ON THE EXISTENCE OF SELF-SIMILAR CONVERGING SHOCKS IN NON-IDEAL MATERIALS

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1 ON THE EXISTENCE OF SELF-SIMILAR CONVERGING SHOCKS IN NON-IDEAL MATERIALS by Z. M. BOYD Los Alamos National Laboratory, New Mexico, USA and Mathematics Department, UCLA, Los Angeles, CA, USA) by S. D. RAMSEY and R. S. BATY Los Alamos National Laboratory, New Mexico, USA) [Received 7 October Revise 24 February Accepted 10 May 2017] Summary We extend Guderley s problem of finding a self-similar scaling solution for a converging cylindrical or spherical shock wave from the ideal gas case to a generalized class of equation of state closure models, giving necessary conditions for the existence of a solution. The necessary condition is a thermodynamic one, namely that the adiabatic bulk modulus, K S, of the fluid be of the form pf ρ) where p is pressure, ρ is mass density, and f is any function. Although this condition has appeared in the literature before, here we give a more rigorous and extensive treatment. Of particular interest is our novel analysis of the governing ordinary differential equations ODEs), which shows that, in general, the Guderley problem is always an eigenvalue problem. The need for an eigenvalue arises from basic shock stability principles an interesting connection to the existing literature on the relationship between self-similarity of the second kind and stability. We also investigate a special case, usually neglected by previous authors, where assuming constant shock velocity yields a reduction to ODEs for every material, but those ODEs never have a bounded, differentiable solution. This theoretical work is motivated by the need for more realistic test problems in the verification of inviscid compressible flow codes that simulate flows in a variety of non-ideal gas materials. 1. Introduction 1.1 Motivation While much is understood concerning inviscid compressible flows in ideal gases, less is known about counterpart flows in non-ideal materials. For instance, in explosion and high-speed impact phenomena, metals, plastics and other non-gaseous materials experience pressures far exceeding the limits of stress strain models and are instead well-modeled by inviscid compressible flow Euler) equations. However, because of the structure of these materials, constitutive relations other than the ideal gas law are needed to correctly model the thermodynamics involved. Some equations of state EOS) that find use in these situations are very dissimilar to that of an ideal gas and may only be valid in a limited range of pressures, energies and densities. These EOS models can arise <zach.boyd@math.ucla.edu> <ramsey@lanl.gov> <rbaty@lanl.gov> Q. Jl Mech. Appl. Math, Vol. 70. No. 4 The Author, Published by Oxford University Press; all rights reserved. For Permissions, please journals.permissions@oup.com Advance Access publication 7 August doi: /qjmam/hbx015

2 402 Z. M. BOYD et al. from theoretical considerations, empirical measurements or a combination of both. Understanding these types of flows on a theoretical and computational level leads to improvements in safety and reliability in explosives handling, aids in the design of blast-resistant materials and enables enhanced experimental design. Inviscid compressible flow codes or hydrocodes ) are widely used to simulate explosive or highspeed impact phenomena, and must be subject to rigorous programs of verification and validation [as reviewed among many others by Oberkampf et al. 1, 2), Roy 3) and Kamm et al. 4)] while being so employed. One important code verification tool is comparison to exact solutions of the underlying equations otherwise solved approximately by a hydrocode. In the ideal gas case the Noh 5 8), Sedov 9), Guderley 10 12), Kidder 13) and Coggeshall 14 16) solutions are example test problems that may be used to reveal strengths and weaknesses in a hydrocode for example, wall-heating errors, symmetry breaking). Unfortunately, in the case of non-ideal gas flows, exact solutions are at best less well-understood, if not altogether unavailable. There has been quite a bit of effort to export some of the aforementioned ideal gas solutions to the inviscid compressible flow of non-ideal materials, mostly by identifying EOS models with special properties and mimicking the derivations from the corresponding ideal gas problems 17 21). There have also been attempts at treating these issues in generality 22 24, 8), some of which have been successful. On the other hand, it has become clear that there is something special about the ideal gas constitutive law that allows for the existence of more exact solutions than other, more realistic EOS models: namely, a lack of inherent dimensional scales in both the Euler equations and EOS, and a corresponding abundance of symmetries. Thus, for each problem, it seems that the best we can do is find all EOS models that possess the necessary symmetries for the Euler equations to have exact solutions with desired properties. This method essentially amounts to finding forms of an EOS that lead to a reduction of the Euler equations from partial differential equations PDEs) to ordinary differential equations ODEs), and then testing whether the ODEs have a solution with desirable properties. While the reduction to ODEs has been well studied, the solvability of those ODEs has received less attention [with the notable exceptions of Guderley 10), Sedov 9) and Lazarus 11)]. One of the contributions of this work is to initiate that study in an important special case. Existence in these problems is not trivial and often reduces to a nonlinear eigenvalue problem, whose solution properties appear to not be rigorously understood. In addition to the need for useful code verification problems, there are two other motivations for finding analytical solutions for non-ideal, inviscid compressible flows. The first discussed principally by Barenblatt 25) is that such solutions frequently express the intermediate asymptotic behavior of physical systems. Additional effort has gone into the closely related problem of showing that different kinds of exact solutions are stable to perturbation and in some sense will attract nearby solutions to them 26 32). Our final motivation is the fact that in the context of inviscid compressible flow, many exact solutions belong to the sub-classes of self-similar or scale-invariant solutions. Understanding the spatial, temporal or other scaling behavior of fluid flow scenarios allows experimentalists to choose ostensibly without penalty the dimensional scale on which to perform work and then extrapolate according to simple scaling laws. This is the main idea behind scale modeling in engineering, and it often helps achieve significant cost savings, increases the amount of data that can be obtained, and in some cases even makes the difference between an experiment being feasible or not.

