Simulation Of Compressible Flows Research Articles

Bridging Large Eddy Simulation and Reduced-Order Modeling of Convection-Dominated Flows through Spatial Filtering: Review and Perspectives

Reduced-order models (ROMs) have achieved a lot of success in reducing the computational cost of traditional numerical methods across many disciplines. In fluid dynamics, ROMs have been successful in providing efficient and relatively accurate solutions for the numerical simulation of laminar flows. For convection-dominated (e.g., turbulent) flows, however, standard ROMs generally yield inaccurate results, usually affected by spurious oscillations. Thus, ROMs are usually equipped with numerical stabilization or closure models in order to account for the effect of the discarded modes. The literature on ROM closures and stabilizations is large and growing fast. In this paper, instead of reviewing all the ROM closures and stabilizations, we took a more modest step and focused on one particular type of ROM closure and stabilization that is inspired by large eddy simulation (LES), a classical strategy in computational fluid dynamics (CFD). These ROMs, which we call LES-ROMs, are extremely easy to implement, very efficient, and accurate. Indeed, LES-ROMs are modular and generally require minimal modifications to standard (“legacy”) ROM formulations. Furthermore, the computational overhead of these modifications is minimal. Finally, carefully tuned LES-ROMs can accurately capture the average physical quantities of interest in challenging convection-dominated flows in science and engineering applications. LES-ROMs are constructed by leveraging spatial filtering, which is the same principle used to build classical LES models. This ensures a modeling consistency between LES-ROMs and the approaches that generated the data used to train them. It also “bridges” two distinct research fields (LES and ROMs) that have been disconnected until now. This paper is a review of LES-ROMs, with a particular focus on the LES concepts and models that enable the construction of LES-inspired ROMs and the bridging of LES and reduced-order modeling. This paper starts with a description of a versatile LES strategy called evolve–filter–relax (EFR) that has been successfully used as a full-order method for both incompressible and compressible convection-dominated flows. We present evidence of this success. We then show how the EFR strategy, and spatial filtering in general, can be leveraged to construct LES-ROMs (e.g., EFR-ROM). Several applications of LES-ROMs to the numerical simulation of incompressible and compressible convection-dominated flows are presented. Finally, we draw conclusions and outline several research directions and open questions in LES-ROM development. While we do not claim this review to be comprehensive, we certainly hope it serves as a brief and friendly introduction to this exciting research area, which we believe has a lot of potential in the practical numerical simulation of convection-dominated flows in science, engineering, and medicine.

An efficient hybrid WENO scheme with minimized dispersion and adaptive dissipation properties for compressible flows

A numerical scheme with good spectral properties is important for the simulation of compressible flows with various of length scales for fine flow scales resolving. The MDAD-HY scheme (Li et al., 2022) using a discontinuity detector and scale sensor achieves the minimized dispersion and adaptive dissipation property. However, the discontinuity detector is devised based on the ratio of the 1st-order and 2nd-order derivatives on two sides of the interface introducing excessive numerical cost. To address this issue, an efficient hybrid WENO scheme with minimized dispersion and adaptive dissipation properties is proposed in this work. Based on the characteristic-decomposition approach, the numerical flux of the present hybrid scheme is achieved by switching between the linear MDAD scheme and the MDAD-WENO scheme according to a new efficient non-dimensional discontinuity detector. The linear flux is reconstructed in a component-wise method to decrease the characteristic-projection operations. To further improve the spectral property of the present scheme, an adaptive parameter controlling the contribution of the optimal linear scheme according to the discontinuity indicator is introduced. Several benchmark test cases involving broadband of length scales and discontinuities are adopted to verify the efficiency and the high-resolution capability of the present scheme.

High-order compact gas-kinetic scheme in arbitrary Lagrangian-Eulerian formulation

This study proposes an extension of the high-order compact gas-kinetic scheme (CGKS) to compressible flow simulation in an arbitrary Lagrangian-Eulerian (ALE) formulation in unstructured mesh. The ALE method is achieved by subdividing arbitrary mesh into tetrahedrons and integrating flux function in a local coordinate system at the cell interface to ensure geometric conservation law. The scheme incorporates a compact reconstruction with third-order accuracy for updating both cell-averaged conservative flow variables and their gradients. HWENO-type nonlinear reconstruction and gradient compression factors are adopted to improve the accuracy and robustness of the scheme. A multi-stage multi-derivative (MSMD) time-stepping method is also implemented to achieve high-order time accuracy with fewer middle stages. The scheme is used to study problems involving moving boundaries. The numerical experiments demonstrate the effectiveness of the scheme in capturing the accurate solutions of both low-speed smooth flow and highly compressible ones with strong shock waves.

