Computation of Fractional Order Derivative and Integral via Power Series Expansion and Signal Modelling

The three techniques of s-to-z transform, power series expansion (PSE) and signal modelling are combined to develop a new procedure for efficiently computing the fractional order derivatives and integrals of discrete-time signals. A mapping function between the s-plane and the z-plane is first chosen, and then a PSE of this mapping function raised to fractional order is performed to get the desired infinite impulse response of the ideal digital fractional operator. Finally, the desired impulse response is modelled as the impulse response of a linear invariant system whose rational transfer function is determined using deterministic signal modelling techniques. Three non-iterative techniques, namely Padé, Prony and Shanks’ methods have been considered in this paper. Using Al-Alaoui’s rule as s-to-z transform, computation examples show that both Prony and Shanks’ method can achieve more accurate fractional differentiation and integration than Padé method which is equivalent to continued fraction expansion technique.     

Some generalizations of the Schwarz-Christoffel mapping formula

Summary The Schwarz-Christoffel formula for the mapping of a polygon in thez-plane on an upper half-plane (thew-plane) is extended to deal with singlyconnected domains of quite general shape. The mapping problem in the general ease is shown to depend on the solution of an awkward integrodifferential equation and an iterative method of finding this solution is indicated. Two further generalizations are made to the formula; these are (i) the boundary of the singly-connected domain in thez-plane is mapped on to afinite interval of the real axis of thew-plane instead of the whole of it, and (ii) the formula is extended to deal with doubly-connected domains.Paper, read at the first annual general meeting of the Australian Mathematical Society at Sydney, August, 1957.     

Moderately Strong Vorticity in a Bathtub-Type Flow

Theoretical and numerical analysis is performed for an inviscid axisymmetric vortical bathtub-type flow. The level of vorticity is kept high so that the image of the flow on the radial–axial plane (r–z plane)is not potential. The most significant findings are: (1) the region of validity of the strong vortex approximation is separated from the drain by a buffer region, (2) the power-law asymptote of the stream function, specified by Δψ∼r ^4/3Δz, appears near the axis when vorticity in the flow is sufficiently strong and (3) the local Rossby number in the region of the power-law is not very sensitive to the changes of the initial vorticity level in the flow and the global Rossby number. Received 3 April 2000 and accepted 29 September 2000     

Analytical solution to 2D problem for an anticline-diverted brine flow with a floating hydrocarbon trap

Explicit rigorous solution to a steady-state J. Bear (Dynamics of Fluids in Porous Media. Elsevier, New York, 1972) problem of brine flow in an anticline homogeneous isotropic rock with a gas cap under the anticline crest is obtained by the methods of complex analysis. The stagnant hydrocarbon volume is separated from the subjacent moving brine by a sharp interface, which is a free boundary “hanging” in the formation with the loci of the anticline roof-attachment and roof-detachment points, as well as of an inflexion point, a priori unknown. Mathematically, a conformal mapping of the complex potential strip and integral representation of the Hilbert problem for the inversed complex Darcian velocity are used to obtain the physical coordinate, complex velocity and complex potential as functions of an auxiliary variable. The interfaces are plotted for various incident brine flow rates, angles of dipping anticline flanks and gas pressure. For a gas trap comparisons with the interface calculated by the Dupuit–Forchheimer approximation are carried out.     

Shock wave structures ahead of nonuniform fan cascades

We study shock wave structures (SWS), consisting of shock waves and expansion waves between them, that occur in supersonic flow past nonuniform fan cascades when the velocity component normal to their front (“axial” component) is subsonic. The cascade nonuniformity is due to the scatter in the setting angles of identical blades, either sharp or blunt. A result of the uniformity is the generation of combined noise, whose frequencies are much smaller than the fundamental frequency of the uniform cascade, and slower nonlinear SWS attenuation. The accurate and fast “simple wave method” and “nonlinear acoustics approximation”, together with numerical algorithms for integrating Euler equations on overlapping grids (in calculating flow past blunt edges) and on SWS-adapted grids, are applied to determine the “guiding” action of nonuniform cascades and to describe the SWS evolution. The application of the Fourier analysis gives the sound field spectrum. The use of blades with rectilinear initial regions of the “backs” for reducing supersonic fan blade noise is efficient only at small (less than 0.25°) scatter in the setting angles. The shock wave structures attenuate more rapidly ahead of nonuniform cascades composed of blunt blades than ahead of those with sharp blades. For uniform cascades the blade bluntness effect is not large.     

