Linearized Navier-Stokes Equations Research Articles

Effect of slit length on linear and non-linear acoustic transfer impedance of a micro-slit plate

The effect of the slit length on the acoustic transfer impedance of micro-slit plates (MSPs) is investigated in the linear and non-linear regime for a specific slit geometry. This geometry is inspired by slits obtained by cutting and bending the plate. MSPs are plates with arrays of slit-shaped perforations, with the width of the order of the acoustic viscous boundary layer thickness. Impedance tube measurements on two accurately manufactured plates are compared to numerical solution of the Linearized Navier-Stokes equations and to analytical limits. The impedance of the plate is obtained by the impedance of a single slit divided by the plate porosity. The resistance of a slit is independent on the slit length and on the plate porosity. In the linear regime the resistance is accurately predicted by a two-dimensional numerical model. In the non-linear regime, the resistance is strongly dependent on the amplitude of the acoustic waves. The inertance of the slit is weakly dependent on the slit length and on the plate porosity, for low and moderate amplitudes. For high amplitudes, a complicated amplitude dependency of the inertia of short slits is found. One expects that most of the conclusions obtained can be generalised to other slit geometries.

Global Lipschitz stability for an inverse source problem for the Navier–Stokes equations

For linearized Navier–Stokes equations, we consider an inverse source problem of determining a spatially varying divergence-free factor. We prove the global Lipschitz stability by interior data over a time interval and velocity field at over the spatial domain. The key machinery are Carleman estimates for the Navier–Stokes equations and the operator rot.

Nonlinear vibrations and stability of rotating cylindrical shells conveying annular fluid medium

This study aims to investigate the nonlinear dynamic characteristics of annular cylinders coupled with a fluid medium to identify the effective parameters on stability margins at different rotation speeds. To achieve this, stability and nonlinear vibrations of rotating cylindrical shells containing incompressible annular fluid are considered. The fluid medium is bounded by a rigid external cylinder. Sanders–Koiter kinematic assumptions are utilized to determine the geometrically nonlinear structural equations of motion. These equations are then used to investigate finite amplitude vibrations of various fluid-loaded shells at different states. A penalty approach is introduced by using boundary springs with arbitrary stiffness to simulate any desired set of end conditions. Both driven and companion modes of vibration are included in the solution algorithm to capture the traveling wave response. Fluid equations are determined by using the linearized Navier–Stokes equations. The two sets of governing equations are coupled through the flow impermeability condition on the internal shell’s interface and the zero fluid particle velocity condition on the external boundary. Nonlinear numerical solutions to rotating shells with extremal fluid loading are conducted by the Runge–Kutta direct integration technique. The minimum number of driven/companion modes is determined and the results are verified with the available literature. Results show a significant exchange of energy between the driven and companion modes which consequently initiates a traveling wave response in rotating shell configurations. Moreover, the centrifugal loading and external excitation are shown to be effective on the oscillation amplitude as rotation speed changes leading to alteration of the response quality from periodic to chaotic.

Finite Element Iterative Methods for the 3D Steady Navier--Stokes Equations.

In this work, a finite element (FE) method is discussed for the 3D steady Navier–Stokes equations by using the finite element pair . The method consists of transmitting the finite element solution of the 3D steady Navier–Stokes equations into the finite element solution pairs based on the finite element space pair of the 3D steady linearized Navier–Stokes equations by using the Stokes, Newton and Oseen iterative methods, where the finite element space pair satisfies the discrete inf-sup condition in a 3D domain . Here, we present the weak formulations of the FE method for solving the 3D steady Stokes, Newton and Oseen iterative equations, provide the existence and uniqueness of the FE solution of the 3D steady Stokes, Newton and Oseen iterative equations, and deduce the convergence with respect to of the FE solution to the exact solution of the 3D steady Navier–Stokes equations in the norm. Finally, we also give the convergence order with respect to of the FE velocity to the exact velocity u of the 3D steady Navier–Stokes equations in the norm.