Hybrid LES/RANS Simulations of Compressible Flow in a Linear Cascade of Flat Blade Profiles

AbstractThe paper reports on 3D numerical simulations of unsteady compressible airflow in a blade cascade consisting of flat profiles using a hybrid LES/RANS approach including a transition model. As a first step towards simulation of blade flutter in turbomachinery, various incidence angle offsets of the middle blade were modeled. All simulations were run for the flow regime characterized by outlet isentropic Mach number Mis=0.5 and zero incidence. The results of the LES/RANS simulations (pressure and Mach number distributions) were compared to a baseline RANS model, and to experimental data measured in a high-speed wind tunnel. The numerical results show that both methods overpredict flow separation taking place at the leading edge. In this regard, the hybrid LES/RANS method does not provide superior results compared to the traditional RANS simulations. Nevertheless, the LES/RANS results also capture vortex shedding from the blunt trailing edge. The frequency of the trailing edge vortex shedding in CFD simulations matches perfectly the spectral peak recorded during wind tunnel measurements.

CFD approach for determining the losses in two-phase flows through the last stage of condensing steam turbine

Analyzing and simulating wet steam flows within steam turbines holds significant importance, especially due to the presence of a liquid phase in the low-pressure stages of the turbine, resulting in consequential losses that demand careful consideration. Considering these conditions, this study employs advanced predictive CFD models to simulate the wet steam flow within the actual 3D geometry of the final stage in a 200 MW steam turbine, which is observed in the least research due to the complexity of modeling and analysis of three-dimensional flows. This advancement promises more precise and reliable results. Employing the ANSYS CFX software, this study engages in numerical simulations of steady-state compressible two-phase flow. Furthermore, the study offers practical equations to quantify the losses within the entire stage, which serves as a valuable metric for assessing loss levels and offers a benchmark for design improvements aimed at loss reduction. Based on the results, 10 % of losses occur along the stator and 1 % along the rotor, and a total of 11 % of losses occur along the entire stage. Finally, the efficiency of the stage is calculated to be 89 %. Steam condensation strengthens the shock waves, changes their location, diminishes Mach number and the flow velocity, and alters the flow angle. It reduces the efficiency of the turbine by 1.27 % and also reduces the work done by the turbine by 13.97 %. This emphasizes the significant influence of condensation on flow characteristics, emphasizing the necessity of conducting comprehensive investigations into this phenomenon.

High-resolution boundary variation diminishing scheme for two-phase compressible flow with cavitation and evaporation

We propose high-resolution numerical schemes for two-phase compressible flow simulations to reproduce dynamically created gas/vapor-liquid interfaces with phase change. In the Godunov-type finite volume framework, suppressing numerical dissipation errors in numerical schemes is crucial for capturing the discontinuous solutions. The MUSCL scheme has second-order accuracy for smooth solutions and non-oscillatory behavior near discontinuous solutions. However, the MUSCL scheme introduces excessive numerical dissipation and diffuses the discontinuities nonphysically, leading to difficulties in distinguishing gas/vapor and liquid phases, and thus causing a blur in the interfaces during the multi-phase flow simulations.The hybrid-type boundary variation diminishing (BVD) scheme in this paper combines the MUSCL scheme and the THINC scheme to reduce the numerical dissipation errors near the discontinuities. The MUSCL-THINC-BVD scheme applies the MUSCL scheme for smooth solutions and the THINC scheme for discontinuous solutions, resulting in the successful capture of the discontinuities including the dynamically created gas/vapor-liquid interfaces. The Adaptive THINC-BVD scheme, which switches two types of THINC schemes with different values of gradient parameter, also captures the discontinuities clearly. The numerical results of the benchmark tests show that the proposed BVD schemes can lucidly reproduce the vapor-liquid interfaces newly created during the dynamical process of phase change.

