Urban Hydraulics
Hydraulics for Urban Storm Drainage Learning objectives: understanding of basic concepts of fluid flow and how to analyze conduit flows, free surface flows. to analyze, hydrostatic pressure force on a surface steady pressurized flows through conduits force exerted by steady fluid flows steady open channel flows to derive mathematical formulation for estimation of flood water levels and inundation areas due to floods.
What is Fluid? Fluid Deforms continuously under the action of an applied shear stress. Both liquids and gases Conforms to the shape of its container. Liquid retains its own volume, gas takes the full volume of the container Solid When subjected to a shear stress deforms depending on the force and attains a final equilibrium position.
Continuum concept Fluid is considered as continuous substance The conditions at a point is the average of a very large number of molecules surrounding the point within a radius large compared to the intermolecular distance The variation of fluid and flow properties from point to point is considered to be smooth Any property at a point (x,y,z) at time t can be expressed as φ(x,y,z,t).
Properties of Fluids Density ρ Mass per unit volume [M/L 3 ] Bulk Modulus K Ratio between volumetric stress and volumetric strain [ ML 1 T ] Viscosity μ Property of a fluid that enables it to develop resistance to deformation [ML 1 T 1 ] Surface tension Measured as the force acting across a unit length of line drawn in the surface [MT ]
Fluid statics All the particles of the fluid are motionless. No Shear stresses Pressure at a point is the same in all directions Pressures at the same level in a continuous expanse of a static fluid are same e.g. two points at the same elevation in a U tube manometer.
Hydrostatic forces Hydrostatic pressure on the surface increases linearly with depth
Hydrostatic forces
Steady flow Types of fluid flows Properties at a point do not change with time. (at a point (x,y,z) at time t any property is φ(x,y,z,t) = φ(x,y,z) only) Uniform flow Properties at a given instant are same in magnitude and direction at every point in the fluid flow. (at a point (x,y,z) at time t any property is φ(x,y,z,t) = φ(t) only)
Types of fluid flows Fluid flows Steady flows Unsteady flows Fluid flows Steady uniform flows Steady nonuniform flows Unsteady uniform flows Unsteady nonuniform flows
Control volume Control Volume Concept A definite volume in space with fixed boundaries through which matter is allowed to cross. The effect of fluid flow on its boundaries are of more interest The conservation laws are applied to control volumes to describe changes of flow properties
Reynolds Transport Equation N is an extensive property At t, N S, t N C, t At t + Δt, N N C, t t S,, tt dn dt C dn dt S N N in in N N out out System boundary & CV (a) Control volume and system at t System boundary CV Reynolds Transport Equation (b) Control volume and system at t =t +δt
Mass Continuity Equation Let N be the mass m. Since the mass in the system is constant, m in dm C dt m out dm S dm dt dt C 0 m in m out For steady flow m m in out
Mass Continuity Equation CV Q Unsteady flow continuity equation:
Force Momentum Equation Let N be the linear momentum of fluid in X direction, M X. From Newton s Second law M Xin dm XC dt F XS M Xout F XS F XS dm dt dm XC dt XS M X out M X in For Steady flow F XS M X out M X in
Steady flow Speed of incoming jet = v 1 Speed of outgoing jet = v Diameter of jet = D Force on the vane? Force Momentum Equation Impact force on a vane Continuity Equation:
Application of Force Momentum Equation: In X direction: In Y direction: Force Momentum Equation Impact force on a vane
Energy Equation Let N be the total energy (E) de dt des dt E C des E in E dt dqs dw S dt dt m (u + V /+gz) out For steady, adiabatic flow, E in de C dt d E out W dt S no change in Internal Energy
Bernoulli s Equation When there is no shaft work, viscous work, shear work, electromagnetic work, change in internal energy Application of Bernoulli s Equation limits to steady, inviscid, incompressible flow along a stream line.
