CIVE 530 - OPEN-CHANNEL HYDRAULICS

LECTURE 2:   PROPERTIES OF OPEN CHANNELS

2.1  KINDS OF OPEN CHANNELS

  • There are two kinds of channels:

    1. Artificial (prismatic)

    2. Natural (non-prismatic)


      Tinajones feeder canal, Chiclayo, Peru


      Rio La Silla Natural Park, Monterrey, Mexico.

  • Other water-resources-related fields:

    • Hydrology: the study of water in the hydrologic cycle.

    • Hydroclimatology: the study of climate and the hydrologic cycle.

    • Fluvial geomorphology: the study of the shape of streams and rivers.

    • River mechanics: the study of the mechanical properties and behavior or rivers.

    • Sedimentology: the study of sediment.

    • Potamology: the study of rivers.


      Lower Mississippi river near Baton Rouge, Louisiana.

  • Uses of artificial channels:

    • Navigation channels

    • Water conveyance channels (Example: the Dulzura conduit, which links Barrett reservoir with drainage to Lower Otay reservoir, in San Diego County)


      The Dulzura conduit, San Diego County, California.

    • Power canals

    • Irrigation canals (Example: the All-American canal, in the Imperial valley)


      The All-American canal, Imperial County, California.


      Location of the All-American canal, Imperial County, California.


      Location of the All-American canal, Imperial County, California.


      Indus Basin Link Canals, Pakistan.

    • Flood control channels, floodways.


      Rio Santa Catarina, Monterrey, Mexico.

    • Drainage ditches (drainage ditches in Imperial valley, draining to the Salton Sea).


      Imperial Valley irrigation drain.

  • Names for channels:

    • Canal: long, mild-sloped, lined/unlined, ground-supported, masonry/concrete/wood/asphalt.

    • Flume: supported above ground (lab flume), wood/metal/concrete/masonry.

    • Chute: channel with steep slope, usually supercritical.

    • Drop: Chute with very short distance.

    • Culvert: covered channel of comparatively short length, flowing partially full.


      Drainage drop structure in Tijuana, Baja California.


      Tinajones drop structure, Chiclayo, Peru.


      Chute at Taymi Canal, Chiclayo, Peru.


      Canal with falls, La Joya, Arequipa, Peru.


      Junction of main irrigation canal with main drain, La Joya, Arequipa, Peru.


      Crossing of Tinajones Feeder Canal with Chiriquipe Wash, Chiclayo, Peru.


      Crossing of Arroyo Rosa de Castilla with Mexico Highway 2, Tecate, Baja, California.

    2.2  CHANNEL GEOMETRY


    • A prismatic channel has a constant cross section and constant bottom slope.

    • The channel section is the cross section normal to the direction of flow.

    • A trapezoid is the most common cross-sectional shape. It provides bank stability.


      http://onlinechannel.sdsu.edu

    • A rectangular section is used in laboratory flumes.

    • Channels built of stable materials (rock, concrete) can be rectangular.

    • Triangular channels are used in ditches, roadside gutters.

2.3  GEOMETRIC ELEMENTS


  • Depth of flow y:   vertical distance from the free surface to the lowest point in channel cross section.

  • Depth of flow section d:   height of channel cross section; depth normal to direction of flow.

  • For a channel with a longitudinal slope angle θ:

    cos θ = d/y

    y = d / cos θ


    Fig. 2-9 (Chow)

  • Stage y:   elevation of the free surface.

  • Top width T:   width of channel cross section at free surface.

  • Flow area A:   area of flow in channel cross section.

  • Wetted perimeter P:   important in determining friction.

  • Hydraulic radius R:   ratio A/P.

  • Hydraulic depth D:   ratio A/T.

  • For hydraulically wide channels:   T ≈ P

  • For hydraulically wide channels:   D ≈ R

2.4  VELOCITY DISTRIBUTION IN A CHANNEL SECTION


  • In a channel section, the velocities near the surface and near the bottom differ.

  • Velocities near the boundary are close to zero (no-slip condition).

  • Large velocity gradients near the boundary produce large shear stresses (which entrain and transport sediment).

  • The maximum velocity occurs near the surface, at a distance of 0.05 to 0.25 of the flow depth.

  • Velocities also vary transversally along horizontal bends; they are larger on the outside of the bend.

  • Channel roughness will cause the curvature of the vertical velocity profile to increase.

2.5  WIDE OPEN CHANNEL


  • In a wide open channel, the sides have no influence on the velocity profile.

  • The flow has a tendency to be 2-D instead of 3-D.

  • Ratio T/D > 10 will assure wide-channel condition.

  • Often a hydraulic analysis is carried out per unit of channel width.

  • In a rectangular channel:

    Q = v A = v d B = v d T

    q = Q / B = v d

2.6  MEASUREMENTS OF VELOCITY


  • Measurements are taken with a current meter positioned at 0.6 d, measured from the surface.

  • Also, at 0.2 d and 0.8 d, and then find the average of these two values.


    Price AA current meter (Courtesy of the U.S. Geological Survey).


    USGS gaging stating at Campo Creek, San Diego County, California.

2.7  VELOCITY DISTRIBUTION COEFFICIENTS


  • Due to nonuniform distribution of velocities over a cross section, the true velocity head is usually greater than the value computed based on the mean (average) velocity.

  • The true velocity head is:

    hv = α [Vm2/(2g)]

  • α is the energy coefficient or Coriolis coefficient.

