STOCHASTIC OPTIMAL CONTROL OF STRONGLY NONLINEAR SYSTEMS UNDER WIDE-BAND RANDOM EXCITATION WITH ACTUATOR SATURATION

A bounded optimal control strategy for strongly non-linear systems under non-white wide-band random excitation with actuator saturation is proposed. First, the stochastic averaging method is introduced for controlled strongly non-linear systems under wide-band random excitation using generalized harmonic functions. Then, the dynamical programming equation for the saturated control problem is formulated from the partially averaged Itō equation based on the dynamical programming principle. The optimal control consisting of the unbounded optimal control and the bounded bang-bang control is determined by solving the dynamical programming equation. Finally, the response of the optimally controlled system is predicted by solving the reduced Fokker-Planck-Kolmogorov （FPK） equation associated with the completed averaged Itō equation. An example is given to illustrate the proposed control strategy. Numerical results show that the proposed control strategy has high control effectiveness and efficiency and the chattering is reduced significantly comparing with the bang-bang control strategy.

Nanoparticle Transport and Coagulation in Bends of Circular Cross Section via a New Moment Method

Transport of nanoparticles and coagulation is simulated with the combination of CFD in a circular bend. The Taylor-expansion moment method(TEMOM)is employed to study dynamics of nanoparticles with Brownian motion,based on the flow field from numerical simulation.A fully developed flow pattern in the present simulation is compared with previous numerical results for validating the model and computational code.It is found that for the simulated particulate flow system,the particle mass concentration,number concentration,particle polydispersity, mean particle diameter and geometric standard deviation over cross-section increase with time.The distribution of particle mass concentration at different time is independent of the initial particle size.More particles are concen-trated at outer edge of the bend.Coagulation plays more important role at initial stage than that in the subsequent period.The increase of Reynolds number and initial particle size leads to the increase of particle number concentration.The particle polydispersity,mean particle diameter and geometric standard deviation increase with decreasing Reynolds number and initial particle size.

Stability and Drag Reduction in Transient Channel Flow of Fibre Suspension

Drag reduction features in the transition regime of channel flow with fibre suspension were analyzed in terms of the linear stability theory. The modified stability equation was obtained based on the slender-body theory and natural closure approximation. Results of the stability analysis show attenuating effects of fibre additives to the flow instability. For the cases leading to transition, drag reduction rate increases with the characteristic parameter H of fibres. The mechanism of drag reduction by fibres is revealed through the variation of velocity profile and the decrease of wall shear stress. The theoretical results are qualitatively consistent with some typical experiments.

FORMATION AND EVOLUTION OF THE STREAMWISE VORTICES IN MIXING LAYERS

The evolution of the three-dimensional time-developing mixing layer was simulated numerically using the pseudo-spectral method. The initial perturbations consisted of the two-dimensional fundamental wave and the streamwise-invariant three-dimensional disturbance. A comparison of the formations of the streamwise vortices with different amplitude functions for three-dimensional disturbances was made. In one case the results are similar to that of Rogers and Moser (1992), whereas a different way in which the quadrupole forms and sudden expansion of the rib were observed in another case. The simulation also confirms that stretching by the forming roller rather than Rayleigh centrifugal instability is responsible for the formation of the rib. Finally, numerical flow visualization results were presented. (Edited author abstract) 9 Refs.

Science Letters: Equivalent thickness of materials of fused silica and stainless steel in the flow of microtubes

The deviation of flow characteristics from the predictions of the conventional theory for microtubes was attrib- uted to the change of fluid viscosity resulted from the interactions between the molecules on solid wall and in fluid. The degree of this departure is dependent on the microtubes materials. A concept of equivalent thickness with which conventional theory can be used to predict the flow in microtubes without modifying the fluid viscosity was put forward. The values of equivalent thickness for fused silica and stainless steel materials were determined as 1.8 μm and 1.5 μm, respectively, by repeated numeri- cal simulation.

