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1、arXiv:physics/0702211v3 physics.class-ph 23 Jul 2007 ScalingofHuygens-frontspeedupinweaklyrandommedia Jackson R. Mayo, Alan R. Kerstein Combustion Research Facility, Sandia National Laboratories, Livermore, CA 94551, USA Abstract Front propagation described by Huygens principle is a fundamental mech
2、anism of spatial spreading of a property or an eff ect, occurring in optics, acoustics, ecology and combustion. If the local front speed varies randomly due to inhomogeneity or motion of the medium (as in turbulent premixed combustion), then the front wrinkles and its overall passage rate (turbulent
3、 burning velocity) increases. The calculation of this speedup is subtle because it involves the minimum-time propagation trajectory. Here we show mathematically that for a medium with weak isotropic random fl uctuations, under mild conditions on its spatial structure, the speedup scales with the 4/3
4、 power of the fl uctuation amplitude. This result, which verifi es a previous conjecture while clarifying its scope, is obtained by reducing the propagation problem to the inviscid Burgers equation with white-in-time forcing. Consequently, fi eld-theoretic analyses of the Burgers equation have signi
5、fi cant implications for fronts in random media, even beyond the weak- fl uctuation limit. Key words: Front propagation, Random media, Geometrical optics, Turbulent combustion, Burgers equation PACS: 02.50.Ey, 42.15.Dp, 47.70.Fw 1. Introduction Phenomena from combustion 1 to seismic waves 2 to popul
6、ation spreading 3 can be modeled by Huygens prin- ciple of front propagation, fi rst stated as a law of geometri- cal optics. In each application, the boundary of the aff ected region (idealized as a sharp front) advances normal to itself at a locally specifi ed speed. In a uniform medium, where thi
7、s speed is constant, an initially wrinkled front fl attens out over time; but in a spatially varying medium, a com- petition occurs as wrinkling is continually reintroduced 4. A central problem for the latter case is to determine the overall statistically steady propagation rate of the wrinkled fron
8、t, which exceeds the average local speed because the front is defi ned by the fastest paths through the medium (fi rst passage). Our main result, establishing under general conditions the proportionality of the speedup to the 4/3 power of the amplitude of weak fl uctuations, agrees with previousheur
9、isticanalysis4andnumericalsimulations4 7, and eliminates ambiguities 8 concerning the physical relevance of the scaling. In the important case of premixed fl uid combustion, the result describes the weak-advection scaling of the turbulent burning velocity 9. Corresponding author. Email addresses: jm
10、ayosandia.gov (Jackson R. Mayo), arkerstsandia.gov (Alan R. Kerstein). Our analysis relates the front dynamics to the evolu- tion of a pressure-free fl uid obeying the white-noise-driven Burgers equation, itself a widely studied model of turbu- lence 10. The results reported here enable adaptation o
11、f existing treatments of Burgers turbulence 11,12 to esti- mate the prefactor of the speedup scaling and its depen- dence on medium structure, with implications even for the opposite, practical limit of strongly advected fl ames. The propagation of a Huygens front in a general nonuni- form medium is
12、 governed by the local advecting velocity of the medium asafunction of time and space,u(t,x), and the local speed of propagation relative to the medium, v(t,x). Huygens principle can then be stated as follows (using three-dimensional language for defi niteness): If at time t0 a point x0 lies in the
13、aff ected region (including its bound- ary, the front), then at a slightly later time t1= t0+ dt all points in a ball of radius v(t0,x0)dt about the point x0+u(t0,x0 )dt are aff ected. The boundary of the aff ected region at t1(the new front) thus consists of certain points on the surfaces of balls
14、originating from the initial front. If we consider all points on these spherical surfaces, then we automatically include those on the new front and more. Hence the front at any later time will be found among the aff ected points reached by arbitrary trajectories x(t) that start on the initial front
15、and always move at the local speed v (in any direction) relative to the medium, so that they Preprint submitted to Phys. Lett. ASAND2007-1087J obey ? ? ? ? dx dt u(t,x) ? ? ? ? = v(t,x).(1) Two simplifi cations of this general framework are phys- ically important. First, for geometrical optics in a
16、static or “quenched” medium with nonuniform refractive index, or for combustion of a solid propellant with nonuniform burning rate, we have v = v(x) (time-independent fl uctu- ations) and u 0 (no advection). Note that our nonrel- ativistic equations do not correctly describe advection of light anywa
17、y, but are adequate for an optical medium “at rest” (where fl uctuations of v dominate those of u) and for advection of sound in geometrical acoustics. Second, for idealized combustion of a premixed turbulent fl uid in the limit of a very thin fl ame front 1, v is a constant (the lami- nar fl ame sp
