Application of the Ffowcs Williams/Hawkings equation to two-dimensional problems

2000 ◽  
Vol 403 ◽  
pp. 201-221 ◽  
Author(s):  
Y. P. GUO
Keyword(s):  
Turbulent Flows ◽  
Functional Dependence ◽  
Pressure Fluctuations ◽  
Three Dimensional ◽  
The Body ◽  
Far Field ◽  
Two Dimensional ◽  
Retarded Time ◽  
Practical Applications ◽  
Dimensional Version

This paper discusses the application of the Ffowcs Williams/Hawkings equation to two-dimensional problems. A two-dimensional version of this equation is derived, which not only provides a very efficient way for numerical implementation, but also reveals explicitly the features of the source mechanisms and the characteristics of the far-field noise associated with two-dimensional problems. It is shown that the sources can be interpreted, similarly to those in three-dimensional spaces, as quadrupoles from turbulent flows, dipoles due to surface pressure fluctuations on the bodies in the flow and monopoles from non-vanishing normal accelerations of the body surfaces. The cylindrical spreading of the two-dimensional waves and their far-field directivity become apparent in this new version. It also explicitly brings out the functional dependence of the radiated sound on parameters such as the flow Mach number and the Doppler factor due to source motions. This dependence is shown to be quite different from those in three-dimensional problems. The two-dimensional version is numerically very efficient because the domains of the integration are reduced by one from the three-dimensional version. The quadrupole integrals are now in a planar domain and the dipole and monopole integrals are along the contours of the two-dimensional bodies. The calculations of the retarded-time interpolation of the integrands, a time-consuming but necessary step in the three-dimensional version, are completely avoided by making use of fast Fourier transform. To demonstrate the application of this, a vortex/airfoil interaction problem is discussed, which has many practical applications and involves important issues such as vortex shedding from the trailing edge.

scholarly journals Geometry of the 3D Pythagoras' Theorem

2016 ◽  
Vol 8 (6) ◽  
pp. 78 ◽  
Author(s):  
Luis Teia
Keyword(s):  
Three Dimensional ◽  
Transformation Process ◽  
Two Dimensional ◽  
Natural Evolution ◽  
Dimensional Version ◽  
Pythagoras Theorem

This paper explains step-by-step how to construct the 3D Pythagoras' theorem by geometric manipulation of the two dimensional version. In it is shown how $x+y=z$ (1D Pythagoras' theorem) transforms into $x^2+y^2=z^2$ (2D Pythagoras' theorem) via two steps: a 90-degree rotation, and a perpendicular extrusion. Similarly, the 2D Pythagoras' theorem transforms into 3D using the same steps. Octahedrons emerge naturally during this transformation process. Hence, each of the two dimensional elements has a direct three dimensional equivalent. Just like squares govern the 2D, octahedrons are the basic elements that govern the geometry of the 3D Pythagoras' theorem. As a conclusion, the geometry of the 3D Pythagoras' theorem is a natural evolution of the 1D and 2D. This interdimensional evolution begs the question -- Is there a bigger theorem at play that encompasses all three?

Two- and Three-Dimensional Simulations of the Flow Around a Cylinder Fitted With Strakes

2012 ◽  
Author(s):  
Bruno S. Carmo ◽  
Rafael S. Gioria ◽  
Ivan Korkischko ◽  
Cesar M. Freire ◽  
Julio R. Meneghini
Keyword(s):  
Reynolds Number ◽  
Vortex Shedding ◽  
Free Stream ◽  
Three Dimensional ◽  
The Body ◽  
Vortex Wake ◽  
Two Dimensional ◽  
Flow Around A Cylinder ◽  
Three Dimensional Simulations ◽  
Angle Of Rotation

Two- and three-dimensional simulations of the flow around straked cylinders are presented. For the two-dimensional simulations we used the Spectral/hp Element Method, and carried out simulations for five different angles of rotation of the cylinder with respect to the free stream. Fixed and elastically-mounted cylinders were tested, and the Reynolds number was kept constant and equal to 150. The results were compared to those obtained from the simulation of the flow around a bare cylinder under the same conditions. We observed that the two-dimensional strakes are not effective in suppressing the vibration of the cylinders, but also noticed that the responses were completely different even with a slight change in the angle of rotation of the body. The three-dimensional results showed that there are two mechanisms of suppression: the main one is the decrease in the vortex shedding correlation along the span, whilst a secondary one is the vortex wake formation farther downstream.

