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PPT – MECH 221 FLUID MECHANICS (Fall 06/07) Chapter 10: OPEN CHANNEL FLOWS PowerPoint presentation | free to view - id: 4382d5-MjViZ

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MECH 221 FLUID MECHANICS (Fall 06/07) Chapter 10

OPEN CHANNEL FLOWS

- Instructor Professor C. T. HSU

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

10.1 General Concept of Flows in Open Channel

- Hydraulic radius
- Hydraulic depth

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

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

10.1 General Concept of Flows in Open Channel

- The open channel flows as classified by Froude

number Fr are

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

10.2 Propagation of Surface Waves

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,

10.2 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

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

10.3.1 Friction Loss

- Similar to pipe flow,
- so,
- Empirical values of C were determined by Manning,

who suggested that

10.3.1 Friction Loss

- Now,
- For a rectangular channel,

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

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

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

10.3.1 Specific Energy

- The velocity at the critical condition
- Since , we have
- Continuity
- Then

10.3.1 Specific Energy

- For non-rectangular channels, the channel depth

varies across the width. At the minimum specific

energy,

10.3.1 Specific Energy

- Consider,

10.4 Gradually Varies Flow

- The energy equation for the differential C.V. is
- For a rectangular channel UQ/by, so

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

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

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

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

10.5.1 Depth Change Across a Hydraulic Jump

- Eliminate U2 from the momentum equation by using

the continuity equation to get

10.5.2 Head Loss Across a Hydraulic Jump

- Head loss through a jump is just the difference

in specific energy,

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

10.6 Flow Over a Bump

- Along the free surface, p1ppatm, thus

10.6 Flow Over a Bump