Monroe L. Weber-Shirk S chool of Civil and Environmental Engineering Dimensional Analysis and Similitude CEE 331 June 28, 2015 CEE 331 June 28, 2015 

Why? ä “One does not want to have to show and relate the results for all possible velocities, for all possible geometries, for all possible roughnesses, and for all possible fluids...” Wilfried Brutsaert in “Horton, Pipe Hydraulics, and the Atmospheric Boundary Layer.” in Bulletin of the American Meteorological Society

On Scaling... ä “...the writers feel that they would well deserve the flood of criticism which is ever threatening those venturous persons who presume to affirm that the same laws of Nature control the flow of water in the smallest pipes in the laboratory and in the largest supply mains running over hill and dale. In this paper it is aimed to present a few additional arguments which may serve to make such an affirmation appear a little less ridiculous than heretofore.” Saph and Schoder, 1903

Why? ä Suppose I want to build an irrigation canal, one that is bigger than anyone has ever built. How can I determine how big I have to make the canal to get the desired flow rate? Do I have to build a section of the canal and test it? ä Suppose I build pumps. Do I have to test the performance of every pump for all speed, flow, fluid, and pressure combinations? ä Suppose I want to build an irrigation canal, one that is bigger than anyone has ever built. How can I determine how big I have to make the canal to get the desired flow rate? Do I have to build a section of the canal and test it? ä Suppose I build pumps. Do I have to test the performance of every pump for all speed, flow, fluid, and pressure combinations?

Dimensional Analysis ä The case of Frictional Losses in Pipes (NYC) ä Dimensions and Units   Theorem ä Assemblage of Dimensionless Parameters ä Dimensionless Parameters in Fluids ä Model Studies and Similitude ä The case of Frictional Losses in Pipes (NYC) 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).

Two Opposing Theories ä agrees with the “law of a falling body” ä f varies with velocity and is different for different pipes ä agrees with the “law of a falling body” ä f varies with velocity and is different for different pipes ä ä Fits the data well for any particular pipe ä ä Every pipe has a different m and n. ä ä What does g have to do with this anyway? “In fact, some engineers have been led to question whether or not water flows in a pipe according to any definite determinable laws whatsoever.” Saph and Schoder, 1903 h l is mechanical energy lost to thermal energy expressed as p.e.

Research at Cornell! ä Augustus Saph and Ernest Schoder under the direction of Professor Gardner Williams ä Saph and Schoder had concluded that “there is practically no difference between a 2-in. and a 30- in. pipe.” ä Conducted comprehensive experiments on a series of small pipes located in the basement of Lincoln Hall, (the principle building of the College of Civil Engineering) ä Chose to analyze their data using ________ ä Augustus Saph and Ernest Schoder under the direction of Professor Gardner Williams ä Saph and Schoder had concluded that “there is practically no difference between a 2-in. and a 30- in. pipe.” ä Conducted comprehensive experiments on a series of small pipes located in the basement of Lincoln Hall, (the principle building of the College of Civil Engineering) ä Chose to analyze their data using ________

Saph and Schoder Conclusions Oh, and by the way, there is a “critical velocity” below which this equation doesn’t work. The “critical velocity” varies with pipe diameter and with temperature. Check units... h l is in ft/1000ft V is in ft/s d is in ft Oops!!

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 usually significant in fluids problems (except some very slow flows) ä Nondimensionalize all other forces 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 usually significant in fluids problems (except some very slow flows) ä Nondimensionalize all other forces by creating a ratio with the inertial force ä The magnitudes of the force ratios for a given problem indicate which forces govern

ä Forceparameter ä Mass (inertia)______ ä Viscosity______ ä Gravitational______ ä Surface Tension______ ä Elasticity______ ä Pressure______ ä Forceparameter ä Mass (inertia)______ ä Viscosity______ ä Gravitational______ ä Surface Tension______ ä Elasticity______ ä Pressure______ Forces on Fluids   g pp  K Dependent variable

Ratio of Forces ä Create ratios of the various forces ä The magnitude of the ratio will tell us which forces are most important and which forces could be ignored ä Which force shall we use to create the ratios? ä Create ratios of the various forces ä The magnitude of the ratio will tell us which forces are most important and which forces could be ignored ä Which force shall we use to create the ratios?

