MODAL SYNTHESIS METHOD FOR NORM COMPUTATION OF H∞ DECENTRALIZED CONTROL SYSTEMS （ I ）

When using H∞ techniques to design decentralized controllers for large systems,the whole system is divided into subsystems, which are analysed using H∞ control theorybefore being recombined. An analogy was established with substructural analysis instructural mechanics, in which H∞ decentralized control theory corresponds to substructuralmodal synthesis theory so that the optimal H∞ norm of the whole system corresponds to thefundamental vibration frequency of the whole structure. Hence, modal synthesismethodology and the extended Wittrick-Williams algorithm were transplanted from structuralmechanics to compute the optimal H∞ norm of the control system. The orthogonality and theexpansion theorem of eigenfunctions of the subsystems H∞ control are presented in part(I) of the paper. The modal synthesis method for computation of the optimal H∞ norm ofdecentralized control systems and numerical examples are presented in part (Ⅱ).     

MODAL SYNTHESIS METHOD FOR NORM COMPUTATION OF H∞ DECENTRALIZED CONTROL SYSTEMS （II）

When using H∞ techniques to design decentralized controllers for large systems, the whole system is divided into subsystems, which are analysed using H∞ control theorybefore being recombined. An analogy was established with substructural analysis instructural mechanics, in which H∞ decentralized control theory corresponds to substructuralmodal synthesis theory so that the optimal H∞ norm of the whole system corresponds to thefundamental vibration frequency of the whole structure. Hence, modal synthesismethodology and the extended Wittrick-Williams algorithm were transplanted from structuralmechanics to compute the optimal H∞ norm of the control system. The orthogonality and theexpansion theorem of eigenfunctions of the subsystems H~ control are presented in part (I)of the paper. The modal synthesis method for computation of the optimal H∞ norm ofdecentralized control systems and numerical examples are presented in part (Ⅱ).     

Effects of solid and slotted baffles on the convection characteristics of backward-facing step flow in a channel

This study investigates the enhancement of the laminar forced convection characteristics of backward-facing step flow in a two-dimensional channel through the installation of solid and slotted baffles onto the channel wall. The effects of the height of baffle H _b, inclination of baffle installation ϕ_b, height of slot in baffle H _t, inclination of slot in baffle ϕ_t, and distance between the backward-facing step and baffle D on the flow structure, temperature distribution and Nusselt number variation for the system at various Re are numerically explored. Results show that a slotted baffle can enhance the average Nusselt number for the heating section of channel plate by the maximum 190% when Pr=0.7, H _s=0.5, L=5, H _b ≤ 0.3, W _b ≤ 0.2, 0.1 ≤ D ≤ 0.5, 0° ≤ ϕ_b ≤ 45°, H _t ≤ 0.1, 0° ≤ ϕ_t ≤ 45° and 50 ≤ Re ≤ 400. As for the solid baffle, the enhancement may be up by 230%. The solid baffle might cause the re-separation of main stream, and consequently result in poor local heat transfer coefficient in the end region of heating section. This disadvantage can be obviously improved as the baffle is slotted. Besides the penalty of increase in pressure drop due to the baffle installation is much higher for the situation with solid baffle.     

Measurement and scaling analysis of critical energy for direct initiation of gaseous detonations

In this paper, the critical energies required for direct initiation of spherical detonations in four gaseous fuels (C₂H₂, C₂H₄, C₃H₈ and H₂)–oxygen mixtures at different initial pressures, equivalence ratios and with different amounts of argon dilution are reported. Using these data, a scaling analysis is performed based on two main parameters of the problem: the explosion length R _o that characterizes the blast wave and a characteristic chemical length that characterizes the detonation. For all the undiluted mixtures considered in this study, it is found that the relationship is closely given by R_o ? 26 l{R_{\rm o} \approx 26 \lambda} , where λ is the characteristic detonation cell size of the explosive mixture. While for C₂H₂–2.5O₂ mixtures highly diluted with argon, in which cellular instabilities are shown to play a minor role on the detonation propagation, the proportionality factor increases to 37.3, 47 and 54.8 for 50, 65 and 70% argon dilution, respectively. Using the ZND induction length Δ _I as the characteristic chemical length scale for argon diluted or ‘stable’ mixtures, the explosion length is also found to scale adequately with R_o ? 2320 D_I{R_{\rm o} \approx 2320 \Delta_I} .     