3 EXISTENCE OF SELF-SIMILAR CONVERGING SHOCKS IN NON-IDEAL MATERIALS Contributions of this work In this work, we focus on the Guderley s idealized implosion problem 10, 11, 12, 33, 34), which considers an infinitely strong cylindrically or spherically symmetric shock wave moving toward the one-dimensional 1D) curvilinear origin, focusing at the origin, and then reflecting back into the surrounding once-shocked medium. In addition to being solved by the aforementioned authors in the ideal gas case, the Guderley problem has also been solved in some other scenarios 17 19), and Holm 22), Axford and Holm 23) and Hutchens 24) even provide a general class of EOS models for which there exists a reduction from the PDEs of inviscid compressible flow to ODEs. For the purposes of code verification, a reduction to ODEs is often considered an exact solution because numerical ODE solvers are usually considered to be more robust than the hydrocodes in general, numerical PDE solvers) they are often used to verify. Moreover, Ramsey et al. 8) have shown that certain classical ideal gas test problems can be solved for very broad classes of EOS closures, using a choice of similarity variables and associated reduction-to-odes that applies equally well regardless of the material in which the compressible flow occurs. On the other hand, there is an obstacle in the case of curvilinear flows that prevents the existence of counterpart universal solutions. For example, Ramsey et al. 8) find that the 1D cylindrical or spherical Noh problems only have solutions for certain choices of EOS. Such issues with curvilinear symmetry do not arise because of incompatible boundary conditions, but rather from the fact that the reduced ODEs do not have a non-trivial solution with the properties that define a generalized Noh flow. Indeed, while the presence of enough symmetries to reduce the governing Euler PDEs to ODEs has been extensively studied [see, for example, Ovsiannikov 35), Andreev et al. 36), Cantwell 37), Holm 22), Axford and Holm 23) and Hutchens 24)], less is known about when the resulting ODEs actually have a solution that satisfies conditions specific to a given class of problems. Thus, in this work, we obtain some first results concerning the aforementioned ODEs and various conditions inherent to the Guderley problem. Using similarity analysis, we quickly reduce to two types of similarity variables corresponding to either accelerating and constant-velocity shock waves) which respect all the boundary conditions of the problem. We then show that for accelerating shocks, either the solutions to the ODEs are 1) nondifferentiable, 2) unbounded or 3) arise as the solution to a nonlinear eigenvalue problem. For constant-velocity shocks, we are able to show that any solutions must be unbounded even though the reduction to ODEs can be performed for EOS models with very broad properties. Such unbounded solutions may still be valuable for code verification, but must be applied with caution, see for example 38). Our analysis thus provides evidence that the existence of suitable solutions is a non-trivial consideration, especially when working with problems exhibiting Zel dovich and Raizer s and Barenblatt s self-similarity of the second kind 39, 25). Moreover, our presentation is more rigorous and thorough than previous presentations, and we dwell especially on the ODE-related aspects of the problem. For completeness, we have included appendices which treat the general initial and boundary conditions necessary to treat the Guderley problem in arbitrary materials. Some of the results in these appendices are counterintuitive and of independent interest. We also note that the case of constant-velocity shocks has been neglected in the literature, but is nonetheless of great interest because of its broader applicability than the accelerating case an important consideration in verification of hydrocodes intended to simulate multiple materials. Finally, we note that by approaching the classical Guderley s problem from this more abstract perspective, we are able to get an intuitive understanding of the ODEs and their existence properties,

4 404 Z. M. BOYD et al. which is lacking in previous studies. The derivation clarifies that the stability of compression shocks and the requirement of boundedness combine to necessitate the solution of nonlinear eigenvalue problems in the case of accelerating shocks, and prevents the existence of such solutions in the constant-velocity case. In particular, investigating unbounded solutions, such as those in 38) is a promising avenue for future research, as the eigenvalue problem can be avoided in such cases. The rest of this work consists of two sections: Section 2 contains the necessary background and assumptions, and Section 3 contains all the main results, together with their derivations and interpretation. We conclude in Section 4. Other aspects of the problem, such as boundary conditions and the shock trajectory, are deferred to appendices. 2. Mathematical model In this section, we introduce the governing equations, define the Guderley problem and state the assumptions underlying this work. 2.1 Governing equations Specializing to 1D symmetry, the equations of adiabatic motion for an inviscid compressible fluid are given by [see, for example, Landau and Lifschitz 40), Courant and Friedrichs 41), Ovsiannikov 35), Axford 42) and Harlow and Amsden 43)] ρ u t + u ρ r + ρ r + ku ) = 0, 2.1) r u t + u u r + 1 p = 0, 2.2) ρ r p t + u p r + K S u r + ku r ) = 0, 2.3) where u is the velocity field, p is the scalar) pressure, ρ is the mass density, and k = 1or2 for cylindrical or spherical symmetry, respectively. The planar symmetry case k = 0) will not be considered in this work, as its properties are different, and a wide variety of solutions can be shown to exist. Moreover, in 2.3), K S is the adiabatic bulk modulus, defined as K S ρ p ρ ), 2.4) S where S is the fluid entropy. The adiabatic bulk modulus is also related to the local speed of sound c by K S = ρc 2, 2.5) and is in general a material-dependent function of p and ρ indeed, the introduction of the adiabatic bulk modulus is the only source of information about the thermodynamic properties of the specific material under consideration. Moreover, given an EOS of the form p = pe,ρ) where e is the energy per unit mass or specific internal energy; SIE), K S is obtained from the relation K S = ρ p ρ + p p e ρ e. 2.6) ρ

