Gradually Varied Flow

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Chapter FourGRADUALLY VARIED FLOW (GVF)

• 4.1 Introduction
• A steady non uniform flow in a prismatic channel
  with gradual changes in its water surface
  elevation is termed as gradually varied flow
  (GVF).
• The back water produced by a dam or a weir across
  a river and the draw down produced at a sudden
  drop in a channel are few typical examples of Gvf.

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Gradually Varied Flow
Fig.4.1
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Gradually Varied Flow

• In GVF, the velocity varies along the channel and
  consequently the bed slope ,water surface slope,
  and energy slope will all differ from each other.
• Regions of high curvature are excluded in the
  analysis of this flow.
• The two basic assumptions involved in the
  analysis of GVF are

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Gradually Varied Flow

• The pressure distribution at any section is
  assumed to be hydrostatic.
• The resistance to flow at any depth is assumed to
  be given by corresponding uniform-flow
  equation,such as the Mannings formula,with the
  condition that the slope term to be used in the
  equation is the energy slope and not the bed
  slope.
• Thus, if in a gradually varied flow the depth of
  low at any section is y, the energy slope Sf is
  given by

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Gradually Varied Flow

• Sf n2v2/R4/3 --------------------(4.1)
• Where R hydraulic radius of the section at
  depth y.
• 4.2 DIFFERENTIAL EQUATION OF GVF
• Consider the total energy H of gradually varied
  flow in a channel of small slope and ? 1 as
• H Z E Z Y V2/2g -----4.2
• Where E specific energy,

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4.2 DIFFERENTIAL EQUATION OF GVF

• A schematic sketch of a gradually varied flow is
  shown in fig.4.1 Since the water surface in
  general ,varies in the longitudinal (x)
  direction, the depth of flow and total energy are
  functions of x. Differentiating Eq.(4.2) with
  respect to x

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4.2 DIFFERENTIAL EQUATION OF GVF
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4.2 DIFFERENTIAL EQUATION OF GVF
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4.2 DIFFERENTIAL EQUATION OF GVF
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4.2 DIFFERENTIAL EQUATION OF GVF

• The second form of the equation of gradually
  varied flow can be derived if it is recognized
  that dE/dx dE/dy.dy/dx and that from chapter
  three dE/dy 1-F2.
• Provided that the Froude number is properly
  defined.
• Then Equation 4.8a becomes

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4.2 DIFFERENTIAL EQUATION OF GVF
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4.2 DIFFERENTIAL EQUATION OF GVF

• The definition of the Froude number in equation
  4.8 b depends on the channel geometry in which
  Froude number is given by the formula
• While for a regular prismatic channel in
  which
• negligible it assumes the conventional energy
  definition given by .

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Reading Assignment

• For non uniform gradually varied flow friction
  slope is not parallel to bottom channel slope,
  but is evaluated using mannings the chezs
  equation.
• There is no general clear solution although
  particular solutions are available for prismatic
  channels. Numerical methods are normally used.

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4.3 Classification of flow profiles

• The general profile equation is written as

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4.3Classification of flow profiles
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4.3 Classification of flow profiles
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4.3 Classification of flow profiles

• Depending up on the channel category and region
  of flow ,the water surface profiles will have
  characteristic shapes.
• Whether the given GVF profile will have an
  increasing or decreasing water depth in the
  direction of flow will depend up on the term
  dy/dx in Eq.(4.8) being positive or negative.
• It can be seen from Eq.4.12 that dy/dx is positive

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4.3 Classification of flow profiles

• If numerator is gt0 and denominator gt0 .
• If numerator is lt0 and denominator lt0

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4.3 Classification of flow profiles
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4.3 Classification of flow profiles
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4.3 Classification of flow profiles
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4.3 Classification of flow profiles
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4.3 Classification of flow profiles
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EXAMPLE 4.1

• A rectangular channel with a bottom width of 4.0m
  and a bottom slope of 0.0008 has a discharge of
  1.5m3/s .In a gradually varied flow in this
  channel, the depth at a certain location is found
  to be 0.30m. Assuming n0.016.Determine the type
  of GVF profile.
• Solution

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Solution

• (a) To find the normal depth yo
• F Qn/S00.5 B8/3
• 1.500.016/0.008 0.5(40)8/3
• Referring to table 3A.1, the value of YO/B for
  this value of F ,by interpolation,
• YO/B 0.1065
• Yo 0.426m

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solution

• b)Critical depth Yc
• q Q/B 1.5/4.0 0.375 m3/S/m
• Yc (q2/g2)1/3 ( 0.3752/9.81) 0.243m
• C0 Type of Profile
• Since yo gtyc, the channel is a mild slope
  channel.Also the given y0.30m such that
• Yo gt ygtyc
• As such the profile is of the M2 type table 4.2

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Summary of flow profiles

Backwater curve Uniform flow curve Draw down curve
y Sf ltSo So-Sfgt0 Gradually varied
Y yn Sf So So-Sf 0 Uniform flow
Y lt yn SfgtSo So-Sflt0 Gradually varied
ygtyc Frlt1 1-fr2gt0 Sub-critical
Yyc Fr1 1-fr2 0 Critical
Yltyc Frgt1 1-fr2lt0 Supercritical
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GVF Computation

• Almost all major hydraulic-engineering activities
  in free surface flow involve the computation of
  GVF PROFILES.
• Considerable computational effort is involved in
  the analysis of problems, such as
• Determination of the effect of a hydraulic
  structure on the channels,
• Inundation of lands due to a dam or weir
  construction

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GVF Computation

• c) Estimation of flood zone
• The various available procedures for computing
  GVF can be classified as
• Direct integration
• Numerical method
• Graphical method

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GVF Computation

• A GVF computation procedures for use in an
  artificial channel may or may not be applicable
  to natural channels of irregular cross sections
  since in a natural channel, the cross sectional
  properties are known only at specified locations
  while in a prismatic artificial channel the cross
  sections are constant all along the channel .
• Because of this ,some methods have been
  developed, particularly for use in natural
  channels.

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GVF Computation
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GVF Computation

• The direct integration and graphical methods are
  suprseded by numerical integration methods due to
  the wide spread use of computers.
• Numerical Integration Method

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Numerical Integration method
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Numerical Integration method
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Computation procedure through Example
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Computation procedure through Example
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solution
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solution
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solution
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Computation steps for flow profiles
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Computation steps for flow profiles
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Computation steps for flow profiles
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Example

• A rectangular channel is 3.0m wide ,has
  0.01slope,discharge of 5.3 m3/s, andn0.011.Find
  Yn and Yc .If the actual depth of flow is 1.7m,
  what type of profile exists?

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Solution
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Exercise

• A rectangular channel has a gradient of 2 in 1000
  and ends in a free out fall. At a discharge of
  4.53m3/s with mannings
• n 0.012 and b 1.83m, how far from the outfall
  is the depth equal to 99 of normal depth if five
  depth steps are used in the calculation ?

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