Urban Hydraulics - un-ihe.org · PDF fileThe Hazen –Williams Equation ... V= flow...
Transcript of Urban Hydraulics - un-ihe.org · PDF fileThe Hazen –Williams Equation ... V= flow...
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 anapplied shear stress.
– Both liquids and gases– Conforms to the shape of its container.Liquid retains its own volume, gas takes the fullvolume of the container
Solid– When subjected to a shear stress deformsdepending on the force and attains a finalequilibrium 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 FluidsDensity – ρ
– Mass per unit volume [M/L3]Bulk Modulus – K
– Ratio between volumetric stress and volumetric strain [ ML‐1T‐2 ]
Viscosity – μ– Property of a fluid that enables it to develop resistance to deformation [ML‐1T‐1 ]
Surface tension– Measured as the force acting across a unit length of line drawn in the surface [MT‐2 ]
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 – 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
Types of fluid flows
Fluid flows
Fluid flows
Steady flows Unsteady flows
Steady uniformflows
Steady nonuniform
flows
Unsteady uniformflows
Unsteady nonuniform
flows
Control Volume Concept
Control volume• 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
CVSystem boundary
System boundary & CV
(a) Control volume and system at t
(b) Control volume and system at
t =t +δt
At t,
tCtS NN ,,
At t + Δt,
outinttSttC NNNN ,,,
outinSC NN
dtdN
dtdN
Reynolds Transport Equation
N is an extensive property
Mass Continuity EquationLet N be the mass m .Since the mass in the system is constant,
dtdmCinm outm 0
dtdmS
outinC mm
dtdm
For steady flow
outin mm
Unsteady flow continuity equation:
Mass Continuity Equation
Q
CV
Force‐Momentum Equation
Let N be the linear momentum of fluid in X‐ direction,MX.
dtdM XCXinM
XSF
XoutM dtdMF XS
XS
inXoutXXC
XS MMdt
dMF
For Steady flow
inXoutXXS MMF
From Newton’s Second law
Steady flowSpeed of incoming jet = v1Speed of outgoing jet = v2Diameter of jet = D
Force on the vane?
Continuity Equation:
Force Momentum Equation‐ Impact force on a vane
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)
EnergyInternalinchangenoflowadiabaticsteadyFor
mEdtWd
dtdQ
dtdE
EEdt
dEdt
dE
SSS
outinSC
,,gz)+/2V+u( 2
dtdEC
dtWd S
inE outE
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 tosteady, inviscid, incompressible flow along a stream line.
Bernoulli’s Equation
• 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
H (Total Head) = p/ρ (pressure head) + V2/2(velocity head)+ gz (elevation head)
= constant
H z
2
2V
g
Elevation datum
gp
Constant2
2
zg
VgpH
Bernoulli’s Equation
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 RzNo 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:
Force Momentum Equation‐ Force on a bend
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.
• 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]= 2000, 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
Laminar and turbulent flows
Laminar
Shear stress = du/dy
Turbulentw w
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,hL = head loss due to friction f = f(Re, ε/D) is the friction factorRe = ρvD/µε/D = relative roughness ε = equivalent sand grain roughness of the pipe L = pipe length D = pipe diameter V = cross-sectionally averaged velocity of the flowg = gravitational acceleration
• For Re < 2,000, where the flow is laminar flow, fdepends only on the Re. f = 64/Re
• For large Re where the flow is fully turbulent fdepends 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
The Darcy‐Weisbach equation Moody diagram
103 2(10 )3 4 6 81042(10 )4 4 6 8105 2(10 )5 4 6 8 106 2(10 )6 4 6 8 107 2(10 )7 4 6 8 108
0.00005
0.0002
0.00040.00060.001
0.002
0.006
0.004
0.01
0.02
0.03
0.05
4 610.60.40.20.1 2 10 20 40 60 100 200 400 600 1000 2000 4000 10,000
0.008
0.009
0.010
0.015
0.020
0.025
0.03
0.04
0.05
0.06
0.07
0.080.09
0.10
Values of (VD) for water at 60 F [Diameter (D) in in., Velocity (V) in ft/sec]0
Rel
ativ
e ro
ughn
ess,
/De
Reynolds number, Re = VD n
Smooth pipes
e, ft.Riveted steel 0.003 - 0.03 0.9 - 9Concrete 0.001 - 0.01 0.3 - 3Wood stave 0.0006 - 0.003 0.18 - 0.9Cast iron 0.00085 0.25Galvanized iron 0.0005 0.15Asphal ted cas t iron 0.0004 0.12Steel or wrought iron 0.00015 0.045Drawn Tubing 0.000005 0.0015
e, mm.
Fric
ti on
fact
o r, f
=
h L
VD
2
g
L2
Laminar flow, f = 64Re
The Hazen –Williams Equation
Where V = flow velocity Cf = a unit conversion factor
(0.849 for SI units)Ch = Hazen –Williams resistance coefficient R = hydraulic radius Sf = Energy gradient
V = Cf Ch R0.63 Sf0.54• Primarily used for water distribution design
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
Where V = velocity at the downstreamK = loss coefficient
gVKhL 2
2
Loss coefficients
Fitting KFlanged 90o elbow 0.22‐0.31
Globe valve fully open 10
Flange T‐joint Line flowBranch flow
0.140.69
Sudden expansion‐ referred to upstream velocity head.
D1 and D2 : upstream and downstream velocities respectively
,
Pipe Flow1
2Z2 = 130 m
150 m
10m
75 m
f = .035
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?
Kexit=1
Oil density = 9.0 kN/m3
Loss Coefficients : Kbend = 0.19, Kentrance = 0.5, Kexit = 1
Pipe Flow
1
2Z2 = 130 m
150 m
10 m
75 m
Z1 = ?
