Computers & Fluids最新文献_第4页

High-order gas-kinetic scheme with two-derivative-based time discretization for compressible flows 可压缩流动的高阶气体动力学双导数时间离散格式

IF 3 3区工程技术 Q3 COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS

Computers & Fluids

Pub Date : 2025-12-18 DOI: 10.1016/j.compfluid.2025.106950

Qiushuo Qin , Jie Wu , Lan Jiang

In this paper, an efficient high-order gas-kinetic scheme for compressible flow simulation is presented. It combines the fifth-order targeted essentially non-oscillatory (TENO) reconstruction with a circular-function-based gas kinetic flux solver (C-GKFS), which enhances the resolution of flow field details while maintaining stability and high accuracy. To further improve the efficiency of time advancement, this work introduces two-derivative, multi-step or multi-stage time discretization schemes with strong stability preserving time coefficients. These schemes are combined with the Lax-Wendroff spatio-temporal coupling strategy, which effectively reduce the number of function evaluations, lowers the computational complexity, and improves the overall robustness. The numerical results of several typical examples verify the advantages of the proposed method in terms of resolution, stability and efficiency, especially in the simulations of the shock-bubble interaction and the inviscid 3D Taylor-Green vortex, which show good potential for applications in complex flows.

本文提出了一种高效的高阶气体动力学可压缩流动模拟格式。它将五阶定向非振荡（TENO）重建与基于圆函数的气体动力学通量求解器（C-GKFS）相结合，提高了流场细节的分辨率，同时保持了稳定性和高精度。为了进一步提高时间推进的效率，本文引入了具有强保稳时间系数的二阶、多步或多阶段时间离散化方案。这些方案与Lax-Wendroff时空耦合策略相结合，有效减少了函数求值次数，降低了计算复杂度，提高了整体鲁棒性。几个典型算例的数值结果验证了该方法在分辨率、稳定性和效率方面的优势，特别是在激波-气泡相互作用和无粘三维泰勒-格林涡的模拟中，显示出良好的应用潜力。

引用次数: 0

Non-orthogonal plane-marching parabolized stability equations for the secondary instability of crossflow vortices 横流涡旋二次不稳定性的非正交平面推进抛物化稳定性方程

IF 3 3区工程技术 Q3 COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS

Computers & Fluids

Pub Date : 2025-12-17 DOI: 10.1016/j.compfluid.2025.106947

Koen J. Groot , Jordi Casacuberta , Stefan Hickel

A detailed derivation, analysis, and verification is given for the non-orthogonal, plane-marching Parabolized Stability Equations (PSE) approach. In applying the approach to a flow distorted by a medium-amplitude crossflow vortex, we determine its linear secondary instability mechanisms. We show that converged solutions can be achieved for a broad frequency range with an existing stabilization method for the line-marching PSE approach. We verify that 1) solutions converge versus grid size in all dimensions, 2) primary disturbance solutions agree with line-marching PSE results, and 3) secondary disturbance solutions match amplitude and growth-rate evolution of reference Direct Numerical Simulation (DNS) results. We show how and why the type-II instability displays a delayed neutral point when modeled with the plane-marching approach versus the considered local stability approaches, whether the streamwise evolution of the distorted base flow is accounted for or not. This may explain why the type-II disturbance is scarcely captured by DNS in the literature.

对非正交的平面行进抛物稳定方程（PSE）方法进行了详细的推导、分析和验证。将该方法应用于中幅横流涡畸变的流动，确定了其线性二次不稳定机制。我们证明了用现有的线推进PSE方法的稳定方法可以在很宽的频率范围内获得收敛解。我们验证了1)解决方案在所有维度上与网格大小收敛，2)主要干扰解决方案与线推进PSE结果一致，3)次要干扰解决方案与参考直接数值模拟（DNS）结果的振幅和增长率演变相匹配。我们展示了ii型不稳定性如何以及为什么在使用平面行进方法与考虑的局部稳定方法建模时显示延迟中性点，无论是否考虑了扭曲基流的流向演变。这也许可以解释为什么在文献中，DNS几乎没有捕捉到ii型干扰。

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引用次数: 0

Aerodynamic shape optimization with discretization error control for high-order compressible flow simulations 高阶可压缩流动模拟的离散误差控制气动形状优化

