Multibody Formulation with Large Bending Finite Elements (LBFE)

The goal of this paper is to present a flexible multibody formulation for Euler-Bernoulli beams involving large displacements. This method is based on a discretisation of internal and kinetic energies. The beam is represented by its line of centroids and each section is oriented by a frame defined by three Euler angles. We apply a finite element formulation to describe the evolution of these angles along the neutral fibre and describe the internal energy. The kinetic energy is approximated as the one of two rigid bars tangent to the neutral fibre at the ends of the beam. We derive the equations of motion from a Lagrange formulation. These equations are solved using the Newmark method or/and the Newton-Raphson technique. We solve some very classic problems taken from the literature as the curved beam presented by Simo [Simo, J. C., ‘A three-dimensional finite-strain rod model. the three-dimensional dynamic problem. Part I’, Comput. Meths. Appl. Mech. Engrg. 49, 1985, 55–70; Simo, J. C. and Vu-Quoc, L., ‘A three-dimensional finite-strain rod model, Part II: Computationals aspects’, Comput. Meths. Appl. Mech. Engrg. 58, 1988, 79–116] and Lee [Lee, Kisu, ‘Analysis of large displacements and large rotations of three-dimensional beams by using small strains and unit vectors’, Commun. Numer. Meth. Engrg. 13, 1997, 987–997] or the rotational rod presented by Avello [Avello, A., Garcia de Jalon, J., and Bayo, E., ‘Dynamics of flexible multibody systems using cartesian co-ordinates and large displacement theory’, Int. J. Num. Methods in Engineering 32, 1991, 1543–1563] and Simo [Simo, J. C. and Vu-Quoc, L., ‘On the dynamics of flexible beams under large overall motions – the planar case. Part I’ Jour. of Appl. Mech. 53, 1986, 849–854; Simo, J. C. and Vu-Quoc, L., ‘On the dynamics of flexible beams under large overall motions – the planar case. Part II’, Jour. of Appl. Mech. 53, 1986, 855–863].     

Phase locking control in the Circle Map

The phase-locking between two oscillators occurs when the ratio of their frequencies becomes locked in a ratio p/q of integer numbers over some finite domain of parameters values. Due to it, oscillators with some kind of nonlinear coupling may synchronize for certain set of parameters. This phenomenon can be better understood and studied with the use of a well-known paradigm, the Circle Map, and the definition of the winding number. Two diagrams related to this map are especially useful: the ‘Arnold tongues’ and the ‘devil’s staircase’. The synchronization that occurs in this map is described by the ‘Farey Series’. This property is the starting point for the development of control algorithms capable of locking the system under the action of an external excitation into a desired winding number. In this work, we discuss the main characteristics of the phase-locking phenomenon and consider three control algorithms designed to drive and keep the Circle Map into a desired winding number.     

Computational and experimental study of supersonic flow over axisymmetric cavities

Laminar and turbulent computations are presented for annular rectangular-section cavities, on a body of revolution, in a Mach 2.2 flow. Unsteady ‘open cavity flows’ result for all laminar computations for all cavity length-to-depth ratios, L/D (1.33, 10.33, 11.33 and 12.33). The turbulent computations produce ‘closed cavity flows’ for L/D of 11.33 and 12.33. Surface pressure fluctuations at the front corner of the L/D = 1.33 cavity are periodic in some cases depending on the cavity length and depth, the boundary layer at the cavity front lip and the cavity scale. The turbulent computations are supported by experimental schlieren images, obtained using a spark light source, and time-averaged surface pressure data.     

Unsteady Flux-Vector-Based Green Element Method

This article extends the mathematical formulation and solution procedure of the modified ‘q-based’ GEM to unsteady situations, namely to the modified unsteady ‘q-based’ GEM. Solutions that provide information on the evolution of the pressure and the flux over long time intervals are available by incorporating the additional dimension of time into steady problems. This approach is first tested by solving an example for which an analytical solution is available. The numerical results for this example is found to be in excellent agreement with the analytical solution. Several problems involving geological features, such as wells and faults, are then investigated, with different properties applying to the faults. A strong influence of the low permeability faults is in evidence in these problems.     

From Rate-Dependent to Rate-Independent Brittle Crack Propagation

On the base of many experimental results, e.g., Ravi-Chandar and Knauss (Int. J. Fract. 26:65–80, 1984), Sharon et al. (Phys. Rev. Lett. 76(12):2117–2120, 1996), Hauch and Marder (Int. J. Fract. 90:133–151, 1998), the object of our analysis is a rate-dependent model for the propagation of a crack in brittle materials. Restricting ourselves to the quasi-static framework, our goal is a mathematical study of the evolution equation in the geometries of the ‘Single Edge Notch Tension’ and of the ‘Compact Tension’. Besides existence and uniqueness, emphasis is placed on the regularity of the evolution making reference also to the ‘velocity gap’. The transition to the rate-independent model of Griffith is obtained by time rescaling, proving convergence of the rescaled evolutions and of their energies. Further, the discontinuities of the rate-independent evolution are characterized in terms of unstable points of the free energy. Results are illustrated by a couple of numerical examples in the above mentioned geometries.     

