Abstract A firm understanding and control of viscous fingering (VF) and miscible displacement will be vital to a wide range of industrial, environmental, and pharmaceutical applications, such as geological carbon dioxide sequestration, enhanced oil recovery, and drug delivery. We restrict our study to VF, a well-known hydrodynamic instability, in miscible fluid systems but consider double-diffusive (DD) effects—the combined effect of compositional changes because of solute transport and temperature. One often uses numerical formulations to study VF with DD effects. The primary aim of the current study is to show that popular formulations have limitations to study VF with DD effect. These limitations include exhibiting node-to-node spurious oscillations, violating physical constraints such as the nonnegativity of the concentration field or mathematical principles such as the maximum principle, and suppressing physical instabilities. We will use several popular stabilized finite element formulations—the SUPG formulations and three modifications based on the SOLD approach—in our study. Using representative numerical results, we will illustrate two critical limitations. First, we document that these formulations do not respect the nonnegative constraint and the maximum principle for the concentration field. We will also show the impact of these violations on how viscous fingers develop. Second, we show that these stabilized formulations, often used to suppress numerical instabilities, may also suppress physical instabilities, such as viscous fingering. Our study will be valuable to practitioners who use existing numerical formulations and to computational mathematicians who develop new formulations. Graphic Abstract This figure shows the unphysical concentration values got using the SUPG stabilized formulation on the quarter five-spot problem. The numerical solution for the concentration field violated the nonnegative constraint (left) and the maximum principle (right). The mathematical model comprises coupled flow-thermal-transport equations.

中文翻译：

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模拟双扩散粘性指法中的数值稳定性

摘要 对粘性指法 (VF) 和混相驱替的深入理解和控制对于广泛的工业、环境和制药应用至关重要，例如地质二氧化碳封存、提高石油采收率和药物输送。我们将我们的研究限制在混相流体系统中的 VF，一种众所周知的流体动力学不稳定性，但考虑双扩散 (DD) 效应——由于溶质传输和温度引起的成分变化的综合效应。人们经常使用数值公式来研究具有 DD 效应的 VF。当前研究的主要目的是表明流行的配方在研究具有 DD 效应的 VF 方面存在局限性。这些限制包括表现出节点到节点的虚假振荡，违反物理约束（例如浓度场的非负性）或数学原理（例如最大值原理），并抑制物理不稳定性。在我们的研究中，我们将使用几种流行的稳定有限元公式——SUPG 公式和三个基于 SOLD 方法的修改。使用具有代表性的数值结果，我们将说明两个关键限制。首先，我们证明这些公式不遵守非负约束和浓度场的最大值原则。我们还将展示这些违规行为对手指粘稠发展的影响。其次，我们展示了这些通常用于抑制数值不稳定性的稳定公式，也可以抑制物理不稳定性，例如粘性指法。我们的研究对使用现有数值公式的从业者和开发新公式的计算数学家都很有价值。图形摘要 此图显示了在四分之一五点问题上使用 SUPG 稳定公式获得的非物理浓度值。浓度场的数值解违反了非负约束（左）和最大值原理（右）。数学模型包括耦合的流动-热-传输方程。浓度场的数值解违反了非负约束（左）和最大值原理（右）。数学模型包括耦合的流动-热-传输方程。浓度场的数值解违反了非负约束（左）和最大值原理（右）。数学模型包括耦合的流动-热-传输方程。