Computers & Fluids最新文献_第4页

Time integration of cut-cell surfaces for geometric conservation, applied to scalar transport with moving interfaces 用于几何守恒的切割单元曲面的时间积分，应用于带移动界面的标量输运

IF 2.5 3区工程技术

Computers & Fluids Pub Date : 2025-04-19 DOI: 10.1016/j.compfluid.2025.106627

D. Dupuy , A. Toutant , A. du Cluzeau , G. Bois

{"title":"Time integration of cut-cell surfaces for geometric conservation, applied to scalar transport with moving interfaces","authors":"D. Dupuy , A. Toutant , A. du Cluzeau , G. Bois","doi":"10.1016/j.compfluid.2025.106627","DOIUrl":"10.1016/j.compfluid.2025.106627","url":null,"abstract":"<div><div>In cut-cell methods, interface-resolved simulations of two-phase flows are performed by cutting a fixed nonconforming mesh at the interface boundary. The cells which are cut to conform to the interface use modified discretisation schemes that account for the modified cell volume and face areas of cut cells, which evolve dynamically with the motion of the interface. This article investigates the effect of the method used for time integration of cut-face areas in a cut-cell method, for the convection–diffusion of a passive scalar in a two-phase flow with moving interfaces. The cut-cell method, based on a finite-volume approach and a three-dimensional staggered Cartesian grid, naturally enforces strict conservation laws and ensures numerical stability in small cells using a flux-redistribution strategy. The simulation of heat diffusion in and around a spherical interface under a uniform velocity field is addressed. A semi-implicit time-integration method taking into account initial and final cut-face areas provides significant improvements at a negligible cost compared to an explicit time-integration method.</div></div>","PeriodicalId":287,"journal":{"name":"Computers & Fluids","volume":"295 ","pages":"Article 106627"},"PeriodicalIF":2.5,"publicationDate":"2025-04-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143864898","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Validation of a low-dissipation Finite-Volume solver for implicit Large-Eddy Simulation 隐式大涡模拟低耗散有限体积求解器的验证

IF 2.5 3区工程技术

Computers & Fluids Pub Date : 2025-04-12 DOI: 10.1016/j.compfluid.2025.106623

Giove De Cosmo , Luca di Mare , Mauro Carnevale

{"title":"Validation of a low-dissipation Finite-Volume solver for implicit Large-Eddy Simulation","authors":"Giove De Cosmo , Luca di Mare , Mauro Carnevale","doi":"10.1016/j.compfluid.2025.106623","DOIUrl":"10.1016/j.compfluid.2025.106623","url":null,"abstract":"<div><div>In the field of turbulent flow modelling, implicit Large-Eddy Simulation (iLES) is appealing for its low cost and ease of implementation. Such advantages rely on the absence of a sub-grid scale model, since the dissipation of the numerical scheme is assumed to match the behaviour of unresolved turbulence. The implementation of an iLES model in traditional Unsteady-RANS codes for Computational Fluid Dynamics is not a straightforward exercise, as most of the classical schemes used for the discretisation of the Navier–Stokes equations prove too dissipative. This work presents a low-dissipation fix for the traditional Flux-Difference Splitting scheme of Roe in the context of Finite-Volume discretisations. The fix consists in selectively scaling the eigenvalues of the Roe matrix to lower the numerical dissipation as needed, by means of a scalar parameter. The low-dissipation version of the Roe scheme is implemented in an existing Finite-Volume compressible wall-resolved URANS code, to obtain an iLES model. The solver is first verified on a fundamental test case, i.e. vortex transport in uniform flow. The scalar parameter is then properly calibrated on the decay of Homogeneous Isotropic Turbulence, to ensure physical meaningfulness. A robust validation of the iLES model is finally presented on realistic turbulent flows. Results show that a relatively simple fix can achieve excellent agreement with the benchmark DNS data on a flat-wall channel flow and a bumped-wall channel flow.</div></div>","PeriodicalId":287,"journal":{"name":"Computers & Fluids","volume":"295 ","pages":"Article 106623"},"PeriodicalIF":2.5,"publicationDate":"2025-04-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143844706","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Implementation of the DADI algorithm with implicit source term treatment into an unsteady Eulerian droplet solver for in-flight icing simulation 带隐式源项处理的DADI算法在非定常欧拉液滴解算器中的实现