Unsteady free convection on a vertical cylinder with variable heat and mass flux

The unsteady natural convection boundary layer flow over a semi-infinite vertical cylinder is considered with combined buoyancy force effects, for the situation in which the surface temperature T ^′ _w(x) and C ^′ _w(x) are subjected to the power-law surface heat and mass flux as K(T ^′/r) = −ax ⁿ and D(C ^′/r) = −bx ^m . The governing equations are solved by an implicit finite difference scheme of Crank-Nicolson method. Numerical results are obtained for different values of Prandtl number, Schmidt number ‘n’ and ‘m’. The velocity, temperature and concentration profiles, local and average skin-friction, Nusselt and Sherwood numbers are shown graphically. The local Nusselt and Sherwood number of the present study are compared with the available result and a good agreement is found to exist. Received on 7 July 1998     

Symmetry, Optima and Bifurcations in Structural Design

Motivated by optimization problems in structural engineering, we study the critical points of symmetric, ‘reflected', one-parameter family of potentials U(p, x) = max (f(p,x), f(p, −x)), yielding modest generalizations of classical bifurcations, predicted by elementary catastrophe theory. One such generalization is the ‘five-branch pitchfork’, where the symmetric optimum persists beyond the critical parameter value. Our theory may help to explain why symmetrical structures are often optimal.     

Flutter suppression of a plate-like wing via parametric excitation

The present work is motivated by the well known stabilizing effect of parametric excitation of some dynamical systems such as the inverted pendulum. The possibility of suppressing wing flutter via parametric excitation along the plane of highest rigidity in the neighborhood of combination resonance is explored. The nonlinear equations of motion in the presence of incompressible fluid flow are derived using Hamilton's principle and Theodorsen's theory for modeling aerodynamic forces. In the presence of air flow, the bending and torsion modes possess nearly the same frequency. Under parametric excitation and in the absence of air flow, each mode oscillates at its own natural frequency. In the neighborhood of combination resonance, the nonlinear response is determined using the multiple scales method at the critical flutter speed and at slightly higher airflow speed. The domains of attraction and bifurcation diagrams are obtained to reveal the conditions under which the parametric excitation can provide stabilizing effect. The basins of attraction for different values of excitation amplitude reveal the stabilizing effect that takes place above a critical excitation level. Below that level, the response experiences limit cycle oscillations, cascade of period doubling, and chaos. For flow speed slightly higher than the critical flutter speed, the response experiences a train of spikes, known as ‘firing,’ a term that is borrowed from neuroscience, followed by ‘refractory’ or recovery effect, up to an excitation level above which the wing is stabilized. The results of the multiple scales method are verified using numerical simulation of the original nonlinear differential equations.     

Density stratified environments: the double-tank method

We present an alternative method of producing density stratifications in the laboratory based on the ‘double-tank’ method proposed by Oster (Sci Am 213:70–76, 1965). We refer to Oster’s method as the ‘forced-drain’ approach, as the volume flow rates between connecting tanks are controlled by mechanical pumps. We first determine the range of density profiles that may be established with the forced-drain approach other than the linear stratification predicted by Oster. The dimensionless density stratification is expressed analytically as a function of three ratios: the volume flow rate ratio n, the ratio of the initial liquid volumes λ and the ratio of the initial densities ψ. We then propose a method which does not require pumps to control the volume flow rates but instead allows the connecting tanks to drain freely under gravity. This is referred to as the ‘free-drain’ approach. We derive an expression for the density stratification produced and compare our predictions with saline stratifications established in the laboratory using the ‘free-drain’ extension of Oster’s method. To assist in the practical application of our results we plot the region of parameter space that yield concave/convex or linear density profiles for both forced-drain and free-drain approaches. The free-drain approach allows the experimentalist to produce a broad range of density profiles by varying the initial liquid depths, cross-sectional and drain opening areas of the tanks. One advantage over the original Oster approach is that density profiles with an inflexion point can now be established.