Numerical modeling of gas flow around oscillating compressor blades

The problem of gas flow around a plane cascade of oscillating blades is numerically solved using the ANSYS CFX package. The blade surface displacement is taken into account using a movable grid generated before the beginning of the calculations at each time step. The calculated and experimental data are compared. The calculated results are used for determining the blade stability against flutter.     

Flow structure and skin friction in the vicinity of a streamwise-angled injection hole fed by a short pipe

The velocity field and skin friction distribution around a row of five jets issuing into a crossflow from short (L/D ≃ 1) pipes inclined by 35° with respect to the streamwise direction, (i.e., “short holes”) are presented for two different jet supply flow directions. Velocity was measured using PIV, while the skin friction was measured with oil-film interferometry. The flow features are compared with previously published data for jets issuing through holes oriented normal to the crossflow and with numerical simulations of similar geometries. The distinguishing features of the flow field include a reduced recirculation region in comparison to the 90° case and markedly different in-hole flow physics. The jetting process caused by in-hole separations force the bulk of the jet fluid to issue from the leading half of the streamwise-angled injection hole, as previously reported by Brundage et al. (Tech Rep ASME 99-GT-35, 1999) and predicted by Walters and Leylek (ASME J Turbomach 122:101–112, 2000). The flow structure impacts the skin friction distribution around the holes, resulting in higher near-hole shear stress for a counter-flow supply plenum (jet fluid supplied by a high speed plenum flowing opposite to the free stream direction). In contrast, the counter-flow supply plenum was previously found to have the lowest near-hole wall shear stress for normal injection holes (Peterson and Plesniak in Exp Fluids 37:497–503, 2004b). Streamwise-angled injection generally reduces the near-hole skin friction due to the reduced jet trajectory resulting from the lower wall-normal jet momentum. Far downstream, the skin friction distributions are similar for the two injection angle cases.     

Accuracy of out-of-plane vorticity measurements derived from in-plane velocity field data

 A study of the errors in out-of-plane vorticity (ω _z ) calculated using a local χ² fitting of the measured velocity field and analytic differentiation has been carried out. The primary factors of spatial velocity sampling separation and random velocity measurement error have been investigated. In principle the ω _z error can be decomposed into a bias error contribution and a random error contribution. Theoretical expressions for the transmission of the random velocity error into the random vorticity error have been derived. The velocity and vorticity field of the Oseen vortex has been used as a typical vortex structure in this study. Data of different quality, ranging from exact velocity vectors of analytically defined flow fields (Oseen vortex flow) sampled at discrete locations to computer generated digital image frames analysed using cross-correlation DPIV, have been investigated in this study. This data has been used to provide support for the theoretical random error results, to isolate the different sources of error and to determine their effect on ω _z measurements. A method for estimating in-situ the velocity random error is presented. This estimate coupled with the theoretically derived random error transmission results for the χ² vorticity calculation method can be used a priori to estimate the magnitude of the random error in ω _z . This random error is independent of a particular flow field. The velocity sampling separation is found to have a profound effect on the precise determination of ω _z by introducing a bias error. This bias error results in an underestimation of the peak vorticity. Simple equations, which are based on a local model of the Oseen vortex around the peak vorticity region, allowing the prediction of the ω _z bias error for the χ² vorticity calculation method, are presented. An important conclusion of this study is that the random error transmission factor and the bias error cannot be minimised simultaneously. Both depend on the velocity sampling separation, but with opposing effects. The application of the random and bias vorticity error predictions are illustrated by application to experimental velocity data determined using cross-correlation DPIV (CCDPIV) analysis of digital images of a laminar vortex ring. Received: 31 October 1997/Accepted: 6 February 1998     

