From global to local bifurcations in a forced Taylor–Couette flow

The unfolding due to imperfections of a gluing bifurcation occurring in a periodically forced Taylor–Couette system is analyzed numerically. In the absence of imperfections, a temporal glide-reflection Z₂ symmetry exists, and two global bifurcations occur within a small region of parameter space: a heteroclinic bifurcation between two saddle two-tori and a gluing bifurcation of three-tori. As the imperfection parameter increase, these two global bifurcations collide, and all the global bifurcations become local (fold and Hopf bifurcations). This severely restricts the range of validity of the theoretical picture in the neighborhood of the gluing bifurcation considered, and has significant implications for the interpretation of experimental results. PACS 47.20.Ky, 47.20.Lz, 47.20.Ft     

Development of the Rayleigh–Taylor instability due to interaction of a diffusion mixing layer of two gases with compression waves

A mathematical model of mechanics of a two-velocity and two-temperature mixture of gases is developed. Based on this model, the evolution of the mixing layer of two gases of different densities, which are accelerated by a compression wave, is considered by methods of numerical simulation. A solution of an initial-boundary problem is obtained in a one-dimensional approximation. This solution describes the formation of a diffusion layer between the two gases. The problem of interaction of this layer with the compression wave, the heavy medium being accelerated by the light medium, is solved numerically. Problems of instability development in a sine-perturbed mixing layer accelerated by a compression wave are solved numerically in a two-dimensional unsteady formulation. The calculated width of the mixing region is in reasonable agreement with experimental data.     

Instability analysis of modulated Taylor vortices

This study investigates the instability analysis of modulated Taylor vortices flow by utilising a numerical method. Based on the consideration that the outer cylinder is fixed and the inner cylinder rotates at a non-zero averaged speed under varying modulated amplitudes and frequencies, the flow is converted from one-dimension Couette flow to Taylor vortices. When the modulated amplitude is greater than 1 and the rotation speed of the inner cylinder exceeds the threshold value for one-dimensional flow, the flow will be more stable at intermediate and high frequencies. When the modulated amplitude is sufficiently large and the inner cylinder rotates at medium frequency, subharmonic flow arises.     

A Two-Time-Scale Model for Turbulent Mixing Flows Induced by Rayleigh–Taylor and Richtmyer–Meshkov Instabilities

A two-time-scale closure model for compressible flows previously developed is extended to turbulent Rayleigh-Taylor and Richtmyer-Meshkov driven flows where mixing coexists with mean pressure gradients. Two model coefficients are calibrated with the help of Canuto-Goldman's model. For several Rayleigh-Taylor configurations, it is shown that the characteristic lengths scale as t ² while the kinetic energies and spectral transfers behave as t ² and t, respectively. The computed phenomenological coefficients of Youngs' scaling law are compared with experimental data ones. Comparisons with Youngs' three-dimensional numerical simulation (The Physics of Fluids A 3 (1991) 1312) are also performed. Finally three shock tube experiments, where the Richtmyer-Meshkov instability initiates the mixing, are simulated. The mixing thickness evolution is well reproduced while the turbulence levels seem to be overestimated with such first order models. The capability of the two-time-sale model to recover available data for different turbulent flows allows us to conclude to a more universal behavior in comparison with single-time-scale models. This revised version was published online in July 2006 with corrections to the Cover Date.     

Effect of inertia on thermoelastic flow instability

Thermal effects induced by viscous heating cause thermoelastic flow instabilities in curvilinear shear flows of viscoelastic polymer solutions. These instabilities could be tracked experimentally by changing the fluid temperature T₀ to span the parameter space. In this work, the influence of T₀ on the stability boundary of the Taylor–Couette flow of an Oldroyd-B fluid is studied. The upper bound of the stability boundary in the Weissenberg number (We)–Nahme number (Na) space is given by the critical conditions corresponding to the extension of the time-dependent isothermal eigensolution. Initially, as T₀ is increased, the critical Weissenberg number, We_c, associated with this upper branch increases. Increasing T₀ beyond a certain value T^* causes the thermoelastic mode of instability to manifest. This occurs in the limit as We/Pe → 0, where Pe denotes the Péclet number. In this limit, the fluid relaxation time is much smaller than the time scale of thermal diffusion. T₀ = T^* represents a turning point in the We_c–Na_c curve. Consequently, the stability boundary is multi-valued for a wide range of Na values. Since the relaxation time and viscosity of the fluid decrease with increasing T₀, the elasticity number, defined as the ratio of the fluid relaxation time to the time scale of viscous diffusion, also decreases. Hence, O(10) values of the Reynolds number could be realized at the onset of instability if T₀ is sufficiently large. This sets limits for the temperature range that can be used in experiments if inertial effects are to be minimized.     