IF 3 3区工程技术 Q3 COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS

Computers & Fluids

Pub Date : 2025-12-16 DOI: 10.1016/j.compfluid.2025.106949

Kush Pandya, Aravind Balan

The shape of an aerospace vehicle significantly influences its performance, and Aerodynamic Shape Optimization (ASO) refines the shape to achieve objectives such as drag reduction and lift improvement. ASO typically combines Computational Fluid Dynamics (CFD) simulations with gradient-based optimization. Adjoint methods can compute gradients independently of the number of shape parameters, making it computationally efficient when optimizing for a large number of shape parameters. For high reliability on the results of ASO, we need accurate evaluation of the objective functions and their gradients. High-order numerical methods, such as the Discontinuous Galerkin (DG) method, provide superior accuracy for compressible flow simulations at a computational cost comparable to traditional methods, making them ideal for ASO. The present study highlights the impact of discretization errors on optimal geometry, emphasizing the need for controlling them in an efficient way during optimization. A novel strategy is proposed that efficiently integrates adjoint-based mesh optimization within the ASO framework that uses a high-order DG method for the flow simulations. The approach ensures that discretization errors in the objective function remain bounded throughout the ASO process by dynamically adapting the meshes based on adjoint-based error estimates. The mesh adaptation is formulated in such a way that one gets optimum meshes with minimal number of elements for a prescribed error tolerance. The strategy results in a multi-fidelity approach where there is an effective utilization of computation cost by progressively refining the mesh as ASO progresses. The effectiveness of the proposed methodology is demonstrated through a comparative analysis with conventional ASO performed on a fixed fine mesh, using a benchmark test case from the AIAA Aerodynamic Design Optimization Discussion Group (ADODG). The results highlight significant improvements in the optimization process’s reliability and computational efficiency.

航空航天飞行器的外形对其性能有很大的影响，气动外形优化（ASO）对外形进行优化，以达到减少阻力和提高升力等目标。ASO通常将计算流体动力学（CFD）模拟与基于梯度的优化相结合。伴随方法可以独立于形状参数的数量计算梯度，使其在优化大量形状参数时计算效率高。为了保证ASO结果的高可靠性，我们需要对目标函数及其梯度进行准确的评估。高阶数值方法，如不连续Galerkin （DG）方法，为可压缩流动模拟提供了更高的精度，而计算成本与传统方法相当，使其成为ASO的理想选择。本研究强调了离散误差对最优几何的影响，强调了在优化过程中有效控制离散误差的必要性。提出了一种新的策略，将基于自伴随的网格优化有效地集成到ASO框架中，使用高阶DG方法进行流动模拟。该方法通过基于伴随误差估计的网格动态自适应，确保目标函数的离散化误差在整个ASO过程中保持有界。网格自适应是以这样一种方式制定的，即在规定的误差容忍度下，使用最少数量的元素获得最佳网格。该策略是一种多保真度方法，通过随着ASO的进行逐步细化网格，有效地利用了计算成本。采用AIAA气动设计优化讨论组（ADODG）的基准测试用例，与在固定细网格上执行的传统ASO进行了对比分析，证明了该方法的有效性。结果表明，优化过程的可靠性和计算效率有了显著提高。

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引用次数: 0

Predicting flow-induced vibration in isolated and tandem cylinders using hypergraph neural networks 用超图神经网络预测孤立和串联式气缸的流激振动

