PROPAGATION OF A LONG WAVE IN A CURVED DUCT (Ⅱ)APPLICATIONS OF MATCHED EXPANSION TO LONG WAVE PROPAGATION THROUGH A HOLE WITH VARIABLE CROSS-SECTION

Only the case in which the parameterε=ka《1 is considered in this paper,where k is the wave number and a is the characteris-tic radius of the cross-section of the hole.The general asymp-totic expansion of the complex velocity potential of a long wave propagating in the hole with variable cross-section is obtained by regular perturbation:The methods of matched asymptotic ex-pansion are employed to calculate the reflection coefficients,scattering coefficients and radiation coefficients at the open ends of the hole when a long wave propagates through it,which may be open at both ends or only at one end.Three examples of different kinds of holes are given to show the way to solve such two-dimensional or three-dimensional problems.     

Wave propagation in a fluid flowing through a curved thin-walled elastic tube

The pulsatile flow in a curved elastic pipe of circular cross section is investigated. The unsteady flow of a viscous fluid and the wall motion equations are written in a toroidal coordinate system, superimposed and linearized over a steady state solution. Being the main application relative to the vascular system, the radius of the pipe is assumed small compared with the radius of curvature. This allows an asymptotic analysis over the curvature parameter. The model results an extension of the Womersley's model for the straight elastic tube. A numerical solution is found for the first order approximation and computational results are finally presented, demonstrating the role of curvature in the wave propagation and in the development of a secondary flow.     

Rayleigh wave propagation in curved waveguides

A JWKB asymptotic expansion describing inplane elastic waves is used to approximate a Rayleigh-like wave guided within a curved elastic waveguide whose curvature is small and changes slowly over a wavelength. The two lowest eigenmodes in a curved guide, taken together, constitute the Rayleigh-like wave. It is shown that this wave lies in the shadows of four, closely spaced, virtual caustics, two caustics per constituent eigenmode. If the curvature becomes too large one or more of the caustics ceases to be virtual and enters the guide after which a Rayleigh-like wave cannot propagate. The overall disturbance is shown to have an amplitude that is modulated because the wavenumbers of the constituent eigenmodes differ by a small amount. Moreover, the disturbance is shown to propagate with a wavenumber that, to leading order, has a linear dependence on the curvature causing the phase to be modulated, as well. Passing from a thin guide to a very thick one suppresses the amplitude modulation, making the phase modulation evident. Propagation into an environment of increasing curvature, for both thin and thick, shallowly curved guides is studied so that the modulations may be observed.     

Propagation of Rayleigh waves on curved surfaces

The propagation and properties of Rayleigh waves on curved surfaces are investigated theoretically. The Rayleigh wave dispersion equation for propagation on a curved surface is derived as a parabolic equation, and its penetration depth is analyzed using the curved surface boundary. Reciprocity is introduced to model the diffracted Rayleigh wave beams. Simulations of Rayleigh waves on some canonical curved surfaces are carried out, and the results are used to quantify the influence of curvature. It is found that the velocity of the surface wave increases with greater concave surface curvature, and a Rayleigh wave no longer exists once the surface wave velocity exceeds the bulk shear wave velocity. Moreover, the predicted wave penetration depth indicates that the energy in the Rayleigh wave is transferred to other modes and cannot propagate on convex surfaces with large curvature. A strong directional dependence is observed for the propagation of Rayleigh waves in different directions on surfaces with complex curvatures. Thus, it is important to include dispersion effects when considering Rayleigh wave propagation on curved surfaces.     

Prediction of wake in a curved duct

Experimental data on the development of an aerofoil wake in a curved stream are compared with calculations based on the k-ε model of turbulence with standard constants and with the model constant C_μ dependent on the local curvature. The mean velocity profile is asymmetric, the half-width of the wake is more on the inner side of the curved duct than on the outer side, and the turbulent shear stress decreases rapidly on the outer side. The standard k-ε model is able to satisfactorily reproduce this behaviour. Making C_μ dependent on the local radius improves the agreement on the inner side but slightly worsens it on the outer side.     

Study of the propagation of a torsional wave in a curved wire

Summary The problem of the propagation of a torsional wave in a curved wire is examined. The displacements with a natural and with a forced curvature are calculated and the corresponding dispersion relations are obtained on the basis of a purely elastic model.The results are discussed in connexion with the problem of the propagation of a signal in a wire-type acoustic delay line and compared with the results previously obtained by other workers.

