Unsteady Ballistic Code for Performance Prediction of Solid Propellant Rocket Engines Luca d’Agostino1 and Mariano Andrenucci2 Dept. of Aerospace Engineering, University of Pisa, I-56122, Pisa, Italy Solid Propellant Rockets (SPRs) by their very nature embody in their chemical properties and geometry the whole history of grain combustion from ignition to burnout. Thus, the development of robust and reliable ballistic codes for accurate prediction of SPR performance is critical for their cost-effective utilization in space propulsion applications, especially in the case of launch vehicle boosters where it can greatly contribute to limit the number of extremely expensive experimental tests. In this perspective the present work illustrates a numerical code (compiled with Fortran 77) based on the unsteady quasi-onedimensional conservation laws for the flow in SPR thrust chambers. By means of an ENO scheme and a Roe solver the code is capable of simulating the whole history of solid propellant grain combustion from ignition to burnout, thereby eliminating the matching problems associated with the use of two different codes for the transient and steady-state combustion phases. After successful validation in a number of test cases, sample results are presented in order to illustrate the effectiveness and accuracy of the program for the prediction of the internal ballistic of real SPRs .
Nomenclature A a αpr β Chc cp Dh e hc hf hig keb kpr . m ig Pb p rb ρ ρpr Tps Tpi t u vinj x 1 2
= = = = = = = = = = = = = = = = = = = = = = = = =
channel area pre-exponential factor of the nonerosive burning rate solid propellant thermal diffusivity erosive burning exponent heat transfer corrective coefficient constant pressure specific heat hydraulic diameter volumetric specific energy local heat transfer coefficient specific reaction enthalpy specific igniter gas enthalpy erosive burning constant solid propellant thermal conductivity igniter mass flow burning perimeter pressure burning rate gas density propellant grain density propellant surface initial propellant surface temperature time gas velocity igniter mass injection velocity axial coordinate
Professor,
[email protected]. Member AIAA. Professor,
[email protected]. Member AIAA. 1 American Institute of Aeronautics and Astronautics
I. Introduction
I
N less than a second, the environment of a solid propellant rocket launch pad transforms into an intricate network of shock waves propagating in the external flow of the engine nozzle and reflecting on the surrounding structures, while the exhaust jet generates high pressure and heat loads on the rocket vehicle. This is the result of the complex, rapid and intense physical and chemical events involved in the development of rocket propellant ignition transients. The main phases of this time interval can be identified1,2 as follows: a) igniter delay, b) lag time, c) flame spread and d) chamber filling. While the igniter delay is repeatable, being a function of ignition system design, the lag time, which is defined as the time period from the end of igniter delay to the beginning of propellant ignition, is more difficult to predict. Generally it is controlled by several complex thermal, physical and chemical processes, such as heat transfer by conduction, convection and radiation, igniter flow impingement on the propellant grain, simultaneous chemical homogeneous and heterogeneous reactions in different regions, and more. All of these events are sometimes accompanied by additional undesired phenomena3, such as overpressures, hangfires, detonations, combustion oscillations and combustion extinction, which make the ignition transient extremely complex to study. These considerations make the simplification of the physical problem important and, once the numerical model has been chosen, great importance and care must be paid to the efficient matching between the main core solver and the external routines, such as grain geometry and input data acquisition, igniter modeling, boundary conditions, etc. In the last decades, considerable efforts have been developed in both experimental and theoretical studies of ignition transients in SRMs, with the main goal of understanding the complexity of the whole phenomena in order to improve the ability to design solid propellant rockets and predict their performance. Unfortunately, nowadays detailed knowledge of ignition transients is still relatively poor and we have to trade-off our limited understanding of reality with the need for generating reliable predictions. Spatially uniform approaches (p(t) models)4 can be used for quick, simple and preliminary analyses, but cannot be used for high aspect ratio channels because in this case, even in the steady-state phase, all thermodynamic quantities vary with axial coordinate in a nonnegligible way. This limitation makes these models inadequate for the majority of solid rocket motors like, for instance, common boosters used for first stages of space launchers. For similar reasons, onedimensional steady models (p(x) models), which sometimes can be used for low port-to-throat area ratio motors, are not adequate for attaining the accuracy required from today’s rocket ballistic simulations of SPR ignition transients. Multidimensional models5,6 can reach high accuracy levels but at the cost of long computational times. For these reasons, onedimensional unsteady approaches (p(x,t) models)7,8,9 usually represent the best compromise between computational cost and accuracy when considering rocket motors with high volumetric loading densities, small port-to-throat area ratios and large length-to-diameter ratios. This type of models is the focus the present study, and has been used for predicting the whole firing time of solid propellant motors. In the simulation of SPR combustion it is worth noticing that the relatively smaller magnitude of the temporal and spatial gradients of the flow variables during quasi-steady phase does not necessarily simplify the numerical simulation as much as one might expect. In fact, if on the one hand the time history of the solution is smoother, on the other hand the unavoidable simplifications of the computational model (boundary conditions, interfaces, input data available, etc.) introduce errors of various nature whose effects are negligible during the transient phase but can grow in time and, in the worst case, lead to uncontrollable instabilities during the quasi-steady phase of propellant combustion. This danger is especially manifest in grains with highly irregular geometries, which therefore have been subject to accurate analysis in this work in order to test and validate the reliability of the numerical code for application to real SPR motors.
