Phase-Plane Analysis Application to Guderley s Problem

Transcription

1 Journal of Emerging Trends in Engineering and Applied Sciences (JETEAS) 4(3): Scholarlink Research Institute Journals, 2013 (ISSN: ) jeteas.scholarlinkresearch.org Journal of Emerging Trends in Engineering and Applied Sciences (JETEAS) 4(3): (ISSN: ) Phase-Plane Analysis Application to Guderley s Problem 3 Narsimhulu Dunna, 1 Addepalli Ramu, 2 Dipak Kumar Satpathi and 4 G Sudha 3 Research scholar Department of Mathematics, BITS- Pilani, Hyderabad Campus, India. 1,2 Department of Mathematics, BITS- Pilani, Hyderabad Campus, India. 4 Faculty and Research Scholar Department of Mathematics, M.G.I.T, Hyderabad, India. Corresponding Author: Addepalli Ramu Abstract The self-similar converging-diverging shock wave problem of Guderley in 1942 has been the major source of considerable mathematical and physical interest. We investigate an application of the Guderley s solution with reference to phase-plane analysis and a code has been developed for the verification test problem for compressible flow. In this an effort has been made in understanding the problem s mathematical and computational approach. The simplifications and group invariance properties reduce the compressible flow equations (for a medium whose equation of state (EOS) is of Mie-Gruneisen type), to two coupled s nonlinear equation whose solution is an eigenvalue problem. The information we provide, together with previously published material, gives a complete description to construct a semi-analytic Guderley solution. The assumption (1 φ(g) k) made led to developing an EOS and thus the pahse-plane analysis method employed provides a global numerical solution. Such assumption alone helped in obtaining the numerical solution. Taking k γ, we developed the Guderley s polytropic gas results as a special case from our investigation and the results are matching well with the results of other authors and taking k physically meaningful values led to condensed matter EOS and results. Thus k is a very general quantity which leads in obtaining different EOS. Keywords: compressible flow; shock waves; exact fluid flow solution; convergent flow, Mie-Gruneisen INTRODUCTION The strong shock wave problem in converging cylindrically or spherically in a gas is well known in hydrodynamics and is considered important in several areas such as, converging compressible flow to the laser induced shockwaves ( S. N. Luoa, D. C. Swift et al.2004) and in astrophysical applications to double detonation supernovae (Fink et al. 2007). Guderley (1942) was the first to investigate the basic problem of a strong cylindrically or spherically symmetric shock wave converging into an inviscid, nonradiating, non-heat-conducting, perfect gas. This type of problem was also solved independently by Landau and Stanyukovich (Stanyukovich 1970). Guderley observed that certain physical assumptions lead to a self-similar problem formulation. The solution of the self-similar problem depended on determining the numerical value of similarity exponent that characterizes the space time path of the infinite strength incoming (converging) and finite-strength reflected (diverging) shock waves in proximity to the location of collapse. Several authors such as Lazarus and Richtmyer (1977), Lazarus (1981) and Hafner (1998) Ranga Rao and Ramana (1976) calculated the numerical value of this similarity exponent (a function of the adiabatic exponent and geometry) with high accuracy using various techniques. The classic Guderley problem, reviewed by Meyerter-Vehn and Schalk (1982), Zel dovich and Raizer (2002) has variations that have been explored in some detail. Axford and Holm (1978) used group theoretic techniques to determine a more general equation of state (represented through the adiabatic bulk modulus) that admits self-similar solutions for a Guderley-type problem. Various authors have found similarity solutions for strong shock waves converging into dusty gases (Jena and Sharma 1999), variable-density gases Toque (2001), Madhumita and Sharma (2003), and radiating gases NiCastro 1970, Hirschler (2002). The present work is presented in this paper is organized as follows. Section 2 a brief review of the Guderley problem and using group-theoretic methodology. Section 3 contains application of Guderley s problem to represent it in a more general form by introducing the non-ideal medium by Mie- Gruneisen type, along with the Hugonite jump conditions and the strong shock conditions. In section 4, solution methodology (the suitable transformations as discussed by Zel dovich and Raizer are presented and also the transformed set of governing equations and the shock conditions are presented.) Section 5, the numerical solution is discussed in detail and the results for the present problem and lastly conclusions remarks and the comparative results are presented. 406