Double distribution function-based lattice Boltzmann flux solver for simulation of compressible viscous flows

In this work, a double distribution function-based lattice Boltzmann flux solver (LBFS) is proposed for simulating compressible viscous flows. This approach utilizes the double distribution function compressible lattice Boltzmann model and employs Chapman–Enskog expansion analysis to connect the lattice Boltzmann equation (LBE) with the Navier–Stokes (N–S) equations. Unlike conventional computational fluid dynamics methods that compute inviscid and viscous fluxes separately, the present method simultaneously evaluates both types of fluxes at the cell interface by locally reconstructing the solution of the LBE. Recognizing the significance of considering the non-equilibrium part of distribution functions for viscous flows, a straightforward method is introduced to calculate this component. This facilitates the derivation of computational expressions for macroscopic conservative variables and fluxes in the N–S equations. To validate the accuracy and stability of the present numerical scheme, various benchmark problems, including shock tube problem, Couette flow, lid-driven cavity flow, and flow around the NACA0012 airfoil, are tested. The obtained numerical results are compared with analytical solutions or existing reference data, confirming the capability of the proposed LBFS to deliver accurate and stable numerical results for compressible flows. Moreover, this method demonstrates effectiveness in handling viscous flow problems on non-uniform grids and with curved boundaries.

Method for scalable and performant GPU-accelerated simulation of multiphase compressible flow

Multiphase compressible flows are often characterized by a broad range of space and time scales, entailing large grids and small time steps. Simulations of these flows on CPU-based clusters can thus take several wall-clock days. Offloading the compute kernels to GPUs appears attractive but is memory-bound for many finite-volume and -difference methods, damping speedups. Even when realized, GPU-based kernels lead to more intrusive communication and I/O times owing to lower computation costs. We present a strategy for GPU acceleration of multiphase compressible flow solvers that addresses these challenges and obtains large speedups at scale. We use OpenACC for directive-based offloading of all compute kernels while maintaining low-level control when needed. An established Fortran preprocessor and metaprogramming tool, Fypp, enables otherwise hidden compile-time optimizations. This strategy exposes compile-time optimizations and high memory reuse while retaining readable, maintainable, and compact code. Remote direct memory access realized via CUDA-aware MPI and GPUDirect reduces halo-exchange communication time. We implement this approach in the open-source solver MFC [1]. Metaprogramming results in an 8-times speedup of the most expensive kernels compared to a statically compiled program, reaching 46% of peak FLOPs on modern NVIDIA GPUs and high arithmetic intensity (about 10 FLOPs/byte). In representative simulations, a single NVIDIA A100 GPU is 7-times faster compared to an Intel Xeon Cascade Lake (6248) CPU die, or about 300-times faster compared to a single such CPU core. At the same time, near-ideal (97%) weak scaling is observed for at least 13824 GPUs on OLCF Summit. A strong scaling efficiency of 84% is retained for an 8-times increase in GPU count. Collective I/O, implemented via MPI3, helps ensure the negligible contribution of data transfers (<1% of the wall time for a typical, large simulation). Large many-GPU simulations of compressible (solid-)liquid-gas flows demonstrate the practical utility of this strategy.

A general multi-objective Bayesian optimization framework for the design of hybrid schemes towards adaptive complex flow simulations

Achieving accuracy with underresolved simulation of complex compressible flows with multiscale flow structures is a challenge. Either the numerical dissipation or the resolution and thereby the numerical cost is impractically high. Also, in the design of numerical solvers, the application of a solver for specific flow classes is balanced by robustness allowing the study of a broad range of flows. In this study, we propose a hybrid fifth-order targeted essentially non-oscillatory (TENO5)-based scheme tailored to optimally simulate compressible flows with underresolved dilatational and vortical multiscale structures. For optimal design, three data-driven objectives are defined. A novel objective that derives from the numerical dissipation rate analysis is a key element to deal with underresolved complex flows in practical applications. The optimization process employs a multi-objective Bayesian optimization framework with an expected hypervolume improvement and three flow configurations representative for a broad range of two- and three-dimensional flows with genuine and non-genuine subgrid scales. The optimized hybrid scheme is validated by comparing with shock-capturing schemes of the weighted essentially non-oscillatory (WENO)- and TENO- families with flows of complex shock interactions, Kelvin-Helmholtz instabilities, shock-vortex interactions, vortical and turbulent flows

Simulation of a two-phase loop thermosyphon using a new interface-resolved phase change model

