Slightly Two or Three Dimensional Self-Similar Solutions
Re'em Sari, J. Nate Bode, Almog Yalinewich, Andrew MacFadyen
SSlightly Two or Three Dimensional Self-Similar Solutions
Re’em Sari, Nate Bode, Almog Yalinewich, Andrew MacFadyenReceived ; accepted a r X i v : . [ a s t r o - ph . H E ] S e p ABSTRACT
Self similarity allows for analytic or semi-analytic solutions to many hydro-dynamics problems. Most of these solutions are one dimensional. Using linearperturbation theory, expanded around such a one-dimensional solution, we findself-similar hydrodynamic solutions that are two- or three-dimensional. Sincethe deviation from a one-dimensional solution is small, we call these slightly two-dimensional and slightly three-dimensional self-similar solutions, respectively. Asan example, we treat strong spherical explosions of the second type. A strongexplosion propagates into an ideal gas with negligible temperature and densityprofile of the form ρ ( r, θ, φ ) = r − ω [1 + σF ( θ, φ )], where ω > σ (cid:28)
1. An-alytical solutions are obtained by expanding the arbitrary function F ( θ, φ ) inspherical harmonics. We compare our results with two dimensional numericalsimulations, and find good agreement. Subject headings: hydrodynamics, shock waves, instabilities
1. Introduction
Astrophysics supplies ample examples of hydrodynamic problems that admit self-similarsolutions. In supernovae explosions (Koo and McKee 1990; Chevalier 1976) a shock waveis created by the release of an immense amount of energy during a short time in the centerof an exploding star. When the shock wave propagates into the surrounding medium, thehydrodynamics is described by the the Sedov-Taylor solutions (Sedov 1946; Von-Neumann1947; Taylor 1950; Waxman and Shvarts 1993). Gamma-ray bursts provide a relativisticanalog of that (Blandford and McKee 1976; Best and Sari 2000; Sari 2006; Pan and Sari2006). If the external medium is spherical, these are one-dimensional solutions. However, ifthe external density has angular dependence, it will cause the shape of the shock, and theflow behind it, to deviate from sphericity.An inherently two-dimensional version of this problem is the explosion in half space.Here, space is assumed to be empty on one side of a plane, while the other side is filled withan ideal gas with constant density. A large amount of energy is then released at a point onthe surface. This describes the propagation of shockwaves in the process of cratering causedby large impacts on a planetary surface. Qualitatively, this problem and its self-similarnature was described by Zel’Dovich and Raizer (1967), but a two-dimensional self-similarsolution was not developed there.Here, we obtain two-dimensional and three-dimensional self-similar solutions thatdeviate only slightly from some known one-dimensional solution. We show that whentreating such solutions as perturbations, the analysis is analogous to the treatment ofstability (Ryu and Vishniac 1987; Goodman 1990; Chevalier 1990; Sari et al. 2000; Kushniret al. 2005). We call these solutions slightly two-dimensional or slightly three-dimensionalself-similar solutions. As a working example, we analyze small deviations from sphericityin the case of the strong explosion problem with external density falling as a power law of 4 –distance ρ ∝ r − ω , where ω >
3. Solutions with these values of ω are known to be self-similarsolutions of the second type (Waxman and Shvarts 1993).In Sec. 2 we briefly review the main features of the one-dimensional solutionwhich serves as the unperturbed solution for our analysis. In Sec. 3, we discuss theperturbation formalism for this problem and find the slightly two-dimensional and slightlythree-dimensional self-similar solutions for density perturbations proportional to a sphericalharmonic. These solutions are then demonstrated using the values of ω and γ for which theunperturbed solution is analytic in Sec. 4. The solution is then given for small deviationsfrom sphericity with arbitrary angular dependence in Sec. 5, while in Sec. 6 the small l limit is investigated. Our semi-analytic solutions are then favorably compared with fullfluid-dynamic simulations in Sec. 7. Finally, in Sec. 8 we give our concluding remarks.