5 EXISTENCE OF SELF-SIMILAR CONVERGING SHOCKS IN NON-IDEAL MATERIALS 405 The reverse relation e K S p + ρ e ρ ρ = p p ρ, 2.7) shows that an inverted EOS of the form e = ep,ρ) can also be recovered from K S [as shown in detail by Axford 42)]. Equations 2.1) 2.3) express the conservation of mass, momentum and energy of the fluid, neglecting heat conduction, body forces, material strength, anisotropy of material structure and viscosity. These assumptions are appropriate in many contexts where shock waves form, including supersonic flows 41), explosions 39), shock tube experiments 39) and space reentry 44). 2.2 Guderley s problem We seek solutions to a natural extension of Guderley s idealized implosion problem 10), which is a cylindrically or spherically symmetric converging-reflecting shock solution of 2.1) 2.3). For the case of an ideal gas, this problem has also been solved by Stanyukovich 45), Butler 46), Lazarus 11) and many others. Some variations on this problem have also been considered and solved 17, 18, 23, 19). In this work, we develop the theory of existence of Guderley solutions with any material whose corresponding K S satisfies mild conditions. The objective of the Guderley problem is to determine the flow in a scenario where a strong cylindrically or spherically symmetric shock generated far from the origin converges at the origin and is reflected back. Following authors such as Stanyukovich 45), Zel dovich and Raizer 39) and Chisnell 34), in this work we restrict to the converging regime. As shown in detail by Lazarus 11) and Ramsey et al. 12), analysis of the reflected regime follows naturally from that of the converging regime, and will be deferred to a future study. For the sake of mathematical precision, we define the Guderley problem as the problem of finding functions ρ,u, p, and r s the shock wave location as a function of t) satisfying the following requirements: r s :, 0] [0, ) and lim r st) =, r s 0) = 0. r s is monotonic. The time t = 0 t corresponds to the moment of focusing, so t < 0 throughout this work. ρ,u, p :, 0) 0, ) R are solutions to 2.1) 2.3), and are differentiable except at r s t), where a shock occurs. u and p are negligibly small for r < r s. We set them formally to zero. The unshocked SIE takes whatever value is determined by our choice of ρ and p, and the EOS closure model see Appendix A for more detail). Density ρ is a prescribed constant ρ 0 > 0 for r < r s t). ρ, u and p are bounded at each fixed t < 0. K S :[0, ) [0, ) [0, ) is bounded and continuous, and K S > 0 whenever ρ>0and p > 0. ρ>0, p > 0, and u < 0 for r > r s t). The shock travels subsonically relative to the fluid immediately behind it, that is, u s u p ) 2 < c 2 immediately post-shock. Here, u s and u p are the shock and particle velocities, respectively. This is a stability condition and is implied by either of the following two common assumptions: Convex P ρ adiabats: 2 p ρ 2 > 0, which is usually true of physical systems far away from S

6 406 Z. M. BOYD et al. phase transitions, see 41). 1 Perturbative stability, also known as evolutionary stability: if the unshocked and shocked regions are perturbed by the addition of small acoustic waves, the shock will remain a shock over time and only adjust in speed to accommodate the perturbations. This prevents the shock from devolving into a rarefaction, as discussed by Jeffrey 50) and Burgess 51). Moreover, the Guderley solution is an example of a self-similar scaling solution of 2.1) 2.3), of the form ρr, t) = t β ρ Dξ), 2.8) ur, t) = t β u Vξ), 2.9) pr, t) = t β p ξ), 2.10) where ξ = r t 1 λ), 2.11) D, V and are functions solely of the indicated argument, and β ρ,β u, β p and λ are constants. The constant λ is referred to as the similarity exponent, and it must satisfy λ [0, 1) as shown in Appendix A). The constants appearing in 2.8) 2.10) must be chosen so that 2.1) 2.3) reduce to ODEs in the independent variable ξ. The choice of these similarity variables can be motivated in several ways. They are perhaps most frequently used as an ansatz or based on dimensional considerations indeed, the initial solutions of the Guderley problem were motivated by such reasoning. From a more principled standpoint, and consistent with the interpretation of self-similar variables as provided by numerous authors 52, 12, 42, 23, 24), 2.8) 2.11) represent an example of the invariant functions of the scaling group admitted by 2.1) 2.3). Symmetry analyses of 2.1) 2.3) have been completed by many of the same authors, from which 2.8) 2.11) explicitly and naturally emerge as the so-called canonical coordinates corresponding to scaling phenomena in the context of the inviscid Euler equations. Thus, while the introduction of 2.8) 2.11) may be rigorously motivated using symmetry analysis techniques, a complete discussion of this methodology is outside the scope of the current work. 3. Existence theorem The main result of this work is: Theorem 1. All adiabatic bulk moduli for which the Guderley problem has a self-similar scaling solution are of the form K S = pf ρ), for some function f. In all such solutions, the shock is accelerating as r t 1 λ for λ 0, 1). The similarity exponent, λ, if it exists, solves a nonlinear eigenvalue problem. We now prove this result in two subsections. The first investigates the choice of similarity variables, and the second considers solvability. In a third subsection, we focus on a special case which seems not to have been previously considered, but which is nonetheless of great interest. 1 We note the fact that substances exist that violate this assumption, see for example, 47, 48, 49). Such materials necessarily have non-convex p-v isentropes and can sometimes exhibit stable, entropy-increasing rarefaction shocks rather than compression shocks. We explicitly exclude such materials from our analysis, as a sufficiently detailed investigation of them likely constitutes an entirely separate work and potentially fruitful avenue of future study.