The remarkable progress made in power electronics has been coupled with the miniaturization of systems, requiring high-performance cooling. A solution that has garnered significant interest involves the use of two-phase loops, in which heat transfer is associated with the phase change of the working-fluid. This paper aims at modeling two-phase flow with separated phases in a gravity-driven two-phase loop. The Volume-of-Fluid method is employed to capture the moving interface. Amid the variety of the phase change models available in the literature, we have chosen to employ a hybrid model: a subgrid-scale model to generate phase change at the wall in the absence of an interface, and a interface-resolved model for phase change at pre-existing liquid-vapor interfaces. These models have been implemented in the OpenFOAM computational code. The developed solver makes possible the simulation of compressible laminar two-phase flow with diabatic liquid-vapor phase change. Conjugate heat transfer between the wall and the fluid is also taken into account. Two-dimensional numerical simulations demonstrate that the developed solver is capable of reproducing flow regimes obtained from experiments, including the formation of Taylor bubbles and the propulsion of liquid from the evaporator to the condenser during geyser boiling. Furthermore, this study examines how the filling ratio affects flow regimes and performance.

Direct numerical simulation of compressible turbulent channel flows over porous boundaries

The impact of porous boundaries on skin friction and heat transfer in compressible turbulent channel flows is explored through direct numerical simulations. Utilizing the volume-averaged Navier–Stokes equations for the porous media, we examine flows at Mach numbers Mb=0.3 and Mb=1.5, with friction Reynolds numbers ranging from 176 to 220, across three permeability levels: 0 (impermeable), 1.6×10−5, and 1.0×10−4. Our findings reveal a dual impact of porous boundaries on skin friction: a direct reduction due to averaged slip velocity at the interface and an indirect increase from amplified Reynolds shear stress due to enhanced wall-normal permeability. This permeability weakens the wall-blocking effect, leading to intensified wall-normal velocity fluctuations and consequently higher Reynolds stress. Additionally, the Mach number significantly influences flow structures, particularly at higher permeabilities, where it disrupts near-wall streaks at Mb=0.3 but has a diminished effect at Mb=1.5. Thermal analysis confirms the validity of the generalized Reynolds analogy for the mean temperature in these flows, with the turbulent Prandtl number approximating 1, in line with the strong Reynolds analogy. Moreover, a strong spatial correlation between heat flux and wall shear stress fluctuations underscores the intertwined dynamics of momentum and thermal transport at the porous–fluid interface.

Scale-Resolving Simulation of Shock-Induced Aerobreakup of Water Droplet

Two different scale-resolving simulation (SRS) approaches to turbulence modeling and simulation are used to predict the breakup of a spherical water droplet in air, due to the impact of a traveling plane shock wave. The compressible flow governing equations are solved by means of a finite volume-based numerical method, with the volume-of-fluid technique being employed to track the air–water interface on the dynamically adaptive mesh. The three-dimensional analysis is performed in the shear stripping regime, examining the drift, deformation, and breakup of the droplet for a benchmark flow configuration. The comparison of the present SRS results against reference experimental and numerical data, in terms of both droplet morphology and breakup dynamics, provides evidence that the adopted computational methods have significant practical potential, being able to locally reproduce unsteady small-scale flow structures. These computational models offer viable alternatives to higher-fidelity, more costly methods for engineering simulations of complex two-phase turbulent compressible flows.

Physics-constrained and flow-field-message-informed graph neural network for solving unsteady compressible flows

With the increasing use of deep neural networks as surrogate models for accelerating computational simulations in mechanics, the application of artificial intelligence in computational fluid dynamics has seen renewed interest in recent years. However, the application of deep neural networks for flow simulations has mainly concentrated on relatively simple cases of incompressible flows. The strongly discontinuous structures that appear in compressible flows dominated by convection, such as shock waves, introduce significant challenges when approximating the nonlinear solutions or governing equations. In this work, we propose a novel physics-constrained, flow-field-message-informed (FFMI) graph neural network for spatiotemporal flow simulations of compressible flows involving strong discontinuities. To enhance the nonlinear approximation capability of strong discontinuities, a shock detector method is leveraged to extract the local flow-field messages. These messages are embedded into the graph representation to resolve the discontinuous solutions accurately. A new FFMI sample-and-aggregate-based message-passing layer, which aggregates the edge-weighted attributes with node features on different hop layers, is then developed to diffuse and process the flow-field messages. Furthermore, an end-to-end paradigm is established within the encoder–decoder framework to transform the extracted information from the flow field into latent knowledge about the underlying fluid mechanics. Finally, a variety of one- and two-dimensional cases involving strong shock waves are considered to demonstrate the effectiveness and generalizability of the proposed FFMI graph neural network.