A multi-layer overlapping structure for continuous broadband acoustic wave absorption at lower-frequencies

This paper is concerned with the design of a multi-layer overlapping structure consisting of a series of cavities and necks for acoustic wave absorption within continuous broadband at lower frequencies. The acoustic absorber is designed as a periodic structure, whose fundamental units can be regarded as Helmholtz resonators. A finite element model based on the linearized Navier-Stokes equations is developed to evaluate the thermal and viscous dissipation of the absorber. To verify the acoustic absorption characteristics of the structure, a series of absorbers fabricated by 3D printing are tested in an impedance tube. The numerical solutions of the acoustic absorption coefficient are in good agreement with the experimental results. It is found that the coupling effect of the cavities and the necks strengthens the sound absorption performance of the absorber in a wideband at lower frequencies. Physical insights into the absorption mechanism of the overlapping structure due to the energy dissipation of the necks are provided. The effects of the geometrical parameters including length, width and radius of the absorber on the acoustic wave absorption of the structure are investigated. In order to verify the efficiency of the designed overlapping structure for noise control in practical applications, a number of absorbers are mounted inside a scaled payload fairing of a space launch vehicle. The results show that the inner noise of the payload fairing can be significantly reduced in a broadband with a maximum value of 40 dB.

Electric and viscous correction for viscous potential flow analysis of electrohydrodynamic instability of an electrified leaky-dielectric jet

The electric and viscous correction of viscous potential flow (EVCVPF) is developed for analyzing the electrohydrodynamic instability of an electrified leaky-dielectric viscous jet. The EVCVPF model is based on the viscous potential flow (VPF) and the viscous correction of VPF (VCVPF), proposed by Joseph and Wang [“The dissipation approximation and viscous potential flow,” J. Fluid Mech. 505, 365–377 (2004)]. The purpose is to resolve the discrepancy between the non-zero irrotational viscous and the electric tangent stresses. The power of the pressure correction is introduced to compensate the neglected viscous dissipation in the flow bulk in VPF, which is equal to the average power of the irrotational viscous and the electric tangent stresses. The model has been validated by comparing it to the exact normal-mode solution of the linearized Navier–Stokes equations (fully viscous flow, FVF). The energy budget is also performed to assist in understanding underlying mechanisms. Results show that EVCVPF is accurate for charged jets with low and moderate viscosities, i.e., the Ohnesorge number approximately Oh ≤ 0.1. The inaccuracy for highly viscous jets are the limitations of VPF itself. The electric field has less influence compared to the fluid viscosity. To achieve more accurate approximations, VCVPF and VPF are chosen for axisymmetric and non-axisymmetric modes under weak electric fields. EVCVPF is in remarkably good agreement with FVF under moderate and strong electric fields. In general, as VCVPF extends the applicability in fluid viscosity of VPF, EVCVPF further improves the adequacy when studying the electrohydrodynamic instability.

Space-time energy spectra in turbulent shear flows

This article reviews the recent theories and models of space-time energy spectra in turbulent shear flows. The review is based on the picture of turbulent passage proposed by Taylor's frozen-flow hypothesis and Kraichnan-Tennekes random sweeping hypothesis: convection of small-scale eddies by large-scale eddies with a certain distortion, which determines the peaks and bandwidths of space-time energy spectra. The data-refined stochastic models and data-based reconstruction models are examined. The linearized Navier-Stokes equations with random forcing for space-time energy spectra are also discussed.

Modal Decomposition and Linear Modeling of Swirl Fluctuations in the Mixing Section of a Model Combustor Based on Particle Image Velocimetry Data