A return toward equilibrium in a 2D Rayleigh–Taylor instability for compressible fluids with a multidomain adaptive Chebyshev method

We numerically simulate a single-mode Rayleigh–Taylor instability between compressible miscible fluids with a highly accurate self-adaptive pseudospectral Chebyshev multidomain method in two two-dimensional boxes at small aspect ratios. The simulations are started from rest and pursued until the return toward mechanical equilibrium of the mixing. Four regimes—linear and weakly nonlinear, nonlinear steady bubble rise, return toward equilibrium, and finally a system of acoustic waves—can be identified. We show that this one-dimensional system of stationary acoustic waves is damped by the physical viscosity. This provides a reference solution.      

Solution of Müller's paradox of frame dependence in mixtures of polymeric liquid crystals

Recently, there appeared in this journal a short review article by Müller ((2002), 14: 227–229), in which it was argued that the internal energy of a reacting mixture of liquid crystals should not be an objective quantity (i.e., a quantity independent of the referential frame). Such a paradoxical conclusion has revealed the urging necessity for a better comprehension of the interactions taking place in structured mixtures, specially when referred to non-inertial observers. This work shows that Müller's paradox is avoided when all inertial effects are carefully accounted for. Further, it predicts interesting phenomena without analogue in reacting mixtures of structureless fluids: internal inertial effects, produced by a combination of mass exchanges (e.g. by chemical reactions or phase changes) with the extra degrees of freedom posed by the microstructure. Such effects clarify the reasons why inertial forces and couples acting on a mixture do not always coincide with the sum of the inertial forces and couples exerted upon its individual constituents. The present conclusions establish a reinterpretation of some fundamental concepts of continuum mechanics and thermodynamics, including a deeper understanding of the manner in which total energy and momenta are conserved in complex media. Received: July 25, 2002 / Published online: February 17, 2003 Communicated by Kolumban Hutter, Darmstadt e-mail: faria@mechanik.tu-darmstadt.de     

A non‐linear dynamical system approach to finite amplitude Taylor‐Vortex flow of shear‐thinning fluids

The effect of shear thinning on the stability of the Taylor–Couette flow is explored for a Carreau–Bird fluid in the narrow‐gap limit. The Galerkin projection method is used to derive a low‐order dynamical system from the conservation of mass and momentum equations. In comparison with the Newtonian system, the present equations include additional non‐linear coupling in the velocity components through the viscosity. It is found that the critical Taylor number, corresponding to the loss of stability of the circular Couette flow, becomes lower as the shear‐thinning effect increases. That is, shear thinning tends to precipitate the onset of Taylor vortex flow, which coincides with the onset of a supercritical bifurcation. Comparison with existing measurements of the effect of shear thinning on the critical Taylor and wave numbers show good agreement. The Taylor vortex cellular structure loses its stability in turn, as the Taylor number reaches a critical value. At this point, an inverse Hopf bifurcation emerges. In contrast to Newtonian flow, the bifurcation diagrams exhibit a turning point that sharpens with shear‐thinning effect. Copyright © 2004 John Wiley & Sons, Ltd.     

A computationally efficient solution of the wave equation for the transient response of infinite reservoirs

In this paper, the transient response of an infinite reservoir is analyzed using the dual-reciprocity boundary element method. A vertical and an inclined-face rigid dam are analyzed under a transient loading. Sharan-type boundary-condition transmission is implemented in the formulation. The results are compared with the exact solution and those obtained by using the finite element method. It is seen that the application of the dual-reciprocity boundary element method is simpler and the results are in very good agreement with the exact solution and those obtained by using the finite element method.     

