Wavenumber Of Mode Research Articles

Strong fluid-structure coupling in a two-dimensional waveguide bounded by elastic plates

The typical model for compliant walls in a waveguide take them to be locally reacting. The corresponding boundary conditions for the transverse mode functions are Robin-type, which leads to an analysis that is a straightforward modification of the conventional analyses for rigid and pressure-release walls. The present investigation takes up the case where the walls are elastic plates, and the fluid loading is heavy. The consequence is strong coupling of the fluid and structural responses. The fact that the walls are not rigid obviates the possibility of a planar mode, so it is necessary to consider the transverse dependence of modes at any frequency. The analytically derived characteristic equation for the coupled response vividly displays the interaction of elastodynamics and acoustics. The eigenvalues are the transverse wavenumbers of the various modes at a specified frequency. They can be obtained through numerical methods, but implementing such a procedure to obtain dispersion curves of transverse wavenumber vs. frequency for each mode is cumbersome. An alternative procedure based on contours of the characteristic equation is exceptionally easy to implement, and extensible to other problems. One of the results that follow from the analysis is the observation that there is a minimum frequency below which all modes evanesce.

Acoustic features of sharp and continuous three-dimensional curved coastal fronts

Three-dimensional curved shelf-slope fronts have been modeled with sharp [Lin and Lynch, J. Acoust. Soc. Am. (2012)] and continuous [DeCourcy et al., J. Acoust. Soc. Am. (2016)] sound speed changes. For the sharp front, expressions were found for determining dependence of acoustic quantities such as modal speeds on feature-model parameters. Asymptotic approximations are used to simplify terms in the dispersion relation for the horizontal wavenumber, and further analysis leads to convenient and accurate formulas for the parameter dependence. Corresponding results will be presented for the continuous front, which possesses a different class of acoustic modes known as tunneling modes that carry larger amounts of energy across and along the front than other modes. Results for both frontal models will be discussed, comparing and contrasting characteristics including wavenumber distributions, mode types, and parameter dependences. An important goal is to examine the similarities and differences for sharp or continuous fronts in the sensitivity of acoustic quantities to feature parameters, such as front location and width, bottom slope angle, and source frequency. This would permit identification of situations where the more convenient sharp front model is appropriate. [Work supported by ONR Grants N00014-14-1-0372 and N00014-11-1-0701.]

Leak detection and location in gas pipelines by extraction of cross spectrum of single non-dispersive guided wave modes

The gas-leak-induced acoustic emission propagates along the pipe wall as a multimodal guided wave. Different modes possess variable propagation velocities and different degrees of dispersion. Thus if the cross correlation of full leak acoustic signals are used to locate a leakage, the location error is inevitably large. It will reduce the detection uncertainty and location error if using a single non-dispersive mode for leak location. The single-mode component in the cross spectrum of a leak acoustic emission can be extracted by applying the weighing window with parameters in terms of the wavenumber of a special mode. The cross spectrum of single-mode guided wave is then inversely Fourier-transformed to obtain the single-mode cross-correlation for estimating the time delay. This way, the more accurate acoustic velocity of a single mode can be used for locating a suspected gas leakage. The experimental results indicate that, for a gas pipeline, compared with the leak location by correlating the undecomposed leak acoustic emission, the average relative leak location errors are reduced by more than 7% at a distance of more than 80 m due to the enhancement of correlativity of a single mode in leak acoustic emission and the adoption of a more accurate acoustic velocity.

Vibrational properties of FeCoCrMoCBY bulk metallic glasses and their correlation with glass-forming ability

Here, the authors focus on the relationships between the vibrational properties and glass-forming ability (GFA) of an Fe-Co-Cr-Mo-C-B-Y alloy system. Among the alloy systems studied, the best glass former Fe41Co7Cr15Mo14C15B6Y2 bulk metallic glass (BMG) (Co7 alloy) showed the maximum vibrational wavenumber of acoustic and optical modes. The origin of the excellent GFA of the Co7 alloy was discussed in terms of the atomic cluster size and bonding energy.