Bernoulli s Equation H (Total Head) = p/ρ (pressure head) + V /(velocity head) + gz (elevation head) = constant The total head which is the energy per unit weight of the fluid is constant along a streamline in a steady, incompressible, inviscid fluid flow
Bernoulli s Equation H p g V g z Constant p g V g H z Elevation datum
Force Momentum Equation Force on a bend Steady incompressible flow through a bend Force on the bend? Continuity equation: Bernoulli equation:
Force Momentum Equation Force on a bend In the vertical direction, force is Rz No change in momentum Force Momentum Equation in Z direction,
Force Momentum Equation Force on a bend Forces in x direction: Rate of change of Momentum in x direction: Force momentum equation in x direction:
Laminar and turbulent flows In laminar flow Fluid particles move in layers, with one layer sliding smoothly over an adjacent layer. Random fluctuations in particle velocities are damped by the viscous forces and orderly flow is maintained.
Laminar and turbulent flows In turbulent flow, Fluid particles deviate to move from orderly manner and viscous shear stresses are not sufficient to eliminate the random fluctuations.
Laminar and turbulent flows Reynolds Number (Re = ρvl) / µ ) proportional to ratio of forces inertia force/viscous force criterion to determine whether flow is laminar or turbulent when the Reynolds Number is below a critical value of [( ρvd) / µ]= 000, pipe flow is normally laminar.
Laminar and turbulent flows In turbulent flows, compared to laminar flow, mixing of fluid (transfer of momentum) results in more even velocity profile at a pipe section Wall shear stress is greater Energy loss rate is higher Pipe roughness is also an important factor Shear stress = du/dy w Laminar w Turbulent
Pressurized flow in conduits Flow is driven by the total head difference at the two ends of the conduit Head loss between two sections is equal to difference in total head at the sections
Pressurized flow in conduits Total headline or total energy grade line (EGL) referred to the datum Hydraulic grade line (HGL) or piezometric head line referred to the datum
Friction losses in pipe flows Friction loss depends on geometric properties, fluid properties and flow properties semi empirical or empirical equations established based on experimental investigations to estimate friction loss Darcy Weisbach equation Hazen Williams equation Manning s equation Chezy s equation.
The Darcy Weisbach equation Where, h L = head loss due to friction f = f(re, ε/d) is the friction factor Re = ρvd/µ ε/d = relative roughness ε = equivalent sand grain roughness of the pipe L = pipe length D = pipe diameter V = cross-sectionally averaged velocity of the flow g = gravitational acceleration
The Darcy Weisbach equation For Re <,000, where the flow is laminar flow, f depends only on the Re. f = 64/Re For large Re where the flow is fully turbulent f depends only on the relative roughness of the pipe. In the transitional region between laminar and fully turbulent flow, f depends on both Re and relative roughness.
The Darcy Weisbach equation Moody diagram 0 Values of (VD) for water at 60 F [Diameter (D) in in., Velocity (V) in ft/sec] 0.1 0. 0.4 0.6 1 4 6 10 0 40 60 100 00 400 600 1000 000 4000 10,000 0.10 0.09 0.08 0.07 Laminar flow, f = 64 Re 0.05 h L L V D g 0.06 0.05 0.03 0.0 Friction factor, f = 0.04 0.03 0.05 0.00 0.015 0.010 0.009 0.008 10 3 (10 ) e, ft. e, mm. Riveted steel 0.003-0.03 0.9-9 Concrete 0.001-0.01 0.3-3 Wood stave 0.0006-0.003 0.18-0.9 Cast iron 0.00085 0.5 Galvanized iron 0.0005 0.15 Asphalted cast iron 0.0004 0.1 Steel or wrought iron 0.00015 0.045 Drawn Tubing 0.000005 0.0015 Smooth pipes 3 4 6 810 4 (10 4 ) 4 6 810 5 (10 5 ) 4 6 8 10 6 (10 6 ) 4 6 8 10 7 (10 7 ) 4 6 8 10 8 0.01 0.006 0.004 0.00 0.001 0.0006 0.0004 0.000 Relative roughness, e /D 0.00005 Reynolds number, Re = VD n
The Hazen Williams Equation Primarily used for water distribution design V = C f C h R 0.63 S f 0.54 Where V = flow velocity C f = a unit conversion factor (0.849 for SI units) C h = Hazen Williams resistance coefficient R = hydraulic radius S f = Energy gradient
Minor losses energy losses at fittings in pipelines, entrance and exits of reservoirs/man holes, pipe expansion and contractions, changes in pipe alignment Head loss at a fitting is expressed as V h L K g Where V K = velocity at the downstream = loss coefficient