First instabilities in the wake past a circular cylinder. Comparison of transient regimes with landau's model

Velocity measurements were carried out in the flow past a circular cylinder at Reynolds number ranging from 30 to 180 to study the genesis and the evolution of the first instabilities in the wake. Since the occurrence of aeroelastic coupling between flow and cylinder has been cause of debate about the results obtained in recent experiments on the subject, a great deal of care has been taken to avoid that coupling. We focused our attention on two points: the onset of the first instability and the transition from the low-speed to the high-speed oscillation mode. With regard to the second topic we can confirm the existence of a discontinuity in the relation between St and Re at Re90 even in the absence of any aeroelastic coupling. However, no band enlargements in the velocity spectrum, indicating the appearance of highly disorganized motion, were observed for Re varying through that discontinuity interval. To investigate the onset of the first periodic oscillation we performed measurements with a slowly varying Reynolds number by means of very small positive or negative accelerations. Our results indicate that in the neighborhood of the onset of the first oscillation the amplitude of the transient velocity fluctuations does not obey the so-called Landau model.

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Sommario Al fine di studiare la genesi e l'evoluzione delle prime instabilitá nelle scie si sono eseguite delle misure della velocitá nel flusso a valle di un cilindro circolare variando il numero di Reynolds tra 30 e 180. Poiché esiste, per questo tipo di esperimento, la possibilitá di ritrovarsi in una situazione di accoppiamento aeroelastico tra il cilindro ed il flusso, situazione tale da inficiarne l'attendibilitá e che é stata causa di dispute scientifiche sull'interpretazione dei risultati ottenuti nei piú recenti esperimenti sul soggetto, si é accuratamnete progettato l'esperimento in modo da escludere qualisiasi condizione suscettibile di innescare un tale accoppiamento. Si é concentrata l'attenzione su due punti principali: l'insorgere della prima instabilitá e la transizione tra il low-speed ed il high-speed oscillation mode. Per quel che concerne il secondo punto noi possiamo confermare l'esistenza di una discontinuitá nella relazione St=f(Re) quando il Re 90 anche in assenza di qualsiasi accoppiamneto aeroelastico. Inoltre, al variare del Re attraverso questa discontinuitá, non si é osservato nessun allargamento di banda nello spettro delle velocitá che potesse indicare l'apparire di un moto altamente disorganizzato. Per studiare la genesi della prima oscillazione periodica si sono eseguite delle misure al lento variare del Re per mezzo di piccolissime accelerazioni, sia positive che negative, impartite al campo di moto. I risultati ottenuti indicano che nelle vicinanze dell' insorgere della prima oscillazione l'ampiezza delle fluttuazioni transienti non obbedisce al modello di Landau.

     

Transient analysis of piles and pile groups in non-homogeneous soil

In this paper, the transient analysis of a single pile and a 3 × 3 pile group is presented for Gibson type non-homogeneous soil by using a hybrid type of boundary and finite element formulation for the soil domain and pile domain, respectively. The formula is presented for a transient point force acting in the interior of a non-homogeneous, isotropic half space. A time stepping boundary element algorithm for soil domain is used together with an implicit time integration scheme for finite pile domain. To investigate the validity of this formulation, a single pile and a pile group are analyzed under Heaviside loading and triangular transient loading. In the analyses, it can be concluded that the results agree well for all cases of the inhomogeneity index by comparing the Laplace domain solutions.     

Dynamics of trailing vortices in the wake of a generic high-speed train

The three-dimensional dynamics of a pair of counter-rotating streamwise vortices that are present in the wake of an ICE3 high-speed train typical of modern, streamlined vehicles in operation, is investigated in a 1/10th-scale wind-tunnel experiment. Velocity mapping, frequency analysis, phase-averaging and proper orthogonal decomposition of data from high-frequency multi-hole dynamic pressure probes, two-dimensional total pressure arrays and one-dimensional multi-hole arrays was performed. Sinusoidal, antisymmetric motion of the pair of counter-rotating streamwise vortices in the wake is observed. These unsteady characteristics are proposed to be representative of full-scale operational high-speed trains, in spite of the experimental limitations: static floor, reduced model length and reduced Reynolds number. This conclusion is drawn from favourable comparisons with numerical literature, and the ability of the identified characteristics to explain phenomena established in full-scale and scaled moving-model experiments.     