A Study of Water Surface Deformation Due to Tip Vortices Wing-in-Ground Effect

2007 ◽  
Vol 51 (02) ◽  
pp. 182-186
Author(s):  
Tracie J. Barber
Keyword(s):  
Water Surface ◽  
Surface Deformation ◽  
Three Dimensional ◽  
Ground Effect ◽  
Aerodynamic Performance ◽  
The Body ◽  
Vehicle Design ◽  
Two Dimensional ◽  
Tip Vortices ◽  
Wing Tip

The accurate prediction of ground effect aerodynamics is an important aspect of wing-in-ground (WIG) effect vehicle design. When WIG vehicles operate over water, the deformation of the nonrigid surface beneath the body may affect the aerodynamic performance of the craft. The likely surface deformation has been considered from a theoretical and numerical position. Both two-dimensional and three-dimensional cases have been considered, and results show that any deformation occurring on the water surface is likely to be caused by the wing tip vortices rather than an increased pressure distribution beneath the wing.

scholarly journals A New Two-Dimensional Mutual Coupled Logistic Map and Its Application for Pseudorandom Number Generator

2019 ◽  
Vol 2019 ◽  
pp. 1-10 ◽  
Author(s):  
Xuan Huang ◽  
Lingfeng Liu ◽  
Xiangjun Li ◽  
Minrong Yu ◽  
Zijie Wu
Keyword(s):  
Statistical Tests ◽  
Chaotic Map ◽  
Logistic Map ◽  
Security Analysis ◽  
Three Dimensional ◽  
Pseudorandom Number ◽  
Pseudorandom Number Generator ◽  
Number Generator ◽  
Two Dimensional ◽  
Practical Applications

Given that the sequences generated by logistic map are unsecure with a number of weaknesses, including its relatively small key space, uneven distribution, and vulnerability to attack by phase space reconstruction, this paper proposes a new two-dimensional mutual coupled logistic map, which can overcome these weaknesses. Our two-dimensional chaotic map model is simpler than the recently proposed three-dimensional coupled logistic map, whereas the sequence generated by our system is more complex. Furthermore, a new kind of pseudorandom number generator (PRNG) based on the mutual coupled logistic maps is proposed for application. Both statistical tests and security analysis show that our proposed PRNG has good randomness and that it can resist all kinds of attacks. The algorithm speed analysis indicates that PRNG is valuable to practical applications.

scholarly journals Projection of compact fractal sets: application to diffusion-limited and cluster-cluster aggregates

2008 ◽  
Vol 15 (4) ◽  
pp. 695-699 ◽  
Author(s):  
F. Maggi
Keyword(s):  
Fractal Dimension ◽  
Three Dimensional ◽  
Fractal Dimensions ◽  
Cohesive Sediment ◽  
Mathematical Approach ◽  
Two Dimensional ◽  
Atmospheric Sciences ◽  
Practical Applications ◽  
Fractal Sets ◽  
Diffusion Limited

Abstract. The need to assess the three-dimensional fractal dimension of fractal aggregates from the fractal dimension of two-dimensional projections is very frequent in geophysics, soil, and atmospheric sciences. However, a generally valid approach to relate the two- and three-dimensional fractal dimensions is missing, thus questioning the accuracy of the method used until now in practical applications. A mathematical approach developed for application to suspended aggregates made of cohesive sediment is investigated and applied here more generally to Diffusion-Limited Aggregates (DLA) and Cluster-Cluster Aggregates (CCA), showing higher accuracy in determining the three-dimensional fractal dimension compared to the method currently used.

THREE‐DIMENSIONAL INDUCED POLARIZATION AND ELECTROMAGNETIC MODELING

Geophysics ◽  
1975 ◽  
Vol 40 (2) ◽  
pp. 309-324 ◽  
Author(s):  
Gerald W. Hohmann
Keyword(s):  
Integral Equation ◽  
Green's Functions ◽  
Green’S Functions ◽  
Three Dimensional ◽  
Induced Polarization ◽  
The Body ◽  
Scale Model ◽  
Two Dimensional ◽  
Electromagnetic Modeling ◽  
The Earth

