Navier-Stokes Research Articles

Simulation of deformation and flow of pulmonary acinus using morphing compared with self-similar deformation

AbstractIn previous studies of the flow inside the pulmonary acinus, the deformation of acinus was limited because a relatively simple deformation methods, such as self-similar deformation, is used. However, acinus deformation may be complex and heterogeneous. The purpose of this study is to evaluate the influence of the difference in wall deformation between the morphed model and the self-similar deformation model on flow field. The heterogeneous deformation achieved in this study is more actual by using the morphing technology that continuously changes from one shape to another. STAR-CCM + Ver. 13.06 (Siemens PLM Software) was used for numerical calculation, and the governing equations of working fluid are continuous equations and Navier–Stokes equations (NS equations). As a result, the differences in deformation influenced the formation of streamlines at the distal end of the alveolar regions. In the self-similar deformation model, the flow does not change throughout a cycle, whereas in the morphed model, the flow changes during a cycle. Graphic abstract

Can isothermal plane Couette flow in fluid overlying porous layer be linearly unstable?

The present study aims to examine the temporal linear stability analysis of isothermal plane Couette flow over a porous layer using the two-domain approach. The flow in the porous layer is described by the unsteady Darcy–Brinkman equations, whereas it is characterised by the Navier–Stokes equations in the fluid layer. In contrast to the Darcy model, it is observed that the isothermal plane Couette flow becomes unstable for such a superposed system on the inclusion of the Brinkman term. From the stability analysis, the two-dimensional mode is found to be least stable, and two modes of instability, namely porous mode and mixed mode are obtained under the consideration of the Darcy–Brinkman model along with advection term (DBA model). For Darcy number $(\delta )=0.01$ , depending on the value of the stress-jump coefficient, mixed mode controls the instability of the system at small values of depth ratio $(\hat {d})$ , and it disappears for relatively high values of $\hat {d}$ , where the porous mode dominates. In addition, it has been observed that when $\hat {d}=0.1$ , the critical mode of instability is found to be mixed for $\delta >0.02$ and porous for $\delta \le 0.02$ . The stress-jump coefficient destabilises the flow in terms of energy production through perturbed stresses at the interface. As observed in the case of isothermal plane Poiseuille flow studied by Chang, Chen & Straughan (J. Fluid Mech., vol. 564, 2006, pp. 287–303), here also depth ratio (Darcy number) stabilises (destabilises) the flow. However, this characteristic does not remain valid when the advection term is eliminated from the considered momentum equation. For a certain range of $\hat {d} (\delta )$ , the destabilising (stabilising) characteristic of the respective parameters are encountered when the fluid mode of instability prevails.

Weakly Compressible Approximation of the Taylor–Green Vortex Solution

ABSTRACTThe Taylor–Green vortex represents an exact solution of the Navier–Stokes equations in . In this work, an approximation of this solution in two spatial dimensions is proposed for weakly compressible flows. These flows are characterized by small compressibility (or, equivalently, by a small Mach number) and are often employed in computational fluid dynamics to approximate the behaviour of incompressible Newtonian fluids. In this framework, the proposed solution is expected to be a useful benchmark for numerical solvers that implement the weakly compressibility approximation. To this end, some numerical examples are reported in the final section of this work.

Quantum spin representation for the Navier-Stokes equation

We develop a quantum representation for Newtonian viscous fluid flows by establishing a mapping between the Navier-Stokes equation (NSE) and the Schrödinger-Pauli equation (SPE). The proposed nonlinear SPE incorporates the two-component wave function and the imaginary diffusion. Consequently, classical fluid flow can be interpreted as a non-Hermitian quantum spin system. Using the SPE-based numerical simulation of viscous flows, we demonstrate the quantum/wavelike behavior in flow dynamics. Furthermore, the SPE equivalent to the NSE can facilitate the quantum simulation of fluid dynamics. Published by the American Physical Society 2024

Assessing the Effect of Coriolis Acceleration on the Coherent Structures in the Wake of a Wind Turbine using DMD