Kexit=1
• Head balance between (1) and (2):
0 + 0 + Z1 = 0 + 0 + 130m + 9.06m + Hminor
Hminor= 2KbendV2/2g + KentV2/2g + KoutV2/2g
• From Loss Coefficients : Kbend = 0.19 Kentrance = 0.5 Kout = 1
Hminor = (0.19x2 + 0.5 + 1) * (1.82/2*9.8) = 0.31 m
Datum
Pipe Flow1
2Z2 = 130 m
150 m
10 m
75 m
Z1 = ?
Kout=1
0 + 0 + Z1 = 0 + 0 + 130m + Hmajor + Hminor
0 + 0 + Z1 = 0 + 0 + 130m + 9.06m + 0.31m
Z1 = 139.4 meters
Types of open channel flows
Open Channel Flow
Steady uniform Flow
Steady Flow
Unsteady Flow
Unsteady nonuniform
Flow
Unsteady uniform Flow
Steady nonuniform
Flow
Gradually Varied Flow
Rapidly Varied Flow
Rapidly Varied Flow
Gradually Varied Flow
Open channel geometry factors
T
A
d
P Hydraulic radius, R = A/PHydraulic depth, Dh = A/T
A = cross‐sectional area P = wetted perimeterT = top width
Pressure Variation in Open Channel Flow
• Force‐Momentum Equation in the direction perpendicular to the flow,
Assumption: the acceleration of flow in the direction is negligible
. cos 0 cos
• 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.
Energy Relationships
gV
dZH2
cos2
LhgVdZ
gVdZ
2cos
2cos
222
22
211
11
Where,Z = channel bottom elevationd = depth of flow normal to the
channel bottomθ = channel slope angle, So = sin θα = a velocity distribution coefficient defined by
A = cross sectional areaV = average flow velocity
13
3
Total Head
Specific Energy• Specific energy, E is the energy head relative to channel bottom elevation
2
22
22 gAQy
gVyE
EEmin = EC
y = yc
y
y
A AT
Alternate depths
Critical depth
Specific Energy
22
22
221
21
1 22 gyqy
gyqyE zEE 21
A) Channel width decreases, discharge per unit width
q2 q1
B) Channel bed level decreases E2 E1
Critical Flow DepthE becomes minimum at the critical flow Critical depth yc
1
011
3
2
3
22
gATQ
gATQ
dydA
gAQ
dydE
1; No., Froude2/1
3
2
C
h
FrgDV
gATQFr
22
2
minch
cc
c
Dy
gV
yE
Rectangular channelT= B, A = B.y and Dh = y
At critical flow,
q = discharge per unit width
gyV
gABQFr
2/1
3
2
32
3 2
2
gq
gBQyc
22
2
minc
cc
cyy
gVyE
2
22
22 gyqy
gVyE
Uniform Flow
motivating forces = resistive forces
τ = γRnS0
S f = S0
W sinθ = γALS0
yn = Depth is called normal depth or uniform depth
Flow Resistance
Constitutive relationships for uniform flow• The Manning Equation (for metric units)
V = cross‐sectional averaged flow velocityn = Manning’s roughness coefficient
• The Chezy Equation
C = Chezy’s Constant (m0.5/s)
2/13/21on SR
nV
fRSCV
Momentum forces
v2
v1
P1
P2
W
Wsin
Rf
L
f21s RPsinWPF
)vv(qRsinW2y
2y
12f
22
21
1̅ 1
2
12̅ 2
2
2 sin
For steady flow
,
2
v1
P1
P2
W
Rf
L
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
Hydraulic jump
12
2
2
1
22
2
2
2
2
1 2
12 1 2
Continuity equation,1 1 2 2
2
1
12 1 8Fr12 1
1
2
12 1 8Fr22 1
Head loss at the hydraulic jump,
1 2 1 22 1
3
4 1 2
Momentum equation neglecting friction force,
If y1 and y2 are conjugate depths
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,
2
2
2gAQyZEZH
2
2
12
3
1 Fr2
/
/
0
1 Fr2
Gradually varied flow
Computation of gradually varied flow profiles
∆ 1 112
2
∆ 2 222
2
∆ 2 1
The direct-step method• a simple method applicable to prismatic channels
E1 and E2 are specific energy at sections 1 and 2 respectively
In the computations Sf is calculated for depths y1 and y2 and theaverage 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.
• If y ≥ yc ‐ hydraulically mild channel slope. (M curve) • If y = yc ‐ critical slope. (C curve •If y ≤ yc ‐ 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
K = proportionality coefficient
S = travel time through the reachX = a dimensionless weighting factor (0.1 ‐ 0.5)I = inflow discharge into the reach (m3/s)Q = outflow discharge from the reach (m3/s)
Continuity equation to the reach
])1([ QXXIKS
tQQtIISS
22
212112
QCICICQ 211202
tXKKXtC
5.0)1(
5.00
tXKtKXC
5.0)1(5.01
tXKtXKC
5.0)1(5.0)1(2
• Muskingum coefficients
River routing ‐ Muskingum-Cunge method
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 modelsSaint Venant equations,
Continuity Eqn
0
tA
xQ
Momentum Eqn
02
fgASxHgA
AQ
xtQ
• St Venant Eqns are approximated to
Continuity Eqn 0
tA
xQ
Momentum Eqn 0SS f
• Velocity c, called kinematic wave celerity, along the channel dy
dQTdA
dQc 1.
• 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.
Kinematic wave model
• St Venant Eqns are approximated to
The diffusion wave model is applied when the slopes are not large and when backwater effect is dominant
Continuity Eqn
0
tA
xQ
Momentum Eqn
0
fSx
H
Diffusion wave model
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 2‐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 2‐D domain using finite volume method, finite element method, etc.