incoming Mach number was thus explored by analyzing the flow around the projectile using a CFD calculation [17]. Moreover, the choice of the equation of state, namely, the Boltzmann virial development might be questionable, or simply improved [18]. IV. Applicable Equations of state Computing the compressibility factor for a given equation of state is the basis for incorporating the real gas corrections. Numerous equations of state have been developed based on the generalized empirical and theoretical considerations. Detailed formulations Unknown Code de champ modifié Pascal BAUER 10/6/y 07:30 PM Supprimé: α σ γ1r α I = + M1, RT 1 1 σ1 (3) Γ R1 RM 1 1T1 R1γ Pascal BAUER 10/6/y 07:31 PM Mis en forme: Police :Non Italique Pascal BAUER 10/6/y 07:31 PM Mis en forme: Interligne : simple Unknown Code de champ modifié Pascal BAUER 10/6/y 07:31 PM Mis en forme: Police :Non Italique Pascal BAUER 10/6/y 07:33 PM Supprimé: Pascal BAUER 10/6/y 07:33 PM Supprimé: the 6

7 about each equation of state (EoS) can be found in the work of Heuzé [19]. Here we only give the general forms of each EoS that is incorporated in the computer code TARAM. The Boltzmann EoS [0] adequately predicts the Chapman-Jouguet properties when the pressure of combustion products does not exceed 00 MPa [1]. This equation of state treats the individual molecules as hard spheres and the mixing rule only accounts for interactions of similar species. The compressibility factor is computed by the formula:, (4) where is defined as, in which B i is the co-volume, X i is the mole fraction, and v i is the specific volume of species i. The Percus-Yevick equation of state [] can be considered as a summation of the virial development such as that of the Boltzman EoS, in which the compressibility factor is computed by:, (5) where a non-dimensional factor is given by: where is Avogadro's number, is the number of moles, and V is the volume. The characteristic distance term is derived from the interaction law: where and are the molecular diameters of the two interacting species and is an adjustable constant. The Becker-Kistiakowsky-Wilson (BKW) EoS was introduced in 191 by Becker, and later modified by Kistiakowsky and Wilson [3]. It can be presented as follows:, (6) 7

8 with and where are semi-empirical constants that must be adjusted. In particular, are the co-volumes and there is no link to the co-volume defined by Boltzmann equation of state. This form of EoS is used for condensed explosive and it was shown that this EoS could be used for the calculation of detonation characteristics at extremely elevated pressures [18]. Nevertheless, in this specific case, all the adjustable parameters must be set accordingly. We have used the BKW EoS in some cases for the ram accelerator modeling as a reference to show the capability of the TARAM computer program for incorporating any equation of state. Moreover, in order to evaluate the role of the equation of state, a calculation was conducted in the quasi-steady assumption. Figure 4 presents the non-dimensional thrust vs Mach number for both experiment and theory. As expected the ideal gas EoS underpredicts the thrust, whereas the Boltzmann and PY EoS modeled the experimental thrust behavior within 3% over the Mach range of 3. to 4.6. The thrust predictions of the BKW EoS were 35% greater than experiment, implying that this EoS was not appropriate at this relative low fill pressure. In order to improve theoretical thrust predictions, some attempts were made to account for the compressibility effects of the initial gaseous mixture. For this purpose, another equation of state, i.e., the Redlich-Kwong equation of state was used [4]. Its choice was dictated by detonation velocity measurements conducted at extremely elevated initial pressures [1]. A slight reduction of the deviation between experimental data and calculation was observed in this case. Pascal BAUER 10/6/y 07:33 PM Supprimé: Boltzman Pascal BAUER 10/6/y 07:34 PM Supprimé: behaviour Figure 4. Non-dimensional thrust-mach plot for.95ch 4 +O +5.7N propellant with 1-D modelling. Ideal gas EoS used at station 1 and various real gas EoS at station. 8

9 V. Analysis of present data and perspectives The corresponding values of the non-dimensional thrust were compared to those in the quasi-steady case (Fig. 5). 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 to evaluate the properties of the combustion products. It was legitimately assumed that the length within which the combustion is completed would to decrease as the Mach number increases due to the Arrhenius reaction rates. Therefore, the influence of variable control volume length on the non-dimensional thrust was studied. This aspect was explored in previous studies, utilizing the data provided by the CFD calculation. In other words, the CFD simulation has been used to determine the control volume length variation at various incoming velocities [17]. Using the CFD-predicted control volume length for the unsteady 1-D modelling, results are in significantly better agreement with experimental data. However, beyond velocities of 1600 m/s, i.e., at Mach number 4.6 and above, a small deviation between the predicted and experimental thrust curves remains. Note that both the quasi-steady and unsteady 1-D models predict zero thrust at the CJ detonation Mach number of the propellant. Experiments with finned projectiles that were allowed to accelerate up to near the CJ speed, and also showed that acceleration at and above the CJ speed is feasible which indicates a cessation of thermal choking of the flow behind the projectile []. This operational characteristic was not adequately modeled in these CFD simulations, most likely because they were constrained to an axi-symmetric geometry. For this reason, a modeling of the chemically reactive 3-D ram accelerator flow field arising from the inclusion of fins on the projectile was undertaken, using a similar CFD simulation approach. This aspect is currently investigated and will be reported in the final paper. After validation against available experimental measurements, the CFD-predicted combustion zone lengths are used in the unsteady 1-D modelling, which leads to excellent agreement with the experimental data [5]. 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 sub-detonative velocity regime, without having to further resort on more complicated -D or 3-D computational schemes. However, the total reliability of this modeling requires a further demonstration of the pertinent choice of the equation of state. For this purpose, a systematic and extensive check of all available equations of state in the TARAM code is being performed on the basis of the unsteady assumption of the flow. A complete set of results with comparison to the experimental will be detailed and discussed in the final paper. 9

10 40 35 D Steady state RANS solution Experimental data 5.5 Thrust (kn) D Quasi-Steady assumption 1D Unsteady assumption Thrust Coeff (F/pA) Velocity (m/s) -1.5 Figure 5. Comparison of CFD and 1-D predicted thrusts with experimemtal measurement. VI. Conclusion 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 using various the equations of state of real-gas. CFD simulations in the subdetonative velocity regime were carried out by solving the steady Reynolds-Averaged Navier-Stokes equations to determine the combustion zone length 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. 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 sub-detonative velocity regime, without having to resort on more complicated -D or 3-D computational schemes. Pascal BAUER 10/6/y 07:35 PM Supprimé: r Pascal BAUER 10/6/y 07:35 PM Supprimé: t Pascal BAUER 10/6/y 07:36 PM Supprimé: modelling Pascal BAUER 10/6/y 07:36 PM Supprimé: modelling 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, 6,, pp [] Hertzberg, A., Bruckner, A. P., and Knowlen, C., 1991, Experimental Investigation of Ram Accelerator Propulsion Modes, Shock Waves, 1, 1, pp

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