Potential flow theory allows for the modeling of inviscid and irrotational fluid flows using potential functions. Key points covered in the document include:
- The Laplace equation relates the potential function to the velocity field for incompressible, inviscid flows.
- Common potential flows include uniform flow, source/sink flows, and irrotational vortex flows. Potential functions can be defined for each case.
- For a line source/sink, the potential function is proportional to the logarithm of radius. For an irrotational vortex, the tangential velocity is inversely proportional to radius.
- Streamlines and equipotential lines are perpendicular for potential flows. Circulation can be calculated from
This chapter discusses differential analysis of fluid flow. It introduces the concepts of stream function and vorticity. The key equations derived are:
1) The differential equations of continuity, linear momentum, and mass conservation which relate the time rate of change of fluid properties like density and velocity within an infinitesimal control volume.
2) The Navier-Stokes equations which model viscous flow using Newton's laws and relate stresses to strain rates via viscosity.
3) Equations for inviscid, irrotational flow where viscosity and vorticity are neglected.
4) The stream function, a potential function whose contour lines represent streamlines, allowing 2D problems to be solved using a
This document discusses key concepts in fluid kinematics and dynamics. It defines streamlines, pathlines, and streaklines as field lines that describe the motion of fluid particles. Streamlines show instantaneous velocity, pathlines show trajectories over time, and streaklines show where particles have passed. The document also classifies fluid flows as steady or unsteady, uniform or non-uniform, laminar or turbulent, rotational or irrotational, and one, two, or three-dimensional. Finally, it discusses momentum equations and their application to forces on pipe bends, as well as Bernoulli's theorem.
This document provides an overview of fluid kinematics, which is the study of fluid motion without considering forces. It discusses key concepts like streamlines, pathlines, and streaklines. It describes Lagrangian and Eulerian methods for describing fluid motion. It also covers various types of fluid flow such as steady/unsteady, laminar/turbulent, compressible/incompressible, and one/two/three-dimensional flow. Important topics like continuity equation, velocity, acceleration, and stream/velocity potential functions are also summarized. The document is intended to outline the syllabus and learning objectives for a course unit on fluid kinematics.
This document outlines introductory concepts in fluid dynamics, including:
- Streamlines represent the velocity field at a specific instant, while particle paths and streaklines show the velocity field over time.
- Equations relate the components of velocity to the tangential displacement along streamlines.
- Fluids are treated as continuous media and are often assumed to be incompressible and homogeneous.
- For incompressible flow, the mass flux across any stream tube section is constant. This leads to the continuity equation relating velocity and fluid density.
This document discusses key concepts in fluid dynamics, including:
(i) Fluid kinematics describes fluid motion without forces/energies, examining geometry of motion through concepts like streamlines and pathlines.
(ii) Fluids can flow steadily or unsteadily, uniformly or non-uniformly, laminarly or turbulently depending on properties of the flow and fluid.
(iii) The continuity equation states that mass flow rate remains constant for an incompressible, steady flow through a control volume according to the principle of conservation of mass.
1) The document discusses fluid kinematics, which deals with the motion of fluids without considering the forces that create motion. It covers topics like velocity fields, acceleration fields, control volumes, and flow visualization techniques.
2) There are two main descriptions of fluid motion - Lagrangian, which follows individual particles, and Eulerian, which observes the flow at fixed points in space. Most practical analysis uses the Eulerian description.
3) The Reynolds Transport Theorem allows equations written for a fluid system to be applied to a fixed control volume, which is useful for analyzing forces on objects in a flow. It relates the time rate of change of an extensive property within the control volume to surface fluxes and the property accumulation.
Topics:
1. Introduction to Fluid Dynamics
2. Surface and Body Forces
3. Equations of Motion
- Reynold’s Equation
- Navier-Stokes Equation
- Euler’s Equation
- Bernoulli’s Equation
- Bernoulli’s Equation for Real Fluid
4. Applications of Bernoulli’s Equation
5. The Momentum Equation
6. Application of Momentum Equations
- Force exerted by flowing fluid on pipe bend
- Force exerted by the nozzle on the water
7. Measurement of Flow Rate
a). Venturimeter
b). Orifice Meter
c). Pitot Tube
8. Measurement of Flow Rate in Open Channels
a) Notches
b) Weirs
This document provides an introduction to fluid mechanics. It begins with definitions of mechanics, statics, dynamics, and fluid mechanics. It then discusses different categories of fluid mechanics including fluid statics, fluid kinematics, fluid dynamics, hydrodynamics, hydraulics, gas dynamics, and aerodynamics. The document also defines what a fluid is, discusses the properties of fluids including density, specific weight, specific volume, and specific gravity. It concludes by explaining viscosity, kinematic viscosity, and Newton's law of viscosity.
This chapter discusses differential analysis of fluid flow. It introduces the concepts of stream function and vorticity. The key equations derived are:
1) The differential equations of continuity, linear momentum, and mass conservation which relate the time rate of change of fluid properties like density and velocity within an infinitesimal control volume.
2) The Navier-Stokes equations which model viscous flow using Newton's laws and relate stresses to strain rates via viscosity.
3) Equations for inviscid, irrotational flow where viscosity and vorticity are neglected.
4) The stream function, a potential function whose contour lines represent streamlines, allowing 2D problems to be solved using a
This document discusses key concepts in fluid kinematics and dynamics. It defines streamlines, pathlines, and streaklines as field lines that describe the motion of fluid particles. Streamlines show instantaneous velocity, pathlines show trajectories over time, and streaklines show where particles have passed. The document also classifies fluid flows as steady or unsteady, uniform or non-uniform, laminar or turbulent, rotational or irrotational, and one, two, or three-dimensional. Finally, it discusses momentum equations and their application to forces on pipe bends, as well as Bernoulli's theorem.
This document provides an overview of fluid kinematics, which is the study of fluid motion without considering forces. It discusses key concepts like streamlines, pathlines, and streaklines. It describes Lagrangian and Eulerian methods for describing fluid motion. It also covers various types of fluid flow such as steady/unsteady, laminar/turbulent, compressible/incompressible, and one/two/three-dimensional flow. Important topics like continuity equation, velocity, acceleration, and stream/velocity potential functions are also summarized. The document is intended to outline the syllabus and learning objectives for a course unit on fluid kinematics.
This document outlines introductory concepts in fluid dynamics, including:
- Streamlines represent the velocity field at a specific instant, while particle paths and streaklines show the velocity field over time.
- Equations relate the components of velocity to the tangential displacement along streamlines.
- Fluids are treated as continuous media and are often assumed to be incompressible and homogeneous.
- For incompressible flow, the mass flux across any stream tube section is constant. This leads to the continuity equation relating velocity and fluid density.
This document discusses key concepts in fluid dynamics, including:
(i) Fluid kinematics describes fluid motion without forces/energies, examining geometry of motion through concepts like streamlines and pathlines.
(ii) Fluids can flow steadily or unsteadily, uniformly or non-uniformly, laminarly or turbulently depending on properties of the flow and fluid.
(iii) The continuity equation states that mass flow rate remains constant for an incompressible, steady flow through a control volume according to the principle of conservation of mass.
1) The document discusses fluid kinematics, which deals with the motion of fluids without considering the forces that create motion. It covers topics like velocity fields, acceleration fields, control volumes, and flow visualization techniques.