Aeolian tones of a plate cascade

A model for the aeroacoustic resonance effects (aeolian tones) excited around a plate cascade in a gas flow is suggested. Methods of calculating the frequencies of natural acoustic oscillations near the cascade are developed. The effect of the cascade geometry and the Mach number of the main flow on the frequencies, abundance, and modes of the natural oscillations is investigated. Anomalous acoustic oscillations near a cyclic plate cascade are shown to exist and are studied. It is shown that there always exist no less than two natural oscillation frequencies in the gas flow near any nontrivial cyclic plate cascade. It has been found that the natural oscillation frequencies can be combined in bundles such that in the case where the number of plates in a period is large the frequencies pertaining to each bundle occupy a certain interval with arbitrary density. The natural oscillations are classified with respect to the form of the eigenfunctions; the classification is based on the theory of representations of groups of locally plane symmetries of the cyclic plate cascade in the solution space. The correctness of the proposed model of the aeroacoustic resonance effects (aeolian tones) excited near a plate cascade in a gas flow is supported by a comparison with the available experimental and theoretical data. On the basis of the investigation performed, some previously unknown physical phenomena are predicted. Thus, the existence of frequency zones or main-flow Mach number ranges on which aeroacoustic resonance phenomena exist near a cyclic cascade with a large number of plates in a period is proved; it is shown that for certain frequencies of the natural oscillations near the cyclic plate cascade the resonance oscillations may be localized in the vicinity of the source; and the existence of narrow-band wave packets slowly propagating along the cascade is demonstrated. Novosibirsk, e-mail: sukhinin@hydro.nsc.ru. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 2, pp. 171–186, March–April, 2000.     

On the generalized Cauchy function and new Conjecture on its exterior singularities

This article studies on Cauchy’s function f (z) and its integral, (2πi)J[ f (z)] ≡■C f (t)dt/(t z) taken along a closed simple contour C, in regard to their comprehensive properties over the entire z = x + iy plane consisted of the simply connected open domain D + bounded by C and the open domain D outside C. (1) With f (z) assumed to be C n (n < ∞-times continuously differentiable) z ∈ D + and in a neighborhood of C, f (z) and its derivatives f (n) (z) are proved uniformly continuous in the closed domain D + = [D + + C]. (2) Cauchy’s integral formulas and their derivatives z ∈ D + (or z ∈ D ) are proved to converge uniformly in D + (or in D = [D +C]), respectively, thereby rendering the integral formulas valid over the entire z-plane. (3) The same claims (as for f (z) and J[ f (z)]) are shown extended to hold for the complement function F(z), defined to be C n z ∈ D and about C. (4) The uniform convergence theorems for f (z) and F(z) shown for arbitrary contour C are adapted to find special domains in the upper or lower half z-planes and those inside and outside the unit circle |z| = 1 such that the four general- ized Hilbert-type integral transforms are proved. (5) Further, the singularity distribution of f (z) in D is elucidated by considering the direct problem exemplified with several typ- ical singularities prescribed in D . (6) A comparative study is made between generalized integral formulas and Plemelj’s formulas on their differing basic properties. (7) Physical sig- nificances of these formulas are illustrated with applicationsto nonlinear airfoil theory. (8) Finally, an unsolved inverse problem to determine all the singularities of Cauchy function f (z) in domain D , based on the continuous numerical value of f (z) z ∈ D + = [D + + C], is presented for resolution as a conjecture.     

Near-wake flow structure downwind of a wind turbine in a turbulent boundary layer