IF 2.5 3区工程技术

Computers & Fluids Pub Date : 2025-04-12 DOI: 10.1016/j.compfluid.2025.106628

Younghyo Kim , Yoonpyo Hong , Soonho Shon , Soo Hyung Park , Kwanjung Yee

{"title":"Implementation of the DADI algorithm with implicit source term treatment into an unsteady Eulerian droplet solver for in-flight icing simulation","authors":"Younghyo Kim , Yoonpyo Hong , Soonho Shon , Soo Hyung Park , Kwanjung Yee","doi":"10.1016/j.compfluid.2025.106628","DOIUrl":"10.1016/j.compfluid.2025.106628","url":null,"abstract":"<div><div>Recent aircraft designs incorporate multiple rotating elements, such as rotors and propellers, that are highly susceptible to icing-induced performance issues and vibrations. This vulnerability highlights the critical need for thorough icing impact assessments, given the significant flow unsteadiness caused by their moving bodies. Therefore, considering this unsteadiness is critical in precise icing analysis. A comprehensive analysis must cover both the unsteady airflow and droplet behavior, which demands extensive computational resources. To address these challenges, this paper introduces a robust and efficient unsteady droplet solver. It utilizes the Diagonalized Alternating Direction Implicit (DADI) algorithm, renowned for its computational efficiency on the structured grids. The implementation of the DADI algorithm unfolded in two stages. Initially, a relaxation method was applied to enhance hyperbolicity for the diagonalization of the Jacobian matrix. Subsequently, a factor for source term was incorporated into the implicit operator for the treatment of stiff source terms. Verification and validation processes were conducted for one-dimensional Riemann problems, water impingement experiments on the NACA 23012 airfoil and NACA 64A008 swept horizontal tail wing, and icing experiments on an oscillating SC2110 airfoil. The computational results demonstrate high accuracy and efficiency in both steady-state and unsteady computations when compared to experimental results and previous numerical analyses. Notably, the solver employing the DADI method with source term treatment achieved solutions 16.4% faster than the TVD 2nd order Runge–Kutta method in steady-state calculations. Moreover, for unsteady computations, it obtained solutions 8.4% more quickly than the DADI method without the implicit treatment of the source term.</div></div>","PeriodicalId":287,"journal":{"name":"Computers & Fluids","volume":"294 ","pages":"Article 106628"},"PeriodicalIF":2.5,"publicationDate":"2025-04-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143829444","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Shape optimization for fluid flow with parametric level set method and deep neural networks 基于参数水平集和深度神经网络的流体流动形状优化

IF 2.5 3区工程技术

Computers & Fluids Pub Date : 2025-04-11 DOI: 10.1016/j.compfluid.2025.106626

Wrik Mallik , Rajeev K. Jaiman , Jasmin Jelovica

{"title":"Shape optimization for fluid flow with parametric level set method and deep neural networks","authors":"Wrik Mallik , Rajeev K. Jaiman , Jasmin Jelovica","doi":"10.1016/j.compfluid.2025.106626","DOIUrl":"10.1016/j.compfluid.2025.106626","url":null,"abstract":"<div><div>This study presents a novel application of the parametric level-set (PLS) method to develop a shape optimization process by directly modifying flow dynamics. The present method employs linear superimposition of polynomial perturbations to traditional PLS as shape optimization parameters. This enables smooth shape changes without any change in topology and limits the design variables to only the number of polynomials required for arbitrary hydrofoil morphing. During optimization, deep convolutional neural networks are integrated with the point clouds of the uniform level set to provide a surrogate model for flow dynamics. The present shape optimization method is employed here to delay stall via mitigation of flow separation on the suction surface of the NACA66 hydrofoil at high angles of attack. Shape optimization mitigates the forward movement of trailing edge flow reversal via changes in hydrofoil thickness and camber forward of the maximum hydrofoil thickness point. The optimized design shows more than two order reductions in mean flow reversal compared to NACA66 under the design condition angle of attack of <span><math><mrow><mn>11</mn><mo>.</mo><msup><mrow><mn>5</mn></mrow><mrow><mo>∘</mo></mrow></msup></mrow></math></span>. At 14°, NACA66 shows complete flow separation while the optimized design exhibits almost three orders lower mean reversal magnitude of top surface flow than that of NACA66, indicating significantly delayed flow separation characteristics. The surrogate-based optimization is performed at four orders of magnitude lower computation time than full-order flow solvers. The results demonstrate the potential of the proposed PLS and deep neural network methodology to perform fast data-driven (non-intrusive) shape optimization of fluid flow.</div></div>","PeriodicalId":287,"journal":{"name":"Computers & Fluids","volume":"295 ","pages":"Article 106626"},"PeriodicalIF":2.5,"publicationDate":"2025-04-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143834974","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