The laminar hole pressure for Newtonian fluids

For flows with wall turbulence the hole pressure, P _H , was shown empirically by Franklin and Wallace (J Fluid Mech, 42, 33–48, 1970) to depend solely on R ₊, the Reynolds number constructed from the friction velocity and the hole diameter b. Here this dependence is extended to the laminar regime by numerical simulation of a Newtonian fluid flowing in a plane channel (gap H) with a deep tap hole on one wall. Calculated hole pressures are in good agreement with experimental values, and for two hole sizes are well represented by: (P _H −P _HS )/τ _w = √(k ² + c ² R ₊²)−k, where the Stokes hole pressure P _HS /τ _w = s (b/H)³, k, c, s are fitted constants, and τ _w is the wall shear stress.     

Robust H∝ filtering for discrete-time impulsive systems with uncertainty

This paper investigates robust filter design for linear discrete-time impulsive systems with uncertainty under H∞ performance. First, an impulsive linear filter and a robust H∞ filtering problem are introduced for a discrete-time impulsive systems. Then, a sufficient condition of asymptotical stability and H∞ performance for the filtering error systems are provided by the discrete-time Lyapunov function method. The filter gains can be obtained by solving a set of linear matrix inequalities （LMIs）. Finally, a numerical example is presented to show effectiveness of the obtained result.     

Incompressible Fluids in Thin Domains with Navier Friction Boundary Conditions (I)

This study is motivated by problems arising in oceanic dynamics. Our focus is the Navier–Stokes equations in a three-dimensional domain Ω_ɛ, whose thickness is of order O(ɛ) as ɛ → 0, having non-trivial topography. The velocity field is subject to the Navier friction boundary conditions on the bottom and top boundaries of Ω_ɛ, and to the periodicity condition on its sides. Assume that the friction coefficients are of order O(ɛ^3/4) as ɛ → 0. It is shown that if the initial data, respectively, the body force, belongs to a large set of H¹(Ω_ɛ), respectively, L²(Ω_ɛ), then the strong solution of the Navier–Stokes equations exists for all time. Our proofs rely on the study of the dependence of the Stokes operator on ɛ, and the non-linear estimate in which the contributions of the boundary integrals are non-trivial.     

Thermal convection in a porous layer: Effects of anisotropy and surface boundary conditions

The theory describing the onset of convection in a homogeneous porous layer bounded above and below by isothermal surfaces is extended to consider an upper boundary which is partly permeable. The general boundary condition p + λ ∂p/∂n = constant is applied at the top surface and the flow is investigated for various λ in the range 0 ⩽ λ < ∞. Estimates of the magnitude and horizontal distribution of the vertical mass and heat fluxes at the surface, the horizontally-averaged heat flux (Nusselt number) and the fraction of the fluid which recirculates within the layer are found for slightly supercritical conditions. Comparisons are made with the two limiting cases λ → ∞, where the surface is completely impermeable, and λ = 0, where the surface is at constant pressure. Also studied are the effects of anisotropy in permeability, ξ = K _H /K _V , and anisotropy is thermal conductivity, η = k _H /k _V , both parameters being ratios of horizontal to vertical quantities. Quantitative results are given for a wide variety of the parameters λ, ξ and η. In the limit ξ/η → 0 there is no recirculation, all fluid being converted out of the top surface, while in the limit ξ/η → ∞ there is full recirculation.     

Non-linear Surface Superconductivity for Type II Superconductors in the Large-Domain Limit

 The Ginzburg-Landau model for superconductivity is considered in two dimensions. We show, for smooth bounded domains, that superconductivity remains concentrated near the surface when the applied magnetic field is decreased below H _C3 as long as it is greater than H _C2 . We demonstrate this result in the large-domain limit, i.e, when the domain's size tends to infinity. Additionally, we prove that for applied fields greater than H _C2 , the only solution in ℝ² satisfying normal-state conditions at infinity is the normal state. The above results have been proved in the past for the linear case. Here we prove them for non-linear problems. (Accepted May 7, 2002) Published online November 12, 2002 Communicated by D. KINDERLEHRER     

Fully developed flow of a viscoelastic film down a vertical cylindrical or planar wall