2 Review of the Guderley problem The classical Guderley problem begins with the consideration of an infinitely strong, symmetric shock wave focusing perfectly on an infinite axis (cylindrical geometry) or point (spherical geometry). The source of the shock wave is not discussed in this scenario, but the initial state of the gas into which the wave is propagating is well-defined. Denoting physical flow variables in this unshocked region by the subscript 0 the initial state is given by: u (r, t) 0 1 ρ (r, t) constant 2 P (r, t) 0 3 Figure 1. Notional representation of converging shock trajectory R s - (t) ðtþ, reflected shock trajectory R s + (t) and space time regions 0, 2a, 2b, and 3. [Zel dovich, Y. and Raizer] where r denotes position (r 0), t time ( < t < 0 ) for the converging mode, for the reflected mode (0<t ), u velocity, r mass density, and P material pressure. For a one-dimensional (1-D Cartesian, cylindrical or spherical), smooth flow free of viscosity, heat conduction, radiation and body forces, the Euler equations describe fluid motion at all continuous (i.e.) non-shock) locations + () + u + + u + (m 1) c (γ 1)c + (m 1) 0 6 where c denotes the local sound speed, defined through the pressure and density by: c γp/ρ 7 Here, we consider polytropic gas with the equation of state (EOS) given by: 407 P(ρ, e) (γ 1)ρe 8 where e is the specific internal energy (SIE), γ denotes the adiabatic exponent (1<γ< ) and m the space dimension (m1, 2, or 3 for 1-D planar, cylindrical or spherical symmetry). In particular, since the converging shock wave is assumed to be infinitely strong, the strong limit of the Rankine Hugoniot jump conditions may be used to connect the flow just upstream to that just downstream: 9a u R (t) 9b u [R (t)] 9c where the subscript 2a (see Figure 1) denotes the state just downstream of (behind) the converging shock, and R (t) denotes the converging shock speed. The gasdynamic equations admit several transformation groups. These gasdynamic equations contain five dimensional quantities ρ, p, u, r and t, the dimensions of three of which are independent, c can be represented in p. If we represent density, velocity and time as the basic dimensional quantities then the dimensions of the velocity and pressure can be represented as [u] r t and [p] [ρ][ r ]/[t ]. As result of existence of three independent dimensional quantities the equations allow three independent similarity transformation groups. Let the functions ρ f (r, t), p f (r, t) and u f (r, t) represent the solution of the equations for some specific motion. Without changing the coordinate and time scale changing the density by introducing the new variables viz., ρ / kρ and p / kp and other variables are unchanged. This transformation does not change the equations. Now multiplying the density and pressure by k and the new motion is described by the functions ρ / kf (r, t), p / kf (r, t), u / kf (r, t) The new motion is similar to the old one differing only in density and pressure scale. The equations remain unchanged if we transform to new variables r / mr, u / mu and p / m p keeping ρ and t unchanged. The solution of the new motion is expressed by the functions ρ / f (r, t), p / m f (r, t), u / mf (r, t). Let the time scale be now changed, leaving length and density unchanged. Then the equations allow following transformations t / nt, u /, p/, ρ/ ρ, r / r. Thus by using a suitable set of transformations the basic gasdynamic equations along with the boundary conditions transform to ordinary differential equations and are solved numerically for various values of γ and the shock decay coefficient is