EFFECTS OF TENSOR CLOSURE MODELS AND 3-D ORIENTATION ON THE STABILITY OF FIBER SUSPENSIONS IN A CHANNEL FLOW

Three different kinds of closure model of fiber orientation tensors were applied to simulate numerically the hydrodynamic stability of fiber suspensions in a channel flow. The effects of closure models and three_dimensional (3_D) orientation distribution of fibers on the results of stability analysis were examined. It is found that the relationship of the behavior in hydrodynamic stability and the parameter of the fiber given by all the three models are the same. However, the attenuation of flow instability is most distinct using 3_D hybrid model because the orientation of the fiber departures from the flow direction, and least apparent using its 2_D counterpart for that the fibers show a tendency towards alignment with the flow direction in this case.

Study on the mixing of fluid in curved microchannels with heterogeneous surface potentials

In this paper the mixing of a sample in the curved microchannel with heterogeneous surface potentials is analysed numerically by using the control-volume-based finite difference method. The rigorous models for describing the wall potential and external potential are solved to get the distribution of wall potential and external potential, then momentum equation is solved to get the fully developed flow field. Finally the mass transport equation is solved to get the concentration field. The results show that the curved microchannel has an optimized capability of sample mixing and transport when the heterogeneous surface is located at the left conjunction between the curved part and straight part. The variation of heterogeneous surface potential ψn has more influence on the capability of sample mixing than on that of sample transport. The ratio of the curved microchanners radius to width has a comparable effect on the capability of sample mixing and transport. The conclusions above are helpful to the optimization of the design of microfluidic devices for the improvement of the efficiency of sample mixing.

Nonlinear stochastic optimal bounded control of hysteretic systems with actuator saturation

A modified nonlinear stochastic optimal bounded control strategy for random excited hysteretic systems with actuator saturation is proposed. First, a controlled hysteretic system is converted into an equivalent nonlinear nonhysteretic stochastic system. Then, the partially averaged It? stochastic differential equation and dynamical programming equation are established, respectively, by using the stochastic averaging method for quasi non-integrable Hamiltonian systems and stochastic dynamical programming principle, from which the optimal control law consisting of optimal unbounded control and bang-bang control is derived. Finally, the response of optimally controlled system is predicted by solving the Fokker-Planck-Kolmogorov (FPK) equation associated with the fully averaged It? equation. Numerical results show that the proposed control strategy has high control effectiveness and efficiency.

Stochastic optimal control of cable vibration in plane by using axial support motion

A stochastic optimal control strategy for a slightly sagged cable using support motion in the cable axial direction is proposed. The nonlinear equation of cable motion in plane is derived and reduced to the equations for the first two modes of cable vibration by using the Galerkin method. The partially averaged Ito equation for controlled system energy is further derived by applying the stochastic averaging method for quasi-non-integrable Hamiltonian systems. The dynamical programming equation for the controlled system energy with a performance index is established by applying the stochastic dynamical programming principle and a stochastic optimal control law is obtained through solving the dynamical programming equation. A bilinear controller by using the direct method of Lyapunov is introduced. The comparison between the two controllers shows that the proposed stochastic optimal control strategy is superior to the bilinear control strategy in terms of higher control effectiveness and efficiency.

Large eddy simulation of a planar jet flow with nanoparticle coagulation

Coagulation and growth of nanoparticles subject to large coherent structures in a planar jet has been explored by using large eddy simulation. The particle field is obtained by employing a moment method to approximate the nanoparticle general dynamic equa- tion. An incompressible fluid containing particles of 1 nm in diameter is projected into a particle-free ambient. The results show that the coherent structures dominate the evolution of the nanoparticle number intensity, diameter and polydispersity distributions as the jet develops. In addition, the coherent structures act to increase the diffusion of particles, and the vortex rolling-up makes the particles distributing more irregularly while the vortex pairing causes particle distributions to become uniform. As the jet travels downstream, the time-averaged particle number concentration becomes lower in the jet core and higher in the outskirts, whereas the time- averaged particle mass over the entire flow field maintains unaltered, and the time-averaged particle diameter and geometric standard deviations grow and reach their maximum on the interface of the jet region and the ambient.

