Far-field Formulation Research Articles

Ear-to-Ear Propagation Model—Antenna Radiation Pattern Optimization and On-Body Far-Field Formulation

Ear-to-Ear Propagation Model—Antenna Radiation Pattern Optimization and On-Body Far-Field Formulation

On the spurious effects in Lamb-vector-based force decomposition methods

Recent methods based on Lamb vector integration allow for a far-field formulation and decomposition of the whole aerodynamic force, with identification of local flow structures associated with the force generation. Different exact Lamb-vector-based force formulations were proposed in the past and the decomposition of total drag into lift-induced and parasite contributions was subject of very recent studies. However, inconsistencies were also reported, raising questions on the physical role of a non-zero induced drag found in two-dimensional flows. Moreover, the effect of the arbitrary pole (appearing in Lamb-vector-based formulas) on the numerical accuracy of such methods was never investigated. Additionally, the separation of parasite drag into viscous and wave drag contributions by Lamb vector field analysis is still questioned. In this paper, the accuracy of different force formulas and of the drag decomposition in lift-induced and parasite contributions is first analyzed, for steady flows. Two-dimensional numerical solutions around RAE2822 and NACA0012 airfoils are considered, the latter in both inviscid and viscous flows (at low and high Reynolds number), to analyze consistency between lift-induced/parasite drag definitions and the drag generated in isentropic flow conditions (reversible drag) or due to entropy production (irreversible drag). A spurious induced drag contribution (related to the streamwise momentum perturbation) is identified, resulting in negative induced drag values in two-dimensional flows. The effect of a shift in the pole location is also analysed, with reference to its impact on the total force and on drag decomposition accuracy, then a new approach to parasite drag breakdown into physical constituents (viscous and wave contributions) is proposed and results are compared against predictions obtained using consolidated thermodynamic-based methods and against available results from recent literature.

Added resistance using Salvesen–Tuck–Faltinsen strip theory and the Kochin function

It is now more than half a century since the development of the renowned and well-established strip theory of Salvesen–Tuck–Faltinsen (STF), Salvesen et al. (1970). One of the existing, and widely used, methods for calculating the wave drift force (zero-speed case), or added resistance (forward-speed case), within this strip theory, is a near-field formulation which was developed by Salvesen himself in Salvesen (1974), Salvesen (1978). This method invokes a weak-scatterer assumption to drop one term, and also approximates the exponential terms as constant in the integrals over each two-dimensional ship section. These approximations are essentially a long-wave assumption Salvesen (1974). The main objective of this paper, which has been entirely inspired by the method of Kashiwagi et al. (2010), is to apply a far-field formulation, in the context of STF strip theory, without the above-mentioned assumptions. This far-field method is implemented following Maruo (1960b), Maruo (1963) and employs the Kochin function Kochin (1936) to express the wave kinematics in the far field. The necessary ingredients for this method are obtained by an implementation of STF theory using a low-order Boundary Element Method (BEM) and the two-dimensional free-surface Green function. No weak-scatterer assumption is required as the Kochin function explicitly includes the interaction of the disturbance potentials. Sectional integrals of the Kochin function are calculated analytically, assuming piecewise-constant variation of the potentials. Within this far-field framework all three integrals from Maruo’s relation are treated, and the line integral along the vessel’s length is calculated exactly assuming either a linear or a Fourier-mode decomposition of the non-exponential terms. The outcome is a far-field calculation procedure inside STF theory which compares very well with experimental measurements and relevant reference solutions. This is verified by the calculation of wave drift force and added resistance for six different ship hulls. One notable advantage of the presented framework, in comparison to Salvesen’s method (and also to that of Gerritsma and Beukelman (1972)), is its ability to accurately predict the zero-speed wave drift force. In addition, the long-wave approximate form of the Kochin function method is also discussed. Similar to Salvesen’s method, the wave drift force using this approximation, is expressed in terms of the body motion, the hydrodynamic coefficients and the sectional geometric constants. The approximate form performs well at zero-speed, but deteriorates quickly as the speed increases.