IF 3 3区工程技术 Q3 COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS

Computers & Fluids

Pub Date : 2025-12-15 DOI: 10.1016/j.compfluid.2025.106930

Shayan Heydari, Rui Gao, Rajeev K. Jaiman

We present a finite element-inspired hypergraph neural network framework for predicting flow-induced vibrations in freely oscillating cylinders. The surrogate architecture transforms unstructured computational meshes into node-element hypergraphs that encode higher-order spatial relationships through element-based connectivity, preserving the geometric and topological structure of the underlying finite-element discretization. The temporal evolution of the fluid-structure interaction is modeled via a modular partitioned architecture: a complex-valued, proper orthogonal decomposition-based sub-network predicts mesh deformation using a low-rank representation of Arbitrary Lagrangian-Eulerian (ALE) grid displacements, while a hypergraph-based message-passing network predicts the unsteady flow field using geometry-aware node, element, and hybrid edge features. High-fidelity ALE-based simulations provide training and evaluation data across a range of Reynolds numbers and reduced velocities for isolated and tandem cylinder configurations. The framework demonstrates stable rollouts and accurately captures the nonlinear variation of oscillation amplitudes with respect to reduced velocity, a key challenge in surrogate modeling of flow-induced vibrations. In the tandem configuration, the model successfully resolves complex wake-body interactions and multi-scale coupling effects, enabling prediction of pressure and velocity fields under strong wake interference conditions. Our results show high fidelity in reproducing force statistics, dominant frequencies, and flow-field dynamics, supporting the framework’s potential as a robust surrogate model for digital twin applications.

我们提出了一个有限元启发的超图神经网络框架，用于预测自由振荡圆柱体中流动引起的振动。代理架构将非结构化计算网格转换为节点-元素超图，通过基于元素的连通性编码高阶空间关系，保留底层有限元离散化的几何和拓扑结构。流固耦合作用的时间演化通过模块化分区架构建模：基于复杂值、适当正交分解的子网络使用任意拉格朗日-欧拉（ALE）网格位移的低秩表示来预测网格变形，而基于超图的消息传递网络使用几何感知节点、元素和混合边缘特征来预测非定常流场。基于ale的高保真仿真为隔离和串联气缸配置提供了一系列雷诺数和降低速度的训练和评估数据。该框架展示了稳定的滚动，并准确捕获了振动幅度与降低速度相关的非线性变化，这是流激振动替代建模的关键挑战。在串联配置下，该模型成功地解决了复杂的尾迹-体相互作用和多尺度耦合效应，能够预测强尾迹干扰条件下的压力场和速度场。我们的研究结果显示，在再现力统计、主导频率和流场动力学方面具有高保真度，支持该框架作为数字孪生应用的健壮替代模型的潜力。

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引用次数: 0

A Markov matrix iterative splitting algorithm for incompressible flow 不可压缩流的马尔可夫矩阵迭代分裂算法

IF 3 3区工程技术 Q3 COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS

Computers & Fluids

Pub Date : 2025-12-15 DOI: 10.1016/j.compfluid.2025.106943

Manuel A. Ramirez-Cabrera , Eduardo Ramos , Manira E. Narvaez-Saucedo , Patricio J. Valades-Pelayo

We present a Monte Carlo method for the incompressible Navier-Stokes equations, enforcing divergence-free solutions through a probabilistic projection framework. Using short random walks embedded in Markov matrices, the method sequentially solves diffusion, convection, and pressure projection steps at each timestep. The method achieves near-linear CPU scaling (O(N^1.18)) for transient simulations through pre-computed transition probability matrices for linear operators, with Multi-Level Monte Carlo acceleration improving steady-state convergence to (O(N^1.58)). Validation on lid-driven cavity flows (Re=100, 1000) shows differences below 3 % versus benchmarks. Additionally, the mesh-free nature of the Monte Carlo approach handles complex geometries simply by tagging random walkers within non-conforming obstacles, bypassing traditional meshing requirements. The method combines accuracy, unconditional stability, and inherent parallelizability, offering a compelling alternative to deterministic approaches.

我们提出了不可压缩Navier-Stokes方程的蒙特卡罗方法，通过概率投影框架强制无散度解。使用嵌入在马尔可夫矩阵中的短随机漫步，该方法在每个时间步上依次解决扩散，对流和压力投影步骤。该方法通过预先计算线性算子的转移概率矩阵，实现暂态模拟的近线性CPU缩放（O(N1.18)），多级蒙特卡罗加速将稳态收敛提高到（O(N1.58)）。对盖子驱动的空腔流动（Re= 100,1000）的验证表明，与基准相比，差异低于3%。此外，蒙特卡罗方法的无网格特性仅通过在不符合障碍物内标记随机步行者来处理复杂的几何形状，绕过传统的网格划分要求。该方法结合了准确性、无条件稳定性和固有的并行性，为确定性方法提供了令人信服的替代方案。

