The influence of dispersion or equivalently of the Péclet number (Pe) on miscible viscous fingering in a homogeneous porous medium is examined. The linear optimal perturbations maximizing finite-time energy gain is demonstrated with the help of the propagator matrix approach based non-modal analysis (NMA). We show that onset of instability is a monotonically decreasing function of Pe and the onset time determined by NMA emulates the non-linear simulations. Our investigations suggest that perturbations will grow algebraically at early times, contrary to the well-known exponential growth determined from the quasi-steady eigenvalues. One of the over-arching objective of the present work is to determine whether there are alternative mechanisms which can describe the mathematical understanding of the spectrum of the time-dependent stability matrix. Good agreement between the NMA and non-linear simulations is observed. It is shown that within the framework of $$L^2$$ L 2 -norm, the non-normal stability matrix can be symmetrizable by a similarity transformation and thereby we show that the non-normality of the linearized operator is norm dependent. A framework is thus presented to analyze the exchange of stability which can be determined from the eigenmodes.

中文翻译：

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直线混合粘性指法中的瞬时增长和对称性

检查了分散或等效的 Péclet 数 (Pe) 对均质多孔介质中可混溶粘性指法的影响。借助基于传播器矩阵方法的非模态分析 (NMA)，证明了最大化有限时间能量增益的线性最优扰动。我们表明不稳定的开始是 Pe 的单调递减函数，并且由 NMA 确定的开始时间模拟非线性模拟。我们的研究表明，扰动会在早期以代数方式增长，这与由准稳态特征值确定的众所周知的指数增长相反。目前工作的首要目标之一是确定是否有替代机制可以描述对时间相关稳定性矩阵频谱的数学理解。观察到 NMA 和非线性模拟之间的良好一致性。结果表明，在$$L^2$$L 2 -范数的框架内，非正态稳定性矩阵可以通过相似变换对称化，从而表明线性化算子的非正态性是范数相关的。因此提出了一个框架来分析可以从本征模式确定的稳定性交换。