7 EXISTENCE OF SELF-SIMILAR CONVERGING SHOCKS IN NON-IDEAL MATERIALS Similarity variables The unshocked state of the Guderley problem is ρr, t) = ρ 0 = const., 3.1) ur, t) = 0, 3.2) pr, t) = 0, 3.3) for r < r s t) and all t. Using 2.8) 2.11) to write these conditions in terms of similarity variables immediately yields β ρ = 0, since the constant unshocked density cannot scale as a function of time with ξ held constant. There are no such constraints on β u and β p appearing in 2.9) and 2.10) since the u and p variables feature no characteristic scales in the unshocked region. Now, we apply 2.8) 2.11) to 2.3) to yield [ ] 1 λ)ξ + t λ+β u V D + t λ+β u V + kv ) = ) ξ This equation depends on ξ alone if either β u = λ. Applying 2.8) 2.11) to 2.2) yields λv + 1 λ)ξv + VV + D t βp+2λ = 0, 3.5) which depends only on ξ if β p = 2λ. With the above determinations, 2.8) 2.11) become Finally, applying 2.8) 2.11) to 2.3) yields ρr, t) = Dξ), 3.6) ur, t) = t λ Vξ), 3.7) pr, t) = t 2λ ξ), 3.8) ξ = r t 1 λ). 3.9) 2λ + 1 λ)ξ + V + t 2λ K S V + kv ξ To cancel the dangling t term from this relation, either λ = 0, or K S must obey ) = ) K S t 2λ, 3.11) leaving λ a free parameter. Since K S can depend only on p = t 2λ ξ) and ρ = Dξ), it then follows that K S = pf ρ), 3.12) where f is an arbitrary function of the indicated argument. The derivation culminating in 3.12) is similar to those provided by Holm 22), Axford and Holm 23), Hutchens 24) and Boyd et al. 53) using symmetry analysis arguments. In any event, one of λ = 0 or 3.12) are the necessary conditions to cancel the dangling t appearing in 3.10) see Appendix A for a formal derivation of this outcome).

8 408 Z. M. BOYD et al. Equation 3.12) has a significant physical interpretation. We observe that the output of f must be dimensionless [since K S has units of pressure, as may be ascertained from inspection of 2.4)]. This means there must be a characteristic density incorporated into f that allows the units of ρ to be canceled; this is the only characteristic scale permitted to exist in f. The reason that a characteristic density is allowed is by making the unshocked density fixed and non-zero, there is already one characteristic density present in the problem definition, so no symmetry is lost by incorporating another. Indeed, we have already seen this characteristic density destroy one degree of symmetry in 2.8) when we were forced to exclude scaling on density in the construction of similarity variables. In Section 3.2, we will assume 3.12) holds. Otherwise, it is notable that there is no restriction on K S in addition to those laid out in Section 2.2) in the case λ = 0, so if a suitable solution of the associated ODEs can be found, it will apply to any flow that can be modeled by 2.1) 2.3) and an EOS closure model expressible in terms of bounded, continuous and positive K S. 3.2 Analysis of ODEs Equations 3.4), 3.5) and 3.10) may be rewritten as where we have substituted XD + DV = k DV, 3.13) ξ XV + 1 D = λv, 3.14) X + K S t 2λ V + k ) ξ V = 2λ, 3.15) X = 1 λ)ξ + V, 3.16) for notational brevity. As noted in Section 3.1, this system collapses to ODEs only if λ = 0 or 3.12) is satisfied. Assuming the latter condition, isolating the derivatives in 3.13) 3.15) gives the system D V V = D X 2 C 2) D k Xξ ) λ + 2λ XV λ X 2 ) kc2 DV ξ DC 2 k X ) ξ λ + 2λ X V 3.17) where Cξ) t λ c may be interpreted as a scaled sound speed that replaces K S via 2.5); this substitution helps provide a more intuitive physical interpretation to the denominator appearing in 3.17). Consistent with the formal definition of the Guderley problem provided in Section 2.2, we seek a solution of 3.17) that is everywhere bounded in the variables D, V and a direct consequence of the boundedness condition on the physical variables ρ, u and P). Since we assume K S is bounded, by its definition C is also bounded. We are also only interested in differentiable solutions of 3.17), and will discard all choices of λ and C that cause D, V and to become infinite. We therefore analyze whether or not the denominator appearing in 3.17) vanishes. Since this denominator is a

