Gary A. Glatzmaier
- Published in print:
- 2013
- Published Online:
- October 2017
- ISBN:
- 9780691141725
- eISBN:
- 9781400848904
- Item type:
- chapter
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691141725.003.0010
- Subject:
- Physics, Particle Physics / Astrophysics / Cosmology
This chapter examines how boundary and geometry affect convection. It begins with a discussion of how one can implement “absorbing” top and bottom boundaries, which reduce the large-amplitude ...
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This chapter examines how boundary and geometry affect convection. It begins with a discussion of how one can implement “absorbing” top and bottom boundaries, which reduce the large-amplitude convectively driven flows within shallow boundary layers or the reflection of internal gravity waves off these boundaries in a stable stratification. It then considers how to replace the impermeable side boundary conditions with permeable periodic side boundary conditions to allow fluid flow through these boundaries and nonzero mean flow. It also introduces “two and a half dimensional” geometry within a cartesian box geometry and describes how a fully 3D cartesian box model could be constructed. Finally, it presents a model of convection in a fully 3D spherical-shell and shows how it can be easily reduced to a 2.5D spherical-shell model. The horizontal structures are represented in terms of spherical harmonic expansions.Less
This chapter examines how boundary and geometry affect convection. It begins with a discussion of how one can implement “absorbing” top and bottom boundaries, which reduce the large-amplitude convectively driven flows within shallow boundary layers or the reflection of internal gravity waves off these boundaries in a stable stratification. It then considers how to replace the impermeable side boundary conditions with permeable periodic side boundary conditions to allow fluid flow through these boundaries and nonzero mean flow. It also introduces “two and a half dimensional” geometry within a cartesian box geometry and describes how a fully 3D cartesian box model could be constructed. Finally, it presents a model of convection in a fully 3D spherical-shell and shows how it can be easily reduced to a 2.5D spherical-shell model. The horizontal structures are represented in terms of spherical harmonic expansions.
Juan Luis Vazquez
- Published in print:
- 2006
- Published Online:
- September 2007
- ISBN:
- 9780198569039
- eISBN:
- 9780191717468
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198569039.001.0001
- Subject:
- Mathematics, Mathematical Physics
The heat equation is one of the three classical linear partial differential equations of second order that form the basis of any elementary introduction to the area of PDEs, and only recently has it ...
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The heat equation is one of the three classical linear partial differential equations of second order that form the basis of any elementary introduction to the area of PDEs, and only recently has it come to be fairly well understood. This book provides a presentation of the mathematical theory of the nonlinear heat equation usually called the Porous Medium Equation (PME). This equation appears in a number of physical applications, such as to describe processes involving fluid flow, heat transfer, or diffusion. Other applications have been proposed in mathematical biology, lubrication, boundary layer theory, and other fields. Each chapter contains a detailed introduction and is supplied with a section of notes, providing comments, historical notes or recommended reading, and exercises.Less
The heat equation is one of the three classical linear partial differential equations of second order that form the basis of any elementary introduction to the area of PDEs, and only recently has it come to be fairly well understood. This book provides a presentation of the mathematical theory of the nonlinear heat equation usually called the Porous Medium Equation (PME). This equation appears in a number of physical applications, such as to describe processes involving fluid flow, heat transfer, or diffusion. Other applications have been proposed in mathematical biology, lubrication, boundary layer theory, and other fields. Each chapter contains a detailed introduction and is supplied with a section of notes, providing comments, historical notes or recommended reading, and exercises.
Gary A. Glatzmaier
- Published in print:
- 2013
- Published Online:
- October 2017
- ISBN:
- 9780691141725
- eISBN:
- 9781400848904
- Item type:
- chapter
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691141725.003.0001
- Subject:
- Physics, Particle Physics / Astrophysics / Cosmology
This chapter presents a model of Rayleigh–Bénard convection. It first describes the fundamental dynamics expected in a fluid that is convectively stable and in one that is convectively unstable, ...
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This chapter presents a model of Rayleigh–Bénard convection. It first describes the fundamental dynamics expected in a fluid that is convectively stable and in one that is convectively unstable, focusing on thermal convection and internal gravity waves. Thermal convection and internal gravity waves are the two basic types of fluid flows within planets and stars that are driven by thermally produced buoyancy forces. The chapter then reviews the equations that govern fluid dynamics based on conservation of mass, momentum, and energy. It also examines the conditions under which the Boussinesq approximation simplifies conservation equations to a form very similar to that of an incompressible fluid. Finally, it discusses the key characteristics of the model of Rayleigh–Bénard convection.Less
This chapter presents a model of Rayleigh–Bénard convection. It first describes the fundamental dynamics expected in a fluid that is convectively stable and in one that is convectively unstable, focusing on thermal convection and internal gravity waves. Thermal convection and internal gravity waves are the two basic types of fluid flows within planets and stars that are driven by thermally produced buoyancy forces. The chapter then reviews the equations that govern fluid dynamics based on conservation of mass, momentum, and energy. It also examines the conditions under which the Boussinesq approximation simplifies conservation equations to a form very similar to that of an incompressible fluid. Finally, it discusses the key characteristics of the model of Rayleigh–Bénard convection.