Abstract The present work aims to study the effect of veer, namely, the height-dependent lateral deflection of wind velocity due to Coriolis acceleration, on the coherent structures in the wake of the NREL 5-MW reference wind turbine using the sparsity-promoting dynamic mode decomposition (SP-DMD) method for the detection of dynamically-relevant flow structures. Large eddy simulation of the flow impacting the wind turbine is carried out by solving the filtered Navier-Stokes equations for incompressible flows. The effects of both tower and nacelle are included in the simulation using the immersed boundary method. Simulations are performed at Re ≈ 108, generating the inlet velocity profile by a precursor simulation of the atmospheric boundary layer subjected to Coriolis acceleration. The analysis of the DMD results allows to study the effects of the Coriolis acceleration on the most relevant dynamic modes in the turbine wake and to understand the basic mechanisms by which the wind veer significantly affects the wake recovery rate. Moreover, as a result of the SP-DMD methodology, the most relevant modes are extracted from the wake and a limited subset of relevant flow features that optimally approximates the original data sequence is identified. This small number of modes represent the kernel that can be employed for developing an accurate reduced order model of the wake.

A consistent diffuse-interface model for two-phase flow problems with rapid evaporation

AbstractWe present accurate and mathematically consistent formulations of a diffuse-interface model for two-phase flow problems involving rapid evaporation. The model addresses challenges including discontinuities in the density field by several orders of magnitude, leading to high velocity and pressure jumps across the liquid–vapor interface, along with dynamically changing interface topologies. To this end, we integrate an incompressible Navier–Stokes solver combined with a conservative level-set formulation and a regularized, i.e., diffuse, representation of discontinuities into a matrix-free adaptive finite element framework. The achievements are three-fold: First, we propose mathematically consistent definitions for the level-set transport velocity in the diffuse interface region by extrapolating the velocity from the liquid or gas phase. They exhibit superior prediction accuracy for the evaporated mass and the resulting interface dynamics compared to a local velocity evaluation, especially for strongly curved interfaces.Second, we show that accurate prediction of the evaporation-induced pressure jump requires a consistent, namely a reciprocal, density interpolation across the interface, which satisfies local mass conservation. Third, the combination of diffuse interface models for evaporation with standard Stokes-type constitutive relations for viscous flows leads to significant pressure artifacts in the diffuse interface region. To mitigate these, we propose to introduce a correction term for such constitutive model types. Through selected analytical and numerical examples, the aforementioned properties are validated. The presented model promises new insights in simulation-based prediction of melt–vapor interactions in thermal multiphase flows such as in laser-based powder bed fusion of metals.

A Pore-Scale Simulation of the Effect of Heterogeneity on Underground Hydrogen Storage

Using underground hydrogen storage technology has been recognized as an effective way to store hydrogen on a large scale, yet the physical mechanisms of hydrogen flow in porous media remain complex and challenging. Studying the heterogeneity of pore structures is crucial to enhance the efficiency of hydrogen storage. In order to better understand the pore-scale behavior of hydrogen in underground heterogeneous porous structures, this paper investigates the effects of wettability, pore–throat ratio, and pore structure heterogeneity on the behavior of the two-phase H2–brine flow using pore-scale simulations. The results show that the complex interactions between wettability, heterogeneity, and pore geometry play a crucial role in controlling the repulsion pattern. The flow of H2 is more obstructed in the region of the low pore–throat ratio, and the obstructive effect is more obvious when adjacent to the region of the high pore–throat ratio than that when adjacent to the region of the medium pore–throat ratio. In high-pore–throat ratio structures, the interfacial velocity changes abruptly as it passes through a wide pore and adjacent narrower throat. Interfacial velocities at the local pore scale may increase by several orders of magnitude, leading to non-negligible viscous flow effects. It is observed that an increase in the pore–throat ratio from 6.35 (low pore–throat ratio) to 12.12 (medium pore–throat ratio) promotes H2 flow, while an increase from 12.12 (medium pore–throat ratio) to 23.67 (high pore–throat ratio) negatively affects H2 flow. Insights are provided for understanding the role of the heterogeneity of pore structures in H2–brine two-phase flow during underground hydrogen storage.

The velocity tracking problem for Navier–Stokes equations with pointwise-integral control constraints in time-space

In this work, we study the velocity tracking control problem of the evolutionary two-dimensional Navier–Stokes system. The regularizing Tikhonov term is not involved in the cost functional and pointwise-integral control constraints in time-space are considered. First- and second-order optimality conditions are established using a suitable extended cone for the sufficient ones. We also address the numerical approximation of the control problem, proving the convergence of the discrete problems and establishing error estimates between the optimal discrete and continuous states.

