Propagation and quenching in a reactive Burgers-Boussinesq system.

Transcription

1 Propagation and quenching in a reactive system., University of New Meico, Los Alamos National Laboratory, University of Chicago October 28

2 Between Math and Astrophysics Motivation The model Reaction Compression Flame bubble emerging on the surface of white dwarf in supernova eplosion, simulation by Cal Jordan and Flash Team,

3 Between Math and Astrophysics Motivation The model Reaction Compression Rayleigh-Taylor instability in Boussinesq approimation. Model is missing compressibility effects!

4 u t + uu = νu + gφ φ t + uφ = κφ + τ R(φ) Motivation The model Reaction Compression A weird model for combustion... Can it be useful for anything else?

5 The Model Burgers equation: u t + uu = νu + gφ φ t + uφ = κφ + τ R(φ) compressibility and shocks; force term gφ, where φ is the reaction progress variable. Motivation The model Reaction Compression Advection-reaction-diffusion equation: front propagation with speed s κ/τ; etinction/quenching: R(φ) = for φ < φth ; well-studied for prescribed velocity. Units: ν = κ =, τ =, s =, l front, l Kanel, l shock /U. Force g is the only parameter in the system.

6 Reaction 3 2 Reaction rate R(φ) φ th =.75 φ th =.5 φ th = Reaction front φ t = φ + R(φ) φ th =.75 φ th =.5 φ th =.25 Motivation The model Reaction Compression R φ.5.25 s = < φ < φ th : R(φ) = φ th < φ < : R(φ) = φ th ( φ th ) ( φ) 2 φ < t : φ = ( φ th ) e > t: φ = φ th e ( t) φ φ ( t) th

7 : φ t =.8.6 φ th = φ t = φ + R(φ) φ t = φ th = Motivation The model Reaction Compression Initial data of small size can be quenched be the diffusion alone. The critical size is called Kanel s length. The shape of the initial data needs to be taken into account. In a prescribed velocity field, φ t + uφ = φ + R(φ), quenching also depends on the size initial data.

8 Effect of stationary compression on quenching φ t + uφ = φ + R(φ) u() = U tanh U 2 Large initial data form two outward propagating fronts, if U < ; speed = U; shrink to a steady profile, if U U cr ; quench, if U > Ucr. Small initial data might shrink to a different steady profile or quench at lower U. u φ φ th = U =. U =.2 U =.3 U =.39 U =. U =.2 U =.3 U =.39 Motivation The model Reaction Compression

9 Coupled system: Weak force u 2 g = /4, φ th =.5 6 t = 2 t = 24 2 t = 36 t = Weak force Moderate force Strong force Structure φ wave-shock structure reaction wave epands in both directions no quenching

10 Coupled system: Moderate force 3 g =, φ th =.5 t = 2 u 2 t = 24 t = 36 t = 48 Weak force Moderate force Strong force Structure φ.5 ramp-wave-shock structure shock and the point of ramp-wave transition move in the direction of force shock moves faster than ramp epands - no quenching

11 Coupled system: Strong force 4 g = 4, φ th =.5 t = 8 u 3 2 t = 6 t = 24 t = 32 Weak force Moderate force Strong force Structure φ.5 ramp-wave-shock structure; the shock and the point of ramp-wave transition move in the direction of force; eventually shock slows down, ramp catches up - quenching.

12 on ramp-wave-shock structure in the ramp region, u() and φ() are linear; in the wave, φ = and u() ; in the shock, u() matches Burger s shock profile; the shock propagates with speed 2 u ma; in the case of the weak force, the wave epands to the left with a constant speed; if present, the ramp originates at the location of initial data; the transition between the ramp and the wave occurs at φ < φ th, which is independent of time; the late-time quenching occurs independently of the size of the initial data; Weak force Moderate force Strong force Structure

13 Notations weak force strong force u f u f u b Notations The ramp The wave The shock b f b f v b v f φ * v b v f f b u f = u( f ) u b = u( b ) v f = ẋ f v b = ẋ b b f b f location of the shock, the front of the wave location of the ramp-wave transition, the back of the wave velocity at the front of the wave velocity at the back of the wave group velocity of f, shock speed group velocity of b

14 The ramp - approimated solution u t + uu = gφ φ t + uφ = In comparison with advection, dissipation effects are negligible on the scale of the ramp: u uu φ uφ u/l 2 u 2 /L /L 2 u/l Notations The ramp The wave The shock In the ramp φ < φ < φ th, therefore R(φ) =. solving the approimated equation, we obtain φ(t, ) = 2 gt 2, 2 u(t, ) = t Assuming that we know φ we can estimate the location of the transition to the ramp b, and corresponding velocities, b = 2 gφ t 2, u b = gφ t, v b = gφ t.

