Dimensional Analysis and SimilitudeDimensional AnalysisFrictional Losses in Pipes circa 1900The Buckingham P TheoremAssemblage of Dimensionless ParametersForces on FluidsInertia as our Reference ForceViscous ForceGravitational ForcePressure ForceDimensionless parametersApplication of Dimensionless ParametersExample: Pipe FlowSlide 14Frictional Losses in Straight PipesWhat did we gain by using Dimensional Analysis?Model Studies and Similitude: Scaling RequirementsRelaxed Similitude RequirementsSimilitude ExamplesScaling in Open Hydraulic StructuresFroude similarityExample: Spillway ModelShip’s ResistanceReynolds and Froude Similarity?Slide 25Closed Conduit Incompressible FlowExample: Valve CoefficientSlide 28Example: Valve Coefficient (Reduce Vm?)Slide 30Approximate Similitude at High Reynolds NumbersPressure Coefficient for a Venturi MeterHydraulic Machinery: PumpsDimensional Analysis SummaryShip’s Resistance: We aren’t done learning yet!Port ModelHoover Dam SpillwayIrrigation Canal ControlsSpillwaysDamsSpillwayKinematic ViscosityKinematic Viscosity of WaterDimensional Analysis and SimilitudeDimensional Analysis and SimilitudeCVEN 311CVEN 311Dimensional AnalysisDimensional AnalysisDimensions and Units TheoremAssemblage of Dimensionless ParametersDimensionless Parameters in FluidsModel Studies and SimilitudeDimensions and Units TheoremAssemblage of Dimensionless ParametersDimensionless Parameters in FluidsModel Studies and SimilitudeFrictional Losses in Pipescirca 1900Frictional Losses in Pipescirca 1900Water distribution systems were being built and enlarged as cities grew rapidlyDesign of the distribution systems required knowledge of the head loss in the pipes (The head loss would determine the maximum capacity of the system)It was a simple observation that head loss in a straight pipe increased as the velocity increased (but head loss wasn’t proportional to velocity). Water distribution systems were being built and enlarged as cities grew rapidlyDesign of the distribution systems required knowledge of the head loss in the pipes (The head loss would determine the maximum capacity of the system)It was a simple observation that head loss in a straight pipe increased as the velocity increased (but head loss wasn’t proportional to velocity).The Buckingham  TheoremThe Buckingham  Theorem“in a physical problem including n quantities in which there are m dimensions, the quantities can be arranged into n-m independent dimensionless parameters”We reduce the number of parameters we need to vary to characterize the problem!“in a physical problem including n quantities in which there are m dimensions, the quantities can be arranged into n-m independent dimensionless parameters”We reduce the number of parameters we need to vary to characterize the problem!Assemblage of Dimensionless ParametersAssemblage of Dimensionless ParametersSeveral forces potentially act on a fluidSum of the forces = ma (the inertial force)Inertial force is always present in fluids problems (all fluids have mass) Nondimensionalize by creating a ratio with the inertial forceThe magnitudes of the force ratios for a given problem indicate which forces governSeveral forces potentially act on a fluidSum of the forces = ma (the inertial force)Inertial force is always present in fluids problems (all fluids have mass) Nondimensionalize by creating a ratio with the inertial forceThe magnitudes of the force ratios for a given problem indicate which forces governForc e para meter dime nsionlessMass (inertia) ______Viscosity ______ ______Gravitational ______ ______Pressure ______ ______Surface Tension ______ ______Elastic ______ ______Forc e para meter dime nsionlessMass (inertia) ______Viscosity ______ ______Gravitational ______ ______Pressure ______ ______Surface Tension ______ ______Elastic ______ ______Forces on FluidsForces on FluidsRFpCpWKMDependent variableDependent variableInertia as our Reference ForceInertia as our Reference ForceF=maFluids problems always (except for statics) include a velocity (V), a dimension of flow (l), and a density ()F=maFluids problems always (except for statics) include a velocity (V), a dimension of flow (l), and a density ()F a Fa ff ML T2 2f ML T2 2L lL lT T M M fifilVlVl3l3Vl2Vl2Viscous ForceViscous ForceWhat do I need to multiply viscosity by to obtain dimensions of force/volume?What do I need to multiply viscosity by to obtain dimensions of force/volume?CfCffCfCLTMTLMC22LTMTLMC22LTC1LTC1μiffμiff2lVC 2lVC VlμiffVlμiffVlRVlRReynolds numberL lTlVM l3fiVl22lVlV2Gravitational ForceGravitational ForcegCggfgCggf222TLTLMCg222TLTLMCg3LMCg3LMCggiffgiffgCgCglV2giffglV2giffglVFglVFFroude numberL lTlVM l3fiVl2lV2gPressure ForcePressure ForcepCppfpCppf222LTMTLMCp222LTMTLMCpLCp1LCp1piffpifflCp1lCp1pV2piffpV2piff22CVpp22CVppPressure CoefficientL lTlVM l3fiVl2lV2lpDimensionless parametersDimensionless parametersReynolds NumberFroude NumberWeber NumberMach NumberPressure Coefficient(the dependent variable that we measure experimentally)Reynolds NumberFroude NumberWeber NumberMach NumberPressure Coefficient(the dependent variable that we measure experimentally)VlRVlRglVFglVF22CVpp22CVpplVW2lVW2cVM cVM AVd2Drag2CAVd2Drag2CApplication of Dimensionless ParametersApplication of Dimensionless ParametersPipe FlowPump characterizationModel Studies and Similitudedams: spillways, turbines, tunnelsharborsriversships...Pipe FlowPump characterizationModel Studies and Similitudedams: spillways, turbines, tunnelsharborsriversships...Example: Pipe FlowExample: Pipe