The one-dimensional, gravity-driven film flow of a linear (l) or exponential (e) Phan-Thien and Tanner (PTT) liquid, flowing either on the outer or on the inner surface of a vertical cylinder or over a planar wall, is analyzed. Numerical solution of the governing equations is generally possible. Analytical solutions are derived only for: (1) l-PTT model in cylindrical and planar geometries in the absence of solvent, b o [(h)\tilde]_s/([(h)\tilde]_s +[(h)\tilde]_p)=0\beta\equiv {\tilde{\eta}_s}/\left({\tilde{\eta}_s +\tilde{\eta}_p}\right)=0, where [(h)\tilde]_p\widetilde{\eta}_p and [(h)\tilde]_s\widetilde{\eta}_s are the zero-shear polymer and solvent viscosities, respectively, and the affinity parameter set at ξ = 0; (2) l-PTT or e-PTT model in a planar geometry when β = 0 and x 1 0\xi \ne 0; (3) e-PTT model in planar geometry when β = 0 and ξ = 0. The effect of fluid properties, cylinder radius, [(R)\tilde]\tilde{R}, and flow rate on the velocity profile, the stress components, and the film thickness, [(H)\tilde]\tilde{H}, is determined. On the other hand, the relevant dimensionless numbers, which are the Deborah, De=[(l)\tilde][(U)\tilde]/[(H)\tilde]De={\tilde{\lambda}\tilde{U}}/{\tilde{H}}, and Stokes, St=[(r)\tilde][(g)\tilde][(H)\tilde]²/([(h)\tilde]_p +[(h)\tilde]_s )[(U)\tilde]St=\tilde{\rho}\tilde{g}\tilde{\rm H}^{2}/\left({\tilde{\eta}_p +\tilde{\eta}_s} \right)\tilde{U}, numbers, depend on [(H)\tilde]\tilde{H} and the average film velocity, [(U)\tilde]\widetilde{U}. This makes necessary a trial and error procedure to obtain [(H)\tilde]\tilde{H} a posteriori. We find that increasing De, ξ, or the extensibility parameter ε increases shear thinning resulting in a smaller St. The Stokes number decreases as [(R)\tilde]/[(H)\tilde]{\tilde{R}}/{\tilde{H}} decreases down to zero for a film on the outer cylindrical surface, while it asymptotes to very large values when [(R)\tilde]/[(H)\tilde]{\tilde{R}}/{\tilde{H}} decreases down to unity for a film on the inner surface. When x 1 0\xi \ne 0, an upper limit in De exists above which a solution cannot be computed. This critical value increases with ε and decreases with ξ.     

Optimization of the deflagration to detonation transition: reduction of length and time of transition

The aim of this experimental investigation is the study of Deflagration to Detonation Transition (DDT) in tubes in order to (i) reduce both run-up distance and time of transition (L _DDT and t _DDT) in connection with Pulsed Detonation Engine applications and to (ii) attempt to scale L _DDT with λ_CJ (the detonation cellular structure width). In DDT, the production of turbulence during the long flame run-up can lead to L _DDT values of several meters. To shorten L _DDT, an experimental set-up is designed to quickly induce highly turbulent initial flow. It consists of a double chamber terminated with a perforated plate of high Blockage Ratio (BR) positioned at the beginning of a 26 mm inner diameter tube containing a “Shchelkin spiral” of BR ≈ 0.5. The study involves stoichiometric reactive mixtures of H₂, CH₄, C₃H₈, and C₂H₄ with oxygen and diluted with N₂ in order to obtain the same cell width λ_CJ≈10 mm at standard conditions. The results show that a shock-flame system propagating with nearly the isobaric speed of sound of combustion products, called the choking regime, is rapidly obtained. This experimental set-up allows a L _DDT below 40 cm for the mixtures used and a ratio L _DDT/λ_CJ ranging from 23 to 37. The transition distance seems to depend on the reduced activation energy (E _a/RT _c) and on the normalized heat of reaction (Q/a ₀ ²). The higher these quantities are, the shorter the ratio L _DDT/λ_CJ is. PACS 47.40.Rs · 47.60.+i · 47.70.Pq · 47.80.CbThis paper was based on the work that was presented at the 19th International Colloquium on the Dynamics of Explosions and Reactive Systems, Hakone, Japan, July 27–August 1, 2003.     