Simulation of Droplet-gas Flow in the Effervescent Atomization Spray with an Impinging Plate

微滴气体流动的一个全面三维的模型被介绍与一个侵犯的盘子在冒泡的粉碎成原子水花学习水花的进化。为煤气的阶段，与 k- 骚乱模型一起的 N-S 方程被解决，考虑双向联合在微滴和煤气的阶段之间的相互作用。当 Lagrangian 实体，为从主要、第二等的决裂的微滴产生的物理的财务，微滴碰撞和结合，微滴动量和热转，驱散的微滴阶段被建模。吝啬的尺寸和在各种各样的 nozzle-to-plate 距离的把分裂成原子的微滴的统计分发是计算的。一些模拟结果与试验性的数据被作比较很好。结果证明侵犯的板的存在在微滴平均数尺寸，尺寸分发和微滴上有显著影响空间分发。air-to-liquid 比率在微滴尺寸和分发上有明显的效果。

WAVELET ANALYSIS OF COHERENT STRUCTURES IN A THREE-DIMENSIONAL MIXING LAYER

Wavelet analysis is applied to the results obtained by the direct numerical simulation of a three-dimensional (3D) mixing layer in order to investigate coherent structures in dimension of scale. First, 3D orthonormal wavelet bases are constructed, and the corresponding decomposition algorithm is developed. Then the Navier-Stokes equations are transformed into the wavelet space and the architecture for multi-scale analysis is established. From this architecture, the coarse field images in different scales are obtained and some local statistical quantities are calculated. The results show that, with the development of a mixing layer, the energy spectrum densities for different wavenumbers increase and the energy is transferred from the average flow to vortex structures in different scales. Due to the non-linear interactions between different scales, cascade processes of energy are very complex. Because vortices always roll and pair at special areas, for a definite scale, the energy is obtained from other scales at some areas while it is transferred to other scales at other areas. In addition, energy dissipation and transfer always occur where an intense interaction between vortices exists.

Orientation distribution of fibres in a channel flow of fibre suspension

The orientation and concentration distributions of fibres in laminar and turbulent channel flows are investigated numerically. The obtained results are in good agreement with the experimental data. In the laminar flow regime, more fibres orient to the flow direction as the Reynolds number increases. The shear rate of fluid around a fibre plays an important role in determining the orientation distribution of fibres, while the fibre density and the fibre aspect-ratio have marginal influence on the orientation distribution. In the turbulent regime, the orientation distribution of fibres becomes more homogeneous with the increase of Reynolds number, and the concentration profile is flatter than that in the laminar regime. The fluctuating intensity of fibre velocity in the downstream direction is larger than that in the lateral directions.

On the hydrodynamic stability of a particle-laden flow in growing flat plate boundary layer

The parabolized stability equation (PSE) was derived to study the linear stability of particle-laden flow in growing Blasius boundary layer. The stability characteristics for various Stokes numbers and particle concentrations were analyzed after solving the equation numerically using the perturbation method and finite difference. The inclusion of the nonparallel terms produces a reduction in the values of the critical Reynolds number compared with the parallel flow. There is a critical value for the effect of Stokes number, and the critical Stokes number being about unit, and the most efficient instability suppression takes place when Stokes number is of order 10. But the presence of the nonparallel terms does not affect the role of the particles in gas. That is, the addition of fine particles (Stokes number is much smaller than 1) reduces the critical Reynolds number while the addition of coarse particles (Stokes number is much larger than 1) enhances it. Qualitatively the effect of nonparallel mean flow is the same as that for the case of plane parallel flows.

Sedimentation of short cylindrical pollutants with mechanical contacts

Lattice Boltzmann method and elastic particle collision model were used to investigate the sedimentation of short cylindrical pollutant particles with mechanical contacts for varying bulk number density ε and terminal Reynolds number ReT. The corresponding experiments were performed as a comparison. The clusters of pollutant particles with an inverted “T” structure were observed, the pollutant particles for high ε and large ReT scattered wider than that for low ε and small ReT. The sedimentation velocities increased suddenly at the initial stage, then decreased drastically, after that swayed around and approached to a steady value. The time to steady state did not depend on ε and ReT. The effect of particle interactions was to hinder the average sedimentation velocity, and hindrance was directly proportional to ε. The orientation distributions of pollutant particles depended on ReT and ε, especially on the former. Both the standard deviations of vertical and horizontal velocity, the former was larger than the latter, were nearly independent on ε and ReT.