Abstract In order to determine the flame transfer function of a combustion system, different mechanisms have been identified that need to be modeled. This study focuses on the generation and propagation of one of these mechanisms, namely, the swirl fluctuations downstream of a radial swirl combustor under isothermal conditions. Swirl fluctuations are generated experimentally by imposing acoustic perturbations. Time-resolved longitudinal and crosswise particle image velocimetry (PIV) measurements are conducted inside the mixing tube and combustion chamber to quantify the evolution of the swirl fluctuations. The measured flow response is decomposed using spectral proper orthogonal decomposition to unravel the contributions of different dynamical modes. In addition a resolvent analysis is conducted based on the linearized Navier–Stokes equations to reveal the intrinsically most amplified flow structures. Both, the data-driven and analytic approach, show that inertial waves are indeed present in the flow response and an inherent flow instability downstream of the swirler, which confirms recent theoretical works on inertial waves. However, the contribution of the identified inertial waves to the total swirl fluctuations turns out to be very small. This is suggested to be due to the very structured forcing at the swirler and the additional amplification of shear-driven modes. Overall, this work confirms the presence of inertial waves in highly turbulent swirl combustors and evaluates its relevance for industry-related configurations. It further outlines a methodology to analyze and predict their characteristics based on mean fields only, which is applicable for complex geometries of industrial relevance.

Onset of absolutely unstable behaviour in the Stokes layer: a Floquet approach to the Briggs method

For steady flows, the Briggs (Electron-Stream Interaction with Plasmas. MIT Press, 1964) method is a well-established approach for classifying disturbances as either convectively or absolutely unstable. Here, the framework of the Briggs method is adapted to temporally periodic flows, with Floquet theory utilised to account for the time periodicity of the Stokes layer. As a consequence of the antiperiodicity of the flow, symmetry constraints are established that are used to describe the pointwise evolution of the disturbance, with the behaviour governed by harmonic and subharmonics modes. On coupling the symmetry constraints with a cusp-map analysis, multiple harmonic and subharmonic cusps are found for each Reynolds number of the flow. Therefore, linear disturbances experience subharmonic growth about fixed spatial locations. Moreover, the growth rate associated with the pointwise development of the disturbance matches the growth rate of the disturbance maximum. Thus, the onset of the Floquet instability (Blennerhassett & Bassom, J. Fluid Mech., vol. 464, 2002, pp. 393–410) coincides with the onset of absolutely unstable behaviour. Stability characteristics are consistent with the spatio-temporal disturbance development of the family-tree structure that has hitherto only been observed numerically via simulations of the linearised Navier–Stokes equations (Thomas et al., J. Fluid Mech., vol. 752, 2014, pp. 543–571; Ramage et al., Phys. Rev. Fluids, vol. 5, 2020, 103901).

Experimental and numerical investigation of standing-wave thermoacoustic instability under transcritical temperature conditions.

This manuscript describes an experimental and numerical investigation of transcritical thermoacoustic instability in a standing-wave setup using the refrigerant octafluoropropane (R-218) as the working fluid. Thermoacoustic instability is excited by two microtube heat exchangers separated by a vacuum-jacketed microtube stack. R-218 is allowed to flow axially through the microtubes into a closed resonator while heating and cooling fluids flow radially over the microtubes to create a temperature gradient. The fluid achieved pressure amplitudes up to 669 kPa (97 psi) at a temperature difference ΔT=Thot-Tcold of 150 K and a base pressure, P0, of 1.3 times the critical pressure (3.43 MPa). The high pressure amplitudes obtained are attributed to the strong density variations near the critical point of the working fluid. The thermoacoustic response was characterized in a set of parametric studies in which ΔT, base pressure, and resonator length were varied. A modeling approach based on linearized Navier-Stokes equations reproduces the experimental results with fair agreement. This work demonstrates promising application of transcritical working fluids to thermoacoustic engines as devices for energy extraction and waste heat removal.

An adjoint approach for computing the receptivity of the rotating disc boundary layer to surface roughness