On the convergence of successive Galerkin approximation for nonlinear output feedback <Emphasis Type="Italic">H</Emphasis><Subscript>∞</Subscript> control

Although the formulation of the nonlinear theory of H _∞ control has been well developed, solving the Hamilton–Jacobi–Isaacs equation remains a challenge and is the major bottleneck for practical application of the theory. Several numerical methods have been proposed for its solution. In this paper, results on convergence and stability for a successive Galerkin approximation approach for nonlinear H _∞ control via output feedback are presented. An example is presented illustrating the application of the algorithm.     

Asymptotic approximation for the solution of a boundary-value problem with varying type of boundary conditions in a thick two-level junction

We consider a mixed boundary-value problem for a Poisson equation in a plane two-level junction Ω_ε that is the union of a domain Ω₀ and a large number 3N of thin rods with thickness of order . The thin rods are divided into two levels depending on their length. In addition, the thin rods from each level are ε-periodically alternated. The homogeneous Dirichlet conditions and inhomogeneous Neumann conditions are given on the sides of the thin rods from the first level and the second level, respectively. Using the method of matched asymptotic expansions and special junction-layer solutions, we construct an asymptotic approximation for the solution and prove the corresponding estimates in the Sobolev space H ¹(Ω_ε) as ε → 0 (N → +∞). Published in Neliniini Kolyvannya, Vol. 9, No. 3, pp. 336–355, July–September, 2006.     

Influence of unsteady wake on a turbulent separation bubble

 An experimental study was made of turbulent separated and reattaching flow over a blunt body, where unsteady wake was generated by a spoked-wheel type of wake generator with cylindrical rods. The influence of unsteady wake was scrutinized by altering the rotation direction (clockwise and counter-clockwise) and the normalized passing frequency (0 ≤ St _H  ≤ 0.20). The Reynolds number based on the cylindrical rod was Re _d =375. A phase-averaging technique was employed to characterize the unsteady wake. The effect of different rotation directions, which gave a significant reduction of x _R , was examined in detail. The wall pressure fluctuations on the blunt body were analyzed in terms of the spectrum and the coherence. Received: 15 January 2001 / Accepted: 17 July 2001     

Exact Connections of Warping Fields and Torsional Rigidities of Symmetric Composite Bars

Saint-Venant's torsion of symmetric cylindrical bars consisting of two or four homogeneous phases is studied. A symmetric section is meant that the cross section of the cylindrical bar possesses reflectional symmetry with respect to one or more axes. Each constituent region may have different shear modulus. The idea of the analysis is to superimpose suitably reflected potentials to obtain the torsion solution of the same composite section but with different moduli. For two-phase sections, we show that, if the warping fields for a given symmetric section with phase shear moduli μ₁ and μ₂ are known a priori, then the warping fields for the same configuration but with a different set of constituent moduli μ₁ ^′ and μ₂ ^′ are readily found through simple linear superpositions. Further, suppose that the torsional rigidities T(μ₁,μ₂) and T(μ₁ ^′,μ₂ ^′) for any two sets of phase moduli can be measured by some experimental tests or evaluated through numerical procedures, then the torsional rigidity for any other combinations of constituent moduli T(μ₁ ^′′,μ₂ ^′′) can be exactly determined without any recourse to the field solutions of governing differential equations. Similar procedures can be applied to a 4-phase symmetric section. But the coefficients of superposition are only found for a few branches. Specifically, we find that depending on the conditions of μ and μ^′, admissible solutions can be divided into three categories. When the correspondence between the warping field is known to exist, a link between the torsional rigidities can be established as well. This revised version was published online in June 2006 with corrections to the Cover Date.     