, Loss coefficients Fitting Flanged 90 o elbow 0. 0.31 K Globe valve fully open Flange T joint Line flow Branch flow Sudden expansion 10 0.14 0.69 referred to upstream velocity head. D 1 and D : upstream and downstream velocities respectively
Pipe Flow 1 75 m 10m Z = 130 m K exit =1 150 m f =.035 Oil density = 9.0 kn/m 3 Oil flows from the upper reservoir to lower reservoir through a pipe with the diameter of 150mm. If the velocity in the pipe is 1.8m/s, find the elevation of the oil surface in the upper reservoir? Loss Coefficients : K bend = 0.19, K entrance = 0.5, K exit = 1
Pipe Flow 1 Z 1 =? 75 m 10 m Z = 130 m K exit =1 Datum 150 m Head balance between (1) and (): 0 + 0 + Z 1 = 0 + 0 + 130m + 9.06m + H minor H minor = K bend V /g + K ent V /g + K out V /g From Loss Coefficients : K bend = 0.19 K entrance = 0.5 K out = 1 H minor = (0.19x + 0.5 + 1) * (1.8 /*9.8) = 0.31 m
Pipe Flow 1 Z 1 =? 75 m 10 m Z = 130 m K out =1 150 m 0 + 0 + Z 1 = 0 + 0 + 130m + H major + H minor 0 + 0 + Z 1 = 0 + 0 + 130m + 9.06m + 0.31m Z 1 = 139.4 meters
Types of open channel flows Steady uniform Flow Gradually Varied Flow Open Channel Flow Steady Flow Unsteady Flow Steady nonuniform Flow Unsteady uniform Flow Rapidly Varied Flow Gradually Varied Flow Unsteady nonuniform Flow Rapidly Varied Flow
Open channel geometry factors T d A P Hydraulic radius, R = A/P Hydraulic depth, D h = A/T A = cross sectional area P = wetted perimeter T = top width
Force Momentum Equation in the direction perpendicular to the flow, Assumption: the acceleration of flow in the direction is negligible. cos 0 cos Pressure Variation in Open Channel Flow In an open channel flow with small bottom slope and no flow acceleration in the direction perpendicular to the flow, the pressure distribution is hydrostatic.
Total Head V H Z d cos g Where, Z = channel bottom elevation d = depth of flow normal to the channel bottom θ = channel slope angle, S o = sin θ α = a velocity distribution coefficient defined by 1 3 3 A = cross sectional area V = average flow velocity Energy Relationships Z 1 d 1 1V1 cos g Z d V cos g h L
Specific Energy Specific energy, E is the energy head relative to channel bottom elevation E y V Q y g ga A T A y Alternate depths y y = y c Critical depth E min = E C E
Specific Energy A) Channel width decreases, discharge per unit width q1 q B) Channel bed level decreases E1 E E y 1 1 1 q q y gy gy E 1 E z
Critical Flow Depth E becomes minimum at the critical flow Critical depth y c de dy 1 Q ga da dy 1 Q T ga 3 0 Q T ga 3 1 Froude No., Fr Q T 3 ga 1/ V gd h ; Fr C 1 E min y c c V g y c D h c
Rectangular channel T= B, A = B.y and D h = y At critical flow, q = discharge per unit width gy V ga B Q Fr 1/ 3 3 3 g q gb Q y c min c c c c y y g V y E gy q y g V y E
Uniform Flow motivating forces = resistive forces S f =S 0 W sinθ = γals 0 τ = γr n S 0 y n = Depth is called normal depth or uniform depth
Flow Resistance Constitutive relationships for uniform flow The Manning Equation (for metric units) 1 / 3 1/ V R n S o n V = cross sectional averaged flow velocity n = Manning s roughness coefficient The Chezy Equation V C RS f C = Chezy s Constant (m 0.5 /s)
Momentum forces L P 1 Wsin For steady flow F s P 1 Wsin P, R y y 1 W sin R f q(v f v 1 ) v 1 W v R f P 1 1 1 sin
Hydraulic jump The hydraulic jump is a phenomenon that occurs when the flow in an open channel changes abruptly from supercritical flow to subcritical flow, with a considerable loss of energy
Hydraulic jump If y 1 and y are conjugate depths Momentum equation neglecting friction force, 1 1 Continuity equation, 1 1 1 1 1 1 1 18Fr 1 1 Head loss at the hydraulic jump, 1 1 18Fr 1 1 1 1 3 4 1
Gradually varied flow occurs in an open channel reach when the motivating force and the resistance forces are not balanced Hydrostatic pressures can be assumed to exist in the flow and uniform flow, H Z E Z y Q ga
Gradually varied flow 1 3 1 Fr / / 0 1Fr
Computation of gradually varied flow profiles The direct-step method a simple method applicable to prismatic channels 1 1 1 1 E 1 and E are specific energy at sections 1 and respectively In the computations S f is calculated for depths y 1 and y and the average of two values are taken in the equation.
