# Hydromechanics: Course Summary

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1 Hydromechanics: Course Summary Hydromechanics VVR090 Material Included; French: Chapters to 9 and 4 + Sample problems Vennard & Street: Chapters 8 + 3, and (part of it) Roberson & Crowe: Chapter Collection of sample problems in open channel flow Exam: 8th of May in MA0G

2 Fundamental Equations Conservation of mass: Q = ua Conservation of momentum: F =ρq( u u ) Conservation of energy: u H = z+ y+ g H = H + h L Laboratory Experiments Often difficult to solve fluid flow problems by analytical or numerical methods. Also, data are required for validation. The need for experiments Difficult to do experiment at the true size (prototype), so they are typically carried out at another scale (model). Develop rules for design of experiments and interpretation of measurement results.

3 Similitude and Dimenisional Analysis Similitude: how to carry out model tests and how to transfer model results to prototype (laws of similarity) Dimensional analysis: how to describe physical relationships in an efficient, general way so that the extent of necessary experiments is minimized (Buckingham s P-theorem) Basic Types of Similitude geometric kinematic dynamic d d p m lp = =λ l m All of these must be obtained for complete similarity between model and prototype. 3

4 Important Forces for the Flow Field pressure (F P ) inertia (F I ) gravity (F G ) viscosity (F V ) elasticity (F E ) surface tension (F T ) Dimensionless Numbers Reynolds Froude Cauchy (Mach) Weber Vl Vl Re = = μ/ ρ ν V Fr = gl V V C = = = M E/ ρ c ρlv W = σ Euler E = V ρ Δp Dimensionless numbers same in prototype and model produces dynamic similarity. 4

5 Dimensional Analysis Dimensions (e.g., length, mass, time, temperature) Units (e.g., m, kg, s, K) Three independent dimensions of primary interest: length (L) mass (M) time (t) Metric system F = ML t Force: [ ] Buckingham s P-Theorem Buckingham provided a systematic approach to dimensional analysis through his theorem expressed as:. If n variables are involved in the problem, then k equations of their exponents can be written. In most cases k is the number of independent dimensions (e.g., M, L, t) 3. The functional relationship may be expressed in terms of n- k distinct dimensionless groups 5

6 Example of Dimensional Analysis Drag force (D) on a ship. Assume that D is related to length (l), density (ρ), viscosity (m), speed (V), and acceleration due to gravity (g): { } f D,, l ρμ,, V, g = 0 Problem involves n = 6 variables and k = 3 fundamental dimensions Æ k - n = 6 3 = 3 dimensionless groups can be formed: { } f ' Π, Π, Π = 0 3 Many different ways to combine the variables into dimensionless groups rational approach needed. Method for Deriving Dimensionless Groups. Find the largest number of variables which do not form a dimensionless P-group. Determine the number of P-groups to be formed 3. Combine sequentially the variables in. with the remaining variables to form P-groups. Present example: select ρ, V, and l and combine with remaining variables: 3 3 {, ρ,, } { μ, ρ, Vl, } {, ρ,, } Π = f D V l Π = f Π = f g V l 6

7 First P-group: Π = D ρ V l a b c d Analyze dimensions: M Lt a b c ML M L = 3 t L t ( L) d M : 0= a+ b L: 0= a 3b+ c+ d t: 0= a+ c b= a c= a d = a Result: D Π = ρlv a Fluid Flow About Immersed Objects Flow about an object may induce: drag forces lift forces vortex motion Asymmetric flow field generates a net force Drag forces arise from pressure differences over the body (due to its shape) and frictional forces along the surface (in the boundary layer) 7

8 Drag force: o A D= psin θ da+ τ cosθda A Pressure drag (D p ) Frictional drag (D f ) (form drag) Pressure drag function of the body shape and flow separation Frictional drag function of the boundary layer properties (surface roughness etc) Results of Dimensional Analysis Total drag force: D = CD ρav C = f D 3 o { Re,M} Total lift force: L= CL ρav C = f L 4 o { Re,M} 8

