Linearized Navier-Stokes Equations Research Articles

Navier–Stokes Equation in a Cone with Cross-Sections in the Form of 3D Spheres, Depending on Time, and the Corresponding Basis

The work establishes the unique solvability of a boundary value problem for a 3D linearized system of Navier–Stokes equations in a degenerate domain represented by a cone. The domain degenerates at the vertex of the cone at the initial moment of time, and, as a consequence of this fact, there are no initial conditions in the problem under consideration. First, the unique solvability of the initial-boundary value problem for the 3D linearized Navier–Stokes equations system in a truncated cone is established. Then, the original problem for the cone is approximated by a countable family of initial-boundary value problems in domains represented by truncated cones, which are constructed in a specially chosen manner. In the limit, the truncated cones will tend toward the original cone. The Faedo–Galerkin method is used to prove the unique solvability of initial-boundary value problems in each of the truncated cones. By carrying out the passage to the limit, we obtain the main result regarding the solvability of the boundary value problem in a cone.

Numerical study of thermo-viscous effects in acoustic materials using linearized Navier-Stokes equations

Acoustic materials, characterized by small-scale structures, require a detailed understanding of dissipation effects within the viscous and thermal boundary layers where most of the dissipation takes place. Despite the prevalent assumption of lossless or isentropic conditions in many applications, a comprehensive study of regions where thermal and viscous losses occur is crucial, particularly in scenarios where wall-induced losses are significant. Bridging this gap, the study aims to investigate these losses in various structures to guide the development of innovative acoustic materials. To achieve this objective, the Linearized Navier-Stokes Equations (LNSE) solver from the finite element software COMSOL is employed to determine the absorption coefficients and to identify predominant regions of thermal and viscous losses. The model is validated using experimental and published data. The study effectively reveals specific regions where these losses significantly dominate, highlighting their critical influence on acoustic material performance. Although the structures are simplified to 2D models to balance physical details with computational feasibility, the ability to identify dominant loss regions marks a significant contribution to the field and paves the way to refine the design of acoustic materials.

A quantum algorithm for the lattice-Boltzmann method advection-diffusion equation

We present a versatile and efficient quantum algorithm based on the Lattice Boltzmann method (LBM) approximate solution of the linear advection-diffusion equation (ADE). We emphasize that the LBM approximation modifies the diffusion term of the underlying exact ADE and leads to a modified equation (mADE). Due to its versatility in terms of operator splitting, the proposed quantum LBM algorithm for the mADE provides a building block for future quantum algorithms to solve the linearized Navier-Stokes equation on quantum computers. We split the algorithm into four operations: initialization, collision, streaming, and calculation of the macroscopic quantities. We propose general quantum building blocks for each operator, which adapt intrinsically from the general three-dimensional case to smaller dimensions and apply to arbitrary lattice-velocity sets. Based on (sub-linear) amplitude data encoding, we propose improved initialization and collision operations with reduced complexity and efficient sampling-based simulation. Quantum streaming algorithms are based on previous developments. The proposed quantum algorithm allows for the computation of successive time steps, requiring full state measurement and reinitialization after every time step. It is validated by comparison with a digital implementation and based on analytical solutions in one and two dimensions. Furthermore, we demonstrate the versatility of the quantum algorithm for two cases with non-uniform advection velocities in two and three dimensions. Various velocity sets are considered to further highlight the flexibility of the algorithm. We benchmark our optimized quantum algorithm against previous methods employed in sampling-based quantum simulators. We demonstrate sampling efficiency, with sampling accelerated convergence requiring fewer shots.

Theoretical model for the elastic properties of cracked fluid-saturated rocks considering the crack connectivity

SUMMARY Cracks are a common rock microstructure and have a large effect on elastic properties during wave propagation. The fluid flow between a crack and its adjacent pore space can cause wave attenuation and dispersion. In this work, we introduce a crack connectivity parameter which is meant to improve the expression of local flow by weighting the contributions of fully connected and isolated cracks. We then update the analytical expression for frequency-dependent moduli by modifying the boundary conditions of the linearized Navier–Stokes equation and mass conservation equation. The proposed model contains the effect of cracks and stiff pores, in which the attenuation and dispersion are determined by squirt-flow and stiff-pore relaxations. The resulting model shows the squirt-flow relaxation frequency depends on not only the crack aspect ratio but also the crack connectivity. However, their contributions are different. The crack connectivity has little effect on the attenuation amplitude of shear modulus, but affects the attenuation amplitude of bulk modulus when multiple sets of cracks exist in the rock. The attenuation frequency band is also affected by the crack connectivity. As the crack connectivity deteriorates, the attenuation peak moves to low frequencies. In addition, by comparing the crack connectivity with the fluid viscosity coefficient, it is observed that the crack connectivity only affects the attenuation frequency band of cracks, whereas the fluid viscosity coefficient affects the attenuation frequency bands of cracks and stiff pores simultaneously. Thus, the introduction of crack connectivity is a supplement to the theoretical model of cracked fluid-saturated rocks. It helps understand the local fluid flow induced by seismic waves and provides a reasonable variation analysis of moduli and attenuation, especially for tight reservoirs.