Application of Triple- and Quadruple-Generalized Laplace Transform to (2+1)- and (3+1)-Dimensional Time-Fractional Navier–Stokes Equation

In this study, the solution of the (2+1)- and (3+1)-dimensional system of the time-fractional Navier–Stokes equations is gained by utilizing the triple-generalized Laplace transform decomposition method (TGLTDM) and quadruple-generalized Laplace transform decomposition method (FGLTDM). In addition, the results of the offered methods match with the exact solutions of the problems, which proves that, as the terms of the series increase, the approximate solutions are closer to the exact solutions of each problem. To verify the appropriateness of these methods, some examples are offered. The TGLTDM and FGLTDM results indicate that the suggested methods have higher evaluation convergence as compared to the ADM and HPM.

The Navier-Stokes equation and a fully developed turbulence

In fairly general conditions we give explicit (smooth) solutions for the potential flow. We show that, rigorously speaking, the equations of the fluid mechanics have not rotational solutions. However, within the usual approximations of an incompressible fluid and an isentropic flow, the Navier-Stokes equation has approximate vorticial (rotational) solutions, generated by viscosity. However, in general, these vortices are unstable, and a discrete distribution of vorticial solutions is not in me chanical equilibrium; it forms an unstable vorticial liquid. On the other hand, these solutions may exhibit turbulent, fluctuating instabilities for large variations of the velocity over short distances. We represent a fully developed turbulence as a homogeneous, isotropic and highly-fluctuating distribution of singular centres of turbulence. A regular mean flow can be included. In these circumstances the Navier-Stokes equation exhibits three time scales. The equations of the mean flow can be disentangled from the equations of the fluctuating part, which is reduced to a vanishing inertial term. This latter equation is not satisfied after averaging out the temporal fluctuations. For a homogeneous and isotropic distribution of non-singular turbulence centres the equation for the inertial term is satisfied trivially, i.e. both the average fluctuating velocity and the average fluctuating inertial term are zero. If the velocity is singular at the turbulence centres, we are left with a quasi-ideal classical gas of singularities, or a solution of singularities (solute) in quasi thermal equilibrium in the background fluid (solvent). This is an example of an emergent dynamics. We give three examples of vorticial liquids.

New a Priori estimate for stochastic 2D Navier–Stokes equations with applications to invariant measure

AbstractThe paper deals with the stochastic two-dimensional Navier–Stokes equations for homogeneous and incompressible fluids, set in a bounded domain with Dirichlet boundary conditions. We consider additive noise in the form $$G\,\textrm{d}W$$ G d W , where W is a cylindrical Wiener process and G a bounded linear operator with range dense in the domain of $$A^\gamma $$ A γ , A being the Stokes operator. While it is known that existence of invariant measure holds for $$\gamma >1/4$$ γ > 1 / 4 , previous results show its uniqueness only for $$\gamma > 3/8$$ γ > 3 / 8 . We fill this gap and prove uniqueness and strong mixing property in the range $$\gamma \in (1/4, 3/8]$$ γ ∈ ( 1 / 4 , 3 / 8 ] by adapting the so-called Sobolevskiĭ-Kato-Fujita approach to the stochastic N–S equations. This method provides new a priori estimates, which entail both better regularity in space for the solution and strong Feller and irreducibility properties for the associated Markov semigroup.

An Improved Direct Forcing Immersed Boundary Method With Integrated Mooring Algorithm for Floating Offshore Wind Turbines

Abstract An upgraded direct forcing immersed boundary method is implemented in the open-source hydrodynamic framework REEF3D::CFD for simulating the six-degrees-of-freedom (6DOF) motions of a floating offshore wind turbine (FOWT) based on the OC5 semi-submersible design. The direct forcing method is enhanced with a new density interpolation method across the fluid-structure interface that removes unphysical spurious phenomena and ensures stable and accurate wave load calculations on floating objects. A quasi-static algorithm is used for modeling the mooring system of the OC5 platform and restraining its motions in waves. The Navier-Stokes equations are solved on a staggered structured rectilinear grid for the hydrodynamic simulations. The level-set method is used to capture the free surface of the ocean waves. A ray-casting algorithm is employed to get inside-outside information near the fluid-solid interface while maintaining the underlying Cartesian grid in the hydrodynamic domain. The performance and accuracy of the mooring algorithm are compared to the widely-used mooring model MoorDyn, which is coupled to the hydrodynamic solver in REEF3D::CFD. The study demonstrates that the enhanced direct forcing method with the integrated quasi-static mooring algorithm in REEF3D::CFD provides a robust and accurate tool, suitable for the numerical analysis of the state-of-the-art FOWT in ocean waves.