2. The One-Dimensional Self-Similar Solution
Here we summarize the formalism leading to the one-dimensional self-similar solution(Waxman and Shvarts 1993). The discussion here follows Sari et al. (2000). Consider theStrong Explosion Problem in which a large amount of energy is released at the center ofa sphere of ideal gas with a density profile decreasing with the distance from the originaccording to ρ = Kr − ω , forming a strong outgoing shock wave.This problem was first investigated by Sedov (1946), Von-Neumann (1947), and Taylor(1950), who found the solutions for ω <
5, known as the Sedov-Taylor solutions. Waxmanand Shvarts (1993) showed that the Sedov-Taylor solutions are valid only for ω <
3, wherethe solutions are known as self-similar solutions of Type-I, and contain decelerating shockwaves. New, Type-II, self-similar solutions for almost all the range ω > ω >
3. The hydrodynamicequations for an ideal gas with adiabatic index γ in spherical symmetry are given by:( ∂ t + u∂ r ) ρ + ρr − ∂ r ( r u ) = 0 ,ρ ( ∂ t + u∂ r ) u + ∂ r ( ρc /γ ) = 0 , ( ∂ t + u∂ r )( c ρ − γ /γ ) = 0 , (1)where the dependent variables u , c , and ρ are the fluid velocity, sound velocity, and density,respectively. We now seek a self-similar solution to the hydrodynamic equations (Eqn. 1) ofthe form: u ( r, t ) = ˙ RξU ( ξ ) , c ( r, t ) = ˙ RξC ( ξ ) ,ρ ( r, t ) = BR (cid:15) G ( ξ ) , p ( r, t ) = BR (cid:15) ˙ R P ( ξ ) , (2)where ξ = r/R ( t ) is the dimensionless spatial coordinate, and the length scale R ( t )(frequently abbreviated as simply R ) is the shock radius and satisfies (Zel’Dovich andRaizer 1967; Waxman and Shvarts 1993)¨ RR ˙ R = δ ⇒ ˙ R ∝ R δ , (3)where δ is a constant. The quantities G, C, U, and P, which are defined by Eqns. 2, give thespatial dependence of the hydrodynamic quantities. The diverging (exploding) solutions ofEqn. 3 are R ( t ) = A ( t − t ) α , δ < Ae t/τ , δ = 1 A ( t − t ) α , δ > α = 1 / (1 − δ ).Solutions with δ < t represents the time of the pointexplosion, which is usually taken to be t = 0. For δ < < δ it accelerates. For δ > t represents the time of divergence rather than the explosion time. Thetransition between finite and infinite divergence occurs at δ = 1 where we have exponentialtime dependance (Simonsen and Meyer-Ter-Vehn 1997).Substituting Eqn. 2 into the hydrodynamic equations (Eqns. 1) and using Eqn. 3, onegets regular differential equations for the similarity quantities U , C , and G (see for exampleLandau & Lifshitz) with two free constants, the similarity parameters (cid:15) and δ : dUd log ξ = ∆ ( U, C )∆(
U, C ) , dCd log ξ = ∆ ( U, C )∆(
U, C ) (5)and an explicit expression for the density G: C − (1 − U ) λ G γ − λ ξ λ − = const . (6)The functions ∆, ∆ , and ∆ are given by:∆ = C − (1 − U ) , ∆ = U (1 − U )(1 − U − δ ) − U C − C ( (cid:15) + 2 δ ) /γ , ∆ = C (1 − U )(1 − U − δ ) − ( γ − CU (1 − U + δ/ −− C + 2 δ − ( γ − (cid:15) γ C − U , (7)and the parameter λ is λ = 2 δ − ( γ − (cid:15) (cid:15) . (8)The similarity parameter (cid:15) can be found from the boundary conditions at the strongshock, the Hugoniot jump conditions (Landau and Lifshitz 1987). Applying these relationsto a strong shock one gets (cid:15) = − ω , and also U (1) = 2 γ + 1 , C (1) = (cid:112) γ ( γ − γ + 1 , G (1) = γ + 1 γ − . (9) 7 –The boundary conditions on the shock do not state any limits on the possible values ofthe similarity parameter δ . In order to determine the value of this parameter one shoulddistinguish two kinds of similarity flows: Type-I and Type-II, defined first by Zel’dovich(Zel’Dovich and Raizer 1967). A solution of Type-I describes the flow in all space andtherefore conservation laws must be obeyed by the self-similar solution. One can thendeduce δ = ( ω − /
