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ENGR-320 Fluid Mechanics – Chapter 8: Dimensional Analysis & Similitude Fluid Mechanics Summer 2005 Prof. Mesut Pervizpour Office: KH #349B Ph: x4046 1 Modified from original notes of Dr. Lennon, CE Dept. Lehigh University Dimensional Analysis • Assume quantities are related by multiplication and division • Dimensions and Units P Theorem • Assemblage of Dimensionless Parameters • Dimensionless Parameters in Fluids • Model Studies and Similitude 2 Pendulum Example What is the expression for period of the pendulum? • Important parameters L, W, g • One way plot Tp-Lp, Tp-W, Tp-g • Relationship: Tp = f (L, W, g) Lp • Dimensions: [Tp] = T [Lp] = L W [g] = L/T 2 [W] = F or ML/T2 Note W has only F dimensions (can’t divide W by anything to eliminate F, so Tp is not a function of W) Tp = f (L, g) By observation What is the only way to combine L & g to have same dimensions as Tp? L L T Tp f g g 3 Pendulum Example (cont.) L • Run one experiment and plot Tp vs. g • Match the slope: Tp = a + mx (where x = L g ) • a = 0, and m = 2 • The full relationship: L Tp 2 g Tp L g 4 The Buckingham P 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” No. Dimensionless Groups = No. Variables Involved – No. Basis Dimensions (3 max) • We reduce the number of parameters we need to vary to characterize the problem! 5 Example – Drag on a falling body Drag on a model blimp 1530 N FD What is the force on prototype? FD = fnc ( , , V, D) W Dimensions: [FD] = ML/T2 [] = FT/L2 = M/LT V [] = M/L3 or FT2/L4 [V] = L/T [D] = L Our variables: FD, , , V, D n=5 Our dimensions: M, L, T m=3 Number of groups = n – m = 2 Choose 3 repeating variables (usually one fluid property, one flow property, and one scale): , V, D 6 Example – (cont.) FD = fcn (, , V, D) Two equations for groups: 1 = a Vb Dc FD Eqn-1 2 = A VB DC Eqn-2 Recognize that i = L0 T0 F0 L0 T0 F0 = (FT2L-4)a (LT-1)b (L)c F L0 T0 F0 = L(-4a+b+c) T(2a-b) F(a+1) F0 : 0 = a+1 a = -1 T0 : 0 = 2a – b b = 2a = -2 L0 : 0 = -4a + b + c c = -2 FD 1 V D FD 1 2 2 (Normalized drag force) Similarly for 2: V D2 2 1 2 V D 1 1 1 VD Re 7 Example (cont.) 1 = f (2) f Re FD Where f(Re) = CRe : Drag coefficient V D 2 2 FD C Re V D 2 2 FD CRe V 2 D2 For sphere projected Area Ap = D2 /4 = const. D2 1 2 FD C D Ap V Formula for drag of an object in air 2 8 Dimensional Analysis • Want to study pressure drop as function of p1 velocity (V1) and diameter (do ) p0 • Carry out numerous experiments with different values of V1 and do and plot the V1 V0 data A0 A1 V12 V02 p1 p0 2 2 p 2 V02 V12 d 4 p V12 1 1 5 parameters: 2 d0 p, , V1, d1, do 4 p d1 1 2 d0 2 dimensionless parameters: V1 p/(V2/2), (d1/do) 2 d Much easier to Cp f 1 d 0 establish functional relations with 2 9 parameters, than 5 Exponent Method - revisit 1. List all n variables involved in the problem • Typically: all variables required to describe the problem geometry (D) or define fluid properties (, ) and to indicate external effects (dp/dx) 2. Express each variables in terms of MLT dimensions (j) 3. Determine the required number of dimensionless parameters (n – j) 4. Select a number of repeating variables = number of dimensions • All reference dimensions must be included in this set and each must be dimensionalls independent of the others 5. Form a dimensionless parameter by multiplying one of the nonrepeating variables by the product of the repeating variables, each raised to an unknown exponent 6. Repeat for each nonrepeating variable 7. Express result as a relationship among the dimensionless parameters 10 Example (8.7) • Find: Drag force on rough sphere is function of D, , , V and k. Express in form: 3 f ( 1 , 2 ) FD D V k 1 ( D