An adjoint approach is developed to compute the receptivity of the rotating disc boundary layer to surface roughness. The adjoint linearised Navier–Stokes equations, in cylindrical coordinates, are derived and receptivity characteristics are computed for a broad range of azimuthal mode numbers using a fully equivalent velocity–vorticity formulation. For each set of flow conditions (i.e. azimuthal mode number), the adjoint method only requires that the linear and adjoint solutions be computed once. Thus, the adjoint approach offers significant computational and time advantages over alternative receptivity schemes (i.e. direct linearised Navier–Stokes) as they can be used to instantaneously compute the receptivity of boundary layer disturbances to many environmental mechanisms. Stationary cross-flow disturbances are established by randomly distributed surface roughness that is periodic in the azimuthal direction and modelled via a linearisation of the no-slip condition on the disc surface. Each roughness distribution is scaled on its respective root-mean-square. A Monte-Carlo type uncertainty quantification analysis is performed, whereby mean receptivity amplitudes are computed by averaging over many thousands of roughness realisations with variable length and wavelength filters. The amplitude of the cross-flow instability is significantly larger for roughness distributions near the conditions for neutral linear instability, while roughness elements radially outboard have a negligible effect on the receptivity process. Furthermore, receptivity increases sharply for roughness distributions that encompass wavelength scales equivalent to that associated with the cross-flow instability. Finally, mean receptivity characteristics are used to predict the radial range that stationary cross-flow vortices achieve amplitudes sufficient to invalidate the linear stability assumptions.

SPECFEM2D-DG, an open-source software modelling mechanical waves in coupled solid–fluid systems: the linearized Navier–Stokes approach

SUMMARYWe introduce SPECFEM2D-DG, an open-source, time-domain, hybrid Galerkin software modelling the propagation of seismic and acoustic waves in coupled solid–fluid systems. For the solid part, the visco-elastic system from the routinely used SPECFEM2D software is used to simulate linear seismic waves subject to attenuation. For the fluid part, SPECFEM2D-DG includes two extensions to the acoustic part of SPECFEM2D, both relying on the Navier–Stokes equations to model high-frequency acoustics, infrasound and gravity waves in complex atmospheres. The first fluid extension, SPECFEM2D-DG-FNS, was introduced in 2017 by Brissaud, Martin, Garcia, and Komatitsch; it features a nonlinear Full Navier–Stokes (FNS) approach discretized with a discontinuous Galerkin numerical scheme. In this contribution, we focus only on introducing a second fluid extension, SPECFEM2D-DG-LNS, based on the same numerical method but rather relying on the Linear Navier–Stokes (LNS) equations. The three main modules of SPECFEM2D-DG all use the spectral element method (SEM). For both fluid extensions (FNS and LNS), two-way mechanical coupling conditions preserve the Riemann problem solution at the fluid–solid interface. Absorbing outer boundary conditions (ABCs) derived from the perfectly matched layers’ approach is proposed for the LNS extension. The SEM approach supports complex topographies and unstructured meshes. The LNS equations allow the use of range-dependent atmospheric models, known to be crucial for the propagation of infrasound at regional scales. The LNS extension is verified using the method of manufactured solutions, and convergence is numerically characterized. The mechanical coupling conditions at the fluid–solid interface (between the LNS and elastodynamics systems of equations) are verified against theoretical reflection-transmission coefficients. The ABCs in the LNS extension are tested and prove to yield satisfactory energy dissipation. In an example case study, we model infrasonic waves caused by quakes occurring under various topographies; we characterize the acoustic scattering conditions as well as the apparent acoustic radiation pattern. Finally, we discuss the example case and conclude by describing the capabilities of this software. SPECFEM2D-DG is open-source and is freely available online on GitHub.

Hydrodynamic model of blood flow in major arteries pulsing in various modes.