2, which gives the well-known Sedov-Taylor solutions. However, for ω > δ contains an infiniteamount of energy and therefore can not describe the flow over the whole space. Therefore,the flow must be Type-II.In Type-II solutions, there is a region, whose scale relative to the flow characteristiclength R ( t ) goes to zero with time, in which the similarity solution does not describethe physical system. Therefore, for this kind of solution the energy does not have to beconserved in the self-similar solution since this solution does not describe the whole flow.In order that the region which is not self-similar (located around the origin) does notinfluence the self-similar solution, the solution must pass through the singular point definedby (Zel’Dovich and Raizer 1967; Waxman and Shvarts 1993): U + C = 1 . (10)From this singular point requirement, the dependence of δ upon the parameters ω and γ can be found. It was found (Waxman and Shvarts 1993) that for ω > ω g ( γ ) > δ for which the solution passes through a singular point, and therefore a secondtype self-similar solution exists.A fully analytic solution to Eqns. 5–8 exists for the case where ω = ω a ( γ ) ≡ γ − / ( γ + 1): C ( ξ ) = (cid:112) γ ( γ − γ + 1 ξ , U ( ξ ) = 2 γ + 1 ,G ( ξ ) = γ + 1 γ − ξ − , P ( ξ ) = 2 γ + 1 . (11)For this analytical case the parameter δ is given by δ = ( γ − / ( γ + 1).
3. Slightly Two- and Three-Dimensional Self-Similar Solutions
We now consider small deviations from the spherically symmetric problem discussedin Sec. 2. In general, we wish to solve the problem for an external density perturbation ofarbitrary angular dependence, which we parameterize by ρ ( r, θ, φ ) = r − ω [1 + σF ( θ, φ )] , (12)where σ (cid:28) F is an arbitrary function of θ and φ , and ω > F ( θ, φ ) = Y l,m . In Sec. 5 these solutions will be used to construct the solutionfor an arbitrary F ( θ, φ ).We shall use here the Eulerian perturbation approach. We define the perturbedquantities as the difference between the perturbed solution (i.e., the slightly two-dimensionalself-similar solution) and the unperturbed one-dimensional solution at the same spatialpoint. The derivation of the perturbation equation is similar to the one given by Ryu andVishniac (1987), Chevalier (1990), and Sari et al. (2000). The perturbed hydrodynamic 9 –quantities are defined as δ(cid:126)v ( r, θ, φ, t ) = (cid:126)v ( r, θ, φ, t ) − v ( r, t )ˆ r ,δρ ( r, θ, φ, t ) = ρ ( r, θ, φ, t ) − ρ ( r, t ) ,δp ( r, θ, φ, t ) = p ( r, θ, φ, t ) − p ( r, t ) , (13)where (cid:126)v , p , and ρ are the velocity, pressure, and density in the perturbed solution, while v ˆ r , p , and ρ are the same quantities as in the unperturbed solution.We consider perturbations that can be written in a separation of variables form (Cox1980): δ(cid:126)v ( r, θ, φ, t ) = ξ ˙ R [ δU r ( ξ ) Y lm ( θ, φ )ˆ r + δU T ( ξ ) ∇ T Y lm ( θ, φ )] f ,δρ ( r, θ, φ, t ) = BR (cid:15) δG ( ξ ) Y lm ( θ, φ ) f ,δp ( r, θ, φ, t ) = BR (cid:15) ˙ R δP ( ξ ) Y lm ( θ, φ ) f , (14)where ∇ T ≡ ˆ θ ∂∂θ + ˆ φ θ ∂∂φ (15)are the tangential components of the gradient and R ( t ) is the