3 evaluated which is self-similar problem and an eigen value problem of second kind. CONSTRUCTION OF THE GUDERLEY PROBLEM In this section the Guderley s problem is attempted for the converging wave generated either the spherical piston moving into the gas imparting to it a certain amount of energy, or by an instantaneous release of energy on rigid cylindrical wall respectively. A possible method of achieving high degree of compression is to launch successive shock waves from the ablating spherical surface. In this process a sequence of shocks of increasing strength can be produced such that the successive shocks do not overtake each other before they converge to the centre. Thus for these converging shocks energy becomes concentrated at the front and the wave is assumed to a strong shock. This will be adiabatic until the shock arrive simultaneously at the point of convergence. It is assumed that the motion will be self-similar as the wave converges to the centre and the energy in the self-similar region decreases with time, following a power law. The self-similar solutions are investigated for converging cylindrical and spherical strong shock waves into a medium satisfying the equation of state of Mie-Gruneisen type. The shock is assumed to be strong and obeys a power law x (t) A( t) where x 1 is the shock position at the time t and A and α are constants. The converging wave problem is investigated by adopting an appropriate approximation on the Gruneisen coefficient to reduce the governing system of equations to first order ordinary differential equations of Poincare type. The self-similar solution and the similarity exponent α are obtained numerically by phase-plane analysis (as discussed by Zel dovich and Raizer) and the determination of α is an Eigen value problem. The results are shown along with the other results. BASIC EQUATIONS The basic conservation equations of mass, momentum and energy governing adiabatic flow in Eulerian co-ordinates are [Zel dovich, Y. and Raizer, Y] + (υ 1) u u + u ln 0 12 Where υ 2, 3 denotes the cylindrical and spherical geometry of the shock wave, ρ, u, p are density, velocity and pressure of the medium. The governing equations discussed in the previous section are the same as discussed above, except that the velocity of sound c is replaced by e the specific internal energy. The medium of flow is assumed to be obeying the EOS of Mie-Gruneisen type: p ρeγ( ) 13 Where e is the specific internal energy, Γ( ) is the Gruneisen coefficient. By replacing Γ by (γ 1) we obtain the EOS defined earlier, where γ is the ratio of the specific heats. At the shock front, r x 1 (t), the boundary conditions are the Hugoniot-Jump conditions such as, u 1 x 14 p p ρ u x 15 e e u The suffixes 1 and 0 denote the quantities just behind and ahead of the shock front respectively. The dot denotes differentiation with respect to time. The strong shock conditions for the problem under consideration are u (1 β)x 17 e p (1 β)/(2ρ ) 18 p (1 β)ρ x 20 β is the shock density ratio which is obtained from the strong shock conditions and the equation of state as, Γ ( 1 β) 2β 21 TRANSFORMATIONS Several transformation groups as discussed earlier are applicable to the governing equations and a set of suitable similarity transformations are u v(λ)x ρ ρ g(λ), p ρ x Π(λ) 22 Where λ, x A( t ) Along with these transformations a set of transformations are considered for convenience P(λ) α α () (), G(λ) g(λ), U(λ) and Y () () Using the defined similarity transformations the differential equations governing the flow (10 to 12) are transformed into a system of ordinary differential equations. Hence the transformed system of equations of mass, momentum and energy governing the adiabatic flow are (U α) + + νu 0 24 Y + (U α) + + 2Y + U(U 1) Y φ(g) + () 0 26 Where

4 () φ(g) Γ(G) 27 The analytical solution to the transformed set of equations is difficult and hence an appropriate numerical method is applied in order to obtain the similarity solution. the problem of obtaining the similarity exponent for the problem under investigation is for a known value of the shock density ratio is an Eigen value problem. NUMERICAL SOLUTION METHOD The transformed set of equations are written as 1 (U α) 0 (U α) Y 1 0 Yφ(G) 1 28 and are solved for the derivatives existing in the column matrix. In order to apply the numerical solution the above equations in the matrix form are rewritten as the derivatives,,, Where Δ, Δ, and Δ are obtained from the above non-singular matrix as Δ Y[ 1 φ(g)] (U α) 30 Δ υuy[1 φ(g)] + U(U 1)(U α) + 2Y(1 α) 31 Δ υu(u α) U(U 1) + 2Y () () Δ 2Y φ(g) + Y(U(U 1)φ(G) + 2Y(U 1)(U α) 33 The phase plane analysis method is applied on these equations to solve numerically and for this the differential equations and the determinants must be functions of (U,Y) only. In order to reduce it to this form we consider the approximation 1 φ(g) k, where the constant k depends on the materials considered T.Neal (1976). Thus the determinants with the above assumption can be written as Δ Yk (U α) 34 Δ υuyk + U(U 1)(U α) + 2Y(1 α) 35 Δ υu(u α) U(U 1) + 2Y () 36 () Δ 2Y (1 k) + Y(U(U 1)(1 k) + A 37 A 2Y(U 1)(U α) 38 Writing a Poincare type differential equation in (U,Y) phase plane by dividing the third equation by the first equation we get υu 2Y U(U 1) 2 () () Y (,) (,) (,) () (,) Where M(Y,U) 2Y (1 k) + Y(U(U 1)(1 k) + A 40 N(U,Y) υuyk + U(U 1)(U α) + 2Y(1 α) 41 Thus these equations are solved numerically, subject to boundary conditions: On the image of the shock front U(1) (1 β)α, Y(1) β(1 β)α 42 and image of the free boundary, a singularity of saddle type, U μ (), Y 43 With respct to the type of boundary conditions the differential equation is undefined and to overcome this difficulty we introduce a new variable X U(λ)/Y(λ) then the boundary conditions become U(1) (1 β)α, X () 44 and U μ (), X 0 45 The variable λ must increase monotonically upon moving from image of free boundary to the shock front. It is difficult to satisfy the se conditions for any arbitrary value of α because on of the boundary condition is a saddle type singularity. But then there exists a unique value of α for which the problem reduces to a nonlinear eigenvalue problem. Thus the differential equation is solved in obtaining the integral curve in the (U,X) phase-plane. The singularity depends on α anf k. The slope of the integral curve at the saddle type singular point is obtained from equation (39) using L Hospital s rule as,