Wind turbines operate in the surface layer of the atmospheric boundary layer, where they are subjected to strong wind shear and relatively high turbulence levels. These incoming boundary layer flow characteristics are expected to affect the structure of wind turbine wakes. The near-wake region is characterized by a complex coupled vortex system (including helicoidal tip vortices), unsteadiness and strong turbulence heterogeneity. Limited information about the spatial distribution of turbulence in the near wake, the vortex behavior and their influence on the downwind development of the far wake hinders our capability to predict wind turbine power production and fatigue loads in wind farms. This calls for a better understanding of the spatial distribution of the 3D flow and coherent turbulence structures in the near wake. Systematic wind-tunnel experiments were designed and carried out to characterize the structure of the near-wake flow downwind of a model wind turbine placed in a neutral boundary layer flow. A horizontal-axis, three-blade wind turbine model, with a rotor diameter of 13 cm and the hub height at 10.5 cm, occupied the lowest one-third of the boundary layer. High-resolution particle image velocimetry (PIV) was used to measure velocities in multiple vertical stream-wise planes (x–z) and vertical span-wise planes (y–z). In particular, we identified localized regions of strong vorticity and swirling strength, which are the signature of helicoidal tip vortices. These vortices are most pronounced at the top-tip level and persist up to a distance of two to three rotor diameters downwind. The measurements also reveal strong flow rotation and a highly non-axisymmetric distribution of the mean flow and turbulence structure in the near wake. The results provide new insight into the physical mechanisms that govern the development of the near wake of a wind turbine immersed in a neutral boundary layer. They also serve as important data for the development and validation of numerical models.     

Interaction between elliptical and ellipsoidal inclusions under bending stress fields

Summary  This paper deals with interaction problems of elliptical and ellipsoidal inclusions under bending, using singular integral equations of the body force method. The problems are formulated as a system of singular integral equations with Cauchy-type or logarithmic-type singularities, where unknown functions are densities of body forces distributed in the x,y and r,θ,z directions in infinite bodies having the same elastic constants as those of the matrix and inclusions. In order to satisfy the boundary conditions along the elliptical and the ellipsoidal boundaries, the unknown functions are approximated by a linear combination of fundamental density functions and polynomials. The present method is found to yield the exact solutions for a single elliptical or spherical inclusion under a bending stress field. It yields rapidly converging numerical results for interface stresses in the interaction of inclusions. Received 9 September 1999; accepted for publication 15 January 2000     

Fourier Representation of Intermittent Flow in a Porous Medium

The field measurements and numerical results for intermittent flow regime in a sandy soil show that the time distributions of the soil water flux q(z, t), and the soil water content θ(z, t)at various depths are periodic in nature, where t is time and z is the depth (i.e., at the surface z = 0 and at depths z = − 5, − 10, − 15 cm, etc). The period of q(z, t) and θ(z, t) variations are generally determined by the sum of the duration of pulse and the duration between the initiation of two consecutive pulses of water at the soil surface. Fourier series models have been given for q(z, t) and θ(z, t) variations. The predicted Fourier results for these variations have been compared with the experimentally verified numerical results—designated as observed values. The results show that the amplitudes of these variations were damped exponentially with depth, and the phase shift increased linearly with depth.     

Simulation of Wind Flow Around a Building with a k–ε Model

The three-dimensional numerical simulation of airflow around a building using a k–ε two-equation turbulence model is presented in this paper. Several cases of numerical simulation of airflow around a building are carried out to estimate the influence of mesh spacing on simulated results. The accuracy of simulations is examined by comparing the predicted results with wind-tunnel experiments. It is confirmed that numerical simulations by means of the k–ε model reproduce the velocity fields well when using fine mesh resolution. In the latter part of the paper, the simulation method is applied to predict the flow field around a building with different width-to-height ratios, under light wind conditions. Received 16 June 1999 and accepted 20 July 2000     

Predicting the Collapse of Turbulence in Stably Stratified Boundary Layers

The collapse of turbulence in a plane channel flow is studied, as a simple analogy of stably stratified atmospheric flow. Turbulence is parameterized by first-order closure and the surface heat flux is prescribed, together with the wind speed and temperature at the model top. To study the collapse phenomenon both numerical simulations and linear stability analysis are used. The stability analysis is nonclassical in a sense that the stability of a parameterized set of equations of a turbulent flow is analyzed instead of a particular laminar flow solution. The analytical theory predicts a collapse of turbulence when a certain critical value of the stability parameter δ/L (typically O(0.5–1)) is exceeded, with δ the depth of the channel and L the Obukhov length. The exact critical value depends on channel roughness to depth ratio z ₀/δ. The analytical predictions are validated by the numerical simulations, and good agreement is found. As such, for the flow configuration considered, the present framework provides both a tool and a physical explanation for the collapse phenomenon.     

