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Monroe L. Weber-Shirk S chool of Civil and Environmental Engineering Fluid Kinematics Fluid Mechanics July 15, 2015 Fluid Mechanics July 15, 2015

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Fluid Flow Concepts and Reynolds Transport Theorem ä Descriptions of: ä fluid motion ä fluid flows ä temporal and spatial classifications ä Analysis Approaches ä Lagrangian vs. Eulerian ä Moving from a system to a control volume ä Reynolds Transport Theorem ä Descriptions of: ä fluid motion ä fluid flows ä temporal and spatial classifications ä Analysis Approaches ä Lagrangian vs. Eulerian ä Moving from a system to a control volume ä Reynolds Transport Theorem

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Defined as particle moves (over time) Defined instantaneously Descriptions of Fluid Motion ä streamline ä has the direction of the velocity vector at each point ä no flow across the streamline ä steady flow streamlines are fixed in space ä unsteady flow streamlines move ä pathline ä path of a particle ä same as streamline for steady flow ä streakline ä tracer injected continuously into a flow ä same as pathline and streamline for steady flow ä streamline ä has the direction of the velocity vector at each point ä no flow across the streamline ä steady flow streamlines are fixed in space ä unsteady flow streamlines move ä pathline ä path of a particle ä same as streamline for steady flow ä streakline ä tracer injected continuously into a flow ä same as pathline and streamline for steady flow Draw Streamlines and Pathlines

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Streamlines Ideal flow machine V 1, b 1 V 2, b 2

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Descriptors of Fluid Flows ä Laminar flow ä fluid moves along smooth paths ä viscosity damps any tendency to swirl or mix ä Turbulent flow ä fluid moves in very irregular paths ä efficient mixing ä velocity at a point fluctuates ä Laminar flow ä fluid moves along smooth paths ä viscosity damps any tendency to swirl or mix ä Turbulent flow ä fluid moves in very irregular paths ä efficient mixing ä velocity at a point fluctuates Transition to turbulence movie

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If averaged over a suitable time Temporal/Spatial Classifications ä Steady - unsteady ä ä Uniform - nonuniform ä ä Steady - unsteady ä ä Uniform - nonuniform ä Can turbulent flow be steady? _______ ________________ ________________ Changing in time Changing in space

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Analysis Approaches ä Lagrangian (system approach) ä Describes a defined _____ (position, velocity, acceleration, pressure, temperature, etc.) as functions of time ä Track the location of a migrating bird ä Eulerian ä Describes the flow ______ (velocity, acceleration, pressure, temperature, etc.) as functions of position and time ä Count the birds passing a particular location ä Lagrangian (system approach) ä Describes a defined _____ (position, velocity, acceleration, pressure, temperature, etc.) as functions of time ä Track the location of a migrating bird ä Eulerian ä Describes the flow ______ (velocity, acceleration, pressure, temperature, etc.) as functions of position and time ä Count the birds passing a particular location If you were going to study water flowing in a pipeline, which approach would you use? ____________ Eulerian mass field

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The Dilemma ä The laws of physics in their simplest forms describe systems (the Lagrangian approach) ä Conservation of Mass, Momentum, Energy ä It is impossible to keep track of the system in many fluids problems ä The laws of physics must still hold in a Eulerian world! ä We need some tools to bridge the gap ä The laws of physics in their simplest forms describe systems (the Lagrangian approach) ä Conservation of Mass, Momentum, Energy ä It is impossible to keep track of the system in many fluids problems ä The laws of physics must still hold in a Eulerian world! ä We need some tools to bridge the gap

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Reynolds Transport Theorem ä A moving system flows through the fixed control volume ä The moving system transports extensive properties across the control volume surfaces ä We need a bookkeeping method to keep track of the properties that are being transported into and out of the control volume ä A moving system flows through the fixed control volume ä The moving system transports extensive properties across the control volume surfaces ä We need a bookkeeping method to keep track of the properties that are being transported into and out of the control volume

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per unit mass Total amount of some property Control Volume Conservation Equation B =__________________________ in the system b = Amount of the property ___________ =+ Rate of increase of the property in the system Rate of increase of the property in the control volume Rate of efflux of the property across the control volume boundary Example

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Application of Reynolds Transport Theorem ä Conservation of mass (for all species) ä Newton’s 2 nd law of motion (momentum) _______ ä First law of thermodynamics (energy) ä Conservation of mass (for all species) ä Newton’s 2 nd law of motion (momentum) _______ ä First law of thermodynamics (energy) F = ma

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Summary ä Reynolds Transport Theorem can be applied to a control volume of finite size ä We don’t need to know the flow details within the control volume! ä We do need to know what is happening at the control surfaces. ä We will use Reynolds Transport Theorem to solve many practical fluids problems ä Reynolds Transport Theorem can be applied to a control volume of finite size ä We don’t need to know the flow details within the control volume! ä We do need to know what is happening at the control surfaces. ä We will use Reynolds Transport Theorem to solve many practical fluids problems

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Control Volume Conservation Equation 0 = -1 + (-0 + 1) 0 = 1 + (-1 + 0) 0 = 0 + (-0 + 0)

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Mt. St. Helens

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Application of Reynold’s Transport Theorem ä Chemical with concentration C in enters container with flow rate Q. ä Chemical decays at rate kC ä What is b? What is B? What is b? ä What is left side of equation? ä What is C? ä Chemical with concentration C in enters container with flow rate Q. ä Chemical decays at rate kC ä What is b? What is B? What is b? ä What is left side of equation? ä What is C?

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