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1982 Numerical Gas Dynamics of Internal and External Flows in Gun Barrels. Oktay Baysal Louisiana State University and Agricultural & Mechanical College
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Universfc/ Microfilms International 300 N. Zeeb Road Ann Arbor, Ml 48106 8229490
Baysal, Oktay
NUMERICAL GAS DYNAMICS OF INTERNAL AND EXTERNAL FLOWS IN GUN BARRELS
The Louisiana State University and Agricultural and Mechanical Col PH.D. 1982
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University Microfilms International NUMERICAL GAS DYNAMICS OF INTERNAL AND EXTERNAL FLOWS IN GUN BARRELS
A Dissertation
Submitted to the Graduate Faculty of the Louisiana State University and Agricultural and Mechanical College In partial fulfillment of the requirements for the degree of Doctor of Philosophy in
Mechanical Engineering
Oktay Baysal B.5., The Technical University of Istanbul, Turkey, 1976 M.Sw, The University of Birmingham, U.K., 1978 August 1982 ACKNOWLEDGEMENTS
I am deeply indebted to my research advisor, Dr. Robert W. Courter, for his helpful advice, supervision and encouragement with upmost sincerity and unseifishness.
Beyond any doubt, the completion of this work is owed to Professor Andrew
3. McPhate*s hours of revealing conversation and discussion. I would also like to express my deepest gratitude to my committee members, Dr. Ozer A. Arnas, Dr.
Magd E. Zohdi, Dr. John A. Hiidebrant and Dr. Daniel W. Yannitell for their guidance and assistance.
Sincere appreciation and thanks are due the department chairman, Dr.
William N. Sharpe, for allowing me to use the computer and departmental facilities freely. I also gratefully acknowledge the patience and punctuality of my manuscript typist, Ms. Dawn A. Tipton.
Last, but certainly not the least, I would like to reiterate my gratitude, respect and deep love to my mother and father who have always been most supportive, encouraging and caring without which all would have been absolutely impossible. TABLE OF CONTENTS
Page
Acknowledgment ii
List of Tables v
List of Figures vi
List of Symbols ix
Abstract xii
Introduction 1
Gun Barrel Flows 2 Muzzle Blast Flows 4
Theory 7
Internal Flow 7
External Flow 11
Numerical Methods of Analysis 18
Internal Flow 18
External Flow 22
Results 30
Discussion and Conclusions 39
Figures 46
References 94
i i i Appendices
A. Analysis of One-dimensional Steep-Fronted Wave
B. Analysis of One-dimensional Non-Steep Wave
C. Program Listing LIST OF TABLES
Page
TABLE 1. Summary of Results for Test Cases 32
TABLE 2. Results for One Dimensional Shock Propagation 36
TABLE 3. Results for Spherical Non-self-similar Shock
Propagation 38
v UST OF FIGURES
A Typical Two Stage Gun 47
Aeroballistic Research Facility 48
Muzzle Performance 49
Experimental Results from Seigel 50
Characteristics Around the Projectile and the
Initiation of Shock on Time-Distance Plot 51
A Schematic of a Barrel and Projectile Assembly 52
Characteristics of an Unsteady, One Dimensional Flow 53
Finite Difference Grid Based on Inverse Marching
Method of Hartree 54
Shock Wave Calculation with the Aid of Characteristics 55
Moving Shock-Stationary Shock Transformation 56
Projectile Motion Calculation with the Aid of
Characteristics 57
Finite Difference Grid Based on MacCormack Scheme 58
Sample Mesh for External Flow Computation 59
Projectile Time Versus Projectile Travel 60
Projectile Base Pressure Versus Projectile Travel 61
Projectile Base Density Versus Projectile Travel 62
Projectile Acceleration Versus Projectile Travel 63
Projectile Velocity Versus Projectile Travel 64 vi Page
19. Nose Pressure of the Projectile Versus Projectile Travel 65
20. Pressure Contours. Cycle 211 66
21. Pressure Contours. Cycle 214 67
22. Flow Field of Cycle 214 68
23. Pressure Contours. Cycle 216 69
24. Flow Field of Cycle 216 70
25. Pressure Contours. Cycle 217 71
26. Flow Field of Cycle 217 72
27. Pressure Contours. Cycle 218 73
28. Flow Field of Cycle 218 74
29. Axial Distribution of Pressure. Cycle 218, Row 7 75
30. Axial Distribution of Pressure. Cycle 218, Row 10 76
31. Pressure Contours. Cycle 219 77
32. Projectile Time Versus Projectile Travel and
Shock Time Versus Shock Travel 78
33. Shock Pressure Versus Shock Travel 79
34. Shock Velocity Versus Shock Travel 80
35. Projectile Base Pressure Versus Projectile Travel 81
36. Projectile Base Density Versus Projectile Travel 82
37. Projectile Acceleration Versus Projectile Travel 83
38. Projectile Velocity Versus Projectile Travel 84
39. Nose Pressure of the Projectile Versus Projectile Travel 85
40. Pressure Contours. Cycle 60 86
41. Flow Field of Cycle 60 87
v ii Page
42. Axial Distribution of Pressure. Cycle 60, Row 10 88 43. Pressure Contours. Cycle 64 89
44. Pressure Contours. Cycle 71 90
45. Flow Field of Cycle 71 91
46. Pressure Contours. Cycle 72 92
47. Flow of Cycle 72. 93
v i i i LIST OF SYMBOLS
A area (m2) a speed of sound (m/s) C a constant c speed of wave propagation (m/s)
D inner diameter (m) h enthalpy (J/kg) K a constant k coefficient of heat conductivity (W/m-
M Mach number m mass (kg)
P pressure (Pa) Q total of viscous terms (m/s2) R gas constant (N-m/i r radial axis
S slope
T temperature <°K) t time (s) u axial component of particle velocity (m/s)
V partical velocity (m/s)
V radial component of particle velocity (m/s)
X axial displacement (m) y radial displacement (m) z axial displacement (m)
ix f ratio of specific heats 0 angle (deg) 2 \ second coefficient of viscosity (N-s/m )
F first coefficient of viscosity (N-s/m )
Superscripts, subscripts b back f front in inner
1 any integer number
L axiaJ subscript
M radial subscript
N time subscript 0 pathline ou outer r reference s shock
Other
D_ DT substantial derivative (1/s)
DT change in time (s)
DX change in axial direction (m)
DY change in radial direction (m)
P predictor value of property p
1 t absolute value
^ change
V gradient
diver gence UPR03 ; velocity of projectile (m/s)
UMUZ : velocity of flow at the muzzle plane (m/s)
Pshock : pressure at shock front (Pa)
Pprecursor : pressure at precursor shock front (Pa)
XBRRL : length of barrel (m)
AO : initial acceleration (m/s2)
PBAS : initial base pressure of projectile (Pa)
RBAS : initial base density of projectile (kg/m3)
PAMB : ambient pressure (Pa)
WMUZ : velocity of shock at the muzzle plane (m/s)
xi ABSTRACT
A computational combined solution to the internal and external flows in gun barrels is devised. A computer code, IEFLOW, is constructed to carry out all the calculations and present a complete package of tables and graphs of the physical property histories. With the input of the initial conditions at the threshold of the barrel, the internal and external flow fields and the simulation of blast is made possible.
The flow inside the barrel, which is unsteady, compressible and one dimensional, is solved by a predictor-corrector technique using the method of characteristics. A two step, explicit, finite difference scheme is used to integrate the non-conservative form of the governing equations of the axisymmetric, unsteady, compressible, viscous flow outside the barrel where explicit artificial dissipative terms are used to capture the shocks. Two sample cases and a listing of the code are included. In an effort to demonstrate the performance of the solution, experimental and numerical results for the internal flow, and spherical blast results for the external flow are superimposed. Agreement in both cases is very good. Suggestions to improve the success of the solution are made. INTRODUCTION
A number of recent and valuable research efforts and publications related to
gun muzzle problems are available (References 1 through 9). Most of these are
experimental which require a range and a prototype to realize. More recent ones
are computational which are more in the nature of approximation and require fast
computers but are easy and relatively inexpensive to use. Therefore, the latter,
with improved accuracies, is more in demand. However, all of these studies
concentrate on either the internal flows or the external flows but never both
combined. The present investigation is directed toward a combined solution. The problem is not limited to any specific design or kind of gun.
Consequently, a simple but general geometry is chosen; a cylindrical barrel with a
projectile in it. The projectile is stationary at time zero. Then, with the impulse
and expansion of propellant gases, it starts accelerating and causes a shock
downstream in front of it. This sudden impulse can be visualized physically as the
ripture of a diaphragm when the pressure behind it reaches a certain level
(Figpre 1). The initial location and time of the shock is computed, and if it is
found to be inside the barrel, the blast analysis outside the barrel is turned on
when this shock reaches the muzzle plane. If the initial shock location is outside
the barrel, the blast analysis is turned on when the nose of the projectile reaches the muzzle plane.
The severity of the blast, parameters that characterize the blast, gun effects on blast pressure, and pressure and velocity histories inside the gun are points of attention. Application of the method can be extended to all kinds of small arms, guns, gas guns, light gas guns, and tube launched projectiles. 2
The research was inspired by a visit and discussion at the Aeroballistic Research Facility*®, Egiin Air Force Base, Florida. Models are shot by a conventional powder gun inside a range (Figure 2). To achieve high velocities, high back pressures and, of course, high blast pressures, are reached. Since such high velocities are needed for the sake of the experiments, a design of a gun which would create blast pressures within bearable ranges, yet with high muzzle velocities, is sought (Figure 3). To achieve this goal a capability of calculating inside and outside parameters is most desirable.
Formulation of the main stages of the numerical solution is as follows:
1. Development of a physical model and mathematical statement of the problem.
2. Development of the numerical algorithm and its theoretical interpretation.
3. Programming and the formal adjustment of the program.
4. Methodical adjustment of the algorithm, i.e., test of its operation,
elimination of the drawbacks found.
5. Serial calculations, accumulation of experience, and estimation of the
effectiveness of the algorithm.
Gun Barrel Flows
The gun barrel problem has been investigated by many researchers, and various solutions have been presented (References II through 20). The Eulerian approach, where the calculation network is fixed in space, is used for constructing flows with large deformation. In the Lagrangian approach, where the coordinate network is related to the fluid particles, the structure of the flow is better 3
defined, and one succeeds in constructing rather accurate numerical schemes for flows with comparatively small relative displacements. The partial differential equations describing the motion of the gas have not been solved analytically in their general form. An analytical solution is possible, however, for the special cases of constant-driving-pressure at the projectile base
(Wilenius**) density-uniform at any time, or gas velocity proportional to distance 12 13 from the breech (Hunt ). Lagrange initiated the study of this problem in 1793, when he presented an approximate solution to the problem. He assumed that 14 density was a function only of time, not location. Love replaced the system of hyperbolic quasi-linear partial differential equations which describe the problem by a single partial differential equation of second order for one dependent variable and reached a solution. Later, K ent^ suggested a special solution. All of these are analytical approaches.
Gun calculation systems based on numerical integration techniques can overcome some of the limitations of these analytical methods and can predict the gun characteristics within reasonable accuracies. There are basically two methods being applied at the present. The first is the method of characteristics, 16 17 which will be discussed in detail later. Vasiliu and Siegel used this method for their solutions. The second method is a Lagrangian scheme in which the gas is broken into small layers, to each of which Newton's Law is applied. The pressures acting are assumed to vary negligibly over a small time interval during which the calculation is made to determine the movement of the sides of each layer. This determines the new volume and, therefore, the new pressure for each layer. The process is then repeated. Piacesi 18 used this method with an ability to take into account shocks. 4
More recent techniques, the fluid-in-cell method, other finite difference methods, and the finite element method, are also applied but would consume more computer time than the method of characteristics applied in this study. Murphy 19 et al used the finite difference method on the Eulerian form of the equations.
They based their method on a scheme developed by Longley in21 which explicit dissipation terms are added to assist in the calculation of shock waves.
In all the aforementioned investigations improvements are made in two categories; first, improvements on the accuracy of the integration of the gas dynamic equations; second, improvements to the barrel model.
Muzzle Blast Flows
The muzzle blast problem is a rather recent one with few experimental or computational investigations. The phenomenon has been photographed and experimented with various set ups, and it has been simulated numerically. The involvment of many fluid mechanical phenomena makes the computational studies rather demanding. This is a time dependent, compressible flow with large gradients of fluid properties that has shocks, contact surfaces, expansion fans, vortex motions, turbulence, and if viscosity is taken into account, boundary layers.
Spark shadowgraphs of muzzle blasts have been taken by Schmidt et al*.
With the initial shock (precursor shock), pressure waves start propagating outside the barrel until the projectile ejects. When the base of the projectile emerges, very high pressures behind it come in contact with the lower outside pressures causing the second shock (blast shock). There are strong secondary compression pulses generated in the viscous, shear layers of the propellant gas jet. Analytical solution to the general form of the flow equations describing this motion is not possible. However, if spherical symmetry is assumed, an approximation using 5
22 Sedov's spherical blast analysis may be obtained. A self-similar solution to a spherical blast is given by Thompson^. 2 Most of the computational studies use differencing methods. Traci developed the SAMS code using conservation equations and the fluid-in-cell 2U differencing technique of the OIL code. SAMS separates the calculations into two distinct phases. First, an explicit Lagrangian calculation for a fluid particle defined by a computational cell in which fluid properties are assumed uniform, is used to compute kinematics. Then, mass, momentum, and energy are transported across the fixed cell boundaries, thereby completing the process of updating flow quantities. This is a first-order accurate method. Romine and Edquist used the
SHELL code (another fluid-in-cell differencing technique) to develop MBLAST.
The muzzle gas flow out of the gun barrel is treated as a transient boundary condition fixed at the muzzle exit, since the inside of the gun barrel is not £i computed. Moore et al used the SHELL code for the computation of the 5"-Gun, and Zoltani^ used the SAMS code for the simulation of the blast of M-16 rifle. A different approach is reported by Erdos et al*\ The muzzle blast field is characterized by a highly underexpanded supersonic exhaust plume, which terminates at a strong shock. The blast wave is assumed spherical and in viscid.
Moretti^ developed a solution based on his shock-fitting technique. Basically, his attempt started with a one-dimensional analysis and expanded to a two- dimensional case using integration scheme that he calls the A-scheme.
The finite-difference forms of the governing equations may be written explicitly or implicitly. Explicit methods have demonstrated their ability to solve a wide range of flow problems. However, the size of the time increment that a solution can be advanced during each step of the calculation is restricted by numerical stability conditions to exceedingly small values when large gradients 6
and velocities are present. Published works (Reference 25) show that explicit methods give solutions with good accuracy. Although implicit methods, unlike explicit methods, can be theoretically stable at all time-step sizes, they have serious difficulty in solving strong-interaction problems (see Reference 26), and the accuracy of the solution is dependent on matrix inversion accuracy. There also exist evolved forms of these two, so-called hybrid methods, such as 27 Mac Cor mack's hybrid scheme. The choice of which method to use largely depends on numerical efficiency.
Some of these schemes, such a Cline's 28 , use artificial dissipative mechanisms, similar to viscosity and heat conduction, to smooth the shock effects 26 on the computations. And yet, some researchers, such as Baldwin et al , use smoothing functions all over the computation zone to handle large gradients and oscillations. THEORY
Internal Flow
Flow inside a barrel can be looked on as two separate flows. Flow behind the projectile is caused by the rapid expansion of pressurized gases which also accelerates the projectile. Flow in front of the projectile is due to this motion and may contain a shock wave. Even though there is a radial motion of gas particles, it is negligible compared to the main stream motion which is axial.
Thus, the flow is assumed one dimensional. Also, viscosity is comparatively small since the fluid is gaseous. Therefore, the ratio of viscous forces to inertial forces is very low, and diffusion and dissipation terms in the momentum and energy equations can be omitted without incurring much error in the results.
The expansion process in the barrel is a rapid one (in the order of a few milliseconds), so one could well assume an adiabatic expansion. Also, it is assumed that the propellant is burned completely before the barrel is reached to avoid a heat source term. To justify the assumption that the flow is isentropic, the process must also be reversible. As any other real process, it is not.
However, it is realized that the irreversibilities associated with the process of rapid expansion are inherently smaller than those associated with the process of rapid compression. Thus, the point is, this expansion process is non-isentropic only to a small degree. The same assumption has been made by Zucrow and 29 17 Hoffman in their piston problem, and by Seigel in his projectile problem solution.
Contributing effects to non-isentropic behavior are the viscous and heat transfer effects between gas layers and the walls, heat loss by radiation, boundary 7 layer information, and finally, projectile-barrel friction. It was found, however, that the discrepancy in velocities between isentropic theory and experimental results was within a few percent (see Michels^® and Seigel^). The irreversibilities may be accounted for by multiplying the final velocities by a factor slightly less than unity. Furthermore, it is assumed that expansion is in a constant cross sectional area tube.