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引用次数: 0

RANS-CNN: A physics-informed convolutional neural network for solving reynolds-averaged Navier-Stokes equations in duct flows ranss - cnn：一个物理信息卷积神经网络，用于解决管道流动中的reynolds-average Navier-Stokes方程

IF 3 3区工程技术 Q3 COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS

Computers & Fluids

Pub Date : 2025-12-15 DOI: 10.1016/j.compfluid.2025.106946

Gaurav Bokil, Sebastian Merbold, Stefanie De Graaf

Classical Computational Fluid Dynamics (CFD) simulations of turbulent flows in aerospace applications are computationally demanding and limit rapid design exploration. Convolutional Neural Networks (CNN) are being employed as surrogate models to overcome this challenge. Physics-informed approaches have also been applied to CNNs albeit only for simple flow fields such as laminar flow and heat conduction. This study advances Physics-Informed Convolutional Neural Networks (PICNN) to solve the steady incompressible Reynolds-Averaged Navier-Stokes (RANS) equations in wall-bounded geometries. The proposed method employs a higher-order finite difference scheme for computing spatial gradients, thus enhancing numerical accuracy. Additionally, the Dirichlet boundary conditions are strongly enforced in the network architecture using custom output layers and boundary masks. Numerical stabilisation is incorporated to enable the CNN to simulate high Reynolds number flows without losing stability. To assess the capabilities of this approach on aerospace use cases, it is tested on three data-free cases: S-shaped duct, a ducted body force heat exchanger, and flow over a forward facing step along with a backward facing step geometry with sparse labelled data. Moreover, a comparison between zero-equation and one-equation turbulence models is presented when employed in this framework. The RANS-CNN models performed with over 95 % accuracy on geometries with attached flow and 80 % on separated flow cases. The results obtained from the case studies confirm the capability of the RANS-CNN method in developing a robust and computationally efficient surrogate model with sparse data for smooth ducts.

经典计算流体动力学（CFD）在航空航天应用中的湍流模拟计算要求很高，限制了快速设计探索。卷积神经网络（CNN）被用作替代模型来克服这一挑战。物理信息的方法也被应用于cnn，尽管只是简单的流场，如层流和热传导。本研究推进了基于物理信息的卷积神经网络（PICNN）在有壁几何中求解稳定不可压缩的reynolds - average Navier-Stokes （RANS）方程。该方法采用高阶有限差分格式计算空间梯度，提高了数值精度。此外，Dirichlet边界条件在使用自定义输出层和边界掩码的网络架构中被强制执行。数值稳定纳入使CNN能够模拟高雷诺数流动而不失去稳定性。为了评估这种方法在航空航天用例中的能力，我们在三种无数据的情况下对其进行了测试：s形管道，导管式体力热交换器，以及通过具有稀疏标记数据的前向台阶和后向台阶几何形状的流动。此外，还比较了零方程和单方程湍流模型在此框架下的应用。ranss - cnn模型在具有附加流的几何形状上的准确率超过95%，在分离流情况下的准确率超过80%。从案例研究中获得的结果证实了ranss - cnn方法在为光滑管道开发具有稀疏数据的鲁棒且计算效率高的代理模型方面的能力。

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引用次数: 0

Stabilization of a hypersonic boundary layer via the oxidized surface of carbon/silicon carbide composite 通过碳/碳化硅复合材料氧化表面的高超声速边界层稳定化

