Unformatted text preview: CEE 331 FINAL REVIEW New Material I. Chapter 8: Pipe Flow Laminar and Turbulent, Minor and Major Losses i. Single Pipes three types of problems 1. Type I specify desired flowrate or average velocity, determine the necessary pressure difference or head loss Examples: pp. 304, 306 2. Type II specify applied driving pressure or head loss, determine the flowrate Example: 307 3. Type III specify the pressure drop and flowrate, determine diameter of pipe needed (iteration required) Example: 309 ii. Multiple Pipe Systems 1. Pipes in series (310) a. Q1 = Q2 = Q3 b. hLtotal = hL1 + hL2 + hL3 2. Pipes in parallel (310) a. Q = Q1 + Q2 + Q3 b. hL1 = hL2 = hL3 iii. Pipe Flowrate Measurement (with ideal situation hL = 0) 1. Orifice Meter  flat plate with a hole b/t 2 flanges of a pipe a. Q = CoQideal *Qideal on page 312 b. A0 = d2/4 is the area of hole c. Orifice discharge coefficient, Co, is a function of = d/D and the Reynolds number 2. Nozzle Meter (312, Example: 314) a. Q = CnQideal b. An = d2/4 is the area nozzle c. Nozzle discharge coefficient, Cn, is a function of = d/D and the Reynolds number 3. Venturi Meter most precise, reduce head loss to minimum a. Q = CvQideal b. AT = d2/4 is the throat area c. Venturi discharge coefficient, Cv, is a function of = d/D, the Reynolds number, shape of meter, etc II. Chapter 9: Boundary Layer Flow Laminar and Turbulent, Drag i. Lift and Drag 1. Drag > resultant force in the direction of upstream velocity a. Eqn. 9.1, 383 b. Mechanisms responsible: shear stress and pressure difference c. Drag coefficient: CD, defined on 330 2. Lift> resultant force normal to the upstream velocity a. Eqn. 9.2, 328 b. Lift Coefficient: CL, defined on 330 3. Area in drag and lift coefficients in the frontal area the projected area seen by a person looking toward the object from a direction parallel to the upstream velocity * Sometimes the planform ("bird's eye") area is used ii. Characteristics of flow past an object 1. Reynolds number (331/332) a. Represents the ratio of inertial effects to viscous effects b. Nature of flow depends whether Re > 1 or Re < 1 c. Re = Ul / , where l is length of plate and U is upstream velocity d. If Reynolds number is large, flow is dominated by inertial effects and viscous effects are negligible everywhere except a boundary layer very close to the plate and in the wake region i. Boundary layer has thickness << l, and is region where fluid velocity changes from upstream value of u = U to zero velocity on plate ii. Wake region is due entirely to viscous interaction b/t fluid and plate e. Large Reynolds number flow past a circular cylinder causes flow separation (333) iii. Boundary Layer Characteristics 1. For an infinitely long plate we use x, the distance along the plate from the leading edge, as the characteristic length 2. Near the leading edge of the plate, there is laminar boundary flow 3. Transition from laminar to turbulent boundary flow occurs at a critical value of the Reynolds number on the order of 2*105 to 3*106, depending on surface roughness we use Rexcr = 5*105 4. Boundary Layer thickness, , is the distance from plate at which the fluid is w/i some value of the upstream velocity = y where u = 0.99U 5. Boundary layer displacement thickness, *, is the amount that the thickness of a body must be increased so that the fictitious uniform inviscid flow has the same flowrate as the actual viscous flow (335) 6. Boundary layer momentum thickness, , is the analogous thickness for the deficit in momentum flux due to the assumption that flow is inviscid 7. Prandtl/Blasius boundary layer solution 336 8. Momentum integral boundary layer equation 339 iv. Turbulent boundary layer flow 1. Shear stress greater than for laminar 2. Surface roughness does not affect shear stress and, hence, the drag coefficient 3. For flat plate, boundary layer governed by inertial and viscous forces, pressure remains constant th/o flow Example: 343 4. For flow past object other than flat plate, pressure field not uniform circular cylinder a. Pressure gradient caused by variation in the freestream velocity, Ufs, the fluid velocity at the edge of the boundary layer b. Decrease in pressure in the direction of flow along the front half of the cylinder is a favorable pressure gradient c. Increase in pressure in the direction of flow along the rear half of the cylinder is adverse pressure gradient d. Because of viscous effects, particle experiences loss of energy as it flows and does not have enough energy to coast up rear half of cylinder, and boundary layer separation occurs i. Because of this separation, pressure on rear half of cylinder is less than that on front half v. Drag total drag coefficient, CD, dependent on shape, Re, Ma, Fr, and roughness, on 347 1. Friction Drag > part of drag due directly to shear stress Equation on 347 2. Pressure Drag > part of drag due directly to pressure Equation on 347 3. Shape dependence a. The more blunt the body, the larger the drag coeff 4. Reynolds number dependence a. Low Re (Re <= 1), Drag = f(U, l , ) i. Sphere: CD = 24/Re ii. Example: 350 b. Moderate Re flows past blunt bodies produce drag coefficients that are relatively constant c. For streamlined bodies, CD increases when flow becomes turbulent; for a relatively blunt object such as a cylinder or sphere, CD decreases when flow becomes turbulent Example: 353 5. Compressibility Effects a. If velocity is sufficiently large, CD