On the streamwise velocity, temperature and passive scalar fields in compressible turbulent channel flows: a viewpoint from multiphysics couplings

It is generally believed that the velocity and passive scalar fields share many similarities and differences in wall-bounded turbulence. In the present study, we conduct a series of direct numerical simulations of compressible channel flows with passive scalars and employ the two-dimensional spectral linear stochastic estimation and the correlation function as diagnostic tools to shed light on these aspects. Particular attention is paid to the relevant multiphysics couplings in the spectral domain, i.e. the velocity–temperature ( $u-T$ ), scalar–temperature ( $g-T$ ) and velocity–scalar ( $u-g$ ) couplings. These couplings are found to be utterly different at a given wall-normal position in the logarithmic and outer regions. Specifically, in the logarithmic region, the $u-T$ and $u-g$ couplings are tight at the scales that correspond to the attached eddies and the very large-scale motions (VLSMs), whereas the $g-T$ coupling is robust in the whole spectral domain. In the outer region, $u-T$ and $u-g$ couplings are only active at the scales corresponding to the VLSMs, whereas the $g-T$ coupling is diminished but still strong at all scales. Further analysis indicates that although the temperature field in the vast majority of zones in a channel can be roughly treated as a passive scalar, its physical properties gradually deviate from those of a pure passive scalar as the wall-normal height increases due to the enhancement of the acoustic mode. Furthermore, the deep involvement of the pressure field in the self-sustaining process of energy-containing motions also drives the streamwise velocity fluctuation away from a passive scalar. The current work is an extension of our previous study (Cheng &amp; Fu, J. Fluid Mech., vol. 964, 2023, A15), and further uncovers the details of the multiphysics couplings in compressible wall turbulence.

FLEW: A DNS Solver for Compressible Flows in Generalized Curvilinear Coordinates

AbstractWe present FLEW, an in-house high-fidelity solver for direct numerical simulation (DNS) of turbulent compressible flows over arbitrary shaped geometries. FLEW solves the Navier–Stokes equations written in a generalized curvilinear coordinate system, in which the surface coordinates are non-orthogonal, whereas the third axis is normal to the surface. Spatial discretization relies on high-order finite-difference schemes. The convective terms are discretized using an hybrid approach, combining the near-zero numerical dissipation provided by central approximations with the robustness of weighted essentially non-oscillatory (WENO) schemes, required to capture shock waves. Central schemes are stabilized using a skew-symmetric-like splitting of convective derivatives, endowing the solver with the energy-preserving property in the inviscid limit. The maximum order of accuracy is eighth for central schemes (also used for viscous terms discretization) and seventh for WENO. The code is oriented to modern high-performance computing (HPC) platforms thanks to message passing interface (MPI) parallelization and the ability to run on graphics processing unit (GPU) architectures. Reliability, accuracy and robustness of the code are assessed in the low-subsonic, transonic and supersonic regimes. We present the results of several benchmarks, namely the inviscid Taylor–Green vortex, turbulent curved channel flow, transonic laminar flow over a NACA 0012 airfoil and turbulent supersonic ramp flow. The results for all configurations proved to be in excellent agreement with previous studies.

An averaged mass correction scheme for the simulation of high subsonic turbulent internal flows using a lattice Boltzmann method

This paper addresses the simulation of internal high-speed turbulent compressible flows using lattice Boltzmann method (LBM) when it is coupled with the immersed boundary method for non-body-fitted meshes. The focus is made here on the mass leakage issue. The recent LBM pressure-based algorithm [Farag et al. Phys. Fluids 32, 066106 (2020)] has shown its superiority on classical density-based algorithm to simulate high-speed compressible flows. Following our previous theoretical work on incompressible flows [Xu et al. Phys. Fluids 34, 065113 (2022)], we propose an averaged mass correction technique to mitigate mass leakage when simulating high-Mach-number compressible flows. It is adapted to deal here with a density, which is decoupled from the zero-moment definition. The simulations focus on two generic but canonical configurations of more complex industrial devices, the straight channel at different angles of inclination at Mach numbers (Ma) ranging from 0.2 to 0.8, and the National Aeronautics and Space Administration Glenn S-duct at Ma = 0.6. The present results show that mass leakage can be a critical issue for the accuracy of the solution and that the proposed correction technique effectively mitigates it and leads to significant improvements in the prediction of the solution.

