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Monroe L. Weber-Shirk S chool of Civil and Environmental Engineering Fluid Mechanics EIT Review

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Shear Stress change in velocity with respect to distance Tangential force per unit area rate of shear

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P 1 = 0 h1h1h1h1 ? h2h2h2h2 Manometers for High Pressures Find the gage pressure in the center of the sphere. The sphere contains fluid with 1 and the manometer contains fluid with 2. What do you know? _____ Use statics to find other pressures. Find the gage pressure in the center of the sphere. The sphere contains fluid with 1 and the manometer contains fluid with 2. What do you know? _____ Use statics to find other pressures. 1 1 2 2 3 3 =P 3 1 2 For small h 1 use fluid with high density. Mercury! + h 1 2 - h 2 1 P1P1 P1P1

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Differential Manometers h1h1 h3h3 Mercury Find the drop in pressure between point 1 and point 2. p1p1 p2p2 Water h2h2 orifice = p 2 p 1 - p 2 = (h 3 -h 1 ) w + h 2 Hg p 1 - p 2 = h 2 ( Hg - w ) p1p1 p1p1 + h 1 w - h 2 Hg - h 3 w

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Forces on Plane Areas: Inclined Surfaces q q A B B O O O O x x y y Free surface centroid center of pressure The origin of the y axis is on the free surface

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Statics ä Fundamental Equations ä Sum of the forces = 0 ä Sum of the moments = 0 ä Fundamental Equations ä Sum of the forces = 0 ä Sum of the moments = 0 p c is the pressure at the __________________ centroid of the area Line of action is below the centroid

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Properties of Areas ycyc b a I xc ycyc b a R ycyc d

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Properties of Areas a ycyc b I xc ycyc R R ycyc

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Inclined Surface Summary ä The horizontal center of pressure and the horizontal centroid ________ when the surface has either a horizontal or vertical axis of symmetry ä The center of pressure is always _______ the centroid ä The vertical distance between the centroid and the center of pressure _________ as the surface is lowered deeper into the liquid ä What do you do if there isnt a free surface? ä The horizontal center of pressure and the horizontal centroid ________ when the surface has either a horizontal or vertical axis of symmetry ä The center of pressure is always _______ the centroid ä The vertical distance between the centroid and the center of pressure _________ as the surface is lowered deeper into the liquid ä What do you do if there isnt a free surface? coincide below decreases

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An elliptical gate covers the end of a pipe 4 m in diameter. If the gate is hinged at the top, what normal force F applied at the bottom of the gate is required to open the gate when water is 8 m deep above the top of the pipe and the pipe is open to the atmosphere on the other side? Neglect the weight of the gate. hinge water F 8 m 4 m Solution Scheme Magnitude of the force applied by the water Example using Moments ä ä ä ä ä ä ä ä Location of the resultant force Find F using moments about hinge

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Magnitude of the Force b = 2 m a = 2.5 m p c = ___ F r = ________ h = _____ hinge water F F 8 m 4 m FrFr FrFr 10 m Depth to the centroid 1.54 MN

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Location of Resultant Force hinge water F F 8 m 4 m FrFr FrFr 12.5 m Slant distance to surface 0.125 m b = 2 m a = 2.5 m cp

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Force Required to Open Gate How do we find the required force? F = ______ b = 2 m 2.5 m l cp =2.625 m l tot hinge water F F 8 m 4 m FrFr FrFr Moments about the hinge =Fl tot - F r l cp 809 kN cp

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Example: Forces on Curved Surfaces Find the resultant force (magnitude and location) on a 1 m wide section of the circular arc. F V = F H = water 2 m 3 m W1W1 W2W2 W 1 + W 2 = (3 m)(2 m)(1 m) + p/4 (2 m) 2 (1 m) = 58.9 kN + 30.8 kN = 89.7 kN = (4 m)(2 m)(1 m) = 78.5 kN y x

