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 pr opagates at a constant burning rate, self-similar multiple scale fields, wh ether fractal or nonfractal, induce anomalous rates of burning determined b y the space-filling properties of the initial field. We compare the regimes induced by (line-cuts through) three specific geometries with distinct spa ce-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 Cant er dust which has D-k=H>0. The (nondimensional) burning rate Us induced by all three geometries takes the general form U(B)similar toF(tau (-zeta)) wh ere F is a function whose form depends on the specific geometry, zeta is an exponent that contains the space-filling characteristic of the geometry, a nd 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 Canter dust, F-mu( x) =1(x), and zeta =H (for the envelope); F itself is a step-like discontin uous 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 zet a-->1; and as R-->1 (space filling) then zeta-->infinity. Two numerical met hods, a fundamental (Eulerian) solution to the equation of motion and a Lag rangian model for flamelet propagation, confirm these theoretical predictio ns. The Lagrangian model is based on the idealized flamelet as a "point" wi th finite flame thickness Delta (L), (which is determined by the two-flamel et collision process), propagating with a given flame speed U-L. The Lagran gian model allows simulations in parameter ranges not easily accessible by the fundamental method (such as the case for the fractal Canter dust. Inter estingly, the linear regime of scalar diffusion in an algebraic spiral fiel d displays the same dependence on Dk as in the present reaction-diffusion c ase. The nonlinear regime of advection-diffusion (Burger turbulence) shows a different dependence on D-k.