Local Enhancements of the Mean Drift Wave Force on a Vertical Column Shielded by an Exterior Thin Porous Shell

The wave interaction with a vertical column shielded by an exterior porous shell is studied within the framework of potential flow theory. The structures are fixed rigidly at the sea bottom. The interior cylinder is impermeable, and the exterior shell is slightly porous and thin. Additionally, the exterior shell is assumed to have fine pores, and a linear pressure drop is adopted at the porous geometry. The mean drift wave force on the system is thereby formulated by two alternative ways, based respectively on the direct pressure integration, i.e., the near-field formulation, and the application of the momentum conservation theorem in the fluid domain, i.e., the far-field formulation. The consistency of the two formulations in calculating the mean drift wave force is assessed for the present problem. Numerical results illustrate that the existence of the porous shell can substantially reduce the mean drift wave force on the interior column. It also appears that the far-field formulation consists of a conventional part as well as an additional part caused by the energy dissipation through the porous geometry. The mean drift wave force on the system is dominated by the first part, which resembles that on an impermeable body. Local enhancements of the mean drift wave force are found at some specific wave frequencies at which certain propagation modes of the fluid satisfy a no-flow condition at the porous shell.

Numerical Simulation of Aeroacoustics of Hovering Helicopter Rotor

The specifics of the problem of estimating the noise of a hovering rotor, allows for some simplification of the Ffowcs Williams-Hawkins (FW-H) equation. Most published works dedicated to helicopter tonal noise estimates use the far-field formulation. This paper estimates the aeroacoustic emissions of a helicopter rotor in hover, for observers placed at different distances using the FW-H equation, including near-field and far-field terms. The blade pressure distribution is obtained from numerical simulations with the RANS equations. To demonstrate this approach, the near- and far-field contributions are analyzed for the model-scale UH-1H main helicopter rotor. For the numerical simulations, the HMB solver of Glasgow University and the ANSYS Fluent13 commercial solver are used. As the rotor blade behaviour is characterized by a complex motion in the computer program it is assumed that the blade is seen in as a rotating rigid body. The most commonly used mathematical model of the FW-H equation corresponds to the classical impermeable formulation. In this case, the source surface corresponds to the blade surface. Then, the acoustic pressure (based on the FW-H 1A formulation) is modified with empirical adjustments, based on the radiation Mach number. This was applied for the near- or far-field thickness noise depending on the rotor-observer distance.

Theoretical modeling of hydrodynamic characteristics of a compound column-plate structure based on a novel derivation of mean drift force formulation

To improve the seakeeping capability, some devices, such as submerged plates, are often installed on floating structures. The attached plate can not only suppress the motion response but also provide an additional immersed body surface that receives fluid action, aggravating the wave loads. In this study, a theoretical model is developed within the context of linear potential theory to study the hydrodynamic characteristics of a floating column with a submerged plate attached at the bottom. The eigenfunction expansion matching method is applied to obtain the velocity potential, based on which the linear wave force and wave runup can be found immediately. A novel derivation of the mean drift force formulation is developed via the application of Green’s second identity to the velocity potential and its derivative in finite fluid volume surrounding the body. Mean drift force formulation that involves control surfaces is then obtained. With the availability of the velocity potential, semi-analytical solution of the mean drift force on the compound column-plate structure is developed based on, respectively, the derived and the classic far-field formulations. After conducting convergence tests and validating the theoretical model, detailed numerical analysis is performed thereafter based on the theoretical model. The influence of the plate size, such as the radius and height, on the wave force and the associated wave runup is assessed.

Hull form optimization in the conceptual design stage considering operational efficiency in waves

This study focuses on the optimization of ship dimensions by considering hydrodynamic performance in waves. In actual seaways, a ship experiences speed loss due to environmental loads by waves and wind. Therefore, along with calm water resistance, speed loss in waves should be considered in the hull form design in order to improve operational efficiency in waves. However, a trade-off may be needed between total resistance on the ship and the speed loss in waves. To address this problem, Non-dominated Sorting Genetic Algorithm II, which is a multi-objective optimization method, is used to minimize the total resistance on a ship in seaways and the speed loss by additional resistance. In the optimization process, added resistance is predicted using a numerical method based on slender-body theory, Maruo’s far-field formulation, and an empirical formula for added resistance in short waves. The speed loss in waves, which can be expressed by a weather factor ( fw), is estimated using power–speed curves. This article introduces some examples of the sensitivity analysis of added resistance and speed loss in waves to the variations of ship dimensions. Finally, the optimization solutions on a Pareto front set are compared to a basis ship in terms of hull form, and the corresponding hydrodynamic performances are evaluated.