becomes a function of the Mach number, Ma = U/c, where c is the speed of sound in the fluid i. If Ma < 0.5, effects not important b. CD increases dramatically at or around Ma = 1 (sonic flow) i. Shock waves come into existence (354) 6. Surface Roughness a. For streamlined, drag increases with surface roughness b. For blunt bodies (sphere or cylinder) increase in surface roughness decreases drag golf ball Example: 355 7. Froude number effects, Fr = U / (gl) a. Froude number is the ratio of freestream speed to a typical wave speed on the interface of two fluids, i.e. the ocean surface 8. Composite body drag > treat body as composite of parts Example: 357 vi. Lift total lift coefficient, CL, on 361 1. The most important parameter that effects lift is the shape of the object 2. Most liftgenerating devices operate in large Re range in which flow 3. Most of lift comes from surface pressure distribution 4. B/c most airfoils are thin, it is customary to use planform area 5. Most important quantity is ratio of lift to drag developed Example: 364 6. Circulation a. Generation of lift is directly proportional to production of circulation or vortex flow around an object b. By symmetry, inviscid flow past a circular cylinder produces zero drag and zero lift, BUT c. If cylinder is rotated about its axis, the rotation will drag some of the fluid around, producing circulation and, thus, lift III. Chapter 10: Open Channel Flow Manning's equation, specific energy, controls, the hydraulic jump, & weirs i. Open Channel Flow > the flow of a liquid in a channel that is not completely filled 1. Uniform flow > UF; depth does not vary along channel (dy/ dx = 0) 2. Nonuniform (varied) flow > depth varies with distance along channel (dy/dx 0) ii. iii. iv. v. vi. a. Rapidly varying flow > RVF; flow depth changes considerably over a short distance (dy/dx ~ 1) b. Gradually varying flow> GVF; flow depth changes slowly with distance along channel (dy/dx < 1) 3. Most open channel flows have large Re; it is rare to have laminar openchannel flow 4. Critical flow: Fr = Fr = U/(gl) = 1 5. Subcritical flow: Fr < 1 6. Supercritical flow: Fr >1 Surface Waves 1. Wave speed, c = (gy) (378) a. If water depth is much greater than wavelength (ocean) wave speed, c = (g / 2), where is wavelength 2. Froude number effects a. Fr = V/c is the ratio of fluid velocity to wave speed b. If Fr < 1 then V < c and wave can move upstream (hydraulic communication b/t upstream and downstream locations) c. If Fr > 1, then V > c and wave only moves downstream d. If Fr = 1, then V = c and upstream propogating wave remains stationary Energy 1. Specific Energy, E, 381 a. For a given flowrate and specific energy, two alternate flow depths are possible (one is supercritical and one is subcritical) b. Critical conditions (Fr = 1) occur at location of Emin, where Emin = 3yc/2 and yc = (q2/g)1/3 Example: 382 c. Specific Energy Diagrams Uniform depth channel flow 1. Chezy Equation  385 2. Manning Equation 386 3. Examples: 387 (determine flow rate), 388 (determine flow depth), 389 (variable roughness), 391 (find best hydraulic crosssection) Gradually Varied Flow 1. If the bottom depth and the energy line slope are not equal, flow depth will vary along the channel, either increasing or decreasing in the flow direction 2. Whether the depth increases or decreases depends on various parameters of the flow...? Rapidly Varied Flow 1. Hydraulic Jump a. Approximated as a discontinuity in the freesurface elevation (dy/dx = ) b. Depth ratio, y2/y1, across the jump is a function of the upstream Froude number (eqn 10.19, 394) c. To have a hydraulic jump, the flow must be supercritical (Fr1 > 1) Example: 395 2. SharpCrested Weir > a vertically sharpedged plate placed across the channel in a way such that the fluid must flow across the sharp edge into a pool downstream a. Assume velocity profile upstream is uniform and pressure within the nappe (flow falling over weir) is atmospheric use Bernoulli b. Flowrate depends on cross channel width of strip of weir area (constant for rectangular, function of h for triangular and circular) i. Q for rectangular 398 ii. Q for triangular 399 3. BroadCrested Weir > structure in open channel that has a horizontal crest over which fluid pressure may be considered to be hydrostatic a. Nearly uniform flow is achieved over weir block b. As flow passes, it reaches critical conditions i. Fr = 1 ii. y2 = yc = 2H/3 iii. Flow equation on 401 c. Example: 401 4. Underflow gates (402) a. Flow under gate is said to be free outflow when the fluid issues as a jet of supercritical flow with a free surface open to the atmosphere Chart Locations Friction drag coefficient for a flat plate parallel to upstream flow, 342 Empirical equations for flat plate drag coefficient, 342 Drag coefficients for an ellipse, 348 Drag coefficients for smooth sphere and smooth cylinder, 351 Drag coefficients for various shapes, 352 Drag coefficients as a function of Mach number for supersonic flow, 354 Sphere drag coefficients as a function of surface roughness, 355 Typical drag coefficients for regular 2D, 3D, and interest objects, 358360 Values of the Manning Coefficient, 386 Head loss across a hydraulic jump as a function of upstream Froude number, 395 Weir coefficient for triangular sharpcrested weirs, 400 Typical discharge coefficients for underflow gates, 403 ...
View
Full Document
 Fall '07
 COWEN
 Fluid Dynamics

Click to edit the document details