Electromagnetic propagation in periodic porous structures

A variational technique is employed to compute approximate propagation constants for electromagnetic waves in a dielectric structure which is periodic in the X−Y plane and translationally invariant in the Z-direction. The fundamental cell, in the periodic structure, is composed of a pore and the surrounding host media. The pore is a circle of radius R₀ filled with a dielectric ε₁ and the host dielectric characterized by ε₂. The size of the cell is characterized by the length A which is R₀.

Two limiting cases are considered. In the first, the pore size is assumed to be much smaller than the wavelength; this limit is motivated by microwave heating of porous material. The approximate propagation constants are explicitly computed for this case and are shown to depend upon the two dielectric constants, the relative areas of the two regions in the cell, and on a modal number. They are not given by a simple mixture formula.

In the second limit, the pore size is taken to be of the same order as the wavelength; this limit is motivated by the propagation of light in a holey fiber. In this case our argument directly yields the dispersion relationship recently derived by Ferrando et al. [Opt. Lett. 24 (1999) 276], using intuitive and physical reasoning. Thus, our method puts theirs into a mathematical framework from which other approximations might be deduced.     

Flow in a cylinder driven by rotating split-disk endwalls

Flow of an incompressible viscous fluid contained in a cylindrical vessel (radius R, height H) is considered. Each of the cylinder endwalls is split into two parts which rotate steadily about the central axis with different rotation rates: the inner disk (r < r₁) rotating at Ω₁, and the outer annulus (r₁ < r < R) rotating at Ω₂. Numerical solutions to the axisymmetric Navier-Stokes equations are secured for small system Ekman numbers E ( v/(ΩH²)). In the linear regime, when the Rossby number Ro

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, the numerical results are shown to be compatible with the theoretical prediction as well as the available experimental measurements. Emphasis is placed on the results in the nonlinear regime in which Ro is finite. Details of the structures of azimuthai and meridional flows are presented by the numerical results. For a fixed Ekman number, the gross features of the flow remain qualitatively unchanged as Ro increases. The meridional flows are characterized by two circulation cells. The shear layer is a region of intense axial flow toward the endwall and of vanishing radial velocity. The thicknesses of the shear layer near r = r₁ and the Ekman layer on the endwall scale with E and E , respectively. The numerical results are consistent with these scalings.     

Traveling disturbance appearing in boundary layer transition in a Yawed cylinder

Boundary layers that develop over a body in fluid flow are in most cases three-dimensional owing to the spin, yaw, or surface curvature of the body. Therefore, the study of three-dimensional (3D) boundary-layer transition is essential to work in practical aerodynamics. The present investigation is concerned with the problem of 3D boundary layers over a yawed body. A yawed cylinder model that represents the leading edge portion of a swept wing and the mechanism of crossflow instability are investigated in detail using hot-wire velocimetry and a flow visualization technique. As a result, traveling disturbances having frequencies f₁ and f₂, which differ by about one order of magnitude, are detected in the transition region. The phase velocities and directions of travel of those disturbances are measured. Results for the low-frequency disturbance f₁ show qualitative coincidence with results numerically predicted for a crossflow unsteady disturbance. Nameley, F₁ travels nearly spanwise to the yawed cylinder and very close to the cylinder wall. The results for the high-frequency disturbance f₂ good agreement with the existing experimental results. The ₂ disturbance is found to be the high-frequency inflectional secondary instability that appears in 3D boundary layer transition in general. A two-stage transition process, where stationary crossflow vortices appear as the primary instability and a traveling inflectional disturbance is generated as a secondary instability, was observed. Secondary instability seems to play a major role in turbulent transition.     

Pressure spectra in homogeneous low Reynolds number turbulent shear flow subjected to weak rotation

In this paper, pressure spectra have been derived from the authors’ model (Eur. J. Mech., B/Fluids 12 (1) (1993) 31–42) developed by means of rapid distortion theory (RDT) of homogeneous low Reynolds number turbulent shear flow subjected to weak rotation. The combined effects of uniform shear dU₁/dx₂ and weak rotation Ω₃ on the evolution of pressure spectra have been examined in terms of the rotation number 2Ω₃/(dU₁/dx₂). It is found that the system rotation exhibits the opposite effect on the pressure field as compared with the influence of rotation on the velocity fluctuations.     

Stress state of compound polygonal plate

The constructions made of bars and plates with holes, openings and bulges of various forms are widely used in modern industry. By loading these structural elements with different efforts, there appears concentration (accumulation) of stress whose values sometimes exceeds the admissible one. The durability of the given element is defined according to the quantity of these stresses. Since the failure of details and construction itself begins from the place where the stress concentration has the greatest value.