2) There are two main descriptions of fluid motion - Lagrangian, which follows individual particles, and Eulerian, which observes the flow at fixed points in space. Most practical analysis uses the Eulerian description.
3) The Reynolds Transport Theorem allows equations written for a fluid system to be applied to a fixed control volume, which is useful for analyzing forces on objects in a flow. It relates the time rate of change of an extensive property within the control volume to surface fluxes and the property accumulation.
Topics:
1. Introduction to Fluid Dynamics
2. Surface and Body Forces
3. Equations of Motion
- Reynold’s Equation
- Navier-Stokes Equation
- Euler’s Equation
- Bernoulli’s Equation
- Bernoulli’s Equation for Real Fluid
4. Applications of Bernoulli’s Equation
5. The Momentum Equation
6. Application of Momentum Equations
- Force exerted by flowing fluid on pipe bend
- Force exerted by the nozzle on the water
7. Measurement of Flow Rate
a). Venturimeter
b). Orifice Meter
c). Pitot Tube
8. Measurement of Flow Rate in Open Channels
a) Notches
b) Weirs
This document provides an introduction to fluid mechanics. It begins with definitions of mechanics, statics, dynamics, and fluid mechanics. It then discusses different categories of fluid mechanics including fluid statics, fluid kinematics, fluid dynamics, hydrodynamics, hydraulics, gas dynamics, and aerodynamics. The document also defines what a fluid is, discusses the properties of fluids including density, specific weight, specific volume, and specific gravity. It concludes by explaining viscosity, kinematic viscosity, and Newton's law of viscosity.
This document provides an overview of boundary layer concepts and laminar and turbulent pipe flow. It defines boundary layer thickness, displacement thickness, and momentum thickness. It describes how boundary layers develop on surfaces and transition from laminar to turbulent. It also discusses Reynolds number effects, momentum integral estimates for flat plates, and examples calculating boundary layer thickness in air and water flow. Finally, it introduces concepts of laminar and turbulent pipe flow.
The document discusses the derivation of the Navier-Stokes equations, which describe compressible viscous fluid flow. It derives the continuity, momentum, and energy equations using conservation principles. The equations contain terms for advection, pressure, and viscous forces. Viscous stresses are related to velocity gradients via Newton's law of viscosity. The Navier-Stokes equations, along with appropriate equations of state, form the governing equations for fluid dynamics problems.
The document provides an introduction to fluid dynamics and fluid mechanics. It defines key fluid properties like density, viscosity, pressure and discusses the continuum hypothesis. It also introduces important concepts like the Navier-Stokes equations, Bernoulli's equation, Reynolds number, and divergence. Applications of fluid mechanics in various engineering fields are also highlighted.
This document discusses fluid kinematics, which is the branch of fluid mechanics that deals with the geometry and motion of fluids without considering forces. It defines key concepts like acceleration fields, Lagrangian and Eulerian methods of describing motion, types of flow such as laminar vs turbulent and steady vs unsteady, streamlines vs pathlines vs streaklines, circulation and vorticity, and analytical tools like the stream function and velocity potential function. Flow nets are introduced as a way to graphically study two-dimensional irrotational flows using a grid of intersecting streamlines and equipotential lines.
Dimension less numbers in applied fluid mechanicstirath prajapati
In dimensional analysis, a dimensionless quantity is a quantity to which no physical dimension is assigned. It is also known as a bare number or pure number or a quantity of dimension one[1] and the corresponding unit of measurement in the SI is one (or 1) unit[2][3] and it is not explicitly shown. Dimensionless quantities are widely used in many fields, such as mathematics, physics, chemistry, engineering, and economics. Examples of quantities, to which dimensions are regularly assigned, are length, time, and speed, which are measured in dimensional units, such as meter , second and meter per second. This is considered to aid intuitive understanding. However, especially in mathematical physics, it is often more convenient to drop the assignment of explicit dimensions and express the quantities without dimensions, e.g., addressing the speed of light simply by the dimensionless number 1.
A STUDY ON VISCOUS FLOW (With A Special Focus On Boundary Layer And Its Effects)Rajibul Alam
This document summarizes a study on viscous flow with a focus on boundary layers and their effects. It defines viscosity and describes the boundary layer that forms along a solid surface moving through a fluid. Laminar and turbulent boundary layers are differentiated. The boundary layer equations are presented and used to derive the Navier-Stokes equations that govern viscous fluid flow. Key properties of boundary layers like thickness and velocity profiles are discussed. The interaction of boundary layers and shockwaves is also summarized.
1. This document describes various types of ideal fluid flow, including uniform flow, source/sink flow, vortex flow, and combinations of different flows.
2. Special cases of flow geometry allow the stream function ψ to be related to the distance n along a path between streamlines by ψ = wn. Examples include uniform flow in the x-direction and uniform flow from a line source.
3. Combining different flow types allows modeling of more complex scenarios. A doublet represents a close source-sink pair, and combining it with uniform flow models flow around a cylinder.
This document defines and compares three types of boundary layer thickness:
1. Boundary layer thickness is the distance from the surface where the flow velocity is 99% of the free-stream velocity.
2. Displacement thickness is a theoretical thickness where displacing the surface would result in equal flow rates across sections inside and outside the boundary layer.
3. Momentum thickness is a measure of boundary layer thickness defined as the distance the surface would need to be displaced to compensate for the reduction in momentum due to the boundary layer. It is often used to determine drag on an object.
120218 chapter 8 momentum analysis of flowBinu Karki
The document discusses momentum analysis of fluid flow. It contains the following key points:
1) The momentum equation is based on the law of conservation of momentum, which states that the net force acting on a fluid mass is equal to the rate of change of momentum of the fluid.
2) The momentum principle can be written as an impulse-momentum equation: the impulse of a force acting on a fluid mass over a short time interval is equal to the change in momentum of the fluid.
3) The momentum equation is used to determine the resultant force exerted by a flowing fluid on a pipe bend based on the fluid's velocity, pressure, area, and external forces at two sections of the pipe.
This document discusses various topics related to fluid mechanics including:
1. Fluid statics, hydrostatic pressure variation, and Pascal's law.
2. Different types of pressures like atmospheric pressure, gauge pressure, vacuum pressure, and absolute pressure.
3. The hydrostatic paradox and how pressure intensity is independent of the weight of fluid.
4. Different types of manometers used to measure pressure like piezometers, U-tube manometers, single column manometers, differential manometers, and inverted U-tube differential manometers.
5. How bourdon tubes and diaphragm/bellows gauges can be used to measure pressure by converting pressure differences into mechanical displacements.
This document discusses the different types of fluid flows:
- Steady and unsteady flows, uniform and non-uniform flows, laminar and turbulent flows, compressible and incompressible flows, rotational and irrotational flows, and one, two, and three-dimensional flows. Each type of flow is defined and examples are provided. The key characteristics such as changes in velocity, density, and flow patterns with respect to time and space are outlined for each type of flow. Reynolds number criteria for laminar versus turbulent flow is also mentioned.
This document discusses fluid dynamics and Bernoulli's equation. It begins by defining different forms of energy in a flowing liquid, including kinetic energy, potential energy, pressure energy, and internal energy. It then derives Bernoulli's equation, which states that the total head of a fluid particle remains constant during steady, incompressible flow. The derivation considers forces acting on a fluid particle and uses conservation of energy. Finally, the document presents the general energy equation for steady fluid flow and the specific equation for incompressible fluids using the concepts of total head, head loss, and hydraulic grade line.