How the vibrational frequency varies with temperature

A renormalization of mode wavenumber with temperature in inorganic (silicon, diamond) and molecular crystals (α-S8) is considered. The conventional theory of wavenumber varying is based on phonon decay into other phonons. A new approach based on the process of phonon excitation and analytical expression for ω(T) function is proposed. A numerical quantity of anharmonic contributions from vibration of chemical bonds of different types is discussed. It is shown that anharmonicity of hydrogen bonds is essentially higher than that of the other types of chemical bond. Copyright © 2016 John Wiley & Sons, Ltd.

Transient growth in Poiseuille-Rayleigh-Bénard flows of binary fluids with Soret effect

The transient growth due to non-normality is investigated for the Poiseuille-Rayleigh-Benard problem of binary fluids with the Soret effect. For negative separation factors such as ψ = −0.1, it is found that a large transient growth can be obtained by the non-normal interaction of the two least-stable-modes, i.e., the upstream and downstream modes, which determine the linear critical boundary curves for small Reynolds numbers. The transient growth is so strong that the optimal energy amplification factor G(t) is up to 102 ~ 103. While for positive separation factors such as ψ = 0.1, the transient growth is weak with the order O(1) of the amplification factor, which can even be computed by the least-stable-mode. However, for both cases, the least-stable-mode can govern the long-term behavior of the amplification factor for large time. The results also show that large Reynolds numbers have stabilization effects for the maximum amplification within moderate wave number regions. Meanwhile, much small negative or large positive separation factors and large Rayleigh numbers can enlarge the maximum transient growth of the pure streamwise disturbance with the wavenumber α = 3.14. Moreover, the initial and evolutionary two-dimensional spatial patterns of the large transient growth for the pure streamwise disturbance are exhibited with a plot of the velocity vector, spanwise vorticity, temperature, and concentration field. The initial three-layer cell vorticity structure is revealed. When the amplification factor reaches the maximum Gmax, it develops into one cell structure with large amplification for the vorticity strength.

Isogeometric analysis with trimming technique for quadruple arch‐cut ridged circle waveguide

AbstractThe conventional non‐uniform rational B‐spline (NURBS)‐based isogeometric analysis (IGA) preserves the exact description of geometrical shapes and significantly improves the computational accuracy, but suffers from the topological limitations of a single intact NURBS patch in the parameter space, which renders IGA to be applied in complex topology like the ridged structure inconvenient. In this present work, trimming technique which is the isogeometric approach for handling two‐dimensional topologically complicated geometry with a single patch is employed to demonstrate how this issue can be resolved, and further extended to analyze the quadruple arch‐cut ridged circular waveguide. The main benefit of the proposed approach is that only one patch is sufficient to represent the complex two‐dimensional topology such as circular or any degrees of polygonal shapes. Because the finite element constituents of the trimmed elements are calculated and mapped into underlying global control variables by remodeling the trimmed elements as a single patch, various properties of conventional IGA are maintained. Numerical example illustrates the flexibility for describing complicated two‐dimensional domain, and high computational accuracy and efficiency of present method with much less degree of freedom compared to commercial finite‐element software package. Effects of arch‐cut ridge dimensions on the cut‐off wave numbers of modes are investigated in details. Copyright © 2016 John Wiley & Sons, Ltd.

Hierarchical scale dependence associated with the extension of the nonlinear feedback loop in a seven-dimensional Lorenz model

