On the T-matrix for water-wave scattering problems

A rigid cylinder of infinite length is floating in the free surface of deep water. The cylinder is held fixed and a given time-harmonic wave of small amplitude is incident upon it. The corresponding linear two-dimensional boundary-value problem for a velocity potential φ is treated using the null-field method, and an expression for the T-matrix is obtained. (The T-matrix connects the diffraction potential away from the cylinder to the given incident potential.) Fundamental properties of the T-matrix are derived from considerations of energy and reciprocity. For regular wavetrains incident from the right or from the left, there are well-known relations between the corresponding reflection and transmission coefficients; these relations are recovered by specialising the equations satisfied by the T-matrix. Two extensions to water of constant finite depth are described: one uses multipole potentials whilst the other uses Havelock wavemaker functions; this second approach also leads to a new method for treating the problem of waves in a semi-infinite channel with an end-wall of arbitrary shape.     

Experimental investigation of the turbulent wake past real seabed elements for velocity variations characterization in the water column.

In high flow velocity areas like those suitable for marine energy application, bathymetry variations create strong velocity fluctuations in the water column. It is therefore essential to characterize the turbulence evolution in the wake of seabed elements which may impact the loads on tidal turbines. For that purpose, experiments are carried out in a flume tank with R_e as high as achievable in Froude similitude, with bathymetry variations experimentally represented with various wall-mounted square elements of height H: a cylinder or a cube as unitary obstacles and combinations of these elements followed by an inclined floor to resemble smooth bathymetry changes. The onset flow is a simple boundary layer profile with height 1.3 H and a low turbulence intensity. PIV and LDV measurements are used to investigate the wake past all test cases in order to distinguish high floor elevation cases (unitary obstacles) from mean roughness effect (obstacle combinations). Results show that the obstacle combinations produce a wake less extended than for a single wide cylinder that produces an extended wake and very energetic turbulent events. With a single cube, no downstream development of large turbulent events exist and the wake reduces by a factor of 3 compared to the wake cylinder case. An inclined floor downstream of a single wall-mounted obstacle reduces its wake length but does not alter the turbulent structures shed. Turbulent velocity profiles extracted from every wake topology investigated are also compared. The general conclusion is that: for small aspect ratio cases, the obstacle will not affect the water column. On the contrary, strong energetic turbulent events are emitted from large aspect ratio obstacles. Combinations cases stand in-between.     

Instability and friction

A review on the stability analysis of solids in unilateral and frictional contact is given. The presentation is focussed on the stability of an equilibrium position of an elastic solid in frictional contact with a fixed or moving obstacle. The problem of divergence instability and the obtention of a criterion of static stability are discussed first for the case of a fixed obstacle. The possibility of flutter instability is then considered for a steady sliding equilibrium with a moving obstacle. The steady sliding solution is generically unstable by flutter and leads to a dynamic response which can be chaotic or periodic. This dynamic response leads to the generation of stick–slip–separation waves on the contact surface in a similar way as Schallamach waves in statics. Illustrating examples and principal results recently obtained in the literature are reported. Some problems of friction-induced vibration and noise emittence, such as brake squeal for example, can be interpreted in this spirit. To cite this article: Q.S. Nguyen, C. R. Mecanique 331 (2003).     

On the development of lift and drag in a rotating and translating cylinder

The two-dimensional flow around a rotating cylinder is investigated numerically using a vorticity forces formulation with the aim of analyzing quantitatively the flow structures, and their evolutions, that contribute to the lift and drag forces on the cylinder. The Reynolds number considered, based on the cylinder diameter and steady free stream speed, is Re=200, while the non-dimensional rotation rate (ratio of the surface speed and free stream speed) selected was α=1 and 3. For α=1 the wake behind the cylinder for the fully developed flow is oscillatory due to vortex shedding, and so are the lift and drag forces. For α=3 the fully developed flow is steady with constant (high) lift and (low) drag. Each of these cases is considered in two different transient problems, one with angular acceleration of the cylinder and constant speed, and the other one with translating acceleration of the cylinder and constant rotation. We characterize quantitatively the contributions of individual fluid elements (vortices) to aerodynamic forces, explaining and quantifying the mechanisms by which the lift is generated in each case. In particular, for high rotation (when α=3), we explain the relation between the mechanisms of vortex shedding suppression and those by which the lift is enhanced and the drag is almost suppressed when the fully developed flow is reached.     