Fast solution of incompressible flow problems with two-level pressure approximation

This paper develops efficient preconditioned iterative solvers for incompressible flow problems discretised by an enriched Taylor–Hood mixed approximation, in which the usual pressure space is augmented by a piecewise constant pressure to ensure local mass conservation. This enrichment process causes over-specification of the pressure when the pressure space is defined by the union of standard Taylor–Hood basis functions and piecewise constant pressure basis functions, which complicates the design and implementation of efficient solvers for the resulting linear systems. We first describe the impact of this choice of pressure space specification on the matrices involved. Next, we show how to recover effective solvers for Stokes problems, with preconditioners based on the singular pressure mass matrix, and for Oseen systems arising from linearised Navier–Stokes equations, by using a two-stage pressure convection–diffusion strategy. The codes used to generate the numerical results are available online.

Three-dimensional stochastic dynamical modeling for wind farm flow estimation

Modifying turbine blade pitch, generator torque, and nacelle direction (yaw) are conventional approaches for enhancing energy output and alleviating structural loads. However, the efficacy of such methods is challenged by the lag in adjusting such settings after atmospheric variations are detected. Without reliable short-term wind forecasting tools, current practice, which mostly relies on data collected at or just behind turbines, can result in sub-optimal performance. Data-assimilation strategies can achieve real-time wind forecasting capabilities by correcting model-based predictions of the incoming wind using various field measurements. In this paper, we revisit the development of a class of prior models for real-time estimation via Kalman filtering algorithms that track atmospheric variations using ground-level pressure sensors. This class of models is given by the stochastically forced linearized Navier-Stokes equations around the three-dimensional waked velocity profile defined by a curled wake model. The stochastic input to these models is devised using convex optimization to achieve statistical consistency with high-fidelity large-eddy simulations. We demonstrate the ability of such models in reproducing the second-order statistical signatures of the turbulent velocity field. In support of assimilating ground-level pressure measurements with the predictions of said models, we also highlight the significance of the wall-normal dimension in enhancing two-point correlations of the pressure field between the ground and the computational domain.

Thermocapillary migration of a compound drop in an arbitrary viscous flow

The thermocapillary migration of a concentric compound drop in an arbitrary viscous flow under the consideration of negligible Reynolds number is investigated. The thermocapillary effect refers to the migration of a drop under the influence of a temperature gradient. The thermal and hydrodynamic problems are examined. The thermal field is governed by the heat conduction equation whereas the hydrodynamic fluid velocities are governed by the linearized Navier–Stokes equations. Presence of temperature gradient results in variation of the interfacial tension which is assumed to depend on temperature linearly. Variation of interfacial gradient leads to the coupling of the hydrodynamic problem with the thermal problem through the boundary condition. A complete general solution of Stokes equations is utilized to obtain closed-form expressions for the velocity vector and pressure. The hydrodynamic forces acting on the compound drop are obtained and expressed in terms of Fax́en’s law. Some important asymptotic limiting cases of hydrodynamic drag are also derived. The hydrodynamic drag for cases of uniform flow, shear flow, and heat source with the known ambient flow are derived and it is found that in the case of shear flow, the hydrodynamic drag is contributed only by the thermal component and the shear flow has no effect on it. The obtained results for drag and torque in the limiting cases are in agreement with the existing results in the literature. Furthermore, the migration velocity of the compound drop is obtained by equating the hydrodynamic drag force to zero. The results obtained for migration velocity are explained with the aid of graphs. The migration velocity is found to be a monotonic function of the Marangoni number and the radius of the innermost drop.