II. The Physical Model and the Solution Algorithm The physical model on which the numerical code has been built is based on the following assumptions: 1-D unsteady inviscid flow, negligible mass forces, adiabatic channel walls and constant specific heat ratio. The governing equations originating from these assumptions are10:
∂ ( ρ A) ∂ ( ρuA) + =0 ∂t ∂x
(continuity)
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(1)
2 ∂ ( ρuA) ∂ ⎡⎣( ρu + p) A⎤⎦ ∂A + =p ∂t ∂x ∂x
(momentum)
(2)
∂ (eA) ∂ ⎡⎣(e + p)uA⎤⎦ + =0 ∂t ∂x
(total energy)
(3)
The solution technique used is based on a second order ENO11 (Essentially Non-Oscillatory) scheme, known as a robust tool capable of simulating flow fields where discontinuities exist12,13. A simple explanation of how it works can be given directly through the numeric discretization of the previous governing equations. By integrating them in the generic domain D = {[ − h / 2, h / 2] × [ 0,τ ]} of x-t space, we obtain the following integration scheme:
ρin +1 Ain +1 = ρin Ain −
∆t n +1/ 2 n +1/ 2 n +1/ 2 ⎡ρ u A − ρ in−+1/1/22 uin−+1/1/22 Ain−+1/1/22 ⎤⎦ ∆ x ⎣ i +1/ 2 i +1/ 2 i +1/ 2
ρin +1uin +1 Ain +1 = ρin uin Ain − + pin +1/ 2 ein +1 Ain +1 = ein Ain +1 −
in which
( )i +1/ 2
n +1/ 2
and
( )i −1/ 2
n +1/ 2
(4)
{
}
2 2 ∆t ⎡ n +1/ 2 p + ρ in++1/1/22 ( uin++1/1/22 ) ⎤⎥ Ain++1/1/22 − ⎡⎢ pin−+1/1/22 + ρ in−+1/1/22 ( uin−+1/1/22 ) ⎤⎥ Ain−+1/1/22 + ⎦ ⎣ ⎦ ∆ x ⎣⎢ i +1/ 2
Ain++1/1/22 − Ain−+1/1/22 ∆t ∆x
∆t ∆x
{( p
n +1/ 2 i +1/ 2
(5)
+ ein++1/1/22 ) uin++1/1/22 Ain++1/1/22 − ( pin−+1/1/22 + ein−+1/1/22 ) uin−+1/1/22 Ain−+1/1/22
}
(6)
represent cell interface quantities evaluated at t n+1/2 through the ENO scheme.
