Fluid KinematicsFluid Flow Concepts and Reynolds Transport TheoremDescriptions of Fluid MotionDescriptors of Fluid FlowsTemporal/Spatial ClassificationsAnalysis ApproachesThe DilemmaReynolds Transport TheoremControl Volume Conservation EquationSummarySlide 11Mt. St. HelensApplication of Reynold’s Transport TheoremMonroe L. Weber-Shirk School of Civil and Environmental EngineeringFluid KinematicsFluid KinematicsFluid MechanicsJanuary 14, 2019Fluid Flow Concepts and Reynolds Transport TheoremDescriptions of:fluid motionfluid flowstemporal and spatial classificationsAnalysis ApproachesLagrangian vs. EulerianMoving from a system to a control volumeReynolds Transport TheoremDefined as particle moves (over time)Defined instantaneouslyDescriptions of Fluid Motionstreamlinehas the direction of the velocity vector at each pointno flow across the streamlinesteady flow streamlines are fixed in spaceunsteady flow streamlines movepathlinepath of a particlesame as streamline for steady flowstreaklinetracer injected continuously into a flowsame as pathline and streamline for steady flowDraw StreamlinesUnsteady demoDescriptors of Fluid FlowsLaminar flowfluid moves along smooth pathsviscosity damps any tendency to swirl or mixTurbulent flowfluid moves in very irregular pathsefficient mixingvelocity at a point fluctuatesIf averaged over a suitable timeTemporal/Spatial ClassificationsSteady - unsteady Uniform - nonuniform Can turbulent flow be steady? _______ ________________ ________________Changing in timeChanging in spaceAnalysis ApproachesLagrangian (system approach)Describes a defined _____ (position, velocity, acceleration, pressure, temperature, etc.) as functions of timeTrack the location of a migrating birdEulerianDescribes the flow ______ (velocity, acceleration, pressure, temperature, etc.) as functions of position and timeCount the birds passing a particular locationIf you were going to study water flowing in a pipeline, which approach would you use? ____________EulerianmassfieldThe DilemmaThe laws of physics in their simplest forms describe systems (the Lagrangian approach)Conservation of Mass, Momentum, EnergyIt is impossible to keep track of the system in many fluids problemsThe laws of physics must still hold in a Eulerian world!We need some tools to bridge the gapReynolds Transport TheoremA moving system flows through the fixed control volumeThe moving system transports extensive properties across the control volume surfacesWe need a bookkeeping method to keep track of the properties that are being transported into and out of the control volumeper unit massTotal amount of some propertyControl Volume Conservation Equationˆsyscv csDBbdV b dADt tr r�= + ��� �V nB =__________________________ in the systemb = Amount of the property ___________ = +Rate of increase of the property in the systemRate of increase of the property in the control volumeRate of efflux of the property across the control volume boundarySummaryReynolds Transport Theorem can be applied to a control volume of finite sizeWe don’t need to know the flow details within the control volume!We do need to know what is happening at the control surfaces.Conservation of mass (for all species)Newton’s 2nd law of motion (momentum) _______First law of thermodynamics (energy)F = maControl Volume Conservation Equation0 = -1 + (-0 + 1)0 = 1 + (-1 + 0)0 = 0 + (-0 + 0)ˆsyscv csDBbdV b dADt tr r�= + ��� �V nMt. St. HelensApplication of Reynold’s Transport TheoremChemical with concentration Cin enters reactor with flow rate Q and exits with concentration C.Chemical decays at rate kCWhat is b ? What is B?What is b?What is left side of equation?What is ?ˆsyscv csDBbdV b dADt tr r�= + ��� �V n( )inCVkCV Q C Ct�- = + -�C/CVCkCV-1ˆcsdAV