9 EXISTENCE OF SELF-SIMILAR CONVERGING SHOCKS IN NON-IDEAL MATERIALS 409 continuous function of its arguments, it will vanish if we can identify one coordinate where it is negative and another where it is positive. Far from the origin that is, r ) and for all finite times, ξ by 2.11). As a result, the denominator of 3.17), or D X 2 C 2) = D [1 λ) ξ + V] 2 DC 2, 3.18) is observed by inspection to be strictly positive at the aforementioned coordinate, as all quantities besides ξ appearing in it are bounded. Immediately adjacent to the shock front, the denominator is negative. To see this, observe that if ξ s is the coordinate of the shock wave in ξ-space, then the shock velocity u s satisfies ξ s = u s t λ 1 λ, 3.19) as we show in A.10) in the appendices. With 3.16), it then follows that X = t λ u u s ), 3.20) immediately adjacent to the shock front. The quantity u u s appearing in 3.20) is the shock velocity in a reference frame where the post-shock particles are motionless. With 3.20) and the definition of C, the denominator appearing in 3.17) may be written in terms of physical variables as ρ t 2λ [ u s u) 2 c 2]. 3.21) Thus, to determine the sign of this quantity when evaluated immediately adjacent to the shock front, we need only determine whether the shocked sound speed is bigger or smaller than the aforementioned motionless-particle reference frame shock velocity. The conclusion follows exactly from the shock stability assumption provided in Section 2.2: we assume that the shock travels subsonically relative to a particle immediately behind it. As a result, 3.21) is strictly negative when evaluated immediately adjacent to the shock front. Combined with 3.18) evaluated as ξ, this result ensures that the denominator appearing in 3.17) does indeed change sign between the shock front and r, and will somewhere vanish. Thus, 3.17) does not have a bounded, differentiable solution, except if the numerators appearing within can all be made to vanish simultaneously with the denominator. Since by the definition of the Guderley problem none of the variables D, V, and C can be made zero, only the parameter λ may be selected to in principle) achieve the required condition. This result shows that the counterpart phenomenon appearing in the ideal gas Guderley solution is observed in general: 3.17) is a nonlinear eigenvalue problem. In the ideal gas case, the correct value of λ is determined numerically, as has been done among many others by Guderley 10), Stanyukovich 45), Butler 46), Lazarus 11), Chisnell 34) and Ramsey et al. 12). The preceding proof is notable in that it gives an intuitive meaning to the blowup observed in solving 3.17): the similarity variable derivatives are proportional to the inverse difference between the sound speed and the shock speed in a certain reference frame, at least immediately adjacent to the shock front: [ D, V, u s u) 2 c 2] )

10 410 Z. M. BOYD et al. Adjacent to the shock front, stability requires that this difference have one sign, whereas far from the shock, boundedness requires it to have the opposite sign. Thus, one is forced to either abandon boundedness [as done by Velikovich et al. 38) and Coggeshall 15) in some shock-free solutions] or solve an eigenvalue problem in the style of Guderley and his many successors. As far as the authors are aware, this interpretation is novel; the complete theory for when the eigenvalue problem has a solution or what the solution might be in general) is left as a matter for future work. 3.3 The universal symmetry In Section 3.2 the reduction to ODEs was accomplished by assuming either λ = 0orK S = pf ρ). While the latter case notably includes the ideal gas Guderley and related problems as members, the λ = 0 case is much broader, encompassing all materials modeled by 2.1) 2.3) with an EOS closure model expressible in terms of a bounded, continuous, and positive adiabatic bulk modulus. If λ = 0 solutions can be found, they thus yield code verification test problems for a very broad class of flows and materials. This case has not been studied rigorously in the context of the Guderley solution, although it has been treated by Bethe 54), Menikoff and Plohr 55), Kamm 56) and Ramsey et al. 8) for various Riemann and Noh problems. With 3.9), setting λ = 0 gives the similarity variable ξ = r t, 3.23) which, following arguments presented in Appendix A, implies constant velocity shock waves following the trajectories r s = ξ s t, 3.24) where ξ s is a constant. With this parameterization, the ODEs given by 3.17) become where 3.16) likewise becomes D V V = D [ X 2 C 2] kxd ξ kc2 ξ kxdc 2 ξ, 3.25) X = ξ + V. 3.26) The singular analysis of 3.25) proceeds identically to that of 3.17) presented in Section 3.2, with identical conclusions: the denominator of 3.25) is strictly positive as r, and strictly negative at r = r s t). As before, the denominator of 3.25) must therefore vanish somewhere, resulting in an unbounded solution contrary to the definition of the Guderley problem as outlined in Section 2.2. However, for λ = 0 the zero-denominator pathology cannot be averted simply by requiring the numerators of 3.25) to simultaneously vanish with its denominator: again, none of D, V, and C may be made zero, as ρ, u and c must be non-zero and finite for all t < 0, by the definition of the Guderley problem. Thus, by requiring λ = 0, we have ostensibly removed a degree of freedom that would otherwise allow for at least the possibility of a nonlinear eigenvalue problem, and thus the