The boundary layer is the layer of fluid in immediate contact with a bounding surface, where the effects of viscosity are important. Within the boundary layer, the fluid velocity increases from zero at the surface to 99% of the free-stream velocity. The boundary layer equations allow simplifying the full Navier-Stokes equations by dividing flow into viscous and inviscid regions. Laminar and turbulent boundary layers can form, with laminar producing less drag but being prone to separation in adverse pressure gradients. Boundary layer control techniques influence transition and separation.
Fluid Mechanics Chapter 4. Differential relations for a fluid flowAddisu Dagne Zegeye
Introduction, Acceleration field, Conservation of mass equation, Linear momentum equation, Energy equation, Boundary condition, Stream function, Vorticity and Irrotationality
This document discusses laminar and turbulent fluid flow in pipes. It defines laminar flow as smooth, ordered motion of fluid layers and turbulent flow as irregular motion with velocity fluctuations. The Reynolds number determines the flow regime, with laminar flow below 2000 and turbulent flow above 4000. For fully developed laminar pipe flow, the velocity profile is parabolic and the pressure drop is proportional to flow rate, pipe length, and fluid viscosity, inversely proportional to pipe diameter raised to the fourth power.
This document discusses dimensional analysis and its applications. It begins with an introduction to dimensions, units, fundamental and derived dimensions. It then discusses dimensional homogeneity, methods of dimensional analysis including Rayleigh's method and Buckingham's π-theorem. The document also covers model analysis, similitude, model laws, model and prototype relations. It provides examples of applying Rayleigh's method and Buckingham's π-theorem to define relationships between variables. Finally, it discusses different types of forces acting on fluids and dimensionless numbers, and provides model laws for Reynolds, Froude, Euler and Weber numbers.
1. The document presents an overview of a training package on fluid kinematics for students of environmental engineering.
2. It defines key concepts like fluid velocity, acceleration, types of flow, and the continuity equation.
3. Performance objectives are provided to help students understand fluid kinematics and related concepts after completing the training package.
This document discusses two methods of dimensional analysis: Rayleigh's method and Buckingham π-theorem. Rayleigh's method expresses dependent variables as an exponential function of independent variables. It is useful for problems with 3-4 variables but difficult above that. Buckingham π-theorem states that variables can be grouped into dimensionless π terms, with the number of terms equal to the total variables minus the number of fundamental dimensions. This allows problems with many variables to be solved through grouping and repeating variables. Both methods involve setting up equations using dimensional homogeneity to solve for exponent values.
Fluid Mechanics Chapter 3. Integral relations for a control volumeAddisu Dagne Zegeye
Introduction, physical laws of fluid mechanics, the Reynolds transport theorem, Conservation of mass equation, Linear momentum equation, Angular momentum equation, Energy equation, Bernoulli equation
This document discusses potential flow theory and its applications. It defines irrotational flow, introduces the velocity potential and stream function, and describes several elementary plane flows including uniform flow, source/sink flow, vortex flow, and doublet flow. It also discusses how more complex flows can be modeled through superposition of these elementary flows, providing examples of combining sources, sinks, and uniform flow and combining doublets, vortices, and uniform flow.
This document provides a summary of comments on the book "Vorticity and Vortex Dynamics" by Jie-Zhi Wu, Hui-Yang Ma, and Ming-De Zhou. It notes several typographical errors, inconsistencies in notation, and suggestions for clarifying or improving explanations in equations, figures, and text. In total, 33 numbered comments are provided with specific references to pages, paragraphs, lines, and elements to address. The comments aim to enhance the accuracy and readability of the material presented.
This document provides an overview of boundary layer concepts and laminar and turbulent pipe flow. It defines boundary layer thickness, displacement thickness, and momentum thickness. It describes how boundary layers develop on surfaces and transition from laminar to turbulent. It also discusses Reynolds number effects, momentum integral estimates for flat plates, and examples calculating boundary layer thickness in air and water flow. Finally, it introduces concepts of laminar and turbulent pipe flow.
The document discusses the derivation of the Navier-Stokes equations, which describe compressible viscous fluid flow. It derives the continuity, momentum, and energy equations using conservation principles. The equations contain terms for advection, pressure, and viscous forces. Viscous stresses are related to velocity gradients via Newton's law of viscosity. The Navier-Stokes equations, along with appropriate equations of state, form the governing equations for fluid dynamics problems.
The document provides an introduction to fluid dynamics and fluid mechanics. It defines key fluid properties like density, viscosity, pressure and discusses the continuum hypothesis. It also introduces important concepts like the Navier-Stokes equations, Bernoulli's equation, Reynolds number, and divergence. Applications of fluid mechanics in various engineering fields are also highlighted.
This document discusses fluid kinematics, which is the branch of fluid mechanics that deals with the geometry and motion of fluids without considering forces. It defines key concepts like acceleration fields, Lagrangian and Eulerian methods of describing motion, types of flow such as laminar vs turbulent and steady vs unsteady, streamlines vs pathlines vs streaklines, circulation and vorticity, and analytical tools like the stream function and velocity potential function. Flow nets are introduced as a way to graphically study two-dimensional irrotational flows using a grid of intersecting streamlines and equipotential lines.
Dimension less numbers in applied fluid mechanicstirath prajapati
In dimensional analysis, a dimensionless quantity is a quantity to which no physical dimension is assigned. It is also known as a bare number or pure number or a quantity of dimension one[1] and the corresponding unit of measurement in the SI is one (or 1) unit[2][3] and it is not explicitly shown. Dimensionless quantities are widely used in many fields, such as mathematics, physics, chemistry, engineering, and economics. Examples of quantities, to which dimensions are regularly assigned, are length, time, and speed, which are measured in dimensional units, such as meter , second and meter per second. This is considered to aid intuitive understanding. However, especially in mathematical physics, it is often more convenient to drop the assignment of explicit dimensions and express the quantities without dimensions, e.g., addressing the speed of light simply by the dimensionless number 1.
A STUDY ON VISCOUS FLOW (With A Special Focus On Boundary Layer And Its Effects)Rajibul Alam
This document summarizes a study on viscous flow with a focus on boundary layers and their effects. It defines viscosity and describes the boundary layer that forms along a solid surface moving through a fluid. Laminar and turbulent boundary layers are differentiated. The boundary layer equations are presented and used to derive the Navier-Stokes equations that govern viscous fluid flow. Key properties of boundary layers like thickness and velocity profiles are discussed. The interaction of boundary layers and shockwaves is also summarized.
1. This document describes various types of ideal fluid flow, including uniform flow, source/sink flow, vortex flow, and combinations of different flows.
2. Special cases of flow geometry allow the stream function ψ to be related to the distance n along a path between streamlines by ψ = wn. Examples include uniform flow in the x-direction and uniform flow from a line source.
3. Combining different flow types allows modeling of more complex scenarios. A doublet represents a close source-sink pair, and combining it with uniform flow models flow around a cylinder.