Abstract. In this study, we construct a seven-dimensional Lorenz model (7DLM) to discuss the impact of an extended nonlinear feedback loop on solutions' stability and illustrate the hierarchical scale dependence of chaotic solutions. Compared to the 5DLM, the 7DLM includes two additional high wavenumber modes that are selected based on an analysis of the nonlinear temperature advection term, a Jacobian term (J(ψ, θ)), where, ψ and θ represent the streamfunction and temperature perturbations, respectively. Fourier modes that represent temperature in the 7DLM can be categorized into three major scales as the primary (the largest scale), secondary, and tertiary (the smallest scale) modes. Further extension of the nonlinear feedback loop within the 7DLM can provide negative nonlinear feedback to stabilize solutions, thus leading to a much larger critical value for the Rayleigh parameter (rc ∼ 116.9) for the onset of chaos, as compared to an rc of 42.9 for the 5DLM as well as an rc of 24.74 for the 3DLM. The rc is determined by an analysis of ensemble Lyapunov exponents (eLEs) with a Prandtl number (σ) of 10. To examine the dependence of rc on the value of the Prandtl number, a linear stability analysis is performed near the nontrivial critical point using a wide range of the Rayleigh parameter (40 ≤ r ≤ 195) and the Prandtl number (5 ≤ σ ≤ 25). Then an eLE analysis is conducted using selected values of the Prandtl number. The linear stability analysis is done by solving for the analytical solutions of the critical points, by linearizing the 7DLM with respect to the analytical solutions, and by calculating the eigenvalues of the linearized system. Within the range of (5 ≤ σ ≤ 25), the 7DLM requires a larger rc for the onset of chaos than the 5DLM. In addition to the negative nonlinear feedback illustrated and emulated by the quasi-equilibrium state solutions for high wavenumber modes, the 7DLM reveals the hierarchical scale dependence of chaotic solutions. For chaotic solutions with r = 120, the Pearson correlation coefficients (PCCs) between the primary and secondary modes (i.e., Z and Z1) and between the secondary and tertiary modes (i.e., Z1 and Z2) are 0.988 and 0.998, respectively. Here, Z, Z1, and Z2 represent the time-varying amplitudes of the primary, secondary, and tertiary modes, respectively. High PCCs indicate a strong linear relationship among the modes at various scales and a hierarchy of scale dependence. Future work will be undertaken to examine how higher-dimensional LMs may produce a larger critical value for the Rayleigh parameter for the onset of chaos and reveal stronger hierarchical scale dependence.

Density functional theory study of the interaction of H2O, CO2 and CO with the ZrO2 (111), Ni/ZrO2 (111), YSZ (111) and Ni/YSZ (111) surfaces

The triple phase boundary (TPB), where the gas phase, Ni particles and the yttria-stabilised zirconia (YSZ) surface meet, plays a significant role in the performance of solid oxide fuel cells (SOFC). Indeed, the key reactions take place at the TPB, where molecules such as H2O, CO2 and CO interact and react. We have systematically studied the interaction of H2O, CO2 and CO with the dominant surfaces of four materials that are relevant to SOFC, i.e. ZrO2(111), Ni/ZrO2(111), YSZ(111) and Ni/YSZ(111) of cubic ZrO2 stabilized with 9% of yttria (Y2O3). The study employed spin polarized density functional theory (DFT), taking into account the long-range dispersion forces. We have investigated up to five initial adsorption sites for the three molecules and have identified the geometries and electronic structures of the most stable adsorption configurations. We have also analysed the vibrational modes of the three molecules in the gas phase and compared them with the adsorbed molecules. A decrease of the wavenumbers of the vibrational modes for the three adsorbed molecules was observed, confirming the influence of the surface on the molecules' intra-molecular bonds. These results are in line with the important role of Ni in this system, in particular for the CO adsorption and activation.

Space-time characteristics of wall-pressure and wall shear-stress fluctuations in wall-modeled large eddy simulation.