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Sommario Si esamina il problema della propagazione di un'onda torsionale in un filo curvato. Viene svolto il calcolo degli spostamenti con curvatura sia naturale, sia imposta e si ottengono le corrispondenti relazioni di dispersione sulla base di un modello puramente elastico. Si discutono inoltre i risultati ottenuti, in relazione al problema della propagazione di un segnale in una linea di ritardo meccanica, confrontandoli con quelli precedentemente acquisiti da altri Autori.

     

On the validity of planar, thick curved beam models derived with respect to centroidal and neutral axes

Most dynamic analyses of planar curved beams found in the literature are carried out based on a curved beam model which assumes that the neutral axis coincides with the centroidal axis of the curved beam. This assumption leads to governing equations of motion which are relatively simple with analysis results that have acceptable accuracy for shallow curved beams. However, when a curved beam is not shallow and/or its cross section is not doubly symmetric, the offset distance between the neutral and centroidal axes may be large enough to influence the in-plane dynamics of the curved beam even for small motion. In this paper, the validity of this underlying assumption for modeling a linear curved beam is examined. To this end, two sets of equations of motion governing the in-plane dynamics of a planar curved beam are derived, in a consistent manner for comparison, based on the linear strain-displacement relations and Hamilton’s principle. The first set of equations is derived from the displacement components measured with reference to the neutral axis of the curved beam while the second set is derived with respect to the centroidal axis of the cross section. The curved beam is considered extensional and the effects of rotary inertia and radial shear deformation are included. In addition to the curvature parameter that characterizes the wave motion for both curved beam models, an eccentricity parameter is introduced in the first model to account for the offset between the neutral and centroidal axes. The dynamic behavior predicted by each curved beam model is compared in terms of the dispersion relations, frequency spectra, cutoff frequencies, natural frequencies and modeshapes, and frequency responses. In order to ensure that the comparison is accurate, the wave propagation technique is applied to obtain exact wave solutions. It is shown that, when the curvature parameter is not small, the underlying assumption has a substantial impact on the accuracy of the linear dynamic analysis of a curved beam.     

Three-dimensional compressible potential flow in a curved duct

Summary The three-dimensional problem of compressible potential flow in a bend of a circular duct is studied by means of a perturbation method. As perturbation parameter is used the quotient between the radius of the duct and the radius of curvature of the duct, and the solution includes terms of the first and second order of this parameter. Numerical calculations have been performed for the values 0.2 and 0.1 of the perturbation parameter, and the plotted result shows how the axial velocity distribution changes from the straight to the curved region of the duct.

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Übersicht Mit Hilfe einer Störungsmethode wird die dreidimensionale Strömung einer kompressiblen Flüssigkeit in einer gekrümmten kreisförmigen Rohrleitung untersucht. Als Störungsparameter wird das Verhältnis von Querschnittradius zum Krümmungsradius des Leitungsbogens verwendet. Die Lösung wird bis zu Gliedern von zweiter Ordnung angegeben. Numerische Berechnungen werden für die Werte 0,1 und 0,2 des Störungsparameters durchgeführt. In Diagrammen wird gezeigt, wie sich die axiale Geschwindigkeitsverteilung beim Übergang vom geraden zum gekrümmten Teil der Leitung verändert.

     

Bifurcation and stability analysis of laminar flow in curved ducts

The development of viscous flow in a curved duct under variation of the axial pressure gradient q is studied. We confine ourselves to two‐dimensional solutions of the Dean problem. Bifurcation diagrams are calculated for rectangular and elliptic cross sections of the duct. We detect a new branch of asymmetric solutions for the case of a rectangular cross section. Furthermore we compute paths of quadratic turning points and symmetry breaking bifurcation points under variation of the aspect ratio γ (γ=0.8…1.5). The computed diagrams extend the results presented by other authors. We succeed in finding two origins of the Hopf bifurcation. Making use of the Cayley transformation, we determine the stability of stationary laminar solutions in the case of a quadratic cross section. All the calculations were performed on a parallel computer with 32×32 processors. Copyright © 2009 John Wiley & Sons, Ltd.     