Interface values of the local density, speed and total energy are preliminary calculated by a Taylor’s expansion, in accordance with the following example referring to left and right values of the density at “i +1/2-th” interface:
( )
n+1/2 ρi+1/2 |L = ρ xi ,t n +
n
n
∂ρ ∆ x ∂ρ ∆t + ; ∂ x i 2 ∂t i 2
(
)
n+1/2 ρi+1/2 = ρ xi+1 ,t n − R
n
n
∂ρ ∆ x ∂ρ ∆t + . ∂ x i+1 2 ∂ t i+1 2
(7)
In these and other similar equations, all time derivatives can be expressed by means of the conservation equations as only functions of space derivatives. In turn, the space derivatives are evaluated through the function minmod, which is typical of ENO schemes and has the main feature of avoiding the introduction of local peaks in the data reconstruction. In particular, the general space derivative sni can be written as : sin =
1 min mod (Vi n − Vi −n1 , Vi +n1 − Vi n ) , h
where
if xy < 0 ⎧⎪0 min mod ( x, y ) ≡ ⎨ ⎪⎩sign ( x ) min ( x , y ) otherwise
The ambiguity created on each cell interface represents a Riemann problem, that has been solved through a Roe14 solver. It can be demonstrated15 that this kind of linear data reconstruction leads to a second order-accurate scheme.
III. Validation Runs and Simple Fictitious Cases As anticipated earlier, this basic code has been preliminarily tested on simple problems whose exact solutions are known, in order to assess its capability of numerically reconstructing a number of relevant flow configurations. The code yielded excellent results in the simulation of all of the following problems: double expansions, double shocks, transonic expansions and contact discontinuities in constant port area channels. These findings can be summarized here by analyzing the simulation of the Sod problem, where an initial discontinuity evolves in three separate waves, an expansion, a contact discontinuity and a shock wave. As an example we report in Fig. 1 the graphs of pressure and density, which show the exact Riemann solution in red and the two simulations for Courant numbers cfl =0.2 and cfl =0.9 in blue and light blue, respectively. It is evident that the code is capable of accurately predicting both the expansion and shock wave, while it has a slight difficulty in keeping the contact discontinuity compact. By close analysis of the graphs it can also be seen that the simulation at cfl =0.2 is slightly worse than the other one. This can be explained as a consequence of the accumulation of numerical viscosity errors over a larger number of iterations.
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Before introducing the source terms, additional tests have been successfully carried out, after complicating the physical problem with more realistic modeling of the propellant grain combustion. For instance, a wall boundary condition has been introduced in order to verify the correct reflection of a shock or an expansion wave. We encounter this kind of BC during all of the SPR burning time at the head section of the combustion chamber and at the end of the port channel during the first part of the ignition transient, when the nozzle is still corked. In addition, the variation of the port area has been assigned by specifying the A(x,t) law and the code behavior has been tested
Figure 1. Channel pressure and density in Sod problem. All variables are dimensionless. against various boundar and initial condition problems whose exact solutions are unknown, but can be easily and qualitatively estimated by common sense. For conciseness, we report only two20 of them in Figures 2 and 3. In the first test (see Fig. 2), a shock wave, originating from a discontinuity at x=5, propagates and reflects on the right wall, which as been modeled by creating a fictitious cell (m+1) after the last one (m) and by imposing16 that n = −umn . In these graphs no units are shown because the flow variables have been nondimensionalized with um+1 respect to the quantities:
[u ] ≡ m s ; [ p ] ≡ Pa ; [ a ] ≡ m s ; [ ρ ] ≡ kg
m3 ;
[t ] ≡ s ;
γ = 1.4 ; R = 287 J kmole K
also used in all of later simulations. It is worth noting, however, that these test problems do not represent any real physical situations and that our interest is only focused on the comparison between their exact mathematical
Figure 2. Channel pressure and velocity in shock reflection. The wall boundary condition is at x=10. Cfl=0.8. Initial Conditions: p=1.8, ρ=1, u(left)=1, u(right)=0. solutions and the corresponding results of their simulations. The last example is based on a conical channel geometry whose cross-sectional area is increasing in the axial direction, with wall boundary conditions at both ends. Each cell area varies in time according to the following formula: area(i)=area(i) + 0.02 dt, where dt is the time step. Initially, the internal fluid is uniform, without discontinuities. By properly manipulating the code output data with another software created to this purpose, we can obtain a threedimensional graph (in x and t) for each variable. For this test, we only report the velocity diagram in Fig. 3, because it clearly shows what we can expect from the solution. In time, each cell area varies of the same amount, and consequently the percent increase is greater where the axial coordinate is lower (lower area). Here the density decreases more than in larger cells, so a negative velocity appears. Finally, we can observe that this velocity 4 American Institute of Aeronautics and Astronautics
is greater in absolute value at low axial coordinates. This can be explained once again by considering the relative increase in time of the port area, which is maximum near the left side of the channel. This example, as well as other variable area runs carried out during the validation of the code, are of particular importance because the variation of the port area is one of the main peculiarities of solid propellant grain combustion; in fact, even if a number of efficient codes for the prediction of ignition transients are based on the assumption of a constant port area, we cannot introduce this simplification if we want to extend the calculation to the quasi-steady combustion phase.