unperturbed shock radiuswhich still satisfies Eqn. 3. The perturbed shock radius, R ( t, θ, φ ), is given by R ( t, θ, φ ) − R ( t ) ≡ δR ( t, θ, φ ) = Y l,m ( θ, φ ) R ( t ) f . (16)Eqns. 14 and 16 define the quantities δU r , δU T , δP , δG , and f . The quantity f measuresthe fractional amplitude of the perturbation to the shock wave radius. Here we deviatefrom the standard treatment of stability. There, f is a function of time: if the function f increases with time then the solution is unstable, while if f decreases with time thenthe solution is stable. However, here, since we demand that the perturbed solution beself-similar, f has to be independent of time. 10 –We linearize the hydrodynamic equation around the unperturbed self-similar solutionto get a linear set of equations: M Y (cid:48) = N Y (17)where Y = δGδU R δU T δP ,M = ξ ( U –1) Gξ U –1) ξ G U –1) ξ G γξ ( U –1) G ξ ( U –1) P ,N = ω –3 U – ξU (cid:48) – ξG (cid:48) –3 G l ( l + 1) G P (cid:48) G – (1– δ –2 U – ξU (cid:48) ) Gξ δ – U ) Gξ – ξ – − ξγ ( U –1) G (cid:48) G − ξ (cid:16) P (cid:48) P − γ G (cid:48) G (cid:17) ξ ( U –1) P (cid:48) P , and G , U , and P are defined by Eqn. 2.Unlike the perturbation equations for stability, the equations above do not contain anunknown parameter. They are, in fact, a special case of the equations used in Sari et al.(2000), but with the perturbation growth rate set to q = 0. In that sense they are similarto the equations of Oren and Sari (2009) for discretely self-similar solutions. Instead, anew parameter d = σ/f appears in the shock boundary conditions (the linearized Hugoniot 11 –jump conditions): δG (1) = γ + 1 γ − d − ω ) − G (cid:48) , δU r (1) = − U (cid:48) δU T (1) = − γ + 1 , δP (1) = 2 γ + 1 (2 + d − ω ) − P (cid:48) . (18)For any value of the parameter d one can integrate Eqn. 17 beginning at the shockfront using the shock boundary conditions. However, the singular point of the unperturbedsolution, ξ c , where C + U = 1, is also a singular point of the perturbed solution. Therefore,in general, such integration will diverge at the sonic point ξ c . Only for specific values of theparameter d , where an additional boundary condition at the singular point is satisfied, isthe solution regular. These are the physical values for the parameter d .Technically, solving these equations is easier than the equivalent perturbation case.The reason is that the unknown parameter d appears only in the shock boundary condition,and is absent from the differential equations. We can therefore solve these equationsstarting from the sonic point outward, and find the three independent solutions that arenonsingular at ξ c . Then we can find a linear combination of these three solutions, and thevalue of d that can solve the four boundary conditions at the shock.
4. Results
For convenience we investigate the case γ = 5 / ω = 17 /
4, where the unperturbedsolution is analytic. For l = 1 we obtain d = − .
2. This means that the fractionalamplitude of perturbations in the shock wave position, f , are an order of magnitude smallerthan the fractional amplitude of perturbations in the external density σ . The negativesign implies that at angles where the external density is higher, the shock wave position isretarded. This is expected intuitively. From the shock boundary conditions, we infer thatthe pressure at these angles is also lower. For l = 2 we find d = − .
6, and for l = 3 we find 12 – d = − .
