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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.

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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.

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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.

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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.

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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

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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

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Gradually Varied Flow ¢ Differentiating H with respect to x gives

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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.

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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

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dH dx

= d dx

zb + y + v 2

2g ⎛ ⎝⎜

⎞ ⎠⎟

dH dx

= dzb dx

+ dy dx

+ v g

dv dx

Gradually Varied Flow ¢ Therefore

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dH dx

= dzb dx

+ dy dx

+ v g

dv dx

dH dx

= −Sf

dzb dx

= −S0

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Gradually Varied Flow ¢ Substituting

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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.

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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

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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

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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

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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

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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.

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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.

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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.

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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.

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S0 −Sf 1− Fr 2( ) =

dy dx

Gradually Varied Flow ¢ The Froude number is always positive and

so is the friction slope Sf .

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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