Numerical study on evolution of subharmonic varicose low-speed streaks in turbulent channel flow

The evolution of two spanwise-aligned low-speed streaks in a wall turbulent flow, triggered by the instability of the subharmonic varicose （SV） mode, is studied by a direct numerical simulation （DNS） method in a small spatial-periodic channel. The results show that the SV low-speed streaks are self-sustained at the early stage, and then transform into subharmonic sinuous （SS） low-speed streaks. Initially, the streamwise vortex sheets are formed by shearing, and then evolve into zigzag vortex sheets due to the mutual induction. As the intensification of the SV low-speed streaks becomes prominent, the tilted streamwise vortex tubes and the V-like streamwise vortex tubes can be formed simultaneously by increasing ＋~. When the SV low-speed streaks break down, new zigzag streamwise vortices will be generated, thus giving birth to the next sustaining cycle of the SV low-speed streaks. When the second breakdown happens, new secondary V-like streamwise vortices instead of zigzag streamwise vortices will be generated. Because of the sweep motion of the fluid induced by the secondary V-like streamwise vortices, each decayed low-speed streak can be divided into two parts, and each part combines with the part of another streak, finally leading to the formation of SS low-speed streaks.

The viscoelasticity of lipid shell and the hysteresis of subharmonic in liquid containing microbubbles

The viscoelasticity and subharmonic generation of a kind of lipid ultrasound contrast agent are investigated. Based on the measurement of the sound attenuation spectrum, the viscoelasticity of the lipid shell is estimated by use of an optimization method. Shear modulus Gs=10MPa and shear viscosity μs=1.49N.S/m^2 are obtained. The nonlinear oscillation of the encapsulated microbubble is studied with Church＇s model theoretically and experimentally. Especially, the dependence of subharmonic on the incident acoustic pressure is studied. The results reveal that the development of the subharmonic undergoes three stages, i.e. occurrence, growth and saturation, and that hysteresis appears in descending ramp insonation.

ON SECONDARY INSTABILITY WITH RESPECT TO THREE DIMENSIONAL SUBHARMONIC DISTURBANCES IN BOUNDARY LAYER

The general equations of secondary instability with respect to three-dimensional subharmonic disturbances are derived and applied to Blasius boundary layer in the present paper.The theoretical results of evolution and spatial distribution of subharmonic disturbances are compared with experimental data.The re- suits show the important role of the process of route to transition in low-disturbance environments,and indi- cate that spatial mode is more rational than temporal mode.

1/2 SUBHARMONIC RESONANCE OF A SHAFT WITH UNSYMMETRICAL STIFFNESS

The 1/2 subharmonic resonance of a shaft with unsymmetrical stiffness is studied. By means of the Hamilton's principle the nonlinear differential equations of motion of the rotating shaft are derived in the rotating rectangular coordinate system. Transforming the equations of motion from rotating coordinate system into stationary coordinate system and introducing a complex variable, the equation of motion in complex variable form is obtained, in which the stiffness coefficient varies periodically with time. It presents a nonlinear oscillation system under parametric excitation. By applying the method of multiple scales (MMS) the averaged equation, the bifurcating response equations and local bifurcating set are obtained. Via the theory of singularity, the stability of constant solutions is analyzed and bifurcating response curves are obtained. This study shows that the rotating shaft has rich bifurcation phenomena.

STABILITY OF SUBHARMONIC SOLUTIONS OF FIRST-ORDER HAMILTONIAN SYSTEMS WITH ANISOTROPIC GROWTH

Using the dual Morse index theory, we study the stability of subharmonic solutions of first-order autonomous Hamiltonian systems with anisotropic growth, that is, we obtain a sequence of elliptic subharmonic solutions(that is, all its Floquet multipliers lying on the unit circle on the complex plane C).

Subharmonic resonance of a clamped-clamped buckled beam with 1:1 internal resonance under base harmonic excitations

The subharmonic resonance and bifurcations of a clamped-clamped buckled beam under base harmonic excitations are investigated.The nonlinear partial integrodifferential equation of the motion of the buckled beam with both quadratic and cubic nonlinearities is given by using Hamilton’s principle.A set of second-order nonlinear ordinary differential equations are obtained by spatial discretization with the Galerkin method.A high-dimensional model of the buckled beam is derived,concerning nonlinear coupling.The incremental harmonic balance(IHB)method is used to achieve the periodic solutions of the high-dimensional model of the buckled beam to observe the nonlinear frequency response curve and the nonlinear amplitude response curve,and the Floquet theory is used to analyze the stability of the periodic solutions.Attention is focused on the subharmonic resonance caused by the internal resonance as the excitation frequency near twice of the first natural frequency of the buckled beam with/without the antisymmetric modes being excited.Bifurcations including the saddle-node,Hopf,perioddoubling,and symmetry-breaking bifurcations are observed.Furthermore,quasi-periodic motion is observed by using the fourth-order Runge-Kutta method,which results from the Hopf bifurcation of the response of the buckled beam with the anti-symmetric modes being excited.