An Euler–Lagrange coupling strategy for pressurized liquid jet atomization in Large Eddy Simulation 大涡模拟中高压液体射流雾化的欧拉-拉格朗日耦合策略

IF 2.5 3区工程技术

Computers & Fluids Pub Date : 2025-04-11 DOI: 10.1016/j.compfluid.2025.106625

Thibault Gioud , Thomas Laroche , Thomas Schmitt , Bénédicte Cuenot , Odier Nicolas

{"title":"An Euler–Lagrange coupling strategy for pressurized liquid jet atomization in Large Eddy Simulation","authors":"Thibault Gioud , Thomas Laroche , Thomas Schmitt , Bénédicte Cuenot , Odier Nicolas","doi":"10.1016/j.compfluid.2025.106625","DOIUrl":"10.1016/j.compfluid.2025.106625","url":null,"abstract":"<div><div>The atomization process is crucial for numerous industrial and engineering applications, ranging from fuel injection systems to spray coating technologies and remains a challenge for numerical prediction. This study presents a coupled Euler/Lagrange methodology to model the atomization processes, at affordable numerical costs. An Eulerian framework is used to resolve the liquid core and the largest liquid structures, while a Lagrangian approach tracks the smallest droplets and their interactions, addressing transport, secondary breakup, and evaporation phenomena. The Eulerian two-phase flow method considered here is a multi-fluid diffuse interface approach that assumes equilibrium of temperature, pressure, velocity, and Gibbs potentials at the liquid–gas interface. An originality of this paper is the transfer of liquid from the under resolved Eulerian liquid structures to the Lagrangian formalism when user-defined geometrical criteria are met. A validation of the coupling strategy is performed on a jet in cross flow configuration. Although the spatial distribution of the droplet volumetric flux is not exactly retrieved, other key characteristics of the spray, such as the Sauter Mean Diameter (SMD) and the outer trajectory, show very encouraging results. Finally, this paper shows the importance of taking into account the liquid core in the simulation, rather than directly injecting already atomized droplets.</div></div>","PeriodicalId":287,"journal":{"name":"Computers & Fluids","volume":"294 ","pages":"Article 106625"},"PeriodicalIF":2.5,"publicationDate":"2025-04-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143835227","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Application of the quantum Fourier transform in a harmonic balance solver for Burgers’ equation 量子傅立叶变换在Burgers方程谐波平衡解算器中的应用

