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S. Maskey (2011) 1
Free Surface Hydrodynamics
Water Science and Engineering
y yA part of Module 2: Hydraulics and Hydrology
Water Science and Engineering
Dr. Shreedhar MaskeySenior Lecturer
UNESCO-IHE Institute for Water Education
S. Maskey (2011) 1
About Module 2 Hydraulics and Hydrology
Free surface hydrodynamics (35%)
Engineering hydrology (35%)
GIS and remote sensing (30%)
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S. Maskey (2011) 2
Part-1: Fundamental principles and equations
Continuity and Momentum principles Euler and Navier-Stokes
equations Bernoulli equation and applications Bernoulli equation
and applications
3
Continuity principleZ
xuAx
xuAuA xx
Expresses the conservation of mass in a control volume occupied
by a fluid.
Obtained by equating the change in X
Y
y q g gfluid mass to the difference between the rate of mass IN
and OUT in a given time.
Change in mass in time t is Rate of fluid mass inflow and
outflow for the continuity equation.
zyxtt
zyxtt
))(( Change in mass due to the change in
and u (in x-direction)
tt
zyxt
xut
xuxAtuAx
xuAuA xxxx
)(4
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S. Maskey (2011) 3
Continuity principle X
Z
Y
xuAx
xuAuA xx
General form of continuity equation:
0
wvuwvut
For incompressible fluid with constant :
zyxzyxt
0
zw
yv
xu
For a steady and incompressible flow in a pipe:
2211 AVAVQ Also valid for a steady continuous flow in an open
channel.
5
Continuity principle For unsteady flow in an open channel:
0
thB
xQ 0
tAQ
tx tx
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S. Maskey (2011) 4
Momentum principle Inertia force: To change an existing motion
of a body of
mass M, it is necessary to apply a force F to this mass, which
causes an acceleration a = dV/dt, such that
dV dV
This is the Newton's Equation of Motion. The product M(dV/dt) is
the inertia force.
Two types of inertia forces are considered
dtdM VF )volumeunit(
dtdVF
Two types of inertia forces are considered Due to local
accelation change in velocity in time
Due to convective acceleration change in velocity over a
distance
dtdV
dxdVV
7
Momentum principle Inertia force components in 3 dimensions (x,
y, z axes):
zuw
yuv
xuu
tu
zww
ywv
xwu
tw
zvw
yvv
xvu
tv
y
Local ConvectiveLocalacceleration
term
Convectiveacceleration
terms
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S. Maskey (2011) 5
Momentum principle Applied forces: Gravity force (Fg = Mg)
Pressure force
Z
)( zyp zyx
xppp
Pressure forces acting on two opposite faces on the
yz-plane.
Temperature - t -(oC)
Dynamic Viscosity- -(N s/m2) x 10-3
Kinematic Viscosity- -(m2/s) x 10-6
0 1.787 1.7875 1.519 1.51910 1.307 1.30720 1.002 1.004
Viscous force (recall Newtons Law of Viscosity)
X
Y
zyxxpzyx
xppzyp
)(
d 30 0.798 0.80140 0.653 0.65850 0.547 0.55360 0.467 0.47570
0.404 0.41380 0.355 0.36590 0.315 0.326100 0.282 0.294
9
dydu
Shear stress
Coefficient of viscosity
Momentum principle In 3 dimensions
X
Z
xyxz
zyzx
yzyx
)()()( yxzz
zxyy
zyxx
zxyxxx
X
Y
)()()(
)()()(
yxzz
zxyy
zyxx
yxzz
zxyy
zyxx
zyx
zzyzxz
zyyyxy
Convention used to represent shear stresses on various planes.
The first subscript represents the axis normal to the plane and the
second subscript represents the axis along which the force is
acting.
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S. Maskey (2011) 6
Momentum equation Obtained from balancing inertia
and applied forces
Euler equation neglects viscous X
Z
xyxz
zyzx
yzyx
Euler equation neglects viscous forces
X
Y
Convention used to represent shear stresses on various planes.
The first subscript represents the axis normal to the plane and the
second subscript represents the axis along which the force is
acting.y
pzvw
yvv
xvu
tv
xp
zuw
yuv
xuu
tu
1
1
11
gzp
zww
ywv
xwu
tw
1
Momentum equation Navier-Stokes equation also includes
viscous forces
X
Z
xyxz
zyzx
yzyx
pdu zxyxxx
1 X
Y
Convention used to represent shear stresses on various planes.