This document defines and compares three types of boundary layer thickness:
1. Boundary layer thickness is the distance from the surface where the flow velocity is 99% of the free-stream velocity.
2. Displacement thickness is a theoretical thickness where displacing the surface would result in equal flow rates across sections inside and outside the boundary layer.
3. Momentum thickness is a measure of boundary layer thickness defined as the distance the surface would need to be displaced to compensate for the reduction in momentum due to the boundary layer. It is often used to determine drag on an object.
120218 chapter 8 momentum analysis of flowBinu Karki
The document discusses momentum analysis of fluid flow. It contains the following key points:
1) The momentum equation is based on the law of conservation of momentum, which states that the net force acting on a fluid mass is equal to the rate of change of momentum of the fluid.
2) The momentum principle can be written as an impulse-momentum equation: the impulse of a force acting on a fluid mass over a short time interval is equal to the change in momentum of the fluid.
3) The momentum equation is used to determine the resultant force exerted by a flowing fluid on a pipe bend based on the fluid's velocity, pressure, area, and external forces at two sections of the pipe.
This document discusses various topics related to fluid mechanics including:
1. Fluid statics, hydrostatic pressure variation, and Pascal's law.
2. Different types of pressures like atmospheric pressure, gauge pressure, vacuum pressure, and absolute pressure.
3. The hydrostatic paradox and how pressure intensity is independent of the weight of fluid.
4. Different types of manometers used to measure pressure like piezometers, U-tube manometers, single column manometers, differential manometers, and inverted U-tube differential manometers.
5. How bourdon tubes and diaphragm/bellows gauges can be used to measure pressure by converting pressure differences into mechanical displacements.
This document discusses the different types of fluid flows:
- Steady and unsteady flows, uniform and non-uniform flows, laminar and turbulent flows, compressible and incompressible flows, rotational and irrotational flows, and one, two, and three-dimensional flows. Each type of flow is defined and examples are provided. The key characteristics such as changes in velocity, density, and flow patterns with respect to time and space are outlined for each type of flow. Reynolds number criteria for laminar versus turbulent flow is also mentioned.
This document discusses fluid dynamics and Bernoulli's equation. It begins by defining different forms of energy in a flowing liquid, including kinetic energy, potential energy, pressure energy, and internal energy. It then derives Bernoulli's equation, which states that the total head of a fluid particle remains constant during steady, incompressible flow. The derivation considers forces acting on a fluid particle and uses conservation of energy. Finally, the document presents the general energy equation for steady fluid flow and the specific equation for incompressible fluids using the concepts of total head, head loss, and hydraulic grade line.
The boundary layer is the layer of fluid in immediate contact with a bounding surface, where the effects of viscosity are important. Within the boundary layer, the fluid velocity increases from zero at the surface to 99% of the free-stream velocity. The boundary layer equations allow simplifying the full Navier-Stokes equations by dividing flow into viscous and inviscid regions. Laminar and turbulent boundary layers can form, with laminar producing less drag but being prone to separation in adverse pressure gradients. Boundary layer control techniques influence transition and separation.
Fluid Mechanics Chapter 4. Differential relations for a fluid flowAddisu Dagne Zegeye
Introduction, Acceleration field, Conservation of mass equation, Linear momentum equation, Energy equation, Boundary condition, Stream function, Vorticity and Irrotationality
This document discusses laminar and turbulent fluid flow in pipes. It defines laminar flow as smooth, ordered motion of fluid layers and turbulent flow as irregular motion with velocity fluctuations. The Reynolds number determines the flow regime, with laminar flow below 2000 and turbulent flow above 4000. For fully developed laminar pipe flow, the velocity profile is parabolic and the pressure drop is proportional to flow rate, pipe length, and fluid viscosity, inversely proportional to pipe diameter raised to the fourth power.
This document discusses dimensional analysis and its applications. It begins with an introduction to dimensions, units, fundamental and derived dimensions. It then discusses dimensional homogeneity, methods of dimensional analysis including Rayleigh's method and Buckingham's π-theorem. The document also covers model analysis, similitude, model laws, model and prototype relations. It provides examples of applying Rayleigh's method and Buckingham's π-theorem to define relationships between variables. Finally, it discusses different types of forces acting on fluids and dimensionless numbers, and provides model laws for Reynolds, Froude, Euler and Weber numbers.
1. The document presents an overview of a training package on fluid kinematics for students of environmental engineering.
2. It defines key concepts like fluid velocity, acceleration, types of flow, and the continuity equation.
3. Performance objectives are provided to help students understand fluid kinematics and related concepts after completing the training package.
This document discusses two methods of dimensional analysis: Rayleigh's method and Buckingham π-theorem. Rayleigh's method expresses dependent variables as an exponential function of independent variables. It is useful for problems with 3-4 variables but difficult above that. Buckingham π-theorem states that variables can be grouped into dimensionless π terms, with the number of terms equal to the total variables minus the number of fundamental dimensions. This allows problems with many variables to be solved through grouping and repeating variables. Both methods involve setting up equations using dimensional homogeneity to solve for exponent values.
Fluid Mechanics Chapter 3. Integral relations for a control volumeAddisu Dagne Zegeye
Introduction, physical laws of fluid mechanics, the Reynolds transport theorem, Conservation of mass equation, Linear momentum equation, Angular momentum equation, Energy equation, Bernoulli equation
This document discusses potential flow theory and its applications. It defines irrotational flow, introduces the velocity potential and stream function, and describes several elementary plane flows including uniform flow, source/sink flow, vortex flow, and doublet flow. It also discusses how more complex flows can be modeled through superposition of these elementary flows, providing examples of combining sources, sinks, and uniform flow and combining doublets, vortices, and uniform flow.
This document provides a summary of comments on the book "Vorticity and Vortex Dynamics" by Jie-Zhi Wu, Hui-Yang Ma, and Ming-De Zhou. It notes several typographical errors, inconsistencies in notation, and suggestions for clarifying or improving explanations in equations, figures, and text. In total, 33 numbered comments are provided with specific references to pages, paragraphs, lines, and elements to address. The comments aim to enhance the accuracy and readability of the material presented.
Astronauts from the International Space Station have observed and documented cloud formations caused by Karman vortices around islands and coastal regions using photos shared on Twitter. The tweets and photos show swirling cloud patterns formed over islands off the coasts of Mexico, the Canary Islands, Cape Verde, and Central America as well as formations shaped by volcanic islands. Studying these cloud patterns from space provides insights into fluid dynamics on Earth.
This document discusses fluid kinematics, which is the branch of fluid mechanics that deals with the geometry of motion of fluids without considering forces or energies. It describes Lagrangian and Eulerian methods for describing fluid motion, defines types of flow such as laminar, turbulent, steady, and unsteady. It also discusses concepts like acceleration fields, circulation, vorticity, streamlines, pathlines, streaklines, stream functions, and velocity potential functions. Flow nets, which use a grid of streamlines and equipotential lines, are introduced as a way to study two-dimensional irrotational flows.
Abstract: Vorticity is a curl of fluid velocity and the Absolute ξ_a = ξ +f where ξ is the relative
vorticity while f is the coriolis parameter. North word moving air acquires and increasing value for f
and hence decreasing value for ξ. The concept of circulation is related that of vorticity, and it has a
number of important applications horizontal closed curves in either hemisphere will have positive
circulation vorticity is negative while the curvature is negative the vorticity is positive.