Boundary-layer instability on a highly swept fin on a cone at Mach 6

The growth and characteristics of linear, oblique instabilities on a highly swept fin on a straight cone in Mach 6 flow are examined. Large streamwise pressure gradients cause doubly inflected cross-flow profiles and reversed flow near the wall, which necessitates using the harmonic linearized Navier–Stokes equations. The cross-flow instability is responsible for the most-amplified disturbances, however, not all disturbances show typical cross-flow characteristics. Distinct differences in perturbation structure are shown between small ( $\sim$ 3–5 mm) and large ( $\sim$ 10 mm) wavelength disturbances at the unit Reynolds number $Re' = 11 \times 10^6$ m $^{-1}$ . As a result, amplification measurements based solely on wall quantities bias a most-amplified disturbance assessment towards larger wavelengths and lower frequencies than would otherwise be determined by an off-wall total-energy approach. A spatial-amplification energy-budget analysis demonstrates (i) that wall-normal Reynolds-flux terms dictate the local growth rate, despite other terms having a locally larger magnitude and (ii) that the Reynolds-stress terms are responsible for large-wavelength disturbances propagating closer to the wall compared with small-wavelength disturbances. Additionally, the effect of free-stream unit Reynolds number and small yaw angles on the perturbation amplification and energy budget is considered. At a higher Reynolds number ( $Re' = 22 \times 10^6$ m $^{-1}$ ), the most-amplified wavelength shrinks. Perturbations do not behave self-similarly in the thinner boundary layer, and the shift in most-amplified wavelength is due to decreased dissipation relative to the lower-Reynolds-number case. Small yaw angles produce a streamwise shift in the boundary layer and disturbance amplification. The yaw results quantify a potential uncertainty source in experiments and flight.

Flow–acoustic resonance mechanism in tandem deep cavities coupled with acoustic eigenmodes in turbulent shear layers

This study presents the interplay of flow and acoustics within tandem deep cavities, focusing on the resonance mechanism occurring between turbulent shear layers and acoustic eigenmodes. The arrangement inside the tandem deep cavities includes both close and remote configurations. A combined fully coupled and decoupled aeroacoustic simulation strategy was devised. Employing an advanced high-order spectral/hp element method in conjunction with implicit large eddy simulation, the nonlinear compressible Navier–Stokes equations were solved to acquire internal flow–acoustic resonant field. In parallel, the linearized Navier–Stokes equations were tackled to determine coherent shear layer perturbations with external acoustic forcing. Based on acoustic measurements, the mainstream Reynolds number approaches approximately $R{e_{in}} = {O}({10^5})$ , where we identified the presence of frequency lock-in and a resonance range. Aeroacoustic noise sources were examined by implementing spectral proper orthogonal decomposition to decompose the pressure fields into hydrodynamic and acoustic components. As feedback intensified, the flow characteristics by the acoustic forcing effect and the flow-interactive effect were categorized according to the development of concurrent turbulent shear layers. Subsequently, the alternating and synchronous behaviours of concurrent shear layers resonated with the out-of-phase and in-phase acoustic eigenmodes were identified, and the corresponding large-scale counter-rotating vortex pairs and co-rotating vortex structures at the cavity entrances were extracted. The acoustic power generated by the Coriolis force was calculated using Howe's vortex-sound analogy, and the aeroacoustic energy transfer mechanism between large-scale shear layer vortices with acoustic eigenmodes was further explored. Finally, a linear response of coherent perturbations of the concurrent shear layers by external acoustic forcing was established. The amplification of flow in the streamwise direction toward the main duct led to the formation of coherent vortex structures, accompanied by separation bubbles into the main duct.

Numerical investigation on acoustic damping characteristics of dual Helmholtz resonators in presence of a grazing flow