Robert C. Hilborn
- Published in print:
- 2000
- Published Online:
- January 2010
- ISBN:
- 9780198507239
- eISBN:
- 9780191709340
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198507239.003.0011
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics
When a dynamical system has significant spatial extent, its nonlinear dynamics can lead to the spontaneous formation of spatial patterns. Such systems provide models for how nature might have ...
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When a dynamical system has significant spatial extent, its nonlinear dynamics can lead to the spontaneous formation of spatial patterns. Such systems provide models for how nature might have developed ordered, spatial structures from disordered states. Examples are given from fluid flow, transport models, coupled-oscillator modes, cellular automata, transport models, and reaction-diffusion systems. Diffusion-limited aggregation, viscous fingering, and dielectric breakdown provide further examples of pattern formation. Fractal structures make another appearance in this new context. This chapter also explores the somewhat controversial topic of self-organized criticality which has been put forward as an explanation for the occurrence of fractal structures in nature.Less
When a dynamical system has significant spatial extent, its nonlinear dynamics can lead to the spontaneous formation of spatial patterns. Such systems provide models for how nature might have developed ordered, spatial structures from disordered states. Examples are given from fluid flow, transport models, coupled-oscillator modes, cellular automata, transport models, and reaction-diffusion systems. Diffusion-limited aggregation, viscous fingering, and dielectric breakdown provide further examples of pattern formation. Fractal structures make another appearance in this new context. This chapter also explores the somewhat controversial topic of self-organized criticality which has been put forward as an explanation for the occurrence of fractal structures in nature.
H. A. Stone
- Published in print:
- 2017
- Published Online:
- November 2017
- ISBN:
- 9780198789352
- eISBN:
- 9780191831201
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198789352.003.0001
- Subject:
- Physics, Soft Matter / Biological Physics
The topics discussed are all related to basic fluid mechanics. In these introductory notes I highlight some of the main features of fluid flows and their mathematical characterization. There is much ...
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The topics discussed are all related to basic fluid mechanics. In these introductory notes I highlight some of the main features of fluid flows and their mathematical characterization. There is much physical intuition encapsulated in the differential equations, and one of our goals is to gain more experience (i) understanding the governing equations and various related principles of kinematics, (ii) developing intuition with approximating the equations, (iii) applying the principles to a wide range of problems, which includes (iv) being able to rationalize scaling laws and quantitative trends, often without having a detailed solution in hand. Where possible we provide examples of the ideas with ‘soft interfaces’ in mind.Less
The topics discussed are all related to basic fluid mechanics. In these introductory notes I highlight some of the main features of fluid flows and their mathematical characterization. There is much physical intuition encapsulated in the differential equations, and one of our goals is to gain more experience (i) understanding the governing equations and various related principles of kinematics, (ii) developing intuition with approximating the equations, (iii) applying the principles to a wide range of problems, which includes (iv) being able to rationalize scaling laws and quantitative trends, often without having a detailed solution in hand. Where possible we provide examples of the ideas with ‘soft interfaces’ in mind.
Bijan Mohammadi and Olivier Pironneau
- Published in print:
- 2009
- Published Online:
- February 2010
- ISBN:
- 9780199546909
- eISBN:
- 9780191720482
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199546909.003.0001
- Subject:
- Mathematics, Mathematical Physics
This is a general introduction with prospective issues. This book discusses shape optimization problems for fluids and presents the state of the art in shape optimization for an extended range of ...
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This is a general introduction with prospective issues. This book discusses shape optimization problems for fluids and presents the state of the art in shape optimization for an extended range of applications involving fluid flows.Less
This is a general introduction with prospective issues. This book discusses shape optimization problems for fluids and presents the state of the art in shape optimization for an extended range of applications involving fluid flows.
J. N. Reddy
- Published in print:
- 2014
- Published Online:
- June 2015
- ISBN:
- 9780199641758
- eISBN:
- 9780191789557
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199641758.003.0010
- Subject:
- Physics, Condensed Matter Physics / Materials
This chapter presents finite element models of flows of viscous incompressible fluids for Newtonian and non-Newtonian (power-law) fluids. It develops two different finite element models, namely, the ...
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This chapter presents finite element models of flows of viscous incompressible fluids for Newtonian and non-Newtonian (power-law) fluids. It develops two different finite element models, namely, the mixed finite element model involving velocities and pressure, and the penalty finite element model that involves only velocity degrees of freedom. The formulation of the equations of motion for transient analysis is also discussed. Finally, finite element models of coupled fluid flow and heat transfer, using the penalty method, accounting for buoyancy effects are presented and a number of numerical examples are provided.Less
This chapter presents finite element models of flows of viscous incompressible fluids for Newtonian and non-Newtonian (power-law) fluids. It develops two different finite element models, namely, the mixed finite element model involving velocities and pressure, and the penalty finite element model that involves only velocity degrees of freedom. The formulation of the equations of motion for transient analysis is also discussed. Finally, finite element models of coupled fluid flow and heat transfer, using the penalty method, accounting for buoyancy effects are presented and a number of numerical examples are provided.