IF 3 3区工程技术 Q3 COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS

Computers & Fluids

Pub Date : 2025-12-15 DOI: 10.1016/j.compfluid.2025.106948

Liangzhong Fan , Qingyong Zhu , Zhihui Li , Dongheng Lai

The evolution of second-mode waves in hypersonic boundary layers may be affected by the surface morphology of thermal protection systems. In this study, passive oxidation experiment is conducted on carbon/silicon carbide (C/SiC) thermal protection materials, yielding oxide-layer thicknesses of approximately 0.6–0.9 mm. Furthermore, the relationship between oxide layer thickness and composite material composition is analyzed. A strategy is proposed to mitigate second mode waves by tailoring the compositional distribution of C/SiC composites to form a wavy oxidized surface. Numerical simulations are performed to investigate the effect of the wavy surfaces with wave heights of 0.6 mm and 1.8 mm on the evolution of near-wall disturbances at Mach 6. When the wavy surface extends downstream of the synchronization point, it significantly suppresses second mode wave amplitude. However, when positioned only upstream of the synchronization point, a wave height of 0.6 mm instead amplifies disturbance waves. Increasing the wave height to the average boundary layer thickness (1.8 mm) shifts the peak disturbance location upstream but does not reduce its amplitude. Additionally, an idealized oxidized surface with a linearly graded composite composition is examined. Compared to an arc-shaped wavy surface, the idealized variable-curvature wavy surface provides more sustained suppression of the disturbance wave amplitude. Further analysis suggests that the modulation of disturbance waves by pressure oscillations induced by the wavy surface may be a key mechanism for reducing the second mode amplitude.

高超声速边界层中第二模波的演化可能受到热防护系统表面形貌的影响。本研究对碳/碳化硅（C/SiC）热防护材料进行被动氧化实验，得到氧化层厚度约0.6-0.9 mm。进一步分析了氧化层厚度与复合材料成分的关系。提出了一种通过调整C/SiC复合材料的成分分布来形成波纹状氧化表面的策略。通过数值模拟研究了波高分别为0.6 mm和1.8 mm的波浪面对6马赫数下近壁扰动演化的影响。当波面向同步点下游延伸时，会显著抑制第二模态波幅值。然而，当仅位于同步点的上游时，0.6 mm的波高反而放大了扰动波。当波高增加到边界层平均厚度（1.8 mm）时，扰动峰值位置会向上游移动，但不会减小其振幅。此外，一个理想化的氧化表面与线性梯度复合成分进行了检查。与圆弧型波形面相比，理想变曲率波形面对扰动振幅的抑制更持久。进一步分析表明，波浪表面引起的压力振荡对扰动波的调制可能是降低第二模态振幅的关键机制。

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引用次数: 0

Artificial viscosity based on the velocity gradient components in a 3D Lagrangian Lax-Wendroff scheme 三维拉格朗日Lax-Wendroff格式中基于速度梯度分量的人工粘度

IF 3 3区工程技术 Q3 COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS

Computers & Fluids

Pub Date : 2025-12-13 DOI: 10.1016/j.compfluid.2025.106944

Matěj Klíma, Milan Kuchařík, Richard Liska

In this paper, a new type of artificial viscosity, based on the divergence of velocity gradient, is developed for the Lagrangian Lax-Wendroff cell-centered schemes. Similar approaches led previously to the development of tensor viscosity in methods with staggered grids. The velocity gradient can be split into the compressible, symmetric, and rotational components that each have a different effect on shock waves or rotational flows. We demonstrate the effects of using each component or their combinations on a set of typical hydrodynamics testing problems.

The new artificial viscosity leads to far greater robustness and symmetry compared to the original HLL-type one for the Saltzman test. For treating radial shocks (Noh), the best results are achieved by combination of symmetric and rotational terms, while for the tests involving rotational flow (Taylor-Green vortex) or instability formation (Richtmyer-Meshkov), the compressible part of the velocity gradient is more suitable, as it does not suppress the vorticity of the flow.

本文针对拉格朗日Lax-Wendroff胞心格式，提出了一种基于速度梯度散度的新型人工黏度。类似的方法导致了先前在交错网格方法中的张量粘度的发展。速度梯度可以分为可压缩、对称和旋转三部分，每一部分对激波或旋转流有不同的影响。我们演示了使用每个组件或它们的组合对一组典型的水动力学测试问题的影响。与萨尔茨曼测试中原始的hhl型相比，新的人工粘度具有更大的鲁棒性和对称性。对于处理径向激波（Noh），将对称项和旋转项结合使用可获得最佳结果，而对于涉及旋转流动（Taylor-Green vortex）或不稳定形成（richmyer - meshkov）的测试，速度梯度的可压缩部分更合适，因为它不会抑制流动的涡度。

引用次数: 0

On the spurious modes associated with the pressure centred low Mach number fix for compressible flows 可压缩流中与压力中心低马赫数固定相关的伪模态

IF 3 3区工程技术 Q3 COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS

Computers & Fluids

Pub Date : 2025-12-13 DOI: 10.1016/j.compfluid.2025.106945

Jonathan Jung , Ibtissem Lannabi , Vincent Perrier

Density based finite volume schemes usually used for the computation of compressible flows are known to be, in general, not accurate in the low Mach number limit. A very popular low Mach number fix that has been proposed in different forms consists in centring the gradient of the pressure. In this article, we first perform some numerical experiments for proving that this low Mach number fix may lead to non-convergence on the momentum, especially if unstructured meshes are used. We also show that on the long time limit of the wave system, the same kind of problem may occur.