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Example: Forces on Curved Surfaces The vertical component line of action goes through the centroid of the volume of water above the surface. water 2 m 3 m A W1W1 W2W2 Take moments about a vertical axis through A. = 0.948 m (measured from A) with magnitude of 89.7 kN

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Example: Forces on Curved Surfaces water 2 m 3 m A W1W1 W2W2 The location of the line of action of the horizontal component is given by b h y x (1 m)(2 m) 3 /12 = 0.667 m 4 4 m

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Example: Forces on Curved Surfaces 78.5 kN 89.7 kN 4.083 m 0.948 m 119.2 kN horizontal vertical resultant

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C (78.5kN)(1.083m) - (89.7kN)(0.948m) = ___ 0 0 0.948 m 1.083 m 89.7kN 78.5kN Cylindrical Surface Force Check ä All pressure forces pass through point C. ä The pressure force applies no moment about point C. ä The resultant must pass through point C. ä All pressure forces pass through point C. ä The pressure force applies no moment about point C. ä The resultant must pass through point C.

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Curved Surface Trick ä Find force F required to open the gate. ä The pressure forces and force F pass through O. Thus the hinge force must pass through O! ä All the horizontal force is carried by the hinge ä Hinge carries only horizontal forces! (F = ________) ä Find force F required to open the gate. ä The pressure forces and force F pass through O. Thus the hinge force must pass through O! ä All the horizontal force is carried by the hinge ä Hinge carries only horizontal forces! (F = ________) water 2 m 3 m A W1W1 W2W2 F F O O W 1 + W 2 11.23

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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)

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Model Studies and Similitude: Scaling Requirements 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 MachReynoldsFroudeWeber

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Froude similarity 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 difficult to change g 11.33

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Control Volume Equations ä Mass ä Linear Momentum ä Moment of Momentum ä Energy ä Mass ä Linear Momentum ä Moment of Momentum ä Energy

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Conservation of Mass 1 2 v1v1v1v1 A1A1A1A1 V = spatial average of v If mass in cv is constant [M/t] If density is constant [L 3 /t] Area vector is normal to surface and pointed out of cv

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Conservation of Momentum

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Energy Equation laminarturbulent Moody Diagram

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z z Example HGL and EGL z = 0 pump energy grade line hydraulic grade line velocity head pressure head elevation datum

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Smooth, Transition, Rough Turbulent Flow ä Hydraulically smooth pipe law (von Karman, 1930) ä Rough pipe law (von Karman, 1930) ä Transition function for both smooth and rough pipe laws (Colebrook) ä Hydraulically smooth pipe law (von Karman, 1930) ä Rough pipe law (von Karman, 1930) ä Transition function for both smooth and rough pipe laws (Colebrook) (used to draw the Moody diagram)

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Moody Diagram 0.01 0.10 1E+031E+041E+051E+061E+071E+08 R friction factor laminar 0.05 0.04 0.03 0.02 0.015 0.01 0.008 0.006 0.004 0.002 0.001 0.0008 0.0004 0.0002 0.0001 0.00005 smooth 0.02 0.03 0.04 0.05 0.06 0.08

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l find head loss given (D, type of pipe, Q) l find flow rate given (head, D, L, type of pipe) l find pipe size given (head, type of pipe,L, Q) Solution Techniques

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Power and Efficiencies ä Electrical power ä Shaft power ä Impeller power ä Fluid power ä Electrical power ä Shaft power ä Impeller power ä Fluid power IE T T T T QH p Motor losses bearing losses pump losses

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Manning Formula The Manning n is a function of the boundary roughness as well as other geometric parameters in some unknown way... Hydraulic radius for wide channels

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Drag Coefficient on a Sphere 0.1 1 1 10 100 1000 0.1 1 1 10 10 2 10 3 10 4 10 5 10 6 10 7 Reynolds Number Drag Coefficient Stokes Law

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