This document discusses dimensional analysis and conversion factors for converting between different units of measurement. It provides examples of converting between units like inches and centimeters, hours and minutes, meters and kilometers. It emphasizes setting up conversion factors so that the initial and final units cancel out properly. Finally, it lists some common conversion factors between English and metric units like grams to pounds, centimeters to inches, and liters to quarts.
Second application of dimensional analysis
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Dimensional analysis, also known as the factor-label or unit factor method, is a technique for solving problems involving conversion between units. It works by setting up a mathematical equation where the units cancel out, leaving the converted value with the correct units. If the units work out properly, the converted number will be right, unless a mistake is made in calculations. Conversion factors allow changing between units, and can be written as fractions with the initial and final units in the numerator and denominator.
The document discusses metrics, dimensional analysis, and examples of unit conversions. It introduces the metric system which uses base SI units and prefixes that are multiples of ten. Dimensional analysis allows simple unit conversions by identifying conversion factors that relate different units and cancelling units to isolate the desired unit. Examples provided include converting between feet to inches, inches to feet, days to hours, minutes to seconds, centimeters to meters, meters per second to hours, and miles per hour to feet per second.
Line integral,Strokes and Green TheoremHassan Ahmed
The document defines key concepts related to line integrals of vector fields. It defines a vector as having both magnitude and direction. It then defines a line integral as integrating a function along a line, and defines a vector field as a region where a vector quantity (like magnetic field) assigns a unique vector value to each point. Finally, it discusses the definition of a line integral of a vector field, and three fundamental theorems relating line integrals to other integrals: the gradient theorem relating it to differences of a potential function at endpoints, Green's theorem relating it to a double integral, and Stokes' theorem relating it to a surface integral.
This document discusses fluid mechanics concepts including Newton's second law applied to fluid flows and the Bernoulli equation. It provides examples of using the Bernoulli equation to solve problems involving fluid flow, pressure, velocity, and height. The examples calculate pressure differences, flow rates, and maximum jet heights. The document also briefly introduces flowrate measurement using a Pitot tube.
Bernoulli's principle states that an increase in the speed of a fluid occurs simultaneously with a decrease in pressure. Bernoulli discovered that the static pressure plus the dynamic pressure is equal to the total pressure throughout a fluid flow. Applications of Bernoulli's principle include explaining how blood vessels and airplanes are able to fly through the air due to variations in fluid pressure caused by changes in flow speed.
Business Information Systems covers dimensional analysis and dimensional models. Dimensional models organize data into fact and dimension tables for understandability and ease of reporting rather than update efficiency. Facts are measures associated with business processes, while dimensions provide context. Dimensional modeling involves selecting fact tables, determining granularity, adding surrogate keys, date dimensions, and other necessary dimensions. This provides a standardized framework that responds well to changing reporting needs.
This presentation discusses Bernoulli's equation and the Bernoulli family of mathematicians. It introduces Jacob Bernoulli, who published the first differential equation in 1690. While talented, the Bernoulli family members had ego conflicts that strained their relationships. The document then presents Bernoulli's equation, noting it is true for any real number n except 0 or 1, for which it becomes linear. It is solved using a method developed by Leibniz.
This document discusses the application of vector integration in various domains. It begins by defining vector calculus concepts like del, gradient, curl, and divergence. It then presents several theorems of vector integration. Next, it explains how vector integration can be used to find the rate of change of fluid mass and analyze fluid circulation, vorticity, and the Bjerknes Circulation Theorem regarding sea breezes. It also discusses using vector calculus concepts in electricity and magnetism.
The document discusses principles and equations for measuring flow rate using Bernoulli's equation. It covers:
1) Bernoulli's equation relates pressure, velocity, and elevation in fluid flow, and is used in many flow measurement devices.
2) Flow through sluice gates can be calculated using Bernoulli's equation, relating flow rate to gate width and height difference.
3) Weirs can also be used to measure flow rate, with the flow rate proportional to the water height above the weir raised to the 3/2 power.
We discussed most of what one wishes to learn in vector calculus at the undergraduate engineering level. Its also useful for the Physics ‘honors’ and ‘pass’ students.
This was a course I delivered to engineering first years, around 9th November 2009. But I have added contents to make it more understandable, eg I added all the diagrams and many explanations only now; 14-18th Aug 2015.
More such lectures will follow soon. Eg electromagnetism and electromagnetic waves !
The use of Calculus is very important in every aspects of engineering.
The use of Differential equation is very much applied in the concept of Elastic beams.
This document discusses the different types of fluid flows. It describes 6 main types of fluid flows: 1) steady and unsteady, 2) uniform and non-uniform, 3) laminar and turbulent, 4) compressible and incompressible, 5) rotational and irrotational, and 6) one-, two-, and three-dimensional flows. For each type of flow, it provides a brief definition and examples to explain the differences between the types.
Potential flow theory assumes flows are inviscid and irrotational. A potential function φ can be defined to represent the velocity field, satisfying continuity and irrotationality. The Laplace equation governs φ. Potential and streamlines are perpendicular. Simple flows include uniform flow, source/sink, and vortex flows, with characteristic potential functions. The Bernoulli equation relates pressure, velocity, and elevation for steady, inviscid flows.
The document provides an overview of fluid kinematics and dynamics concepts over 12 hours. It discusses types of fluid flow such as steady, unsteady, uniform, laminar, turbulent and more. It also covers fluid motion analysis using Lagrangian and Eulerian methods. Key concepts covered include velocity, acceleration, streamlines, pathlines, continuity equation, and momentum equation. Circulation and vorticity are also defined. The document aims to equip readers with fundamental understanding of fluid motion characteristics and governing equations.
This document discusses several methods for approximating solutions to the Navier-Stokes equations (NSE), including nondimensionalization, creeping flow, inviscid flow, irrotational flow, and potential flow. It explains how these approximations simplify the NSE by removing terms to create linear, analytically solvable forms. Elementary flows like source/sink, vortex, and doublet are introduced that can be combined using superposition to model more complex flows.
This document summarizes key concepts from Chapter 9 of a fluid mechanics textbook on flow over immersed bodies. It discusses:
1) Boundary layer flow and bluff body flow, which are characterized by high or low Reynolds numbers and possible flow separation.
2) Drag is decomposed into form and skin friction contributions, with implications for streamlining bodies.
3) The boundary layer equations, which provide a simplified description of viscous flow and can be solved using methods like the Blasius solution for laminar flat plate boundary layers.
This document discusses two-dimensional ideal fluid flow. It begins by defining an ideal fluid as having no viscosity, compressibility, or surface tension. The continuity equation is then derived, stating that the net flow out of a control volume must equal the change in mass within the volume. Euler's equations are also derived, forming a set of partial differential equations that can be solved to determine pressure and velocity fields. Bernoulli's equation is obtained by integrating the Euler equations, relating total pressure, velocity, and elevation. The concepts of rotational and irrotational flow are introduced, with irrotational flow defined as having zero rotation of any fluid element.
This document provides an overview of fluid kinematics concepts including:
1. The types of fluid flow are defined such as real vs ideal, laminar vs turbulent, steady vs unsteady, uniform vs non-uniform, and one, two, and three dimensional flows.
2. Fluid kinematics variables like velocity, acceleration, and pressure fields are introduced. Streamlines, streamtubes, vorticity, and circulation are also defined.
