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Fluid Kinematics Fluid Dynamics

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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

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Streamlines Ideal flow machine V 1, b 1 V 2, b 2 Tom Hsu’s numerical simulation

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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

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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

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

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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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Mt. St. Helens

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