Researchers can incorporate uncertainties in computational fluid dynamics (CFD) that go beyond the inaccuracies caused by numerical discretization thanks to stochastic simulations. This study confirms the validity of current stochastic modeling tools by providing examples of stochastic simulations in conjunction with numerical solutions for incompressible flows. A numerical technique for solving deterministic and stochastic models is developed in this work. Our approach employs the Euler-Maruyama method for stochastic modeling, representing a stochastic version of the third-order explicit-implicit scheme. For the deterministic model, the scheme is third-order accurate. The consistency and stability of the constructed scheme are provided in the mean square sense. The scheme is the predictor–corrector type that is built on two time levels. Moreover, a mathematical model of the Casson nanofluid flow with variable thermal conductivity is given with the effect of the chemical reaction. The appropriate transformations are used to condense the set of partial differential equations (PDEs) down to one that is dimensionless. The scheme is applied for the deterministic and stochastic models of dimensionless flow problems. The velocity profile's deterministic and stochastic behavior are shown using contour plots. Results show that growing values of the thermal mixed convection parameter enhance the velocity profile. This article presents the progress made in stochastic computational fluid dynamics (SCFD) and highlights the energy-related aspects of our discoveries. Our computational approach and stochastic modeling techniques provide new insights into the energy properties of Casson nanofluid flow, specifically regarding the variability of thermal conductivity and chemical processes. Our objective is to clarify the complex interaction of these factors on energy dynamics. This article presents a contemporary summary of the latest SCFD advancements. Additionally, it highlights potential directions for future research and unresolved issues that require attention from the members of the field of computational mathematics.

中文翻译：

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Casson纳米流体随机非定常混合对流的两级可靠计算方案

研究人员可以将计算流体动力学 (CFD) 中的不确定性纳入其中，这些不确定性超出了随机模拟造成的数值离散化造成的不准确性。本研究通过提供随机模拟示例以及不可压缩流的数值解，证实了当前随机建模工具的有效性。这项工作开发了一种用于求解确定性和随机模型的数值技术。我们的方法采用 Euler-Maruyama 方法进行随机建模，代表三阶显式-隐式方案的随机版本。对于确定性模型，该方案是三阶精度的。所构造方案的一致性和稳定性在均方意义上被提供。该方案是建立在两个时间水平上的预测-校正器类型。此外，还给出了具有可变热导率的卡森纳米流体流动的数学模型以及化学反应的影响。使用适当的变换将偏微分方程组 (PDE) 压缩为无量纲方程组。该方案适用于无量纲流动问题的确定性和随机模型。使用等高线图显示速度剖面的确定性和随机行为。结果表明，热混合对流参数值的不断增大会增强速度分布。本文介绍了随机计算流体动力学 (SCFD) 方面取得的进展，并重点介绍了我们的发现与能量相关的方面。我们的计算方法和随机建模技术为卡森纳米流体流的能量特性提供了新的见解，特别是关于热导率和化学过程的变化。我们的目标是阐明这些因素对能源动态的复杂相互作用。本文对 SCFD 的最新进展进行了当代总​​结。此外，它还强调了未来研究的潜在方向和需要计算数学领域成员关注的未解决问题。