Dimensional Analysis and Similitude
Dimensional Analysis ä Dimensions and Units Theorem ä Assemblage of Dimensionless Parameters ä Dimensionless Parameters in Fluids ä Model Studies and Similitude ä Dimensions and Units Theorem ä Assemblage of Dimensionless Parameters ä Dimensionless Parameters in Fluids ä Model Studies and Similitude
Frictional Losses in Pipes circa 1900 ä Water distribution systems were being built and enlarged as cities grew rapidly ä 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) ä 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). ä Water distribution systems were being built and enlarged as cities grew rapidly ä 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) ä 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 Theorem ä “in a physical problem including n quantities in which there are m dimensions, the quantities can be arranged into n-m independent dimensionless parameters” ä We reduce the number of parameters we need to vary to characterize the problem! ä “in a physical problem including n quantities in which there are m dimensions, the quantities can be arranged into n-m independent dimensionless parameters” ä We reduce the number of parameters we need to vary to characterize the problem!
Assemblage of Dimensionless Parameters ä Several forces potentially act on a fluid ä Sum of the forces = ma (the inertial force) ä Inertial force is always present in fluids problems (all fluids have mass) ä Nondimensionalize by creating a ratio with the inertial force ä The magnitudes of the force ratios for a given problem indicate which forces govern ä Several forces potentially act on a fluid ä Sum of the forces = ma (the inertial force) ä Inertial force is always present in fluids problems (all fluids have mass) ä Nondimensionalize by creating a ratio with the inertial force ä The magnitudes of the force ratios for a given problem indicate which forces govern
ä Forceparameterdimensionless ä Mass (inertia)______ ä Viscosity____________ ä Gravitational____________ ä Pressure____________ ä Surface Tension____________ ä Elastic____________ ä Forceparameterdimensionless ä Mass (inertia)______ ä Viscosity____________ ä Gravitational____________ ä Pressure____________ ä Surface Tension____________ ä Elastic____________ Forces on Fluids R F pp CpCp W K M Dependent variable
Inertia as our Reference Force ä F=ma Fluids problems always (except for statics) include a velocity (V), a dimension of flow (l), and a density ( ) ä F=ma Fluids problems always (except for statics) include a velocity (V), a dimension of flow (l), and a density ( )
Viscous Force ä What do I need to multiply viscosity by to obtain dimensions of force/volume? Reynolds number
Gravitational Force Froude number
Pressure Force Pressure Coefficient
Dimensionless parameters ä Reynolds Number ä Froude Number ä Weber Number ä Mach Number ä Pressure Coefficient ä (the dependent variable that we measure experimentally) ä Reynolds Number ä Froude Number ä Weber Number ä Mach Number ä Pressure Coefficient ä (the dependent variable that we measure experimentally)
Application of Dimensionless Parameters ä Pipe Flow ä Pump characterization ä Model Studies and Similitude ä dams: spillways, turbines, tunnels ä harbors ä rivers ä ships ä... ä Pipe Flow ä Pump characterization ä Model Studies and Similitude ä dams: spillways, turbines, tunnels ä harbors ä rivers ä ships ä...
Example: Pipe Flow Inertial diameter, length, roughness height Reynolds l/D viscous /D ä What are the important forces? ______, ______. Therefore _________ number. ä What are the important geometric parameters? _________________________ ä Create dimensionless geometric groups ______, ______ ä Write the functional relationship ä What are the important forces? ______, ______. Therefore _________ number. ä What are the important geometric parameters? _________________________ ä Create dimensionless geometric groups ______, ______ ä Write the functional relationship
Example: Pipe Flow C p proportional to l f is friction factor ä How will the results of dimensional analysis guide our experiments to determine the relationships that govern pipe flow? ä If we hold the other two dimensionless parameters constant and increase the length to diameter ratio, how will C p change? ä How will the results of dimensional analysis guide our experiments to determine the relationships that govern pipe flow? ä If we hold the other two dimensionless parameters constant and increase the length to diameter ratio, how will C p change?