The effect of a wake-mounted splitter plate on the flow around a surface-mounted finite-height circular cylinder

The influence of a wake-mounted splitter plate on the flow around a surface-mounted circular cylinder of finite height was investigated experimentally using a low-speed wind tunnel. The experiments were conducted at a Reynolds number of Re=7.4×10⁴ for cylinder aspect ratios of AR=9, 7, 5 and 3. The thickness of the boundary layer on the ground plane relative to the cylinder diameter was δ/D=1.5. The splitter plates were mounted on the wake centreline with negligible gap between the base of the cylinder and the leading edge of the plate. The lengths of the splitter plates, relative to the cylinder diameter, ranged from L/D=1 to 7, and the plate height was always equal to the cylinder height. Measurements of the mean drag force coefficient were obtained with a force balance, and measurements of the vortex shedding frequency were obtained with a single-component hot-wire probe situated in the wake of the cylinder–plate combination. Compared to the well-studied case involving an infinite circular cylinder, the splitter plate was found to be a less effective drag-reduction device for finite circular cylinders. Significant reduction in the mean drag coefficient was realized only for the finite circular cylinder of AR=9 with intermediate-length splitter plates of L/D=1–3. The mean drag coefficients of the other cylinders were almost unchanged. In terms of its effect on vortex shedding, a splitter plate of sufficient length was able to suppress Kármán vortex shedding for all of the finite circular cylinders tested. For AR=9, vortex shedding suppression occurred for L/D≥5, which is similar to the case of the infinite circular cylinder. For the smaller-aspect-ratio cylinders, however, the splitter plate was more effective than what occurs for the infinite circular cylinder: for AR=3, vortex shedding suppression occurred for all of the splitter plates tested (L/D≥1); for AR=5 and 7, vortex shedding suppression occurred for L/D≥1.5.     

Free convection heat transfer from a horizontal fin attached cylinder between confined nearly adiabatic walls

Steady state two-dimensional free convection heat transfer from a horizontal, isothermal fin attached cylinder, located between nearly two adiabatic walls is studied experimentally using a Mach–Zehnder interferometer. Effects of the walls inclination angel (θ) on heat transfer from the cylinder is investigated for Rayleigh number ranging from 1000 to 15,500. Two cylinders with different diameters of D = 10 and 20 mm are used to cover wide Rayleigh range. Results indicate that, heat transfer phenomena differ for different Rayleigh number. For Rayleigh numbers lower than 5500, heat transfer rate from cylinder surface is lower than the heat transfer from a single cylinder. In this range by the use of walls, heat transfer from the cylinder decreases slightly and walls’ inclination does not change heat transfer rate from the cylinder surface. For Rayleigh number ranging from 5500 to 15,500, amount of heat transfer from the cylinder surface is less than that of a single cylinder. However, by adding nearly adiabatic walls to experimental model heat transfer mechanism differs and chimney effect between fin and walls increases the heat transfer rate from the cylinder surface. By increasing the walls inclination angel from 0° to 20°, the chimney effect between walls and fin diminishes and heat transfer rate from the cylinder surface is approaching to the heat transfer rate of fin attached cylinder without adiabatic walls.     

Transient thermoelastic waves in an anisotropic hollow cylinder due to localized heating

This paper presents a theoretical study of transient ultrasonic guided waves generated by concentrated heating of the outer surface of an infinite anisotropic hollow circular cylinder. Generalized thermoelastic theory proposed by Lord and Shulman is adopted to model the dynamic thermoelastic behavior of the cylinder. The concentrated heat source model used is to represent heating due to a pulsed laser beam, which is focused on the outer surface of the cylinder. A semi-analytical finite element (SAFE) method is employed to evaluate guided wave modes in the cylinder. Using integral transform techniques, the modal wave forms are obtained in frequency and wave number domains. Time histories of the propagating modes are then calculated by applying inverse Fourier transformation in the time domain. Numerical results showing the dispersion curves for the group velocities of the propagating modes and transient radial displacements are presented. For this purpose it is assumed that the cylinder is made of transversely isotropic silicon nitride (Si₃N₄). Attention is focused on the propagation characteristics of longitudinal and flexural modes separately.     