IV. The Complete Model
Figure 3. Flow velocity in a cone-shaped channel.
The initial version of the code illustrated in the previous paragraph has then been upgraded by means of the introduction of the necessary source terms and the igniter flow effects, the inclusion of heat transfer, grain ignition and burning speed models, and the implementation of the relevant boundary conditions. The following main assumptions have been accepted in order to simplify the introduction of source terms: 1-D unsteady inviscid flow, negligible stresses and thermal conduction due to axial velocity and temperature gradients in the propellant gas, negligible friction coefficient on the grain surface, negligible gas density when compared to the density of the solid propellant grain. The numerical equations are the same as in Eq’s (4), (5), and (6) with the addition of the following source terms: n +1/ 2 n +1/ 2 ⎡ . ⎛ . ⎞ ⎛ ⎞ m v A ig inj n +1/ 2 mig A ⎟ ⎢ ⎟ +⎜ S = ⎢( rb Pb ρ pr ) ,⎜ , ( rb Pb ρ pr h f i ⎜⎜ V ⎟⎟ ⎜ V ⎟ ⎢ ⎝ ⎠i ⎝ ⎠i ⎣
n +1/ 2 T
⎛ . ⎞ n +1/ 2 ⎜ mig hig A ⎟ + )i ⎜⎜ V ⎟⎟ ⎝ ⎠i
⎤ ⎥ ⎥ . ⎥ ⎦
(8)
With reference to the heat transfer and grain ignition models, the propellant surface temperature17 before propellant ignition is obtained by solving the 1-D unsteady heat equation for the solid phase in the direction normal to the grain surface by means of a boundary integral method3. The local convective heat transfer coefficient is deduced from the conventional Dittus-Boelther correlation for turbulent flow in pipes. The code can be run using three different burning rate laws: the De Vielle-Saint Robert law, the Lenoir and Robillard law, or the modified Lenoir and Robillard law18,19. The last two ones take erosive burning into account. Just like in similar solid propellant rocket simulation programs, three multiplicative correction factors have used in order to account for the grain deformation during the burning phase (Scale Factors), the irregularities introduced during the production of the propellant (Hump), and the so named “residual correction” or Burning Rate Correction Factor (BRCF), due to the remaining deviations from the real combustion process. Their effects are significant especially during the quasi-steady phase. The head boundary condition used in the numerical model is the rigid wall condition described in Par. 3, corrected when necessary in order to account for the igniter combustion. In particular, for better reproduction of the real situation, the igniter mass flow has been spread among the front cells of the port channel in adjustable percentages. For the nozzle boundary condition two different routines can be used, both expressing the relevant flow properties at the inlet of the convergent nozzle, and their dependence on the nozzle geometry, the external ambient conditions and the chocked/unchocked status of the nozzle throat, by means of the steady quasi-1D analysis of an isentropic gas flow in a channel with variable cross-section. The first routine makes use of the method of characteristics3, but suffers from some modeling inaccuracies, including the problematic assumption that the last cell of the combustion chamber must be considered as isentropic even if it contains burning propellant. Its use resulted in the need of a large number of iterations for convergence. For these reasons, the second boundary routine has been created, based on the following steps20: - As illustrated in Fig. 4, two fictitious cells have been created (named LLEN2F and LLEN2S in the figure). Their port area is the same (updated at each time step) as the previous one (i.e. the last real cell, LLEN2 in the figure), which is the last one whose walls are bounded by the propellant grain surface. The reason of this choice consisted 5 American Institute of Aeronautics and Astronautics