  • The value of α is typically in the range 1.03-1.36 for fairly straight prismatic channels.

  • The value is greater for small channels, and smaller for large channels.

  • The true momentum flux is:

    F = β ρ Q Vm

  • β is the momentum coefficient or Boussinesq coefficient.

  • The value of β is typically in the range 1.01-1.12 for fairly straight prismatic channels.

  • The values of α and β are slightly greater than 1.

  • α is always greater than β.

  • In channels of complex cross section, the values of α and β can easily get to be 1.6 and 1.2, respectively.

  • Values of α greater than 2 have been observed in very irregular cross sections.

2.8  CALCULATION OF VELOCITY DISTRIBUTION COEFFICIENTS


  • The true velocity head is usually greater than the value computed based on the average velocity.

  • Assume:

    • A = total area of the cross section [L2]

    • ΔA = incremental area [L2]

    • Vm = mean velocity of the cross section [L T-1]

    • V = velocity through ΔA [L T-1]

  • The weight flux through ΔA is:

    γ V ΔA

    		   [F T-1]

  • The weight flux through A is:

    γ Vm A

    		   [F T-1]

  • Kinetic energy = force × distance = (mass × acceleration) × distance = (1/2) mV2    [M L2 T-2]

  • Momentum = force × time = (mass × acceleration) × time = mV    [M L T-1]  

     

  • Velocity head = kinetic energy per unit of weight

  • Velocity head = [(1/2) m V2] / (mg) = V2/(2g)        [L]

  • Kinetic energy flux [through incremental area ΔA] =

    kinetic energy per unit of weight × weight flux =

    [V2/(2g)] [γ V ΔA]= γ V3 ΔA /(2g)

  • For all the increments of area ΔA:

    ∑ γ V3 ΔA /(2g)

  • Kinetic energy flux [through total area A] =

    kinetic energy per unit of weight × weight flux =

    [αVm2/(2g)] [γ Vm A] = α γ Vm3 A /(2g)

  • Therefore:

    ∑ V3 ΔA = α Vm3 A

    α = ∑ V3 ΔA / (Vm3 A)

  • Momentum β coefficient:

  • The mass flux through ΔA is:

    ρ V ΔA

    		   [M T-1]

  • The mass flux through A is:

    ρ Vm A

    		   [M T-1]

  • Momentum = force × time = mass × velocity = m V    [M L T-1]

  • Momentum flux = mass flux × velocity    [F]

  • The momentum flux through ΔA is:

    ρ V2 ΔA

    		   [F = M L T-2]

  • For all the increments of area ΔA:

    ∑ ρ V2 ΔA

  • The momentum flux through A is:

    β ρ Vm2 A

    		   [F = M L T-2]

  • Therefore:

    ∑ V2 ΔA = β Vm2 A

    β = ∑ V2 ΔA / (Vm2 A)

  • Energy flux = (weight flux) × (velocity head)    [(F/T) L = FL/T]

  • Momentum flux (force) = (mass flux) × (velocity)    [(M/T) (L/T) = M (L/T2)]

  • Note that the mean velocity is defined as:

    Vm = ∑ V ΔA / ∑ ΔA

  • For approximate values, α and β can be computed as follows:

    α = 1 + 3ε2 - 2ε3

    β = 1 + ε2

    with

    ε = (Vmax/Vm) - 1

 

2.9  PRESSURE DISTRIBUTION IN A CHANNEL CROSS SECTION


  • The pressure is measured by the height of the water column at any point in the vertical.

  • The pressure at any point is directly proportional to the depth of the point and equal to the hydrostatic pressure corresponding to this depth.

  • The distribution is linear, and is known as the hydrostatic law of pressure distribution.

  • This assumes no vertical accelerations.

  • This type of flow is known as parallel flow.

  • The streamlines have no substantial curvature.

  • Uniform flow is practically parallel flow.

  • Gradually varied flow may be regarded as parallel flow.

  • If the curvature is substantial, the flow is curvilinear flow.

  • In curvilinear flow, the pressure distribution is not hydrostatic.


    Fig. 2-7 (Chow)

  • The centrifugal pressure p is [mass (per unit of area) × centrifugal acceleration]:

    p = (γ/g) d (Vm2/r)

  • The pressure rise c is:

    c = p/γ = (d/g) (Vm2/r)

  • The rise is positive for concave flow, and negative for convex flow.

2.10  EFFECT OF SLOPE


  • With reference to a channel of slope θ, the weight of the shaded element of length dL is:

    γ d (1) dL = γy cos θ (1) dL

  • The pressure due to this weight is:

    p = γ h = γ d cos θ = γy cos2 θ

  • The pressure head is:

    h = d cos θ = y cos2 θ.

  • If θ is small, the factor cos2 θ will be very close to unity.

  • The correction is less than 1% when θ is less than 6o (slope of 1/10).


    Fig. 2-8 (Chow)

  • A channel with slope greater than 1/10 is called a channel of large slope.

  • If a channel of large slope has an appreciable curvature in the longitudinal vertical profile, the pressure head should be corrected for the curvature of the streamlines.

  • High flow velocities entrain air, producing a volume swell and depth increase.

  • The pressure obtained with y cos2 θ has been shown to be higher than the actual measured pressure obtained by model testing.

  • Spillway photo gallery.


    Fig. 2-9 (Chow)

    Emergency spillway at Sheep Creek Barrier Dam, Utah.
     

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