The first subscript represents the axis normal to the plane and the
second subscript represents the axis along which the force is
acting.
gzyxz
pdtdw
zyxyp
dtdv
zyxxdt
zzyzxz
zyyyxy
1
1
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Bernoulli equation Can be derived from Euler equation Steady
flow assumption is used Has wide application in hydraulics
Represents the total energy and signifies that total the Represents
the total energy and signifies that total the
energy remains constantConstant
2
2
zgp
gV
SfgV 22
1Energyline
TOTALENERGYLINE
13
S0
V1
h1 h2
x
V2
SfgV 21gV 222Freesurface
Channelbottom
Part-2: 1D Channel Flow
Steady-uniform flow Friction coefficient and velocity
distribution Specific discharge critical depth Specific discharge,
critical depth Non-uniform gradually varied flow (backwater
curve) Unsteady flow
St-Venant equationKinematic wave Kinematic wave
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Steady-uniform flow Velocity is constant (i.e. V/t = 0) and is
uniform in
space (i.e. in one dimension V/x = 0). The water surface line
remains parallel to the channel
bed profile. p Gravity force to be balanced by the shear force
due to
the bed resistance.
Pb dPxxAg 0sin sinsin gRPAgb
15
Remember:
P is wetted perimeter
Hydraulic radius, R = A / P
Steady-uniform flow For small slope
0tansin Szb 0gRSb
For turbulent flows, the shear stress can be assumed to be
related to the average velocity V and friction coefficient Cf,
as
Finally
x
2VC fb Finally
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0RSCgVf
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Steady-uniform flow Chezys formula
0RSCV 3
Mannings formula121
21
021
2
0 SP
ACRSCAQ
135
12 11 A
Remember:
Discharge, Q = A x V
Hydraulic radius, R = A / P
17
20
31 SRn
V 2032
320
3 11 SP
An
SARn
Q
Steady-uniform flow Normal depth depth based on uniform flow
From Mannings formula Remember:F id t l
03/2
3/5
SnQ
PA
3/5)( nQBhn 53
nQh
Rectangular channel Wide rectangular channel
21
032
35
1 SP
An
Q For a wide rectangular channel (i.e. B >> h)
P BR h
18
03/2)2( ShB n
0
SB
nQhn
Can you work out similarly the relationship for the normal depth
using the Chezys formula?
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S. Maskey (2011) 10
Uniform flow computation in a compound channel The flow velocity
and the carrying capacity of the channel are
normally different for the main channel and the over bank flow
or the floodplain.
The roughness coefficient may also vary in the main channel The
roughness coefficient may also vary in the main channel and the
floodplain.
h1
h2 h3
b22 b21 b31 b32A1
A2 A31
2 3
b11b12 b13
2/13/2
321total
1
isapproach usual The
SRAn
QQQQ
iii
That is, the total discharge is the sum of the
discharges through each segment of the cross section.
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Specific energy The energy (in water head)
due to the depth of water (h) plus its velocity (the velocity
head) is the specific energy.
Differentiating w.r.t. h
Supecritical flowrangehc
Emin
2
22
22 gAQh
gVhE
dAQdE3
2
1 VdE2
1 With dA/dh = B and D = A/B
For a given discharge there exists a min. specific energy Emin
(at dE/dh = 0):
20
dhgAdh 31
gDdh1
1gDV
1ghV
With dA/dh = B and D = A/BT.
For a rectangular channel
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Critical depth, Froude number Water depth at Emin is called
a
critical depth, hc.
1V QDA2
2
F d b (F ) t
Supecritical flowrangehc
Emin
gD g
31
231
2
2
gq
gBQhc
Froude number (Fr) represents the ratio of inertia forces to
gravity forces, given by
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gDVFr At critical depth (hc), Fr = 1.
Subcritical and supercritical flowsSubcritical flow: Fr <
1
Critical flow: Fr = 1
V < (gD) V = (gD)
Supercritical flow: Fr > 1Subcritical flow
hchn
V > (gD)
Supercritical flowhc
hn
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Celerity, subcritical and supercritical flow The critical
velocity (gD) is also the speed, or more precisely,
"celerity" of small gravity waves that occur in shallow water in
open channels as a result of a disturbance to the free surface.
A gravity wave can be propagated upstream in water of
subcritical flow (c > V), but not in water of supercritical flow
(c < V).
gDc ghc For a rectangular channel
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Wave patterns created by disturbances: (a) Still water, V = 0;
(b) subcritical flow, c > V; (c) critical flow, c = V; and (d)
supercritical flow, c < V.
Non-uniform, steady flow
Gradually varied flow- backwater curve computation
(non-uniform, steady flow)
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Gradually varied (non-uniform) steady flow
Gradually varied (non-uniform) steady flow equations can be
derived either by
(1) Taking / t terms equal to zero (because steady flow) in the
St. Venant equation, or
(2) From the consideration of total energy (Bernoulli equation)
between two sections in a uniform channel.