Research on the behavior of fiber orientation probability distribution function in the planar flows

The equation of two-dimensional fiber direction vector was solved theoretically to give the fiber orientation distri- bution in simple shear flow, flow with two direction shears, extensional flow and arbitrary planar incompressible flow. The Fok- ker-Planck equation was solved numerically to validify the theoretical solutions. The stable orientation and orientation period of fiber were obtained. The results showed that the fiber orientation distribution is dependent on the relative not absolute magnitude of the matrix rate-of-strain of flow. The effect of fiber aspect ratio on the orientation distribution of fiber is insignificant in most conditions except the simple shear case. It was proved that the results for a planar flow could be generalized to the case of 3-D fiber direction vector.

STABILITY ANALYSIS IN SPATIAL MODE FOR CHANNEL FLOW OF FIBER SUSPENSIONS

Different from previous temporal evolution assumption, the spatially growing mode was employed to analyze the linear stability for the channel flow of fiber suspensions. The stability equation applicable to fiber suspensions was established and solutions for a wide range of Reynolds number and angular frequency were given numerically . The results show that, the flow instability is governed by a parameter H which represents a ratio between the axial stretching resistance of fiber and the inertial force of the fluid. An increase of H leads to a raise of the critical Reynolds number, a decrease of corresponding wave number, a slowdown of the decreasing of phase velocity , a growth of the spatial attenuation rate and a diminishment of the peak value of disturbance velocity. Although the unstable region is reduced on the whole, long wave disturbances are susceptible to fibers.

Orientational Distribution of Fibres in Sheared Fibre Suspensions

Motion of fibres in sheared fibre suspensions is simulated numerically by using the lattice Boltzmann method. The orientational distributions of the fibres are presented for different Reynolds numbers, Stokes numbers, shear rate and fibre aspect ratio. Some computational results are compared with the experimental data of pipe flow, and the qualitative agreement is achieved. The results show that the orientational distributions are greatly affected by the Reynolds numbers, while relatively insensitive to the fibre aspect ratio. The Stokes number and shear rate have obvious influence on the orientation distribution.

Flow instability of nanofuilds in jet

The flow instability of nanofluids in a jet is studied numerically under various shape factors of the velocity profile, Reynolds numbers, nanoparticle mass loadings,Knudsen numbers, and Stokes numbers. The numerical results are compared with the available theoretical results for validation. The results show that the presence of nanoparticles enhances the flow stability, and there exists a critical particle mass loading beyond which the flow is stable. As the shape factor of the velocity profile and the Reynolds number increase, the flow becomes more unstable. However, the flow becomes more stable with the increase of the particle mass loading. The wavenumber corresponding to the maximum of wave amplification becomes large with the increase of the shape factor of the velocity profile, and with the decrease of the particle mass loading and the Reynolds number. The variations of wave amplification with the Stokes number and the Knudsen number are not monotonic increasing or decreasing, and there exists a critical Stokes number and a Knudsen number with which the flow is relatively stable and most unstable,respectively, when other parameters remain unchanged. The perturbation with the first azimuthal mode makes the flow unstable more easily than that with the axisymmetric azimuthal mode. The wavenumbers corresponding to the maximum of wave amplification are more concentrated for the perturbation with the axisymmetric azimuthal mode.

On the stochastic dynamics of molecular conformation

An important functioning mechanism of biological macromolecules is the transition between different conformed states due to thermal fluctuation. In the present paper,a biological macromolecule is modeled as two strands with side chains facing each other,and its stochastic dynamics including the statistics of stationary motion and the statistics of conformational transition is studied by using the stochastic averaging method for quasi Hamiltonian systems. The theoretical results are confirmed with the results from Monte Carlo simulation.