Nonequilibrium Molecular Dynamics Method to Generate Poiseuille-Like Flow between Lipid Bilayers.

There are various flows inside and outside cells in vivo. Nonequilibrium molecular dynamics (NEMD) simulation is a useful tool for understanding the effects of these flows on the dynamics of biomolecules. We propose an NEMD method to generate a Poiseuille-like flow between lipid bilayers. We extended the conventional equilibrium MD method to produce a flow by adding constant external force terms to the water molecules. Using the Lagrange multiplier method, the center of mass of the lipid bilayer is constrained so that the flow does not sweep away the lipid bilayer, but the individual lipid molecules fluctuate. The temperature of the system is controlled properly in the solution and membrane by using the Nosé-Hoover thermostat. We found that the flow velocity increases linearly as the applied external force term increases. It is possible to estimate the appropriate value of acceleration to generate a flow with an arbitrary velocity using this proportional relation once a single short MD simulation is performed. We also found that the flow between two lipid bilayers is slower than the analytical solution of the Navier-Stokes equations between rigid parallel plates due to the interactions between water molecules and the membrane. This method can be applied not only to a flow on lipid membranes but also to a flow on soft surfaces generally.

STUDY OF THE INFLUENCE OF HDPE AND UHMWPE ON THE PROPERTIES OF ETHYLENE PROPYLENE DIENE VULCANIZATES

Introduction of small additives of thermoplastic polymer into the elastomer matrix is one of the simple and effective ways to improve the complex of physical and mechanical properties of rubbers. This article is devoted to the study of the effect of low-density polyethylene (HDPE) grade 273-83 and ultra-high molecular weight polyethylene (UHMWPE) grade GUR 4113 on the vulcanization characteristics of rubber compounds, physical and mechanical properties and resistance to thermal aging of vulcanizates based on ethylene propylene diene rubber grades SKEP-40 and SKEP-50. The use of HDPE and UHMWPE as additives is of interest not only due to their properties, but also due to their similar chemical nature to SKEP. It is shown that the introduction of polyethylene additives leads to some increase in the duration of the rubber compounds in the viscous flow state and the vulcanization optimum. Vulcanizates containing polyethylene additives are charac-terized by an increase in the conventional stress at 100 % elongation and a decrease in the relative elongation at break, which indicates an increase in the rigidity of the vulcanizates. In the presence of PE, wear resistance and the ability to elastic-elastic recovery of vulcanizates after compression also increase significantly. The frost resistance coefficient of rubber samples containing UHMWPE remains at the level of the original vulcanizates, and in the case of HDPE it decreases significantly, which can be explained by the better ability of UHMWPE to maintain its flexibility at low temperatures. According to the results of accelerated thermal aging, it was found that the presence of polyethylene additives leads to a significant increase in the stability of the elastic-strength properties of vulcanizates based on SKEP-50. Moreover, the higher the molecular weight of polyethylene, the higher its efficiency. It should be noted that in the DOT-4 environment, vulcanizates are characterized by a greater degree of preservation of elastic-strength indicators compared to the air environment, which is due to their high resistance to this working environment. It was found that UHMWPE, compared to HDPE, results in the production of an elastomeric material based on EPDM with a more optimal set of physical and mechanical properties and increased stability of elastic strength indicators after thermal aging.

