6.6 gradually varied flow and... 55 6.6 gradually varied flow • non-uniform flow is a flow for...

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  • 55

    6.6 Gradually Varied Flow

    • Non-uniform flow is a flow for which the depth of flow is varied.

    • This varied flow can be either Gradually varied flow (GVF) or

    Rapidly varied flow (RVF).

    • Such situations occur when:

    - control structures are used in the channel or,

    - when any obstruction is found in the channel,

    - when a sharp change in the channel slope takes place.

  • 56

    Classification of Channel-Bed Slopes

    The slope of the channel bed is very important in determining the

    characteristics of the flow.

    Let

    • S0 : the slope of the channel bed ,

    • Sc : the critical slope or the slope of the channel that

    sustains a given discharge (Q) as uniform flow at the critical

    depth (yc).

    • yn is is the normal depth when the discharge Q flows as

    uniform flow on slope S0.

  • 57

    S S or y yc n c0  

    S S or y yc n c0  

    S S or y yc n c0  

    S0 0 0 .

    S negative0 

    The slope of the channel bed can be classified as:

    1) Critical Slope C : the bottom slope of the channel is equal to the critical slope.

    2) Mild Slope M : the bottom slope of the channel is less than the critical slope.

    3) Steep Slope S : the bottom slope of the channel is greater than the critical slope.

    4) Horizontal Slope H : the bottom slope of the channel is equal to zero.

    5) Adverse Slope A : the bottom slope of the channel rises in the direction of the

    flow (slope is opposite to direction of flow).

  • 58

  • 59

    Classification of Flow Profiles (water surface profiles)

    • The surface curves of water are called flow profiles (or water surface

    profiles).

    • The shape of water surface profiles is mainly determined by the slope of

    the channel bed So.

    • For a given discharge, the normal depth yn and the critical depth yc

    may be calculated. Then the following steps are followed to classify the

    flow profiles:

    1- A line parallel to the channel bottom with a height of yn is drawn and is

    designated as the normal depth line (N.D.L.)

    2- A line parallel to the channel bottom with a height of yc is drawn and is

    designated as the critical depth line (C.D.L.)

    3- The vertical space in a longitudinal section is divided into 3 zones

    using the two lines drawn in steps 1 & 2 (see the next figure)

  • 60

    4- Depending upon the zone and the slope of the bed, the water profiles

    are classified into 13 types as follows:

    (a) Mild slope curves M1 , M2 , M3 .

    (b) Steep slope curves S1 , S2 , S3 .

    (c) Critical slope curves C1 , C2 , C3 .

    (d) Horizontal slope curves H2 , H3 .

    (e) Averse slope curves A2 , A3 .

    In all these curves, the letter indicates the slope type and the subscript

    indicates the zone. For example S2 curve occurs in the zone 2 of the

    steep slope.

  • 61

    Flow Profiles in Mild slope

    Flow Profiles in Steep slope

  • 62

    Flow Profiles in Critical slope

    Flow Profiles in Horizontal slope

    Flow Profiles in Adverse slope

  • 63

    Dynamic Equation of Gradually Varied Flow

    Objective: get the relationship between the water surface slope and other

    characteristics of flow.

    The following assumptions are made in the derivation of the equation

    1. The flow is steady.

    2. The streamlines are practically parallel (true when the variation in

    depth along the direction of flow is very gradual). Thus the hydrostatic

    distribution of pressure is assumed over the section.

    3. The loss of head at any section, due to friction, is equal to that in the

    corresponding uniform flow with the same depth and flow

    characteristics. (Manning’s formula may be used to calculate the slope

    of the energy line)

    4. The slope of the channel is small.

    5. The channel is prismatic.

    6. The velocity distribution across the section is fixed.

    7. The roughness coefficient is constant in the reach.

    .

  • 64

    H Z y V

    g   

    2

    2

    dH

    dx

    dZ

    dx

    dy

    dx

    d

    dx

    V

    g   

     

     

    2

    2

    Consider the profile of a gradually varied flow in a small length dx of an open channel the channel as shown in the figure below.

    The total head (H) at any section is given by:

    Taking x-axis along the bed of the channel and differentiating the equation with

    respect to x:

  • 65

         

     

     S S

    dy

    dx

    d

    dx

    V

    g f 0

    2

    2

    dy

    dx

    dy

    dy

    d

    dx

    V

    g S S f

     

       

    2

    0 2

    dy

    dx

    d

    dy

    V

    g S S f1

    2

    2

    0 

     

     

     

     

     

    dy

    dx

    S S

    d

    dy

    V

    g

    f 

     

     

     

    0

    2

    1 2

    • dH/dx = the slope of the energy line (Sf).

    • dZ/dx = the bed slope (S0) .

    Therefore,

    Multiplying the velocity term by dy/dy and transposing, we get

    or

    This Equation is known as the dynamic equation of gradually varied flow. It gives the variation of depth (y) with respect to the distance along the bottom of the channel (x).

  • 66

    dy

    dx

    S S

    Q T

    g A

    f 

    0

    2

    3 1

    dy

    dx

    dE dx

    Q T

    g A

    /

    1

    2

    3

    The dynamic equation can be expressed in terms of the discharge Q:

    The dynamic equation also can be expressed in terms of the specific energy E :

  • 67

    dy

    dx  0

    dy

    dx positive

    dy

    dx negative

    Depending upon the type of flow, dy/dx may take the values:

    The slope of the water surface is equal to the bottom

    slope. (the water surface is parallel to the channel bed)

    or the flow is uniform.

    The slope of the water surface is less than the bottom slope

    (S0) . (The water surface rises in the direction of flow) or the profile obtained is called the backwater curve.

    The slope of the water surface is greater than the bottom

    slope. (The water surface falls in direction of flow) or the

    profile obtained is called the draw-down curve.

    (a)

    (c)

    (b)

  • 68

    Notice that the slope of water surface with respect to horizontal (Sw) is different

    from the slope of water surface with respect to the bottom of the channel (dy/dx).

    A relationship between the two slopes can be obtained:

    S bc

    ab

    cd bd

    abw   

     sin

    S cd

    ad

    cd

    ab0   sinq

    • Consider a small length dx of the open channel.

    • The line ab shows the free surface,

    • The line ad is drawn parallel to the bottom at a slope of S0 with the horizontal.

    • The line ac is horizontal.

    Let q be the angle which the bottom makes with the horizontal. Thus

    The water surface slope (Sw) is given by

  • 69

    dy

    dx

    bd

    ad

    bd

    ab  

    dx

    dy SS

    w 

    0

    dy

    dx S Sw 0

    The slope of the water surface with respect to the channel bottom is given by

    This equation can be used to calculate the water

    surface slope with respect to horizontal.

  • 70

    Water Profile Computations (Gradually Varied Flow)

    • Engineers often require to know the distance up to which a surface

    profile of a gradually varied flow will extend.

    • To accomplish this we have to integrate the dynamic equation of

    gradually varied flow, so to obtain the values of y at different locations

    of x along the channel bed.

    •The figure below gives a sketch of calculating the M1 curve over a

    given weir.

  • 71

    Direct Step Method

    • One of the most important method used to compute the water profiles is

    the direct step method.

    • In this method, the channel is divided into short intervals and the

    computation of surface profiles is carried out step by step from one section

    to another.

    For prismatic channels:

    Consider a short length of channel, dx , as shown in the figure.

    dx

  • 72

    S dx y V

    g y

    V

    g S dxf0 1

    1 2

    2 2 2

    2

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