E+031E+041E+051E+061E+071E+08 R friction factor laminar smooth Each curve one geometryCapillary tube or 24 ft diameter tunnelWhere is temperature?Compare with real data!Where is “critical velocity”?Where do you specify the fluid? At high Reynolds number curves are flat. Frictional Losses in Straight Pipes
What did we gain by using Dimensional Analysis? ä Any consistent set of units will work ä We don’t have to conduct an experiment on every single size and type of pipe at every velocity ä Our results will even work for different fluids ä Our results are universally applicable ä We understand the influence of temperature ä Any consistent set of units will work ä We don’t have to conduct an experiment on every single size and type of pipe at every velocity ä Our results will even work for different fluids ä Our results are universally applicable ä We understand the influence of temperature
Model Studies and Similitude: Scaling Requirements MachReynoldsFroudeWeber ä dynamic similitude ä geometric similitude ä all linear dimensions must be scaled identically ä roughness must scale ä kinematic similitude ä constant ratio of dynamic pressures at corresponding points ä streamlines must be geometrically similar ä _______, __________, _________, and _________ numbers must be the same ä dynamic similitude ä geometric similitude ä all linear dimensions must be scaled identically ä roughness must scale ä kinematic similitude ä constant ratio of dynamic pressures at corresponding points ä streamlines must be geometrically similar ä _______, __________, _________, and _________ numbers must be the same
Relaxed Similitude Requirements same size ä Impossible to have all force ratios the same unless the model is the _____ ____ as the prototype ä Need to determine which forces are important and attempt to keep those force ratios the same ä Impossible to have all force ratios the same unless the model is the _____ ____ as the prototype ä Need to determine which forces are important and attempt to keep those force ratios the same
Similitude Examples ä Open hydraulic structures ä Ship’s resistance ä Closed conduit ä Hydraulic machinery ä Open hydraulic structures ä Ship’s resistance ä Closed conduit ä Hydraulic machinery
Scaling in Open Hydraulic Structures ä Examples ä spillways ä channel transitions ä weirs ä Important Forces ä inertial forces ä gravity: from changes in water surface elevation ä viscous forces (often small relative to gravity forces) ä Minimum similitude requirements ä geometric ä Froude number ä Examples ä spillways ä channel transitions ä weirs ä Important Forces ä inertial forces ä gravity: from changes in water surface elevation ä viscous forces (often small relative to gravity forces) ä Minimum similitude requirements ä geometric ä Froude number NCHRP Request For Proposal on “Effects of Debris on Bridge-Pier Scour “
Froude similarity difficult to change g ä Froude number the same in model and prototype ä ________________________ ä define length ratio (usually larger than 1) ä velocity ratio ä time ratio ä discharge ratio ä force ratio ä Froude number the same in model and prototype ä ________________________ ä define length ratio (usually larger than 1) ä velocity ratio ä time ratio ä discharge ratio ä force ratio
Example: Spillway Model ä A 50 cm tall scale model of a proposed 50 m spillway is used to predict prototype flow conditions. If the design flood discharge over the spillway is 20,000 m 3 /s, what water flow rate should be tested in the model?
Ship’s Resistance Viscosity, roughness gravity Reynolds Froude ä Skin friction ______________ ä Wave drag (free surface effect) ________ ä Therefore we need ________ and ______ similarity ä Skin friction ______________ ä Wave drag (free surface effect) ________ ä Therefore we need ________ and ______ similarity
Water is the only practical fluid Reynolds and Froude Similarity? ReynoldsFroude L r = 1
Ship’s Resistance ä Can’t have both Reynolds and Froude similarity ä Froude hypothesis: the two forms of drag are independent ä Measure total drag on Ship ä Use analytical methods to calculate the skin friction ä Remainder is wave drag ä Can’t have both Reynolds and Froude similarity ä Froude hypothesis: the two forms of drag are independent ä Measure total drag on Ship ä Use analytical methods to calculate the skin friction ä Remainder is wave drag empirical analytical
Closed Conduit Incompressible Flow viscosity inertia velocity ä Forces ä __________ ä If same fluid is used for model and prototype ä VD must be the same ä Results in high _________ in the model ä High Reynolds number (R) ä Often results are independent of R for very high R ä Forces ä __________ ä If same fluid is used for model and prototype ä VD must be the same ä Results in high _________ in the model ä High Reynolds number (R) ä Often results are independent of R for very high R
Example: Valve Coefficient ä The pressure coefficient,, for a 600-mm-diameter valve is to be determined for 5 ºC water at a maximum velocity of 2.5 m/s. The model is a 60-mm-diameter valve operating with water at 5 ºC. What water velocity is needed?