18、eed) and u(t,x) is the turbulent fl ow (assumed to be unaff ected by the fl ame). Following the derivation of our key results in Sections 2 and 3, further implications for combustion are discussed in Section 4. 2. Fronts and particles In the general case, Eq. (1) defi nes a large family of “vir- tua
19、l” trajectories x(t) of which only a subset actually form the front at a given time. The criterion for the relevant trajectories is simplest if |u| v everywhere, as we now assume (in Section 4 we discuss relaxing this assumption). This inequality, which is trivially satisfi ed for a quenched medium,
20、 ensures that advection can never sweep the front backward and thus that each point y is crossed by the front only once. The time of this crossing, which we call T0(y), is simply the time when the fi rst virtual trajectory reaches y, hence defi ning a fi rst-passage problem. To reduce the number of
21、extraneous trajectories, we can exploit the fi rst-passage criterion (minimization of travel time) and obtain constraints on relevant trajectories. Let us parametrize the solutions of Eq. (1) by dx dt = u + vn,(2) with n(t) a unit vector. Then (see Appendix A) a necessary condition for a fi rst-pass
22、age trajectory is dn dt = Pn(A n + v),(3) where Pndenotes the projection orthogonal to n and A is the velocity gradient tensor Aij= iuj; a further neces- sary condition is that the trajectory starts out with n nor- mal to the initial front, from which it followsthat n remains normal to the evolving
23、front. In a quenched medium, where u and A are zero, Eqs. (2) and (3) reduce to the ray equa- tions of geometrical optics, and there it is well known that rayspropagatenormalto fronts.But with advection,we see that atrajectorystangent vectordx/dt is no longeraligned with the front normal n. As discu
24、ssed in Section 4, Eqs. (2) and (3) govern front-tracking trajectories even when A B C D i Fig. 1. Irrelevance of particles after collision. Two particles, starting from A and B on the initial front i, collide at C; the continuation of trajectory AC (dashed) reaches D. If it were the fi rst arrival,
25、 its travel time would be an absolute minimum over all virtual trajectories from i to D, including BCD. Having the same minimum time, BCD must also obey the law of motion, but this deterministic law does not permit the time-reversed trajectory DC to split at C. Hence neither ACD nor BCD represents f
26、i rst passage to D, and the particles can be discarded upon colliding. |u| v, but we assume |u| v so that the front evolu- tion is described by a simple fi rst-passage problem and a single-valued function T0(y). For fi rst-passage purposes, then, we consider a contin- uum of “particles” starting sim
27、ultaneously from all points on the initial front (with initial n givenby the unit normal), and obeying Eqs. (2) and (3). If the Huygens front repre- sents a wave phenomenon in the geometrical-optics limit of very short wavelength, then we are describing physical quasiparticles: photons for light or
28、phonons for sound. Mo- tivated by this physical case, we call Eqs. (2) and (3) the “law of motion” for fi rst-passage trajectories. In premixed combustion, where the front represents a thin fl ame, the “particle” trajectories are mathematical constructs known as ignition curves 13. Even with the con
29、dition (3), not all of these particles re- main on the front. The departure of particles from the front (into the interior of the aff ected region) is associated with “cusps” at which the normal n is not unique. Such cusps develop during propagation in a random medium even if the initial front is sm
30、ooth 14,15. When two distinct parti- cles reachthe same point at the sametime (necessarilywith diff erent n), both colliding particles fall behind the front and can then be discarded, as shown in Fig. 1. (In optics and acoustics, the corresponding photons and phonons re- main observable as second an
31、d later arrivals, but our scope is limited to fi rst passage.) The collision rule implies a correspondence between our continuum of particles and a model “fl uid” without pres- sure or viscosity, consisting of fl uid elements that move in- dependently in response to external forces but disappear whe
32、n they collide. Such a fl uid is described by the inviscid Burgers equation (with suitable forcing) and the collisions are known as shocks 13,16. Because our particles fi ll a sur- 2 Fig. 2. Complementary descriptions of fi rst passage. A front propa- gates upward amid possibly time-dependent variat
33、ions in u (arrows) and v (grayscale, dark for slow). We show u and v at the local time T0. Snapshots of the front (thick curves) are contours of T0; parti- cles (thin curves) track the front until they collide at cusps. Two circles expanding from aff ected points illustrate Huygens principle. The fl