Usually the propellant gas thermodynamic data are not accurately known. For simplicity an ideal gas assumption can be made. However, for high relative pressure and temperature effects, the Nobel-Abel gas equation of state may give slightly better results. Though this equation will be used outside the barrel, the ideal gas law will be employed inside. An experimental study (from Reference 17) shows the incurred discrepancy for a certain case in Figure 4. This assumption implies that the following conditions are true during the expansion of the gas: (1) the number of gas particles does not change, and their volume is negligible, (2) the forces between the gas particles are negligible. In some instances an examination of the relative merits of propellant gases and the determination of the real gas effects on their performance is essential. Then, the tabular values fitted to empirical equations may be used. The ideal gas assumption also leads to writing the enthalpy and the speed of sound in terms of pressure and density as
making it possible to write the derivatives of internal energy in the conservation of energy equation in terms of the derivatives of pressure and density. 9
Equations applicable to unsteady, compressible, isentropic, inviscid, one dimensional flow of an ideal gas in a constant area tube derived from the general conservation equations are as follows:
Conservation of Mass
— -*> -2— ( P u'l **0 9 t a * ' r *
Conservation of Axial Momentum
JO. + u 2 L ♦ -L -2L dt ax ? ax
Conservation of Energy
J Z + u 2 L _ a1 (.?£■ + u -? L 1 « at a* at »*
Equation of State
_L = RT P
There is gas (air) in the barrel in front of the projectile which will resist its forward motion. The projectile compresses this gas as it moves, and compression impulses are sent forward from the projectile's leading surface. Each compression travels faster than the one preceding it (as the pressure and the temperature increase, both the gas particle velocity and the local speed of sound increase, and 10
their sum is the speed of wave propagation). The compressions converge and a shock forms at some point downstream (Figure 5). The coordinates at which the shock waves begin relative to the projectile's initial position may be obtained 31 26 analytically as suggested by Gourant and Corner
- Z a amw«i\t / / I ^rrojccUte initial
Thus, the unsteady one dimensional flow equations stated above are applied to the propellant gas in the regions behind and in front of the projectile. If the shock location, x$, is found to be inside the barrel, the shock equations (Rankine-
Hugoniot) are applied across the shock.
The interaction of a shock wave, or any compression wave for that matter, with the open end of the muzzle yields different results depending on the Mach number of the ejecting flow. For subsonic flow, a widely used approximation is to assume the outside pressure to remain constant. This will result in rarefactive reflections into the barrel. However, if the pressure in the back of the shock is considerably larger than the outside pressure, a flow with a supersonic Mach number at the exit will result. In such a case, the assumption that the external pressure at the muzzle remains constant is very questionable, and the unsteady external flow field should be computed, as is done in this study.
Newton's second law is applied for the motion of the projectile,
(^projectile ^ ^projectile/ The gun in the limit of projectile mass going to zero, is actually a shock tube. The significant difference is, in the shock tube the equalization of the pressure between the front and the back of the interface occurs instantaneously, whereas in the gun, between the front and back of the projectile, this requires an infinite time. The pressure drop behind the projectile results from the fact that the projectile has accelerated, and in turn each 'layer' of gas has been accelerated. The more quickly each layer of gas moves into its neighbor's evacuated space, the less is the pressure drop and the better able is the gas to push on the projectile. Thus, a good propellant gas would be one of low inertia in this process of successive movement. During the entire movement of the projectile, rarefaction waves are produced and propagate toward the breech. When these rarefaction impulses reach the change of area of the transition (Figure 6), they are partially reflected as compression disturbances and travel toward the back of the projectile. Therefore, there is a drop in pressure from accelerating projectile motion, and there is a rise in pressure from the compressions reflected from the change of area at the back of the barrel. In the latter stages of the projectile motion, rarefaction waves reflected all the way from the breech (closed end at the very end of the barrel) will tend to drop the back pressure of the projectile. External Flow Outside the muzzle, unlike the inside, there are no physical bounds other than the outer surface of the barrel and the projectile, if the projectile is aiready out. In order to effect a numerical solution a domain of computation must be 12 established. Of course, the main criterion in doing so is the range of effects. Either the domain is extended until the gradients are no longer significant, as in 2 Traci et al , or the value of the gradients are approximated at these transmittive boundaries. Axial symmetry helps the formulation of the problem tremendously since it reduces the number of independent variables from four to three. Viscous effects are taken into consideration here, but gases will still be assumed to be ideal. The governing equations of the external flow are the continuity, momentum and energy equations written in cylindrical coordinates for an unsteady, compressible, axisymmetric flow. Since the external flow is axisymmetric, all derivatives with respect to the azimuthal (0) axis, and the azimuthal component of velocity (Vg) vanish. Also, the ideal gas assumption, as mentioned above, makes it possible to write the derivatives of internal energy in the energy equation in terms of the derivatives of pressure and density. Therefore, the governing equations for unsteady, compressible, axisymmetric flow of an ideal gas, where the axial direction is denoted by x, and the radial direction is denoted by y, are Conservation of Mass 2 L + — (fu'j + -£- (fv'j + PJL * o at ax v ' ay v 1 1 b Conservation of Axial Momentum 3 0 . + + v i 5i + - L 2 - = - i - - a - r at ax ay p ax pa* ** + A — 1 , t* i - ( .2!£. , + J _ t (A ,* * )— + 1 ay J p ay v ax ay • *y 1 v ax ayJ 13 Conservation of Radial Momentum at ax as P as { ay ay ax J + (Jtl * 2!L \ + ( A+2H ^ - - M p ax ax ay py ay y ' Conservation of Energy 3P -v u i L + v 2L - a*(-iE. + u 2L + v-2£-\ at ax ay at ax ay ax ay H [ (IL)* ♦ (-is .)1 ] t a* £2.-2*. *• ax' ' ay ; J ax ay + 2 i l 2 1 + ( A + 2 V* ^ (JLN 1 ay ax 1 v y ' + 2 XV , av t au > + _k_ 9T_ y ay ax y ay + _a_ ( k aT ^ + a_ ^ k ai_\ 1 ax v ax * ay v ^y / J 14 Depending on the relative order of magnitudes of diffusion terms (right-hand sides), these partial differential equations are hyperbolic or parabolic. For low viscosities, they are hyperbolic. Also, since they are linear in the highest-order partial derivatives, they are quasi-iinear. The compensation for high relative pressure and temperature effects may be made by using the Nobel-Abel gas law for the equation of state fp'jRT where pa 0.001 e In supersonic low Reynolds number (high viscosity) flows, which is not the case here, shock waves can not exist as lines of discontinuities, but they appear as regions of strong compressions. The gradients of flow properties in such layers are high, but they are not of an order of magnitude greater than in the rest of the flow. At large Reynolds numbers, the thickness of the shock becomes quite small compared to the scale of flow gradients outside the shock. If the Reynolds number is large enough, the shock structure has only local influence on the flow field, with only the jumps of flow properties across the shock being of interest. In this case the flow is practically inviscid in the vicinity of the shock and the jump relations are the usual Rankine-Hugoniot relations. In many cases, however, it may be convenient to solve the Navier-Stokes equations in the whole flow field when this inviscid flow region is of small extent. If the shock is considered as a discontinuity, the question of the determination of correct jump conditions arises. As will be discussed later, the shock is treated as a strong compression wave rather than as a discrete discontinuity demanding special jump conditions. A theory has been established to take into account the dissipative effects in the jump relations. This analysis rests upon the use of the Navier-Stokes 15 32 equations with approximate viscous jump relations. Neumann and Richtmyer devised an approximate method of calculation in which shocks are taken care of automatically, when and where they arise. Dissipative mechanisms, like viscosity and heat conduction, have a smoothing effect on a shock, so that the surface of discontinuity is replaced by a thin transition layer in which quantities change rapidly, but not discontinuously. Computational details of such a mechanism used in this investigation will be explained in a later section. This mechanism is turned on when the divergence of the velocity vector is found to be negative. From the continuity equation ap ^ a p -X(JL + u — + v — ^ * ? at ax ay •ax 'd 'i a or --i-l— ) - V-V p bt DP and since a compression wave means is positive, and density is always a positive quantity, V » V is negative. In addition to the shock waves, there exist three zones which need special care; namely, solid (impermeable) walls, transmittive boundaries, and the axis of symmetry. The problem of the treatment of boundary conditions on an impermeable wall in viscous compressible flow reduces to that of the calculation of density and pressure from continuity and energy equations, since the velocity is equal to the velocity of the wall (no-slip wall), and the temperature can be found as a dependent variable from the equation of state. The external surfaces of the barrel and the projectile are such boundaries. 16 At the axis of symmetry, the radial component of velocity as well as the radial gradients vanish. Also, in the governing equations, terms with the (-^-) factor can not be taken as they are due to the fact that y=0 on the center line (a singularity). These terms are approximated by (^). None of the properties vanish at the transmittive boundaries, and as the name suggests, they are all transmitted. According to the theory, the maximum turning angle of a supersonic flow of an inviscid diatomic gas around a sharp corner is 130°. The more viscous the gas is, the less this angle becomes. Therefore, separation at the muzzle and projectile corners is expected. Projectile motion is based on the rules of dynamics. The projectile initially accelerates with respect to the muzzle due to the high ejecting-gas pressure at its base. At later times it decelerates as these gases expand around the projectile, thereby decreasing the base pressure. Since the flow is viscous, in addition to the pressure forces and gravitational forces there exists a drag force. This can be taken into consideration by an experimentally determined coefficient. The effect of gravitational forces is negligible considering the duration of blast. The dynamic equation can be written = ( V r d* ) - o™9 The thermal conductivity coefficient and the first and second viscosity coefficients need to be computed along with the other flow properties. It can be shown from the second law of thermodynamics, in the absence of internal relaxation phenomena, that Stokes' relation holds. Thus, As far as the 17 first coefficient of viscosity and the conductivity coefficient are concerned, Sutherland's formula is as good as any. Hence H / T f Tr + * | s, = ( ------) V ------) » T r T Ki k = ( J l _ f 12 ( Tr »Ka \ Tr T + K2 33 where Kj and K2 are constants found from the physical property tables . The existence of pressure and temperature limits for the dependability of this formula is overlooked for the sake of simplicity. NUMERICAL METHODS OF ANALYSIS Internal Flow The governing equations for the barrel flow are hyperbolic partial differential equations in two independent variables, namely x and t. One way of solving them is by converting them into ordinary differential equations along their characteristics and approximating the values of the dependent variables through iterations. This is the method of characteristics. It gives better and quicker results than any other numerical technique. One of the most thorough discussions of the method as applied to multidimensional flows can be found in Zucrow and H offm an^. Multiplying the continuity, momentum, and energy equations of the internal flow by the unknown coefficients K. and adding yields Kj (Continuity) + ^(Momentum) + K^(Energy) = 0. Factoring out the coefficients of the x- derivatives gives the compatibility equation f + u K2) otu ( K2 + u K3) dp + ( uKi - a* J f ■ o which is valid along the characteristics. The slopes of the characteristic an 4b Ap curves,-^, are the coefficients of the partial derivatives — , ~ , such that dt _ m K» ^ Ki - a* K3 dx K(fuK 2 + u Kj uV 18 19 solving for Kj, K^, and yields the characteristic equations, at d x * at ax _ u - o at a These are the slopes of the characteristic equations, first for the right running characteristic, second for the left running characteristic, and the third for the pathiine (Figure 7). Writing the compatibility equation along each characteristic and the pathiine using the corresponding slope, gives the ordinary differential equations (compatibility equations), dp + ( fcO du sO along the right running characteristic, Jp - (fa) du = O along the left running characteristic, dp - a* J P » o along the pathiine Finite difference forms of the characteristic equations are A t . = (— -— } Ax U - a At0 - (— ) AX. U0 20 and those of the compatibility equations are + a) &u+ « o “ (fa) Au_ = o Ape - (a2) * o . 