5 () () ()( ) 46 From we obtain he Gruneisen coefficient in the form Г(G) ()Г () () Г, 47 Thus taking k 1.42, Г 2.12 and k 1.298, Г 1.017, T.Neal (1976), we obtain the Mie-Gruneisen type equation of states. The value of β is obtained for each of these cases from the the strong shock condition. Thus with the Gruneisen coefficient the for different materials in condensed matter, the system of equations are reduced to one of Poincare type in a (U,X) phase-plane. Thus using the phaseplane analysis we solve the differential equation numerically (Runge Kutta 4 th order method) from the singular point of saddle type and the solution must pass through the two singular points. Thus the results are shown in the following tables. CONCLUSION AND REMARKS AND RESULTS The similarity exponents are obtained for the spherical and cylindrical converging shock waves propegating in a non-ideal media. The assumption considered on the Gruneisen coefficient alone resulted in obtaining the solution using phase-plane analysis. The numerical solution discussed is useful in determining the similarity exponent α for a class of problems using different Gruneisen coefficients. For different values of the equation(47) represents different EOS of Mie-Gruneisen type. The perfect gas solutions are recovered by substituting k γ and Г (γ -1). The results obtained are compared with the results obtained by Lazarus (1981,1982) and found to be in full agreement. Table 1. Selected results of the similarity exponent α for perfect gas and the condensed matter EOS. Cylindrical Geometry Perfect gas Condensed matter γ α Г, k, β, α Lazarus Values Present work (1981, 1982) , 1.42, , / , 1.298, , Table - 2 Selected results of the similarity exponent α for perfect gas and the condensed matter EOS. Spherical Geometry Perfect gas Condensed matter γ α Г, k, β, α Lazarus Values Present work (1981, 1982) , 1.42, , / , 1.298, , REFERENCES S. N. Luo D C. Swifta,. T E. Tierney, D L. Paisley, A. Kyralaa, R. P. Johnsona, G A. Hauera, R. P. Johnson a. A. Hauer O Tschauner and. P D. Asimow Laser-induced shock waves in condensed Matter: some techniques and applications High Pressure Research Vol. 24, No. 4 pp Fink, M., Hillebrandt, W., and Ro pke, F., Doubledetonation supernovae of sub-chandrasekhar mass white dwarfs. Astronomy & Astrophysics, 476, Guderley, G., Starke kugelige und zylindrische Verdichtungssto ße in der Na he des Kugelmittelpunktes bzw. der Zylinderachse. Luftfahrtforschung, 19, Stanyukovich, K., Unsteady motion of continuous media. New York: Pergammon Press. Lazarus, R. and Richtmyer, R., Similarity solutions for converging shocks. Technical report LA-6823-MS, Los Alamos Scientific Laboratory. Lazarus, R., Self-similar solutions for converging shocks and collapsing cavities. SIAM Journal of Numerical Analysis, 18, Hafner, P., Verdichtung und Reaktivita tsaufbau bei der Zu ndung von Kernspaltungswaffen, Teil 1: nalytischena herung zum einfallenden Guderleyschen Verdichtungsstoß.Technical report 109, Fraunhofer- Institut fu r Naturwissenschaftlich-Technische Trendanalysen (INT). 410

6 Ranga Rao M P and Ramanna BV., Unsteady flow of a gas behind an exponential shock, Jr., of Math, Physical., Sci.,10: Meyer-ter-Vehn, J. and Schalk, C., Selfsimilar spherical compression waves in gas dynamics. Zeitschrift fu r Naturforschung A, 37, Zel dovich, Y. and Raizer, Y., Physics of shock waves and high-temperature hydrodynamic phenomena. Mineola, NY: Dover Publications Axford, R. and Holm, D., Spherical shock collapse in a non-ideal medium. In: Proceedings of the joint IUTAM/IMU symposium, group theoretical methods in mechanics, August, Novosibirsk, USSR. IUTAM, Jena, J. and Sharma, V., Self-similar shocks in a dusty gas. International Journal of Non-Linear Mechanics, 34, Toque, N., Self-similar implosion of a continuous stratified medium. Shock Waves, 11, Madhumita, G. and Sharma, V., Propagation of strong converging shock waves in a gas of variable density.journal of Engineering Mechanics, 46, NiCastro, J., Similarity analysis of the radiative gas dynamic equations with spherical symmetry. Physics of Fluids, 13, Hirschler, T., A parametric study of selfsimilar collapsing shock waves in radiating gas. Physics of Fluids, 14, Neal, T., 1976., Dynamic determination of Gruneisen coefficient in aluminum and aluminum alloys for density up to 6 Mg/ m 3, Phy. Rev-B 14,