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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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¢  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).



¢ 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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¢  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.



¢  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.


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


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


dHdx

= ddx

zb + y + v 2

2g⎛⎝⎜

⎞⎠⎟

dHdx

=dzb

dx+ dy

dx+ v

gdvdx

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.


dHdx

= ddx

zb + y + v 2

2g⎛⎝⎜

⎞⎠⎟

dHdx

=dzb

dx+ dy

dx+ v

gdvdx

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


dHdx

= ddx

zb + y + v 2

2g⎛⎝⎜

⎞⎠⎟

dHdx

=dzb

dx+ dy

dx+ v

gdvdx

Gradually Varied Flow ¢ Therefore


dHdx

=dzb

dx+ dy

dx+ v

gdvdx

dHdx

= −Sf

dzb

dx= −S0

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Gradually Varied Flow ¢ Substituting


dHdx

=dzb

dx+ dy

dx+ v

gdvdx

−Sf = −S0 +dydx

+ vg

dvdx

S0 −Sf =dydx

+ vg

dvdx

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.


dHdx

=dzb

dx+ dy

dx+ v

gdvdx

−Sf = −S0 +dydx

+ vg

dvdx

S0 −Sf =dydx

+ vg

dvdx

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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.


S0 −Sf =dydx

+ vg

dvdx

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.


S0 −Sf =dydx

+ vg

dvdx

Q = vA = vbydQdx

= ddx

vby( ) = bydvdx

+ bvdydx

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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


S0 −Sf =dydx

+ vg

dvdx

Q = vA = vbydQdx

= ddx

vby( ) = bydvdx

+ bvdydx

Gradually Varied Flow ¢ Substituting


S0 −Sf =dydx

+ vg

dvdx

dQdx

= bydvdx

+ bvdydx

0 = bydvdx

+ bvdydx

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Gradually Varied Flow ¢ Solving for dv/dx


S0 −Sf =dydx

+ vg

dvdx

−bydvdx

= +bvdydx

dvdx

= − vy

dydx

Gradually Varied Flow ¢ Substituting into the original expression


S0 −Sf =dydx

+ vg

dvdx

dvdx

= − vy

dydx

S0 −Sf =dydx

+ vg

− vy

dydx

⎛⎝⎜

⎞⎠⎟

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Gradually Varied Flow ¢ Collecting terms


S0 −Sf =dydx

+ vg

dvdx

S0 −Sf =dydx

− v 2

gydydx

S0 −Sf = 1− v 2

gy⎛⎝⎜

⎞⎠⎟

dydx

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

recognized as the square of the Froude number of the flow


S0 −Sf = 1− v 2

gy⎛⎝⎜

⎞⎠⎟

dydx

S0 −Sf = 1− Fr 2( )dydx

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Gradually Varied Flow ¢  Isolating the differential term we have


S0 −Sf

1− Fr 2( ) =dydx

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

surface profile as you move down the channel.


S0 −Sf

1− Fr 2( ) =dydx

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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.


S0 −Sf

1− Fr 2( ) =dydx

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.


S0 −Sf

1− Fr 2( ) =dydx

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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.


S0 −Sf

1− Fr 2( ) =dydx

Gradually Varied Flow ¢ The Froude number is always positive and

so is the friction slope Sf .


S0 −Sf

1− Fr 2( ) =dydx

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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 channel (adverse flow).


S0 −Sf

1− Fr 2( ) =dydx

Gradually Varied Flow ¢ The flow depth increases when dy/dx > 0,

decreases when dy/dx < 0, and remains constant (and thus the free surface is parallel to the channel bottom, as in uniform flow) when dy/dx = 0 and thus S0 = Sf.


S0 −Sf

1− Fr 2( ) =dydx

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Gradually Varied Flow ¢ For specified values of S0 and Sf, the term

dy/dx may be positive or negative, depending on whether the Froude number is less than or greater than 1.


S0 −Sf

1− Fr 2( ) =dydx

Gradually Varied Flow ¢ Therefore, the flow behavior is opposite in

subcritical and supercritical flows. For S0 - Sf > 0, for example, the flow depth increases in the flow direction in subcritical flow, but it decreases in supercritical flow.


S0 −Sf

1− Fr 2( ) =dydx

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Gradually Varied Flow ¢ Open-channel systems are designed and

built on the basis of the projected flow depths along the channel.

¢ Therefore, it is important to be able to predict the flow depth for a specified flow rate and specified channel geometry.


S0 −Sf

1− Fr 2( ) =dydx

Gradually Varied Flow ¢ A plot of flow depths gives the surface

profile of the flow. ¢ The general characteristics of surface

profiles for gradually varied flow depend on the bottom slope and flow depth relative to the critical and normal depths.


S0 −Sf

1− Fr 2( ) =dydx

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Gradually Varied Flow ¢ A typical open channel involves various

sections of different bottom slopes S0 and different flow regimes, and thus various sections of different surface profiles.


S0 −Sf

1− Fr 2( ) =dydx

Gradually Varied Flow ¢ For example, the general shape of the

surface profile in a downward-sloping section of a channel is different than that in an upward-sloping section.