FLOW-INDUCED VIBRATION OF A CIRCULAR CYLINDER AT LIMITING STRUCTURAL PARAMETERS

Transverse oscillation of a dynamically supported circular cylinder in a flow at Re=100 has been numerically simulated using a high-resolution viscous-vortex method, for a range of dynamical parameters. At the limiting case with zero values of mass, damping and elastic force, the cylinder oscillates sinusoidally at amplitudeA /D=0·47 and frequency fD/U_∞=0·156. For zero damping, the effects of mass and elasticity are combined into a new, “effective” dynamic parameter, which is different from the classic “reduced velocity”. Over a range of this parameter, the response exhibits oscillations at amplitudes up to 0·6 and frequencies between 0·15 and 0·2. From this response function, the classic response in terms of reduced velocity can be obtained for fixed values of the cylinder/fluid ratio m^*. It displays “lock-in” at very high values of m^*.     

The mathematical model of reflection and refraction of longitudinal waves in thermo-piezoelectric materials

In this paper, the basic equations of motion, of Gauss and of heat conduction, together with constitutive relations for pyro- and piezoelectric media, are presented. Three thermoelastic theories are considered: classical dynamical coupled theory, the Lord–Shulman theory with one relaxation time and Green and Lindsay theory with two relaxation times. For incident elastic longitudinal, potential electric and thermal waves, referred to as qP, φ-mode and T-mode waves, which impinge upon the interface between two different transversal isotropic media, reflection and refraction coefficients are obtained by solving a set of linear algebraic equations. A case study is investigated: a system formed by two semi-infinite, hexagonal symmetric, pyroelectric–piezoelectric media, namely Cadmium Selenide (CdSe) and Barium Titanate (BaTiO₃). Numerical results for the reflection and refraction coefficients are obtained, and their behavior versus the incidence angle is analyzed. The interaction with the interface give rises to different kinds of reflected and refracted waves: (i) two reflected elastic waves in the first medium, one longitudinal (qP-wave) and the other transversal (qSV-wave), and a similar situation for the refracted waves in the second medium; (ii) two reflected potential electric waves and a similar situation for the refracted waves; (iii) two reflected thermal waves and a similar situation for the refracted waves. The amplitudes of the reflected and refracted waves are functions of the incident angle, of the thermal relaxation times and of the media elastic, electric, thermal constants. This study is relevant to signal processing, sound systems, wireless communications, surface acoustic wave devices and military defense equipment.     

GROUND EFFECT OF FLOW AROUND AN ELLIPTIC CYLINDER IN A TURBULENT BOUNDARY LAYER

The flow characteristics around an elliptic cylinder with an axis ratio of AR=2 located near a flat plate were investigated experimentally. The elliptic cylinder was embedded in a turbulent boundary layer whose thickness is larger than the cylinder height. For comparison, the same experiment was carried out for a circular cylinder having the same vertical height. The Reynolds number based on the height of the cylinder cross-section was 14000. The pressure distributions on the cylinder surface and on the flat plate were measured for various gap distances between the cylinder and the plate. The wake velocity profiles behind the cylinder were measured using hot-wire anemometry. In the near-wake region, the vortices are shed regularly only when the gap ratio is greater than the critical value of G/B=0·4. The critical gap ratio is larger than that of a circular cylinder. The variation of surface pressure distributions on the elliptic cylinder with respect to the gap ratio is much smaller than that on the circular cylinder. This trend is more evident on the upper surface than the lower one. The surface pressures on the flat plate recover faster than those for the case of the circular cylinder at downstream locations. As the gap ratio increases, the drag coefficient of the cylinder itself increases, but the lift coefficient decreases. For all gap ratios tested in this study, the drag coefficient of the elliptic cylinder is about half that of the circular cylinder. The ground effect of the cylinder at small gap ratio constrains the flow passing through the gap, and restricts the vortex shedding from the cylinder, especially in the lower side of the cylinder wake. This constraint effect is more severe for the elliptic cylinder, compared to the circular cylinder. The wake region behind the elliptic cylinder is relatively small and the velocity profiles tend to approach rapidly to those of a flat plate boundary layer     