Characteristics of guided waves in graded spherical curved plates

An elastodynamic solution for the stress wave propagation in spherical curved plates composed of functionally graded materials (FGM) is presented. Properties of materials are assumed to vary in the direction of the thickness according to a known radial variation law (gradient field). The formulation is based on the linear three-dimensional elasticity. The Legendre orthogonal polynomial series expansion approach is used for determining the guided waves dispersion curves in functionally graded spherical curved plates. Our results from a homogeneous anisotropic spherical curved plate are compared with those published earlier to confirm the accuracy and range of applicability of this polynomial approach. Guided wave dispersion curves for graded spherical curved plates with different gradient fields are calculated, and the effects of the gradient field on the characteristics of guided waves are illustrated. Finally, dispersion curves for graded spherical curved plates at different ratios of inner radius to thickness are calculated to discover the influences of that ratio on the wave characteristics.     

环形截面螺旋管道内二次流动特性的研究

从曲线柱坐标系下的N-S方程出发,以曲率和挠率为小参数,采用摄动法求解了环形截面螺旋管道内的黏性流动,给出了完全二阶摄动解,结果表明：当挠率为零时,二次流表现为上下对称的四个涡;当挠率不为零,涡的对称性遭到破坏,二次涡的强度和个数受De数和环形截面内外径之比δ的影响,轴向速度最大值在De数较小时靠近管道的内侧,随着De数的增加,其最大值向外侧移动。     

Elastic wave propagation in anisotropic spherical curved plates

Spherical plate-like structures are used in pressure vessels, spherical domes of power plants, and in many other industrial applications. For non-destructive evaluation of such spherical structures, the mechanics of elastic wave propagation in spherical curved plates must be understood. The current literature shows some valuable studies on Rayleigh surface wave propagation in isotropic solids with spherical boundaries. However, the guided wave propagation problem in an anisotropic spherical curved plate, which has not been studied before, is solved for the first time in this paper.The wave propagation, in both isotropic and anisotropic spherical curved plates, is investigated. The differential equations of motion and the stress-free boundary conditions on the inner and outer surfaces of a hollow sphere are approximately solved by a general solution technique. This solution technique was successfully utilized by the authors for solving the wave propagation problem in cylindrical plates, in their earlier works. Dispersion curves for spherical plates made of isotropic aluminum, steel, and anisotropic composite material are presented as well.     

Electrokinetic flow and energy conversion in a curved microtube

Curved channels are ubiquitous in microfluidic systems. The pressuredriven electrokinetic flow and energy conversion in a curved microtube are investigated analytically by using a perturbation analysis method under the assumptions of the small curvature ratio and the Reynolds number. The results indicate that the curvature of the microtube leads to a skewed pattern in the distribution of the electrical double layer (EDL) potential. The EDL potential at the outer side of the bend is larger than that at the inner side of the bend. The curvature shows an inhibitory effect on the magnitude of the streaming potential field induced by the pressure-driven flow. Since the spanwise pressure gradient is dominant over the inertial force, the resulting axial velocity profile is skewed into the inner region of the curved channel. Furthermore, the flow rate in a curved microtube could be larger than that in a straight one with the same pressure gradient and shape of cross section. The asymptotic solutions of the axial velocity and flow rate in the absence of the electrokinetic effect are in agreement with the classical results for low Reynolds number flows. Remarkably, the curved geometry could be beneficial to improving the electrokinetic energy conversion (EKEC) efficiency.     

Determination of separated region in a curved tube

The formation of atherosclerosis in a curved aorta is closely related to the existence of separated vortex region. This paper deals with the steady laminar motion of an incompressible Newtonian fluid through a curved tube with circular cross-section whose curvature is small and whose curvature gradient is not too large. Using the momentum integral method and the approximation of quasi-constant curvature, an equation which determines the location of separation and reattachment is derived. From this equation the earliest point of separation and the corresponding critical Reynolds number are obtained, and the relation between the position of separation and reattachment and Reynolds number R_e for different azimuthal angle are revealed. It is concluded that the separation first emerges at the position whose curvature gradient has the maximum absolute value. With increasing R_e, the separation region extends in the direction of mainstream, azimuthal angle and radius vector, and then forms a three-dimensional separated vortex, which gradually enlarges in all three directions with the increase of Reynolds number. The theoretical results also very clearly demonstrate the following striking experimental fact: if a symmetrical curved tube exhibits a separated vortex at the outside of the upstream, then it must have another one symmetrically placed at the inside of its downstream.     

Nonlinear hyperbolic wave propagation in a one-dimensional random medium

We use an asymptotic expansion introduced by Benilov and Pelinovski to study the propagation of a weakly nonlinear hyperbolic wave pulse through a stationary random medium in one space dimension. We also study the scattering of such a wave by a background scattering wave. The leading-order solution is non-random with respect to a realization-dependent reference frame, as in the linear theory of O’Doherty and Anstey. The wave profile satisfies an inviscid Burgers equation with a nonlocal, lower-order dissipative and dispersive term that describes the effects of double scattering of waves on the pulse. We apply the asymptotic expansion to gas dynamics, nonlinear elasticity, and magnetohydrodynamics.     