in keeping the geometry of this fictitious section of the channel as smooth and regular as possible. In this way the assumption to consider isentropic the last two flow cells is also consistent with the absence of heat release from grain combustion. - In order to calculate the updated values of the flow variables from the initial cell i =1 to i = n at time t +dt, at each time step the numeric scheme adopted in the code employes the values of all of the flow variables at time t at least until i = n +1. As a consequence of this observation, if we assume to know the solution at time t, the code uses the cells from 1 to LLEN2S in order to calculate the updated values of the flow variables in the cells from 1 to LLEN2F. - From the thermodynamic properties of LLEN2F cell, assuming the flow from LLEN2F to the nozzle exit to be isentropic and applying the area law from the nozzle throat, the flow variables at the LLEN2S cell are calculated. In this Figure 4. Nozzle BC scheme. way, all of the physical quantities updated at time t +dt are stored for the last cell and the calculation cycle is closed. An important feature of the code is the possibility to refine the mesh (also in a localized way) where required for the accuracy of the computation in the presence of sudden changes of the flow geometry20. For instance, in the first intersegment region of Ariane 5 solid rocket boosters, between the first and the second segment, the port area changes abruptly by a factor of seven over a distance significantly smaller than the port radius. The mesh routine works in this way: assuming that the cell m of length l has to be divided in 3 parts, each of the new cells m1, m2, m3 is assigned a length l/3; their port areas, which are used only for fluid-dynamic calculations, are obtained by linear interpolation on the cell lengths, while their wet and burning perimeters are kept equal to those of the original cell, in order to retain the coherence on source terms.
Figure 5. Zefiro 16 and Ariane 5 EAP. Not in scale.
V. Ariane 5 and Vega Simulations The final code (indicated by the acronym UBLG) has been validated by comparison with the test data of two different motors, Zefiro 16 (an experimental motor ancestor of the second and third stages of the VEGA launcher) and Ariane 5 EAP (Etage d’Acceleration à Poudre), whose grain sections are shown in Fig. 5. The results are presented in Figures 6 and 7, respectively20. Arbitrary units have been used when reporting all of the experimental validations in order to rule out the possibility of disclosing potentially sensitive information. In Fig. 6, we also show for comparison the reference simulation of the engine manufacturer, because Zefiro 16 is an experimental motor whose BRCF has not been calculated yet. The agreement between the code simulation and the experimental data is
Experimental UBLG
Experimental
UBLG
0
Forward Chamber Pressure
ForwardCham berPressure
Old Simulation
0
0
0 Time
Time
Figure 6. Zefiro 16 forward chamber pressure in transient (left) and steady-state (right) conditions. 6 American Institute of Aeronautics and Astronautics
UBLG mesh x 3 Experimental
0
Forward Chamber Pressure
Forward Chamber Pressure
quite satisfactory during the whole development of grain combustion from ignition to burn-out, with just a slight discrepancy in the tail-off phase, probably due to the inaccuracy of the steady model used to implement the downstream boundary conditions, and possibly also to the neglect of ablative phenomena in the thermal protections. We point out that the simulations of the Ariane 5 boosters (Fig. 7) have been carried out using a mesh size (labeled “mesh x 3”) three times thicker than the original one (“mesh x 1”) of the engine manufacturer. The reason of this choice is that when the original grid is used unacceptable errors arise during the quasi-steady combustion from the conclusion of the ignition transient to the time when the chamber pressure reaches its maximum value. This is presumably due to the presence of a star-shaped grain in the first segment of the Ariane 5 EAP, which burns its propellant completely in the first phase of quasi-steady combustion (star phase) and has a great influence on the solution, thereby magnifying the influence of the errors introduced by insufficiently fine grids. It is worth noting that also the Zefiro motor has a star-shaped grain but, being located in the rear part of the propellant grain, generates a more favorable thermo-fluid-dynamic flow field from the standpoint of simulation accuracy. In order to better show this behavior and to prove that the error tends to zero as the mesh is refined, we compare in Fig. 8 the simulations of the transition from the ignition transient to the quasi-steady combustion obtained with the original mesh x 1 and with increasingly finer meshes (x 3, 4, 5, 6, 7).