3. The conservation of mass principle (continuity equation) is presented for one, two, and three dimensional steady and unsteady compressible/incompressible flows.
Consider the incompressible and irrotational flow around a Rankine h.pdfatulkapoor33
Consider the incompressible and irrotational flow around a Rankine half-body... Consider the
incompressible and irrotational flow around a Rankine half-body (cf. Figure 1). The flow can be
modeled using the superposition of a source at point (a. b) and a uniform flow in the redirection.
Suppose that we would like to model the flow around a halfbody that is inclined (relative to the
horizontal) at an angle alpha (i.e. it is rotated CCW by an angle alpha). Find the velocity
potential function Psi for the inclined half-body in terms of x and y coordinates. Find the stream
function Phi for the inclined half-body in terms of x and y coordinates. Utilizing your answer to
part (a), find expressions for the velocity components u and v. Utilizing your answer to part (b).
find expressions for the velocity components u and v. Show that the Laplacian of Psi is equal to
zero. Show that the Laplacian of Phi is equal to zero. Find an expression for the pressure
coefficient that is valid for all points in the flow. Neglect gravitational forces, and assume that
Bernoulli\'s equation applies.
Solution
ø = øUniform flow(UF) + øSource(S)
= - (q/2) ln(r )- Ux
‘q= volume flow rate per unit width
r- Radius
b) Stream function,
j = jUF + jS
= (q/2) - Vy
c) Velocity component from (a)
Ur= Ucos + m/(2r)
d) Velocity component from (b)
V = Usin
e) The equation for z component of vorticity is
v/ x - u/ y =
For irrotational flows , the condition irrotantionality is = 0
v/ x - u/ y = 0
Now in irrotantionality, the scalar function of ø is as follows
‘u = ø/x v= ø/y ………………(1)
Substituting this in vorticity equation, we get
2 ø/xy - 2 ø/yx
=2 ø/xy -2 ø/xy
= 0
Which illustrates that we can write velocity vector u = (u,v) as gradient scalar function of ø. i.e.
If u = ø, x u = 0
Substituting 1 in continuity equation (u/x + v/y= 0)
2 ø/x2 + 2 ø/y2 = 0
As from above we can see laplacian of ø is zero.
f) The equation for z component of vorticity is
v/ x - u/ y =
For irrotational flows , the condition irrotantionality is = 0
v/ x - u/ y = 0
Now in irrotantionality, the scalar function of ø is as follows
‘u = /x v= /y ………………(1)
Substituting this in vorticity equation, we get
2 /xy - 2 /yx
=2 /xy - 2 /xy
= 0
Which illustrates that we can write velocity vector u = (u,v) as gradient scalar function of ø. i.e.
If u = ø, x u = 0
Substituting 1 in continuity equation (u/x + v/y= 0)
2 /x2 + 2 /y2 = 0
As from above we can see laplacian of is zero.
g) Continuity equation, u/x + v/y= 0
Assuming liquid to be perfect with steady flow& neglecting gravity eulers equation can be
described as
u u/x + v v/y = - (1/p) p/x
u v/x + v v/y = - (1/p) p/y
Simplifying this equations
(u2/2)/x + v u/y = (p/p)/x
Adding & subtracting
/x ( u2/2 + v2/2 ) -/x ( (v2/2) + v u/y = - P/x
Rearranging,
/y (u2/2 + v2/2 + P/) = - u (v/x - u/y)
We know that equation for z component of vorticity is
v/x - u/y = 0 (for irrotational flows)
u2/2 + v2/2 + P/ = H (bernouli equation)
H is constant
If flow is uniform @ speeds U
u2/2 .
1. A qubit is the basic unit of quantum computation, represented as a point on the Bloch sphere. Unlike a classical bit, a qubit can exist in a superposition of states |0 and |1.
2. Qubits are mathematically represented as vectors in a two-dimensional Hilbert space. The Pauli matrices σx, σy, σz represent rotations in this space and form a basis to describe operators on qubits.
3. Single qubit gates like the Hadamard gate, phase gate, and √NOT gate put qubits in superpositions of states and are uniquely possible in quantum computers. Multi-qubit systems are described using the tensor product of individual qubit
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This document provides lecture notes on two-dimensional flow of real fluids. It discusses the following key points in 3 sentences:
The flow field is divided into a viscous sublayer zone close to boundaries, where velocity gradients are high, and a potential flow zone away from boundaries, where gradients are small. The Navier-Stokes equations, which include viscosity effects, govern flow in the sublayer zone, while potential flow equations can be used in the outer zone. Laminar flow between parallel plates and in circular pipes is analyzed using the Navier-Stokes equations, yielding parabolic and paraboloid velocity profiles respectively.
This document discusses coordinate systems and vector calculus concepts needed for electromagnetic field theory. It introduces Cartesian, cylindrical, and spherical coordinate systems. It explains that vector integration requires defining appropriate differential elements (length, area, volume) that vary based on the coordinate system. It also introduces concepts of gradient, divergence, and curl - vector operators used to take derivatives of vector fields. The gradient represents the maximum rate of change, divergence measures flux, and curl represents rotational nature. Expressions for these operators are given in the three coordinate systems.
3. Linear Algebra for Machine Learning: Factorization and Linear TransformationsCeni Babaoglu, PhD
The seminar series will focus on the mathematical background needed for machine learning. The first set of the seminars will be on "Linear Algebra for Machine Learning". Here are the slides of the third part which is discussing factorization and linear transformations.
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1) The document discusses transmission lines and their characteristics. It describes different types of transmission lines including coaxial lines, two-wire lines, and microstrip lines.
2) It presents the telegrapher's equations which model voltage and current on a transmission line as a function of position and time. These equations include parameters like inductance and capacitance per unit length.
3) Waves can propagate down transmission lines, maintaining their shape as they travel at a characteristic velocity. The wavelength depends on the wave velocity and frequency. Phasors are used to represent sinusoidal waves independent of time.
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Since this path is often described in three dimensions, vector analysis will
be used to formulate the particle's position, velocity, and acceleration
This document introduces 4-vectors in special relativity. It defines a 4-vector as a set of four components that transform in the same way as (ct, x, y, z) under a Lorentz transformation. Examples of 4-vectors are given, including velocity, momentum, acceleration, and force 4-vectors. Properties of 4-vectors like linear combinations and invariant inner products are discussed. 4-vectors make calculations and concepts in special relativity simpler and more transparent.
This document discusses the continuity of pressure in quantum statistical mechanics. It proves that under certain conditions on the pairwise interaction potential, the pressure in the canonical ensemble is a continuous function that satisfies a Lipschitz condition. Specifically, it shows that if the interaction potential is twice continuously differentiable and satisfies an additional inequality, then:
1) The pressure exists for all specific volumes and temperatures.
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This document summarizes research using an immersed boundary method to model journal bearings in viscoelastic fluids. Key findings include:
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A SYSTEMATIC RISK ASSESSMENT APPROACH FOR SECURING THE SMART IRRIGATION SYSTEMS
Potential flow
1. 2.016 Hydrodynamics Reading #4
2.016 Hydrodynamics
Prof. A.H. Techet
Potential Flow Theory
“When a flow is both frictionless and irrotational, pleasant things happen.” –F.M.
White, Fluid Mechanics 4th ed.