In this study, the acoustic damping performances of the dual Helmholtz resonators were numerically evaluated using a 3D model. The grazing flow passes tangentially through the resonator neck, with a Mach number range of 0 ≤ Ma ≤ 0.1. The numerical model operates by solving the linearized Navier–Stokes equations. The current model is validated through a comparison with experimental data. The model is then utilized to explore the effects of the dual Helmholtz resonators on acoustic transmission loss performance in the presence of a grazing flow. Three key parameters are examined: 1) different implementation configurations of the dual Helmholtz resonators (including Models (b), (c), and (d)), 2) the mean temperature of the grazing flow, and 3) the axial distance between the dual Helmholtz resonators. For comparison, the acoustic damping performance of these dual Helmholtz resonators is compared to the single Helmholtz resonator case (Model (a)). The maximum transmission loss of Model (c) is significantly higher, recording values of 91%, 89.4%, and 92.5% than those observed for Model (a) at Ma = 0, Ma = 0.05, and Ma = 0.1, respectively. It is observed that the dual Helmholtz resonators dramatically increase the transmission loss. Model (c) is demonstrated to be associated with the most significant damping on the acoustic plane waves in comparison with that of Model (a). Additionally, the maximum transmission loss of Model (c) is 23.23 dB, 30.32 dB, and 34.58 dB at mean temperatures of 300 K, 600 K, and 900 K, respectively. Therefore, increasing the mean temperature is shown to be beneficial to enhance transmission losses in the presence of the grazing flow. Furthermore, under Ma = 0.1, the resonant frequency of Model (c) is 127 Hz, 152 Hz, and 172 Hz, corresponding to mean temperatures of 300 K, 600 K, and 900 K. It can be concluded that increasing the temperature has the effect of broadening the resonant frequency, especially at a high grazing flow Mach number. However, increasing the mean temperature results in a reduction of transmission loss in the absence of the grazing flow. In the case of Model (c), a 32 cm axial distance results in a 5.6% larger transmission loss at Ma = 0 and a 26.4% larger loss at Ma = 0.1 compared to a 16 cm axial distance. This indicates that increasing the axial distance between the dual Helmholtz resonators improves transmission loss.

Synchrotron X-ray phase-contrast imaging of ultrasonic drop atomization

Ultrasonic atomization is employed to generate size-controllable droplets for a variety of applications. Here, we minimize the number of parameters dictating the process by studying the atomization of a single drop pending from an ultrasonic horn. Spatiotemporally resolved X-ray phase-contrast imaging measurements show that the number-median sizes of the ejected droplets can be predicted by the linear Navier–Stokes equations, signifying that the size distribution is controlled by the fluid properties and the driving frequency. Experiments with larger pendant water drops indicate that the fluid–structure interaction plays a pivotal role in determining the ejection onset of the pendant drop. The atomization of viscoelastic drops is dictated by extended ligament formation, entrainment of air, and ejection of drop-encapsulated bubbles. Existing scaling laws are used to explain the required higher input amplitudes for the complete atomization of viscoelastic drops as compared to inviscid drops. Finally, we elucidate the differences between capillary wave-based and cavitation-based atomization and show that inducing cavitation and strong bubble oscillations quickens the onset of daughter drop ejection but impedes their size control.

The harmonic linearized Navier–Stokes equations for transition prediction in three-dimensional flows

The conventional method to predict the onset of laminar-turbulent transition in convectively unstable boundary-layer flows is based on the logarithmic amplification ratio, the N-factor, of the linear instability waves. To calculate the N-factor, the flow variables are decomposed into a laminar basic state solution and the linear disturbances, which are assumed to be harmonic in time. The most commonly used linear stability analysis approaches include the locally parallel linear stability theory (LST) and the nonlocal, weakly nonparallel parabolized stability equations (PSE). However, these methods do not account for strong streamwise gradients that are encountered in several configurations of interest, such as those in the vicinity of roughness elements, steps, gaps, or corners. To compute the linear evolution of disturbances along such strongly nonparallel regions, the harmonic linearized Navier–Stokes equations (HLNSE) need to be solved. The discretization of the HLNSE for spanwise/azimuthally inhomogeneous laminar basic states yields a linear system of complex arithmetic with a leading dimension of the order of 107 to 108 even in relatively simple flows. A combined multithread and multiprocessor algorithm is implemented for the direct solution of such linear systems. Results for a supersonic boundary layer over a three-dimensional roughness patch show good agreement with experimental measurements when the evolution of the instability waves over the roughness patch is included via the HLNSE. Additionally, inflow-resolvent analysis based on the HLNSE for discrete-roughness-induced disturbances in the nose tip of a blunt cone at Mach 6 demonstrates the importance of including the disturbance amplification along the near vicinity of the roughness element and separation region.

