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    Civil Engineering Hydraulics

    Gradually Varying Flow When I look into a mirror..

    Gradually Varied Flow

    ¢ So far in open channels we considered uniform flow during which the flow depth z and the flow velocity v remain constant.

    Friday, November 9, 2012 Gradually Varied Flow 2

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    Gradually Varied Flow

    ¢  In this section we consider gradually varied flow (GVF), which is a form of steady nonuniform flow characterized by gradual variations in flow depth and velocity (small slopes and no abrupt changes) and a free surface that always remains smooth (no discontinuities or zigzags).

    Friday, November 9, 2012 Gradually Varied Flow 3

    Gradually Varied Flow

    ¢ Flows that involve rapid changes in flow depth and velocity, are called rapidly varied flows (RVF).

    ¢ A change in the bottom slope or cross section of a channel or an obstruction in the path of flow may cause the uniform flow in a channel to become gradually or rapidly varied flow.

    Friday, November 9, 2012 Gradually Varied Flow 4

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    Gradually Varied Flow

    ¢  In gradually varied flow, the flow depth and velocity vary slowly, and the free surface is stable.

    ¢ This makes it possible to formulate the variation of flow depth along the channel on the basis of the conservation of mass and energy principles and to obtain relations for the profile of the free surface.

    Friday, November 9, 2012 Gradually Varied Flow 5

    Gradually Varied Flow

    ¢  In uniform flow, the slope of the energy line is equal to the slope of the bottom surface. Therefore, the friction slope equals the bottom slope, Sf = S0.

    ¢  In gradually varied flow, however, these slopes are different.

    Friday, November 9, 2012 Gradually Varied Flow 6

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    Gradually Varied Flow ¢ Consider steady flow in a rectangular open

    channel of width b, and assume any variation in the bottom slope and water depth to be rather gradual.

    Friday, November 9, 2012 Gradually Varied Flow 7

    Gradually Varied Flow ¢ Write the equations in terms of average

    velocity v and assume the pressure distribution to be hydrostatic.

    Friday, November 9, 2012 Gradually Varied Flow 8

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    Gradually Varied Flow ¢ The total head of the liquid at any cross

    section is H = zb + y + v 2/2g, where zb is the vertical distance of the bottom surface from the reference datum.

    ¢ Differentiating H with respect to x gives

    Friday, November 9, 2012 Gradually Varied Flow 9

    Gradually Varied Flow ¢  In this expression, z is the depth of the

    channel bottom from the datum elevation, not the depth of flow in the channel.

    ¢ Differentiating H with respect to x gives

    Friday, November 9, 2012 Gradually Varied Flow 10

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    Gradually Varied Flow ¢ Differentiating H with respect to x gives

    Friday, November 9, 2012 Gradually Varied Flow 11

    dH dx

    = d dx

    zb + y + v 2

    2g ⎛ ⎝⎜

    ⎞ ⎠⎟

    dH dx

    = dzb dx

    + dy dx

    + v g

    dv dx

    Gradually Varied Flow ¢ H is the total energy of the liquid and thus

    dH/dx is the slope of the energy line (negative quantity), which is equal to the friction slope.

    Friday, November 9, 2012 Gradually Varied Flow 12

    dH dx

    = d dx

    zb + y + v 2

    2g ⎛ ⎝⎜

    ⎞ ⎠⎟

    dH dx

    = dzb dx

    + dy dx

    + v g

    dv dx

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    Gradually Varied Flow ¢  dzb/dx is the bottom slope ¢ Both of these slopes are negative

    Friday, November 9, 2012 Gradually Varied Flow 13

    dH dx

    = d dx

    zb + y + v 2

    2g ⎛ ⎝⎜

    ⎞ ⎠⎟

    dH dx

    = dzb dx

    + dy dx

    + v g

    dv dx

    Gradually Varied Flow ¢ Therefore

    Friday, November 9, 2012 Gradually Varied Flow 14

    dH dx

    = dzb dx

    + dy dx

    + v g

    dv dx

    dH dx

    = −Sf

    dzb dx

    = −S0

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    Gradually Varied Flow ¢ Substituting

    Friday, November 9, 2012 Gradually Varied Flow 15

    dH dx

    = dzb dx

    + dy dx

    + v g

    dv dx

    −Sf = −S0 + dy dx

    + v g

    dv dx

    S0 −Sf = dy dx

    + v g

    dv dx

    Gradually Varied Flow ¢ For continuity, the mass flow rate is the

    same at every cross section. ¢ Since this a incompressible fluid, the

    volumetric flow rate must also be the same.