1. A plot of d as function of l is given in Fig. 1.
5. Extension to Arbitrary Angular Dependence
The analysis above is limited to external density perturbations whose angulardependence is a spherical harmonic. This is necessary in order to obtain separationbetween the angular and radial dependencies. However, since we are dealing with linearperturbations, any arbitrary angular dependence can be expanded into a sum of sphericalharmonics, each of which could be solved in the method described in the previous section.Then, the solutions can be summed, leading to the perturbation solution for externaldensity perturbations with arbitrary angular dependence.As an example, we consider the following problem: A strong point-like explosion islaunched into a surrounding which has a density on one side of a plane slightly differentfrom the density on the other side of the plane. In our notation this is ρ ∝ r − / (1 + σH ( θ ))where H ( θ ) = 1 for θ < π/ H ( θ ) = − θ > π/
2. The point explosion in half spacecould be thought of as an extreme version of this density profile with σ = 1. However, oursolution formalism applies only for slightly two-dimensional cases where σ (cid:28) H ( θ ) = ∞ (cid:88) n =0 π √ n + 3Γ(1 / − n )Γ(2 + n ) Y n +1 , ( θ, . (19)The shape of the shock, R + δR ( θ ), deviates from its unperturbed value R by δR ( θ ) R = σ ∞ (cid:88) n =0 π √ n + 3 d (2 n + 1)Γ(1 / − n )Γ(2 + n ) Y n +1 , ( θ, . (20)This shape is plotted in Fig. 2 for both this analytic solution and for the numerical solutiondiscussed in Sec. 7. To make the analytic curve the sum was taken from n = 0 to n = 50. 13 – ! ! ! !
200 spherical harmonic degree l d " f ! Σ Fig. 1.— Dots show d as function of l as obtained by solving the differential equations for γ = 5 / ω = 17 /
4. For small l we have d ∼ = − . , while for large l , i.e., short wavelength,we obtain a linear relation: d = − (cid:112) / l (solid line). 14 – θ δ R / ( σ R ) NumricalAnalytic
Fig. 2.— The analytic (green) and numerical (blue) fractional deviation of the shock positionas function of θ for the Heaviside density distribution (Eqn. 19) in units of σ . The numericalresult is obtained for σ = 0 .
01, and is discussed in Sec. 7, while the analytic solution comesfrom Eqn. 20. It can be seen in both curves that, roughly speaking, the shock is composedof two hemispheres, connected smoothly over a short angular scale of less than 0 .
6. Short Wavelength Limit
Because the flow does not vary in the short wavelength limit, we may treat the matrices M and N as constants close to the shock front. By using the unperturbed values of thestate variable at the shock, we find the four independent modes of the problem: λ = − , , ± (cid:114) γγ + 1 l . (21)The first two are independent of l and indicate that the state functions, close to the shock,vary on the scale R , regardless of the wavelength of the perturbation. However, the othertwo are linear in l meaning that close to the shock the state functions vary over smallscales of order R/l . Therefore, for large l , the positive mode is growing inward very rapidly,and thus can not exist physically. For this reason we demand that the perturbation hasno component along this mode on the shock front by requiring it to be written as a linearcombination of the three eigenvectors associated with the other modes. This provides theextra boundary condition at the shock that allows us to determine d . Performing thiscalculation we find that for general ω and γd = − (cid:114) γγ + 1 l (22)in the limit l (cid:29)
1. We plot the general solution of d ( l ) in Fig. 1 for the case discussed inSec. 4, along with the short wavelength limit described by Eqn. 22. It can be seen that theagreement is good.
7. Comparison with 2D Numerical Hydrodynamical Simulations
To compare the analytic solution presented in Secs. 2–6 to numerical results we usedthe PLUTO hydrodynamic code (Mignone et al. 2007) to simulate an explosion on a weaklydiscontinuous surface with σ = 0 .
01 (see Eqn. 12). Again for convenience we consider thecase where the unperturbed solution is analytic: ω = 17 / γ = 5 /
3. 16 – ξ δ P / P l = 1 NumericAnalytic Fig. 3.— Comparison between the numerical (blue) and analytic (green) solutions for thenormalized l = 1 self-similar pressure perturbation as a function of the self-similar variable ξ for a Heaviside initial density distribution (Eqn. 19). Good agreement between the simulationand the analytic solution is found in the region of self-similarity (0 .