Traditionally, the arteries in mammals are viewed as the tubes with elastic wall, whose elasticity could be slowly (during a minute) tuned to change its diameter thereby regulating regional blood supply. Recent findings showed that an artery is a much more sophisticated organ, which can change elasticity of vascular wall within a fraction of a second during a cardiac cycle due to activation of its smooth muscles manifested by generation of arterial action potentials. The rapid variations in elasticity of vascular wall resulted in three basic modes of arterial pulsing: passive, active, and intermediate. The latter is characterized by counteraction of dilation force of arterial pressure and the contractile one of smooth muscle in arterial wall, which can result in seemingly "rigid" artery of constant diameter. The prevalence of any of these forces results in active or passive pulsing modes. Existence of various pulsing modes raises the question of their effect on the main function of blood vessels, i.e., the transport of blood. The aim of this study is to assess the effect of various modes of arterial pulsing on hydraulic impedance of major arteries. The linearized Navier-Stokes equation was employed to develop a model of pulsatile flow of viscous incompressible fluid at small velocity via a conduit artery with the walls of variable or constant elasticity. An essential feature of the developed model is the shape of variable pressure drop applied to the ends of arterial segment, which simulates the real changes in arterial pressure during the heartbeat. Here, it is modeled by periodic (systolic) positive bell-shaped impulses with maxima corresponding to systolic arterial pressure, while the minimal plateau level refers to diastolic arterial pressure. The model assesses the changes in arterial hydraulic impedance during a cardiac cycle relatively to the stable level corresponding to constant blood flow driven by persistent pressure drop. Within intermediate variety of pulsing modes between the active and passive ones, the approximation of rigid arterial segment with infinite elasticity of arterial wall showed that hydraulic impedance in rigid artery is not constant due to inertial properties of the flowing blood. In passive pulsing mode characterized with constant elasticity of arterial wall, the diameter of artery changes in parallel with systolic pressure applied to the ends of arterial segment. At this, the overall change of hydraulic resistance is negative. In active pulsing mode, elasticity of arterial wall varies at different phase shifts relative to arterial pressure due to periodic contractions and relaxation of the smooth muscles in arterial wall. An important feature of active mode is possibility to decrease the hydraulic impedance during the front of arterial pressure. Various experimental modes of artery pulsing can be mathematically simulated. The passive and active modes of pulsing as well as a broad variety of intermediate pulsing modes with various phase shifts between arterial pressure and its diameter result in potency of the arteries to tune its performance in order to meet the regional circulatory requirements. The model showed that active arterial pulsing can diminish the arterial hydraulic impedance and contribute to the work needed for circulation thereby helping the pumping action of the heart.

Squirt flow in partially saturated cracks: a simple analytical model

SUMMARY Obtaining the seismic response of rocks containing cracks whose scales are much smaller than the prevailing wavelengths is a classic and important problem in rock physics. Seminal analytical models yield the seismic signatures of cracked rocks saturated with a single fluid phase. However, in a wide variety of practically relevant scenarios, cracks may be partially saturated with multiple immiscible fluids of contrasting compressibilities, such as gas and water. When a passing seismic wave deforms the medium, fluid pressure gradients arise within such partially saturated cracks, which, in turn, tend to relax through a process commonly known as squirt flow. The corresponding viscous dissipation may greatly affect the seismic amplitudes and velocities, as well as the anisotropic behaviour of the medium. To date, extensions of classical analytical models to include squirt flow occurring within isolated partially saturated cracks remain limited either in the saturation or in the frequency range. In this work, we present a simple analytical model to compute the seismic response of rocks containing partially saturated aligned cracks accounting for squirt flow effects. First, we solve the linearized Navier–Stokes equations within a partially saturated penny-shaped crack subjected to an oscillatory strain. Then, we obtain a closed analytical expression for a complex-valued frequency-dependent effective fluid bulk modulus which accounts for the stiffness variations of each crack due to squirt flow. Using classic effective medium models, together with such an effective saturating fluid, we retrieve the effective compliance matrix of the probed partially saturated cracked rock. The proposed analytical solution is validated by comparison with corresponding 3-D numerical simulations and existing analytical models.

Efficient computation of global resolvent modes

Abstract

Convergence of coprime factor perturbations for robust stabilization of Oseen systems

<p style='text-indent:20px;'>Linearization based controllers for incompressible flows have been proven to work in theory and in simulations. To realize such a controller numerically, the infinite dimensional system has to be linearized and discretized. The unavoidable consistency errors add a small but critical uncertainty to the controller model which will likely make it fail, especially when an observer is involved. Standard robust controller designs can compensate small uncertainties if they can be qualified as a coprime factor perturbation of the plant. We show that for the linearized Navier-Stokes equations, a linearization error can be expressed as a coprime factor perturbation and that this perturbation smoothly depends on the size of the linearization error. In particular, improving the linearization makes the perturbation smaller so that, eventually, standard robust controller will stabilize the system.</p>

Influence of geometry on acoustic end-corrections of slits in microslit absorbers.