Resolvent-based motion-to-wake modelling of wind turbine wakes under dynamic rotor motion

We propose a linearized deterministic model for predicting coherent structures in the wake of a floating offshore wind turbine subject to platform motions. The model's motion-to-wake predictive capability is achieved through two building blocks: a motion-to-forcing (M2F) part and a forcing-to-wake (F2W) part. The M2F model provides a unified framework to parameterize the effects of arbitrary floating wind turbine motions as unsteady loads of a fixed actuator disk, requiring only the radial distribution of the aerodynamics force coefficient on the blade as input. The F2W model is derived based on a bi-global resolvent model obtained from the linearized Navier–Stokes equations, using the time-averaged wake of a fixed wind turbine as input. In addition to its capability of predicting sensitive frequency ranges, the model excels linear stability analysis by providing spatial modes of the wake response in a motion-specific and phase-resolved manner. The model successfully predicts the wake pulsing mode induced by surge, as well as the similarity and difference of the wake meandering modes caused by sway and yaw. Large-eddy simulations under different inflow turbulence intensities (TIs) and length scales are further conducted to analyse the wake meandering triggered by the simultaneous excitation of free-stream turbulence and sway motion. The results show distinct frequency signatures for the wake dynamics induced by ambient turbulence and sway motion. The inflow TI is found to have a stabilizing effect on the wake, reducing the motion-induced wake responses. Such a stabilizing effect is captured satisfactorily with the proposed model, provided that the effective viscosity is calibrated properly using the data from the fixed turbine wake under the corresponding turbulent inflow.

Enhanced dissipation and transition threshold for the Poiseuille-Couette flow

In this paper, we consider the hydrodynamic stability of the 2D Navier-Stokes equations on T×R around Poiseuille-Couette flow V=(y2,0)+a(y,0) with a∈(0,1). By means of the hypocoercivity method with time dependent weights and introducing a new estimate on the energy functional, we establish the enhanced dissipation estimates for the linearized Navier-Stokes equations. More precisely, for a⩽10−3ν14, we obtain a decay rate proportional to ν12, which is consistent with the Poiseuille flow. For a=10−3νs with s∈[0,14], the decay rate is proportional to ν1−2s. As an application of the linear estimates, we obtain the result of nonlinear stability transition threshold.

Acoustic attenuation prediction and analysis of perforated hybrid mufflers with non-uniform flow based on frequency domain linearized Navier-Stokes equations

The three-dimensional frequency-domain linearized Navier-Stokes equations (LNSEs) with consideration of eddy viscosity are developed to evaluate the acoustic attenuation performance of perforated hybrid mufflers in the presence of non-uniform flow. The computations are performed in two steps: time averaged flow variables are acquired by using steady-state computational fluid dynamics (CFD) method and then mapped into the acoustic mesh, and the acoustic perturbation variables are obtained by solving frequency-domain LNSEs, where the sound-absorbing material is treated as an equivalent fluid with complex sound speed and density. The predictions of transmission losses of the two-pass perforated hybrid mufflers in the presence of non-uniform flow are in good consistencies with the measurements, which verifies the correctness of LNSEs. The effect of Mach numbers on acoustic attenuation performance of the mufflers with different filling densities and perforated components is investigated in detail. The transmission loss of the mufflers with various filling densities are increased by complex airflow in the lower frequency range. For the mufflers where not all components have been perforated, the flow lowers low-frequency resonance peak. The influence of complex airflow on acoustic attenuation performance of all configurations is weakened at higher frequencies.

Optimal shaping of acoustic black holes for sound absorption in air

This article presents a systematic numerical study of the absorbing properties of acoustic black holes (ABHs) serving as an anechoic termination of waveguides. The study focuses on the sensitivity of ABHs’ absorbing performance to their profile and internal-structure parameters. The article compares numerical predictions from 1D model based on the Riccati equation with a detailed 2D model based on the linearized Navier-Stokes equations and the finite element method, finding good agreement among all results, especially for ABHs with fine internal structures. The mean value of the reflection coefficient modulus is used to quantify the ABH’s absorbing performance, and the article introduces the use of power-law functions and cubic splines to define the ABH’s shape function. An evolutionary algorithm is employed to optimize the ABH’s profile, resulting in improved absorbing performance. The numerical results suggest that the optimum shape is simple and more-or-less insensitive to other geometrical and internal-structure parameters.