IF 2.5 3区工程技术

Computers & Fluids Pub Date : 2025-04-09 DOI: 10.1016/j.compfluid.2025.106619

Loïc Dewitte , Jérémie Roland , Frank Eulitz

{"title":"Application of the quantum Fourier transform in a harmonic balance solver for Burgers’ equation","authors":"Loïc Dewitte , Jérémie Roland , Frank Eulitz","doi":"10.1016/j.compfluid.2025.106619","DOIUrl":"10.1016/j.compfluid.2025.106619","url":null,"abstract":"<div><div>This paper explores the potential of quantum computing for computational fluid dynamics (CFD) applications, with a focus on turbomachinery CFD. A harmonic balance solver is developed for Burgers’ equation, in which discrete Fourier transforms are approximated using a hybrid quantum-classical algorithm based on the quantum Fourier transform (QFT). Three novel algorithms are presented to estimate Fourier coefficients, providing complete knowledge of their amplitudes and phases, which is not possible with the standard QFT. Their behaviour is studied theoretically and numerically, under noiseless and noisy conditions. The algorithms, whose performance is limited by the extensive sampling required to achieve a small error on the Fourier coefficients, tolerate low levels of depolarising noise. The behaviour of the hybrid solver is investigated for a baseline case with a Reynolds number of 1000, 7 harmonics and 100 grid cells, using the best performing algorithm in a noiseless setting with up to 10<sup>8</sup> samples per QFT. Residuals decrease until errors introduced by statistical uncertainty dominate the total error on the solution. Nevertheless, the hybrid solutions match the classical one closely, with RMS residuals as low as 10<sup>–4</sup>. The impact of several solver parameters on convergence and solution quality is also assessed, including the effect of noise. Although the latter rapidly degrades the solution, the solver achieves satisfactory approximations to the classical solution with 0.001% of depolarising noise. Without seeking to demonstrate a quantum advantage, this work offers valuable insights into the opportunities and challenges of quantum computing, helping readers understand how to design, implement and study quantum algorithms, as well as evaluate their impact in CFD applications.</div></div>","PeriodicalId":287,"journal":{"name":"Computers & Fluids","volume":"295 ","pages":"Article 106619"},"PeriodicalIF":2.5,"publicationDate":"2025-04-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143860235","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Locally energy-stable finite element schemes for incompressible flow problems: Design and analysis for equal-order interpolations 不可压缩流动问题的局部能量稳定有限元方案：等阶插值的设计与分析

IF 2.5 3区工程技术

Computers & Fluids Pub Date : 2025-04-09 DOI: 10.1016/j.compfluid.2025.106622

Hennes Hajduk , Dmitri Kuzmin , Gert Lube , Philipp Öffner

引用次数: 0

Divergence cleaning for weakly compressible smoothed particle hydrodynamics 弱可压缩光滑粒子流体力学的散度清理

IF 2.5 3区工程技术

Computers & Fluids Pub Date : 2025-04-08 DOI: 10.1016/j.compfluid.2025.106638

G. Fourtakas , R. Vacondio , B.D. Rogers

{"title":"Divergence cleaning for weakly compressible smoothed particle hydrodynamics","authors":"G. Fourtakas , R. Vacondio , B.D. Rogers","doi":"10.1016/j.compfluid.2025.106638","DOIUrl":"10.1016/j.compfluid.2025.106638","url":null,"abstract":"<div><div>This paper presents a divergence cleaning formulation for the velocity in the weakly compressible smoothed particle hydrodynamics (SPH) scheme. The proposed hyperbolic/parabolic divergence cleaning, ensures that the velocity divergence, <em>div</em>(<strong>u</strong>), is minimised throughout the simulation. The divergence equation is coupled with the momentum conservation equation through a scalar field <em>ψ</em>. A parabolic term is added to the time-evolving divergence equation, resulting in a hyperbolic/parabolic form, dissipating acoustic waves with a speed of sound proportional to the local Mach number in order to maximise dissipation of the velocity divergence, preventing unwanted diffusion of the pressure field. The <em>div</em>(<strong>u</strong>)-SPH algorithm is implemented in the open-source weakly compressible SPH solver DualSPHysics. The new formulation is validated against a range of challenging 2-D test cases including the Taylor-Green vortices, patch impact test, jet impinging on a surface, and wave impact in a sloshing tank. The results show that the new formulation reduces the divergence in the velocity field by at least one order of magnitude which prevents spurious numerical noise and the formation of unphysical voids. The temporal evolution of the impact pressures shows that the <em>div</em>(<strong>u</strong>)-SPH formulation virtually eliminates unwanted acoustic pressure oscillations. Investigation of particle resolution confirms that the new <em>div</em>(<strong>u</strong>)-SPH formulation does not reduce the spatial convergence rate.</div></div>","PeriodicalId":287,"journal":{"name":"Computers & Fluids","volume":"295 ","pages":"Article 106638"},"PeriodicalIF":2.5,"publicationDate":"2025-04-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143870853","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Large Eddy simulation of a supersonic starting impinging jet 超音速起动撞击射流的大涡模拟