The Dean instability in power-law and Bingham fluids in a curved rectangular duct

The laminar flow of power-law and yield-stress fluids in 180° curved channels of rectangular cross section was studied experimentally and numerically in order to understand the effect of rheological fluid behavior on the Dean instability that appears beyond a critical condition in the flow. This leads to the apparition of Dean vortices that differ from the two corner vortices created by the channel wall curvature.Flow visualizations showed that the Dean vortices develop first in the near-wall zone on the concave (outer) wall, where the shear rate is higher and the viscosity weaker; then they penetrate into the centre of the channel cross section where power-law fluids have high viscosity and Bingham fluids are unyielded in laminar flow. Based on the complete formation on the concave wall of the new pairs of counter-rotating vortices (Dean vortices), the critical value of the Dean number decreases as the power-law index increases for the power-law fluids, and the Bingham number decreases for the Bingham fluids. For power-law fluids, a diagram of critical Dean numbers, based on the number of Dean vortices formed, was established for different axial positions. For the same flow conditions, the critical Dean number obtained using the axial velocity gradient criterion was smaller then that obtained with the visualization technique.     

Acoustic waves in a cylindrical duct with periodically varying cross section

The null field approach is used to study the propagation of acoustic waves in a rotationally symmetric hard-walled duct with a periodically varying cross section. For a radius that varies sinusoidally along the axial distance, numerical computations give the axial wave number and the passbands and stopbands of the modes of the duct. In particular, small passbands are seen to exist even for very large variations in radius, and probably all the way to the point where the duct is cut off. We also present some plots which show the pressure pattern inside the duct.     

Solution of the laminar duct flow problem by a boundary integral equation method

Summary A boundary integral equation method is proposed for approximate numerical and exact analytical solutions to fully developed incompressible laminar flow in straight ducts of multiply or simply connected cross-section. It is based on a direct reduction of the problem to the solution of a singular integral equation for the vorticity field in the cross section of the duct. For the numerical solution of the singular integral equation, a simple discretization of it along the cross-section boundary is used. It leads to satisfactory rapid convergency and to accurate results. The concept of hydrodynamic moment of inertia is introduced in order to easily calculate the flow rate, the main velocity, and the fRe-factor. As an example, the exact analytical and, comparatively, the approximate numerical solution of the problem of a circular pipe with two circular rods are presented. In the literature, this is the first non-trivial exact analytical solution of the problem for triply connected cross section domains. The solution to the Saint-Venant torsion problem, as a special case of the laminar duct-flow problem, is herein entirely incorporated.     

Fluid flow in a rotating curved rectangular duct

The fluid flowing in a rotating curved duct is subjected to both the Coriolis force due to a rotation and the centrifugal force due to a curvature. In this paper, the combined effects of the two forces on the flows in rotating curved rectangular ducts are examined numerically. According to the aspect ratio of the cross-section, the rectangular ducts are divided into three types: η>1, η=1, η<1, where η is the aspect ratio. The variations of the flow structures with the force ratio F (the ratio of the Corislis force to the centrifugal force) are studied in detail and many hitherto unknown flow patterns are found. The effects of the force ratio and the aspect ratio of the cross-section on the friction factor are also examined. Present results show both the characteristics of the secondary flow, axial flow and the natures of the friction factor.     

Shock wave propagation in a branched duct

The propagation of a planar shock wave in a 90° branched duct is studied experimentally and numerically. It is shown that the interaction of the transmitted shock wave with the branching segment results in a complex, two-dimensional unsteady flow. Multiple shock wave reflections from the duct's walls cause weakening of transmitted waves and, at late times, an approach to an equilibrium, one-dimensional flow. While at most places along the branched duct walls calculated pressures are lower than that existing behind the original incident shock wave, at the branching segment's right corner, where a head on-collision between the transmitted wave and the corner is experienced, pressures that are significantly higher than those existing behind the original incident shock wave are encountered. The numerically evaluated pressures can be accepted with confidence, due to the very good agreement found between experimental and numerical results with respect to the geometry of the complex wave pattern observed inside the branched duct. Received 15 July 1996 / Accepted 20 February 1997