Two-dimensional buoyant plumes in a uniform co-flow

Journal of Fluid Mechanics ◽

10.1017/jfm.2021.1004 ◽

2021 ◽

Vol 932 ◽

Author(s):

Gary R. Hunt ◽

Jamie P. Webb

Keyword(s):

Theoretical Investigation ◽

Analytical Solutions ◽

Richardson Number ◽

Relative Magnitude ◽

Dynamical Behaviour ◽

Finite Width ◽

Two Dimensional ◽

Buoyant Plumes ◽

The Subject ◽

Flow Case

The behaviour of turbulent, buoyant, planar plumes is fundamentally coupled to the environment within which they develop. The effect of a background stratification directly influences a plumes buoyancy and has been the subject of numerous studies. Conversely, the effect of an ambient co-flow, which directly influences the vertical momentum of a plume, has not previously been the subject of theoretical investigation. The governing conservation equations for the case of a uniform co-flow are derived and the local dynamical behaviour of the plume is shown to be characterised by the scaled source Richardson number and the relative magnitude of the co-flow and plume source velocities. For forced, pure and lazy plume release conditions the co-flow acts to narrow the plume and reduce both the dilution and the asymptotic Richardson number relative to the classic zero co-flow case. Analytical solutions are developed for pure plumes from line sources, and for highly forced and highly lazy releases from sources of finite width in a weak co-flow. Contrary to releases in quiescent surroundings, our solutions show that all classes of release can exhibit plume contraction and the associated necking. For entraining plumes, a dynamical invariance spatially only occurs for pure and forced releases and we derive the co-flow strengths that lead to this invariance.

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Kelvin–Helmholtz billows above Richardson number

Journal of Fluid Mechanics ◽

10.1017/jfm.2019.725 ◽

2019 ◽

Vol 879 ◽

Cited By ~ 3

Author(s):

J. P. Parker ◽

C. P. Caulfield ◽

R. R. Kerswell

Keyword(s):

Dynamical System ◽

Richardson Number ◽

Finite Amplitude ◽

Linear Instability ◽

Finite Width ◽

Simple Explanation ◽

Dimensional Manifold ◽

Two Dimensional ◽

Hyperbolic Tangent ◽

Uniform Background

We study the dynamical system of a two-dimensional, forced, stratified mixing layer at finite Reynolds number $Re$, and Prandtl number $Pr=1$. We consider a hyperbolic tangent background velocity profile in the two cases of hyperbolic tangent and uniform background buoyancy stratifications, in a domain of fixed, finite width and height. The system is forced in such a way that these background profiles are a steady solution of the governing equations. As is well known, if the minimum gradient Richardson number of the flow, $Ri_{m}$, is less than a certain critical value $Ri_{c}$, the flow is linearly unstable to Kelvin–Helmholtz instability in both cases. Using Newton–Krylov iteration, we find steady, two-dimensional, finite-amplitude elliptical vortex structures – i.e. ‘Kelvin–Helmholtz billows’ – existing above $Ri_{c}$. Bifurcation diagrams are produced using branch continuation, and we explore how these diagrams change with varying $Re$. In particular, when $Re$ is sufficiently high we find that finite-amplitude Kelvin–Helmholtz billows exist when $Ri_{m}>1/4$ for the background flow, which is linearly stable by the Miles–Howard theorem. For the uniform background stratification, we give a simple explanation of the dynamical system, showing the dynamics can be understood on a two-dimensional manifold embedded in state space, and demonstrate the cases in which the system is bistable. In the case of a hyperbolic tangent stratification, we also describe a new, slow-growing, linear instability of the background profiles at finite $Re$, which complicates the dynamics.