UBLG mesh x 3 Experimental
0 0
0 Time
Time
Figure 7. Ariane 5 EAP forward chamber pressure in transient (left) and steady-state (right) conditions. This is an example of the difficulties we mentioned at the end of Section I in connection with the extension of the code to the simulation of the quasi-steady combustion phase. It is also noteworthy to point out another problem that occurred during the simulations of the Ariane 5 EAP with the intermediate versions of the code: we noticed that after the burn-out of the star-shaped grain in the upper booster section, the output file was indicating an undesirable run instability due to nonphysical self-sustained oscillations. By means of a thorough analysis of a large number of simulations, monitoring the time histories of the relevant flow variables, we realized that a very small oscillation was occurring in the first intersegment region when the last cell of the first segment was burning out after the extinction of all of the upstream cells. In this state, the gas velocity tends to vanish and eventually become negative. This behavior is then followed by the increase of the flow velocity, originating in a very slow but undamped oscillation, which negatively affects the simulation. The nonphysical nature of these self-sustained oscillations led us to implement a special routine, which identifies the occurrence of this phenomenon and prevents it from developing. We point out that, during our investigation of this problem, other real propellant geometries (such as the Ariane 4 boosters) have been simulated together with a number of configurations similar to grains of common use, but none of them manifested the occurrence of the above problem. This further underlines how strong and potentially troublesome the geometric discontinuities of the Ariane 5 solid rocket boosters are for the Figure 8. Mesh refinement effects on the chamber stability of internal ballistic simulations. pressure simulation. 7 American Institute of Aeronautics and Astronautics
The robustness of UBLG code can be highlighted by the 3-D transient graphs of the Ariane 5 EAP (Fig. 9), where the large and sudden variations of the flow velocity in the intersegment zones do not introduce any instability in the simulations. These variations are the (spurious) consequences in 1-D simulations of the cross-sectional area changes at the intersegments. In reality, flow recirculation effects make the axial velocity profile much smoother than in current simulations.
Figure 9. Ariane 5 EAP flow velocity (left) and forward chamber pressure (right) transients.
VI. Conclusions As shown by the very good accuracy obtained in the simulations of Zefiro 16 and Ariane 5 EAP, the code described herein provides an accurate and reliable tool for the prediction of both the transient and quasi-steady combustion phases of SPR grain combustion. The computational time is relatively short and strictly dependent on the mesh type. For instance, the complete simulation of the Ariane 5 booster combustion carried out using the original number of cells (79) and a cfl=0.8 requires 3 hours with an ordinary 1 GHz CPU. The time needed for the same simulation increases to 27 hours if the number of cells is globally tripled. The most significant cause of the longer duration of the computation is not only the larger number of cells to be calculated, but rather the increased number of time steps needed to carry out the simulation, which is dictated by the value of the cfl parameter required to satisfy the stability condition of the numerical scheme. This aspect underlines the importance of having the possibility of locally thickening the computational grid; for example, if we want to make the geometry 3 time smoother in a certain section of the channel whose cells have a similar length l and we apply the thickening to all of the channel length, it can happen that a hypothetical l/10 cell located somewhere else would cause the time step to decrease uselessly by a factor of 3 with respect to the original grid. Local thickening solves this problem that actually occurs in the case of the Ariane 5 input data. An additional possibility (tested separately but not implemented in the present work) in order to further reduce the computational time consists in starting the simulation with a relatively fine mesh (to be used, for example, in the transient and star phases) and switching later, after a suitable time, to a coarser grid. Finally, the reliability of the simulations is enhanced by a number of factors, namely: (i) the use of different chemical species for the flows generated by the igniter and the main propellant grain, (ii) the inclusion of jetimpingement effects, (iii) the choice of the desired burning model and the consideration of erosive burning effects, (iv) the inclusion of throat erosion, and (v) the more favorable handling of geometric discontinuities within the 1-D approximation. All of these aspects are significant, together with the continuous monitoring of the mass-budget, which has been verified to remain within strict tolerances during the entire simulation of the combustion history.
References 1
Bolieau, C. W., Eriksson, T. L., “SRM Ignition Transient Study”, Thiokol/NASA, TWR-10283, 1975.
