Elementary Fluid Dynamics: The Bernoulli EquationStreamlinesBernoulli Along a StreamlineSlide 4Integrate F=ma Along a StreamlineBernoulli EquationSlide 7Hydraulic and Energy Grade Lines (neglecting losses for now)Bernoulli Equation: Simple Case (V = 0)Bernoulli Equation: Simple Case (p = 0 or constant)Bernoulli Equation Application: Stagnation TubeSlide 12Stagnation TubeBernoulli Normal to the StreamlinesSlide 15Integrate F=ma Normal to the StreamlinesPressure Change Across StreamlinesPitot TubesPitot TubeRelaxed Assumptions for Bernoulli EquationBernoulli Equation ApplicationsPing Pong BallSummaryJet ProblemJet SolutionExample: VenturiSlide 27Example VenturiElementary Fluid Dynamics:The Bernoulli EquationElementary Fluid Dynamics:The Bernoulli EquationCVEN 311 Fluid DynamicsCVEN 311 Fluid DynamicsStreamlinesStreamlinesSteady StateBernoulliAlong a StreamlineBernoulliAlong a Streamlinejizzyyxxkˆsˆnˆp r g- � = +a kSeparate acceleration due to gravity. Coordinate system may be in any orientation!spassdzdr g�- = +�Component of g in s directionNote: No shear forces! Therefore flow must be frictionless. Steady state (no change in p wrt time)(eqn 2.2)BernoulliAlong a StreamlineBernoulliAlong a StreamlinesdV Vadt s�= =�sdV Vadt s�= =�sp dzas dsr g�- = +�p pdp ds dns n� �= +� �0 (n is constant along streamline)dp dV dzVds ds dsr g- = +( )212d VdVVds ds=Write acceleration as derivative wrt schain ruledsdt=dsdt=dp ds p s\ =� �dV ds V s=� �andCan we eliminate the partial derivative?VVs��VVs��Integrate F=ma Along a StreamlineIntegrate F=ma Along a Streamline( )212d Vdp dzds ds dsr g- = +( )2102dp d V dzr g+ + =( )2102dpd V g dzr+ + =� � �212dpV gz Cr+ + =�212p V z Cr g+ + =If density is constant…Along a streamlineBut density is a function of ________.pressureEliminate dsNow let’s integrate…Inviscid (frictionless)Bernoulli EquationBernoulli EquationAssumptions needed for Bernoulli EquationEliminate the constant in the Bernoulli equation? _______________________________________Bernoulli equation does not include ___________________________ ___________________________Assumptions needed for Bernoulli EquationEliminate the constant in the Bernoulli equation? _______________________________________Bernoulli equation does not include ___________________________ ___________________________Apply at two points along a streamline.Mechanical energy to thermal energyHeat transfer, shaft workSteadyConstant density (incompressible)Along a streamlineBernoulli EquationBernoulli EquationThe Bernoulli Equation is a statement of the conservation of ____________________Mechanical Energyp.e.k.e.212pgz V Cr+ + =22p Vz Cgg+ + =Pressure headpg=z =pzg+ =22Vg=Elevation headVelocity headPiezometric head22p Vzgg+ + =Total headHydraulic and Energy Grade Lines (neglecting losses for now)Hydraulic and Energy Grade Lines (neglecting losses for now)The 2 cm diameter jet is 5 m lower than the surface of the reservoir. What is the flow rate (Q)?zz22VgElevation datumPressure datum? __________________Atmospheric pressureWhat about the free jet?pBernoulli Equation: Simple Case (V = 0)Bernoulli Equation: Simple Case (V = 0)Reservoir (V = 0)Put one point on the surface, one point anywhere elseReservoir (V = 0)Put one point on the surface, one point anywhere elsehElevation datum21 2pz zg- =21 2pz zg- =Pressure datum12Same as we found using statics 22p Vz Cgg+ + =1 21 2p pz zg g+ = +We didn’t cross any streamlines so this analysis is okay!Bernoulli Equation: Simple Case (p = 0 or constant)Bernoulli Equation: Simple Case (p = 0 or constant)What is an example of a fluid experiencing a change in elevation, but remaining at a constant pressure? ________What is an example of a fluid experiencing a change in elevation, but remaining at a constant pressure? ________2 21 1 2 21 22 2p V p Vz zg gg g+ + = + +2 21 1 2 21 22 2p V p Vz zg gg g+ + = + +( )22 1 2 12V g z z V= - +( )22 1 2 12V g z z V= - +2 21 21 22 2V Vz zg g+ = +2 21 21 22 2V Vz zg g+ = +Free jetBernoulli Equation Application:Stagnation TubeBernoulli Equation Application:Stagnation TubeWhat happens when the water starts flowing in the channel?Does the orientation of the tube matter? _______How high does the water rise in the stagnation tube?How do we choose the points on the streamline?What happens when the water starts flowing in the channel?Does the orientation of the tube matter? _______How high does the water rise in the stagnation tube?How do we choose the points on the streamline?2C2p Vzgg+ + =2C2p Vzgg+ + =Stagnation pointYes!2 21 1 2 21 22 2p V p Vz zg gg g+ + = + +x2C2p Vzgg+ + =2C2p Vzgg+ + =2a1a2b1bBernoulli Equation Application:Stagnation TubeBernoulli Equation Application:Stagnation Tube1a-2a_______________1b-2a_______________ 1a-2b____________________________1a-2a_______________1b-2a_______________ 1a-2b____________________________Same streamlineCrosses streamlinesDoesn’t cross streamlinesz1. We can obtain V1 if p1 and (z2-z1) are known2. z2 is the total energy!Stagnation TubeStagnation TubeGreat for measuring __________________How could you measure Q?Could you use a stagnation tube in a pipeline?What problem might you encounter?How could you modify the stagnation tube to solve the problem?Great for measuring __________________How could you measure Q?Could you use a stagnation tube in a pipeline?What problem might you encounter?How could you modify the stagnation tube to solve the problem?EGL (defined for a point)EGL (defined for a point)Q V dA= ��Q V dA= ��Bernoulli Normal to the StreamlinesBernoulli Normal to the Streamlineskˆsˆnˆp r g- � = +a kSeparate acceleration due to gravity. Coordinate system may be in any orientation!npanndzdr g�- = +�Component of g in n directionBernoulli Normal to the StreamlinesBernoulli Normal to the Streamlines2nVa =R2nVa =Rnp dzan dnr g�- = +�p pdp ds dns n� �= +� �0 (s is constant along streamline)2dp V dzdn dnr g- = +Rcentrifugal force. R is local radius of curvaturedp dn p n\ =� � dV dn V n=� �andn is toward the center of the radius of curvatureIntegrate F=ma Normal to the StreamlinesIntegrate F=ma Normal to the Streamlines2dp Vdn gdz Cr+ + =�������RIf density is constant…Normal to streamline2dp V dzdn dnr g- = +R2p Vdn gz Cr+ + =���R2Vp dn z Cr