Kinematic problem of an ideal incompressible fluid flow around an arbitrarily moving airfoil

An unsteady kinematic problem for arbitrary two-dimensional motion of an airfoil in an ideal incompressible fluid with formation of one and two vortex wakes is solved. The problem is solved by the method of conformal mapping of the flow domain onto a circle exterior; solution singularities in the vicinity of a sharp edge are analyzed, and the initial asymptotics of the solution is taken into account. The calculated results are found to be in good agreement with available experimental data on visualization of the flow pattern. The necessity of correct modeling of the initial stage of vortex-wake formation is demonstrated. A regular flow pattern is found to form after three and more periods of oscillations. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 50, No. 2, pp. 120–128, March-April, 2009.     

Large chord turbine cascade testing at engine Mach and Reynolds numbers

A technique is presented for producing a flow through a linear cascade of turbine blades of large chord which gives the pressure distribution around a blade the same as that obtained in an infinite cascade for Mach and Reynolds numbers typical of gas turbine operating conditions. Results of experiments with a cascade of three blades of large chord are compared with results from a cascade of nine blades of smaller chord to confirm the validity of the technique. Experiments are performed on the large-chord cascade to examine surface phenomena with high spatial resolution. Boundary layer scales are also increased and profiles on both suction and pressure surfaces of the blade are obtained.     

Three-dimensional wake structure measurement using a modified PIV technique

Three-dimensional vortical structures have been measured in a circular-cylinder wake using particle imaging velocimetry (PIV) for the Reynolds number range of 2×10³ to 1×10⁴. The PIV was modified, compared with the conventional one, in terms of its light sheet arrangement to capture reliably streamwise vortices. While in agreement with previous reports, the presently measured spanwise structures complement the data in the literature in the streamwise evolution of the near-wake spanwise vortex in size, strength, streamwise and lateral convection velocities, shedding new light upon vigorous interactions between oppositely signed spanwise structures. The longitudinal vortices display mushroom patterns in the (x, z)-plane in the immediate proximity to the cylinder. Their most likely inclination in the (x, y)-plane is inferred from the measurements in different (x, z)-planes. The longitudinal vortices in the (y, z)-plane show alternate change in sign, though not discernible at x/d > 15. They decay in the maximum vorticity and circulation rapidly from x/d = 5 to 10 and slowly for x/d > 10, and are further compared with the spanwise vortices in size, strength and rate of decay.     

Development of a novel flow metering system using ultrasonic velocity profile measurement

 A new flow metering system using an ultrasonic Doppler method has been developed. By this method, an instantaneous velocity profile is measured and its integral is evaluated to give flow rate. Instead of making multidimensional flow mapping to evaluate its performance, the method was assessed in laboratory experiments where single line measurements were used over the whole pipe diameter or radius to determine the flow rate. The system shows a good capability of tracking transient flows. Its accuracy was investigated using the National Institute of Standards and Technology (NIST) flow standards in Gaithersburg, MD, USA; averaged results show its deviation from NIST results to be 0.18% for Re=400 K and 0.58% for Re=2.6 M. Good repeatability and reproducibility were also confirmed. Received: 28 April 2000/Accepted: 1 November 2000 Published online: 29 November 2001     

Numerical Fluid Flow Analysis for Aerodynamic Response Characteristics of Tandem Circular Cylinders

The aerodynamic characteristics of tandem cables of cable-stayed bridges have become an increasingly serious problem with increments in span length. In order to reduce the construction cost and maintenance of cables, tandem cables have been adopted for cable-stayed bridges. These cables, however, have aerodynamic response characteristics such as wake-galloping. Therefore, a method to suppress wake-galloping in tandem cables is required. The purpose of this study is to investigate the characteristics of the wake-galloping phenomenon of tandem cables of cable-stayed bridges using numerical fluid flow analysis. The flow around the oscillating tandem circular cylinders modeled on tandem cables is calculated. The flow field is treated as an incompressible viscous flow. The Arbitrary Lagrangian-Eulerian (ALE) method is employed to solve the flow field around the cylinders, and the three-step Taylor-Galerkin method, which is based on a fractional step finite element method, is adopted for discretization.