Receptivity of the rotating disk boundary layer to traveling disturbances

An adjoint approach is developed to undertake a receptivity study of the rotating disk boundary layer. The adjoint linearized Navier–Stokes equations are first derived in cylindrical coordinates. A receptivity formula is then formulated that specifies the response of stationary and traveling linear perturbations to an external force, including sources of momenta and mass and unsteady wall motion. Using the parallel flow approximation, in which the radial dependence of the undisturbed flow is ignored, receptivity characteristics are computed for a broad range of temporal frequencies, radial wavenumbers, azimuthal mode numbers, and Reynolds numbers. The type-I crossflow instability attains a maximum amplitude for external forces fixed near the wall-normal location of the critical layer (i.e., α¯rF+βG=ω), and the type-II Coriolis instability achieves larger amplitudes when external forces are located in the vicinity of a vanishing effective shear stress (i.e., α¯rF′+βG′=0). Sources of radial momenta fixed about these wall-normal locations establish larger-sized disturbances than equivalent-sized sources of azimuthal momenta, wall-normal momenta, and mass. At the disk surface, motion along the wall-normal direction induces a stronger receptivity response than wall motions acting along the radial and azimuthal directions. In general, the crossflow instability achieves larger-sized amplitudes than the Coriolis instability, with the peak response realized for Reynolds numbers near the critical conditions for linear instability.

Numerical prediction of the whistling potentiality of a turbulent channel flow with corrugated walls

This study explores the turbulent flow-induced whistling phenomena in a channel with corrugated wall surfaces, which is crucial for mitigating the acoustic fatigue problem in the aerospace field. By solving a compressible linearized Navier–Stokes equation in the frequency domain, the interference between the turbulent flow field along the corrugated wall and the incident acoustic field is studied, including the acoustic wave scattering phenomenon caused by turbulence and the fluid perturbation induced by acoustic waves. In conjunction with this, the acoustic two-ports method is utilized to investigate the transfer-function model and predict the whistling potentiality of the turbulent flow along corrugated walls. Experimental validations through the literature results confirm the numerical accuracy of this aeroacoustic simulation strategy. Subsequently, the investigation extends to different cavity configurations with different cavity profiles and numbers, and the two-port scattering matrix is applied to quantify the acoustic transmission and damping coefficients caused by the background flow field and turbulent eddy viscosity. The acoustic power conversion mechanism between the turbulent flow field and the incident acoustic field is established, allowing for quick prediction and effective analysis of the generation frequency range of the whistling phenomenon. Furthermore, the modulation effect of sound waves on the fluid is studied by analyzing the response of the incident sound wave frequency to the phase interference momentum and shear layer of different configurations of corrugated cavities. The results show that compared with the right-edge configuration, the rounded-edge configuration produces whistling at a lower frequency due to the turbulence effect, and the number of cavities adjusts the intensity, not the frequency, of the sound power generated. In addition, the oscillation of the shear layer caused by sound waves is related to the cavity configuration and the sound wave frequency.

Data assimilation finite element method for the linearized Navier-Stokes equations with higher order polynomial approximation

In this article, we design and analyze an arbitrary-order stabilized finite element method to approximate the unique continuation problem for laminar steady flow described by the linearized incompressible Navier–Stokes equation. We derive quantitative local error estimates for the velocity, which account for noise level and polynomial degree, using the stability of the continuous problem in the form of a conditional stability estimate. Numerical examples illustrate the performances of the method with respect to the polynomial order and perturbations in the data. We observe that the higher order polynomials may be efficient for ill-posed problems, but are also more sensitive for problems with poor stability due to the ill-conditioning of the system.

Acoustic Attenuation Prediction of Perforated Reactive and Dissipative Mufflers With Flow by Using Frequency-Domain Linearized Navier-Stokes Equations

To comprehensively consider the impact of complex airflow on sound propagation, a numerical approach by solving the three-dimensional (3-D) frequency-domain linearized Navier-Stokes equations (LNSEs) is proposed to accurately predict the transmission losses of perforated tube reactive and dissipative mufflers in the presence of complex airflow. The numerical computations are performed in two steps: (1) solve the time averaged flow by steady-state computational fluid dynamics (CFD) simulation and then map the results into acoustic mesh; (2) treat sound-absorbing material as an equivalent fluid and obtain acoustic perturbation variables by solving the frequency-domain LNSEs. The transmission losses of perforated tube reactive and dissipative mufflers are measured in the presence of flow and compared with the numerical predictions. The excellent consistencies between predictions and measurements validate the present approach. The influence of Mach number on acoustic attenuation performance of perforated tube reactive and dissipative mufflers is then investigated. With Mach number increases, the transmission loss of perforated tube reactive muffler increases somewhat, while the transmission loss of dissipative mufflers decreases obviously in the lower frequency range and change is limited at higher frequencies.