The induced polarization (IP) and electromagnetic (EM) responses of a three‐dimensional body in the earth can be calculated using an integral equation solution. The problem is formulated by replacing the body by a volume of polarization or scattering current. The integral equation is reduced to a matrix equation, which is solved numerically for the electric field in the body. Then the electric and magnetic fields outside the inhomogeneity can be found by integrating the appropriate dyadic Green’s functions over the scattering current. Because half‐space Green’s functions are used, it is only necessary to solve for scattering currents in the body—not throughout the earth. Numerical results for a number of practical cases show, for example, that for moderate conductivity contrasts the dipole‐dipole IP response of a body five units in strike length approximates that of a two‐dimensional body. Moving an IP line off the center of a body produces an effect similar to that of increasing the depth. IP response varies significantly with conductivity contrast; the peak response occurs at higher contrasts for two‐dimensional bodies than for bodies of limited length. Very conductive bodies can produce negative IP response due to EM induction. An electrically polarizable body produces a small magnetic field, so that it is possible to measure IP with a sensitive magnetometer. Calculations show that horizontal loop EM response is enhanced when the background resistivity in the earth is reduced, thus confirming scale model results.

Flow past a rotationally oscillating cylinder

2013 ◽  
Vol 735 ◽  
pp. 307-346 ◽  
Author(s):  
S. Kumar ◽  
C. Lopez ◽  
O. Probst ◽  
G. Francisco ◽  
D. Askari ◽  
...  
Keyword(s):  
Reynolds Number ◽  
Vortex Shedding ◽  
Three Dimensional ◽  
Far Field ◽  
Two Dimensional ◽  
Flow Past ◽  
Vortex Shedding Frequency ◽  
Shedding Frequency ◽  
Flow Visualizations ◽  
Hydrogen Bubble Technique

AbstractFlow past a circular cylinder executing sinusoidal rotary oscillations about its own axis is studied experimentally. The experiments are carried out at a Reynolds number of 185, oscillation amplitudes varying from $\mathrm{\pi} / 8$ to $\mathrm{\pi} $, and at non-dimensional forcing frequencies (ratio of the cylinder oscillation frequency to the vortex-shedding frequency from a stationary cylinder) varying from 0 to 5. The diagnostic is performed by extensive flow visualization using the hydrogen bubble technique, hot-wire anemometry and particle-image velocimetry. The wake structures are related to the velocity spectra at various forcing parameters and downstream distances. It is found that the phenomenon of lock-on occurs in a forcing frequency range which depends not only on the amplitude of oscillation but also the downstream location from the cylinder. The experimentally measured lock-on diagram in the forcing amplitude and frequency plane at various downstream locations ranging from 2 to 23 diameters is presented. The far-field wake decouples, after the lock-on at higher forcing frequencies and behaves more like a regular Bénard–von Kármán vortex street from a stationary cylinder with vortex-shedding frequency mostly lower than that from a stationary cylinder. The dependence of circulation values of the shed vortices on the forcing frequency reveals a decay character independent of forcing amplitude beyond forcing frequency of ${\sim }1. 0$ and a scaling behaviour with forcing amplitude at forcing frequencies ${\leq }1. 0$. The flow visualizations reveal that the far-field wake becomes two-dimensional (planar) near the forcing frequencies where the circulation of the shed vortices becomes maximum and strong three-dimensional flow is generated as mode shape changes in certain forcing parameter conditions. It is also found from flow visualizations that even at higher Reynolds number of 400, forcing the cylinder at forcing amplitudes of $\mathrm{\pi} / 4$ and $\mathrm{\pi} / 2$ can make the flow field two-dimensional at forcing frequencies greater than ${\sim }2. 5$.

scholarly journals The Biomechanics of Erections: Modeling the Penis as a One-Comparment Pressurized Vessel vs. Modeling it as a Two-Compartments Pressurized Vessel

2009 ◽  
Vol 3 (2) ◽  
Author(s):  
A. Mohamed ◽  
A. Erdman ◽  
G. Timm
Keyword(s):  
Structural Integrity ◽  
Tunica Albuginea ◽  
Three Dimensional ◽  
The Body ◽  
Physical Models ◽  
Von Mises Stress ◽  
Two Dimensional ◽  
Biomechanical Models ◽  
Significant Difference ◽  
The One

