Malik NA, Fung JC
Department of Mechanical Engineering, Imperial College of Science, Technology, and Medicine, Exhibition Road, London SW7 2BX, England and Department of Mathematics, Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong.
Phys Rev E Stat Phys Plasmas Fluids Relat Interdiscip Topics. 2000 Nov;62(5 Pt A):6636-47. doi: 10.1103/physreve.62.6636.
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).
与单个理想化火焰面的经典问题（由非线性反应扩散运动方程描述，以恒定燃烧速率传播）不同，自相似多尺度场，无论其是分形的还是非分形的，都会引发由初始场的空间填充特性决定的异常燃烧速率。我们比较了由具有不同空间填充特性的三种特定几何形状（通过线切割）所引发的状态：（1）一种代数螺旋线，其容量（盒计数维数）D(k)>0，分形维数H = 0；（2）一种指数螺旋线，其D(k)=0且H = 0，几何比R>1；（3）一种分形康托尘，其D(k)=H>0。由这三种几何形状所引发的（无量纲）燃烧速率U(B)具有一般形式U(B)≈F(τ^(-ζ))，其中F是一个函数，其形式取决于特定几何形状，ζ是一个包含几何形状空间填充特性的指数，τ是一个无量纲时间。（1）对于代数螺旋线，F(x)=1(x)，且ζ = D(k)；F是连续的。（2）对于指数螺旋线，F(x)=ln(x)，且ζ = 1/(R - 1)；F是连续的。（3）对于分形康托尘，F(u)(x)=1(x)，且ζ = H（对于包络）；F本身是一个阶梯状不连续函数。因此，当D(k)→0，或当H→0，或当R→∞时，那么ζ→0且U(B)→常数；当D(k)→1，或当H→1（空间填充）时，那么ζ→1；当R→1（空间填充）时，那么ζ→∞。两种数值方法，运动方程的基本（欧拉）解和火焰面传播的拉格朗日模型，证实了这些理论预测。拉格朗日模型基于将理想化火焰面视为具有有限火焰厚度Δ(L)的“点”（由双火焰面碰撞过程确定），以给定的火焰速度U(L)传播。拉格朗日模型允许在基本方法不易达到的参数范围内进行模拟（例如分形康托尘的情况）。有趣的是，代数螺旋线场中标量扩散的线性状态对D(k)的依赖性与当前反应扩散情况相同。平流扩散的非线性状态（伯格斯湍流）对D(k)显示出不同的依赖性。