We report the space-time characteristics of the wall-pressure fluctuations and wall shear-stress fluctuations from wall-modeled large eddy simulation (WMLES) of a turbulent channel flow at Re τ = 2000. Two standard zonal wall models (equilibrium stress model and nonequilibrium model based on unsteady RANS) are employed, and it is shown that they yield similar results in predicting these quantities. The wall-pressure and wall shear-stress fields from WMLES are analyzed in terms of their r.m.s. fluctuations, spectra, two-point correlations, and convection velocities. It is demonstrated that the resolution requirement for predicting the wall-pressure fluctuations is more stringent than that for predicting the velocity. At least δ/Δx > 20 and δ/Δz > 30 are required to marginally resolve the integral length scales of the pressure-producing eddies near the wall. Otherwise, the pressure field is potentially aliased. Spurious high wave number modes dominate in the streamwise direction, and they contaminate the pressure spectra leading to significant overprediction of the second-order pressure statistics. When these conditions are met, the pressure statistics and spectra at low wave number or low frequency agree well with the DNS and experimental data. On the contrary, the wall shear-stress fluctuations, modeled entirely through the RANS-based wall models, are largely underpredicted and relatively insensitive to the grid resolution. The short-time, small-scale near-wall eddies, which are neither resolved nor modeled adequately in the wall models, seem to be important for accurate prediction of the wall shear-stress fluctuations.

Wave and Jet Maintenance in Different Flow Regimes

Abstract The wave spectrum and zonal-mean-flow maintenance in different flow regimes of the jet stream are studied using a two-layer modified quasigeostrophic (QG) model. As the wave energy is increased by varying the model parameters, the flow transitions from a subtropical jet regime to a merged jet regime and then to an eddy-driven jet regime. The subtropical jet is maintained at the Hadley cell edge by zonal-mean advection of momentum, while eddy heat flux and eddy momentum flux convergence (EMFC) are weak and concentrated far poleward. The merged jet is narrow and persistent and is maintained by EMFC from a narrow wave spectrum, dominated by zonal wavenumber 5. The eddy-driven jet is wide and fluctuating and is maintained by EMFC from a wide wave spectrum. The wave–mean flow feedback mechanisms that maintain each regime are explained qualitatively. The regime transitions are related to transitions in the wave spectrum. An analysis of the wave energy spectrum budget and a comparison with a quasi-linear version of the model show that the balance maintaining the spectrum in the merged and subtropical jet regimes is mainly a quasi-linear balance, whereas in the eddy-driven jet regime nonlinear inverse energy cascade takes place. The amplitude and wavenumber of the dominant wave mode in the merged and subtropical jet regimes are determined by the constraint that this mode would produce the wave fluxes necessary for maintaining a mean flow that is close to neutrality. In contrast, the equilibrated mean flow in the eddy-driven jet regime is weakly unstable.

Characterizing Wave Behavior in a Beam Experiment by Using Complex Orthogonal Decomposition

Abstract Complex orthogonal decomposition (COD) is applied to an experimental beam to extract the dispersive wave properties from response measurements. The beam is made of steel and is rectangular with a constant cross section. One end of the beam is free and is hung by a soft elastic cord. An impulse is applied to the free-end. The other end is buried in sand to absorb the wave as it travels from the impact site on the free-end; this effectively prevents reflections of the wave off the buried end and emulates a semi-infinite beam. The beam response is measured with an array of accelerometers, whose signals are integrated to obtain an ensemble of displacement signals. Acceleration responses are also compared in the frequency domain to predictions from the Euler–Bernoulli model. COD is applied to the displacement ensemble to obtain complex modal vectors and associated complex modal coordinates (COCs). The spatial whirl rates of nearly harmonic modal vectors are used to extract the modal wave numbers, and the temporal whirl rates of the modal coordinates are used to estimate the modal frequencies. The dispersion relationship between the frequencies and wave numbers compare favorably to those of the theoretical infinite Euler–Bernoulli beam.