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2 Kulkarni, A. K., Kumar, M. and Kuo, K. K., “Review of Solid Propellant Ignition Studies”, AIAA/SAE/ASME 16th Joint Propulsion Conference, Hartford, Connecticut, June 30-July 2, 1980. 3 d’Agostino, L., Biagioni, L., Lamberti, G., “An ignition Transient Model for Solid Propellant Rocket Motors”, 37th AIAA/ASME/SAE/ASEE Joint Propulsion Conference, Salt Lake City, UT, July 8-11, 2001. 4 Willis, J. E., “An Engineering Approach to Modelling Rocket Motor Ignition”, AIAA Joint Propulsion Conference, Huntsville AL, June16-18, 1986. 5 Le Tanter, G., “Transient Flow in a Solid Rocket Motor During the Ignition Phase”, AIAA Joint Propulsion Conference, Cleveland OH, June 21-23, 1982. 6 Sabnis, J. S., Gibeling, H. J., and Mc Donald, H., “Navier-Stokes Analysis of Solid Propellant Rocket Motor Internal Flows”, Joint Propulsion and Power, Vol.5, No. 6, pp. 657-664, Nov-Dec 1989. 7 Peretz, A., Kuo, K. K., Caveny, L. H., and Summerfield M., “Starting Transient of Solid-Propellant Rocket Motors with High Internal Gas Velocities”, AIAA Journal, Vol. 11, December 1973, pp. 1719-1727. 8 Pardue, B. A., and Han, S., “Ignition Transient Analysis of Solid Rocket Motor Using One Dimensional Two Fluid Model”, AIAA Joint Propulsion Conference and Exhibit, Nasville TN, July 6-8, 1992. 9 Price, E. W., Bradley, H. H., Dehority, G. L., and Ibiricu, M. M., “Theory of Ignition of Solid Propellants”, AIAA Journal, Vol. 4, No. 7, July 1966, , pp. 1153-1181. 10 d’Agostino, L., Rocket Propulsion, Course notes, Dipartimento di Ingegneria Aerospaziale, Università di Pisa, Italy. 11 Harten, A., Engquist, B., Osher, S., Chakravarthy, S.R., “Uniformly High Order Accurate Essentially Non-Oscillatory Schemes”, JCP, Vol. 71, 1987, pp. 231. 12 Dorrepaal J. M., Casper, J., “Finite-Volume Application of High Order ENO Schemes to Two-Dimensional BoundaryValue Problems”, Old Dominion University, Department of Mathematical Science, Norfolk, Virginia, 1990. 13 Serraglia, F. “Modeling and Numerical Simulation of Ignition Transient of Large Solid Rocket Motors”, Ph. D. Dissertation, Scuola di Ingegneria Aerospaziale, Roma, 2003. 14 Roe, P. L., “Approximate Riemann Solvers, Parameter Vectors, and Difference Schemes”, Royal Aircraft Establishment, Bedford, United Kingdom, JCP, Vol. 43, 1981, pp. 357-372. 15 Sabetta F., Lezioni di Gasdinamica, Course notes, Dipartimento di Meccanica e Aeronautica, Università di Roma “La Sapienza”, Italy. 16 LeVeque, R. J., Numerical Methods for Conservation Laws, Birkhauser Verlag, Basel – Boston – Berlin, 1990, pp. 222223. 17 Martin H., “Heat and Mass Transfer between Impinging Gas Jets and Solid Surfaces”, Advances in Heat Transfer, Academic Press, 1977, Vol. 13, pp. 1-60. 18 Lenoir J. M. and Robillard G., “A Mathematical Method to Predict The Effects of Erosive Burning in Solid Propellant Rockets”, Proc. 6th International Symposium on Combustion, Reinhold, NY, 1957, pp. 663-667. 19 Razdan M. K. and Kuo K. K., “Erosive Burning of Solid Propellant”, Progress in Astronautics and Aeronautics, Fundamental of Solid Propellant Combustion, Vol. 90, ed. Kuo. K. K. and Summerfield M., Nov. 1984, pp. 515-598. 20 Giraldi L., “Sviluppo di un Codice di Balistica Interna Non Stazionaria per la Predizione delle Prestazioni di Motori a Razzo a Propellente Solido”, M.S. thesis, Dipartimento di Ingegneria Aerospaziale, Università di Pisa, Pisa, Italy, 2005.
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