29 A modified Euler predictor-corrector method is used for the approximations of the finite difference equations above. At the predictor step, all of the coefficients (parameters in parentheses) are calculated at the known initial data points. The corrector step employs the average values of the three primary dependent variables (density, pressure, and velocity) for each characteristic curve, and the coefficients of the finite difference equations are calculated for the aforementioned average values of the flow properties. One disadvantage of the method is that the solution is smeared by the interpolation required for determining the flow properties at the initial data points. The spatial step is predetermined by the location of initial data points. The time step, however, should be chosen to ensure the presence of the intersection of the characteristics, solution point, and the pathiine within the grid (Figure 8). Hence, from the characteristic equations, At is ( iut + A )maximum 34 A numerical algorithm based on HartreeV inverse marching method is used. A finite difference grid is constructed with the characteristic curves, substituting short lengths of straight lines to approximate the corresponding short sections of the characteristic curves (Figure 8). 21 When the shock is formed inside the barrel, it must be tracked through the flow field, and its location and properties must be determined at the points of intersection of the shock wave and the t-Jines of an x-t plot (Figure 9). But first, the moving shock is transformed into a stationary one (Figure 10) to make use of the Rankine-Hugoniot equations. Then, the immediate vicinity of the shock is looked on and computed as a shock wave point between two isentropic flow points. The soiution point is determined from the path of the shock wave, as dx S gs + u where u is the average velocity of fluid particles on the right hand side of the shock, and U is the velocity of the shock relative to the fluid in front of it. Their values are found iteratively until the difference between the pressure value found by the Rankine-Hugoniot equations and the isentropic equations for the characteristic line behind the shock falls within the desired convergence tolerance. The finite difference grid for a point on the projectile is constructed similar to the one for an isentropic flow point. However, as seen in Figure 11, it is considered to consist of two halves of the finite difference grids of two isentropic flow points, each having the same velocity but having different pressures. The back of the projectile has only a right running characteristic and a pathiine. The front of the projectile has only a left running characteristic and a pathiine. Since the characteristic equations are used to find the location of the solution point, the equation xright = xleft + ^Len8th of projectile) is equivalent to and replaces the missing characteristic equation. The corresponding compatability equations are replaced by the Newton's second law 22 fra)ccttle and U * U s U Uft rtjkt prqtiUte External Flow The governing equations of the external flow, as mentioned above, are for axisymmetric, time-dependent, compressible, viscous flow. These are partial differential equations in three independent and five dependent variables, and they are hyperbolic in nature. An excellent survey of numerical methods for solving such equations was presented by Peyret and Viviand .25 In this study, these equations are integrated by a second-order accurate finite-difference scheme 35 which was first introduced by MacCormack . The equations are written in non conservative form, because the results of some published work, such as Moretti's 36 , show that the conservative form improves shock calculations only slightly, but it increases the computational time significantly for all flows. The scheme used incorporates a two step algorithm. The first step predicts a new solution at tim e t = (n+I)»At at each node (L,M) from the known solution at time t = n«£t» using one sided differences (backward difference in this study) to approximate the first order derivatives and centered differences for the second order derivatives. The second step corrects the predicted values using opposite one-sided differences (forward difference in this study), for the first order 23 derivatives. A typical node (L,M) with its neighboring points is plotted in Figure 12. Viscous terms, which have the second order derivatives, are calculated only in the initial value plane. Calculation of these terms in the second step would help accuracy only slightly in such a high Reynolds number flow while significantly increasing the computation time and storage requirements. The finite difference equations are written as follows: Predictor Step Continuity Equation IH-O b't Axial Momentum Equation 1 i a 14** - U ** \ + u f u14 -U “ \ + Vl^w iu* - u M \ I ui,* “i-t,*1 + — £ • ' i,h ^ n - 1 1 + !— ( P N - F ** 'I = —1— j f u * *L,w ^ 2 Um tUM W ( V * -V** - V M * \i M \ W,M w‘*mt l-l.M-l * * ^ 4 W1.IW t 1 r f u M -u ** \ t m . v •* \ 1 \ 24 Radial Momentum Equation ' t v**' - v w w Jfik ( vN - V M ^ f f 4 v»-.w i..«* L'M ' + S % .. { m * - v M w — !— ( p M - p M \ . ^ ' L»* ?*„ Vf t‘w > I . „ , _z » » ON I h v Z V L . * * ‘ <-.M 1 't.M + ( U* - U M - U H + U M \ ^ py iMil Mi 1 M"1 + ** "" { V M - 2 V M t V M ^ + V **•»* * » p hiS-( U H - u H - u * * u M \ 40X0X LtljM-t t"l> M*1 + t r I # y W _ v W * V*-,K n \ Jvi-v\ KV ^ l.jM-1 lK-rt W J 25 Energy Equation —^ »( L,»' - r" ) r + ^ (p“ L»n - p" 1-1,H j +T y . y r ^ [ p L(H 14 „ prt * H.,) ^ (ciw \a 1 f “——5 t i( rltM-r p M*’ - P ** t(H) \ ♦ -“fin f’* - f M w ^kff*4 -PM « ft* ,M I--1*** 1 W »■•" 't.H-i'J k* t.M M (Y-p> { (M-0 &Y I C * T ^ - ) + k*4■W T ** - * t S h * T,,*-. 1 + 6V* I.N T W U*t, H - 2 T "« * ' M* (X+2t*)MM f - L ( u * - u N + L,K DX* L*r1 H ,M * -1 - t v M - v w «■%— \ ] + W * L.w t.M-1 J lH. 0 i J hh r _j_ ( v 14 - v " \* ♦- L ( u w - u w \*-it L.w L ix» 1 I..M l-I.M* pf« V t,W IM-I1 J ( a N - u “ \ I v M - V *1 ^ + |))l L.M-1 * ' W,M L-l,H • 2 An r - .»... / UM . u N \tvM -v" w l*K 1 bX M L'M L~t(K1 * * \, h J v[ b k _ (— ( VM - v M ) ♦— (a" - u M ) \ 1 \ (m-i\-U wt s ' w ‘••w t-iM-r t* ' t,w 7 7 * J 26 Each of these equations, due to its explicit nature, may be solved for the predictor value (with a bar on top) of a property. Temperature and local speed of sound may be found from ideal gas equations. Corrector Step Continuity Equation Axial Momentum Equation 1 / z u .11* 1 - . u *1 \ i _ £*•♦' \ .WH + (U wl - u M*‘ + Dy 1 U.Mtl L,M I T ?*!♦» U,M 1 ’ L,M W Radial Momentum Equation *< « Energy Equation W t L \ h FL.M *UtH> + w vPul(H ”L)H 1 -^r'“ rL.M 1 / — M*t\* f _L_ i 0% «»*♦> SW1 *** \ Ul m , «H*l pWl \ " ITF I 2 m.,m " U,h ' \ n ) ♦ Pun,K- ) mm Uxl . Vfc.M ( P **V . P»“ N 1 . Q 4 '*M 27 where Q2, Q3, Q4 are the right-hand sides of the continuity, momentum, and energy equations of the predictor step, respectively. Each of these equations may be solved for the corrected values (unbarred values) of f, u, v, and p at time step (N + 1). The temperature and the local speed of sound are found from ideal gas equations. The time derivative term in each of the corrector step equations is actually an average of the predictor value of the property and its value obtained in the corrector step. The method is stable if the time step satisfies the stability criterion where C is the Courant number which is unity for inviscid, shock-free flows, and less than unity otherwise. However, there is no universal criterion for spatial steps. It is a matter of compromise between computation time which requires large grid sizes and better resolution which requires small grid sizes. This may be realized by a fixed but non-uniform grid with fine mesh around the muzzle and a coarser mesh far from the muzzle (Figure 13). An alternative is to use a transformation equation, usually a hyperbolic one, which clusters the grid points where better resolution is necessary. The truncation error involved in a finite difference approximation to a first order derivative term, such as [fu (du/ax)], is a third order term, [-fu(^u/ax5)(AX1/0] For a second order derivative term, such as [ ^ /3X*)], the truncation error is a fourth order term, [-H(84u/3x4)(AX/t2)]. 28 A compression wave (a shock Is a strong compression wave) is detected when the density increases with time, or the divergence of the velocity is negative at a node. To smoothen the property gradients at such a point, artificial dissipative 28 terms (from Cline ) are included. These are the artificial first coefficient of viscosity t*A - U,./cx) , the artificial second coefficient of viscosity \ * < C(v > f i-fr ’ the artificial conductivity coefficient kA - (i/cj (Rt/ct-ni ha , and the artificial right-hand side of the continuity equation i t i J - . e l l , ♦ » _ (« I L w 1 . p 1 ax v * ax ' j>) A sa 1 3 aa i C, Cj,, Cp, are constants. These coefficients are utilized by replacing f*, A,k in the finite difference equations by V - p* * 2 p M - P M determines the pressure at a point on the right-hand transmittive boundary. As in any other numerical scheme, to improve the results, damp the high oscillations, stop density and pressure undershoots, and in short, to overcome local instabilities and divergence of truncation errors, some "patching-up" should be built in. Relaxation techniques, such as, replacing bad values by an averaged, readjusted value, usually at large gradient points or sharp corners, are 37 recommended by researchers such as Roache . RESULTS After considerable effort and experimentation, a computer code, IEFLOW, has been constructed. It has been tested for a wide range of parameters, such as, initial projectile-base conditions, projectile lengths and masses, barrel lengths and diameters. The quality of the results were found to be insensitive to the projectile and barrel parameters. However, the response of the code and the quality of the results varied with the initial projectile-base conditions. Therefore, a representative model has been chosen to present the performance of the code under two separate sets of conditions. The first set of conditions leads to a subsonic exit flow, and the second set of conditions leads to a supersonic exit flow. All dimensional quantities used are SI units. The parameters of the model used are: Length of barrel 2.00m Outer diameter of barrel 0.08m Inner diameter of barrel 0.06m Length of projectile 0.10m Mass of projectile 0.50kg Working fluid Air (specific heat ratio = 1.4) Computational parameters are: Time step for the internal points 0.05ms Time step for the external points substeps varying according to the stability criterion External spatial steps 0.01m or 0.05m 30 31 Dimensions of the external computation zone : 0.34m x 0.12m Coordinates of the outer corner of muzzle : (0.14m, 0.04m) Physical properties are: Initial projectile base pressure : 90 x 105 Pa for the subsonic case 1500 x 105 Pa for the supersonic case Initial projectile base density j 61.50 kg/m3 for the subsonic case 542.65 kg/m3 for the supersonic case Ambient pressure : 1 x 105 Pa Ambient density : 1.21 kg/m3 The results are presented in the \ of tables and graphs. A summary of the output follows in Table 1. TABLE 1. Summary of Results for Test Cases CASE SUBSONIC SUPERSONIC EJECTION EJECTION AO, Initial Acceleration (m/s^) 50 m 811 105 Location of Initial Shock (m) 2.06 0.26 Time of Initial Shock (ms) 5.65 0.35 Properties at the Time (ms) 2.96 muzzle plane when the shock reaches Cycle number - 59 the muzzle (Precursor 5 shock conditions) Pressure (Pa) - Density (kg/m3) - 4.23 Speed of flow (m/s) - 642.48 Mach number - 1.25 Speed of shock (m/s) - 899.59 (continued on the next page) TABLE 1. (continued) CASE SUBSONIC SUPERSONIC EJECTION EJECTION Properties at the Time (ms) 10.45 3.45 muzzle plane when the nose of the Cycle number 209 69 projectile reaches the muzzle Pressure (Pa) 4.58*105 I2.85*105 Density (kg/m3) 3.59 7.49 Speed of flow (m/s) 296.58 768.09 Mach number 0.70 1.57 Properties at the Time (ms) 10.48 3.65 muzzle when the base of the projectile Cycle number 216 73 reaches the muzzle (Blast shock condi Pressure (Pa) 30.74 115.55 tions) Density (kg/m3) 28.54 86.97 Speed of flow (m/s) 303.18 777.34 Mach number 0.78 1.80 34 Plots of each case are presented together. Some are nondimensionalized, and the rest include quantities in SI units. Each set consists of: 1. Time versus projectile displacement (Experimental results from Reference 19 are normalized and superimposed. Also, for the supersonic case, shock time versus shock travel is plotted cm the same graph): Figures 14 and 32. 2. Shock pressure versus shock travel (Supersonic case only): Figure 33. 3. Shock velocity versus shock travel (Supersonic case only): Figure 34 4. Base pressure of the projectile versus projectile travel: Figures 15 and 35. 5. Base density of the projectile versus projectile travel: Figures 16 and 36. 6. Acceleration of the projectile versus projectile travel: Figures 17 and 37. 7. Projectile velocity versus projectile travel (Numerical results from Reference 19 are normalized and superimposed): Figures 18 and 38. 8. Projectile nose pressure versus projectile travel: Figures 19 and 39. 