15 The ramp - verification u φ / t g = 4, φ th =.5 t = 8 t = 6 t = 24 t = / gt 2 Notations The ramp The wave The shock To verify obtained scaling, u(t, ) = 2 t, 2 φ(t, ) = numerical solution in the rescaled coordinates. gt 2, we plot

16 The wave assumptions u t + uu = g We drop out the equation for φ, assuming that the reaction is complete inside the wave, φ =. Using the same argument as for the ramp, we can neglect the dissipation term. Encouraged by observed similarity at different times, we look for velocity in the form u(t, ) = ũ ( b (t)) + u b (t), [(u b b)ũ + u b] + ũũ = g Notations The ramp The wave The shock The time dependent epression in square brackets becomes time-independent if b = 2 gφ t 2, u b = gφ t, v b = gφ t, or (accelerated shift observed for moderate and large g), b = ct, u b = c, v b = c, (constant shift observed with c < for small g).

17 The wave approimated solution The remaining equation for ũ can be solved. For moderate and large g: gφ + ũũ = g ũ() = 2g( φ ) u(t, ) = gφ t + 2g( φ ) ( 2 ) gφ t 2 For small g: Notations The ramp The wave The shock ũũ = g ũ() = 2g u(t, ) = c + 2g( ct) In each case, we have one unknown parameter, c and φ. Both parameters c and φ are functions of gravity. We can find c and φ by fitting numerical data.

18 The wave - verification for small g u 2 g = /4, φ th =.5 6 t = 2 t = 24 2 t = 36 t = Notations The ramp The wave The shock φ Black lines show approimated solution for u() in the wave region fitted with c =.75.

19 The wave - verification for large g u g = 4, φ th =.5 t = 8 t = 6 t = 24 t = 32 u b ( b ) Notations The ramp The wave The shock φ.5 φ th φ * Black lines show approimated solution for u() in the wave region for φ =.95, and velocity at the transition to ramp u b ( b ).

20 The shock u t + uu = u + gφ In a fast, narrow shock, the reaction term is negligible compared to the advection and diffusion terms. The solution is the classical Burgers shock of strength u f propagating with speed v f = u f /2. On the scale of the problem, the shock can be considered as a discontinuity located at f and moving with the speed Notations The ramp The wave The shock v f = ẋ f = u f /2. Since we know velocity in the wave, we know u f ( f, t). Thus, for moderate and large g we obtain [ d f dt = ( gφ t + 2g( φ 2 ) f ) ] 2 gφ t 2. Similar equation can be written for small g.

21 The shock - verification 5 numerical simulation u f 4 3 ODE g = 4 g = 2 g = Notations The ramp The wave The shock 2 g = 8 g = /2 g = / t Notice that the decrease of velocity (slowing down of the shock) and abrupt termination of the curves (quenching) are observed only for high g.

22 When does the quenching occurs? Small g: c <, fronts move in opposite directions, no quenching. Moderate g: φ < /9, fronts move in the direction of force, no quenching. Large g: φ > /9, ramp catches up with the shock, quenching, Notations The ramp The wave The shock It can be shown that φ φ th from below as g. Therefore, φ th > /9 is required for quenching. Notice that we never used the functional form of R(φ). The reaction is important only at ramp-wave transition. R(φ) implicitly present in the model in φ and c.

23 Lenya s argument For moderate and large g, we consider the length of the wave region, y = f b, ẏ = ( ) 2g( φ )y gφ t. 2 This equation has no global in time solution if φ > /9. Indeed, if we write y = ρt 2 z, then we have 2z + tż = [ 2( φ )z φ ]. 2 Notations The ramp The wave The shock Changing the time variable τ = ln t (for t ) this becomes dz = dτ 2 q( z), q(s) = 4s 2 + s 2( φ )s φ. In order to have a global solution we need z to be non-negative and q(s) not to be uniformly negative for all s. This is true only as long as 2( φ ) 6φ >, or φ < /9.

24 Dependence on the reaction threshold φ th φ * / 9. c φ th = / g u f < < u b g φ th = / 4 < u f < u b φ th = / < u b < u f / /4 /2 3/4 φ th With the increase of forcing, the system with a particular R(φ) can ehibit the following regimes: (i) two shocks moving in opposite directions, u b < < u f, no quenching; (ii) stationary left shock and right-propagating right shock, u b = < u f, no quenching; (iii) both shocks move to right, first shock faster, < u b < u f, no quenching; (iv) both shocks move to right, second shock faster, < u f < u b, quenching. Notations The ramp The wave The shock

25 Constantin P., Roquejoffre J.-M., Ryzhik L., and Vladimirova N., Propagation and quenching in a reactive system, Nonlinearity, 2, (28).

26 What do we model? Counter-intuitively, compression facilitate quenching. Our intuition is based on thermodynamics: equation of state, pressure, etc... In a model of a rarefied gas of ecitable particles, φ is the fraction of ecited particles. Only ecited particles feel the force... ecitement spreads as R = R(φ)... why? What kind of particles? What kind of force? What kind of reaction? More wild guesses: spread of panic? migration of species?