A Green's function approach to the deflection of a FGM plate under transient thermal loading

Summary  Green's function approach is adopted for analyzing the deflection and the transient temperature distribution of a plate made of functionally graded materials (FGMs). The governing equations for the deflection and the transient temperature are formulated into eigenvalue problems by using the eigenfunction expansion theory. Green's functions for solving the deflection and the transient temperature are obtained by using the Galerkin method and the laminate theory, respectively. The eigenfunctions of Green's function for the deflection are approximated in terms of a series of admissible functions that satisfy the homogeneous boundary conditions of the plate. The eigenfunctions of Green's function for the temperature are determined from the continuity conditions of the temperature and the heat flux at interfaces. Received 9 October 2000; accepted for publication 3 April 2001     

Effect of Reactive Surface Areas Associated with Different Particle Shapes on Chemical-Dissolution Front Instability in Fluid-Saturated Porous Rocks

In this article, the effect of reactive surface areas associated with different particle shapes on the reactive infiltration instability in a fluid-saturated porous medium is investigated through analytically deriving the dimensionless pore-fluid pressure-gradient of a coupled system between porosity, pore-fluid flow and reactive chemical-species transport within two idealized porous media consisting of spherical and cubic grains respectively. Compared with the critical dimensionless pore-fluid pressure-gradient of the coupled system, the derived dimensionless pore-fluid pressure-gradient can be used to assess the instability of a chemical dissolution front within the fluid-saturated porous medium. The related theoretical analysis has demonstrated that (1) since the shape coefficient of spherical grains is greater than that of cubic grains, the chemical system consisting of spherical grains is more unstable than that consisting of cubic grains, and (2) the instability likelihood of a natural porous medium, which is comprised of irregular grains, is smaller than that of an idealized porous medium, which is comprised of regular spherical grains. To simulate the complicated morphological evolution of a chemical dissolution front in the case of the chemical dissolution system becoming supercritical, a numerical procedure is proposed for solving this kind of problem. The related numerical results have demonstrated that the reactive surface area associated with different particle shapes can have a significant influence on the morphological evolution of an unstable chemical-dissolution front within fluid-saturated porous rocks.     

Study on transient aerodynamic characteristics of parachute opening process

In the research of parachute, canopy inflation process modeling is one of the most complicated tasks. As canopy often experiences the largest deformations and loadings during a very short time, it is of great difficulty for theoretical analysis and experimental measurements. In this paper, aerodynamic equations and structural dynamics equations were developed for describing parachute opening process, and an iterative coupling solving strategy incorporating the above equations was proposed for a small-scale, flexible and flat-circular parachute. Then, analyses were carried out for canopy geometry, time-dependent pressure difference between the inside and outside of the canopy, transient vortex around the canopy and the flow field in the radial plane as a sequence in opening process. The mechanism of the canopy shape development was explained from perspective of transient flow fields during the inflation process. Experiments of the parachute opening process were conducted in a wind tunnel, in which instantaneous shape of the canopy was measured by high velocity camera and the opening loading was measured by dynamometer balance. The theoretical predictions were found in good agreement with the experimental results, validating the proposed approach. This numerical method can improve the situation of strong dependence of parachute research on wind tunnel tests, and is of significance to the understanding of the mechanics of parachute inflation process. The project supported by the National Natural Science Foundation of China (10377006). The English text was polished by Yunming Chen.     