We can treat external flows around bodies as invicid (i.e. frictionless) and irrotational
(i.e. the fluid particles are not rotating). This is because the viscous effects are limited to
a thin layer next to the body called the boundary layer. In graduate classes like 2.25,
you’ll learn how to solve for the invicid flow and then correct this within the boundary
layer by considering viscosity. For now, let’s just learn how to solve for the invicid flow.
We can define a potential function, , , , as a continuous function that satisfies the(x z t)
basic laws of fluid mechanics: conservation of mass and momentum, assuming
incompressible, inviscid and irrotational flow.
There is a vector identity (prove it for yourself!) that states for any scalar, ,
= 0
By definition, for irrotational flow, r
V = 0
Therefore r
V =
where ( , , ) is the velocity potential function. Such that the components of= x y, z t
velocity in Cartesian coordinates, as functions of space and time, are
u
dx
= , v
dy
= and w =
dz
(4.1)
version 1.0 updated 9/22/2005 -1- 2005 A. Techet
2. 2.016 Hydrodynamics Reading #4
Laplace Equation
The velocity must still satisfy the conservation of mass equation. We can substitute in
the relationship between potential and velocity and arrive at the Laplace Equation, which
we will revisit in our discussion on linear waves.
u + v + w = 0 (4.2)
x y z
2 2 2
+ + = 0 (4.3)2 2 2
x y z
LaplaceEquation 2
= 0
For your reference given below is the Laplace equation in different coordinate systems:
Cartesian, cylindrical and spherical.
Cartesian Coordinates (x, y, z)
r ˆ
V = uiˆ+ vjˆ+ wkˆ = iˆ+ j + kˆ =
x y z
2 2 2
2
= + + = 02 2 2
x y z
Cylindrical Coordinates (r, , z)
r2
= x2
+ y2
, = tan y1
( )x
r
V = u eˆ +u eˆ +u eˆ =
eˆ +
1
eˆ + eˆ = r r z z r z
r r z
2 2
2 1 1 2
= + + + = 02 2 2 2
r r r r z1 2434
1
r
r r r
version 1.0 updated 9/22/2005 -2- 2005 A. Techet
3. 2.016 Hydrodynamics Reading #4
version 1.0 updated 9/22/2005 -3- 2005 A. Techet
Spherical Coordinates (r, , )
2 2 2 2
r x y z= + + , ( )1
cos x
r= , or cosx r= , ( )1
tan z
y=
r
V = ur
ˆer +u ˆe +u ˆe =
r
ˆer +
1
r
ˆe +
1
rsin
ˆe =
2
=
2
r2
+
2
r r
1
r2
r
r2
r
1 24 34
+
1
r2
sin
sin
+
1
r2
sin2
2
2
= 0
Potential Lines
Lines of constant are called potential lines of the flow. In two dimensions
d =
x
dx +
y
dy
d = udx + vdy
Since d = 0 along a potential line, we have
dy
dx
=
u
v
(4.4)
Recall that streamlines are lines everywhere tangent to the velocity,
dy
dx
=
v
u
, so potential
lines are perpendicular to the streamlines. For inviscid and irrotational flow is indeed
quite pleasant to use potential function, , to represent the velocity field, as it reduced
the problem from having three unknowns (u, v, w) to only one unknown ( ).
As a point to note here, many texts use stream function instead of potential function as it
is slightly more intuitive to consider a line that is everywhere tangent to the velocity.
Streamline function is represented by . Lines of constant are perpendicular to lines
of constant , except at a stagnation point.
4. 2.016 Hydrodynamics Reading #4
Luckily and are related mathematically through the velocity components:
u = = (4.5)
x y
v = = (4.6)
y x
Equations (4.5) and (4.6) are known as the Cauchy-Riemann equations which appear in
complex variable math (such as 18.075).
Bernoulli Equation
The Bernoulli equation is the most widely used equation in fluid mechanics, and assumes
frictionless flow with no work or heat transfer. However, flow may or may not be
irrotational. When flow is irrotational it reduces nicely using the potential function in
place of the velocity vector. The potential function can be substituted into equation 3.32
resulting in the unsteady Bernoulli Equation.
{ ( )2
} + 1 + p + gz = 0 (4.7)
t 2
or
1 2 p + V + + gz}= 0 . (4.8){t 2
1 2 p ( ) (4.9)UnsteadyBernoulli + V + + gz = c t
t 2
version 1.0 updated 9/22/2005 -4- 2005 A. Techet
5. 2.016 Hydrodynamics Reading #4
Summary
Potential Stream Function
Definition = V = V
v
Continuity
( 0)V =
2
0 = Automatically Satisfied
Irrotationality
( 0)V =
Automatically Satisfied ( ) ( ) 2
=
v v v
= 0
In 2D : 0, 0w
z
= =
2
0 = for continuity 2
0z =
v
for
irrotationality
Cauchy-Riemann Equations for and from complex analysis:
i = + , where is real part and is the imaginary part
Cartesian (x, y)
u
x
=
v
y
=
u
y
=
v
x
=
Polar (r, )
u
r
=
1
v
r
=
1
u
r
=
v
r
=
For irrotational flow use:
For incompressible flow use:
For incompressible and irrotational flow use: and
version 1.0 updated 9/22/2005 -5- 2005 A. Techet
6. 2.016 Hydrodynamics Reading #4
Potential flows
Potential functions (and stream functions, ) can be defined for various simple flows.
These potential functions can also be superimposed with other potential functions to
create more complex flows.
Uniform, Free Stream Flow (1D)
r
V =Uiˆ+ 0 ˆj + 0kˆ (4.10)
u U= = = (4.11)
x y
0v = = = (4.12)
y x
We can integrate these expressions, ignoring the constant of integration which ultimately
does not affect the velocity field, resulting in and
=Ux and =Uy (4.13)
Therefore we see that streamlines are horizontal straight lines for all values of y (tangent
everywhere to the velocity!) and that equipotential lines are vertical straight lines
perpendicular to the streamlines (and the velocity!) as anticipated.
U V• 2D Uniform Flow: V = ( , ,0) ; =Ux +Vy ; =Uy Vx
• 3D Uniform Flow: V = ( , , VyU V W ) ; =Ux + +Wz ; no stream function in 3D
version 1.0 updated 9/22/2005 -6- 2005 A. Techet
7. 2.016 Hydrodynamics Reading #4
Line Source or Sink
Consider the z-axis (into the page) as a porous hose with fluid radiating outwards or
being drawn in through the pores. Fluid is flowing at a rate Q (positive or outwards for a
source, negative or inwards for a sink) for the entire length of hose, b. For simplicity take
a unit length into the page (b = 1) essentially considering this as 2D flow.
Polar coordinates come in quite handy here. The source is located at the origin of the
coordinate system. From the sketch above you can see that there is no circumferential
velocity, but only radial velocity. Thus the velocity vector is
r
V = u eˆ +u eˆ +u eˆ = u eˆ + 0eˆ + 0eˆ (4.14)r r z z r r z
Q m 1
u = = = = (4.15)r
2r r r r
and
1
u = 0 = = (4.16)
r r
Integrating the velocity we can solve for and
ln= m r and = m (4.17)
Q
where m = . Note that satisfies the Laplace equation except at the origin:
2
2 2
r = x + y = 0 , so we consider the origin a singularity (mathematically speaking) and
exclude it from the flow.