Turbulent transport with intermittency: Expectation of a scalar concentration

Scalar transport by turbulent flows is best described in terms of Lagrangian parcel motions. Here we measure the Eulerian distance travel along Lagrangian trajectories in a simple point vortex flow to determine the probabilistic impulse response function for scalar transport in the absence of molecular diffusion. As expected, the mean squared Eulerian displacement scales ballistically at very short times and diffusively for very long times, with the displacement distribution at any given time approximating that of a random walk. However, significant deviations in the displacement distributions from Rayleigh are found. The probability of long distance transport is reduced over inertial range time scales due to spatial and temporal intermittency. This can be modeled as a series of trapping events with durations uniformly distributed below the Eulerian integral time scale. The probability of long distance transport is, on the other hand, enhanced beyond that of the random walk for both times shorter than the Lagrangian integral time and times longer than the Eulerian integral time. The very short-time enhancement reflects the underlying Lagrangian velocity distribution, while that at very long times results from the spatial and temporal variation of the flow at the largest scales. The probabilistic impulse response function, and with it the expectation value of the scalar concentration at any point in space and time, can be modeled using only the evolution of the lowest spatial wave number modes (the mean and the lowest harmonic) and an eddy based constrained random walk that captures the essential velocity phase relations associated with advection by vortex motions. Preliminary examination of Lagrangian tracers in three-dimensional homogeneous isotropic turbulence suggests that transport in that setting can be similarly modeled.

Modal wavenumber estimation using a Bayesian compressed sensing algorithm

In shallow water, low-frequency acoustic propagation is described by modal theory. In this context, the environment acts as a dispersive waveguide, and the geoacoustic properties of the seabed can be inferred from the modal wavenumbers. When considering horizontal aperture (using a horizontal array or a towed source), wavenumber estimation is a well-known problem, equivalent to spectral analysis. In this paper, wavenumber estimation is revisited using Compressed Sensing (CS). Our method benefits from two strong physical hypotheses. Only a few modes are propagating so that the wavenumber spectrum is sparse. Moreover, if the source is broadband, the wavenumbers can be related from one frequency to the next using a general dispersion relationship. Our method resorts to a state-of-the-art Bayesian algorithm exploiting a Bernoulli-Gaussian model. The latter, well-suited to the sparse representations, makes possible a natural integration of the prior dispersion information through a wise choice of the Bernoulli parameters. The whole methodology is assessed on simulated data and successfully applied on marine data.

Excitation of superharmonics by internal modes in non-uniformly stratified fluid

Theory and numerical simulations show that the nonlinear self-interaction of internal modes in non-uniform stratification results in energy being transferred to superharmonic disturbances forced at twice the horizontal wavenumber and frequency of the parent mode. These disturbances are not in themselves a single mode, but a superposition of modes such that the disturbance amplitude is largest where the change in the background buoyancy frequency with depth is largest. Through weakly nonlinear interactions with the parent mode, the disturbances evolve to develop vertical-scale structures that distort and modulate the parent mode. Because pure resonant wave triads do not exist in non-uniformly stratified fluid, parametric subharmonic instability (PSI) is not evident even though noise is superimposed upon the initial state. The results suggest a new mechanism for energy transfer to dissipative scales (from large to small vertical scale and with frequencies larger and smaller than that of the parent mode) through forcing superharmonic rather than subharmonic disturbances.

Lattice dynamics and high-pressure Raman scattering studies of CoTeMoO6 crystal

We have performed lattice dynamics calculations for CoTeMoO6 telluromolybdate and proposed assignment of modes to respective vibrations of atoms in the unit cell. We have also performed pressure-dependent Raman scattering studies that revealed unusual negative pressure dependence of the highest wavenumber modes. We provide explanation of this behavior based on strongly layered nature of CoTeMoO6 crystal. High-pressure data also indicate weak compressibility of the TeO4 groups associated with large changes in the MoOTe bridges. The spectra remain qualitatively the same up to 6.3GPa but some very subtle changes in the spectra have been observed near 2.6–3.0GPa that might indicate onset of a very subtle structural transformation in the studied crystal.

Pattern Formation in Keller–Segel Chemotaxis Models with Logistic Growth

In this paper, we investigate pattern formation in Keller–Segel chemotaxis models over a multidimensional bounded domain subject to homogeneous Neumann boundary conditions. It is shown that the positive homogeneous steady state loses its stability as chemoattraction rate [Formula: see text] increases. Then using Crandall–Rabinowitz local theory with [Formula: see text] being the bifurcation parameter, we obtain the existence of nonhomogeneous steady states of the system which bifurcate from this homogeneous steady state. Stability of the bifurcating solutions is also established through rigorous and detailed calculations. Our results provide a selection mechanism of stable wavemode which states that the only stable bifurcation branch must have a wavemode number that minimizes the bifurcation value. Finally, we perform extensive numerical simulations on the formation of stable steady states with striking structures such as boundary spikes, interior spikes, stripes, etc. These nontrivial patterns can model cellular aggregation that develop through chemotactic movements in biological systems.