Previous biomechanical models of the penis that have attempted to simulate penile erections have either been limited to two-dimensional geometry, simplified three-dimensional geometry or made inaccurate assumptions altogether. Most models designed the shaft of the penis as a one-compartment pressurized vessel fixed at one end, when in reality it is a two-compartments pressurized vessel, in which the compartments diverge as they enter the body and are fixed at two separate points. This study began by designing simplified two-dimensional and three-dimensional models of the erect penis using Finite Element Analysis (FEA) methods with varying anatomical considerations for analyzing structural stresses, axial buckling and lateral deformation. The study then validated the results by building physical models replicating the computer models. Finally a more complex and anatomically accurate model of the penis was designed and analyzed. There was a significant difference in the peak von-Mises stress distribution between the one-compartment pressurized vessel and the more anatomically correct two-compartments pressurized vessel. Furthermore, the two-compartments diverging pressurized vessel was found to have more structural integrity when subject to external lateral forces than the one-compartment pressurized vessel. This study suggests that Mother Nature has favored an anatomy of two corporal cavernosal bodies separated by a perforated septum as opposed to one corporal body, due to better structural integrity of the tunica albuginea when subject to external forces.

Lift and drag in two-dimensional steady viscous and compressible flow

2015 ◽  
Vol 784 ◽  
pp. 304-341 ◽  
Author(s):  
L. Q. Liu ◽  
J. Y. Zhu ◽  
J. Z. Wu
Keyword(s):  
Mach Number ◽  
Viscous Flow ◽  
Compressible Flow ◽  
Velocity Component ◽  
Transverse Velocity ◽  
The Body ◽  
Far Field ◽  
Two Dimensional ◽  
Wide Range ◽  
Lift And Drag

This paper studies the lift and drag experienced by a body in a two-dimensional, viscous, compressible and steady flow. By a rigorous linear far-field theory and the Helmholtz decomposition of the velocity field, we prove that the classic lift formula $L=-{\it\rho}_{0}U{\it\Gamma}_{{\it\phi}}$, originally derived by Joukowski in 1906 for inviscid potential flow, and the drag formula $D={\it\rho}_{0}UQ_{{\it\psi}}$, derived for incompressible viscous flow by Filon in 1926, are universally true for the whole field of viscous compressible flow in a wide range of Mach number, from subsonic to supersonic flows. Here, ${\it\Gamma}_{{\it\phi}}$ and $Q_{{\it\psi}}$ denote the circulation of the longitudinal velocity component and the inflow of the transverse velocity component, respectively. We call this result the Joukowski–Filon theorem (J–F theorem for short). Thus, the steady lift and drag are always exactly determined by the values of ${\it\Gamma}_{{\it\phi}}$ and $Q_{{\it\psi}}$, no matter how complicated the near-field viscous flow surrounding the body might be. However, velocity potentials are not directly observable either experimentally or computationally, and hence neither are the J–F formulae. Thus, a testable version of the J–F formulae is also derived, which holds only in the linear far field. Due to their linear dependence on the vorticity, these formulae are also valid for statistically stationary flow, including time-averaged turbulent flow. Thus, a careful RANS (Reynolds-averaged Navier–Stokes) simulation is performed to examine the testable version of the J–F formulae for a typical airfoil flow with Reynolds number $Re=6.5\times 10^{6}$ and free Mach number $M\in [0.1,2.0]$. The results strongly support and enrich the J–F theorem. The computed Mach-number dependence of $L$ and $D$ and its underlying physics, as well as the physical implications of the theorem, are also addressed.

scholarly journals Optimal control of the two-dimensional Vlasov-Maxwell system

2020 ◽  
Author(s):  
Jörg Weber
Keyword(s):  
Adjoint Equation ◽  
Collisionless Plasma ◽  
Three Dimensional ◽  
Classical Solutions ◽  
Two Dimensional ◽  
Maxwell System ◽  
Global Classical Solutions ◽  
Global Existence Of Solutions ◽  
Dimensional Version ◽  
The One

The time evolution of a collisionless plasma is modeled by the Vlasov-Maxwell system which couples the Vlasov equation (the transport equation) with the Maxwell equations of electrodynamics. We only consider a two-dimensional version of the problem since existence of global, classical solutions of the full three-dimensional problem is not known. We add external currents to the system, in applications generated by coils, to control the plasma properly. After considering global existence of solutions to this system, differentiability of the control-to-state operator is proved. In applications, on the one hand, we want the shape of the plasma to be close to some desired shape. On the other hand, a cost term penalizing the external currents shall be as small as possible. These two aims lead to minimizing some objective function. We restrict ourselves to only such control currents that are realizable in applications. After that, we prove existence of a minimizer and deduce first order optimality conditions and the adjoint equation.

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