S0 −Sf

1− Fr 2( ) =dydx

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Gradually Varied Flow ¢  Likewise, the profile in subcritical flow is

different than the profile in supercritical flow. ¢ Unlike uniform flow that does not involve

inertia forces, gradually varied flow involves acceleration and deceleration of liquid, and the surface profile reflects the dynamic balance between liquid weight, shear force, and inertial effects.


S0 −Sf

1− Fr 2( ) =dydx

Gradually Varied Flow ¢ Each surface profile is identified by a letter

that indicates the slope of the channel and by a number that indicates flow depth relative to the critical depth yc and normal depth yn.


S0 −Sf

1− Fr 2( ) =dydx

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Gradually Varied Flow ¢ The slope of the channel can be mild (M),

critical (C), steep (S), horizontal (H), or adverse (A).

¢ The channel slope is said to be mild if yn > yc, steep if yn < yc, critical if yn = yc, horizontal if S0 = 0 (zero bottom slope), and adverse if S0 < 0 (negative slope).


S0 −Sf

1− Fr 2( ) =dydx


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Gradually Varied Flow ¢ The classification of a channel section

depends on the flow rate and the channel cross section as well as the slope of the channel bottom.


S0 −Sf

1− Fr 2( ) =dydx

Gradually Varied Flow ¢ A channel section that is classified to have a

mild slope for one flow can have a steep slope for another flow, and even a critical slope for a third flow.

¢ Therefore, we need to calculate the critical depth yc and the normal depth yn before we can assess the slope.


S0 −Sf

1− Fr 2( ) =dydx

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Gradually Varied Flow ¢ The number

designation indicates the initial position of the liquid surface for a given channel slope relative to the surface levels in critical and uniform flows.


S0 −Sf

1− Fr 2( ) =dydx


¢ A surface profile is designated by 1 if the flow depth is above both critical and normal depths (y > yc and y > yn).


S0 −Sf

1− Fr 2( ) =dydx

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Gradually Varied Flow ¢  2 if the flow depth is

between the two (yn >y>yc or yn<y<yc).


S0 −Sf

1− Fr 2( ) =dydx

Gradually Varied Flow ¢ And 3 if the flow depth

is below both the critical and normal depths (y < yc and y < yn).


S0 −Sf

1− Fr 2( ) =dydx

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Gradually Varied Flow ¢ Therefore, three

different profiles are possible for a specified type of channel slope.


S0 −Sf

1− Fr 2( ) =dydx

Gradually Varied Flow ¢ For channels with zero

or adverse slopes, type 1 flow cannot exist since the flow can never be uniform in horizontal and upward channels, and thus normal depth is not defined.


S0 −Sf

1− Fr 2( ) =dydx

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Gradually Varied Flow ¢ Also, type 2 flow does

not exist for channels with critical slope since normal and critical depths are identical in this case.


S0 −Sf

1− Fr 2( ) =dydx

Gradually Varied Flow ¢ The five classes of

slopes and the three types of initial positions discussed give a total of 12 distinct configurations for surface profiles in GVF.


S0 −Sf

1− Fr 2( ) =dydx

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S0 −Sf

1− Fr 2( ) =dydx



S0 −Sf

1− Fr 2( ) =dydx

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S0 −Sf

1− Fr 2( ) =dydx



S0 −Sf

1− Fr 2( ) =dydx

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S0 −Sf

1− Fr 2( ) =dydx

Gradually Varied Flow ¢ The prediction of the surface profile y(x) is

an important part of the design of open-channel systems.


S0 −Sf

1− Fr 2( ) =dydx

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Gradually Varied Flow ¢ A good starting point for the determination of

the surface profile is the identification of the points along the channel, called the control points, at which the flow depth can be calculated from a knowledge of flow rate.


S0 −Sf

1− Fr 2( ) =dydx

Gradually Varied Flow ¢ For example, the flow depth at a section of a

rectangular channel where critical flow occurs, called the critical point, can be determined from yc = (Q2/gb2)1/3.

¢ The normal depth yn, which is the flow depth reached when uniform flow is established, also serves as a control point.


S0 −Sf

1− Fr 2( ) =dydx

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Gradually Varied Flow ¢ Once flow depths at control points are

available, the surface profile upstream or downstream can be determined usually by numerical integration of the nonlinear differential equation.


S0 −Sf

1− Fr 2( ) =dydx


Homework 27-1

¢ Water flows uniformly in a rectangular channel with finished-concrete surfaces. The channel width is 3 m, the flow depth is 1.2 m, and the bottom slope is 0.002.

¢ Determine if the channel should be classified as mild, critical, or steep for this flow.

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Homework 27-2

¢ Consider uniform water flow in a wide brick channel of slope 0.4°.

¢ Determine the range of flow depth for which the channel is classified as being steep.


Homework 27-3

¢ Consider the flow of water through a 12-ft-wide unfinished-concrete rectangular channel with a bottom slope of 0.5°.

¢  If the flow rate is 300 ft3/s, determine if the slope of this channel is mild, critical, or steep.

¢ Also, for a flow depth of 3 ft, classify the surface profile while the flow develops.

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