lec fsh smaskey 2011

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%)

2

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

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

6

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

8

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.

10

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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

12

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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

14

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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

16

0RSCgVf

S. Maskey (2011) 9

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?

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.

19

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

S. Maskey (2011) 11

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

21

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

22

S. Maskey (2011) 12

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

23

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)

24

S. Maskey (2011) 13

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.

25


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.

26

dhdEdx

S. Maskey (2011) 14

23 Q


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.

27

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)

28

S. Maskey (2011) 15

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

29

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

30

S. Maskey (2011) 16

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.

31

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

32

S. Maskey (2011) 17

Unsteady, non-uniform flow

Unsteady Flow, St. Venant Equation

Flood Wave Propagation

33

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

34

S. Maskey (2011) 18

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.

35

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.

36

S. Maskey (2011) 19

Saint Venant equations Momentum force:

Gravity force component:

xAxVV

tV

xSgA Gravity force component: Pressure force:

Shear force due to friction:

xAxhg

0xSgA

fxSgA

37

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

38

Local acceleration term

Pressure force term

Convective acceleration term

Gravity force term

Friction force term

S. Maskey (2011) 20

Saint Venant equations Various flood wave approximations from the Saint-

Venant equation:

equationContinuity

0)(

equation Momentum

0

2

fSSgAhgAQQ

0

equation Continuity

tA

xQ

39

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

40

S. Maskey (2011) 21


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.

41


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

42

S. Maskey (2011) 22

Unsteady flow and flood wave propagation

A note on numerical methods for the solution 1D unsteady flow

43

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.

44

S. Maskey (2011) 23

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

45

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).

46

S. Maskey (2011) 24

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.


47

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.


48

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.

49

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).

50

S. Maskey (2011) 26

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

51

End of Presentation

lec fsh smaskey 2011

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