aV b c ) MLT-2 L ML-3 ML-1T-1 LT-1 L M 0 L0T 0 ( ML1T 1 )( L) a ( LT 1 ) b ( ML3 ) c M: 0 1 c c 1 L : 0 1 a b 3c a 1 T : 0 1 b b 1 n=6 No. of dimensional parameters j=3 No. of dimensions VD k = n - j = 3 No. of dimensionless parameters 1 or 1 DV Select “repeating” variables: D, V, and Combine these with nonrepeating variables: F, & k 11 Example (8.7) FD D V k 3 FD ( D aV b c ) MLT-2 L ML-3 ML-1T-1 LT-1 L M 0 L0T 0 ( MLT 2 )( L) a ( LT 1 ) b ( ML3 ) c M: 0 1 c c 1 Select “repeating” variables: D, V, and Combine these with nonrepeating L : 0 1 a b 3c a 2 variables: F, & k T : 0 2 b b 2 2 k ( D aV b c ) FD 3 M 0 L0T 0 ( L)( L) a ( LT 1 )b ( ML3 ) c V 2 D 2 M: 0c c0 L : 0 1 a b 3c a 1 T : 0 b b 0 FD VD k f( , ) 2 k V D 2 2 D D 12 Assemblage 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 13 Forces on Fluids • Force parameterdimensionless • Mass (inertia) ______ • Viscosity ______ ______ R Reynolds# • Gravitational ______g ______ # F Froude • Pressure p ______ ______Pressure Coeff. Cp • s Surface Tension______ ______ # W Weber • Elastic ______ K ______ # M Mach Dependent variable 14 Inertia as our Reference Force F • F=ma F a f a • Fluids problems always (except for statics) include a velocity (V), a dimension of flow (l), and a density () L=l l M= l3 f M T=V L2T 2 2 V f i= l 15 Viscous Force • What do I need to multiply viscosity by to obtain dimensions of force/volume? f l f C C Ll T M l 3 V M V2 L2T 2 C 1 fi C LT l M LT V2 f i Vl V fi Vl C 2 l R l fμ V 2 fμ l Reynolds number 16 Gravitational Force M l fg L2T 2 Ll T M l 3 Cg Cg V g L 2 T 2 fi V l M Cg Cg 3 L V2 fi l fi V 2 V fg g F f g gl gl Froude number 17 Pressure Force M l fp L2T 2 Ll T M l 3 Cp Cp V p M LT 2 V2 fi l 1 Cp Cp 1 L l V2 2p fi l f i V 2 Cp V 2 fp p fp p l Pressure Coefficient 18 Dimensionless parameters Vl R • Reynolds Number V F • Froude Number gl V 2 l • Weber Number W s • Mach Number M V c 2Drag 2p C d • Pressure Coefficient Cp V 2 V 2 A • (the dependent variable that we measure experimentally) 19 Common Dimensionless No’s. • Reynolds Number (inertial to viscous forces) Vd • Important in all fluid flow problems • Froude Number (inertial to gravitational forces) F V • Important in problems with a free surface gh • Euler Number (pressure to inertial forces) p • Important in problems with pressure differences Cp V 2 • Mach Number (inertial to elastic forces) V V • Important in problems with compressibility effects M E/ c • Weber Number (inertial to surface tension forces) LV 2 • Important in problems with surface tension effects W s 20 Application of Dimensionless Parameters • Pipe Flow • Pump characterization • Model Studies and Similitude • dams: spillways, turbines, tunnels • harbors • rivers • ships • ... 21 Example: Pipe Flow • What are the important forces? ______, ______. Therefore _________ Inertial viscous Reynolds number. • What are the important geometric parameters? diameter, length, roughness height _________________________ • Create dimensionless geometric groups l/D e/D ______, ______ • Write the functional relationship l e C p f R , , D D 22 l e Example: Pipe Flow C p f R , , D D • How will the results of dimensional analysis guide our experiments to determine the relationships that govern pipe flow? • If we hold the other two dimensionless parameters constant and increase the length to diameter ratio, how will Cp change? D e 2p Cp proportional to l Cp f ,R Cp l D V 2 D e f Cp f , R f is friction factor l D 23 Frictional Losses in Straight Pipes 0.1 0.05 0.04 0.03 0.02 e 0.015 friction factor 0.01 f 0.008 0.006 0.004 D laminar 0.002 0.001 0.0008 0.0004 0.0002 0.0001 0.00005 0.01 smooth 1E+03 1E+04 1E+05 1E+06 1E+07 1E+08 R 24 What did we gain by using