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Quantum Tunneling

10.1093/oso/9780198808275.003.0014 ◽

2017 ◽

Author(s):

Frank S. Levin

Keyword(s):

Quantum Tunneling ◽

Alpha Particles ◽

Attractive Potential ◽

Scanning Tunneling ◽

Finite Width ◽

Potential Well ◽

Surface Molecules ◽

Repulsive Coulomb ◽

The Subject ◽

The One

Quantum tunneling, wherein a quanject has a non-zero probability of tunneling into and then exiting a barrier of finite width and height, is the subject of Chapter 13. The description for the one-dimensional case is extended to the barrier being inverted, which forms an attractive potential well. The first application of this analysis is to the emission of alpha particles from the decay of radioactive nuclei, where the alpha-nucleus attraction is modeled by a potential well and the barrier is the repulsive Coulomb potential. Excellent results are obtained. Ditto for the similar analysis of proton burning in stars and yet a different analysis that explains tunneling through a Josephson junction, the connector between two superconductors. The final application is to the scanning tunneling microscope, a device that allows the microscopic surfaces of solids to be mapped via electrons from the surface molecules tunneling into the tip of the STM probe.

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Two‐Dimensional V‐VI Binary Nanosheets with Square Lattice: A Theoretical Investigation

physica status solidi (b) ◽

10.1002/pssb.202100178 ◽

2021 ◽

Author(s):

Xin Qiao ◽

Xiaodong Lv ◽

Yinan Dong ◽

Yanping Yang ◽

Fengyu Li

Keyword(s):

Theoretical Investigation ◽

Square Lattice ◽

Two Dimensional

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The lateral force on a keel and rudder

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100016789 ◽

1937 ◽

Vol 33 (1) ◽

pp. 62-69 ◽

Cited By ~ 3

Author(s):

S. D. Daymond ◽

L. Rosenhead

Keyword(s):

Theoretical Investigation ◽

Lateral Force ◽

Inviscid Fluid ◽

Two Dimensional ◽

Naval Architecture ◽

Dimensional Flow ◽

Two Dimensional Flow ◽

The University

The following theoretical investigation of the two-dimensional flow of an inviscid fluid past a keel and rudder, and of the consequent lateral force, follows experiments performed by Prof. T. B. Abell in the Department of Naval Architecture of the University of Liverpool, and we wish to acknowledge our indebtedness to him for the information given in many discussions.

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Transient Two-Dimensional Heat Conduction Problem With Partial Heating Near Corners

Volume 2: Heat Transfer in Multiphase Systems; Gas Turbine Heat Transfer; Manufacturing and Materials Processing; Heat Transfer in Electronic Equipment; Heat and Mass Transfer in Biotechnology; Heat Transfer Under Extreme Conditions; Computational Heat Transfer; Heat Transfer Visualization Gallery; General Papers on Heat Transfer; Multiphase Flow and Heat Transfer; Transport Phenomena in Manufacturing and Materials Processing ◽

10.1115/ht2016-7103 ◽

2016 ◽

Author(s):

Robert L. McMasters ◽

Filippo de Monte ◽

James V. Beck ◽

Donald E. Amos

Keyword(s):

Heat Conduction ◽

Numerical Integration ◽

Analytical Solutions ◽

Numerical Solutions ◽

Thermal Protection ◽

Heat Conduction Problem ◽

Separation Of Variables ◽

Two Dimensional ◽

Rectangular Region ◽

Long Distance

This paper provides a solution for two-dimensional heating over a rectangular region on a homogeneous plate. It has application to verification of numerical conduction codes as well as direct application for heating and cooling of electronic equipment. Additionally, it can be applied as a direct solution for the inverse heat conduction problem, most notably used in thermal protection systems for re-entry vehicles. The solutions used in this work are generated using Green’s functions. Two approaches are used which provide solutions for either semi-infinite plates or finite plates with isothermal conditions which are located a long distance from the heating. The methods are both efficient numerically and have extreme accuracy, which can be used to provide additional solution verification. The solutions have components that are shown to have physical significance. The extremely precise nature of analytical solutions allows them to be used as prime standards for their respective transient conduction cases. This extreme precision also allows an accurate calculation of heat flux by finite differences between two points of very close proximity which would not be possible with numerical solutions. This is particularly useful near heated surfaces and near corners. Similarly, sensitivity coefficients for parameter estimation problems can be calculated with extreme precision using this same technique. Another contribution of these solutions is the insight that they can bring. Important dimensionless groups are identified and their influence can be more readily seen than with numerical results. For linear problems, basic heating elements on plates, for example, can be solved to aid in understanding more complex cases. Furthermore these basic solutions can be superimposed both in time and space to obtain solutions for numerous other problems. This paper provides an analytical two-dimensional, transient solution for heating over a rectangular region on a homogeneous square plate. Several methods are available for the solution of such problems. One of the most common is the separation of variables (SOV) method. In the standard implementation of the SOV method, convergence can be slow and accuracy lacking. Another method of generating a solution to this problem makes use of time-partitioning which can produce accurate results. However, numerical integration may be required in these cases, which, in some ways, negates the advantages offered by the analytical solutions. The method given herein requires no numerical integration; it also exhibits exponential series convergence and can provide excellent accuracy. The procedure involves the derivation of previously-unknown simpler forms for the summations, in some cases by virtue of the use of algebraic components. Also, a mathematical identity given in this paper can be used for a variety of related problems.