    Friday, November 9, 2012 Gradually Varied Flow 16

    dH dx

    = dzb dx

    + dy dx

    + v g

    dv dx

    −Sf = −S0 + dy dx

    + v g

    dv dx

    S0 −Sf = dy dx

    + v g

    dv dx

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    Gradually Varied Flow ¢  If we assume a rectangular channel, then

    the volumetric flow rate at any cross section will be the depth times the channel width times the velocity.

    Friday, November 9, 2012 Gradually Varied Flow 17

    S0 −Sf = dy dx

    + v g

    dv dx

    Q = vA = vby

    Gradually Varied Flow ¢ Differentiating the volumetric flow rate with

    respect to x and remembering that both y and v change along the channel.

    Friday, November 9, 2012 Gradually Varied Flow 18

    S0 −Sf = dy dx

    + v g

    dv dx

    Q = vA = vby dQ dx

    = d dx

    vby( ) = by dv dx

    + bv dy dx

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    Gradually Varied Flow ¢ Since the flow rate does not change with

    respect to x (continuity) the derivative dQ/dx is equal to 0

    Friday, November 9, 2012 Gradually Varied Flow 19

    S0 −Sf = dy dx

    + v g

    dv dx

    Q = vA = vby dQ dx

    = d dx

    vby( ) = by dv dx

    + bv dy dx

    Gradually Varied Flow ¢ Substituting

    Friday, November 9, 2012 Gradually Varied Flow 20

    S0 −Sf = dy dx

    + v g

    dv dx

    dQ dx

    = by dv dx

    + bv dy dx

    0 = by dv dx

    + bv dy dx

  • 11

    Gradually Varied Flow ¢ Solving for dv/dx

    Friday, November 9, 2012 Gradually Varied Flow 21

    S0 −Sf = dy dx

    + v g

    dv dx

    −by dv dx

    = +bv dy dx

    dv dx

    = − v y

    dy dx

    Gradually Varied Flow ¢ Substituting into the original expression

    Friday, November 9, 2012 Gradually Varied Flow 22

    S0 −Sf = dy dx

    + v g

    dv dx

    dv dx

    = − v y

    dy dx

    S0 −Sf = dy dx

    + v g

    − v y

    dy dx

    ⎛ ⎝⎜

    ⎞ ⎠⎟

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    Gradually Varied Flow ¢ Collecting terms

    Friday, November 9, 2012 Gradually Varied Flow 23

    S0 −Sf = dy dx

    + v g

    dv dx

    S0 −Sf = dy dx

    − v 2

    gy dy dx

    S0 −Sf = 1− v 2

    gy ⎛ ⎝⎜

    ⎞ ⎠⎟

    dy dx

    Gradually Varied Flow ¢ The term v squared over gy may be

    recognized as the square of the Froude number of the flow

    Friday, November 9, 2012 Gradually Varied Flow 24

    S0 −Sf = 1− v 2

    gy ⎛ ⎝⎜

    ⎞ ⎠⎟

    dy dx

    S0 −Sf = 1− Fr 2( )dydx

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    Gradually Varied Flow ¢  Isolating the differential term we have

    Friday, November 9, 2012 Gradually Varied Flow 25

    S0 −Sf 1− Fr 2( ) =

    dy dx

    Gradually Varied Flow ¢ The dy/dx term is the slope of the water

    surface profile as you move down the channel.

    Friday, November 9, 2012 Gradually Varied Flow 26

    S0 −Sf 1− Fr 2( ) =

    dy dx

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    Gradually Varied Flow ¢ This relation is derived for a rectangular

    channel, but it is also valid for channels of other constant cross sections provided that the Froude number is expressed accordingly.

    Friday, November 9, 2012 Gradually Varied Flow 27

    S0 −Sf 1− Fr 2( ) =

    dy dx

    Gradually Varied Flow ¢ An analytical or numerical solution of this

    differential equation gives the flow depth y as a function of x for a given set of parameters, and the function y(x) is the surface profile.

    Friday, November 9, 2012 Gradually Varied Flow 28

    S0 −Sf 1− Fr 2( ) =

    dy dx

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    Gradually Varied Flow ¢ The general trend of flow depth—whether it

    increases, decreases, or remains constant along the channel—depends on the sign of dy/dx, which depends on the signs of the numerator and the denominator.

    Friday, November 9, 2012 Gradually Varied Flow 29

    S0 −Sf 1− Fr 2( ) =

    dy dx

    Gradually Varied Flow ¢ The Froude number is always positive and

    so is the friction slope Sf .

    Friday, November 9, 2012 Gradually Varied Flow 30

    S0 −Sf 1− Fr 2( ) =

    dy dx

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    Gradually Varied Flow ¢ The bottom slope S0 is positive for down-

    ward-sloping sections, zero for horizontal sections, and negative for upward-sloping sections of a channe

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