Dimensional Analysis? • Any consistent set of units will work • We don’t have to conduct an experiment on every single size and type of pipe at every velocity • Our results will even work for different fluids • Our results are universally applicable • We understand the influence of temperature 25 Model Studies and Similitude: Scaling Requirements • dynamic similitude (forces) • geometric similitude • all linear dimensions must be scaled identically (LR, AR, ..) • roughness must scale • kinematic similitude • constant ratio of dynamic pressures at corresponding points (VR, QR) • streamlines must be geometrically similar • _______, __________, _________, and _________ Mach Reynolds Froude Weber numbers must be the same Cp = f (M, R, F, W, geometry) 26 Relaxed Similitude Requirements • Impossible to have all force ratios the same same ____ unless the model is the _____ size as the prototype • Need to determine which forces are important and attempt to keep those force ratios the same 27 Similitude Examples • Open hydraulic structures • Ship’s resistance • Closed conduit • Hydraulic machinery 28 Example • Consider predicting the drag on a thin rectangular plate (w*h) placed normal FD f ( w, h, , ,V ) to the flow. • Drag is a function of: w, h, , , V 1 f ( 2 , 3 ) • Dimensional analysis shows: FD w Vw f( , ) • And this applies BOTH to a model w V 2 2 h and a prototype • We can design a model to predict the drag on a prototype. 1m f ( 2m , 3m ) • Model will have: FDm w V w f ( m , m m m) wm mVm 2 2 hm m • And the prototype will have: 1 p f ( 2 p , 3 p ) FDp w p pV p w p f( , ) w p pV p 2 2 hp p 29 Example •Similarity conditions Geometric similarity wm w p hm 2m 2 p wm wp Gives us the size of the model hm h p hp Kinematic similarity mVm wm pV p w p m p w p 3m 3 p Vm Vp m p p m wm Then Gives us the velocity in the model Dynamic similarity 2 2 FDp wp p Vp FDp F FDm 1m 1 p 2 wm mVm 2 2 w p pV p 2 w m V Dm m m 30 Example (8.28) • Given: Submarine moving below surface in sea water ( =1015 kg/m3, n / = 1.4x10-6 m2/s). Model is 1/20th scale in fresh water (20oC). • Find: Speed of water in the test dynamic similarity and the ratio of drag force on model to that on prototype. • Solution: Reynolds number is significant parameter. Re m Re p Fm Fp V L V L mVm lm pV p l 2 2 2 2 p p p m m nm np Fm mVm lm 2 2 Lp n m F p pV p l 2 2 p Vm Vp Lm n p 2 1000 28.6 1 2 20 1 1015 2 20 2m / s 1 1.4 Fm Vm 28.6 m / s 0.504 31 Fp Scaling in Open Hydraulic Structures • Examples • spillways • channel transitions • weirs • Important Forces NCHRP Request For Proposal on “Effects of Debris on Bridge-Pier Scour “ p://www4.trb.org/trb/crp.nsf/All+Projects/NCHRP+24-26 • inertial forces • gravity: from changes in water surface elevation V F • viscous forces (often small relative to gravity forces) gl • Minimum similitude requirements Vl • geometric R • Froude number 32 V F Fm Fp gl Froude similarity V 2 Vp2 • Froude number the same in model and m g mLm g pLp prototype Vm2 Vp2 • ________________________ L m L p difficult to change g Lp • define length ratio (usually larger than 1) Lr Lm • velocity ratio Vr L r Lr • time ratio tr Lr Vr • discharge ratio Qr Vr Ar L r L r L r Lr 5/ 2 • force ratio 3 Lr Fr M r a r r L r 2 L3r tr 33 Example: Spillway Model • 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 m3/s, what water flow rate should be tested in the model? Fm Fp Lr 100 Qr L 5/ 2 r 100,000 20,000 m3 s Qm 0.2 m3 s 100,000 34 Ship’s Resistance • Skin friction ______________ Viscosity, roughness • Wave drag (free surface effect) ________ gravity • Therefore we need ________ and Froude Reynolds ______ similarity 2Drag e , R, F Cd f V A 2 l 35 Reynolds and Froude Similarity? Reynolds