The acoustic behavior of individual slits within microslit absorbers (MSAs) is investigated to explore the influence of porosity, edge geometry, slit position, and plate thickness. MSAs are plates with arrays of slit-shaped perforations, with the height of the order of the acoustic viscous boundary layer thickness, for optimized viscous dissipation. Due to hydrodynamic interaction, each slit behaves as confined in a rectangular channel. The flow within the slit is assumed to be incompressible. The viscous dissipation and the inertia are quantified by the resistive and the inertial end-corrections. These are estimated by using analytical results and numerical solutions of the linearized Navier-Stokes equations. Expressions for the end-corrections are provided as functions of the ratio of the slit height to viscous boundary layer thickness (shear number) and of the porosity. The inertial end-correction is sensitive to the far-field behavior of the flow and for low porosities strongly depends on the porosity, unlike for circular perforations. The resistive end-correction is dominated by the edge geometry of the perforation. The relative position of the slit with respect to the wall of the channel is important for distances to the wall on the order of the slit height. The plate thickness does not have a significant effect on the end-corrections.

Coupled Systems Mechanics

The paper studies the fluid flow profile contained between the orthotropic plate and rigid wall under the action of the moving load on the plate and main attention is focused on the fluid velocity profile in the load moving direction. It is assumed that the plate material is orthotropic one and the fluid is viscous and barotropic compressible. The plane-strain state in the plate and the plane flow of the fluid is considered. The motion of the plate is described by utilizing the exact equations of elastodynamics for anisotropic bodies, however, the flow of the fluid by utilizing the linearized Navier-Stokes equations. For the solution of the corresponding boundary value problem, the moving coordinate system associated with the moving load is introduced, after which the exponential Fourier transformation is employed with respect to the coordinate which indicates the distance of the material points from the moving load. The exact analytical expressions for the Fourier transforms of the sought values are obtained, the originals of which are determined numerically. Presented numerical results and their analyses are focused on the question of how the moving load acting on the face plane of the plate which is not in the contact with the fluid can cause the fluid flow and what type profile has this flow along the thickness direction of the strip filled by the fluid and, finally, how this profile changes ahead and behind with the distance of the moving load.

Dynamic Modeling of Heated Oscillatory Layer of Non-Newtonian Liquid

In this work, we have effectively used the numerical inversion of the Laplace transform to study the time-dependent thin heated film flow of a viscoelastic fluid flowing on an infinitely long flat substrate. Exact and analytical solutions are obtained in some limiting cases. The model describing this problem is a system of equations, coupling the linearized Navier–Stokes equation of the viscoelastic fluid with regard for gravity as an external force and the temperature relation for the energy profile. By assuming that the fluids are incompressible, we first derive a new system of equations, by taking into account additional terms, due to the insoluble surfactants and the viscoelastic properties. The velocity and temperature profiles are shown and the influence of coupling constant, viscoelastic parameters and the interfacial surfactants on the liquid film are discussed in detail. The validity of our solutions is verified by the numerical results to show the effects of different parameters involved and to show how the fluid flow evolves with time.

Fluid Independent Flow Determination by Surface Acoustic Wave Driven Ultrasonic Techniques

A fluid-independent ultrasonic approach for flow determination in microchannels in the harsh environment of an ultra high pressure liquid chromatography (UHPLC) system is presented. Ultrasonic waves in the fluid are excited by separate media surface acoustic waves (SAW) of Rayleigh-Wave type. The LiNbO3 SAW chip being equipped with interdigitated transducers for SAW excitation also marks the bottom of the fluid channel and thus allows for very effective SAW coupling to the fluid. The channel ceiling acts as an acoustical mirror for longitudinal ultrasonic waves propagating through the fluid. To deduce the fluid flow from the ultrasonic transmission after reflection, we employ a combination of time differential phase and time of flight measurements with a two port vector network analyzer. To verify and assign our experimental results, we use an adapted time explicit finite element method. In the simulation, both the piezoelectric single crystal and the fluid are included and we solve the linear Navier-Stokes equation to evaluate the background flow. By changing the ultrasonic propagation direction, we are able to deduce the fluid volume flow over time with very high accuracy, independent of the actual liquid in the channel.