HETEROCLINIC ORBIT AND SUBHARMONIC BIFURCATIONS AND CHAOS OF NONLINEAR OSCILLATOR

Dynamical behavior of nonlinear oscillator under combined parametric and forcing excitation, which includes yon der Pol damping, is very complex. In this paper, Melnikov's method is used to study the heteroclinic orbit bifurcations, subharmonic bifurcations and chaos in this system. Smale horseshoes and chaotic motions can occur from odd subharmonic bifurcation of infinite order in this system-far various resonant cases finally the numerical computing method is used to study chaotic motions of this system. The results achieved reveal some new phenomena.

Subharmonic response of single-degree-of-freedom linear vibroimpact system to narrow-band random excitation

The subharmonic response of a single-degree-of-freedom linear vibroimpact oscillator with a one-sided barrier to the narrow-band random excitation is investigated.The analysis is based on a special Zhuravlev transformation,which reduces the system to the one without impacts or velocity jumps,and thereby permits the applications of asymptotic averaging over the period for slowly varying the inphase and quadrature responses.The averaged stochastic equations are exactly solved by the method of moments for the mean square response amplitude for the case of zero offset.A perturbation-based moment closure scheme is proposed for the case of nonzero offset.The effects of damping,detuning,and bandwidth and magnitudes of the random excitations are analyzed.The theoretical analyses are verified by the numerical results.The theoretical analyses and numerical simulations show that the peak amplitudes can be strongly reduced at the large detunings.

Observations of semidiurnal M 2 internal tidal parametric subharmonic instability in the northeastern South China Sea

Near-diurnal vertically-standing waves with high vertical wavenumbers k z were observed in the velocity and shear fi elds from a set of 75-d long ADCP moored in the northeastern South China Sea(SCS)away from the“critical”latitude of 28.8°.These enhanced near-diurnal internal waves followed a fortnightly spring-neap cycle.However,they always happened during semidiurnal spring tides rather than diurnal springs although strong diurnal internal tides with the fortnightly spring-neap cycle were prevailing,suggesting that they were generated via subharmonic instability(PSI)of dominant semidiurnal M 2 internal tides.When two semidiurnal internal tidal waves with opposite vertical propagation direction intersected,both semidiurnal subharmonic and super harmonic waves were largely intensifi ed.The observed maximum diurnal velocity amplitudes were up to 0.25 m/s.The kinetic energy and shear spectra further suggested that frequencies of daughter waves were not always perfectly equal to M 2/2.The superposition of two daughter waves with nearly equal frequencies and nearly opposite k z in a PSI-triad leaded to the vertically-standing waves.

Research on 1:2 subharmonic resonance and bifurcation of nonlinear rotor-seal system

The 1：2 subharmonic resonance of the labyrinth seals-rotor system is inves- tigated, where the low-frequency vibration of steam turbines can be caused by the gas exciting force. The empirical parameters of gas exciting force of the Muszynska model are obtained by using the results of computational fluid dynamics （CFD）. Based on the multiple scale method, the 1：2 subharmonic resonance response of the dynamic system is gained by truncating the system with three orders. The transition sets and the local bifurcations diagrams of the dynamics system are presented by employing the singular theory analysis. Meanwhile, the existence conditions of subharmonic resonance non-zero solutions of the dynamic system are obtained, which provides a new theoretical basis in recognizing and protecting the rotor from the subharmonic resonant failure in the turbine machinery.

BIFURCATIONS OF SUBHARMONIC SOLUTIONS IN PERIODIC PERTURBATION OF A HYPERBOLIC LIMIT CYCLE

Bifurcations of subharmonic solutions of order m of a planar periodic perturbed system near a hyperbolic limit cycle are discussed. By using a Poincare map and the method of rescaling a discriminating condition for the existence of subharmonic solutions of order m is obtained. An example is given in the end of the paper.

1/3 SUBHARMONIC SOLUTION OF ELLIPTICAL SANDWICH PLATES

The problem of nonlinear forced oscillations for elliptical sandwich plates is dealt with. Based on the governing equations expressed in terms of five displacement components, the nonlinear dynamic equation of an elliptical sandwich plate under a harmonic force is derived. A superpositive-iterative harmonic balance (SIHB) method is presented for the steady-state analysis of strongly nonlinear oscillators. In a periodic oscillation, the periodic solutions can be expressed in the form of basic harmonics and bifurcate harmonics. Thus, an oscillation system which is described as a second order ordinary differential equation, can be expressed as fundamental differential equation with fundamental harmonics and incremental differential equation with derived harmonics. The 1/3 subharmonic solution of an elliptical sandwich plate is investigated by using the methods of SIHB. The SIHB method is compared with the numerical integration method. Finally, asymptotical stability of the 1/3 subharmonic oscillations is inspected.