9. Pressure contours of the external computation zone at some representative instants displaying the growth of flows with non-steep fronted pressure waves: Figures 20, 21, 23, 44 and 46. 10. Pressure contours of the external computation zone at some representative instants displaying the precursor and blast shocks (Shock locations and strengths calculated with spherical blast assumption from Reference 22 are superimposed)): Figures 25, 27, 31,40 and 43. 35 11. Velocity vectors of the external computation zone displaying the flow fields at various instants (The vectors are drawn to scale, the scale being the muzzle velocity. Shock locations calculated with spherical blast assumption from Reference 22 are superimposed): Figures 22, 24, 26, 28, 41, 45 and 47. 12. Axial distribution of pressure at a certain row of the external computation zone at some instants with shock (Pressure distribution over the same row at the same instant of a spherical shock calculated from Reference 22 is superimposed): Figures 29, 30 and 42. To provide further insight and a means of better interpretation of the external flow, two separate computations have been performed. Method 1: A one-dimensional calculation of the initial pressure wave propagation of each case is done assuming there is no attenuation in the strength of the disturbance. An expression for the propagation speed of an adiabatic shock wave through a medium, air in this case, can be derived from continuity, momentum, and energy equations (from Reference 38). Henceforth, time for the disturbance to reach a transmittive boundary is computed (See Appendix A). This computation is conducted for the precursor shock of the supersonic case and the blast shock of the subsonic case to reach the upper (y = 0.09m) and the right (x = 0.20m) transmittive boundaries. Elapsed times, corresponding IEFLOW times and cycle numbers are tabulated (Table 2). 36 TABLE 2. Results for One Dimensional Shock Propagation F'RECURSOR SHOCK OF BLAST SHOCK OF SUPERSONIC CASE SUBSONIC CASE Pressure ratio 7.99 30 ,7k Speed of Propagation (m/s) 899.0 1750.0 Elapsed time to reach the upper transm itt ive boundary (ms) 0.1 0.05 Corresponding IEFLOW time (ms) 3.05 10.85 Corresponding IEFLOW cycle number 61 217 Elapsed time to reach the right- hand transm ittive boundary (ms) 0.22 0.11 Corresponding IEFLOW time (ms) 3.17 10.91 Corresponding IEFLOW cycle number 6 it 219 Method 2: The non-self similar solution of Sedov 22 for a point explosion is used. This technique can only be applied if spherical symmetry exists, hence a one dimensional blast is assumed. Self similarity is lost by the addition of an abstract (nondimensional) constant in the similar (two independent variables, radius and time, replaced by two parameters with independent dimensions) solution. The solution is a numerical one and gives the location and pressure of the shock at any time provided that the initial energy released by the blast is given. It is also amenable to obtaining the property distributions within the radius of shock. Therefore, plots of shock radius versus time, shock pressure versus time, and pressure versus distance from the muzzle are approximated for a few instants after each of the precursor shock of the supersonic case and the blast shock of the subsonic case. 37 The shock location and pressure, according to this theory, are superimposed on the pressure contour, velocity vector, and axial pressure distribution plots as theoretical yields. The center of the blast is assumed to be the intersection of the muzzle plane and the centerline. A summary of these results is tabulated (Table 3). TABLE 3. Results for Spherical Non-self-similar Shock Propagation Precursor Shock of Blast Shock of the Subsonic the Supersonic Case Case Energy released (3) 104.50 102.20 Cycle 60 64 217 218 219 Elapsed time after the shock (ms) 0.05 0.25 0.05 0.10 0.15 Shock radius (m) 0.056 0.11 0.056 0.08 0.093 Pressure on the shock front (Pa) 2.48K105 1.31*105 2.48XI05 1.75X105 1.5X105 Flow speed on the shock front (m/s) 239.2 66.94 239.2 141.50 101.59 Radius where the pressure drops to the ambient value (m) 0.044 0.10 0.044 0.072 0.088 DISCUSSION AND CONCLUSIONS Although some critical assumptions were made in the construction of the model, the results behave very much the same as the theoretical forecasts, e.g. Seigel17. As would be expected, the base pressure, density, and acceleration all decrease (Figures 15, 16, 17, and Figures 35, 36, 37) with the motion of the projectile. Expanding gases behind the projectile push it with a force proportional to the difference in the base and nose pressures. The base pressure drops with this expansion. However, the nose pressure increases with the compression waves from the nose reducing the difference between them causing a sharp decrease in the acceleration. Pressure difference would drop at a smaller rate if the propellant gas were one with less inertia, because then the gas would accelerate better to replace its moving front more rapidly. When the drops in pressure difference in the supersonic and subsonic cases are compared, a significant difference in the rates of the pressure gradients in favor (because a smaller rate in drop means a higher muzzle velocity) of the latter is noticed. This is confirmed by the compatibility equation for the left running characteristics which show the direct proportion between the pressure change and the "acoustic inertia", a. Also, if a diameter increase were made at the back of the barrel, a partial-open- end-effect would occur making the rarefactive waves from the back of the projectile reflect as compressive waves and cause less drop in the back pressure. The initial conditions of the "subsonic case" are prepared to terminate with a subsonic ejection from the muzzle. Therefore, there is not a shock inside the barrel for this case. With the initial conditions of the "supersonic case", however, a shock is located, and its path is superposed on the projectile path plot 39 40 (Figure 32). When carefully inspected, it can be observed that the shock path has a smaller gradient than the projectile path, i.e., the shock moves faster than the projectile and slower than the pressure waves following it. This can be verified from the fact that the Mach number of the flow behind the shock with respect to the shock is less than unity, or U, - u < a or Us < u - o ( - £ 1 > (7 7 } ” S Wav«J btWilrock Normalized experimental results of the CARDE 10/4 gun from the Canadian 19 Armament Research and Development Establishment are overlaid on Figures 14 and 32. These results belong to the barrel portion of a light gas gun. Therefore, the conditions at the barrel inlet are time functions not only of the flow in the barrel but also of the flows in the propellant chamber, pump tube, and the tapered section. Furthermore, the propellant is not pre-bumed as assumed in this investigation, and the working fluid is a low-density gas, not air. Though these experimental conditions differ from the theoretical assumptions used in the present investigation, the discrepancy in time is not greater than 10% at any 19 location for either of the cases. The numerical results of Murphy et al for the same gun are also normalized and superposed on Figures 18 and 38. Figures 33 and 34 show the pressure and velocity distributions of the shock from its initiation point to the muzzle plane. They agree very well with the theory of Seigel^. The sharp increase in both properties flatten as the shock 41 moves down the barrel. Both properties are nondimensionalized with their values at the muzzle. Most of the numerical studies of guns either assume the nose pressure of the projectile to be atmospheric or take it into account with a coefficient. In this investigation, the nose pressure is computed and plotted in Figures 19 and 39. In the supersonic case, after the shock leaves the muzzle, points ahead of the nose are computed with the muzzle plane conditions being extrapolated from the inside of the barrel. In the subsonic case, however, this extrapolation is done in front of the nose all along the barrel, i.e., a point approximately seven diameters ahead of the nose is assumed to be an undisturbed point, and the flow between the nose and this point is computed. This certainly affects the quality of the results in this zone. The farther this point is moved away from the nose, the better the results will be. The results of the external part of the flow simulate the propagation of pressure waves and shocks of a muzzle blast flow successfully. The velocity vector plots and the pressure contours provide a good picture of the developing flow field. In the computer code the "degree" of compression was arbitrarily assigned. A compression with a divergence of less than -10 000 m/s was denoted by SC (strong compression), and a compression with a divergence between 0 and - 10 000 m/s was denoted by WC (weak compression). These SC and WC signs on the vector plots clearly indicate the expected compressive waves. Large pressure gradients are noticeable in the clustering of contour lines. Propagation of the pressure waves is in good agreement with the non-steep, one dimensional wave predictions (Appendix B). The simulations of the shocks are in reasonable 22 agreement with Sedov's one dimensional, non-similar, strong blast solution. 42 Ripples of pressure disturbances start with the nose reaching the muzzle plane (Cycle 209) in the subsonic case (no precursor shock), first iliustrated in Figure 20. Each wave has a higher pressure than its predecessor. The 110 000 Pa contour in this figure is the leading wave, and it has reached x = 0.22m near the centerline. If it were assumed that only the first wave with 4.58 pressure ratio were reieased from the muzzle to propagate one dimensionally with no attenuation, it can be computed to reach x = 0.23m at the time of this figure with the non-steep wave equations (Appendix B). Naturally, this is slightly ahead of its axisymmetric counterpart. If a similar comparison were repeated for Cycle 69 of the supersonic case, the leading wave would be computed to reach x = 0.264m after two time steps (Cycle 71). Figure 44 shows that the 120 000 Pa contour has reached x = 0.22m near the centerline. The difference in the two results is slightly larger because the pressure ratio (12.85) is much higher and the assumption of no attenuation becomes less realistic. In the velocity vector diagram of the same cycle (Figure 45), compression indicators SC and WC are scattered in a bow-like shape starting from the centerline around the axial position of the wave front (120 000 Pa). The speed of the flow around this location is smaller than the projectile speed. The distribution of the vectors in this figure and Figure 47, as well as in Figures 22 and 24, give clear pictures of an axisymmetric flow driven by a projectile. When these successive vector plots are examined, the growth of the flow all over the zone under inspection, boundary layer effects, and corner flows can be demonstrated. Time and space gradients are smaller compared to the cases with shocks. The propagation of the blast shock is tracked and plotted for 0.05ms, 0.10ms, and 0.15ms (Figures 25 through 31). Comparison is possible with 22 Sedov's non-selfsimilar, one dimensional (spherical symmetry) blast wave 43 solution. Agreement is very good 0.10ms and 0.15ms after the blast. However, the same can not be claimed 0.05 ms following the blast. This discrepancy can be attributed to the very close presence of two solid, sharp corners, effects of which are not accounted far in the one dimensional computation. A closer look at the relative shock locations is possible with the plots of axial distribution of pressure on rows that are located 0.03m and 0.045m radially from the centerline for 0.10ms after the blast (Figures 29 and 30). The difference of approximately half a barrel diameter in shock locations could be due the presence of the aforementioned corners. In the muzzle blast flow the waves are generated continuously and in a strengthening manner. Therefore, the pressure increases from the shock point toward the muzzle plane. In the spherical blast wave, there is only the explosion wave which attenuates from the shock point toward the muzzle plane. The ejection of flow from the gap between the muzzle plane and the base of the projectile is very well depicted in the era-responding vector plots (Figures 26 and 28). The precursor shock of the supersonic case is plotted 0.05ms and 0.25ms after it reaches the muzzle plane at Cycle 59 (Figures 40 through 43). The bow shape of the shock is observable in the contour plots and the vector plots. The shock front locations agree within 0.05m with the results of Sedov's one dimensional analysis. In Figure 43 they are almost coincident. Using Sedov's method the flow speed on the shock front is predicted as 239.2 m/s, or 34% of the current flow speed at the muzzle. If a scaled measurement is performed in Figure 41, good agreement can be observed. The one dimensional, no-attenuation, steep-fronted wave (shock) equations (Appendix A) are applied to the blast shock of the subsonic case and the precursor shock of the supersonic case in an effort to test the effectiveness of the transmittive boundary treatment. First-order extrapolation of the property values of the points on these boundaries is surely not more than a coarse approximation. Therefore, effects of non-physical reflections are expected during and after the waves reach these boundaries. Calculations showed that the precursor shock would reach the boundaries before Cycle 61, and the blast shock of the subsonic case would reach