On triangular meshes, we perform a full analysis of the spurious modes of the wave system, and make evidence that the numerical scheme exhibits some oscillatory spurious modes. On simple triangular mesh configurations, these spurious modes are explicitly built. Then on general triangular meshes, we explain how to compute a basis of the spurious modes by relying on the ones built on simple configurations. A major outcome of this article is that the dimension of the spurious modes space is very large, approximately equal to the number of nodes of the mesh.

通常用于可压缩流动计算的基于密度的有限体积格式在低马赫数极限下通常是不准确的。以不同形式提出的一种非常流行的低马赫数固定方法是将压力梯度集中。在本文中，我们首先进行了一些数值实验来证明这种低马赫数固定可能导致动量不收敛，特别是如果使用非结构化网格。我们还表明，在波系的长时间限制下，也可能出现同样的问题。在三角网格上，我们对波系统的杂散模态进行了全面的分析，并证明了数值格式表现出一些振荡杂散模态。在简单的三角形网格结构上，明确地建立了这些伪模态。然后，在一般三角网格上，我们解释了如何通过依赖于建立在简单配置上的模型来计算伪模式的基础。本文的一个主要成果是伪模空间的维数非常大，大约等于网格的节点数。

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引用次数: 0

An improved immersed boundary method for investigating flows over multiple irregular geometries with fractal interpolation 基于分形插值的多不规则几何体流动研究的改进浸入边界法

IF 3 3区工程技术 Q3 COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS

Computers & Fluids

Pub Date : 2025-12-08 DOI: 10.1016/j.compfluid.2025.106942

Dongheng Lai , Xingyu Zhu

In this study, we propose an improved immersed boundary method with dual-layer local triangulation. A novel code was developed for high-order numerical simulations of supersonic flows over multiple complex irregular geometries. A fifth-order weighted essentially non-oscillatory scheme was implemented to capture any steep gradients in the flow created by the geometries. The simulations were carried out on Cartesian grids and the Delaunay triangulation method was implemented twice near the boundary to refine the object boundary discretization and improve the numerical simulation robustness for complex irregular geometries. The proposed method could successfully evaluate various two- and three-dimensional compressible flows with immersed boundaries. Moreover, we studied the flow mechanism over irregularly shaped debris generated by multiple disintegrations during spacecraft re-entry in near-space, with a particular focus on spherical debris objects. We also propose a self-affine fractal interpolation surface method for spherical surfaces to effectively characterize the near-space debris. The improved immersed boundary method with the dual-layer local triangulation was used to simulate the supersonic flow over multiple side-by-side fractal spherical objects. Numerical examples conclusively verified the effectiveness, generality, and robustness of the proposed method.

本文提出了一种改进的双层局部三角剖分浸入边界法。开发了一种新颖的程序，用于多种复杂不规则几何形状的超音速流动的高阶数值模拟。采用了一种五阶加权本质上无振荡的方案来捕捉由几何形状产生的气流中的陡峭梯度。仿真在直角网格上进行，在边界附近采用两次Delaunay三角剖分方法，以细化目标边界离散化，提高复杂不规则几何的数值模拟鲁棒性。该方法可以成功地计算各种浸入边界的二维和三维可压缩流。此外，我们研究了航天器在近空间再入过程中多次解体产生的不规则形状碎片的流动机制，特别关注球形碎片物体。我们还提出了一种球面自仿射分形插值曲面方法来有效地表征近空间碎片。采用改进的浸入边界法和双层局部三角剖分法，模拟了多个并列分形球面物体的超音速流动。数值算例验证了该方法的有效性、通用性和鲁棒性。

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引用次数: 0