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Gradually varied (non-uniform) steady flow
0 fSSdh 3/103/4
22342
22
APQn
RAQnS f
Equation in terms of Froude Number (Fr): Mannings formula
2Fr1f
dx3
22Fr
gAQBT
dhdESS
dxdh f 0OR
Exercise:
Rewrite the formula for Sf for a wide rectangular channel. Can
you workout Sf using Chezys formula?
Sf = friction slope, S0 = channel bed slope, P = wetted
perimeter, A = cross-section area, h = water depth, x = length
along the channel, n = Mannings coefficient, BT = top width of
channel cross section, Fr = Froude number, g = acceleration due to
gravity, Q = discharge, E = specific energy.
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dhdEdx
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S. Maskey (2011) 14
23 Q
Gradually varied (non-uniform) steady flow
Equation in terms of normal and critical depths for a
wide-rectangular channel:
022
3
SBCQhn
33
33
0c
n
hhhhS
dxdh
2
23
gBQhc
C = Chezys coefficient, B = constant bed width, hn = normal
depth, hc = critical depth.
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Gradually varied flow flow profiles
3/10
3/422
APQnS f
M1 (backwater curve)
yn
yc Region 3
Region 2
Region 1M2
M3
(drawdown curve)
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Solution of gradually varied steady flow equation Numerical
solution (direct step solution)
20 fSSdh The problem here is Sf and Fr are dependent on h, i.e.
the 2Fr1dx
fSShx
0
2Fr1 Starting at the downstream boundary, compute the horizontal
distance x that corresponds to a given change in depth h.
pequation is implicit on h.
f0 depth h.
20
Fr1 fSSxh Starting at the downstream boundary, compute change in
depth h for a
chosen horizontal distance x.
OR
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Solution of gradually varied steady flow equation Numerical
solution (improved Eulers method)
20
Fr1 fSS
dxdh Remember!
When starting from downstream towards Fr1dx When starting from
downstream towards upstream, make sure that x is negative.
jh
fjj
SSxhh
202/1 Fr12
SS
Compute Sf and Fr with h = hj.
C t S d F ith h h
2/1
20
1 Fr1
jh
fjj
SSxhh
hj = water depth at section j, hj+1 = water depth the section
next to section j, hj+1/2 = water depth between sections j and
j+1.
Compute Sf and Fr with h = hj+1/2
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Solution of gradually varied steady flow equation Numerical
solution (Predictor-corrector method)
)Fr1( 20
fSS
dxdh Remember!
When starting from downstream towards t k th t i ti
SSSS
)Fr1(dx upstream, make sure that x is negative.
jh
fj
prej
SSxhh
201 Fr1Predictor :
Compute Sf and Fr with h = hj.
prejj h
f
h
fj
corj
SSSSxhh1
20
20
1 Fr1Fr12Corrector :
Repeat Corrector step with a new value of predictor h until the
predictor h is close enough to corrector h. The new value of
predictor is normally taken as the current value of the corrector
h.
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Hydraulic jump Occurs when supercritical flow changes
into a subcritical flow.
1h Hydraulic jump Subcritical
Practical application: to dissipate energy downstream of dams
weirs etc
1Fr8121 2
11
2
hh
h1
h2V1
V2Supercritical
downstream of dams, weirs, etc.
21
312
21 4loss,Enery
hhhhEEE
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S. Maskey (2011) 17
Unsteady, non-uniform flow
Unsteady Flow, St. Venant Equation
Flood Wave Propagation
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H, Q
H, Q
Unsteady non-uniform flow: moving flood wave
Longitudinal view of water surface in an open channel
X
at time T1
at time T2
H, Q
X
X
at time T3
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Unsteady non-uniform flow
T
Q
Longitudinal Profile: Water surface profile along a river
channel at a given time.
X
Represents variation in space
Hydrograph: Discharge at various times through a given section
of a river.
Represents variation in timeComputation of unsteady non-uniform
flow considers variation in both space and time together.
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Saint Venant equations for 1D channel flow Assumptions of
Saint-Venant equation
One-dimensional flow Gradually varying flow in space and time
Hydrostatic pressure prevails vertical acceleration is Hydrostatic
pressure prevails, vertical acceleration is
negligible Stable and relatively small bed slope Longitudinal
axis of the channel is approximated as a
straight line. Incompressible, homogeneous and constant
viscosity fluid
Consists of Continuity and Momentum equations Momentum equation
can be derived from Euler
equation with an added bed shear force.