76 = ξ c ≤ ξ ≤ cells in the radial direction, and 100 cells in thetangential direction. The inner radius was 10 − and the outer radius was 1. The smallestangle was 0 and the largest π . The radius of the initial hot spot was 2 × − , and thepressure there was 10 , whereas outside the hot spot the pressure was 1. The Riemannsolver used was hllc.The numerical and analytic results are compared in Figs. 2–6. Though the self-similarsolution is valid everywhere, the deeper one looks into the flow, the longer it takes forthe physical flow to approach this solution. Therefore, at any finite time, there exists aninner region that is not in agreement with the self-similar solution. Our comparison tendsto reflect these points and in all cases there is agreement to within 10% throughout asignificant fraction of the flow.In particular, in Fig. 2 we compare the deviation of the shock radius from theunperturbed solution to the shock radius in units of σ in both cases. Analytically thisfunction is independent of σ (see Eqn. 20). Near the interface of the media ( θ = π/
2) thetwo solutions match well, while near the poles ( θ = 0 , π ) the two differ by approximately10%.In the following four figures (Fig. 3, Fig. 4, Fig. 5, and Fig. 6) we compare the fractionaldeviation of the pressure and angular velocity for both l = 1 and l = 2. As expected, as isthe case with the general solution plotted in Fig. 2, discrepancies between the numericaland analytic work are always less than 10%. Descriptions of the specific cases are given inthe captions. 18 – ξ δ P / P l = 2 NumericAnalytic Fig. 4.— Comparison between the numerical (blue) and analytic (green) solutions for thenormalized l = 2 self-similar pressure perturbation as a function of the self-similar variable ξ for a Heaviside initial density distribution (Eqn. 19). Good agreement between the simulationand the analytic solution is found in the region of self-similarity (0 .
76 = ξ c ≤ ξ ≤ − − − − − − ξ δ U θ / U l = 1 NumericAnalytic Fig. 5.— Comparison between the numerical (blue) and analytic (green) solutions for thenormalized l = 1 self-similar fractional angular velocity as a function of the self-similarvariable ξ for a Heaviside initial density distribution (Eqn. 19). Good agreement betweenthe simulation and the analytic solution is found even well outside the region of self-similarity(0 .
76 = ξ c ≤ ξ ≤ − − − − − − ξ δ U θ / U l = 2 NumericAnalytic Fig. 6.— Comparison between the numerical (blue) and analytic (green) solutions for thenormalized l = 2 self-similar fractional angular velocity as a function of the self-similarvariable ξ for a Heaviside initial density distribution (Eqn. 19). Good agreement betweenthe simulation and the analytic solution is found even well outside the region of self-similarity(0 .
76 = ξ c ≤ ξ ≤
8. Discussion
We have considered the problem of a strong shock propagating into a slightly asphericalmedium made up of a density with a spherically symmetric radial power-law plus aperturbation of arbitrary angular dependence, and solved for the Type-II self-similarsolution. Such an external medium has a density profile ρ ( r, θ, φ ) = r − ω [1 + σF ( θ, φ )],where ω > σ (cid:28)
1, and F is an arbitrary function of θ and φ . Because the perturbationsare small, the hydrodynamic equations can be linearized around the unperturbed solution.This then allows us to expand F as a series in spherical harmonics, and solve the problemterm by term. In this way the general problem is reduced to one which includes onlyperturbations F ( θ, φ ) ∝ Y lm ( θ, φ ).The linearized self-similar equations are presented for this simpler case, F ( θ, φ ) ∝ Y lm ( θ, φ ), along with the appropriate boundary conditions. There is a unique solutionto these equations which depends on a single parameter d , which is determined by therequirement that the solution would pass smoothly through the sonic point. That theonly dependence on d is in the boundary conditions makes these equations particularlystraight-forward to solve.We demonstrate this process on a specific example which deviates from sphericalsymmetry by a weak step function in the outside density across a plane containing theinitial explosion. As expected, instead of the shock being spherical, it is composed oftwo hemispheres smoothly connected across the plane of the discontinuity. Our 2Dhydrodynamical simulations agree well with this solution.We thank Yonatan Oren for helpful discussions. This research was partially supportedby ERC and IRG grants and by a Packard fellowship. RS is a Guggenheim fellow anda Radcliffe fellow. AIM acknowledges support from NSF grant AST-1009863 and NASA 22 –grant NNX10AF62G. 23 – REFERENCES
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