Fluid Kinematics Fluid Dynamics Fluid Flow Concepts and

Fluid Kinematics Fluid Dynamics

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 ä Moving vs. Eulerian from a system to a control volume ä Reynolds Transport Theorem

Descriptions of Fluid Motion ä streamline Defined instantaneously 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 ä pathline Defined as particle moves (over time) path of a particle ä same as streamline for steady flow ä ä streakline tracer injected continuously into a flow ä same as pathline and streamline for steady flow ä

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

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

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

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

The Dilemma ä The laws of physics in their simplest forms describe systems (the Lagrangian approach) ä Conservation ä It of Mass, Momentum, Energy 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

Reynolds Transport Theorem äA 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

Control Volume Conservation Equation B =_____________ Total amount of some property in the system b = Amount of the property ______ per unit mass 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

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

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

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

Mt. St. Helens