Classification of Flow Profiles Flow profiles are classified based on the relative position of normal depth, y. Ify y c hydraulically mild channel slope. (M curve) Ify = y c critical slope. (C curve If y y c hydraulically steep slope. (S curve)
Flood hydraulics Flood is an unsteady flow phenomenon and is due to unusual discharge Based on different approximations to represent hydrological processes involved Various methods to carry out flood analysis Selection of the method is justified by the objective of the analysis, availability of data and resources. Simple lumped methods, or hydrologic models, based on the principle of conservation of mass fails to consider the influence of downstream flow conditions that control the flow in subcritical flow conditions
Hydraulic models Physically based distributed models (or Hydraulic models) are based on the simultaneous solution of continuity equation and approximated momentum equations. Different models of varying complexities developed with different approximations used to simplify the momentum equations
Channel routing A flood discharge at moderate floods may be carried within the stream cross section and designated flood plain. In this case, the analysis is carried out to determine the behavior of flood hydrograph
River routing Muskingum method Volume stored in channel reach S K[ XI (1 X ) Q] K = proportionality coefficient S = travel time through the reach X = a dimensionless weighting factor (0.1 0.5) I = inflow discharge into the reach (m 3 /s) Q = outflow discharge from the reach (m 3 /s) Continuity equation to the reach S S 1 I 1 I t Q 1 Q Q C0I C1I1 CQ t C 0 0.5t KX K(1 X ) 0.5t KX 0.5t C1 K(1 X ) 0.5t K(1 C K(1 X ) 0.5t X ) 0.5t
River routing Muskingum-Cunge method Muskingum coefficients 0 T = top width, Δx = distance step equal to C. Δt
Hydraulics channel routing models Discharge and water levels are calculated simultaneously by the application of laws of mass and momentum conservation One dimensional models Saint Venant equations, Continuity Eqn Q A 0 x t Momentum Eqn Q t x Q A H ga x gas f 0
Kinematic wave model St Venant Eqns are approximated to Q A Continuity Eqn 0 x t Momentum Eqn S f S 0 Kinematic wave model is applicable when the slope dominates in the momentum equation The flood peak discharge will move downstream at a velocity c with no attenuation. Velocity c, called kinematic wave celerity, along the channel c dq da. 1 T dq dy
Diffusion wave model St Venant Eqns are approximated to Continuity Eqn Q A 0 x t Momentum Eqn H x S f 0 The diffusion wave model is applied when the slopes are not large and when backwater effect is dominant
Two dimensional hydraulic models Two dimensional models are based on Continuity equation Two momentum equations Above equations are depth averaged to derive governing equations of D models e.g. Shallow Water Equations
Solution of hydraulic models Numerical methods are used to solve the governing equations of hydraulic models as analytic methods are not able to solve them. Numerical methods need to solve the equations for both space and time Equations are discretized in the D domain using finite volume method, finite element method, etc.