Spectral Deferred Corrections with Fast-wave Slow-wave Splitting

The paper investigates a variant of semi-implicit spectral deferred corrections (SISDC) in which the stiff, fast dynamics correspond to fast propagating waves (``fast-wave slow-wave problem''). We show that for a scalar test problem with two imaginary eigenvalues $i \lambda_{\text{f}}$, $i \lambda_{\text{s}}$, having $\Delta t ( | \lambda_{\text{f}} | + | \lambda_{\text{s}} | ) < 1$ is sufficient for the fast-wave slow-wave SDC (fwsw-SDC) iteration to converge and that in the limit of infinitely fast waves the convergence rate of the nonsplit version is retained. Stability function and discrete dispersion relation are derived and show that the method is stable for essentially arbitrary fast-wave CFL numbers as long as the slow dynamics are resolved. The method causes little numerical diffusion and its semidiscrete phase speed is accurate also for large wave number modes. Performance is studied for an acoustic-advection problem and for the linearised Boussinesq equations, describing compressible, stratified flow. fwsw-SDC is compared to diagonally implicit Runge--Kutta (DIRK) and implicit-explicit (IMEX) Runge--Kutta methods and found to be competitive in terms of both accuracy and cost.

Numerical Modelling of Sonicated, Continuous Transesterification and Evaluation of Reaction Kinetics for Optimizing Biodiesel Reactor Design

Biodiesel is an alternative and sustainable fuel that can reduce the dependence on fossil diesel. This commodity has not only been promoted in the developed world but also in Indonesia, Brazil and several developing counties. In a general procedure it is a product of a transesterification reaction of vegetable oils or waste cooking oils with an alcohol, in the presence of an acidic or basic catalyst. It is a slow reaction which is conventionally carried out in a mechanically stirred batch process. An advanced method to achieve high yield quality in less time is sonication of the reaction in an integrated continuous process. Sonication causes micro-cavitation in the reactant mixture. The cavitation bubbles can have an internal pressure and temperature as high as 1000 atm and 5000K, respectively. Violent collapse of these bubbles causes tremendous increase in mass transfer, thereby enhancing the reaction rates [1]. To optimize the design of the reactor, high fidelity modeling assisted design is pursued. This enables the effective integration of the reactant transport and the sonication effect in a coupled acoustic, reactive multiple-specie flow. In this work, a cylindrical reactor is considered in which reactant mixture will be circulated and sonicated by a sonotrode type ultrasound equipment. To simulate the sonication effect the linear, time independent wave equation is solved for the fluid domain, which provides us with the acoustic pressure variation in the fluid. To account for the attenuation of the wave due to cavitation bubbles the modified wave number is used. To account for the chemical reactions, laminar reacting flow is assumed for the reactant mixture for which the Navier-Stokes equations and transport equation for dilute species is solved for the fluid. A logical reaction rate coupling model, which is dependent on the acoustic pressure and flow velocity, is used to evaluate the kinetics of the reaction, which are to be used as a judging factor for the reactor design [2].

High-temperature Raman study of L-alanine, L-threonine and taurine crystals related to thermal decomposition

In this work high-temperature Raman spectra are used to compare temperature dependence of the lattice mode wavenumber of L-alanine, L-threonine and taurine crystals. Anharmonic effects observed are associated with intermolecular N-H· · ·O hydrogen bond that plays an important role in thermal decomposition process of these materials. Short and strong hydrogen bonds in L-alanine crystal were associated with anharmonic effects in lattice modes leading to low thermal stability compared to taurine crystals. Connection between thermal decomposition process and anharmonic effects is furnished for the first time.