Anomalous burning rates of flamelets induced by self-similar multiple scale (fractal and spiral) initial fields

In contrast to the classical problem of a single idealized flamelet (which is described by a nonlinear reaction-diffusion equation of motion) which propagates at a constant burning rate, self-similar multiple scale fields, whether fractal or nonfractal, induce anomalous rates of burning determined by the space-filling properties of the initial field. We compare the regimes induced by (line-cuts through) three specific geometries with distinct space-filling characteristics: (1) an algebraic spiral which has capacity (box-counting dimension) D(k)>0, and fractal dimension H=0; (2) an exponential spiral which has D(k)=0 and H=0, and geometric ratio R>1; (3) a fractal Cantor dust which has D(k)=H>0. The (nondimensional) burning rate U(B) induced by all three geometries takes the general form U(B) approximately F(tau(-zeta)), where F is a function whose form depends on the specific geometry, zeta is an exponent that contains the space-filling characteristic of the geometry, and tau is a nondimensional time. (1) For the algebraic spiral, F(x)=1(x), and zeta=D(k); F is continuous. (2) For the exponential spiral, F(x)=ln(x), and zeta=1/(R-1); F is continuous. (3) For the fractal Cantor dust, F(u)(x)=1(x), and zeta=H (for the envelope); F itself is a step-like discontinuous function. Thus, as D(k)-->0, or as H-->0, or as R-->infinity, then zeta-->0 and U(B)-->const; and as D(k)-->1, or as H-->1, (space filling) then zeta-->1; and as R-->1 (space filling) then zeta-->infinity. Two numerical methods, a fundamental (Eulerian) solution to the equation of motion and a Lagrangian model for flamelet propagation, confirm these theoretical predictions. The Lagrangian model is based on the idealized flamelet as a "point" with finite flame thickness Delta(L), (which is determined by the two-flamelet collision process), propagating with a given flame speed U(L). The Lagrangian model allows simulations in parameter ranges not easily accessible by the fundamental method (such as the case for the fractal Cantor dust). Interestingly, the linear regime of scalar diffusion in an algebraic spiral field displays the same dependence on D(k) as in the present reaction-diffusion case. The nonlinear regime of advection-diffusion (Burger turbulence) shows a different dependence on D(k).