before Cycle 217. Some of the plots, such as, of Cycles 217 and 218, show signs of some distortions due to these non-physical reflections. A fixed-nonuniform grid is fashioned which provides the benefit of high density grid spacing in regions of high fluid activity and low grid density in quiescent regions (Figure 13). In the presentation of the results, a uniform grid that proved to give equivalent results is used for purposes of clarity. Variations in the area of the computation zone have been made in an effort to seek optimum dimensions. Making the mesh finer did not change the results significantly. Consequently, it may be claimed that a fine enough mesh and a large enough computation zone are used in obtaining the converged results. This optimization reduced the computation time significantly. The occurence of low pressure, low density zones in the simulation is too frequent to be regarded as a phenomenon similar to the low pressure areas created toward the center of a spherical blast which results in the turbulence and rise of air. Although the trouble is not unique to this investigation, the remedy to the numerical effects giving birth to them is thought to be so. Obtaining weak results for the blast of the supersonic case is the direct consequence of this 39 problem. The same problem has been experienced by Mark , and MacCormack 26 and Baldwin in their computational studies of hyperbolic flow equations. MacCormack states that the change in the sign of velocity during an expansion, 45 and also severe pressure changes cause erroneous drainage of conserved quantities that lead to low and even negative densities and pressures. He has resolved this troubie with fourth-order product damping terms. Such terms were not successful in the present investigation. Mark clusters the nodes at areas of trouble by hyperbolic transformation equations and achieves relatively better solutions. This requires the use of transformed flow equations. Therefore, it is not tried for the 37 present case. However, it is recommended. Roache believes the negative values of density are experienced in the areas of flow separation. It has been noticed in this investigation that this trouble always started around the solid wall corners where separation was expected. In regions near the walls, there exist very slowly varying steep gradients 26 which require more severe resolution. Baldwin suggests replacing the values at such points by averaged values. Rather than the arithmetic averaging applied in this investigation, geometric averaging would achieve better values. Flows around the corners of solid boundaries involve very rapid expansions and vortices. Therefore, more detailed work is required than Roache's neighbor-point averaging which is used for the present study. 28 The artificial viscosity method of Cline proves to have deficiencies in capturing the strong shock waves experienced in this investigation. Therefore, a need is felt to experiment with other forms of artificial dissipative terms and shock capturing, such as, Lax^s technique mentioned in Reference 36. Warming 40 and Beam emphasize the use of the conservative form of governing equations for proper capturing of shocks. The smoothing functions being used do not show the performance expected from them, either. To improve the applicability and to add to the success of code IEFLOW, the points suggested above during further investigation are noteworthy. FIGURES P isto n Diaphragm Muzzle P r o je c tile Piston Tube Chamber FIGURE 1. R TYPICRL TNO STRGE GUN. T -0 WHEN THE DIRPHRRGM SHRTTERS Blast Chamber X /Gun Room Control Room FIGURE E. RER08RLLISTIC RESERHCH FRCILITT. EGLIN RIR FORCE BR3E. FLORIDR. (REFERENCE 10) 49 B la st P ressu re Improved Performance Muzzle V elo city FIGURE 3 . NUZZLE PERFORMANCE. BLRST PRESSURE OF IMPROVEO-PERFORHRNCE GUN IS SMALLER FOR THE SAME NUZZLE VELOCITY 50 340 NITROGEN 300 INITIAL CONDITIONS p0 = 340 ATM T0 = 25°C IDEAL NITROGEN REAL NITROGEN 200 100 1 400 GOO 1000 1200 - (METERS/SEC) FIGURE « . EXPERTNENTRL RESULTS FRBH SETGEL EFFECTS OF IDERL-GRS RSSUHPTION 51 Reflections Rarefactive waves from the back of the y Projectile Path \ projectile y ' y ' Weak Reflections >hock 0, shock Right running compressive waves in front of the p r o je c tile 0, shock FIGURE S . CHARACTERISTICS AROUND THE PROJECTILE RND THE INITIATION OF SHOCK ON TIHE-DISTRNCE PLOT, Compressions Compressions r e f le c te d r e f le c te d from th e Tapered from th e accelerating . transition transition p r o je c tile section se c tio n _____ lllb Mlb ^ e c t i l g b Rarefactions Rarefactions r e f le c te d r e f le c te d T from th e from th e breech accelerating p r o je c tile FIGURE B. II SCHEMATIC BF R BARREL RND PROJECTILE RSSEMBLT. 53 Pathline, — — L eft running Right characteristic running d t characteristics dx u-a dx u+a FIGURE 7 . CHARACTERISTICS OF AN UNSTEADY. ONE DIMENSIONAL FLON THIS CONFIGURATION IS FOR A SUBSONIC FLON 54 S o lu tio n t - l l n e ^ 7 ! At / \ \ Previous t - l i n e FIGURE 8 . FINITE DIFFERENCE GRID BRSED ON INVERSE NRRCHIND NETHOO OF HRRTHEE3* IFROH ZUCHON RND HOFFHRN1* I . STRRIGHT LINE RPPR0X1NRTIONS 5 . 0 . 7 . . . PREVIOUS SOLUTION POINTS 1 . 2 . 0 . . . 1NTERPOLRTED INITIAL ORTR POINTS S...NEW SOLUTION POINT 55 t J , Shock S o lu tio n Wave t - l l n e Previous t - l i n e FIGURE 8 . SHOCK HRVE CALCULATION HITH THE AID OF CHARACTERISTICS. (FROH ZUCHOH AND HOFFHAN2 9 ! SL.SR ..PREVIOUS SHOCK HAVE FOINT 5 . S .7 , 0 . .PREVIOUS SOLUTION POINTS t ' 1 . 2 .3 ., INTERPOLATED INITIAL DATH POINTS UL.VR ..SOLUTION FOINT 56 P4L u 4 E + U 8 %L U,4R fiffi ^ S \ \ \ Moving Shock 4L 4R W U4l U s flffi Stationary Shock FIGURE 10. MOVING SHOCK-STATIONARY SHOCK TRRNSFORMRTION. SUBSCRIPT L IS FOR LEFT. R IS FOR RIGHT RNO S IS FOR SHOCK 57 Projectile Path / S o lu tio n in lin e L eft R ight P revious t - l i n e FIGURE I I . PROJECTILE NOTION CALCULATION NITH THE RIO OF CHARACTERISTICS. (FRON ZUCRON HND H0FFHANZ , l S L .S R .. .PREVIOUS LOCATION OF THE LEFT AND RIGHT HAND SIDES OF THE PROJECTILE S .7 . . . PREVIOUS SOLUTION POINTS 1 ,2 . . . INTERPOLATED INITIAL DATA POINTS UL.UR . . . NEN LOCATION OF THE LEFT RND RIGHT HAND SIDES OF THE PROJECTILE 58 Time, t DY R adial Direction, y A xial Direction, x FIGURE 12. FINITE DIFFERENCE GRID BRSED BN MACCORHRCK3* SCHEME. IN THE FINITE DIFFERENCE EQURTISNSf @ POINTS USED FOR THE FIRST ORDER DERIVATIVE TERMS m, © POINTS USED FOR THE SECOND ORDER DERIVATIVE TERMS * , 0 , © POINTS USED FOR THE SECOND ORDER CROSS DERIVATIVE TERNS • SRMPLE MESH (50x21) TIME (SEC) = RROIRL RROIRL DIRECTION III ,— P M - (D RXIRL DIR :CT ON IX) PRESSURE* (INITII s E FRESSUR E) FIGURE 13. SffltfLE MESH FOR EXTERNRL FLOW COHFUTRTION. t in e IS NERSUREO FROM THE N0NEN1 PROJECTILE STM) ra NOTION 0X*MA1LU“>,Tim t i" D * PD5 m ®XUUU BTu«« "D-010 m 1 60 TINE-LENGTH CURVE OF PROJECTILE PATH Experimental1 results | ! jforTthje CARDE fO/4 gun yj ^J.OO 0.25 0.50 0.75 l'.OO LENGTH OF BARREL (X/XBRRL) FIGURE 14. PROJECTILE TINE VERSUS PROJECTILE TRAVEL. PBAS-90»10* PA. tn -0.5 KG, XBHRL-2.0 N. 0 - 0 .0 6 N 61 tn m to c n ~ d o QQ 00 0.25 0.50 00 LENGTH OF BRRREL tX/XBRRL) FIGURE 15. PROJECTILE BRSE PRESSURE VERSUS PROJECTILE TRAVEL. PBAS-BOn IO* PR. m - 0 .5 KG. XBAHL-2.0 H. D-O.OB N 62 ^ .0 0 0.25 0.50 0.75 1.00 LENGTH OF BARREL (X/XBRRL) FIGUflE IB . PROJECTILE BR3E DENSITY VERSUS PROJECTILE TRRVEL. PBRS-90M10* PR, tn -0.5 KG, XBRRL-2.0 M, RBRS-61.50 KG/H, 0 - 0 .OB N 63 O o CC UJ LU I. CJ(M 0.25 0.50 0.75 1.00 LENGTH OF BRRREL (X/XBRRL) FIGURE 17. PROJECTILE RCCELERRTION VERSUS PROJECTILE TRAVEL. PBAS-BOmIO* PR. m - 0 . 5 KG, XBRRL-2.0 N. D-O.OB N. INITIAL ACCELERATION IA0J-50I7U N/S 64 Numerical Results j bjf Mijrphy ejt ail' UJ iij in cn°i, CEO m a 00 0.25 0.50 0.75 1 .0 0 LENGTH OF BARREL (X/XBRRL) FIGURE 18. FRBJECTILE VEL8CITT VERSUS PROJECTILE TRAVEL. PBAS-80»10r FA. m -O.S KG, XBRRL-2.0 N, 0-0.08 H. PROJECTILE VELOCITY (UHUXI WHEN BASE CLEARS THE HUZZLE-SOS.IB N/S 65 in 00 0.25 0.50 ___ 1.00 LENGTH OF BARREL CX/XBRRL) FIGURE IB . NOSE PRESSURE OF THE PROJECTILE VERSUS PROJECTILE TRAVEL. PBRS-BOiilO* PR. m -O .S KG, XBRRL-2.0 H. PRNB-I h IO PA. 0-0.00 N 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0 .1 0 0 .05 -0 .0 0 -0 .00 0 .05 0.10 0.15 0.20 0 .25 FIGURE 2 0 . PRESSURE CONTOURS. PBR9- 9 0 *1 0 PR, CTCLE-211, TINE-10.55 ms. CONTOUR INTERVRLM mIO * PR, -0.00 0.05 0.10 0.15 0 .50 0.55 0.30 0.10 0 .05 CD oj -0 .0 0 - 0.00 0 .05 0.10 0.15 0 .50 0.30 FIGURE 2 1 . PRESSURE CONTOURS. PBRS- SO nlO * PR. CTCUE-21V. T IM E -1 0 .7 0 m » . CONTOUR IN TER VR L-I m IO 5 PR. VELOCITY VECTORS TIME (SEC.)= 0 .1070E-01 IC PNC PMC N h C KlC HCA HcA HC> C PNC PHC V(C S«C HC> H.tf HIM C JNC PNC ^ *k *& wefl wn-44 uct uf74 n4 I '- f c 't e t e JC PNC £ ^ ^ MC^ , H i/ , H(? He7He7He7 *7 I WC.5 c.JlC YftC Sjc> v . ; 'hcwc I. HC7 HC7 HC7 MC7. . " 5 " \Iic wc \hc vhc 11 nra mct mc7 Ht7 Hly "7 "7 5 5c 5c 5c y I hoJ hcv hc7 hct hC7 * f "7 T 5c 5c 5c 5 hc 5 c 5 c 5 c Hdr hC7 v t f *7 "T 5c 5c 5c 5 hc 5 c 5 c 5 c JfHC^HI^HCT u£7 ue/ vS f 7 7 5c 5c 5c 5 « c. . HC I HC 2 . 2«<5&h£2u&u&u& jfi Tt 5 c 5 c 5 c 5 HC HC HC : HC J*HCiHf7 h(7 n(7 hC7 hIT "7 1 c 5 c 5 c ^ HC : HC HC I HC j HC^, C < H&4 HCUHJ HC HC '!' H C 3 HSX HCO HC : HC : H Q Hi^Hl^ . >Si HC^HC^HC : HC ; h £ * y icv c4 w v 1X& ’ _ R C _ T H C SC . H C ^ _ „ y ..ycyjc^H BRRREL y c *sc 5*c5 hc4 h RXIS OP SYMMETRY' RXIRL DIRECTION (XI PBRS- 0.9000E*07 FIGURE 2 2 . FLOM FIELD OF CYCLE 21U. UPR0J-301.S2 N/S ■SC" STRONG COMPRESSIVE HAVE FOINT "HC" HERK COMPRESSIVE NAVE POINT -0.00 0 .05 0.10 0.15 0.20 0 .25 0.50 0.10 o to o O O °C3 0 .05 - 0.00 0 .20 0 .25 0 .30 - 0 .00 0 .05 0.10 0.15 FIGURE 2 3 . PRESSURE CONTOURS. P8RS- 9 0 -1 0 5 PR, CYC LE-218, T IH E -1 0 .0 0 flU . CONTOUR lNTERVRL-lMlOf PR, VELOCITY VECTORS TINE (SEC.) = 0 .1080E-01 > J m c > c J n c T T v\ WC Yftc VSC T < 1 5 1 1 A 1 ■ J t J# Im nzI i f 51I I j HC x - r V* V T V q @ 3 S f f 3U(Z i | ^^ ! HC ; HC i * .... ^ y ^ i ■ iUlfTlj^Ul^ h(7 k£7uff uffurfhC7 i i t f A t v T Tic Tic T* T V T T HC ! HC T v T TlcT T V 3h<2H rtH C j H ^ H?7 H # 5.5:1:. r HC ! HC I Jm jLH<2 H<7 H t Z V V T T T T T . ; HC i HC ] T V T T T T V T ! tI .... [HC j HC [ -3 3 Ht3 HS hit T T V V T T T T ! u HC HC ; HC | H C ^ T ... V : HC : HC : T T T T ! ; “c ; HC^ 3 i f u&wi? It t IT T 'T -’T -T " i V MC ' HC ; hc : h c 5 J 1 i 2 iHcjH^H^Hy \ HC • HC • HC : : HC |.HC> 3 < T < - ST T V T- ‘ 2 h C ^ H ^ h 2 h Z T -c IT" 'T"T"< < ■■ . : HCS) HC HC |.HC j HC ...... ^ V i HC H £ \ HC : HC T hq 5 t -' < ■ T < < <-■ u ^ HC J H ^ HC ; HC , HC. HC i H ( 5 ' < ■ ■ <■ ■■ hS hThc 1 X. HC I HCyXtlC j Hft^ HC | SC j h 2 HC < - ■ < - ■ ■ < < < -•< ■ ■<-■ H t Hgr HC i : HC i S I^ H * HC :-£ T ilCjyt H r HC ) BRRREL ■ K^XT" ; - J SC ^ XiT^C>Hc5*cJ ^ hc^ t iZ flZ w ->-> > y > > > » > OVt iic 5c ^^j#c3ic'mcS«i 1 r ■N^.&cScSi ^ ^ 3 c 3 c £ j j 3 c 5 c 5 3 c 5 c & iwis t r afmcTnV RXIRL DIRECTION 00 PBRS- 0.B000E+07 FIGURE 2H. FLON FIELD OF CYCLE 216. IIPR0J-S03.IB N/S ■SC STRONG COMPRESSIVE KflVE POINT "NC" HERK COMPRESSIVE HRVE POINT -0.00 0 . OS 0.10 0.15 0.20 0.25 0.30 Non-self-similar spherical blast . wave solution* 0 .05 - 0.00 -0 .0D 0.10 0.15 0.20 0 .25 FIGURE 25. PRESSURE CONTOURS. PBRS- BOmIO5 PR. CTCLE-217. TINE-10.OS ms. CONTOUR INTERVRL-1M05 PR. PHUZ-21.2VM10* PR. PBLRST- 30.74-105 PR, TBLRST* 0.05 ms , SPHERICAL PSH0CK-2.<1Bm10^ PR VELOCITY VECTORS TIME (SEC.) = 0 .1085E-01 a A A a warn *m erm H C : h c A .*• t. A wA HC7 Nw H(7 Her HC7 7 HC ! H t V A ' 7 7 h ( V w C 7 h c 7 h ( 7 h C 7 i t t T H C : H C H C H C V N > “7 V hC7 h£7 h(7 H(7 t f f vX f H C : H C H C , H C H C i HC J f / . T . 'T WCJ HW K 7 HC? H(^ H C ! H C HC . HC I T A 7 “7 V hW hK H ^ vXf I H C i H C 7 hCJ iiR H g h£ h£ i 7 i i f j n f f njT w(7 hTa i h c : hc 2 . H C ' H C 7 m 2 H £ hJt h!^ i 4 hg$hcV H C ' H C 2 ....2.h§ih2.h2.h2 i I H C i HC V "4 HC- Ht» Hcyuc\ 2 Z h2 hZ h £ i «4HC> H C i H C 7 TH^HglH^H^I H O j H C H C ; H C H C H C “2 2 “2 hs H C !: nil H C ■ HI. J w & ni»“2 2 Z h5 HC -SC^HC* 5 4 UCJh “ 2 2 h 2 h Z h C .HCHCO»CA hcQH c2 ^ S c S c ^ H ScSlC^H S cS cth : m is V SYMMETRY AXIRL 0IRECTI0N 00 PBRS- 0.9000E+07 FIGURE 2 6 . FLOH FIELD OF CTCLE 217. UHUZ-S3B.B0 N /S. TBLRST-O.OS m s. ■SC" STRONG COMPRESSIVE HAVE POINT "NC" NERK COMPRESSIVE HAVE POINT SPHERICAL SOLUTION ■ V RT SHOCK FRONT-23 9.20 N /S. SHOCK FRONT 0.30 ' 0.00 0 .05 0.10 0.15 0 .20 o .as Non-self-similar 0 spherical blast 0.10 wave so lu tio n 0.05 -0.00 0 .00 0 .05 0.10 0.15 o .ao o .as o .30 FIGURE 27. PRESSURE CONTOURS. PBRS- 90«!05 PR. CYCLE-21B. TINE-10.80 ms , CONTOUR 1NTERVRL-1h 105 PR, PMUZ-20.85-10* PR, PBLRST- 3 0 .7 1 -1 0 * PR, TBLRST- 0 .1 0 m s , SPHERICAL PSHOCK-1.75-10* PR VELOCITY VECTORS TINE (SEC.)= 0.1090E-01 H C { H C H C H C V A .^ h^ ^ T ^ » W hc1 I MC j .HC HC , HC H C ; N C 3 2 HC7 hC7 Hf7 h & ti c j I H C ; N C H C N C H C j HC 3 2 “S “2 ** hc i i H C I H C > H C 2 2h(V w£7 u ( 7 ' m ( T t i s i LHC : HC j HC 2 2 2 h C / h ^ h !7 h s j H C ! H C 1 * 2 h2 m2 “2 he J HC , NC Z h 2 -m 2 h 2 h ? ’ h c : H C H C ; H C 2 w2 h2 h2 nr A MC i u c HC HC HC HC HCV H C : H C ; H C H C : H C ? i g i f :i Sic Sic H C : H C s c w h c ; h c ® HC 1 “c Urf '7 ur7 H C ; H C u&uZu !.... i- ...... ! BRRREL....I HC^H^H ______u - i — > c^ hc^ h s t i .... j .... | .... !.... I.... :..... t-.... 1...... 1... h ... j - ...... -i-... f... j...... j...... ScQicCi bi::±: ScQcCi .....