9 Drag Coefficient for Various Bodies D 3D Example of Drag Force Calculation parachute jumping sedimentation of particle popcorn popper Basic equation for drag force: D = C D ρav o C D obtained from empirical studies A is the projected area on a plane perpendicular to the flow direction 9

10 Vortex Shedding Under certain conditions vortices are generated from the edges of a body in a flow. Æ Von Karman s vortex street Vortex street behind a cylinder If 6 < Re < 5000, regular vortex sheeding may occur at a frequency n determined by Strouhal s number: nd S = V o (S = 0. over a wide range of Re) Vortices at Aleutian Island Boundary Layer on a Flat Plate Boundary layer: the zone in which the velocity profile is governed by frictional action 0

11 Drag Coefficient for Smooth, Flat Plates Df = Cf ρvo A A: surface area of plate Open Channel Flow Open channel: a conduit for flow which has a free surface Free surface: interface between two fluids of different density Characteristics of open channel flow: pressure constant along water surface gravity drives the motion pressure is approximately hydrostatic flow is turbulent and unaffected by surface tension

12 Flow Classification steady unsteady uniform non-uniform varied flow (= non-uniform): gradually varied rapidly varied Flow Classification subcritical supercritical flow characterized by the Froude number Fr = U gl L taken to be the hydraulic depth D=A/T Fr < subcritical flow Fr = critical flow Fr > supercritical flow

13 Definition of Channel and Flow Properties Hydraulic radius (R): ratio of flow area to wetted perimeter R = A P Hydraulic depth (D): ratio of flow area to top width A D = T Energy Equation Total energy of a parcel of water traveling on a streamline (no friction): H p u = z+ + γ g elevation head pressure head velocity head p z + γ hydraulic grade line 3

14 Critical Flow Specific energy: u Q E = y+α = y+α g ga Minimum specific energy yields: u = gd u Fr = = gd Critical Flow Rectangular channel of width b: Q q = b q u = y y c q = g uc yc = g yc = Ec 3 /3 4

15 Step in Rectangular Channel Bernoulli equation (between upstream and downstream points): u u + y = + y +Δz g g E = E Δz Total energy: Water Surface Variation from the Energy Equation u H = z+ y+ g Differentiating with respect to distance: ( / ) dh dz dy = + + d u g dx dx dx dx Resulting equation: dy dx So Sf = Fr 5

16 Momentum Equation Hydraulic jump Momentum equation (rectangular channel): y y q = u u γ γ γ g ( ) Momentum equation for rectangular section: q = g y y ( y y ) Solutions: y y ( 8Fr ) = + y y ( 8Fr ) = + Energy loss: Δ E = ( y y ) 3 4y y 6

17 Uniform occurs when: Uniform Flow. The depth, flow area, and velocity at every cross section is constant. The energy grade line, water surface, and channel bottom are all parallel: Sf = Sw = So S f = slope of energy grade line S w = slope of water surface S o = slope of channel bed Uniform Flow Formula Mannings equation for velocity: n /3 u = R S Uniform flow rate: n /3 Q= ua= AR S Section factor: /3 AR (increases with depth) Conveyance: K = AR n /3 7

18 Computation of Uniform Flow. Channel cross section and shape, water depth, and slope known => Q or u can be calculated directly. Channel cross section and shape, water velcoity or flow, and slope known => water depth may be calculated through some iterative procedure Roughness known and constant. Manning s Roughness n

19 Gradually Varied Flow Depth of flow varies with longitudinal distance. Occurs upstream and downstream control sections. Governing equation: dy dx So Sf = Fr Classification of Gradually Varied Flow Profiles Water surface profiles may be classified with respect to: the channel slope the relationship between y, y N, and y c Prevailing conditions: If y < y N, then S f > S o If y > y N, then S f < S o If Fr >, then y < y c Profile categories: M (mild) 0 < S o < S c S (steep) S o > S c > 0 C (critical) S o = S c A (adverse) S o < 0 If Fr <, then y > y c If S f = S o, then y = y N 9