---

Asymptotic expansions by Γ-convergence

Our starting point is a parameterized family of functionals (a ‘theory’) for which we are interested in approximating the global minima of the energy when one of these parameters goes to zero. The goal is to develop a set of increasingly accurate asymptotic variational models allowing one to deal with the cases when this parameter is ‘small’ but finite. Since Γ-convergence may be non-uniform within the ‘theory’, we pose a problem of finding a uniform approximation. To achieve this goal we propose a method based on rectifying the singular points in the parameter space by using a blow-up argument and then asymptotically matching the approximations around such points with the regular approximation away from them. We illustrate the main ideas with physically meaningful examples covering a broad set of subjects from homogenization and dimension reduction to fracture and phase transitions. In particular, we give considerable attention to the problem of transition from discrete to continuum when the internal and external scales are not well separated, and one has to deal with the so-called ‘size’ or ‘scale’ effects.      

High-speed laser diagnostics for the study of flame dynamics in a lean premixed gas turbine model combustor

A series of measurements was taken on two technically premixed, swirl-stabilized methane-air flames (at overall equivalence ratios of ϕ = 0.73 and 0.83) in an optically accessible gas turbine model combustor. The primary diagnostics used were combined planar laser-induced fluorescence of the OH radical and stereoscopic particle image velocimetry (PIV) with simultaneous repetition rates of 10 kHz and a measurement duration of 0.8 s. Also measured were acoustic pulsations and OH chemiluminescence. Analysis revealed strong local periodicity in the thermoacoustically self-excited (or ‘noisy’) flame (ϕ = 0.73) in the regions of the flow corresponding to the inner shear layer and the jet-inflow. This periodicity appears to be the result of a helical precessing vortex core (PVC) present in that region of the combustor. The PVC has a precession frequency double (at 570 Hz) that of the thermo-acoustic pulsation (at 288 Hz). A comparison of the various data sets and analysis techniques applied to each flame suggests a strong coupling between the PVC and the thermo-acoustic pulsation in the noisy flame. Measurements of the stable (‘quiet’) flame (ϕ = 0.83) revealed a global fluctuation in both velocity and heat-release around 364 Hz, but no clear evidence of a PVC.     

Time resolved PIV analysis of flow over a NACA 0015 airfoil with Gurney flap

A NACA 0015 airfoil with and without a Gurney flap was studied in a wind tunnel with Re _c = 2.0 × 10⁵ in order to examine the evolving flow structure of the wake through time-resolved PIV and to correlate this structure with time-averaged measurements of the lift coefficient. The Gurney flap, a tab of small length (1–4% of the airfoil chord) that protrudes perpendicular to the chord at the trailing edge, yields a significant and relatively constant lift increment through the linear range of the C _L versus α curve. Two distinct vortex shedding modes were found to exist and interact in the wake downstream of flapped airfoils. The dominant mode resembles a Kàrmàn vortex street shedding behind an asymmetric bluff body. The second mode, which was caused by the intermittent shedding of fluid recirculating in the cavity upstream of the flap, becomes more coherent with increasing angle of attack. For a 4% Gurney flap at α = 8°, the first and second modes corresponded with Strouhal numbers based on flap height of 0.18 and 0.13. Comparison of flow around ‘filled’ and ‘open’ flap configurations suggested that the second shedding mode was responsible for a significant portion of the overall lift increment.     

General mathematical theory of large deflections of thin plates with some holes

The theoretical analysis and the numerical computations for the problem of a thin plate with large deflection and some holes become much more difficult due to the multi-valued properties of the stress functionF and the single-valued demands on the displacements. The necessary and sufficient conditions which can assureF to be single-valued are obtained in this paper. At the same time, we prove that the single-valued demands on the displacements are equivalent to 3m functional constraint equationsDC(w,F)=0, wherem is the number of holes. From these conclusions, the single-valued governing-equations of the problem of plates with large deflection and some holes are derived. It is a system of fourth order partial differential equations with 3m unknown constants and constrained equations. A numerical method for solving this problem is presented. The problem of the critical load is considered and an iterative scheme for computing the buckled states is given when a critical load λ is ‘single’.     

Very high Reynolds number boundary layers over 3D sparse roughness and obstacles: the mean flow

Hot-wire and oil-film interferometry measurements are taken for 3D rough wall boundary layers at very high Reynolds numbers (61,000 < Re θ < 120,000) with low blockage ratios, 10 < δ/H < 135, and high roughness, 100 < H ⁺ < 4,900. The results cover flows over both rough walls and over obstacles and are compared with and provide extension to recent lower Reynolds number results. The validity of the Townsend ‘wall similarity hypothesis’ in relation to consistently increasing 3D roughness is interrogated. In agreement with recent work, Schultz and Flack (J Fluid Mech 580:381–405, 2007) and Castro (J Fluid Mech 585:469–485, 2007) found that, for relatively low roughness, Townsend’s hypothesis holds for the mean velocity field. With increasing roughness, the equilibrium layer diminishes and gradually vanishes. The viscous component of the wall shear stress decreases, while the turbulent component increases as the roughness effects extend across the boundary layer.     