Inertia as our Reference Force ä F=ma  Fluids problems (except for statics) include a velocity (V), a dimension of flow (l), and a density (  )  Substitute V, l,  for the dimensions MLT ä Substitute for the dimensions of specific force ä F=ma  Fluids problems (except for statics) include a velocity (V), a dimension of flow (l), and a density (  )  Substitute V, l,  for the dimensions MLT ä Substitute for the dimensions of specific force

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/Drag Coefficients ä (dependent parameters that we measure experimentally) ä Reynolds Number ä Froude Number ä Weber Number ä Mach Number ä Pressure/Drag Coefficients ä (dependent parameters that we measure experimentally)

Problem solving approach 1. Identify relevant forces and any other relevant parameters 2. If inertia is a relevant force, than the non dimensional Re, Fr, W, M numbers can be used 3. If inertia isn’t relevant than create new non dimensional force numbers using the relevant forces 4. Create additional non dimensional terms based on geometry if there is more than one geometric dimension (choose the repeating geometric dimension) 5. If the problem uses different repeating variables then substitute (for example  d instead of V) 6. Write the functional relationship 1. Identify relevant forces and any other relevant parameters 2. If inertia is a relevant force, than the non dimensional Re, Fr, W, M numbers can be used 3. If inertia isn’t relevant than create new non dimensional force numbers using the relevant forces 4. Create additional non dimensional terms based on geometry if there is more than one geometric dimension (choose the repeating geometric dimension) 5. If the problem uses different repeating variables then substitute (for example  d instead of V) 6. Write the functional relationship

Example ä The viscosity of a liquid can be determined by measuring the time for a sphere to fall a distance L in a cylinder of diameter D. The technique only works if the Reynolds number is less than 1.

Solution 1. viscosity and gravity (buoyancy) 2. Inertia isn’t relevant Substitute d/t for V viscosity and gravity (buoyancy) 2. Inertia isn’t relevant Substitute d/t for V 6.

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 and ______________. 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 and ______________. ä What are the important geometric parameters? _________________________ ä Create dimensionless geometric groups ______, ______ ä Write the functional relationship pressure pressure coefficient

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 Re 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

Same pressure coefficient 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

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 Use C d to get F r and show it is a function of similarity type. 1

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 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 viscosity

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 (Re) simplification ä At high Re viscous forces are small relative to inertia and so Re isn’t important 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 (Re) simplification ä At high Re viscous forces are small relative to inertia and so Re isn’t important

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 to keep similar geometry! ä Reynolds similarity ä Note: roughness height should scale to keep similar geometry! ä Reynolds similarity ν = 1.52 x m 2 /s V m = 25 m/s Use the same fluid

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 Re for high Re ä High Reynolds number means ______ forces are much greater than _______ forces ä Pressure coefficient becomes independent of Re for high Re inertial viscous

Pressure Coefficient for a Venturi Meter E+001E+011E+021E+031E+041E+051E+06 Re 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 As size decreases viscosity becomes important

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

100,000 1,000,000 10,000, year population NYC population CrotonCatskill Delaware New Croton

Supply Aqueducts and Tunnels Catskill Aqueduct Delaware Tunnel East Delaware tunnel West Delaware tunnel Shandaken Tunnel Neversink Tunnel

Delaware Aqueduct 10 km Rondout Reservoir West Branch Reservoir

Flow Profile for Delaware Aqueduct Rondout Reservoir (EL. 256 m) West Branch Reservoir (EL m) 70.5 km Hudson River crossing El m) Sea Level (Designed for 39 m 3 /s)

Ship’s Resistance: We aren’t done learning yet! FASTSHIPS may well ferry cargo between the U.S. and Europe as soon as the year Thanks to an innovative hull design and high-powered propulsion system, FastShips can sail twice as fast as traditional freighters. As a result, valuable cargo should be able to cross the Atlantic Ocean in 4 days.

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)