Therefore the exact determination of stress distribution in details (bars, plates, beams) is of great scientific and practical interest and is one of the important problems of the solid fracture.

Compound details (when the nucleus of different material is soldered to the hole) are often used to decrease the stress concentration.

In the present paper, we study a stress–strain state of polygonal plate weakened by a central elliptic hole with two linear cracks info which a rigid nucleus (elliptic cylinder with two linear bulges) of different material was put in (soldered) without preload.

The problem is solved by a complex variable functions theory stated in papers [Theory of Elasticity, Higher School, Moscow, 1976, p. 276; Plane Problem of Elasticity Theory of Plates with Holes, Cuts and Inclusions, Publishing House Highest School, Kiev, 1975, p. 228; Bidimensional Problem of Elasticity Theory, Stroyizdat, Moscow, 1991, p. 352; Science, Moscow (1996) 708; MSB AH USSR OTH 9 (1948) 1371].

Kolosov–Mushkelishvili complex potential (z) and ψ(z) satisfying the definite boundary conditions are sought in the form of sums of functional series.

After making several strict mathematical transformations, the problem is reduced to the solution of a system of linear algebraic equations with respect to the coefficients of expansions of functions (z) and ψ(z).

Determining the values of (z) and ψ(z), we can find the stress components σ_r, σ_θ and τ_rθ at any point of cross-section of the plate and nucleus on the basis of the known formulae. The obtained solution is illustrated by numerical example.

Changing the parameters A₁, m₁, e, A₂, and m₂ we can get the various contour plates.

For example, if we assume m₁=0, A₁=r, then the internal contour of L₁ becomes the circle of radius r with two rectilinear cracks (for the nucleus––a rectilinear bulges).

Further, if we assume a small semi-axis of the ellipse b₁ to be equal to zero (b₁=0), then a linear crack becomes the internal contour of L₁ (and the nucleus becomes the linear rigid inclusion made of other material). For m₂=0; A₂=R, the external contour L₂ turns into the circle of radius R.

The obtained method of solution may be applied and in other similar problems of elasticity theory; tension of compound polygonal plate, torsion and bending of compound prismatic beams, etc.     

Asymptotic soliton train solutions of Kaup–Boussinesq equations

Asymptotic soliton trains arising from a ‘large and smooth’ enough initial pulse are investigated by the use of the quasiclassical quantization method for the case of Kaup–Boussinesq shallow water equations. The parameter varying along the soliton train is determined by the Bohr–Sommerfeld quantization rule which generalizes the usual rule to the case of ‘two potentials’ h₀(x) and u₀(x) representing initial distributions of height and velocity, respectively. The influence of the initial velocity u₀(x) on the asymptotic stage of the evolution is determined. Excellent agreement of numerical solutions of the Kaup–Boussinesq equations with predictions of the asymptotic theory is found.     

An application of Nagumo''s lemma to some singularly perturbed systems

The existence and asymptotic behavior as ε → 0⁺ of periodic, almost periodic, and bounded solutions of the differential system x = f(t, x, y, ε), Ωy′ = g(t, x, y, ε), are considered where x, f; are n-vectors, y, g are m-vectors and Ω = diag{ε^h₁}…, ε^h_m for integral h_i, h₁ h₂ …, h_m. The principal tools are a lemma of Nagumo which allows the construction of appropriate upper and lower solutions and the asymptotic theory of singularly perturbed linear differential systems.     

Inertial instability of the Stewartson -layer

Inertial stability of a vertical shear layer (Stewartson E^1/4-layer) on the sidewall of a cylindrical tank with respect to stationary axisymmetric perturbations is inverstigated by means of a linear theory. The stability is determined by two non-dimensional parameters, the Rossby number Ro = U/2ΩL and Ekman number E = v/ΩH², where U and L = (E/4)1/4H are the characteristic velocity and width of the shear layer, respectively, Ω the angular velocity of the basic rotation, v the kinematic viscosity and H the depth of the tank.

For a given Ekman number, the flow is more unstable for larger values of the Rossby number. For E = 10⁻⁴, which is a typical value of the Ekman number realized in rotating tank experiments, the critical Rossby number Ro_c for instability and the critical axial wavenumber m_c non-dimensionalized by L⁻¹ are found to be 1.3670 and 8.97, respectively. The value of Ro_c increases and that of m_c decreases with increasing E.     