Wake-induced vibrations of an elastically mounted cylinder located downstream of a stationary larger cylinder at low Reynolds numbers

Two-dimensional numerical simulations of flow past two unequal-sized circular cylinders in tandem arrangement are performed at low Reynolds numbers (Re). The upstream larger cylinder is stationary, while the downstream cylinder has both one (transverse-only) and two (transverse and in-line) degrees of freedom (1-dof and 2-dof, respectively). The Re, based on the free stream velocity U_∞ and the downstream cylinder diameter d, varies between 50 and 200 with a wide range of reduced velocities U_r. The diameter of the upstream cylinder is twice that of the downstream cylinder, and the center-to-center spacing is 5.5d. In general, for the 1-dof case, the calculations show that the wake-induced vibrations (WIV) of the downstream cylinder are greatly amplified when compared to the case of a single cylinder or two equal-sized cylinders. The transverse amplitudes build up to a significantly higher level within and beyond the lock-in region, and the U_r associated with the peak amplitude shifts toward a higher value. The dominant wake pattern is 2S mode for Re=50 and 100, while with the increase of Re to 150 and 200, the P+S mode can be clearly observed at some lower U_r. For the 2-dof vibrations, the transverse response characteristics are similar to those presented in the corresponding 1-dof case. The in-line responses are generally much smaller, except for several significant vibrations resulting from in-line resonance. The obvious in-line vibration may induce a C (chaotic) vortex shedding mode for higher Re (Re=200). With regard to the 2-dof motion trajectories, besides the typical figure-eight pattern, several odd patterns such as figure-double eight and single-looped trajectories are also obtained due to the wake interference effect.     

Elastodynamic behavior of mechanical cloaks designed by direct lattice transformations

Steering waves in elastic solids is more demanding than steering waves in electromagnetism or acoustics. As a result, designing material distributions which are the counterpart of optical invisibility cloaks in elasticity poses a major challenge. Waves of all polarizations should be guided around an obstacle to emerge on the downstream side as though no obstacle were there. Recently, we have introduced the direct-lattice-transformation approach. This simple and explicit construction procedure led to extremely good cloaking results in the static case. Here, we transfer this approach to the dynamic case, i.e., to elastic waves or phonons. We demonstrate broadband reduction of scattering, with best suppressions exceeding a factor of five when using cubic coordinate transformations instead of linear ones. To reliably and quantitatively test these cloaks efficiency, we use an effective-medium approach.     

On a bound for the frequency of surface waves trapped near a cylinder spanning a channel

There has been substantial effort recently put into proving, for a variety of different geometries, the existence of trapped waves, that is unforced time-harmonic motions which do not radiate energy to large distances. Thus it is known that such motions can exist in a deep channel which includes a cylinder spanning the channel, for various shapes of cylinder.The converse problem of proving the absence of such trapped waves has received much less consideration, and the only relevant uniqueness proof for a channel spanned by a cylinder is that of McIver (1991). In an appendix to that paper, McIver demonstrates that no trapped-wave motions can exist for the case in which the cylinder is surface piercing and is entirely contained between vertical planes through the free-surface intersections. This is exactly the same geometrical condition which John (1950) found would ensure uniqueness in water-wave radiation and scattering problems, in finite or infinite depth. Both John and McIver achieved their uniqueness results by consideration of integrals of the potential along vertical lines down from the free surface.John's work was extended by Simon and Ursell (1984) who established uniqueness for a wider class of two-dimensional radiation and scattering problems by consideration of integrals along nonvertical lines. The work presented in this paper is the corresponding extension of McIver's work; although this extension does not rule out trapped waves at all frequencies for any geometry except that already considered by McIver, it does yield an easy lower bound for the ratio of the trapped-mode frequency to the cut-off frequency, in finite or infinite depth.     

Scattering of acoustic waves by elastic and viscoelastic obstacles immersed in a fluid

A scattering or T-matrix approach is presented for studying the scattering of acoustic waves by elastic and viscoelastic obstacles immersed in a fluid. A Kelvin-Voigt model is used to obtain the complex elastic moduli of the viscoelastic solid. The T-matris formulation is somewhat complicated because the wave equations and fields are quite different in the solid and fluid regions and are coupled by continuity conditions at the interface. We have obtained fairly extensive numerical results for prolate and oblate spheroids for a variety of scattering geometries. The backscattering, bistatic, absorption and extinction cross-section are presented as a function of the frequency of the incident wave.     