11 EXISTENCE OF SELF-SIMILAR CONVERGING SHOCKS IN NON-IDEAL MATERIALS 411 only route to the construction of a bounded solution. 2 In return, we have gained one more degree of freedom in the choice of K S, but the above analysis shows that this freedom is not enough to generate any Guderley-like solutions. One potential path to increasing the utility of the λ = 0 case is to investigate it in conjunction with flows featuring multiple shock waves or other discontinuities, as discussed by Lazarus 11). Without a more detailed analysis or rectification of these types of flows, it is difficult to ascertain whether or not they may represent viable intermediate asymptotic entities, as discussed by Barenblatt 25). Otherwise, the restriction on EOS closure models satisfying K S = pf ρ) is, therefore, definitively required in all cases. 4. Conclusion In the foregoing, we have analyzed in detail the Guderley problem for a generalized EOS closure model by showing how to set up the initial and boundary conditions, determining all possible reductions to similarity variables, and showing that the resulting ODE system and ancillary conditions constitute a nontrivial eigenvalue problem. In particular, we have shown that the universal choice of similarity variables always leads to a reduction to ODEs, never conflicts with the initial/boundary conditions, and yet yields ODEs that never have a bounded, differentiable solution. This motivates further study of when the ODEs resulting from the introduction of similarity variables have a solution with reasonable properties. For instance, from the foregoing, it is clear that one must choose between unbounded solutions or an eigenvalue problem. Another novel feature of this work is that, by conducting our analysis of the governing ODEs on a more abstract level than is usually done, we have arrived at an intuitive explanation for the necessity of solving an eigenvalue problem to obtain a self-similar solution; namely, there is a competition between stability of compressive shock waves and boundedness of the shocked driving conditions. Acknowledgements This work was performed under the auspices of the United States Department of Energy by Los Alamos National Security, LLC, at Los Alamos National Laboratory under contract DE-AC52-06NA Z. M. Boyd was also supported by an NDSEG Fellowship. The authors thank J. Schmidt, E. Schmidt, J. Lilieholm, W. Black and J. Ferguson for helpful conversations and valuable insights on these topics. References 1. W. L. Oberkampf, T. G. Trucano and C. Hirsch, Verification, validation, and predictive capability in computational engineering and physics, Appl. Mech. Rev ) W. L. Oberkampf and C. J. Roy. Verification and Validation in Scientific Computing. Cambridge, UK University Press, 2010). 3. C. J. Roy, Review of code and solution verification procedures for computational simulation, J. Comp. Phys ) In the scoping study of Lilieholm et al. 57), it appears there is a global, unbounded solution of 3.25) in some cases, which is a promising avenue for future work.

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13 EXISTENCE OF SELF-SIMILAR CONVERGING SHOCKS IN NON-IDEAL MATERIALS R. A. Axford and D. Holm. Spherical shock collapse in a non-ideal medium. Technical report LA-UR , Los Alamos Scientific Lab, Los Alamos, NM 1978). 24. G. J. Hutchens, Finite-strength Shock Propogation for Alternative Equations of State. PhD thesis, 1990, University of Illinois at Urbana-Champaign, Champaign, IL). 25. G. I. Barenblatt, Similarity, Self-similarity, and Intermediate Asymptotics. Consultants Bureau, 1979). 26. N. Ponchaut, Part I: 3DPTV: Advances and Error Analysis. Part II: Extension of Guderley s Solution for Converging Shock Waves. PhD thesis 2005, California Institute of Technology, Pasadena, CA). 27. N. Ponchaut, H. G. Hornung, D. I. Pullin and C. A. Mouton, On imploding cylindrical and spherical shock waves in a perfect gas, J. Fluid Mech ) H. G. Hornung, D. I. Pullin and N. F. Ponchaut. On the question of universality of imploding shock waves, Acta Mech ) C. Morawetz, Contracting Spherical Shocks Treated by a Perturbation Method. PhD thesis 1951, New York University, New York, NY). 30. W. Häfele, Über die Stabilität des Guderleyschen kugeligen Verdichtungsstoßes, Z. Natuforschung ) K. Brushlinskii, Instability of a convergent spherical shock wave. USSR Comput. Mech. Math. Phys ), J. M. Clarisse, A Godunov-type method in Lagrangian coordinates for computing linearlyperturbed spherically-symmetric flows of gas dynamics. Online. Available from inria.fr/file/index/docid/197498/filename/article.pdf. Accessed 13 September J. Meyer-ter-Vehn and C. Schal, Self-similar spherical compression waves in gas dynamics, Z. Naturforschung A ) R. Chisnell, An analytic description of converging shock waves, J. Fluid Mech ) L. V. Ovsiannikov and W. F. Ames, Group Analysis of Differential Equations. Academic Press 1982). 36. V. K. Andreev, O. V. Kaptsov, V. V. Pukhnachev and A. A. Rodionov, Applications of Group- Theoretical Methods in Hydrodynamics, Vol. 450 Springer Science & Business Media, Dordrecht, Netherlands 1998). 37. B. J. Cantwell, Introduction to Symmetry Analysis. Cambridge University Press, Cambridge, UK 2002). 38. A. L. Velikovich, J. L. Giuliani, S. T. Zalesak, J. W. Thornhill and T. A. Gardiner, Exact selfsimilar solutions for the magnetized Noh Z-pinch problem, Phys. Plasma ). 39. Y. B. Zeldovich and Y. P. Raizer, Physics of Shock Waves and High-Temperature Hydrodynamic Phenomena. Academic Press, New York 1966). 40. L. D. Landau and E. M. Lifschitz. Theoretical Physics, 2nd edn. Pergamon Press, Oxford, UK 1986). 41. R. Courant and K. O. Friedrichs, Supersonic Flow and Shock Waves. Springer, New York 1948). 42. R. Axford, Solutions of the Noh problem for various equations of state using Lie groups, Laser. Particle Beams. 2000). 43. F. H. Harlow and A. A. Amsden, Fluid dynamics: a LASL monograph Mathematical solutions for problems in fluid dynamics) LA 4700, Technical Report Los Alamas National Labratory 1971).