IF 2.5 3区工程技术

Computers & Fluids Pub Date : 2025-04-05 DOI: 10.1016/j.compfluid.2025.106620

Juan Sheng, Feng He, Yitao Wang, Pengfei Hao, Xiwen Zhang, Xiangru Li

{"title":"Large Eddy simulation of a supersonic starting impinging jet","authors":"Juan Sheng, Feng He, Yitao Wang, Pengfei Hao, Xiwen Zhang, Xiangru Li","doi":"10.1016/j.compfluid.2025.106620","DOIUrl":"10.1016/j.compfluid.2025.106620","url":null,"abstract":"<div><div>Supersonic impinging jets have wide applications in many directions, such as rocket exhaust, supersonic combustor mixing, and accidental leakage of pressurized fluids. The steady processes of impinging jets have been extensively studied, but there is little research on the starting impinging jets. Considering the important applications of starting jet in rocket launches, car's airbags, and pulse jets, the under-expanded round jet impinging on a flat plate normally with a nozzle pressure ratio of 2.7 has been investigated using large eddy simulations. The distance between the impinging plate and the nozzle outlet is set to 5 times the nozzle diameter. The simulation results agree well with the experimental data. The variations of flow structures and dominant frequencies of the acoustic resonance at the starting process are presented. The radiation directions of the dominant frequencies exhibit notable variations at different stages of the starting process, and the explanation for this distinction is explored by the two-dimensional correlation analysis. The wavenumber spectra and dispersion relations are employed, showing that at different stages, the upstream-propagating guided jet modes are excited by the interaction between the Kelvin–Helmholtz wavepacket and shock cells of distinct wavenumbers. Finally, based on the instantaneous pressure distribution, the dynamic evolution characteristics of the flow structures at different stages are analyzed.</div></div>","PeriodicalId":287,"journal":{"name":"Computers & Fluids","volume":"294 ","pages":"Article 106620"},"PeriodicalIF":2.5,"publicationDate":"2025-04-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143817814","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Modeling two-phase equilibrium using smoothed particle hydrodynamics 用光滑粒子流体力学建模两相平衡

IF 2.5 3区工程技术

Computers & Fluids Pub Date : 2025-04-05 DOI: 10.1016/j.compfluid.2025.106630

Mohammadreza Bagheri

{"title":"Modeling two-phase equilibrium using smoothed particle hydrodynamics","authors":"Mohammadreza Bagheri","doi":"10.1016/j.compfluid.2025.106630","DOIUrl":"10.1016/j.compfluid.2025.106630","url":null,"abstract":"<div><div>Smoothed Particle Hydrodynamics (SPH) is an emerging particle-based methodology that can also be used for modeling two-phase flows, currently in the early stages of development. This Lagrangian, mesh-free approach utilizes macro-scale formulations at the meso-scale, achieving computational performance comparable to micro-scale methods. This integration allows for efficient computations at larger scales than micro while facilitating detailed analysis at smaller scales than macro. This paper focuses on the study of two-phase equilibrium and droplet formation, employing the Equation of State (EOS) alongside the careful selection of an appropriate smoothing length. The majority of existing SPH literature utilizes the van der Waals (vdW) EOS for two-phase simulations. While the vdW EOS has provided foundational insights, newer models have been developed to accommodate a broader range of fluids. In this study, the Peng-Robinson EOS is employed, which separates the EOS into attractive and repulsive components, thereby enhancing modeling capabilities. This work critically examines the limitations of SPH in simulating two-phase equilibrium, deriving the smoothing length for attractive forces based on surface tension. Furthermore, it contends that employing an updated smoothing length does not accurately reflect physical realities. To the best of the author's knowledge, this research is among the few that directly integrates the Peng-Robinson Equation of State (PR EOS) and a viscosity equation of state within the SPH framework for the simulation of two-phase equilibrium.</div></div>","PeriodicalId":287,"journal":{"name":"Computers & Fluids","volume":"294 ","pages":"Article 106630"},"PeriodicalIF":2.5,"publicationDate":"2025-04-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143824335","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0