Howard C. Elman, David J. Silvester, and Andrew J. Wathen
- Published in print:
- 2014
- Published Online:
- September 2014
- ISBN:
- 9780199678792
- eISBN:
- 9780191780745
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199678792.003.0001
- Subject:
- Mathematics, Numerical Analysis, Computational Mathematics / Optimization
This preliminary chapter gives a brief introduction to the equations studied in depth in the book, as they are used to model incompressible fluids.
This preliminary chapter gives a brief introduction to the equations studied in depth in the book, as they are used to model incompressible fluids.
J. N. Reddy
- Published in print:
- 2014
- Published Online:
- June 2015
- ISBN:
- 9780199641758
- eISBN:
- 9780191789557
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199641758.003.0011
- Subject:
- Physics, Condensed Matter Physics / Materials
This chapter develops finite element formulations for flows of viscous incompressible fluids using high-order spectral/hp finite element technology and least-squares finite element formulation. The ...
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This chapter develops finite element formulations for flows of viscous incompressible fluids using high-order spectral/hp finite element technology and least-squares finite element formulation. The primary objective has been to present novel mathematical models and innovative discretization procedures in the numerical simulation of fluid mechanics problems, wherein the additional benefits of employing high-order spectral/hp finite element technology and least-squares formulations are pronounced. As a result, ad-hoc tricks (e.g., reduced integration and/or mixed interpolations) required to stabilize low-order finite element formulations are unnecessary.Less
This chapter develops finite element formulations for flows of viscous incompressible fluids using high-order spectral/hp finite element technology and least-squares finite element formulation. The primary objective has been to present novel mathematical models and innovative discretization procedures in the numerical simulation of fluid mechanics problems, wherein the additional benefits of employing high-order spectral/hp finite element technology and least-squares formulations are pronounced. As a result, ad-hoc tricks (e.g., reduced integration and/or mixed interpolations) required to stabilize low-order finite element formulations are unnecessary.
Sauro Succi
- Published in print:
- 2018
- Published Online:
- June 2018
- ISBN:
- 9780199592357
- eISBN:
- 9780191847967
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780199592357.003.0001
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics, Condensed Matter Physics / Materials
Fluid flows are a pervasive presence across most branches of human activity, including daily life. Although the basic equations governing the motion of fluid flows are known for two centuries (1822), ...
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Fluid flows are a pervasive presence across most branches of human activity, including daily life. Although the basic equations governing the motion of fluid flows are known for two centuries (1822), since the work of Claude–Louis Navier (1785–1836) and Gabriel Stokes (1819–1903), these equations still set a formidable challenge to the quantitative, and sometimes even qualitative, understanding of the way fluid matter flows in space and time. Meteorological phenomena are among the most popular examples in point, but the challenge extends to many otherinstances of collective fluid motion, both in classical and quantum physics. This Chapter presents the Navier–Stokes equations of fluid mechanics and discuss the main motivations behind the kinetic approach to computational fluid dynamics.Less
Fluid flows are a pervasive presence across most branches of human activity, including daily life. Although the basic equations governing the motion of fluid flows are known for two centuries (1822), since the work of Claude–Louis Navier (1785–1836) and Gabriel Stokes (1819–1903), these equations still set a formidable challenge to the quantitative, and sometimes even qualitative, understanding of the way fluid matter flows in space and time. Meteorological phenomena are among the most popular examples in point, but the challenge extends to many otherinstances of collective fluid motion, both in classical and quantum physics. This Chapter presents the Navier–Stokes equations of fluid mechanics and discuss the main motivations behind the kinetic approach to computational fluid dynamics.
Yaron Oz
- Published in print:
- 2015
- Published Online:
- March 2015
- ISBN:
- 9780198727965
- eISBN:
- 9780191794278
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198727965.003.0008
- Subject:
- Physics, Atomic, Laser, and Optical Physics
This chapter describes how the AdS/CFT correspondence (the Holographic Principle) relates field theory hydrodynamics to perturbations of black hole (brane) gravitational backgrounds. The ...
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This chapter describes how the AdS/CFT correspondence (the Holographic Principle) relates field theory hydrodynamics to perturbations of black hole (brane) gravitational backgrounds. The hydrodynamics framework is first presented from the field theory point of view, after which the dual gravitational description is outlined, first for relativistic fluids and then for the nonrelativistic case. Further details of the fluid/gravity correspondence are then discussed, including the bulk geometry and the dynamics of the black hole horizon.Less
This chapter describes how the AdS/CFT correspondence (the Holographic Principle) relates field theory hydrodynamics to perturbations of black hole (brane) gravitational backgrounds. The hydrodynamics framework is first presented from the field theory point of view, after which the dual gravitational description is outlined, first for relativistic fluids and then for the nonrelativistic case. Further details of the fluid/gravity correspondence are then discussed, including the bulk geometry and the dynamics of the black hole horizon.