14 414 Z. M. BOYD et al. 44. Ames Research Staff, Equations, tables, and charts for compressible flow. Technical report Report 1135, Ames Aeronautical Laboratory 1953). 45. K. Stanyukovich, Unsteady Motion of Continuous Media. Pergammon Press, New York 1970). 46. D. Butler, Converging spherical and cylindrical shocks, Technical Report 54/54, UK Armament Research Establishment 1954). 47. P. A. Thompson, A fundamental derivative in gasdynamics. Phys. Fluid ) P. D. Lax, Hyperbolic systems of conservation laws II, Commun. Pure Appl. Math ) O. A. Oleinik, Construction of a generalized solution of the Cauchy problem for a quasi-linear equation of first order by the introduction of vanishing viscosity, Uspekhi. Mat. Nauk ) A. Jeffrey and T. Taniuti, Non-linear Wave Propogation with Applications to Physics and Magnetohydrodynamics. Academic Press, New York 1964). 51. D. Burgess and M. Scholer, Collisionless Shocks in Space Plasmas. Cambridge University Press, Cambridge, UK 2015). 52. G. W. Bluman and S. C. Anco, Symmetry and Integration Methods for Differential Equations. Springer, New York 2002). 53. Z. M. Boyd, S. Ramsey and R Baty, Symmetries of the Euler compressible flow equations for general equation of state, Technical Report LA-UR , Los Alamos National Laboratory, Available at info:lanl-repo/lareport/la-ur H. Bethe, The theory of shock waves for an arbitrary equation of state, Technical Report PB , Clearinghouse for Federal Scientific and Technical Information, US Department of Commerce, Washington, DC, R. Menikoff and B. J. Plohr. The Riemann problem for fluid flow of real materials. Rev. Mod. Phys ) J. R. Kamm. An exact, compressible one-dimensional Riemann solver for general, convex equations of state, Technical report LA-UR , Los Alamos National Laboratory, Los Alamos, NM 2015). 57. J. F. Lilieholm, E. M. Schmidt, S. D. Ramsey, J. H. Schmidt and Z. M. Boyd. Solving the Guderley implosion problem with a Mie-Gruneisen-like equation of state. Technical report LA-UR , Los Alamos National Laboratory, Los Alamos, NM 2016). 58. R. Axford and D. Holm, Converging finite-strength shocks, Phys. D: Nonlinear Phenomena ) W. J. Rae, Analytical studies of impact-generated shock propagation: survey and new results, High-velocity Impact Phenomena Academic press, New York 1970) B. Lee, Nonuniform propagation of imploding shocks and detonations, AIAA J ) K. Oshima, Blast waves produced by exploding wire, Departmental Bulletin Paper, Aeronautical Research Institute, Report No. 258 University of Tokyo, Tokyo, Japan 1960). 62. E. M. Schmidt, S. D. Ramsey, J. F. Lilieholm, Z. M. Boyd and J. H. Schmidt, A scale invariant equation of state for Gruneisen materials, Technical Report LA-UR , Los Alamos National Laboratory 2016). 63. N. Toque, Self-similar implosion of a continuous stratified medium, Shock Waves )

15 EXISTENCE OF SELF-SIMILAR CONVERGING SHOCKS IN NON-IDEAL MATERIALS G. Madhumita and V. Sharma, Propogation of strong converging shock waves in a gas of variable density, J. Eng. Mech ) P. J. Olver, Applications of Lie Groups to Differential Equations. Springer, New York 1993). 66. P. Hafner, Strong convergent shock waves near the center of convergence: a power series solution, SIAM J. Appl. Math ) APPENDIX A A. The unshocked state and shock trajectory In this appendix, we derive the appropriate unshocked conditions and the motion of the shock front. It will be seen that the traditional assumptions from the ideal gas case are not adequate for our more general situation. In particular, we will consider the case where unshocked SIE e 0 and pressure p 0 are nonzero. Neither of these possibilities has been treated in depth in previous literature, 3 and the details are important if one wishes to apply our results to a hydrocode. The definition of the Guderley problem provided in Section 2.2 includes the typical choice of unshocked state, in particular featuring ρ = ρ 0 and u = 0. In the ideal gas case, one can also assume that e 0 = p 0 = 0, since one should avoid characteristic scales that will destroy self-similarity. In the more general case, however, this is not always an appropriate assumption. To see how this can happen, consider, for example, a simple generalization of the ideal gas EOS provided by p ρe e 0 ), where e 0 is a reference SIE. With A.1), an unshocked p 0 = e 0 = 0 state is only satisfied for ρ 0 = 0 a vacuum state. However, in this case an e = e 0 unshocked state allows for the satisfaction of p 0 = 0 for any unshocked density. Now consider the Rankine Hugoniot jump conditions: ρ 1 u s u 1 ) = ρ 0 u s, A.1) A.2) ρ 1 u s u 1 = p 1 p 0, A.3) ρ 0 u s e 1 e ) 2 u2 1 = p 1 u 1, A.4) where the subscripts 0 and 1 indicate unshocked and shocked quantities, and u s is the shock velocity. If the energy in the unshocked state is e 0, and p 0 is 0, A.2) A.4) reduce to the usual strong shock limit. Since many EOS models are intended to only match experimental data at high pressures and energies, it is reasonable that situations such as the preceding might arise and demand flexibility with the unshocked conditions. Indeed, Schmidt et al. 62) and Lilieholm et al. 57) have completed preliminary work to develop an EOS with a reference SIE feature similar to that included in A.1), which nevertheless exhibits Guderley-like solutions when used in conjunction with the inviscid Euler equations. To see how the scaling can work with nonzero p 0 and e 0, we use a more general self-similar scaling ansatz than given by 2.8) 2.10), namely p p 0 = t 2λ ξ). A.5) 3 Axford and Holm 23) included the case where p 0 = 0 in their analysis, although it seems that in the primary example of their paper, they do set p 0 = 0. In any case, their focus is not on the issues we discuss here.