Coherent vortical and straining structures in the finite wall-mounted square cylinder wake

Topological aspects of the turbulent wake of a finite, surface-mounted, square-cross-section cylinder of h/d = 4 are addressed by decomposing the velocity field into a quasi-periodic coherent part and the unresolved incoherent fluctuations. The three-dimensional large scale structure is educed through a reconstruction of planar phase-averaged PIV measurements using the simultaneously sampled surface pressure difference on opposing sides of the obstacle as a phase reference. A topological model for the vortex structure is educed and mean streamwise wake vorticity is explained in terms of the connections between initially vertical structures shed alternately from either side of the obstacle, rather than previously proposed ‘tip’ vortex structures generated at the obstacle free-end. The coherent structure educed accounts for a significant portion of the fluctuating energy in the wake. The turbulent field is further analyzed by finding Lagrangian straining structures that form by induction of the coherent vorticity field, and these structures are related to the energy transfer from the base phase-averaged flow since they act to stretch incoherent vorticity fluctuations in their neighbourhood.     

Critical effects of a spanwise surface wire on flow past a circular cylinder and the significance of the wire size and Reynolds number

An experimental study is conducted on flow past a circular cylinder fitted with a single spanwise wire on its surface. The work investigates the dependency of the critical wire locations on the wire size and Reynolds number, and examines the near wake and vortex shedding characteristics in an effort to advance the understanding of the critical wire effects beyond the existing literature. The Reynolds number is varied from 5000 to 30 000, and the wire diameter is varied from 2.9% to 5.9% of the cylinder diameter. All wires are larger than the boundary-layer thickness forming around a comparable smooth cylinder. Constant Temperature Anemometry and hydrogen bubble visualization are used as the flow diagnostic tools. The frequency and strength of the Karman instability are shown to vary with the wire location at any given Reynolds number nearly in an inverse fashion. For all the Reynolds numbers and wire sizes considered, two types of critical locations are shown to exist on the cylinder surface for the application of a wire. These locations are associated with the attenuation and amplification of the Karman instability, and in accord with the existing literature, are denoted as θ_c1 and θ_c2, respectively. The present work reveals that θ_c2 consists of a wide range of locations which remains unaffected from the wire size and Reynolds number, while θ_c1 is a relatively distinct location on the cylinder surface and depends on both the Reynolds number and wire size. For a given Reynolds number, increasing the wire size decreases θ_c1. For a given wire size, increasing the Reynolds number from 5000 to 15 000 increases θ_c1, and past 15 000, θ_c1 remains unaffected from the Reynolds number. When a wire is at θ_c1, even though, for the majority of the time the regular formation of Karman vortices ceases, the present data also reveals intermittent, short time periods where the regular shedding resumes.     

Self-switching of displacement waves in elastic nonlinearly deformed materialsSelf-commutation d'ondes de déplacement dans les milieux élastiques non-linéaires

The problem of self-switching plane waves in elastic nonlinearly deformed materials is formulated. Reduced and evolution equations, which describe the interaction of two waves the power pumping wave and the faint signal wave are obtained. For the case of wave numbers matching the pumping and signal waves, a procedure of finding the exact solution of evolution equations is described. The solution is expressed by elliptic Jacobi functions. The existence of the power wave self-switching is shown and commented. To cite this article: J. Rushchitsky, C. R. Mecanique 330 (2002) 175–180.     

Experimental investigation of flow-induced vibration on isolated and tandem circular cylinders fitted with strakes