Asymptotics for Turbulent Flame Speeds of the Viscous G-Equation Enhanced by Cellular and Shear Flows

G-equations are well-known front propagation models in turbulent combustion which describe the front motion law in the form of local normal velocity equal to a constant (laminar speed) plus the normal projection of fluid velocity. In level set formulation, G-equations are Hamilton–Jacobi equations with convex (L ¹ type) but non-coercive Hamiltonians. Viscous G-equations arise from either numerical approximations or regularizations by small diffusion. The nonlinear eigenvalue [`(H)]{\bar H} from the cell problem of the viscous G-equation can be viewed as an approximation of the inviscid turbulent flame speed s <sub>T</sub>. An important problem in turbulent combustion theory is to study properties of s <sub>T</sub>, in particular how s <sub>T</sub> depends on the flow amplitude A. In this paper, we study the behavior of [`(H)]=[`(H)](A,d){\bar H=\bar H(A,d)} as A → + ∞ at any fixed diffusion constant d > 0. For cellular flow, we show that

Measurement of length-scale and solution of cantilever beam in couple stress elasto-plasticity

Owing to the absence of proper analytical solution of cantilever beams for couple stress/strain gradient elasto-plastic theory, experimental studies of the cantilever beam in the micro-scale are not suitable for the determination of material length-scale. Based on the couple stress elasto-plasticity, an analytical solution of thin cantilever beams is firstly presented, and the solution can be regarded as an extension of the elastic and rigid-plastic solutions of pure bending beam. A comparison with numerical results shows that the current analytical solution is reliable for the case of σ0 〈〈 H 〈〈 E, where σ0 is the initial yield strength, H is the hardening modulus and E is the elastic modulus. Fortunately, the above mentioned condition can be satisfied for many metal materials, and thus the solution can be used to determine the material length-scale of micro-structures in conjunction with the experiment of cantilever beams in the micro-scale.     

Numerical study of transient laminar natural convection heat transfer over a sphere subjected to a constant heat flux

Transient laminar natural convection over a sphere which is subjected to a constant heat flux has been studied numerically for high Grashof numbers (10⁵ ≤ Gr ≤ 10⁹) and a wide range of Prandtl numbers (Pr = 0.02, 0.7, 7, and 100). A plume with a mushroom-shaped cap forms above the sphere and drifts upward continuously with time. The size and the level of temperature of the transient cap and plume stem decrease with increasing Gr and Pr. Flow separation and an associated vortex may appear in the wake of the sphere depending on the magnitude of Gr and Pr. A recirculation vortex which appears and grows until “steady state” is attained was found only for the very high Grashof numbers (10⁵ ≤ Gr ≤ 10⁹) and the lowest Prandtl number considered (Pr = 0.02). The appearance and subsequent disappearance of a vortex was observed for Gr = 10⁹ and Pr = 0.7. Over the lower hemisphere, the thickness of both the hydrodynamic (δ_H) and the thermal (δ_T) boundary layers remain nearly constant and the sphere surface is nearly isothermal. The surface temperature presents a local maximum in the wake of the sphere whenever a vortex is established in the wake of the sphere. The surface pressure recovery in the wake of the sphere increases with decreasing Pr and with increasing Gr. For very small Pr, unlike forced convection, the ratio δ_T/δ_H remains close to unity. The results are in good agreement with experimental data and in excellent agreement with numerical results available in the literature. A correlation has also been presented for the overall Nusselt number as a function of Gr and Pr.     

The Effect of Sweep-Angle Variation on the Turbulence Structure in a Separated,Three-Dimensional Flow

A three-dimensional separated flow behind a swept, backward-facing step is investigated by means of DNS for Re _H = C _∞ H/ν = 3000 with the purpose to identify changes in the statistical turbulence structure due to a variation of the sweep angle α from 0° up to 60°. With increasing sweep angle, the near-wall turbulence structure inside the separation bubble and downstream of reattachment changes due to the presence of an edge-parallel mean flow component W. Turbulence production due to the spanwise shear ∂W/∂y at the wall becomes significant and competes with the processes caused by impingement of the separated shear-layer. Changes due to a sweep angle variation can be interpreted in terms of two competing velocity scales which control the global budget of turbulent kinetic energy: the step-normal component U _∞ = C _∞cosα throughout the separated flow region and the velocity difference C _∞ across the entire shear-layer downstream of reattachment. As a consequence, the significance of history effects for the development into a two-dimensional boundary layer decreases with increasing sweep angle. For α ≥50°, near-wall streaks tend to form inside the separated flow region. Received 7 November 2000 and accepted 9 July 2002 Published online 3 December 2002 RID="*" ID="*" Part of this work was funded by the Deutsche Forschungsgemeinschaft within Sfb 557. Computer time was provided by the Konrad-Zuse Zentrum (ZIB), Berlin. Communicated by R.D. Moser