34、 uctuations are weak enough to defi ne a “fl uid” using xkas “time” for the particles. For very weak fl uctuations, the front re- mains nearly fl at, and ubecomes irrelevant as it merely shifts the front without advancing it. face and not all of space, we must defi ne the corresponding Burgers fl ui
35、d more precisely. The particles, by defi nition, always represent the fi rst arrival at their locations; thus if two of their paths reach the same point x, the particles necessarily arrive at the same time T0(x) and are then dis- carded. Let us take the initial front to be planar and use coordinates
36、 x = (xk,x) such that this plane is xk= 0. Then the spatial paths of the particles can be viewed as “trajectories” x(xk) in the “time” xk. In this picture, a Burgers fl uid exists in the lower- dimensional space x, and the physical time information is carried by the function T0(xk,x). Here we must i
37、ntroduce the assumption that the medium fl uctuations are weak, i.e., that |u| and the changes in v are both vanishingly small compared to the average value of v. In this limit, because trajectories cannot deviate signifi cantly from the xk-direction without falling behind the front, the parti- cles
38、 necessarily collide before they are defl ected enough to make the function x(xk ) ill defi ned. The law of motion for these trajectories x(xk) could be derived from Eqs. (2) and (3), but it is simpler to ob- tain the Burgersequationin its conventionalEulerianform. The same minimization principle th
39、at yields Eqs. (2) and (3) also implies that T0 (x) satisfi es a generalized “eikonal” equation (see Appendix B) u?T0(x),x? T0(x) + v?T0(x),x?|T0(x)| = 1.(4) The relation between T0and the particle trajectories is il- lustrated in Fig. 2 for a two-dimensional medium. Because T0is constant over a fro
40、nt, T0lies along the front nor- mal n (in the direction of propagation), and so the physical velocity of particles in Eq. (2) can be written dx dt = u + v T0 |T0|. (5) Wenextshowthat inthelimit ofweakrandomfl uctuations, Eq. (4) reduces to the forced inviscid Burgers equation for a fl uid consisting
41、 of these particles. 3. Weak-fl uctuation limit In a reference frame where the average value of u is zero, and in units such that the average value of v is unity, let us parametrize the weak fl uctuations by u(t,x) = U(t,xk,x),(6) v(t,x) = 1 + V (t,xk,x),(7) where U and V are homogeneous isotropic r
42、andom fi elds and is taken asymptotically to zero. The time dependence of u and v is scaled by because the natural source of time dependence is advection by u itself, which goes to zero with . Heuristic scaling analysis 4,6 yields the following conclusions in the 0 limit: The medium can be con- side
43、red eff ectively frozen (U and V time-independent); the advection component Uorthogonal to the overall prop- agation direction is irrelevant; the front reaches a statisti- cally steady state over a distance of order 2/3; and the steady passage rate exceeds unity by an amount of order 4/3. Guided by
44、these expectations, we defi ne rescaled quan- tities = pxk,(8) (,x) = pT0(xk,x) xk,(9) where we anticipate that p = 2 3 produces a useful 0 limit, but we also consider slightly diff erent p values to see why 2 3 is special. The point of the rescaling is that if reaches a steady-state average value C
45、 after a fi nite- transient, then kT0averages to 1 C2pafter a charac- teristic distance xk p. A planar front propagating at constant speed vwould have kT0= 1/v, and so to lead- ing order the passage rate in the random medium is v= 1 + C2p.(10) Our main result is a demonstration that for p = 2 3, the
46、 rescaledfront is governedby a forced Burgersequation that reaches such a steady state, thus implying 4/3dependence of the speedup. Substituting Eqs. (6)(9) into the eikonal equation (4), we fi nd Uk(1p + 1+p,p,x)(1 + 2p) + U(1p + 1+p,p,x) p + 1 + V (1p + 1+p,p,x) q (1 + 2p)2+ 2p|2= 1.(11) 3 Fig. 3.
47、 Convergence of weak fl uctuations to white noise. Choosing a simple random medium with h(0)(z)i exp(z2), we show the two-point correlation function of , Eq. (13), for = 1 (a) and = 0.2 (b). As 0, the function is infi nitely compressed in the “time” direction , but with the choice of scaling exponen
48、t p = 2 3, its time integral at a given z (shaded area) remains fi xed. This integral gives the z -dependent coeffi cient of a delta function that describes spatially correlated white-in-time noise. For 1 2 p 0 smooths the shocks.) The corresponding variant of Eq. (12), with 2 on the left, is a form
49、 of the Kardar-Parisi-Zhang (KPZ) equation for interface growth 21. Under forcing that is continuous in space and time, the solution of these equations is known to approach as 0 a limiting “viscosity solution” that reproduces the inviscid shocks 22. Under white-noise forc- ing, the steady state of the viscous Burgers equation was analyzed using fi eld theory 10,11, but the 0 limit is more subtle. Fortunately, a recent theorem 23 establishes that the viscous white-noise steady state converges to the inviscid one, making the 0 results of fi eld theory ap-