The effect of varying the geometric parameters of helical strakes on vortex-induced vibration (VIV) is investigated in this paper. The degree of oscillation attenuation or even suppression is analysed for isolated circular cylinder cases. How a cylinder fitted with strakes behaves when immersed in the wake of another cylinder in tandem arrangement is also investigated and these results are compared to those with a single straked cylinder. The experimental tests are conducted at a circulating water channel facility and the cylindrical models are mounted on a low-damping air bearing elastic base with one degree-of-freedom, restricted to oscillate in the transverse direction to the channel flow. Three strake pitches (p) and heights (h) are tested: p=5, 10, 15d, and h=0.1, 0.2, 0.25d. The mass ratio is 1.8 for all models. The Reynolds number range is from 1000 to 10 000, and the reduced velocity varies up to 21. The cases with h=0.1d strakes reduce the amplitude response when compared to the isolated plain cylinder, however the oscillation still persists. On the other hand, the cases with h=0.2, 0.25d strakes almost completely suppress VIV. Spanwise vorticity fields, obtained through stereoscopic digital particle image velocimetry (SDPIV), show an alternating vortex wake for the p=10d and h=0.1d straked cylinder. The p=10d and h=0.2d cylinder wake has separated shear layers with constant width and no roll-up close to the body. The strakes do not increase the magnitude of the out-of-plane velocity compared to the isolated plain cylinder. However, they deflect the flow in the out-of-plane direction in a controlled way, which can prevent the vortex shedding correlation along the span. In order to investigate the wake interference effect on the strake efficiency, an experimental arrangement with two cylinders in tandem is employed. The centre-to-centre distance for the tandem arrangement varies from 2 to 6. When the downstream p=10d and h=0.2d cylinder is immersed in the wake of an upstream fixed plain cylinder, it loses its effectiveness compared with the isolated case. Although the oscillations have significant amplitude, they are limited, which is a different behaviour from that of a tandem configuration with two plain cylinders. For this particular case, the amplitude response monotonically increases for all gaps, except one, a trait usually found in galloping-like oscillations. SDPIV results for the tandem arrangements show alternating vortex shedding and oscillatory wake.     

Passive control of vortex induced vibration in internal flow using body shape

We present a concept for passive control of vortex induced vibration (VIV) that uses the body shape of a prismatic body as the control parameter in 2D internal flow. We consider that the Reynolds number based on the prism cross section height is 200 and that the blockage ratio of the channel is 2.5. The working fluid is water and the solid-to-fluid density ratio is 1, so that the prism presents neutral buoyancy and the body shape parameter γ acts as the only control parameter. Two very different fluid dynamics regimes are observed depending on γ with an abrupt transition between them for γ=γ_c, where γ_c represents a critical value obtained numerically. For γ<γ_c the cylinder oscillation is controlled by vortex shedding and represents a typical case of vortex induced vibration. For γ>γ_c the oscillation is a mixture of galloping and vortex induced vibration that causes the prism motion to shift from a stable periodic motion to a highly irregular pattern. The physical explanation for the change of regime is given based on the cylinder equation of motion.     

Drag reduction using surfactants in a rotating cylinder geometry

Turbulent Couette flow between two circular cylinders has been used for drag reduction experiments using surfactants. In the experiments presented here, only the outer cylinder rotates, the inner cylinder remains at rest and accurate measurements of the torque at the inner cylinder are measured. Water is used as a reference fluid. A drag reducing surfactant called Arquad S-50 (Akzo Nobel Surface Chemistry LLC, Chicago, Ill., USA) (5 mM)+NaSal (12.5 mM) was used as the drag reduction agent. This surfactant can reduce the drag up to 70% (a Reynolds number of about 70,000–150,000) as measured by pressure drop in a pipe flow. Experiments in Couette flow also show drag reduction in the turbulent range. Two arrangements were used, (1) one small trip-wire on the inner cylinder, and (2) four larger trip-wires on the outer cylinder. These trips reduce the critical Reynolds number for transition from laminar to turbulent flow. In case (1), we obtained 18% drag reduction at 5,000<Re<15,000 and in case (2), we obtained an average reduction of about 20% at 2,000<Re<10,000, increasing up to 30% at Re=15,000. The paper also discusses two important problems. First, the shear rate is not constant in the radial gap in circular Couette flow. For non-Newtonian fluids, where the molecular viscosity is a function of the shear rate, this effect must be considered. Second, which viscosity should be used in the Reynolds number? For pipe flow measurements, most authors use the viscosity of the solvent (generally water and Newtonian). For measurements in the Couette flow, we use a different approach, which is described in this paper. We conclude that Couette flow is a useful method for drag reduction investigations. Its advantage is the much smaller geometry in comparison to those of conventional test facilities such as wind tunnels, water, or oil channels or in tubes.