Froude Vl V R F gl mVmlm pV p l p Water is the only Vr L r m p practical fluid Vmlm V pl p Vp lm 1 Vm l p Lr Lr = 1 1 Lr Vr Lr 36 Ship’s Resistance • Can’t have both Reynolds 2D total e C d f , R, F and Froude similarity V A 2 D • Froude hypothesis: the two D total D f D w forms of drag are independent V 2 A e • Measure total drag on Ship Df f ,R 2 D • Use analytical methods to analytical calculate the skin friction V 2 A • Remainder is wave drag Dw f F 2 empirical 37 Closed Conduit Incompressible Flow • Forces • __________ viscosity • __________ inertia • If same fluid is used for model and prototype • VD must be the same velocity • Results in high _________ in the model • High Reynolds number (R) • Often results are independent of R for very high R 38 Example: Valve Coefficient 2p • The pressure coefficient, Cp , for a V 2 600-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? 39 Example: Valve Coefficient • Note: roughness height should scale! • Reynolds similarity Vl VD ν = 1.52 x 10-6 m2/s R R n Vm Dm Vp Dp Vp Dp Vm nm np Dm (2.5m / s)0.6m Vm Vm = 25 m/s 0.06m 40 Example: Valve Coefficient (Reduce Vm?) • What could we do to reduce the velocity in the model and still get the same high Vl R Reynolds number? VD Decrease kinematic viscosity R n Use a different fluid Use water at a higher temperature 41 Example: Valve Coefficient • Change model fluid to water at 80 ºC VD R νm = 0.367 x 10-6 m2/s _____________ n νp = ______________ 1.52 x 10-6 m2/s Vm Dm Vp Dp n mV p D p Vm nm np n p Dm 0.367 x10 m / s (2.5m / s)0.6m 6 2 Vm 1.52 x10 m / s 0.06m 6 2 Vm = 6 m/s 42 Approximate Similitude at High Reynolds Numbers • High Reynolds number means inertial ______ forces are much greater than _______ viscous forces • Pressure coefficient becomes independent of R for high R 43 Pressure Coefficient for a Venturi Meter 10 Cp 2p Cp V 2 1 1E+00 1E+01 1E+02 1E+03 1E+04 1E+05 1E+06 R Vl Similar to rough pipes in R Moody diagram! 44 Hydraulic Machinery: Pumps • Rotational speed of pump or turbine is an additional parameter • additional dimensionless parameter is the ratio of the rotational speed to the velocity of the water _________________________________ streamlines must be geometrically similar • homologous units: velocity vectors scale _____Vr lr • Now we can’t get same Reynolds Number! • Reynolds similarity requires 1 Vr lr • Scale effects 45 Dimensional Analysis Summary Dimensional analysis: • enables us to identify the important parameters in a problem • simplifies our experimental protocol • does not tell us the coefficients or powers of the dimensionless groups (need to be determined from theory or experiments) • guides experimental work using small models to study large prototypes 46 Port Model • A working scale model was used to eliminated danger to boaters from the "keeper roller" downstream from the diversion structure http://ogee.hydlab.do.usbr.gov/hs/hs.html 47 Hoover Dam Spillway A 1:60 scale hydraulic model of the tunnel spillway at Hoover Dam for investigation of cavitation damage preventing air slots. http://ogee.hydlab.do.usbr.gov/hs/hs.html 48 Irrigation Canal Controls http://elib.cs.berkeley.edu/cypress.html 49 Spillways Frenchman Dam and spillway (in use). Lahontan Region (6) 50 Dams Dec 01, 1974 Cedar Springs Dam, spillway & Reservoir Santa Ana Region (8) 51 Spillway Mar 01, 1971 Cedar Springs Spillway construction. Santa Ana Region (8) 52 Kinematic Viscosity 1.00E-03 kinematic viscosity 20C (m2/s) 1.00E-04 1.00E-05 1.00E-06 1.00E-07 53 Kinematic Viscosity (m /s) Kinematic Viscosity of Water 2.0E-06 2 1.5E-06 1.0E-06 5.0E-07 0.0E+00 0 20 40 60 80 100 Temperature (C) 54

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