A 1/3 PURE SUBHARMONIC SOLUTION AND TRANSIENT PROCESS FOR THE DUFFING'S EQUATION

The 1/3 subharmonic solution for the Duffing’s equation is investigated by using the methods of harmonic balance and numerical integration. The sensitivity of parameter variation for the transient process and the transient process for the perturbance initial conditions are studied. Over and above, the precision of numerical integration method is discussed and the numerical integration method is compared with the harmonic balance method. Finally, asymptotical stability of the pure subharmonic oscillations element is inspected.

FIRST SUBHARMONIC OF SOUND IN DISTURBED WATER

It is shown by this experiment that an appreciable first subharmonic component(one-half of driving frequency)can be observed when a traoelling wave of low-intensity sound goes through the disturbed water in a tank.

THE MLP METHOD FOR SUBHARMONIC AND ULTRA-HARMONIC RESONANCE SOLUTIONS OF STRONGLY NONLINEAR SYSTEMS

A new parameter transformation alpha = alpha (epsilon, n omega (0)/m, omega (l)) was defir2ed for extending the applicable range of the modified Lindstedt-Poincare method. It is suitable for determining subharmonic and ultraharmonic resonance solutions of strongly nonlinear systems. The 1/3 subharmonic and 3 ultraharmonic resonance solutions of the Duffing equation and the 1/2 subharmonic resonance solution of the Van der Pol-Mathieu equation were studied. These examples show approximate solutions are in good agreement with numerical solutions.

A Complement to the Valiron-Titchmarsh Theorem for Subharmonic Functions

The Valiron-Titchmarsh theorem on asymptotic behavior of entire functions with negative zeros has been recently generalized onto subharmonic functions with the Riesz measure on a half-line in Rn, n≥3. Here we extend the Drasin complement to the Valiron-Titchmarsh theorem and show that if u is a subharmonic function of this class and of order 0〈ρ〈1, then the existence of the limit limr→∞logu（r）/N（r）, where N（r） is the integrated counting function of the masses of u, implies the regular asymptotic behavior for both u and its associated measure.

SOME EXTENDED RESULTS OF“SUBHARMONIC RESONANCE BIFURCATION THEORY OF NONLINEAR MATHIEU EQUATION”

The authors of [1] discussed the subharmonic resonance bifurcation theory of nonlinear Mathieu equation and obtained six bifurcation diagrams in -plane. In this paper, we extended the results of[1] and pointed out that there may exist as many as fourteen bifurcation diagrams which are not topologically equivalent to each other.

On the Existence of Subharmonic Screech in Choked Circular Jets from a Sharp-Edged Orifice

Experiments are performed in choked circular hot and cold nitrogen jets issuing from a 2.44 cm diameter sharp-edged orifice at a fully expanded jet Mach number of 1.85 in an effort to investigate the character of screech phenomenon. The stagnation temperature of the cold and the hot jets are 299 K and 319 K respectively. The axial distribution of the centerline Mach number was obtained with a pitot tube, while the screech data (frequency and amplitude) at different axial and radial stations were measured with the aid of microphones. The fundamental screech frequency of the hot jet is slightly increased relative to that of the cold jet. It is concluded that temperature effects on the screech amplitude are manifested with regard to the fundamental and the subharmonic even at relatively small temperature range considered.

Infinite Subharmonic Solutions of the Forced Relativistic Oscillators

In this paper, we study the multiplicity of subharmonic solutions of the nonlinear differential equation of the forced relativistic oscillators. By using the generalized Poincaré-Birkhoff fixed point theorem, we prove that the equation has infinite subharmonic solutions provided that g satisfies at most linear growth condition.

P-symmetric Subharmonic Solutions for Nonlinear Hamiltonian Systems

In this paper,we prove that for each positive k≡1 mod m there exists a P-symmetric kmτ-periodic solution xk for asymptotically linear mτ-periodic Hamiltonian systems,which are nonautonomous and endowed with a P-symmetry.If the P-symmetric Hamiltonian function is semi-positive,one can prove,under a new iteration inequality of the Maslov-type P-index,that xk_(1) and xk_(2) are geometrically distinct for k_(1)/k_(2)≥(2n+1)m+1;and xk_(1),xk_(2) are geometrically distinct for k_(1)/k_(2)≥m+1 provided xk_(1) is non-degenerate.