On the dispersion of a solute in Poiseuille flow of a third-grade fluid in a channel

Gill and Sankarasubramanian's analysis of the dispersion of Newtonian fluids in laminar flow between two parallel walls are extended to the flow of non-Newtonian viscoelastic fluid (known as third-grade fluid) using a generalized dispersion model which is valid for all times after the solute injection. The exact expression is obtained for longitudinal convective coefficient K₁(Γ), which shows the effect of the added viscosity coefficient Γ on the convective coefficient. It is seen that the value of the K₁(Γ) for Γ≠0 is always smaller than the corresponding value for a Newtonian fluid. Also, the effect of the added viscosity coefficient on the K₂(t,Γ) (which is a measure of the longitudinal dispersion coefficient of the solute) is explored numerically. Finally, the axial distribution of the average concentration C_m of the solute over the channel cross-section is determined at a fixed instant after the solute injection for several values of the added viscosity coefficient.     

Group properties of utt=[F(u)ux]x

The group properties and the associated Lie algebra are developed for the subject quasilinear wave equation, for arbitrary f[fεC²(R), f > 0, f ≠ 0]. From the resulting information sets of explicit invariant solutions are constructed for wave propagation in gases and for the transonic equation.     

Numerical prediction of locally forced turbulent separated and reattaching flow

An unsteady numerical simulation was performed for locally forced separated and reattaching flow over a backward-facing step. The local forcing was given to the separated and reattaching flow by means of a sinusoidally oscillating jet from a separation line. A version of the k––f_μ model was employed, in which the near-wall behavior without reference to distance and the nonequilibrium effect in the recirculation region were incorporated. The Reynolds number based on the step height (H) was fixed at Re_H=33 000, and the forcing frequency was varied in the range 0St_H2. The predicted results were compared and validated with the experimental data of Chun and Chun. It was shown that the unsteady locally forced separated and reattaching flows are predicted reasonably well with the k––f_μ model. To characterize the large-scale vortex evolution due to the local forcing, numerical flow visualizations were carried out.     

Effusing core at the center of A potential vortex

Experiments were carried out to measure the base pressure distribution of a flow field induced by a potential vortex with its axis normal to a stationary disk. The center base pressure coefficient of the vortex, C₀(0), was found to be proportional to Reynolds number from Re = 2.0 × 10³ to Re > 2.5 × 10⁴, where Re is based on the disk radius and azimuthal velocity at the disk edge. This behavior of C₀(0) is at variance with the experimental results of Phillips (Phys. Fluids, 27, 2215, 1984) and Khoo (M. Eng. Thesis, Natl. Univ. Singapore, 1984), which showed vastly different trends depending on Re. Plausible reasons are suggested for the apparent discrepancies observed. Finally, the extent of the effusing core at the center, r₁ (taken to be the radial position where departure from the outer potential flow took place), was found to be proportional to Re^−1/2 for all Re values considered.     

A study of properties of vortex stretching and enstrophy generation in numerical and laboratory turbulence

Dynamically relevant alignments are used in order to show that regions with weak vorticity are not structureless, non-Gaussian and dynamically not passive. for example, the structure of vorticity in quasi-homogeneous/isotropic turbulent flows is associated with strong alignment between vorticity ω and the eigenvectors of the rate of strain tensor λ_i (especially — but not only — between ω and λ₂) rather than with intense vorticity only. Consequently, much larger regions of turbulent flow than just those with intense vorticity are spatially structured. The whole flow field — even with the weakest measurable enstrophy — is strongly non-Gaussian, which among other things is manifested in strong alignment between vorticity and the vortex stretching vector W_i ≡ ω_jS_ij. It is shown that the quasi-two-dimensional regions corresponding to large cos(ω, λ₂) are qualitatively different from purely two-dimensional ones, e.g. in that they possess essentially nonvanishing enstrophy generation, which is larger than its mean for the whole field.     

Numerical prediction of locally forced turbulent boundary layer

An unsteady numerical simulation was performed to analyze flow structure behind a local suction/blowing in a flat-plate turbulent boundary layer. The local forcing was given to the boundary layer flow by means of a sinusoidally oscillating jet. A version of the unsteady k––f_μ model [Fluid Dyn. Res. 26 (6) (2000) 421] was employed. The Reynolds number based on the momentum thickness was about Re_θ=1700. The forcing frequency was varied in the range 0.011f⁺0.044 with a fixed forcing amplitude A_o=0.4. The predicted results were compared and validated with the experimental data. It was shown that the unsteady locally forced boundary layer flow is predicted well by the k––f_μ model. The time-dependent numerical flow visualizations were demonstrated during one period of the local forcing. The effect of the pitch angle of local forcing on the reduction of skin friction was examined.     