The development of rheometry for strain-sensitive gelling systems and its application in a study of fibrin–thrombin gel formation

This paper reports simulated sequential frequency sweep data which have been reconstructed from time resolved viscoelastic data obtained by Fourier transform mechanical spectroscopy. Comparisons of the results show that the recording of anomalous values of the stress relaxation power law exponent α at the Gel Point under ‘rapid’ gelling conditions may be due to inappropriate rheological techniques. An appropriate rheometrical criterion is established for the application of sequential frequency sweeps in order to obtain accurate values of α in the formation of strain-sensitive, rapidly formed gels. Furthermore, using appropriate rheometry, we report values of α for fibrin–thrombin gels formed by the addition of thrombin to a physiologically relevant level of human fibrinogen, and relate these values to the microstructure of the fibrin gel network in terms of a fractal dimension. The present study is the first to report a modification of the fractal characteristics of incipient clots in fibrin–thrombin gels due to the availability of thrombin. This work confirms the hypothesis that the self-similar (fractal) stress relaxation behaviour recorded at the Gel Point of samples of coagulating blood (Evans et al. 2010a, b) is associated with the microstructural characteristics of the incipient blood clot’s fibrin network.     

Positive solutions to （n-1,1） m-point boundary value problems with dependence on the first order derivative

Using the extension of Krasnoselskii＇s fixed point theorem in a cone, we prove the existence of at least one positive solution to the nonlinear nth order m-point boundary value problem with dependence on the first order derivative. The associated Green＇s function for the nth order m-point boundary value problem is given, and growth conditions are imposed on the nonlinear term f which ensures the existence of at least one positive solution. A simple example is presented to illustrate applications of the obtained results.     

Analysis of periodic-quasiperiodic nonlinear systems via Lyapunov-Floquet transformation and normal forms

In this paper a general technique for the analysis of nonlinear dynamical systems with periodic-quasiperiodic coefficients is developed. For such systems the coefficients of the linear terms are periodic with frequency ω while the coefficients of the nonlinear terms contain frequencies that are incommensurate with ω. No restrictions are placed on the size of the periodic terms appearing in the linear part of system equation. Application of Lyapunov-Floquet transformation produces a dynamically equivalent system in which the linear part is time-invariant and the time varying coefficients of the nonlinear terms are quasiperiodic. Then a series of quasiperiodic near-identity transformations are applied to reduce the system equation to a normal form. In the process a quasiperiodic homological equation and the corresponding ‘solvability condition’ are obtained. Various resonance conditions are discussed and examples are included to show practical significance of the method. Results obtained from the quasiperiodic time-dependent normal form theory are compared with the numerical solutions. A close agreement is found.     

Global dynamics behaviors for new delay SEIR epidemic disease model with vertical transmission and pulse vaccination

A robust SEIR epidemic disease model with a profitless delay and verti- cal transmission is formulated,and the dynamics behaviors of the model under pulse vaccination are analyzed.By use of the discrete dynamical system determined by the stroboscopic map,an‘infection-free’periodic solution is obtained,further,it is shown that the‘infection-free’periodic solution is globally attractive when some parameters of the model are under appropriate conditions.Using the theory on delay functional and impulsive differential equatibn,the sufficient condition with time delay for the perma- nence of the system is obtained,and it is proved that time delays,pulse vaccination and vertical transmission can bring obvious effects on the dynamics behaviors of the model. The results indicate that the delay is‘profitless’.     

Development of some hypersonic benchmark flows using CFD and experiment

Abstract. This paper presents a set of test cases in high speed aerodynamics that describe our perceived relationship between experiment and computation. Computational fluid dynamics, with sensible interpretation, can guide experimental design, so that wind tunnel studies can focus better on fundamental ‘benchmark’ studies. Likewise experimental data may be used as feed back to evaluate codes and to improve their physical modelling. Here we present several test cases, developed in our laboratory, that we regard as basic ‘building blocks’ for high speed aerodynamics. These include: design for boundary-layer/pressure-gradient interaction; cavity flows; shock-wave/boundary-layer interactions; techniques for a graduated and controlled study of three-dimensional separated flows. Received 10 October 2001/ Accepted 19 November 2002 Published online 4 February 2003 Correspondence to: R. Hillier (r.hillier@ic.ac.uk) An abridged version of this paper was presented at the 23rd Int. Symposium on Shock Waves at Fort Worth, Texas, from July 22 to 27, 2001