Example: Valve Coefficient ä Note: roughness height should scale! ä Reynolds similarity ä Note: roughness height should scale! ä Reynolds similarity ν = 1.52 x m 2 /s V m = 25 m/s
Use water at a higher temperature Example: Valve Coefficient (Reduce V m ?) ä What could we do to reduce the velocity in the model and still get the same high Reynolds number? Decrease kinematic viscosity Use a different fluid
Example: Valve Coefficient ä Change model fluid to water at 80 ºC ν m = ______________ ν p = ______________ V m = 6 m/s x m 2 /s 1.52 x m 2 /s
Approximate Similitude at High Reynolds Numbers ä High Reynolds number means ______ forces are much greater than _______ forces ä Pressure coefficient becomes independent of R for high R ä High Reynolds number means ______ forces are much greater than _______ forces ä Pressure coefficient becomes independent of R for high R inertial viscous
Pressure Coefficient for a Venturi Meter E+001E+011E+021E+031E+041E+051E+06 R CpCp Similar to rough pipes in Moody diagram!
Hydraulic Machinery: Pumps streamlines must be geometrically similar ä Rotational speed of pump or turbine is an additional parameter ä additional dimensionless parameter is the ratio of the rotational speed to the velocity of the water _________________________________ ä homologous units: velocity vectors scale _____ ä Now we can’t get same Reynolds Number! ä Reynolds similarity requires ä Scale effects ä Rotational speed of pump or turbine is an additional parameter ä additional dimensionless parameter is the ratio of the rotational speed to the velocity of the water _________________________________ ä homologous units: velocity vectors scale _____ ä Now we can’t get same Reynolds Number! ä Reynolds similarity requires ä Scale effects
Dimensional Analysis Summary ä enables us to identify the important parameters in a problem ä simplifies our experimental protocol (remember Saph and Schoder!) ä does not tell us the coefficients or powers of the dimensionless groups (need to be determined from theory or experiments) ä guides experimental work using small models to study large prototypes ä enables us to identify the important parameters in a problem ä simplifies our experimental protocol (remember Saph and Schoder!) ä does not tell us the coefficients or powers of the dimensionless groups (need to be determined from theory or experiments) ä guides experimental work using small models to study large prototypes Dimensional analysis: end
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Port Model ä A working scale model was used to eliminated danger to boaters from the "keeper roller" downstream from the diversion structure
Hoover Dam Spillway A 1:60 scale hydraulic model of the tunnel spillway at Hoover Dam for investigation of cavitation damage preventing air slots.
Irrigation Canal Controls
Spillways Frenchman Dam and spillway (in use). Lahontan Region (6)
Dams Dec 01, 1974 Cedar Springs Dam, spillway & Reservoir Santa Ana Region (8)
Spillway Mar 01, 1971 Cedar Springs Spillway construction. Santa Ana Region (8)
Kinematic Viscosity 1.00E E E E E-03 mercury carbon tetrachloride water ethyl alcohol kerosene air sae 10W SAE 10W-30 SAE 30 glycerine kinematic viscosity 20C (m 2 /s)
Kinematic Viscosity of Water 0.0E E E E E Temperature (C) Kinematic Viscosity (m 2 /s)