Mixed convection of an incompressible viscous fluid in a porous medium past a hot vertical plate

This paper analyses steady two-dimensional mixed convection of an imcompressible viscous fluid in a porous medium past a hot vertical plate. Assuming Darcy-Brinkman model for the flow in a porous medium, the boundary layer equations are integrated numerically to obtain the non-similar solution for the velocity and temperature distribution for several values of the permeability and viscous dissipation parameters. It is shown that for a fixed value of Prandtl number Pr and dissipation parameter E, the skin-friction at the plate decreases with increase in the permeability parameter K₁. However for the same value or Pr and E, the heat transfer rate at the plate increases with increasing K₁. The dimensionlcss velocity and temperature functions in the flow are plotted for several values of E and K₁ with Pr = 0.73. It is also shown that for fixed values of K₁, and KPr, the skin-friction increases with increase in the dissipation parameter E.     

圆盘近尾迹流场相关谱分析

对Re=22 000 时径厚比D/h=5 圆盘近尾迹开展大涡模拟数值研究. 通过对x/D=1, 2 和8 处脉动速度进行快速傅里叶变换（fast Fourier transformation, FFT）,发现3 个特征频率：斯特劳哈尔数St₂=0.123 为自然脱落频率,与文献结果相符;较小频率St₁=0.035,与回流区伸缩和剪切层附近涡旋脱落点的旋转相关;高频率St₃=1.3～1.7 则与剪切层湍流结构相关. 通过分析截面r/D=2.8 圆周上两点间流向速度相关系数、相干谱和相位谱,发现相关系数受涡旋脱落影响出现以30°或45°为周期的正负波动,表明轴面上涡旋脱落点具有随机性.     

Maximum amplitude of modulated wavetrain

For water wave predictions, it is of practical importance to know the size of the biggest wave that emerges from a given uniform wavetrain when it experiences Benjamin-Feir's modulational instability. As far as we are aware, the series of experiments performed by Su and Green [5] are the only work that has pursued this problem so far. In this work their experimental result is compared with theoretical predictions given by the nonlinear Schrödinger equation as well as with the numerical simulation of the fully nonlinear water wave equation. The result of the fully nonlinear simulation shows that the height of the biggest wave reaches more than 3.5 times the initial amplitude of the wavetrain at the maximum modulation when the initial steepness a₀k₀ of the wavetrain is 0.125.     

Flow regime transitions due to cavitation in the flow through an orifice

This paper presents both experimental and theoretical aspects of the flow regime transitions caused by cavitation when water is passing through an orifice. Cavitation inception marks the transition from single-phase to two-phase bubbly flow; choked cavitation marks the transition from two-phase bubbly flow to two-phase annular jet flow.

It has been found that the inception of cavitation does not necessarily require that the minimum static pressure at the vena contracta downstream of the orifice, be equal to the vapour pressure liquid. In fact, it is well above the vapour pressure at the point of inception. The cavitation number [σ = (P₃ − P_v)/(0.5 pV²); here P₃ is the downstream pressure, P_v is the vapour pressure of the liquid, ρ is the density of the liquid and V is the average liquid velocity at the orifice] at inception is independent of the liquid velocity but strongly dependent on the size of the geometry. Choked cavitation occurs when this minimum pressure approaches the vapour pressure. The cavitation number at the choked condition is a function of the ratio of the orifice diameter (d) to the pipe diameter (D) only. When super cavitation occurs, the dimensionless jet length [L/(D - d); where L is the dimensional length of the jet] can be correlated by using the cavitation number. The vaporization rate of the surface of the liquid jet in super cavitation has been evaluated based on the experiments.

Experiments have also been conducted in which air was deliberately introduced at the vena contracta to simulate the flow regime transition at choked cavitation. Correlations have been obtained to calculate the critical air flow rate required to cause the flow regime transition. By drawing an analogy with choked cavitation, where the air flow rate required to cause the transition is zero, the vapour and released gas flow rate can be predicted.     

Hole diameter effect on flow characteristics of wake behind porous fences having the same porosity

The hole diameter effect on the flow characteristics of wake behind porous fences has been investigated experimentally in a circulating water channel having a test section of 300^w×200^h×1200^l (mm). Three porous fences having different hole diameters of d=1.4,2.1,2.8 mm were tested in this study, but they have the same =38.5% geometric porosity. One thousand instantaneous velocity fields for each fence were measured consecutively by the hybrid PTV system employing a high-speed CCD camera. Free stream velocity was fixed at 10 cm/sec and the corresponding Reynolds number based on the fence height was Re=2,985. Consequently, the fence with the smallest hole diameter d=1.4 mm (d_1.4) decreases the streamwise velocity component and increases the vertical velocity component. Among the three hole diameters tested in this study, the d_1.4 fence has the largest turbulence intensity in the shear layer developed from the fence top. Regardless of the hole diameter, however, all three fences having the same porosity reduce the reduction of turbulent intensity in the lower region below the fence height (y/H<1).