Reconstructing the transient,dissipative dynamics of a bistable Duffing oscillator with an enhanced averaging method and Jacobian elliptic functions

Systems characterized by the governing equation of the bistable, double-well Duffing oscillator are ever-present throughout the fields of science and engineering. While the prediction of the transient dynamics of these strongly nonlinear oscillators has been a particular research interest, the sufficiently accurate reconstruction of the dissipative behaviors continues to be an unrealized goal. In this study, an enhanced averaging method using Jacobian elliptic functions is presented to faithfully predict the transient, dissipative dynamics of a bistable Duffing oscillator. The analytical approach is uniquely applied to reconstruct the intrawell and interwell dynamic regimes. By relaxing the requirement for small variation of the transient, averaged parameters in the proposed solution formulation, the resulting analytical predictions are in excellent agreement with exact trajectories of displacement and velocity determined via numerical integration of the governing equation. A wide range of system parameters and initial conditions are utilized to assess the accuracy and computational efficiency of the analytical method, and the consistent agreement between numerical and analytical results verifies the robustness of the proposed method. Although the analytical formulations are distinct for the two dynamic regimes, it is found that directly splicing the inter- and intrawell predictions facilitates good agreement with the exact dynamics of the full reconstructed, transient trajectory.     

The effect of the slip condition on Stokes and Couette flows due to an oscillating wall: exact solutions

Stokes and Couette flows produced by an oscillatory motion of a wall are analyzed under conditions where the no-slip assumption between the wall and the fluid is no longer valid. The motion of the wall is assumed to have a generic sinusoidal behavior. The exact solutions include both steady periodic and transient velocity profiles. It is found that slip conditions between the wall and the fluid produces lower amplitudes of oscillations in the flow near the oscillating wall than when no-slip assumption is utilized. Further, the relative velocity between the fluid layer at the wall and the speed of the wall is found to overshoot at a specific oscillating slip parameter or vibrational Reynolds number at certain times. In addition, it is found that wall slip reduces the transient velocity for Stokes flow while minimum transient effects for Couette flow is achieved only for large and small values of the wall slip coefficient and the gap thickness, respectively. The time needed to reach to steady periodic Stokes flow due to sine oscillations is greater than that for cosine oscillations with both wall slip and no-slip conditions.     

On the Mutual Penetrations of Two Fluids Whose Interface is Accelerated by a Shock Wave

A shock tube experimental investigation and numerical simulations are undertaken to study the evolution of a perturbed interface of two different gases accelerated by a shock wave. The experimental method is based on a high-speed camera laser sheet diagnostic technique, and simulations are provided by our code CARBUR based on a finite volume discretization of Navier–Stokes’s equations. Two gas pairs are used to illustrate both the heavy/light (air/He) and the light/ heavy (air/SF₆) cases. Two simultaneous large initial perturbations, one positive and one negative, are tested for an incident shock wave Mach number in air of about 1.3. The thin membrane (less than 1 μ) which materializes the initial interface between the two test gases presents 2D perturbations whose wave number is close to 1 in order to rapidly reach the non-linear regime. The development of the perturbations is captured at a frequency of 10 kHz after the interface acceleration, and the experiments are complemented with a numerical simulation to validate the interface deformations. Results show an asymmetric mutual gas penetration increasing with the absolute value of the Atwood’s number. Furthermore, they confirm that the heavier gas penetrates the lighter as thin spikes and the lighter gas penetrates the heavier as large bubbles. Moreover, we show that the spike moves faster than the bubble in the heavy/light case and slightly faster in the light/heavy one. Finally, numerical and experimental results are in agreement.     

Interaction of a reflected shock from a concave wall with a flame distorted by an incident shock

Observations are presented from calculations where a laminar spherical CH₄/air flame was perturbed successively by incident and reflected shock waves reflected from a planar or concave wall. The two-dimensional axi-symmetric Navier–Stokes equations with detailed chemistry were used. The computational results were qualitatively validated with experiments which were performed in a standard shock tube arrangement. Under the influence of the incident shock wave, a Richtmyer–Meshkov instability is induced in the flame, and the distorted flame finally takes the form of two separated elliptical burning bubbles in the symmetric cross plane. Then, under subsequent interactions with the shock wave reflected from the planar or the concave wall, the flame takes a mushroom-like shape. Transverse waves produced by the shock reflection from the concave wall can compress the flame towards the axis, and the focusing shock generated on the concave wall will lead to a larger mushroom-like flame than that induced by the planar reflection.