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Saint Venant equations Momentum force:
Gravity force component:
xAxVV
tV
xSgA Gravity force component: Pressure force:
Shear force due to friction:
xAxhg
0xSgA
fxSgA
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S0
V1
h1 h2
x
V2
SfgV 22
1
gV 222
Energyline
Freesurface
Channelbottom
Saint Venant equations Equating momentum and applied forces:
00 fSSgxhgxVVtV fxxt
002
fSSgAx
hgAAQ
xtQ
V = Q/A
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Local acceleration term
Pressure force term
Convective acceleration term
Gravity force term
Friction force term
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Saint Venant equations Various flood wave approximations from
the Saint-
Venant equation:
equationContinuity
0)(
equation Momentum
0
2
fSSgAhgAQQ
0
equation Continuity
tA
xQ
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0)( 0
fSSgAxgAAxtKinematic wave
Diffusion wave
Full dynamic wave
Kinematic wave approximation
0
tA
xQFrom continuity equation:
Kinematic wave approximation neglects q
tA
xQ
Note: If there is lateral inflow q, the continuity equation
becomes:
e at c a e app o at o eg ectsacceleration and pressure terms of
the momentum equation, i.e.
This assumption allows to use Manning or Chezy equation:
fSS 0
2/12/12/13/2 OR 1 SCARQSARn
Q
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Kinematic wave approximation
QA In both Manning and Chezy equation, A can be expressed in
terms of Q as
Q
6.0and6.0
2/1
3/2
SnP
Exercise:
Verify that the given and are for theManning equation and derive
them for theChezy equation.
Using Manning equation, the coefficients alpha and beta are
given by
Chezy equation.
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Kinematic wave approximation
01
QQQ
In continuity equation, replacing A with Q and simplifying.
tQx
01
tQ
cxQ
kck is the kinematic wave celerity.
VAQ
QQ
Qck 3
53511
Where
1
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Unsteady flow and flood wave propagation
A note on numerical methods for the solution 1D unsteady
flow
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Unsteady flow and flood wave propagation
In practice almost always solved using numerical methods.
Finite Difference Method is one of the widely used numerical
methods.
Discretization is used in space and time to approximate the
partial difference equation to a finite difference the partial
difference equation to a finite difference (algebraic)
equation.
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Finite difference numerical solution Understanding space-time
discretisation of
numerical schemest
n-1
n+1
nnjQ
njQ 1
11njQ1n
jQ
known pointsunknown point
t
xj-1 j j+1
initial conditionboundary condition
n = 0j = 0
x
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Finite difference numerical solution
Explicit and Implicit schemes:
In an explicit scheme the value of the variable at time level
n+1 can be computed directly (or explicitly) from the values n+1
can be computed directly (or explicitly) from the values at time
level n.
In an implicit scheme the computation of the value of a variable
at n+1 involves one or more of the values from the same time level
(i.e. n+1).
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Numerical solution of kinematic wave (an example)
01
tQ
cxQ
k
An example of an explicit schemet
1
Partial differential equation
TO
01 1111
tQQ
cxQQ nj
nj
k
nj
nj
1
xj-1 j j+1
n-1
n+1
nnjQ njQ 1
11njQ1n
jQFinite difference equation
njr
njr
nj QCQCQ 1
11 )1(
xtcC kr
Where
Cr is the Courant Number.
known pointsunknown point
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Numerical solution of kinematic wave (an example)
01
tQ
cxQ
k
An example of an implicit schemet
1
Partial differential equation
TO
01 111
111
tQQ
cxQQ nj
nj
k
nj
nj
1 Cx
j-1 j j+1
n-1
n+1
nnjQ njQ 1
11njQ1n
jQFinite difference equation
Multiply both sides by x
11
11 11
1
nj
r
rnj
r
nj QC
CQC
Q
xtcC kr
Where
Cr is the Courant Number.
known pointsunknown point
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S. Maskey (2011) 25
Finite difference numerical solution
Stability of an scheme
In numerical methods, the choice of the space step (x) and time
step (t) is important!and time step (t) is important!
An explicit scheme becomes unstable when the Courant number (Cr
= ct/x) is greater than 1. Where c is the celerity.
An implicit solution is always stable. However, Cr >> 1 is
t d dnot recommended.
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Finite difference numerical solution Initial and boundary
conditions:
Numerical schemes are based on the assumption that variable
pvalues are known at all points on space at time level zero also
called an initial condition.
A numerical scheme also requires that values at all time levels
at the starting point on space (called upstream boundary condition)
upstream boundary condition) and/or at the end point on space
(called the downstream boundary condition).
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Commonly used initial and boundary conditions
Initial condition: Global or local water depth and steady state
computation
Upstream boundary condition: Discharge hydrograph
Downstream boundary condition: Q-h relationshipp
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End of Presentation