^RKlS OFSTHHETRY...... RX1RL DIRECTION 00 PBRS- 0.9000E+07 FIGURE 28. FLOH FIELD OF CYCLE 210. UHUZ-360.22 M/S. TBLRST-0.10 w s,. ■SC* STRONG CONPRESSIVE HRVE POINT *NC* NERK CONPRESSIVE NRVE POINT SPHERICHL SOLUTION i V RT SHOCK FR0NT-1U1.50 N/S. SHOCK FRONT 75 rtRS- 0.90E+D7 RON-7 Noh-self-simllar spherical blast22 wave so lu tio n D-t=b. DO 2.00 4.00 6.00 RXIRL DIRECTION (X/D) FIGURE 2 8. AXIRL DISTRIBUTION OF PRESSURE. T-O.OS M, D-O.OB H, TBLRST-O.lOm*. NUZZLE AT X /D -2.SS . BR3E RT X/D-S.OO, SPHERICAL SHOCK LOCATION X /0 - 3 ,5 5 76 PBRS- 0.90E+07 RON-10 m CD I—t »-« DC I- Non-self rsimilaa: .. cn spherical] b la st^ Wave sdlutlorj 2.00 U.00 6.00 RX1RL DIRECTION (X/D) FIGURE SO. RXIRL DISTRIBUTION OF PRESSURE. T-0.0V5 N, D-O.OB N. TBLRST-0.10 m s. NUZZLE RT X/D-2.SS. BRSE RT X/D-S.OO, SPHERICRL SHOCK LOCRTI0N X/D-3.4-Z -o .o o o .o s o.ao 0 .30 150000> ^ ^ LNon-self-similar spherical blast y-T 0000 0 ° O wave so lu tio n / ^ 0 . 0S - 0.00 - 0.00 0 .0 5 0 .20 0 .25 0 .30 FIGURE 31. PRESSURE CONTOURS. PBRS- 90*10 PR. CTCLE-219. TINE-10.95 ms . CONTOUR INTERVRL-5H10* PR. PHUZ-23.01NI0* PR. PBLRST- 30.7«m05 PR, T9LRST- 0.15 ms . SPHERICAL PSH0CK-1.50n 10S PR o o O SHOCK TIME (SEC) * 1 0 * _ • 1.00 2.00 3,00 4,00 " o — r j IUE POETL TR VRU POETL TRAVEL RND PROJECTILE VERSUS TIRE PROJECTILE . 2 3 FIGURE EGH F ARL (X/XBRRL) BARREL OF LENGTH for the qABl® i n r i g f / 0 i ® l B A q e h t r o f 4 ^|)erimefitalrfeaiiits t i i i a e f r l a t i f e m i r e ) | ^ I rath f r m jro fS . h t a fr e i l t p e j r f r P i .5 .0 .5 1.00 0.75 0.50 0.25 B H 0B . 0 - 0 PBRS-1500 SHOCK VERSUS TRAVEL. SHOCK TINE 1ELNT O POETL RD SHOCK RND PATHS PROJECTILE OF T1HE-LENGTH h 1 g F R O. G XRL20 H. XBRRL-2.0 KG. .S -O m PR. i .. 78 79 Q_in a - o iu cn 00 0.25 0.50 0.75 1.00 LENGTH OF BRRREL (X/XBRRL) FIGURE 3 3 . SHOCK PRESSURE VERSUS SHOCK TRRVSL. PBR3-1500M10* PR,m-O.S KG. XBRRL-2.0 N. 0- 0. OB n , PHUZ HHEK SHOCK IS RT MUZZLE-7.99E5 PA 80 tu UN DJ- a oo 0.50 0.75 1.00 LENGTH OF BARREL (X/XBRRL) FIGURE 914. SHOCK VELOCITY VERSUS SHOCK TRRVEL. PBR3-1500M105 PR .m -0.5 KG. XBRRL-2.0 N. D -0 .0 0 N. SHOCK VELOCITY (HHUZ1 NKEN SHOCK IS RT NUZZLE-89B.5B N/S 81 o LU c o o LU oo 0.25 0.50 0.75 1.00 LENGTH OF BARREL (X/XBRRL) FIGURE 3 5 . PROJECTILE RRSE PRESSURE VERSUS PROJECTILE TRAVEL. PBR3-1500H105 PA.m-0.5 KG. XBRRL-2.0 N, D -0 .0 0 N 82 M . z LU 00 0.25 50 0.75 1.00 LENGTH OF BRRREL (X/XBRRL) FIGURE SB. PROJECTILE BASE DENSITY VERSUS PROJECTILE TRAVEL. PBA3-1500N10* PA.m-O.S KG. XBRRL-2.0 N. RBRS-542.BS KG/N , D-O.OB N 83 ?.. O l - t n “ - o CC LL) 00 0.25 0.50 0.75 1 .0 0 LENGTH OF BARREL (X/XBRRL) FIGURE 9 7. PROJECTILE ACCELERATION VERSUS PROJECTILE TRAVEL. POAS-1500-10* PA.m-O.S KG. XBRRL-2.0 N, D -0 .0 6 K. INITIAL ACCELERATION (AO!-811105 N/S 84 : ...... i..... > 1 s « rw • 0* : a \ ..\ 0^ y = 3 ° / Nume r l cal re su l ts 1 Q Murphj et a] * X W ..9 ..i._.... / / - 5 r *i DO / . m ..... cn°i_ d o CD ].. 1 j . s 1 ; i LENGTH OF BARREL (X/XBRRL) FIGURE 8 0 . PROJECTILE VELOCITY VERSUS PROJECTILE TRRVEL. PBRS-lOOOMlO5, P R .m - 0 .s KG. XBRRL-2.0 N. D -0 .0 8 H. PROJECTILE VELOCITY (UHUZ) MHEN BASE CLEARS THE HUZZLE-777.aU N/S 85 O IU. m 5-s o _ « Hi =5§ C n ° 10(0 u i cc 0.25 0.50 0.75 1.0000 LENGTH OF BRRREL (X/XBRRL) FIGURE 3 9. NOSE PRESSURE OF THE PROJECTILE VERSUS PROJECTILE TRRVEL. PBRS-1500M109 Pfl.m -0.5 KG. XBRRL-2.0 N, PRMB-In IO* PR. D-O.GS N 0 .00 0.05 0.10 0.15 0.20 0.25 0.50 F I 1------1------1------1------1------1------1------1------1------1------1------1------1------1------1------1------1------1------1------1------1------1------1------1------1------1------1------1------I------I------1------1------1------R 0 . 1 0 - Non-self-similar spherical blast wave so lu tio n 0 .05 - 248000 0.00 kfc. t I :: 1=:. I 1=1 - I I ■ L i ______| ______| ______J ______I— I ______| ______| ______1 = -0 .00 0 .0 5 0.10 0.15 0.20 0 .25 0.50 FIGURE 4 0. PRESSURE CONTOURS. PBRS-1500h 10* PR, CTCLE-60, TINE- 3.00 ms , CONTOUR INTERVflL»5MlO* PR, PHUZ-IO.SS mIO9 PR, PPRECURS0R-7.99N10* PR, TPRECURSOH-0.05 m * . SPKERICRL PSH0CN-2.4B-10y PR VELOCITY VECTORS TIME (SEC.1=0.3000E-02 HC S C 7 H ! 7 { f S f/ H (7 HC s(7 hIT^ h(7 y H T h m i s b p symmetry RX1RL DIRECTION (XI PBKS- 0.1500E+08 FIGURE I I . FLOH FIELD IF CYCLE 60. UHUZ-7W.86 N/S. TPRECURSOR-0.05 m j. ■SC* STRONG COMPRESSIVE IWVE POINT ■NC* HERK COMPRESSIVE HRVE POINT SPHERICRL SOLUTION i V RT SHOCK FRONT-111.50 H /S. SHOCK FRONT 88 PBRS-0.15E+QB now-10 m z CE Q. \ o LO W |o' 00 N on4self-s Im ilaij CO spherical blast^ wave solution C lo iijiu a z CO k z ( n _ L U g C l 00 2 .0 0 U.00 6.00 AXIAL DIRECTION IX/D) FIGURE U2. RXIRL OISTHIBUTIQK OF PRESSURE. T-0.0U5 N, 0-0.00 H. TFRECURSQR-O.0 5 m», HUZZLE RT X /0-2.SS, SPHERICAL SHOCK LOCRTION X /0-2.33 -0.00i » i 0.05 i i i i i 0.10 i ' i i * i—i 0.15 'i "i i ■ i i 0.20 i i i 1 i"i 0.25 i i r i 0.50" i i i i i 0.55 i i i i i Non-self-similar spherical KLastg, wave so lu tio n FIGURE «S. PRESSURE CONTOURS. PBRS-1500Mi05 PR. CTCLE-W. TINE-3.2 ms. CONTOUR INTERVRL-SolO* FA. PNUZ-I2.B0*10^ PR. PPRECUR30R-7.90*10^ PA. TPRECURSOR-O.25 m s, SPHERICAL PSHOCK-I.31*10* PR *0.00 0.0S 0.10 0 .1E 0 .20 0.2S 0.30 0 .1 0 0 .0 5 -0 .00 - 0.00 0 .0 5 0.10 0.15 0 .20 0 .25 FIGURE U . PRESSURE CANTOURS. PBRS-1500 n 105 PR. CTCLE-71. TIME- 3.55 mS, CONTOUR INTERVflL-2iiiO* PR. VELOCITY VECTORS TIME (SEC.) =0.3550E-02 I «0T w Mef mg' mb' uc Sd* f M B' r H HC stf .se* ' h H C ? H C J ...... _ / J rse; s s££ HeZ "L. s " M B ' H t 7 sQ^HW h£ * sc* sc£ h . Js^scf! ,H^ > >■> 3 ' ^sc^eT s si ^sr ry ^ si :x::i BxiaBfamemf RXIRL DIRECTION 0 0 PORS-O.150OE*Ofl FIGURE «5. FLOH FIELD OF CTCLE 71. UPROJ-771.23 N/S •SC* STRONG COMPRESSIVE MOTE POINT ■HC* NERK COMPRESSIVE MOTE POINT -0.00 0.05 0.1 0 0.15 0 .20 0.25 0.30 t— i r i— i— i— i— i— i— i ii—r ■ i— i i i— i— i— i— i—i— i 1 i i i i i i i r j i i= 0.10 0 .0 5 -0 .00 FIGURE W . PRESSURE CONTOURS. PBRS-1500n 105 PR, CTCLE-7Z, TINE- S.60 m*. CONTOUR INTERVRL-2h 105 PR. AADIRL DIRECTION ITI XS F STHHETRT1 OF AXIS (...... I— '( • BARREL EOIY VECTORSVELOCITY ...... FIGURE 117. FLOH FIELD DF CYCLE 7 2 . UPR0J-777.3U N/S N/S UPR0J-777.3U . 2 CYCLE 7 DF FLOH FIELD 117. FIGURE « ...... C H C H j O v C 3 c ) J c S c i 5 ^)c \le vfc.\)c. Sic. ^)c V \le V Xflc\fcc ^c c C :3c. SCHC^.. c 3 ^HCyHC N" NERK COMPRESSIVE NRVE POINT "NC" S* STRONG COMPRESSIVE HAVE POINT"SC* XR DRCIN (X) DIRECTION RXIRL 5 N.nj scucv sc C H V C l S J C 1 c! ^ '* e u S C c i S c c i S N K T c c m ■> > > > > > > A c , c s i A A A c h c h c h h c h c m / ^ / O c h * g m / o h O H * G M / C H V > o h tief c h c h / / / * e H Fsc g m o h * * g m g m | SOT 50*| I MOTHO* C H T O H % ? f H C H T O T H O H * G J H O T M O T S O S " ^.^c' .c^ c^ ‘"^ S# HI H # S , * C s L ^ l I Z l 7 i f IE (SEC.1=0. TIME hc mot I ill sc ' OS O.ISOOE+OS TORS- T“ J "" ' I 3600E-02 ...... ; .. . REFERENCES 1. Schmidt, E. M., E. 3. Gion. "Muzzle Blast of 30-mm Cannon," A RBRL-MR- 02805, USA Ballistic Research Laboratory, Maryland, January 1978. 2. Traci, R. M., 3. L. Farr and C. Y. Liu. "A Numerical Method for the Simulation of Muzzle Gas Flows with Fluid and Moving Boundaries," Science Applications Inc., El Segundo, California, SAI Report No. SAI- 74-513-LA, May 1974. 3. Edquist, C. T. and G. Romine. "Muzzle Blast from Canister Launched Missiles," AIAA Paper No. 80-1187, AIAA-5AI-ASME 3oint Propulsion Conference, Hartford, Connecticut, 3uly 1980. 4. Moore, G. R., F. A. Maillie and G. Soo Hoo. "Calculation of 5"/54 Blast and Post Ejection Environment on Projectile," Naval Weapons Laboratory, Report No. NWL-TR-3000, 3anuary 1974. 5. Zoltani, C. "Computer Simulation of Intermediate Ballistic Environment of a Small Arm, M-16," USA Ballistic Research Laboratory, Report No. ARD-76-1, Maryland, 1975. 6. Erdos, 3 .1,. P. D. Del Guidice. "Calculation of Muzzle Blast Flowfields," AIAA 3ournalt Vol. 13, No. 8 , 1974. 7. Moretti, G. "Muzzle Blast Flow and Related Problems," AIAA Paper No. 78- 1190, AIAA Eleventh Fluid and Plasma Dynamics Conference, 3uly 1978. 8 . Westine, P. S. "The Blast Field About the Muzzle of Guns," The Shock and Vibration Bulletin, Vol. 39-6, March 1969. 9. Schmidt, E. M. and D. D. Shear. "Optical Measurements of Muzzle Blast," AIAA Journal, Vol. 13, No. 8 , 1975. 10. Winchenbach, G. L., 3. S. Kleist, D. G. Galanos and G. E. Parrish. "Description and Capabilities of the Aeroballistic Research Facility," AFATL-TR-78-41. Eglin Air Force Base, Fort Walton, Florida, February 1978. 11. Wilenius, G. P. "The Constant Base Pressure Light Gas Gun," Proceeding of the Third Hypervelocity Techniques Symposium, Denver, Colorado, March 1964. 12. Hunt, F. R. Internal Ballistics. H. M. Stationary Office, 1951. 13. Lagrange, 3. L. Journal Ecole Polytechnic, Vol. 21, 1832. 94 95 14. Love, A. E. and R. P. Pidduck. Philosophical Transactions of the Royal Society, A222, 1921, page 167. 15. Kent, R. H. "Some Special Solutions for the Motion of the Powder Gas," Physics, Vol. 7, 1936, page 319. 16. Vasiliu, 3. "Analysis of Chambered, Finite Length Hypervelocity Launchers by The Method of Characteristics," Republic Aviation Corporation, Report No. RAC 1617, 3uly 1963. 17. Seigel, A. E. The Theory of High Speed Guns. AGARDograph No. 91, May 1965. 18. Piacesi, R. "Computer Analysis of Two Stage Hyper velocity Model Launchers," Naval Ordinance Laboratory, Report No. NOLTR 62-87, February 1963. 19. Murphy, 3. R., L. K. Badhwar and G. A. Lavoie. "Interior Ballistics Calculation Systems for Light Gas Guns and Conventional Guns," Computing Devices of Canada Report, 1970. 20. Corner, 3. Theory of Internal Ballistics of Guns. Wiley, New York, 1950. 21. Longley, H. 3. "Methods of Differencing in Eulerian Hydrodynamics," Los Alamos Scientific Laboratories, Report No. LAMS-2379, New Mexico, 1960. 22. Sedov, L. I. Similarity and Dimensional Methods in Mechanics. Academic Press, New York, 1959. 23. Thompson, P. A. Compressible Fluid Dynamics. McGraw Hill, New York, 1972. 24. Johnson, W. E. "OIL, A Continuous Two-Dimensional Eulerian Hydro- dynamic Code," General Dynamics, Report No. GAMS-5580, AD- 477-240, 3une 1965. 25. Peyret, R. and H. Viviand. "Computation of Viscous Compressible Flows Based on the Navier-Stokes Equations," AGARDograph 212, 1975. 26. Baldwin, B. S. and R. W. Mac Cor mack. "Numerical Solution of the Interaction of a Strong Shock Wave with a Hypersonic Turbulent Boundary Layer," AIAA Paper No. 74-558, Seventh Fluid and Plasma Dynamics Conference, Palo Alto, California, 1974. 27. Lomax, H. and R. W. MacCormack. "Numerical Solution of Compressible Viscous Flows,” Annual Review of Fluid Mechanics, Vol. 11, 1979. 28. Cline, M. C. "VNAP: A Computer Program for Computation of Two- Dimensional, Time-Dependent, Compressible, Viscous Internal Flow," Los Alamos Scientific Laboratory, Report No. LA-7326, New Mexico, 1978. 96 29. Hoffman, 3. D. and M. 3. Zucrow. Gas Dynamics, Volumes I and II, 3ohn Wiley, 1977. 30. Michels, A. "The Rapid Expansion of Compressed Gases Behind a Piston, The Effect of Molecular Interaction, Part II," Physica, Vol. 20, 1954. 31. Courant, R. and K. O. Friedrichs. Supersonic Flow and Shock Waves. Interscience Publishers, New York, 1948. 32. Richtmyer, R. D. and I. Von Neumann. "A Method for the Numerical Calculation of Hydrodynamic Shocks," 3ournal of Applied Physics, Vol. 21, 1950. 33. Hilsentrath, 3. Tables of Thermo and Transport Properties. NBS Circulation, Pergamon, New York, 1960. 34. Hartree, D. R. "Some Practical Methods of Using Characteristics in the Calculation of Nonsteady Compressible Flow," U.S. Atomic Energy Commission, Report No. AECU-2713, 1953. 35. MacCormack, R. W. "The Effect of Viscosity in Hypervelocity Impact Cratering," AIAA Paper No. 69-354, AIAA Hypervelocity Impact Conference, Cincinnati, Ohio, April 1969. 36. Moretti, G. "A Critical Analysis of Numerical Techniques; The Piston Driven Inviscid Flow," Polytechnic Institute of Brooklyn, Report No. PIBAL 69-25, 3uly 1969. 37. Roache, P. S. Computational Fluid Dynamics. Hermosa Publishers, New Mexico, 1972. 38. Trengrouse, 3. Wave Propagation Class Notes, Birmingham, U.K., 1978. 39. Mark, A. "Computational Design of Large Scale Blast Simulators," AIAA Paper No. 81-0159, Aerospace Sciences Meeting, St. Louis, Missouri, 1981. 