20 Gradually Varied Flow Profile Classification II Flow Transition Subcritical to supercritical Supercritical to subcritical 0

21 Strategy for Analysis of Open Channel Flow Typical approach in the analysis:. Start at control points. Proceed upstream or downstream depending on whether subcritical or supercritical flow occurs, respectively Control points typically occur at physical barriers, for example, sluice gates, dams, weirs, drop structures, or changes in channel slope. Uniform Channel Prismatic channel with constant slope and resistance coefficient. Apply energy equation over a small distance Dx: d u y So S dx + = g f Express the equation in difference form: u Δ y+ = ( So Sf ) Δx g S f nu = R 4/3

22 y i y i+ u i u i+ Reach i Computation of Gradually Varied Flow Dx i x Δ x = i ( y+ u / g) ( y+ u /g) i+ i 4/3 So ( n u / R ) i+ / All quantities known at i. Assume y i+ and compute Dx i (u i+ given by the continuity equation). Trial-and-Error Approach Well-suited for computations in non-prismatic channels. Channel properties (e.g., resistance coefficient and shape) are a function of longitudinal distance. Depth is obtained at specific x-locations. Apply energy equation between two stations located Dx apart (z is the elevation of the water surface): u Δ z+ = Sf Δx g u u z+ = z + + Sf Δx g g

23 Estimate of frictional losses: S = S + S ( ) f f f Equation is solved by trial-and-error (from to ):. Assume y Æ u (continuity equation). Compute S f 3. Compute y from governing equation. If this value agrees with the assumed y, the solution has been found. Otherwise continue calculations. Examples of Gradually Varied Flow Flow in channel between two reservoirs (lakes):. Steep slope, low downstream water level. Steep slope, high downstream water level 3. Mild slope, long channel 4. Mild slope, short channel 5. Sluice gate located in the channel Study flow situation that develops + calculation procedure 3

24 Spatially Varied Flow Flow varies with longitudinal distance. Examples: side-channel spillways, side weirs, channels with permeable boundaries, gutters for conveying storm water runoff, and drop structures in the bottom of channels. Two types of flow: discharge increases with distance discharge decreases with distance Different principles govern => different analysis approach Increasing discharge: use momentum equation (hard to quantify energy losses) γyb o γ yb a =ρqx ( a) u( xa) 0 Decreasing discharge: use energy equation Q H = z+ y+ ga 4

25 Weirs Types of weirs (classified according to shape): rectangular V-notch trapezoidal parabolic special type (e.g., Cipoletti, Sutro) Distinguish between: Broad-crested Sharp-crested Discharge Formula for Rectangular Broad-Crested Weir h Apply Bernoulli equation between upstream section and the control section (critical depth occurs here). / Q = CDCv g Th 3 3 3/ 5

26 Discharge Formula for Sharp-Crested Weirs h z Rectangular: Triangular: Q = C g bh 3 e ( ) / 3/ 8 Q = C ( g) tan ( θ/) h 5 e / 5/ Parshall Flume Critical flow occurs in the flume throat followed by a hydraulic jump downstream. General discharge formula: B Q = AWH a 6

27 Venturimeters Involves a constriction in the flow. The constriction produces an accelerated flow and a fall in the hydraulic grade line (pressure) directly related to the flow rate. CA v p p Q = g + z z / γ γ ( A A ) Orifices Used for many purposes in engineering, including measuring the flow rate. A difference compared to nozzles is that the minimum flow section does not occur at the orifice but some distance downstream (in vena contracta). Flow rate is given by: CCA v c p p Q = g z + z γ γ ( A A ) Cc / Area in vena contracta is: A = CA c 7

28 Submerged Orifice Discharge from one large reservoir to another: ( ) Q= C C A g h h c v Discharge to the atmosphere: Q = C C A gh = CA gh c v Sluice Gate Special case of orifice flow: only contraction on the top of the jet. Pressure in vena contracta is assumed to be hydrostatic. CCA v c Q= g y y / ( y y ) ( ) 8

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