16 416 Z. M. BOYD et al. where the independent similarity variable ξ is still given by 2.11). This formulation yields a reduction of 2.1) 2.3) to ODEs provided we also have either λ = 0or K S = p p 0 )f ρ). A.6) Moreover, inserting 3.6), 3.7) and A.6) into A.2) A.4) gives D 0 u s V 1 t λ ) = u s, A.7) D 1 u s V 1 t λ = t 2λ 1, A.8) D 0 u s e 1 e ) 2 V 1 2 t 2λ = V 1 1 t 3λ, A.9) where the 0 and 1 subscripts denote pre- and post-shock values, respectively. In order for these conditions to be expressible solely in terms of similarity variables, from inspection of A.7) it is necessary that the shock velocity assume the form u s = 1 λ)ξ s t λ, A.10) and with 1 λ)ρ 0 ξ s e 1 e V 1 2 t 2λ ) = V 1 1 t 2λ + V 1 p 0. A.11) At this point, it would appear that A.11) is irreducible to similarity variables alone if e 0 and p 0 are not zero. This is not quite true, since by careful arrangement it is possible for the terms involving e 0 and p 0 to cancel. We will provide a specific example of how this can happen, building on work in by Boyd et al. 53). In this work, the authors substituted A.6) into 2.7) from Section 2, yielding a partial differential equation for ep,ρ), which they were able to solve analytically. The general solution is e = p p 0 )gρ) + p ) 0 p p0 ρ + h ρ 2 g, A.12) ρ) where g and h are arbitrary functions of their arguments the latter corresponding to the kernel of a linear PDE), and p 0 /ρ may be loosely interpreted as a reference SIE. Setting h = 0 and substituting into A.11) then yields 1 λ)d 0 ξ s t 2λ gd 1 ) + p 0 p ) D 1 D 0 2 V 1 2 t 2λ = V 1 1 t 2λ + V 1 p 0, A.13) and from A.7) we have so that A.13) becomes V 1 = D 0 ξ s 1 D 0 1 D 1 ), A.14) 1 λ)d 0 ξ s gd 1 ) = V 1, A.15) which is a relation depending only on ξ. This relation shows that if K S is of the form p p 0 )f ρ) with the unshocked pressure given by p 0 ), then self-similar scaling can occur, even in the presence of a characteristic unshocked pressure and energy. As a result, the conventional wisdom concerning characteristic scales and self-similar scaling must be applied with

17 EXISTENCE OF SELF-SIMILAR CONVERGING SHOCKS IN NON-IDEAL MATERIALS 417 caution: a Guderley-like problem may still exist for EOS closures that do not admit the standard initial conditions p 0 = e 0 = 0. 4 B. Formal derivation of the scale-invariant K S A more formal derivation of 3.12) proceeds along these lines: suppose there exists a self-similar scaling solution to the Guderley problem for a particular K S. Such a solution yields functions D, : 0, ) [0, ) that are well-defined. Moreover, as outlined in Section 3.1, the function ] Fξ,t) = t 2λ K S [ t 2λ ξ), Dξ), B.1) depends only on ξ, or, Fξ,t) = Fξ,t 0 ), for any choice of ξ,t, and t 0. Let t 0 be fixed; expanding B.1) then gives ] [ ] t 2λ K S [ t 2λ ξ), Dξ) = t 0 2λ K S t 0 2λ ξ), Dξ). B.3) Rearranging, we have B.2) ] K S [ t 2λ ξ), Dξ) = yξ) t 2λ, B.4) where y is an arbitrary function of the indicated argument. Now, let a > 0 be a constant, and let x 2 be an arbitrary element of the range of D [that is, x 2 = Dξ)]. Then, for any x 1 > 0, we can choose t, 0) so that x 1 = t 2λ ξ), since ξ) is assumed to be positive. Then, we have ] K S αx 1, x 2 ) = K S [α t 2λ ξ), Dξ) = K S [ a 1 = cξ) 2λ t a 1 2λ t 2λ 2λ ] ξ), Dξ) = acξ) t 2λ ] = ak S [ t 2λ ξ), Dξ) = ak S x 1, x 2 ), B.5) and thus, K S is homogeneous in its first argument at least in the domain needed for the Guderley problem). 4 If one instead chooses the standard ansatz p = t 2λ ξ), it is still possible to have a non-zero unshocked energy, but it is very difficult to have a nonzero unshocked pressure, as calculations very similar to the foregoing show.