40. Beam, R. M. and R. F. Warming. "An Implicit Finite Difference Algorithm for Hyperbolic systems in Conservation Law Form," 3ournal of Computational Physics, Vol. 22, September 1976. APPENDICES 97 APPENDIX A 38 ANALYSIS OF ONE-DIMENSIONAL STEEP-FRONTED WAVE Moving shock Stationary shock Let c be the propagation speed of a shock with strength p^/pj and u be the particle velocity behind it. Assume an adiabatic and inviscid flow of ideal gases. Continuity fjC « P,(c-u} (1) Momentum r,-r» = (2) Energy * ft + _C* (3) t - l Z e, 2 where p is pressure, f is density, and f is specific heat ratio. Substituting for u from (2) into (1) and (3), and substituting 1/f from (1) into (3), it may be verified that for air { t = 7/5) — - ( ) u _ 5(P./P» -O (42 (P,/P*} So, if no attenuation is assumed, the location of the wave can be found at any given time span, A t 98 APPENDIX B 38 ANALYSIS OF ONE-DIMENSIONAL NON-STEEP WAVE P .a .u If we regard a non-steep wave profile to be made up of a large number of small pressure increments, then each of these increments may be regarded as a small disturbance propagating with a velocity, relative to the gas particles, equal to that of the local acoustic velocity. If the flow is assumed isentropic, then (1) d« “ f P and * (2) Putting (2) in (1) and integrating from ^ to Uj and from Pj to pj, I d U. °Z [(fi/P.) -I] Now, since the speed of propagation of a non-steep wave is c = Uj = a^, C [-Vt IhIW * --^ -1 (4) 99 100 For air, Tf = 7/5, (3) and (4) become, U, = SOj I (P./P,) '* 1 - l] t = a, I 6 ( p, / F, ^ - 5 ] . APPENDIX C IEFLOW COMPUTER CODE LISTING 101 f t f t f t f t f t f t f t f t f t f t f t f t V s \ N >» A A r o A Vw n r> f t » > > > > o n f t f t f t o p i f t A f t A c r> X a ** *#«# » * *# n D n Z JO X a c o n □ n n c n o a o f t f t o c n r * *# # * * »* * z X X AA A X X X X X X XX OD X O A X a > * * * * XXX (A r r M A X X X X XAX X f t X A A X m p * # * # a n n a XX 0 ) A □ n o o a • n o X o AA n a * # > * *# z z z O' a AA O ' z z z z z D z z z A o z a p * * ** s \ s O o< 01 01 o s s s s < \ N s O a \ P a * p f t A * * o 6 a a o o o o o r i < < < X < < < a O' < m « * * o m * * o> 01 * w o o o A I d n ( 0 O' t n A - 4 CjJ r o O ' o A z p * * c a ** \ \ A a A • \ z s •v V a S s o A s C l X * # m p ** XX T • • • A AA w X o o o a a A a 1 ) A * « * o * # X ■D X > > A c - 1 a A a A A r o X X a < f t X p #*A c * * c • A a X a A A r A A < X A a a A f t a a **O # * N X A A A O ' O ' n f t • A A a X c A A O p * # X * « c -1 X 01 01 0 1 o o a o 01 O A 01 a X O O a a « * a #« N X n o o o o o * A o • (A o a a a a O' X • * c * * a • X o o o A A A •A (11 a A a f t O ' O ' o a # • A ** r *n • A A A• • m Ui It o a a < X o o A r *» m A#* r > X • • • A A a A A A a O a • A A a a * * m # * * a > > > > C X m A a • a o i a < a a a X ** A a ## r > X c a A A A A A > A a X > < f t a * # z p # * r a z * A A A A • O A a A a 0 ) f t f t p * * A o #* >— m • (A r A o A m • A i A a f t m A A a # » m X * * « > > a X A A 01 ro O' O 0 ) O XX A A A X* * r n a * * X • ■o 0 1 A 0> 01 o • o • A a a a A A OC * * > z * * X m n o ttl o o o m A O'A O' a a A O a a ** a > ** A o « o a o o A u It o A o a X C a O' a * « a p # * • A A a n A A • * r A o A H m a O' o X * * tn * * * XX •A• • A m P • a a X f t O' o A a * * p > # * X X > • > A XA t o A O' o a X o A a a # * t n z * * c c > a C A • • X o A a a A a c a * * o * # N p > A A A C l X A A A a A a a f t X * * *# a 0 ) T> A A A n X A • A a X > f t 03 a * # m ** • A A > f t A • 01 A > A a a f t o A a • X A * * X 0 1 0 1 A A 01 o X o a f t A A A a * *A m # * X e o 0 1 O' o r A c a a X A A A X » • m a * * c o o o o o •* • O i a a A A o a « a p * » N a a o o A 0 > > o C a O O a a * » z o *# id • « A A a z A f t A A a a a O' X * * > m ** • > > • a A • f t X a a 03 O ' 0 i o m # » p »* r a ■o > AX o A o a f t o o A X * * ** X p A a a AA A c a X A A a A * * • n *• c 0 ) P A A A• A X < a a a A * * p « »* N A X A A CD C l O UI A C X a a # • a * • a O' A CD A A* A a a A C n X * * m •# • o 01 A O' 01 a « o a 03 a A A # • in « # r o o 0 1 o o c i X < a f t f t A a * * A * # X A o o o o • X X a XX A X » * A m «# c •A o AA C l C l t i t a a O M * *Z a • * N l > •A • • A a a < a a * # p #* r o ■ > • A A a O a A O' P«* « C l a ** * A ■o > a X C l c a a O > #* m o « c X* * z 0 ) A a A A r o A A n A a »» > X * z * » m A a A A A • tn A X a X*•A A* * * # z o » X A A f t a a a a c # * tn > * * * • o A n A A C l o a a a ** > A* * * o o 0 1 A O' 0 1 A < A P#* r * • # X a o 01 e o • a > * * * * * * * * It • o o o o a O l f t a • * * ** * * * • o A o A A C l a X m A • A • • t o a a a a e O o o e o o o o o o o o o O o o O o O O o O o e o o O o o o o o o o o O o o o o o e o o o o 9 o O o o P o O O o O o o o p o o o o o o o o t n CJI 0 1 01 0 1 o r 01 o r 01 A A A A A A A A A A 0 1 u 01 o i u o r O l u 0 1 01 t o r o t o t o r o r o a> >1 0 1 01 A u t o A o 10 ® N 0 1 01 A l i i t o A e o s s O' t i t P O i t o A o 201. n n ft s s r z i« m < ■ n o © m ■Tl a a a a a a a a a a o n ft X ft o ft r n * T n « o x < I ro o ro ro m m z m m m m m ro tm o O 3 o • o i l 2 c < - 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X 4 I ■U U .P P .R R ,P B A .P N O .U E .P E .R E 1 0 0 2 5 0 0 0 2 I*************************** 4*********************** 0 0 2 6 0 0 0 2 0 0 2 7 0 0 0 7 SUBROUTINE EX FLOW COMPUTES THE EXTERNAL PART OF THE FLOW 0 0 2 8< 0 01 0I 7 O 1 n ft o n o ft n © 1 o O O z X “4 z 9 Z 9 9 9 PI c PI 9 pi z o Z m r m > p X r CA z © IX X c O 01 ID 1 O © o 1 o o ID • X • X C 73 zz > ro PI 7 \ < \ 9 ft IX ft IX o ro ro o ft X XX |X X o XX c ftm •4 X XX © o XX XX XXX O XX X XX XX X © O O X xx © ID o 9 XX ro o > XX ft XX ro 9 ft c X* ft ro © < ft ma r XX X o ro o ro o o o XX X XX X XX Xx XX X XX o X Xx X a XX »x ►X o X XXX f t XX xx *x o ID o o X X XX X > < XXXX ro H ID ™C ro ft © «x XX (X O ro o o > ro o z X Xx o o X o XX X XX XX • f t XX a XX X X XX XX XX XX X c • ft Ul © ID ID XX X xx XX ft < X CD X o fz t ft ro n PI z V 9 X X a n >< X ft 9 X ft ID CD f t ft r ft X X a ft H X X 9 © X C < ft 5 K X -< o o p X X X X CD © ft © X X * o o o ID X X o < X H ft z Ul XX o o o 9 9 9 9 9 9 9 9 X 9 9■4 9 X 9 9 ■< m 9 •< 9 < < 9 9 X 9 a n z s ii o □ ro ro IO p p n o X X p o nt ro IV) •a *» •• X 9 X 9 O X P A m • p • r i- • A w 1 * * + * 9 9 x X r 9 X •n •n o o 0 r o 9 z W* ►* X » r x p > X o o® ot oat o ui e o o o o o o o • o o o o m • r • p 9 ro • r a # «■» - 9 » x • N •* N O •* tO o oio ro o io ro o V p i I I I ii ci IT ci ii » » * + x ♦ X X 9 9 9 o o X X r o 9 9 r • r *• it >• 7 r ro ^ z IO PP H • P • 9 nt w a i ' H « " I • 1 II * < X •< p a a z z X ro to ro n £ t— 01 0t O 9 o o o O o o o 0t © 0 01 o m at ui o * o w o o o o o o oro ro o o io io o n Ii X z IO o II 9 9 9 0 9 9 9 p p -t X ra m II c XT X m c c in l/s n n o II 9 9 9 i n n X CE "t| 0t 0t 0k O lO B ^ B lO O ! 1 n n 9 • rn m H > II * N 9 9 X 9 9 9 Cl V %0 V A ll o o o © o o o o o © o o o o oo S o o o o " 0 a t X U M o o o o ooe oo oo o o o > a s o 9 # o o o o o o o o o o o o io io ro io ro ro io 9 > -4 II > 9 XX m II II 9 9 n -4 M 9 > OD a a II -0 <0 -0 -0 <0 -0 XX c c II ■ 0 o t U l ♦ X a a o o oo o o o o o o o o o o o o o o io ro io to > te II > 9 H *0 ■o9 9 > a > X n o > c 0 t p p TO ** « -o s II X at X nt 10 IO • O t0 a a t0 O fr x x O O o o O O oro ro o ro ro o • Z r N Cl O to ( i t o o o oo o o o a> m o o o o P • X 0* c H P r • P C to X A II X 9 o o c ••o II X • > * *■«o o o o II 9 o X X > U) P o • o • H e o o o ■r • rn • m o o ro o> o o oo o o oio au * o aui a a o o o o a oio to o io io o > z a • 9 p o • «« p • p r M M ** A il 10 > m n • w rn 11 * r 9 X X • XXX • r w ii ® o a a o ® ot a m o oo o o o o o o io io io > O O o o o o < r p P OD m < X A * ** 9 m x X • z o • X • a m • x > • rX X N o o • c N Z a • r A • m c r ■ ** to * IN o o o o O o eo o o oIO to r n x r x • o £ • o «■» o o a «o r r V z o • r O m • r X > n n n a * rn w X •M • ro IF( < CL.EQ.LKU2).AND • (M.LE.MMUZ ) .AND.(M.GE*MI MJ.OR*( (M.EQ.MMUZI 0 0 9 1 0 0 0 2 • • A N D *(L.LE.Lf'UZ)>) VC(L.M)=0.0 0 0 9 2 0 0 0 2 IF(M * E Q • 1 ) VC(L.MJ=C« 0 0 9 3 0 0 0 2 d l l u > z z z 0) H4 a • m < ft o ro z w • ** II • X > r z a ft w ts « A -4 ft X ID * ■ II II > > a a > > ft « CD w z z z 10 01 f t ft c m -t r r < h rn Zzc t < □ X x a r» r c a Oi M ft ui X z X m a ft • H a • ** • • II * ui o ft a H • •4 o t- o V ii a m > II a H h ** H c > H a * H > a n > r> r ft in c 00 H ft a ft (0 M a ft N a n n p ft a r m D a o tn m o a z m a a h a o m m r Z H m i n n ft ft ri ft n ft H m a a a < • • o mo ft o *» O IO Cl a z w f t o > z x a a c a r ft o in m r o H Z > H ft II ft a a o < m m H c a oi m m oi ( J A a « a n « • « □ « ro ro • • w • « — z— « * H •— ao * * o ii ii ii h »• ti i • 1 i • 1 c a a a a a c a a c a c 7i a * s \ \ H HHHH W ft m ft "ICDSa* > a C H S 3 (A * * in a o a a ** a a a a a IR in a ft z il a -» OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOft*.UrOH-OiO®-«Jftft#liJf1)»-OlOOD^OiOI*WlO*-01£l®*>IOift> oeoooooooeooooooooooooooooeoooooeooeooooooooooooooooooooooooeoooeoooooooooooooooooooooooooooeoooM h)(oioriJtoioroMioroioroiofo(oiofori)*jfo^roioioloroiororoioro miuiuiuMiiHXH»Hi.>.HNHoooaOoooooifiDiCiAiCifi FP=PPF+NS*SFP . 01260002 liU=(JUP+NS*ELL 01270002 2 = 1 »-RR*BET A 0 1 2 8 0 0 0 2 6TT n n r> o o n ICO 5 CONTINUE *NE) *0 *0 *MC1 (L«M DERIVF CALL CONDITIONS RCCLMLZfM J=RR RST SE CLUAIN (RDCO STEPl (PREDICTOR CALCULATIONS STEP T S IR F ARTVIS CALL ACLTOS F H AXI F YMTY CNE LI ) E IN L (CENTER SYMMETRY OF IS X A THE OF CALCULATIONS ( 8 C A )+ Y )*Z .M (L e C (V S E A )+ X )*Z *K 1 (L E C I*L*M C (U S *M B 0 A * = 1 D ( D C EQUATN CALL M=2*MM1 5 DO CONTINUE T )=T .M Z U (LM C T 0 M=1«MM2 2 00 O *LL1 L 2 L = * L 3 2 2 DO D D 0T=1*/C C 0 L=2.LL1 5 COMPRESSI 00 ANY OR SHOCK SMOOTHING FOR TERMS SITY O ISC V L IA IC IF T R A UC(LMUZ*M)=UU CCNTINUE M A = T D E MC1=M—1 L/AA EXIT=U M ) G yR R /R P *(P Z = T T C(MUZH)=FP Z*H U (LM FC AL HE P. .AA) R (PP.R O M ER TH CALL AC (LMUZ »AC (LMUZ M ) = AA >=C*0 UZ*M VC(LM AND*. 1M.NE.MM2 ) IF((M*GT.M1N AND*L* LBA)* (M*E I 1 GT 500 GOTO 11 MIN *LE* M *( D N *A ) A B *L E *G *(L D N *A 1 O N L * E L * L ( ( F I I EC- 1 OO 2231 GOTO 1 -2 C .E N L T IC ( F I 500 TO GO ) ) Z U M .M E .L H .( D N .A ) Z U M .L E .L L ( ( IF F( EC. N) CL ZM1=0*0 0 = 1 UZ*M UC(LM ) IN .M C .E (M IF F( GT 2230 3 2 2 GOTO ) Z L M L * E L * (L IF NIEOT CDT) T D *C T O I(E IN GOTO ) ) 2 L*M L*M S T * J 2 4 4 Y Z + **2 X Z RT( 1 *SQ 01420002 N D 2 0 0 0 2 3 1 0 2 0 0 0 9 2 1 0 9 0 0 0 1 6 1 0 2 0 0 0 9 4 1 0 2 0 0 0 4 4 1 0 2 0 0 0 9 3 1 0 2 0 0 0 5 3 1 0 2 0 0 0 6 5 1 0 2 0 0 0 4 5 1 0 2 0 0 0 1 5 1 0 2 0 0 0 1 4 1 0 2 0 0 0 4 3 1 0 2 0 0 0 0 3 1 0 2 0 0 0 2 6 1 0 2 0 0 0 0 6 1 0 2 0 0 0 9 5 1 0 2 0 0 0 8 5 1 0 2 0 0 0 3 5 1 0 01S20002 2 0 0 0 8 4 1 0 7 0 0 0 7 4 1 0 2 0 0 0 6 4 1 0 2 0 0 0 5 4 1 0 2 0 0 0 3 4 1 0 2 0 0 0 8 3 1 0 2 0 0 0 7 3 1 0 2 0 0 0 6 3 1 0 2 0 0 0 1 3 1 0 2 0 0 0 3 6 1 0 2 0 0 0 7 5 1 0 2 0 0 0 5 5 1 0 2 0 0 0 0 4 01 2 0 0 0 3 3 1 0 2 0 0 0 0 5 1 0 120 121 (VCMPI PI pi CMPI CMCMpi CMpi pi PI PI PI PJPIPI PI pj PJPI PI pi PI PI |N> PJPI PI PI 1 o o O O o O OOO o o e o OO o o o o o o o o o o o o © o o o O ( o o o o o o o © o o o o o o o o o o o o o o o o o o o o o o o O I o o o o o o o o o o o o o o o o o o e o o o o o o o o o o o o O I ♦ in 10 N <0 O' e •4 pi n * in 9 s <0 O' o *4 pi r»9 in 9 f* 9 O' o 44 PI n 9 in ' <0 CM # > N 01+ a * z * o X o N CM n UJ O' oe H a 4 o in 3 »- * 3 a 8 J9 9 < u UJ O Z 9 a o « 9 a CM Z o < z CD 4 4 4 UJ •4 • m + UJ O ' UJ 44 H UJ p i a X J X W O -J p i > 1 - o • 4 H • N H X J a X o * 4 o 44 44 □ o — P A Q 9 • * 9 h 9 • □ X a 3 9 9 9 Q 9 4 < 4 4 Z a < in * 4 UJ ^ Z J 44 < \ 4 4 < (E -» 4 44 a N 4 UJ 4 DC N • 44 44 9 z 3 4 4 XX 9 4 U 3 <■* 4 UJ 9 < X < • 4 * 9 X 4 9 z > X 9 3 J H X CD _ | • 4 4 44 in 4 J 44 4 4 a 4 • 3 • o X 9 X UJ • 9 9 □ a in U J • UJ 4 4 U l a - J UJ 9 9 a DC Z J UJ U l o _J 4 UJ p9 • 9 a a 4 U • 9 • • 4 + j — 9 X • s 4 in a X • 44 _ l P I O' 44 - 1 4 4 a a - J X o X 9 • 44 X X 3 4 4 • 44 •MC1.0.0 ) 01970002 • X X r*i z U O • < 4 4 X 3 o • ¥ 4 Q 4 * J-H z a J-* 3 J 44 3 3 z a Z 44 44 44 a X O < z UJ W 44 a 4 9 < z 4 • • X u U l • X 9 9 9 UJ • 4 44 «“* 44 *■* • • - 1 44 4 4 o 44 • * 9 9 z >4. w 44 V _1 ►* o PJ 44 - 1 X N 44 a a z 9 -* .4 N u 44 U X • 44 J X 3 u X 44 44 44 X 3 U z U . z UJ UJ If) u X 4 4 X z 4 * □ 9 9 a x X z - I > H 9 a 44 UJ 1“ PJ 44 - 1 J -1 N X a □ UJ • J -J • M < 3 a 44 UJ • a II II •• 44 II a Z j- CM • • UJ a 3 44 9 z UJ U l UJ X a J z UJ U l TO « UJ 9 UJ ► 3 in II UJ UJ J PJ UJ a in s 44 3 3 3 < H XJ 3 L> X JL 3 3 a a x 3 -I • • X Q UJ 9 XZ Z Z -J X VI 4 • a 4 1 - Z Z 4 9 a • f 94 M JX < 4 44 M 44 3 ( 0 TO9 3 3 i 3 *4 u z IN J J «4 4 II 3 j it IIII f - H u 44 J O' W* W 44 • -1 H b* z 3 u *4 44 - 1 j o H H Z Z z J 44 J 44 44 *4 9 3 Z Z 3 3 9 U . II u < < a Q Q o o u 4 IL < □ a IL L. 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I KE. ) ) 2 .6 E .K E .IK C N .A J 1 .6 E .N E N /C PC(L.M)=PAMB PC(L.M)=PAMB . 0 = ) ,M (L C U UUPsUU RC(L«H CN NODES BARREL THE CN NUDES DO AAP=AA *DT I1 = T D 0 33 M=1.MM 3 3 00 LI P. *N OTIE OF OUTLINES *AND .T .P R G IN IL F CLMJ=RAHB RC(L.M TC(L.MJ=TA»»e TC(L.MJ=TA»»e p p = p p p A C ( L . N ) = A A N B s n = s n UC(L.M)=VELC UC(L.M)=VELC fifiP=RR T C (L.M )= T A M e e M A T )= (L.M C B T M A )=P tL.M C P L«M) . 0 )= M « (L C V TTP=TT TRG AC T STATEMENTS ITE R H ANC STORAGE KPT KPT S=S CC 47 GCTC ) T I . E L . S N ( F I ACCLv MACCLv )=AAME . = 0 VCCLvM) , E .G M .C R I.O Z U N .L E .G (L ( F I Ff NE. . ( NE. 9 .5 E .N E .(N D N ).A E .E E .N E N fC IF 2 L= 22 +1 }=RAM8 l.LL L . l THE PROJECTILE PROJECTILE . CM. 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EL 0 n 0 * 0 0 V 0 H H t- f 0 0 N . PP ^5 * ^ W % * *— * *- 0 *- o . 0 h n « m b e t * ° * ClS^O o O f * * . «:% fc-. 00 B o H O » 0 0 • 0 • f 0 e ci M M t" 0 0 0 • e 0 0 ^0 fr; *■» fr; *■» K n n f G 0 K ^ B II 0 0 II e n 00 £ f c 0 t- -* o a e c * 0 0 CSE CSE w H O u o O ■o o o teg 0 fr- • II • • ci H A . 0 **» 0 • O H < **» 0 0 00 M n ufi 0 0 0 0 A ^ B ii 0 W 0 W 0 0 0 0 0 M 0tc C C tc 0 _ 0 K3 00 finr. PPP 0 • • • • JCCCPPS'O'O'PO'PPPP o «o o i 0 0 ^ B B . 0 0 00 0 0 ii H JL JL JL H ii 0 0 0 H 0 0 0 0 0 0 0 0 0 a e e e 0 0 0 0 nnnn nnnn K ) k K i M * % * * * % * « * t * « • PP?P w w w w w w w .M . 