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BIFURCATION SCENARIO OF A THREE-DIMENSIONAL VAN DER POL OSCILLATOR

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127493000283 ◽

1993 ◽

Vol 03 (02) ◽

pp. 399-404 ◽

Cited By ~ 5

Author(s):

T. SÜNNER ◽

H. SAUERMANN

Keyword(s):

Bifurcation Diagram ◽

Bifurcation Analysis ◽

Chaotic Behavior ◽

Three Dimensional ◽

Global Bifurcation ◽

Dynamical Behaviour ◽

Van Der Pol Oscillator ◽

Two Dimensional ◽

Van Der Pol ◽

Dimensional Models

Nonlinear self-excited oscillations are usually investigated for two-dimensional models. We extend the simplest and best known of these models, the van der Pol oscillator, to a three-dimensional one and study its dynamical behaviour by methods of bifurcation analysis. We find cusps and other local codimension 2 bifurcations. A homoclinic (i.e. global) bifurcation plays an important role in the bifurcation diagram. Finally it is demonstrated that chaos sets in. Thus the system belongs to the few three-dimensional autonomous ones modelling physical situations which lead to chaotic behavior.

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Numerical simulation of two-dimensional fractional neutron diffusion model describing dynamical behaviour of sodium-cooled fast reactor

Annals of Nuclear Energy ◽

10.1016/j.anucene.2021.108709 ◽

2022 ◽

Vol 166 ◽

pp. 108709

Author(s):

Pradip Roul ◽

Vikas Rohil ◽

Gilberto Espinosa-Paredes ◽

K. Obaidurrahman

Keyword(s):

Numerical Simulation ◽

Diffusion Model ◽

Fast Reactor ◽

Dynamical Behaviour ◽

Neutron Diffusion ◽

Two Dimensional

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Closed-form solutions of an elliptical crack subjected to coupled phonon–phason loadings in two-dimensional hexagonal quasicrystal media

Mathematics and Mechanics of Solids ◽

10.1177/1081286518807513 ◽

2018 ◽

Vol 24 (6) ◽

pp. 1821-1848 ◽

Cited By ~ 4

Author(s):

Yuan Li ◽

CuiYing Fan ◽

Qing-Hua Qin ◽

MingHao Zhao

Keyword(s):

Closed Form ◽

Integral Equations ◽

Analytical Solutions ◽

Boundary Integral Equations ◽

Boundary Integral ◽

Elliptical Crack ◽

Two Dimensional ◽

Crack Face ◽

Penny Shaped Crack ◽

Crack Problems

An elliptical crack subjected to coupled phonon–phason loadings in a three-dimensional body of two-dimensional hexagonal quasicrystals is analytically investigated. Owing to the existence of the crack, the phonon and phason displacements are discontinuous along the crack face. The phonon and phason displacement discontinuities serve as the unknown variables in the generalized potential function method which are used to derive the boundary integral equations. These boundary integral equations governing Mode I, II, and III crack problems in two-dimensional hexagonal quasicrystals are expressed in integral differential form and hypersingular integral form, respectively. Closed-form exact solutions to the elliptical crack problems are first derived for two-dimensional hexagonal quasicrystals. The corresponding fracture parameters, including displacement discontinuities along the crack face and stress intensity factors, are presented considering all three crack cases of Modes I, II, and III. Analytical solutions for a penny-shaped crack, as a special case of the elliptical problem, are given. The obtained analytical solutions are graphically presented and numerically verified by the extended displacement discontinuities boundary element method.

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