Investigating the Morphology of a Free-Falling Jet with an Accurate Finite Element and Level Set Modeling

This study investigates the morphology of a free-falling liquid jet by using a computational approach with an experimental validation. Numerical simulations are developed by means of the Finite Element Method (FEM) for solving the viscous fluid flow and the level set method in order to track the interface between the fluid and air. Experiments are conducted in order to capture the shape of a free-falling jet of viscous fluid via circular orifice, where the shape is measured optically. The numerical results are found to be in agreement with the experimental data, demonstrating the validity of the proposed approach. Furthermore, we analyze the role of the surface tension by implementing linear as well as nonlinear surface energy models. All computational codes are developed with the aid of open-source packages from FEniCS and made publicly available. The combination of experimental and numerical techniques provides a comprehensive understanding of the morphology of free-falling jets and may be extended to multiphysics problems rather in a straightforward manner.

Design and Computational Modelling of AUV Tunnel Thruster Covers for Efficient Operation

Autonomous underwater vehicles have seen widespread adoption across industrial, scientific, and defence applications. They are typically utilized to perform oceanic mapping, surveillance, and inspection-type missions. Hovering AUVs, used for inspection applications, are over-actuated vehicles incorporating multiple thrusters to enable multiple degrees of freedom control at a low velocity. These vehicles, however, are extremely energy-limited, owing to their restrictive structural design that prohibits large batteries. This necessitates careful hydrodynamic design to best utilize this limited energy storage. Of particular importance are the hydrodynamic propulsion efficiencies of these vehicles. Whilst the external structure of AUV platforms is relatively well-defined and hydrodynamically optimized, one area has seen limited focus and optimization. This is the immediate surroundings of the propulsion geometry and housing. In this body of work, we propose an adaptation to the traditional through-body tunnel thruster geometry of an over-actuated AUV platform. The modification is the inclusion of a retractable internal thruster cover. Subsequently, a comparison is provided between a clean-hull AUV configuration, one with open through-body thrusters, and one fitted with the designed cover geometry. A comprehensive computational fluid dynamics analysis is then converged and assessed using the Reynolds-Averaged Navier–Stokes equations. The drag and local flow fields are determined, where the covers are found to reduce the drag coefficient and total drag of the AUV by 9.51%, primarily due to a reduction of 9.91% in the pressure drag. These findings highlight the increased operational efficiency of the cover geometry and support the adoption of such covers for energy-constrained AUVs.

Study on the Hydrodynamics of a Cownose Ray’s Flapping Pectoral Fin Model near the Ground

Cownose rays typically swim close to the ocean floor, and the nearby substrate inevitably influences their swimming performance. In this research, we numerically investigate the propulsive capability of cownose rays swimming near the ground by resolving three-dimensional viscous unsteady Navier–Stokes equations. The ground effect generally has a favorable impact on swimming. The thrust and lift increase as the near-substrate distance decreases. Nevertheless, a body length is the recommended distance from the ground, at which the flapping efficiency and swimming stability obtain a good trade-off. The increase in lift is due to the pressure difference between the dorsal and ventral surfaces of the ray, and the thrust boost is due to the enhanced shear vortex at the fin’s leading edge when swimming near the substrate. Our results indicate that the ground effect is more noticeable when the fin flaps are symmetrical compared to asymmetrical. In asymmetric flapping, the hydrodynamic performance improves at a smaller value than the half-amplitude ratio (HAR). The frequency of flapping also significantly affects swimming performance. We find that a superior flapping frequency, at which maximum efficiency is reached, occurs when flapping close to the substrate, and this superior frequency is consistent with the behavior of our model’s biological counterpart.

A unified framework for the analysis of accuracy and stability of a class of approximate Gaussian filters for the Navier–Stokes equations