00 tn tc fc tc tc -* -* tc tc fc tc tn c e ce c c ♦ a o O 00 « o Cl H to h IC-JCC'J s C r K p b D< D K 0 w >t*flonnrnc 0M H t-t- t-1 0 0 0 B .MB • n ► 0 0 0 0 0-0 0 0 0 0 ► n • K 0 w « w -PSE “ • 0 0> O*oHfl0H0*oClhi t0 0 ;H 00 ^m. 0 . . II S° II h 0 0 0 0 0 0 •T\ 0 E * * * I * 0^'*' 0 1 i 1 o II II H 0 0 -* n n n n o n p?cP 0 •gRKt“ Chh -* 0 0 tt 0 *0 *0 n n n o 0 0 ii 0 n n ii ieaoff>(f*itilca H JL H CKBDCPOCSPOCK i 0 f n o ♦n 0 0 0 **} % % * W w £ 0 0 £ • > 0 0 0 > _ o<5 a a OO0OOOOC|O0OOOOOOO©OOOOOOOOOOOOOOOaeeooeeeoeOoooooeooooooeeooooooeeIC M M M M o » o • DC e*~ 0 •q •q 5 fr* n * • 1 r t - u» tc £ 0 *— « *— P&OJ£CTILE HOSE 00920010 n nILL CASES n o EXCEPT 00930010 8£l ii ------to a ato to *• ii a ii a — h H H to to e to ft to to ft ft ft ftft b to to to o to to to H to to wft ft *- ft ii ll —« * * * * % - “ r r r r< r. .11 to r r r r. .11 r< * a a . Wr*r* O to to O Wr*r* . . to to • M a a a a e o c ii ti ti ft ft to ft ft to to «r. «r. to to t o t o * - a a ► H H to • • ♦ aa a eo*- to tow — ►* ► a to ft ft » bo a a ► ► to to a a a to to r % r t o ■p t-W w w *' * w * w*' t-W * ■p ♦ * * *-*- ft P t o p to to ft to to p • • to a H ft H to ► ► to H ft H to to % ll * h to to O t-*r- to *E o • to co n • a t3 O H to *- it a it ♦ to to H H to to to II ft to — a• ► g G to ato a c h a a c c to a to h to r-. #«*. r* r* #«*. r-. to to to to to ft - to* a a to to *- ^ to a a to a to H 4 t • • to ft n ft ft ft ft n n ft to to to to o o o to • »to to •-» n it « * to* © a © % a ► r- to to to to * H II * * *- ft ft*~ w ft*~ ft *- to w to ft ft to bo to to a a to to ► ► a a to a to • • « *if *if H H C It ♦ ♦ • ► H a ► n*- ft • to w m «* to to «* m to to h a a a * * * * to wwte e ft ft to ft ft e k to & ^ & to k Hr. «- M to o h « ► n HMT to a* % g S S p n to ►3 ►3 t o 8° 2 " sv e r sf= c H o e a n to e to to K n K’ • * * * * i k h h h ii * «e a to a «e nnfi < a to kj • to to to to to to o r> ft at at at at a at at r* <*>. **• r*<*>. • * * i i i i T i i I N cd S to ato to a aa a to to to to it to « « to it * G* * n t> r £ n rt> to*- to to to*- ft * ft at at a « « « % 5 aa a 5 6- n H to to <■: cH to to to <■: tC BO tP 6? to to to to a a a a w a i i * to to u ^ to »» h H f rirrrr ^ CB * ^CB •• w ^. * ►* S * to to to to ^ ^ r*^ jn CALL THEBHO (PCB(L,HM) , fCB 4L,8N) , ACB ) 01270010 ft TOP SU&FACE OF fiABSEL 01200010 6Cl n n ft ft < c tob h h »- M H M to i3 *b to < to H ft rt to H to rt to to < c to H ►non 2 2 ft ft ft C t o t o h ) * * M * e n ft ft ft ft f t ii to to o to to ft ft ft ft ft ft II to to to to to b: to to to to to to to — to to — — to to to to to to to — — — **• —■ — — 2 . 5 to*-. — — — — to to to — — —■ — — — to to to r h t ^ to to ft ft to to to t* to to C f t f t to to to to to to sB *■ • * * » * » * » * to to * * * * * » to • to to to » « « « % «to f t to to to to to to gpf to to — — at to to to to to • O' — — » to ac to to to r t to to to to to to li- t; rt rt to to*- t * * r w P f t • rt to to %- 5 ^ to to • to to 10 JK» N» • • » % h J« ji fc r t % % II B II 11 C B to • to *" to to » ► to to to £ 'to to to to *a to to to to II *M *H II II H H to to to to w w ft *- ft ft *■' to ft*- ft ft-* - B to to b O c • • to Ii — — n Ii to • • ea ii — — ii li to S88 iRR W O to ■" N h to — rt to to o •4 to to to — rt to o o to to to n to to W *■» • rt rt to n * * < to o£ r t rt» o rt rt t o f t * * • • to — — f t • • to ■ • • % to to to © • Ci • • * to to be o o f t ? e ? ^ h t * m to to to to to to B o C w to w » — ♦ ♦ o * % « to * * to to k e to to to ♦ to •*■* to to f t to to ♦ to — - r t K) to to * to to ►3 to to • to to Ot — rt m at rt * r t rt o ■ rt r t ^to r t w. g •• « • *> • to O w — • — rt to to to — *-’ W to to to to *~w *j * o ^ ** ►* to to to • to to b to to to to to to M f t to • ft ft ft r t r t f t f t t o r t * to • ea a • • n to to • f t to to — *-» to to to m • to to to e> to to to <2 to • to • * * e • % ■ to to to a to to to to ■ £ H w to s a w w »•- e to c h It *•* • It ll to • to to to *o to ■ • ► to to O to to * - * - t o ► to to to to to •e* -' r t w a , ot b to to to to to o • to to to F * ~ • — 8_ n *> to 5 ? 8 r t o to to f t to r t o r t tp ► *■»to to • O o to • to * f t f t to — to to r t r t — w • • • to to ** 2 ■ ■ 5= e e to to to j o j o to w f t 8 8 o r t r t 8 o o •o to «J r t OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOQOO ■# ^ ^ «a totoO'togniPU'trwtrtrU'trtji^toBto^Btoto^BfetetjCifcjU.'WbfbiUtfc) bitJMOtfiBajcirckKirtewwcirctoicrtBifitt'OV'ir^ktiiaoie OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO rtoeeoeeeBOoeooooeeoeeooooeoooBOoooeo rt rt rt rt — rt rt rt rt rt rt rt rt rt rt rt rt rt rt rt rt rt rt rt rt rt rt rt rt rt rt rt rt rt oooceooeoooooooooooQOoeooooooooooeo CW to M fe a to to to to to * II II II • » 4 4 «. • * ♦ * ♦ ♦ 4 ♦ 4 + * ♦ 4 + *o *o to •a to f t f t •o » f t f t to to ft ft to to to to • • f t f t W B B B W W to to to W to to to to to B B • • • to to * * F to m w> tr ♦ K K B a o • to to to e» H O ■ ■ M ■ to to to cs tr cs to to ■ to M to **9 to C**9 ft ^ to ft to w B to MO O to to H ftto to B to ft to O to* to * • M ► to C W wB w B H w to to to a to to B irL E to B to cw to o -» toMS ii ii o to to: ♦ ♦ B a* ft to m * •c B t* W % * ♦ -*% a» -*% ♦ + ♦ w Cc n w o — ft ft o — * 0 * * ♦ % kM » + « W ft »W o ft « n * n « ft Q t» to Q) w w to to M M ■ •B ^B B B •* -* W e» to k • w • w • w • to to i to i t-* w t* w t-* « « W WNr *» | V to V. id to to to to to to to ntntBUjU to to to t3^. c ft n r> ft r> n ft c H to H to to to to n k* * * * Ewr ; n o C B b & B & b B C o e ♦ c ';* : 8 W If 1UIG.J.Z.J.H0Z) GOTO 2 01970010 L-LMO 01960010 TtrT 142 oooooooooooooaooo ooooooooooooooooo oooooooooooooaooo MOOOOOOOOOO*-*-*-*-*-*- OOOOOOOOOOOOOOOOO + * ♦ ♦ A >1 A A % * % * mm mm w mm * * ♦ ♦ 9* Hi 4 4 Nrf w A A A A u u U U A A A A + +■ ♦ + O* A A A A * * % * ^ rnm r rn mm 1 1 ). < 4 4 4 4 '■H u 1^3 c* OOOoOOoSOa300QOeOOOOQoBOoOOOOoeO OOOQeOOQOO o o o o o o o o o S aoaoaoBoaoaoo O I a a a a a o a a o eoaooooaooooa 9 J| 0 X m o M 9 0 0 x a a a *■ x . «*» y y y y u 144 aOooOQooeoooooooeoeooooosooo o o aoo OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO oooooooooooeooooeooooooooooooQOoo a»o*» p« * i)>ois»3 (4 3 O H O C3 N (4 B a cj an a n as B B S \ % < -a ■4 ca n* B B ►4 H II % * o M *4 < 3 09 n * % dd o O *4 i4 *4 r* w w ZZ a CJ CJ CO • •4 o a a a i rj <3 Ot N. » ?• Q an « « » « B B 3 an o 3 n b ra C4 « x O >4 13 (4 C4 c4 (4 tn « >4 r4 Q H • B B B S -.8 B ►4 n ) H O a a a a ♦ B B B B 8 8 13 t4 >4 3 »4 *4 Ctn % %Cd Q* % x B o • B CO CJ r *■ • i ** ns* o« s <^n a • »■ 0« (4 H B yuuu 8 B H — et j u S a a •Oi 4 H 4 Sana i4 3 4 d “ iiS J S ii • 3 B a i 4 4 U II H It II n n ii n S* O '- '- • • w•— 4aH B »9 ^ 13 ^ 3i -3 II H CJ CJ - x < J CJ H 48 an C3,>4 - an sd • aqua a a a a _ a ii a a h 4 -.B • • • 3|• •• « » * J *3 j % « j (4 3 b « ii j ; ii ii B 3 14 b 3 . * *> {.4 l4 c4 .4 o C4 h) e4 (4 3 ia if» % -x >4 4 —« 4 a S3 3 3 3 B f* 3 a a 3 an t a n b *• *4 -4w b b« a a a a a a a a 4 3 3 9 * * CJ CJ » J 3 *4 « 8 '’M M -4 <4 8 44440 4444i< i a H 44 o 40 IFii.MO.Ll.LBOZ) 4 GOTO 00920010 p» l»«O 3 U CJ cj u 1 k5 '0 00ooooooeo°ooood0ooooooooob0oci90 ooooooooooooaoooooooooooooooeoooo oaoaoooooeooeoQoaoaaOoeooaoooeooo O H ta et» o to O a a cj 4 a CM M M M 4 4 4 4 a o* a a* i i i i n ii II H a a a a £7 4* 4 « 4 •4 * % * * % id d d y u * 3 a a St 8 y u M CJ CJ a t i i i (4 Qi CJ aa o« a a o a a a a *8 a a a a a a a a 4 ~»_»a • %* * a a % «* *60 H a at <4 <4 41 «J 01 »• m •* i* Ma a 4 4 M Ot * % a Qi % * O ♦ * «M 04 > W >4'3'- » S4 CJ «“ *” • • *• i" • 0 0-3 4 a — • • H CJ CJ # u H ♦ * • • *— ► at -* s ^ s s *4 »4 *4 s o o *4 S3 * « * * a 3 t» O — • * ■"— II » CJ CJ CJ ^• • 3S • 4 • * a II II CJ CJ ^ ~*CJ y 3 - « ■** 04 H CJ ■N N M •>* • a 64 4 CO w a II H H 4 M 7 3 H II ii » a 3 ♦ 4 a II II * }l| M to ^ % %a “• to O % ■4 •> 53 a a to a e ? l . 4 4 3 “*3 a a a a cj <4 * o o * a ■"•ad a 4 to a a a * w^ag Ma a a a a 'J9 •488 • • {JCJ » % a >o a a j JJ*.JUH a eh % * * » t a II s a ii ►4 a — a 04 h «• 4 4 a a • A 4 fS w ^ ^ a « '- — a 0 4 ^ *» -4 II ^ u u u u m a CJ CJ 3 II CJ CJ (J CJCJCJ UMCJ 04 O O 9 CJCJCJ CJ H -4 .4 h|9 > 4 D| H H 64 4 H _0 *0 a > 04 04 H 4 H H CJ a h a a a a to /EC (L,fta}/fcG) (L«fldJ SZ* (PC XC(L#tttl) IHJiBiiO CALL (PC(L,fltt) ,£C(L,ttH) #AC(L#HHJ) 01200010- 01270010 tJO CJ cj cj a o B to H rt to H ► rt to to < a to h rt I *0 »0c *«b to ft ft ft ft f t ft ii to c * *a *- ^ *» *» (*. Cr.^. ^w~~ Var'«a.^ ^ wB * «~- *». to ^ r* *- *- to*- to wr-h t-V f t- to to •»to to to to to to to to to to to to to to to h« t « « « » u * * f t ft* * * » C f t f t * * * * • % a • m - B B to to B** 2to P rr t o t o _ . * k to to to to to to to*-** a to to a to to ^ ft c ?E£** to w «-- h H • • to to*" to w *- to to rt to h R! C ?. * £*Ek Ki fc- M * U || K BE ft ft r t * * ll *3 II ll ft c • to w w- w- w* w w m H • S1? to *o fc» rt rt to to to ► to. to to to ► II H ll U II U • to £ I" n*~3 *- - ft n w *" * • toww n *- ft ft 3 3 to ft ► rt *0 I elDH to » to to *• it • * *« #a H II to to II H e to nfiftjfind! 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ZXH£~L£HGXH PLOTS 00570003 ft OO 00580003 181 *o ip m •* m* h h to to B «H u to to •• * * •• • to to ww to to to to to Ui to to to to •* ww '— to to to to to k) to to k) * to U i * * • w * * * to ■ • ■ o to * w* * * k) * * k k * * to to I * * * m to « • * « to H o H to « *- to w — w to *- .oooeo o z e to o e e o o o • o k ;. t4 £3 n to m s to • to tn to to to s n n a to H a £ * * % ♦ ♦ to to h fc In in to to in N © »> ► H H ► »> © N in n to to'' to** to to ^ to P f* to to to to to to f* P to to ^ to to to to g nft o to fc « to >J • *—* to » o**<*. ^ n o o n n p * . »- «e Ci to to I o to to to g to © to to ms p h « Hin H to ■! • t9 • t9 * ♦ o 1 *-* 1 II ■ to * * o * * U<» k U<» in s 18 to to o k* k to 8 to H Bis*" to m* to to to O to © in * to ci ci to * to « n n to 0 t o t o * ♦ * © •< to to • to to K k k to ► K to * * g “ * to * * * * •• e m* t* to k « -* in to to to to to to to toto to to •»i «j **OtoOC*Jtoto«Ulkl-tOtOW W IB w IT V «i W IV IV toto » ers Ul DC K ► IB* n * © e © to * e • H • fee * H *k W ta to % * % * * * * • * * * to — ki O O O O O O o O O O O O O lo “S2°S00Sfi°2SS2oooqQooooo©o©©ooeootooooooooe 8 Ul N W fc! 2 fc! to E • k Ul • tote 8 UlUiUlUiUiUlUlUlUlUiUlUiUiUi O O OO O O to W W D W W H W B th r * g£S « « o i b o o o o o a • ► in kJ 8 *k s 8 3 K H *J *J to Ul oo e o o o Ui Ui M non *J *i *J Ui Ul to u* to o o to to *i o o Ul Ul IS 1 1 « M « U M e e e 1 o 1 to os ■ ■ to os I o oUl Ul 1 in in in In In in in ► ► ► *0 K K ? * * ♦ e ♦ H H H M s ► M H £ to a .© % *' .* .tti © e © « « to to UikJ £ 8 2 > IT > B Ik H H H H in 0 j * W a e o e S S a Ul Ui Ui Ul I 09 © to kl •*OlO g o o o o o VO tO © © o eoo s & s In ec 0 n 1 M .a nnn 00920003 CALL PLOX(«5#0.|,-J) 00930003 ZQT fc-1 fr* fc-1 as*-?- t * O to M ft ft 01 ft 4 4 ft * KJ I m _ KJ H H H KJ _ ft ft to ft * * *»** O O ft*0 ft ft M H ft* * 14 M M KJ f t ■ Q • ** 01 01 M K C K t o o n N H C J * • e e ft* M ft ft ft ft ft M ft Cl B! tC«. ft ft 0 • M M ftta ft M ta* * * g► - — 4 ft ft 4 ft — -» M r* £ * 4* M 4 f t t o t o 4 r 4 4 ■* to to 4 to ft ft to to O ft ' • ft to to to e H •Q ft a U> ft t f ft O KJ N M ta. ta. * 4 4' 4 4 ♦ 4 04 4 « p « t ft ft ftft ft fE t ft*-. ft 104 ft to ft ft ft ft M Ol n O M tr) ftft W «— o * - «— w Mw ft ft m* ft to to ft ft to Q KJ 4 4 4 4 * 4 C ** C SE O 6 t t Z t Z P P Z P ft ft ft ft ft 4* 4* ft w Ol HI ft H H non w cv r S a* ft ft 8 £ ft ft K J O ft O ft w j ft ft — m ft ftta ta ft ft — ft ft* ft DBW f t t a ter ti> 0 M s C * 4 4 *» « *» .O 4 0* 4 tO K J f t t a f t ••* ft ft ft • n » Ota re »n Ota K to ► te ► CJ a: CJ te ► ► ft H ft to ft ©4*4 % ft • e * O ft © ft: . ft4 O ft <4 BSta ft H fi <4 as ft W O ft at kj ft H ft ft a frj to to as ft ft m te © •. »- ft ft h n h ft ft »- to u(n to a ♦ O • • ft* • * r to > £ 4 4. 4. ta. ta ta ta. o Q ft ta ftta tO ft ^ ft *•• *-» ft *-» *•• ^ ft \ .a; .4 tpz as ■ H ft 4 t* H H 48*- S o SSi % tim * *4 ft *4 ft as ft to K) ta ta M * as ft o n c o n o n n o on on ft o to n^5 o n n *0 Oc 4 O 14 ft O £»*&»£££H H to fto P p rr S H r* ft n H* ft r* r 1 £ g f t t o ^ j*ft w tfl* m to *e to ft to tn §o ft Ann ae ft ft as as *** B 4 4 4 asww ae ae • K) as as as K) n tad nto tad • I t l R D I I O C * H • t * o k n * U< OOCKO 4 fci% 4 f t BBPPW cj O * * O H H Ota ft o R t « « n s E (0 •* -* W N w N ft n n n n n E £ £ f c £ £ * I H to KJ O' M t * ft e * * * H P * * * tag to % 4 4 * * > A R K i g | g S 8 S to: t* •o (A (A O *0 KJ KJ as 16 m 00 M ft e m * * * I M 4 »o 4 »o 4 ♦ 0 4 m m m m m m E^E K mt •Do (a a: at ► to Eo •0 • « f t o « CD O CO CO O CD m p fJ * ► tn •w h u hx> tn KJ ft ft ft ft ft ft ft ft ft ft ft ft ft to ft ft ft ft ft to ft ft ft ft ft ft ft ft ft ft ft ft ft ft ft ft ft ft ft ft ft ft ft ft ft to ft ft ft ft ft to ft ft ft ft ft ft ft ft ft ft oooooeeooooooooocooeeoeooooo«oo«sft ppppooooooooooppopppppoopooepppppft ft * * * « V N a to * ' w © * - CALL L18E(3X4LB,fiP4Lff,MlfiJl, 1,0,2) 01270003 CALL BOP LOT (0) 01280003 £8T B fc Id tn Gf E 2 ta n o n n n n n 8S ta • P ~ B M I d ta P 2 "in S o t-t* a • ft ft ft Ul 8 k • to t se KS. 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J J ( F I /TABS J) , J =TC( TJ (Ij ViIj=VC(JJ»J) * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * A * * * * * * I= (JJ,J) C (I)=U Q DIV(lj = DI¥¥ ( J J , , J) J J = ( DI¥¥ DIV(lj 0E .08 .00 . .03 2.0 .5000 .00283 E5 .0 1 YIA DATA TYPICAL SET SUBROUTINE DESIBED) EXFLOH IF . 1. 1. y /. 8 2 .01 . . 14 4 1. .4 1 1.4 1.4 1. 8. 8. 001 001 001 .00001 .00001 .00001 .00001 287. 287. 0 .4 14 .8 22 .26 2 .2 .18 4 .1 .04 .02 VITA Oktay Baysal was born (Mi July 24, 1954 in Turkey. Alter graduating from The American High School in Istanbul (Robert Academy) in 1972, he registered to the Technical University of Istanbul where he received his Bachelor of Science degree in Mechanical Engineering in 1976. While pursuing his education he worked in the industry in Turkey and W. Germany. He received his Master of Science degree in Mechanical Engineering in 1978 from The University of Birmingham, England. He entered the graduate school of Louisiana State University, USA, in 1979 where he was a graduate teaching assistant for three and a half years. He was elected to the national honor society Phi Kappa Phi in 1982. He is a member of The Chamber of Mechanical Engineers of Turkey, associate member of The American Society of Mechanical Engineers, student member of The American Institute of Aeronautics and Astronautics. He is presently a candidate for the degree of Doctor of Philosophy in the Department of Mechanical Engineering of Louisiana State University. 191 EXAMINATION AND THESIS REPORT Candidate: Oktay Baysal Major Field: M echanical Engineering Title of Thesis: Numerical Gas Dynamics of Internal and External Flows in Gun Barrels Approved: Major Professor and Chairman .____ Dean of the Gradual School EXAMINING COMMITTEE: Date of Examination: •i Mtl * Vl»* ( VH*1 - v “M ^ ♦ 1 ( »•**» - p“*» Q3**1 bY L,H*1 L,w ? Wl w w,m' L,M ^ U,M » * b | H