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MECH 221 FLUID MECHANICS (Fall 06/07) Chapter 10: OPEN CHANNEL FLOWS

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... Varies Flow 10.4 Gradually Varies Flow 10.5 Hydraulic Jump 10.5 Hydraulic Jump 10.5.1 Depth Change Across a Hydraulic Jump 10.5.2 Head Loss Across a ... – PowerPoint PPT presentation

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Title: MECH 221 FLUID MECHANICS (Fall 06/07) Chapter 10: OPEN CHANNEL FLOWS


1
MECH 221 FLUID MECHANICS(Fall 06/07)Chapter 10
OPEN CHANNEL FLOWS
  • Instructor Professor C. T. HSU

2
10.1 General Concept of Flows in Open Channel
  • Open channel flows are flows with free surface
    that have many applications
  • Rivers, Streams, Aqueducts, Canals, Sewers,
    Irrigation
  • Water Channels
  • Pipe Flow vs. Channel Flow

Pipe Flow Open Channel Flow
Closed with solid boundary Open with free surface
Fixed crossed-section Variable depth
Driven by pressure gradient Driven by gravity
Mostly circular All kind of shapes
3
10.1 General Concept of Flows in Open Channel
  • Hydraulic radius
  • Hydraulic depth

4
10.1 General Concept of Flows in Open Channel
  • Based on the property of yh, the open channel
    flow can be classified as
  • a) Constant depth
  • 1-D model can be used
  • b) Gradually varying
  • Depth changes slowly so that 1-D model remains as
    a good approximation
  • c) Rapidly varying
  • Need a 2-D model to treat the problem

5
10.1 General Concept of Flows in Open Channel
  • For most practical cases involving large and deep
  • channels, the Reynolds number, is
    high
  • Hence, most open channel flows are turbulent
  • With a free surface for open channel flow,
    gravity is important. The important parameter is
    the Froude number

6
10.1 General Concept of Flows in Open Channel
  • The open channel flows as classified by Froude
    number Fr are

7
10.2 Propagation of Surface Waves
  • If the free surface initially calm is perturbed
    by a vertical displacement , there will be an
    associated velocity perturbation in fluid.
  • If the water depth is small compared with the
    length scale of the displacement, the
    displacement perturbation will have a fixed form
    that propagates with a velocity C, which is
    called phase velocity.
  • We now consider the flow in a frame moving with
    the phase velocity. Then, in the moving frame the
    shape of the displacement is fixed and steady

8
10.2 Propagation of Surface Waves
9
10.2 Propagation of Surface Waves
  • For small displacement disturbances,
  • Hence, the displacement can propagate upstream
    and downstream with a speed equal to
  • If the fluid moves at a velocity U, then the
    Froude number,

10
10.2 Propagation of Surface Waves
  • This phenomenon can easily be demonstrated by a
    boat moving at a constant speed U on an initial
    clam water where the disturbance are generated by
    the vertical oscillating of the boat.
  • The wave patterns behind the boat for Frgt 1 are
    called the ship wakes

11
10.3.1 Friction Loss
  • Bernoullis with friction loss,
  • Since the flow is uniform,
  • For free surface, we also have P1-P2 since the
    fluid depths are the same. Therefore,
  • where Sb is the hydraulic slope

12
10.3.1 Friction Loss
  • Similar to pipe flow,
  • so,
  • Empirical values of C were determined by Manning,
    who suggested that

13
10.3.1 Friction Loss
  • Now,
  • For a rectangular channel,

14
10.3.1 Specific Energy
  • Define specific energy, E, at a single section in
    the channel as,
  • Let q be volume flow rate per unit width, Qqb,
    so for rectangular channel

15
10.3.1 Specific Energy
  • The variation of depth as a function of specific
    energy for a given flow rate are summaries in the
    specific energy diagram
  • For a given flow rate (Qgt0) and specific energy,
    there are 2 possible values of depth, y. These
    are called alternate depths.

Frlt1
Ey
Frgt1
16
10.3.1 Specific Energy
  • For any value of E, the horizontal distance from
    the vertical axis to the line, yE, gives the
    depth
  • And the distance from the line, yE, to the Q
    curve is then equal to the K.E., U2/2g.
  • For each curve representing a given flow rate,
    there is a value of depth that gives a minimum E.
    This depth is the critical depth obtained by

17
10.3.1 Specific Energy
  • The velocity at the critical condition
  • Since , we have
  • Continuity
  • Then

18
10.3.1 Specific Energy
  • For non-rectangular channels, the channel depth
    varies across the width. At the minimum specific
    energy,

19
10.3.1 Specific Energy
  • Consider,

20
10.4 Gradually Varies Flow
  • The energy equation for the differential C.V. is
  • For a rectangular channel UQ/by, so

21
10.4 Gradually Varies Flow
  • Since dz-sbdx and similarly, we can define dhL
    sfdx, the energy equation now becomes,
  • For flow at normal depth,
  • The sign of the slope of the water surface
    profile depends on whether the flow is
    subcritical or supercritical, and on sf and sb

22
10.4 Gradually Varies Flow
  • To calculate the surface profile, rewrite the
    equation as
  • As dE/dx sf sb, and in finite difference form,
  • m denotes the mean properties over a channel
    length ,
  • sf can be obtained from the Manning
    correlation, since sb for flow at normal depth
    equal sf

23
10.5 Hydraulic Jump
  • For subcritical flow, disturbances cause by a
    change in bed slope or flow cross section may
    move upstream and downstream. The result is a
    smooth adjustment of the flow
  • However, when the flow is supercritical,
    disturbances cannot be transmitted upstream.
    Thus, a gradual change is not possible.
  • The transition from the supercritical to
    subcritical flow occurs abruptly through the
    hydraulic jump

24
10.5 Hydraulic Jump
  • The abrupt change in depth involves a significant
    loss of mechanical energy through turbulent
    mixing
  • The extent of a hydraulic jump is short, so
    friction is negligible. Assuming horizontal
    surface, gravitational effect of bottom elevation
    can be neglected

25
10.5.1 Depth Change Across a Hydraulic Jump
  • Eliminate U2 from the momentum equation by using
    the continuity equation to get

26
10.5.2 Head Loss Across a Hydraulic Jump
  • Head loss through a jump is just the difference
    in specific energy,

27
10.6 Flow Over a Bump
  • Consider frictionless flow in a horizontal
    rectangular channel of constant width, b, with a
    bump of height, h(x). The flow is assumed
    uniform.
  • Since the flow is steady, incompressible and
    frictionless, applying Bernoullis equation along
    the free surface gives

28
10.6 Flow Over a Bump
  • Along the free surface, p1ppatm, thus

29
10.6 Flow Over a Bump
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