Multiscale modeling and simulation of turbulent flows in porous media

Numerical simulation is an important tool for understanding the physics of flows in porous media and for making predictions. The state of the art of multiscale modeling and simulation of turbulent flows in porous media is reviewed in this paper. Numerical simulations of flows in porous media can be classified as microscopic simulations, in which both macroscopic and pore-scale flows are directly resolved, and macroscopic simulations, in which the pore-scale motions are modeled while the volume-averaged equations are solved. Studies in the past few years have shown that microscopic simulations improve the understanding of turbulent flows in porous media considerably; this motivates the development of more efficient and more accurate turbulence models for macroscopic simulations. On the basis of this review, we believe that simulation of flows with higher Reynolds numbers, understanding the transport of macroscopic turbulence, modeling of turbulent flows in inhomogeneous and anisotropic porous media, simulation of compressible and multiphase turbulent flows in porous media, and fluid–structure interaction in deformable porous matrices are important topics to be studied in the future.

Restraining vocal fold vertical motion reduces source-filter interaction in a two-mass model.

Previous experimental studies suggested that restraining the vocal fold vertical motion may reduce the coupling strength between the voice source and vocal tract. In this study, the effects of vocal fold vertical motion on source-filter interaction were systematically examined in a two-dimensional two-mass model coupled to a compressible flow simulation. The results showed that when allowed to move vertically, the vocal folds exhibited subharmonic vibration due to entrainment to the first vocal tract acoustic resonance. Restraining the vertical motion suppressed this entrainment. This indicates that the vertical mobility of the vocal folds may play a role in regulating source-filter interaction.

A high-efficiency adaptive TENO scheme with optimal accuracy order for compressible flow simulation

In this paper, we propose a new adaptive cut-off function and develop a fifth-order targeted ENO scheme that achieves optimal accuracy at any order of critical points, which performs excellently in conventional compressible gas dynamics. The cut-off value CT is crucial for controlling the dissipation in TENO schemes. Adjusting CT can improve shock capture and increase dissipation for small-scale features. To address this issue, Fu et al. [27] and Peng et al. [28] developed the TENO-A and TENO-LAD schemes, respectively, with adaptive cut-off values CT. However, these adaptive cutoffs CT are computationally expensive due to additional free parameters and complexity. Furthermore, all TENO schemes, including the adaptive versions, can suffer from loss of accuracy at higher-order critical points. In this paper, a new adaptive cut-off function is proposed that uses a global smoothness indicator correction so that it can be automatically tuned between 0 and a given positive constant λd, ensuring that the new TENO scheme achieves optimal accuracy in smooth regions and good shock-capturing in the less smooth parts. A series of benchmark simulations demonstrate that the presented scheme provides optimal accuracy, better robustness, resolution, and numerical stability, and higher resolution than existing TENO schemes while improving computational efficiency. Due to the simplicity of the new cut-off structure, it can be easily extended to improve other TENO schemes.Program files associated to this article may be found at https://github.com/fourhairs/cfd.

High‐order gas kinetic flux solver for viscous compressible flow simulations

AbstractAlthough the gas kinetic schemes (GKS) have emerged as one of the powerful tools for simulating compressible flows, they exhibit several shortcomings. Since the local solution of continuous Boltzmann equation with the Maxwellian distribution function is used to calculate the numerical fluxes at the cell interface, the flux expression in GKS is usually more complicated. In this paper, a high‐order simplified gas kinetic flux solver (GKFS) is presented for simulating two‐dimensional compressible flows. Circular function‐based GKFS (C‐GKFS), which simplifies the Maxwellian distribution function into the circular function, combined with an improved weighted essentially non‐oscillatory (WENO‐Z) scheme is applied to capture more details of the flow fields with fewer grids. As a result, a simple high‐order accurate C‐GKFS is obtained, which improves the computing efficiency and reduce its complexity to facilitate the practical application of engineering. A series of benchmark‐test problems are simulated and good agreement can be obtained compared with the references, which demonstrate that the high‐order C‐GKFS can achieve the desired accuracy.