Engine pre-entry thrust and standard net thrust evaluation based on the far-field method

Computation of engine pre-entry thrust is required to evaluate aircraft configuration drag. Numerical computation of this quantity requires knowledge of the captured streamtube, which depends on the stagnation line location. A new method that uses the far-field formulation is developed so that knowledge of the streamtube properties is no longer required. Using similar techniques, an alternative method to compute the standard net thrust, the basis of most thrust/drag bookkeeping systems, is introduced. The classical formulation to compute the standard net thrust needs interpolation of flow quantities in the nacelle's exit plane which leads to loss of accuracy. Theoretical development of the far-field method in power-on conditions is presented as well as an overview of the bookkeeping technique in CFD. Simulations are performed with ANSYS Fluent 13.0 on the isolated CFM56 nacelle in power-on conditions with varying boundary conditions. Results of the proposed approaches are in agreement with ESDU and classical formulations. Drag decomposition in terms of viscous, wave, spurious, induced, and through-flow drag is presented. Configuration drag is computed with two formulations which are in agreement.

Three-Component Breakdown of Spurious Drag in Computational Fluid Dynamics

A three-component phenomenological breakdown of spurious pressure drag in computational fluid dynamics is derived, following a far-field formulation. The study is purposely restricted to solutions to the Euler equations, in which the spurious drag, also extant in viscous simulations, can best be detected and analyzed. The matter of far-field spurious-drag elimination, which must indeed be handled differently in viscous flow and in inviscid flow, is outside its scope. Each of the three spurious components is produced through a different process. The first two components, already identified in the past, are of an irreversible thermodynamic nature. The first one is related to vorticity normal to the velocity, as it appears in Crocco’s relation. The second one is concomitant with destruction of vorticity parallel to the velocity. The third component, of a reversible nature, is purely due to the effect of the far-field boundary condition. It exists in two-dimensional flow as well as in three-dimensional flow, in which it is shown that it must be negative, while positive in two dimensions.

A coupled far-field formulation for time-periodic numerical problems in fluid dynamics

Consider uniform flow past an oscillating body generating a time-periodic motion in an exterior domain, modelled by a numerical fluid dynamics solver in the near field around the body. A far-field formulation, based on the Oseen equations, is presented for coupling onto this domain thereby enabling the whole space to be modelled. In particular, examples for formulations by boundary elements and infinite elements are described.

Far-field formulation of a Cassegrain reflector using a novel illumination function and aperture field integration

In this paper, the far-field pattern of a Cassegrain reflector is formulated. A novel illumination function is used to approximate the field distribution at the aperture of the reflector. The defined illumination function takes into account the central aperture blockage created by the subreflector. Using the illumination function, a closed-form expression describing the far-field radiation pattern of the Cassegrain reflector is formulated. The radiation pattern obtained from the derived equation is compared with the results obtained from physical optics and physical theory of diffraction. The results are found to be consistent with each other. It is found that the derived results show an impressive accuracy of 99.8% over the main-lobe region. The accuracy is found to be over 91 and 84% for the first and second significant side-lobe region, respectively, which can be considered satisfactory for many applications.

Comprehensive three-dimensional split-field finite-difference time-domain method for analysis of periodic plasmonic nanostructures: near- and far-field formulation

The three-dimensional split-field finite-difference time-domain (SF-FDTD) method is combined with the total-field–scattered-field method for injecting a plane wave. A formulation is derived for calculating the incidence transformed fields of SF-FDTD on a one-dimensional auxiliary grid. The resulting fields obtained in the scattered zone are used to calculate the far fields, based on a proposed fully time-domain near-to-far-field transformation. The far-field information is used to calculate the extinction cross section of the periodic structure under oblique incidence. To analyze metallic periodic structures, a formulation with a reduced number of variables is proposed based on the auxiliary differential equation method for dispersive media.

Evaluation of nacelle drag using computational fluid dynamics

Thrust and drag components must be defined and properly accounted in order to estimate aircraft performance, and this hard task is particularty essential for propulsion system where drag components are functions of engine operating conditions. The present work describes a numerical method used to calculate the drag in different nacelles, long and short ducted. Two- and three-dimensional calculations were performed, solving the Reynolds Averaged Navier-Stokes (RANS) equations with a commercial Computational Fluid Dynamics (CFD) code. It is then possible to obtain four drag components: wave, induced, viscous and spurious drag using a far-field formulation. An expression in terms of entropy variations was shown and drag for different nacelle geometries was estimated.

Attenuation correction factors for cylindrical, disc and box geometry

In the present study, attenuation correction factors have been experimentally determined for samples having cylindrical, disc and box geometry and compared with the attenuation correction factors calculated by Hybrid Monte Carlo (HMC) method [ C. Agarwal, S. Poi, A. Goswami, M. Gathibandhe, R.A. Agrawal, Nucl. Instr. and. Meth. A 597 (2008) 198] and with the near-field and far-field formulations available in literature. It has been observed that the near-field formulae, although said to be applicable at close sample-detector geometry, does not work at very close sample-detector configuration. The advantage of the HMC method is that it is found to be valid for all sample-detector geometries.