• The net outward volume flux can be found by integrating in a closed contour
around the origin of the source (sink):
2
ˆ V n dS = V dS = u r d = Qro
C S
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8. 2.016 Hydrodynamics Reading #4
Irrotational Vortex (Free Vortex)
A free or potential vortex is a flow with circular paths around a central point such that the
velocity distribution still satisfies the irrotational condition (i.e. the fluid particles do not
themselves rotate but instead simply move on a circular path). See figure 2.
Figure 2: Potential vortex with flow in circular patterns around the center.
Here there is no radial velocity and the individual particles do not rotate
about their own centers.
It is easier to consider a cylindrical coordinate system than a Cartesian coordinate system
with velocity vector V = ( , ,u ) when discussing point vortices in a local referenceu ur z
frame. For a 2D vortex, uz = 0 . Referring to figure 2, it is clear that there is also no
radial velocity. Thus,
r
V = u eˆ +u eˆ +u eˆ = 0eˆ +u eˆ + 0eˆ (4.18)r r z z r z
where
1
u = 0 = = (4.19)r
r r
and
1
u = ? = = . (4.20)
r r
Let us derive u . Since the flow is considered irrotational, all components of the
vorticity vector must be zero. The vorticity in cylindrical coordinates is
V =
1 uz
u
er +
ur
u
e
1 ru 1 u z
ez = 0 , (4.21) +
r
r z z r r r r
where
version 1.0 updated 9/22/2005 -8- 2005 A. Techet
9. 2.016 Hydrodynamics Reading #4
2 2 2
u = u + u , (4.22)r x y
u = u cos , (4.23) x
and
u = 0 (for 2D flow).z
Since the vortex is 2D, the z-component of velocity and all derivatives with respect to z
are zero. Thus to satisfy irrotationality for a 2D potential vortex we are only left with the
z-component of vorticity (ez )
ru u
r
= 0 (4.24)
r
Since the vortex is axially symmetric all derivatives with respect must be zero. Thus,
(ru ) u r
= = 0 (4.25)
r
From this equation it follows that ru must be a constant and the velocity distribution for
a potential vortex is
K
u = , ru 0= , zu 0= (4.26)
r
By convention we set the constant equal to , where is the circulation,. Therefore
2
u = (4.27)
2r
U
r
Figure 3: Plot of velocity as a function of radius from the vortex center.
At the core of the potential vortex the velocity blows up to infinity and is
thus considered a singularity.
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You will notice (see figure 3) that the velocity at the center of the vortex goes to infinity
(as r 0 ) indicating that the potential vortex core represents a singularity point. This is
not true in a real, or viscous, fluid. Viscosity prevents the fluid velocity from becoming
infinite at the vortex core and causes the core rotate as a solid body. The flow in this core
region is no longer considered irrotational. Outside of the viscous core potential flow can
be considered acceptable.
Integrating the velocity we can solve for and
ln = K and = K r (4.28)
where K is the strength of the vortex. By convention we consider a vortex in terms of its
circulation, , where = 2 K is positive in the clockwise direction and represents the
strength of the vortex, such that
2
= and ln
2
r
= . (4.29)
Note that, using the potential or stream function, we can confirm that the velocity field
resulting from these functions has no radial component and only a circumferential
velocity component.
The circulation can be found mathematically as the line integral of the tangential
component of velocity taken about a closed curve, C, in the flow field. The equation for
circulation is expressed as
= C
V sd
where the integral is taken in a counterclockwise direction about the contour, C, and
ds is a differential length along the contour. No singularities can lie directly on the
contour. The origin (center) of the potential vortex is considered as a singularity point in
the flow since the velocity goes to infinity at this point. If the contour encircles the
potential vortex origin, the circulation will be non-zero. If the contour does not encircle
any singularities, however, the circulation will be zero.
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11. 2.016 Hydrodynamics Reading #4
To determine the velocity at some point P away from a point vortex (figure 4), we need
to first know the velocity field due to the individual vortex, in the reference frame of the
vortex. Equation Error! Reference source not found. can be used to determine the
tangential velocity at some distance ro from the vortex. It was given up front that ur = 0
everywhere.
Since the velocity at some distance ro from the body is constant on a circle, centered on
the vortex origin, the angle is not crucial for determining the magnitude of theo
tangential velocity. It is necessary, however, to know
o in order to resolve the direction
of the velocity vector at point P. The velocity vector can then be transformed into
Cartesian coordinates at point P using equations (4.22) and (4.23).
Figure 4: Velocity vector at point P due to a potential vortex, with strength
, located some distance ro away.
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12. 2.016 Hydrodynamics Reading #4
Linear Superposition
All three of the simple potential functions, presented above, satisfy the Laplace equation.
Since Laplace equation is a linear equation we are able to superimpose two potential
functions together to describe a complex flow field. Laplace’s equation is
2 2 2
2
= + + = 0. (4.30)2 2 2
x y z
Let 2
= + where 2
= 0 and = 0. Laplace’s equation for the total potential is1 2 1 2
2 2 2
( )+
( ) ( )+ + +2 1 2 1 2 1 2
= + . (4.31)2 2 2
x y z
2 2 2 2 2 2
1
+ 2
+ 1
+ 2
(4.32)2
= 2
1
+ 2
+2 2 2 2 2
x x y y z z
2 2 2 2 2 2
2
2
= 2
1
+ 1
+ 2
+ 2
+ 2 (4.33)2
+ 2
1
2 2
x y z x y z
2 2
0 0 = 0 (4.34) = 2
+ = +1 2
Therefore the combined potential also satisfies continuity (Laplace’s Equation)!
Example: Combined source and sink
Take a source, strength +m, located at (x,y) = (-a,0) and a sink, strength –m, located at
(x,y) = (+a,0).
+ =
1
m ln((x + a)
2
+ y2
) ln((x a)
2
+ y2
) (4.35)= source sink
2
2
This is presented in cartesian coordinates for simplicity. Recall r2
= x + y2
so that
1
2 2 2
m r mln (x + y ) =
1
mln(x + y ) (4.36)ln = 2 2
2
2
1 (x a)
2
+ y+
= mln (4.37)2
2 (x a)
2
+ y
This is analogous to the electro-potential patterns of a magnet with poles at (±a,0).
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Example: Multiple Point Vortices
Since we are able to represent a vortex with a simple potential velocity function, we can
readily investigate the effect of multiple vortices in close proximity to each other. This
can be done simply by a linear superposition of potential functions. Take for example two
vortices, with circulation 1 and , placed at ±a along the x-axis (see figure 5). The2
velocity at point P can be found as the vector sum of the two velocity components V1 and
V2, corresponding to the velocity generated independently at point P by vortex 1 and
vortex 2, respectively.
Figure 5: Formulation of the combined velocity field from two vortices in
close proximity to each other. Vortex 1 is located at point (x, y) = (-a, 0)
and vortex 2 at point (x, y) = (+a, 0).
The total velocity potential function is simply a sum of the potentials for the two
individual vortices
+ = 1
+ 2
= + =T v1 v2 1 2 1 2
2 2
with and taken as shown in figure 5. One vortex in close proximity to another vortex1 2
tends to induce a velocity on its neighbor, causing the free vortex to move.
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