Abstract Bayesian state estimation of a dynamical system utilising a stream of noisy measurements is important in many geophysical and engineering applications. In these cases, nonlinearities, high (or infinite) dimensionality of the state space, and sparse observations pose key challenges for deriving efficient and accurate data assimilation (DA) techniques. A number of DA algorithms used commonly in practice, such as the Ensemble Kalman Filter (EnKF) or Ensemble Square Root Kalman Filter (EnSRKF), suffer from serious drawbacks such as catastrophic filter divergence and filter instability. The analysis of stability and accuracy of these DA schemes has thus far focused either on finite-dimensional dynamics, or on complete observations in case of infinite-dimensional dissipative systems. We develop a unified framework for the analysis of several well-known and empirically efficient DA techniques derived from various Gaussian approximations of the Bayesian filtering schemes for geophysical-type dissipative dynamics with quadratic nonlinearities. We establish rigorous results on (time-asymptotic) accuracy and stability of these algorithms with general covariance and observation operators. The accuracy and stability results for EnKF and EnSRKF for dissipative PDEs are, to the best of our knowledge, completely new in this general setting. It turns out that a hitherto unexploited cancellation property involving the ensemble covariance and observation operators and the concept of covariance localization in conjunction with covariance inflation play a pivotal role in the accuracy and stability for EnKF and EnSRKF. Our approach also elucidates the links, via determining functionals, between the approximate-Bayesian and control-theoretic approaches to DA. We consider the ‘model’ dynamics governed by the two-dimensional incompressible Navier–Stokes equations (NSEs) and observations given by noisy measurements of averaged volume elements or spectral/modal observations of the velocity field. In this setup, several continuous-time DA techniques, namely the so-called 3DVar, EnKF and EnSRKF reduce to a stochastically forced NSEs. For the first time, we derive conditions for accuracy and stability of EnKF and EnSRKF. The derived bounds are given for the limit supremum of the expected value of the L 2 norm and of the H 1 Sobolev norm of the difference between the approximating solution and the actual solution as the time tends to infinity. Moreover, our analysis reveals an interplay between the resolution of the observations (roughly, the ‘richness’ of the observation space) associated with the observation operator underlying the DA algorithms and covariance inflation and localization which are employed in practice for improved filter performance.

Influence of particle density on turbulence characteristics over a rough surface

The objective of the study is to use a 3D Acoustic Doppler Velocimeter (ADV) to gauge the typical flow and turbulence characteristics within a non-uniform open channel. The findings of experimental examinations of the subcritical flow along the channel are presented in this work. The behavior of sand grains in turbulent open channel flow across porous and rough bed surfaces was examined in laboratory research, and the results were obtained. The properties of turbulent flow, i.e., turbulence intensity, turbulent kinetic energy, and Reynolds shear stresses, are determined from ADV data. The continuity equation and the Reynolds equation of open-channel flow have been used to build theoretical formulations for the velocity distribution and Reynolds stress distribution in the vertical direction. Measured profiles of vertical velocity and Reynolds stress are compared to the derived expressions. The impact of the size of particles on the distribution of mean flow characteristics is discussed. This work provides a novel origin for the profile and analyzes the behavior of the vertical velocity distribution in the region where fully formed turbulence is dominating in open channels using the Navier–Stokes equations. In comparison to other sand roughness, Chopan sand bed (with greater density) exhibits the strongest turbulence intensities in both vertical and streamwise direction just next to the bed when away from the channel boundary. In contrast to flow across a rough surface, the variance ranges between 150 and 250% concerning the channel bed’s roughness type, impacting the velocity triple products that signifies transfer of turbulent kinetic energy.

Model analysis and experiment study for effects of thermal viscous and fluid flow on ventilated acoustic metamaterials labyrinth

In many noise scenarios, ventilation is necessary. The practical realization of noise attenuation under the premise of ensuring ventilation is an urgent problem to be solved. In this paper, a ventilated acoustic metamaterial labyrinth (VAML) is proposed, and the corresponding analytical and numerical computational models are developed considering the thermal viscous effect (TVE). The loss in sound transmission of the VAML is quantitatively obtained and experimentally verified. The theoretical results are compared with the experimental results with an error of 0.6%. By comparing the transmission coefficient and the impedance at the entrance of the side branches, the mechanism of the TVE on the performance of VAML is analyzed, and it is clarified that the TVE is responsible for the sound absorption coefficient. The effects of structure parameters on the transmission coefficient of the VAML are analyzed individually. The results show that the resonant frequency of the system decreases as the length of channel 1 l1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$l_1$$\\end{document}, the length of channel 2 l2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$l_2$$\\end{document}, and the width of channel d increase. By adjusting the magnitude of these three parameters, the control of the resonant frequency can be realized. Meanwhile, as l1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$l_1$$\\end{document}, l2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$l_2$$\\end{document} and ventilation radius R decrease the valley of transmission coefficient decreases which means better noise reduction. Finally, the effect of flow velocity on the transmission coefficient is analyzed, revealing the mechanism of fluid action on macroscopic acoustic properties.