Middle-field formulation for the computation of wave-drift loads

New formulations of second-order wave loads contributed by a first-order wave field are developed by applying two variants of Stokes’s theorem and Gauss’s theorem to a formulation consisting of direct pressure integrations on a body’s hull which is called the near-field formulation. In addition to this direct formulation and the formulation derived from the momentum theorem called the far-field formulation, for the computation of drift (surge/sway) forces in horizontal directions and drift (yaw) moment around the vertical axis, one of new formulations is defined on the control surfaces surrounding the body and called the middle-field formulation. After a brief summary of both pressure-integration (near-field) and momentum (far-field) formulations, the development of the middle-field formulation involving control surfaces is described and complemented in detail in the appendices. The application of the new formulation shows that the near-field and far-field formulations are mathematically equivalent for wall-sided, as well as non-wall-sided bodies and under the condition that the mean yaw moments are expressed with respect to a space-fixed reference point. It is shown that the middle-field formulation is as robust as the far-field formulation and as general as the near-field formulation of second-order loads on a single body as well as on multiple bodies. Furthermore, the extension to the computation of a second-order oscillatory load, which is so far accessed only by the near-field formulation, is envisioned.

Analysis of dielectric-loaded annular slot array antenna

A method is established for the analysis of annular slot array antennas loaded with dielectric layers and fed by either radial waveguide or cavity. The analysis is based on the boundary value method. The Greens functions for each region are obtained, and then the induced magnetic current over the slots is expanded into Fourier series with unknown coefficients. Boundary conditions are applied, and a matrix equation for these unknown coefficients is obtained. For narrow slots the number of unknowns equals the number of annular slots, and an extremely rapid solution is obtained. The far-field formulation is derived using the magnetic current on the dielectric layer. The method is confirmed numerically by comparing the simulation results for sample small antennas with a commercial numerical tool (IE3D), and good agreement is achieved. It is shown that adding the dielectric layers can improve the antenna directivity. In comparison with other methods, the proposed method is very efficient, and its computation efforts depend on the number of annular slots and not the size of the antenna. As an example, for a single-slot antenna the number of unknowns to be determined is only one, while in IE3D it is more than 300.

Antenna designer's notebook-design of pyramidal horns for fixed phase center as well as optimum gain

A method for designing a pyramidal horn with optimum gain and a phase center that is insensitive to horn rotation over a wide frequency band is presented. The optimum gain comes out to be about 14 dB; ways to increase the optimum gain are suggested. The design is based on a far-field formulation and is useful for common applications. However, a parallel procedure based on a near-field formulation and using more exact gain expressions can be developed.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Scanning currents in Stokes flow and the efficient feeding of small organisms

The feeding behaviour of many small, free-swimming organisms involves the creation of a scanning current by the coordinated movement of a group of appendages. In this paper, we study the generation of scanning currents in Stokes flow in a number of simple models, utilizing the movement of Stokeslets, spheres, or stalks to set up an average scanning drift in a suitable far-field formulation. Various mechanisms may then be classified by the rate of decay at infinity of the mean scanning current. In addition, optimal scanning can be investigated by minimizing the mean power-required to create a current of prescribed amplitude. The simple mechanisms for scanning described here provide a framework within which the appendage movements of small aquatic organisms can be analysed and the relative merits of scanning and swimming strategies can be investigated.

Noise and Vibration Control

This session consisted of nine contributed papers all dealing with different aspects of noise and vibration control. St. Hilaire and Vaidya described the mechanism of sound generation caused by the metal reed in a reed organ, and in another paper described the response in a small room due to a sound source in a larger room connected by an opening between them. Davies applied statistical energy analysis to systems with strong but point coupling. Tseo presented a farfield formulation for sound pressure and radiation intensity from a simply supported rectangular plate. Jai-Lue Lai showed mean-square displacement responses of an axially moving belt for loadings of special random processes. Prediction methods for sound power radiated by several classes of rigid flow-discontinuities in low-speed air ducts were presented with experimental evidence by Hayden. Requirements on scanning rate and detection time constant were presented for space-averaged acoustic measurements by Wooten and Cary. Nitsche presented a survey of spectral analysis techniques for moving sound sources. The use of Helmholtz resonators for acoustic absorption was presented by Czarnecki.