CIVE 530 - OPEN-CHANNEL HYDRAULICS

LECTURE 5: UNIFORM FLOW

5.1  QUALIFICATIONS FOR UNIFORM FLOW


  • In uniform flow, the depth, area, velocity and discharge are constant.

  • In uniform flow, the slopes are all the same: friction, energy, water-surface, bottom.

    Sf = Se = Sw = So = S

  • There is no unsteady uniform flow; if the flow is unsteady, it is not uniform.

  • At high velocities, the flow becomes unstable and unsteady; this is the instability of uniform flow discussed by Chow.

  • Instability of the free surface leads to roll waves.

  • The Vedernikov number V determines the instability.

  • At V < 1, uniform (or equilibrium in natural channels) flow can occur.
5.2  ESTABLISHMENT OF UNIFORM FLOW


  • In uniform flow, gravitational forces are entirely balanced by frictional resistance.

  • "Nature likes uniform flow."

  • The depth of uniform flow is called "normal depth."

  • When yn > yc, this is normal subcritical flow.

  • When yn < yc, this is normal supercritical flow.


    Fig. 5-1 (Chow)

5.3  VELOCITY OF UNIFORM FLOW


  • The velocity of uniform flow is:

    v = C Rx Sy

  • C is a roughness or friction coefficient.

  • Values of x and y vary with roughness and cross-sectional shape.
5.4  CHEZY FORMULA


  • According to Chezy, the velocity of uniform flow is:

    v = C R1/2 S1/2

    where C = Chezy C.

  • The shear stress τb developed along the bottom is proportional to the square of the mean velocity v (this is the "quadratic resistance law"):

    τb = ρ f v2

    where f is a resistance (or drag) coefficient.


    Fig. 5-2 (Chow)

  • The shear force along the wetted perimeter P of a control volume of length L is:

    Fs = τb P L = ρ f v2 P L

  • The body force (weight) resolved along the direction of motion is Wsinθ.

  • Assuming sinθ &asymp tan θ = S:

    W sin θ = γ A L S

  • Therefore:

    W sin θ = Fs

    γ A L S = ρ f v2 P L

    γ A S = ρ f v2 P

    g R S = f v2

    v = (g/f)1/2 (RS)1/2

    C = (g/f)1/2

    f = g/C2

    S = f v2/(gR)

    S = f (D/R) [v2/(gD)]

  • For D ≈ R (hydraulically wide channel):

    S = f [v2/(gD)]

    S = f F2

5.6  MANNING FORMULA

  • According to Manning, the velocity of uniform flow in SI units is:

    v = (1/n) R2/3 S1/2

    where n = Manning's n.

  • The Manning formula was developed originally in SI units, and later converted to U.S. customary units.

  • The Manning formula in U.S. customary units is:

    v = (1.486/n) R2/3 S1/2

  • The constant 1.486 comes from the unit conversion, and it is equal to 1/(0.3048)1/3 = 1.4859

  • To compare with Chezy, the Manning formula is expressed as follows:

    v = (1/n) R1/6 R1/2 S1/2

  • Therefore:

    C = (1/n) R1/6

  • In the Manning formula, C is not a constant, but a function of R.

  • This implies that while C is not a constant, n is a constant.

  • However, n has been shown to vary in natural channels, with stage and discharge.

  • In natural rivers, n varies as a function of the flow regime: lower, transitional, or upper regime.

  • Lower regime has ripples and dunes, which are absent in upper regime.

  • Lower regime has skin and form friction; upper regime is mostly skin friction.
5.7  DETERMINATION OF MANNING ROUGHNESS

  • There is no exact method for selecting Manning roughness.

  • Recommendations:

    • Understand the factors that affect the value of n.

    • Consult a table of typical values.

      1. Chow (B&W photos): minimum n = 0.008; maximum n = 0.250.

      2. Barnes (color photos): minimum n = 0.024; maximum n = 0.075.

      3. manningsn.sdsu.edu: minimum n = 0.024; maximum n = 0.075.

      4. manningsn2.sdsu.edu: minimum n = 0.1; maximum n = 0.2.

    • Become acquainted with the appearance of typical channels whose n values are known.


      Ash Creek, New Harmony, Utah


      Rachichuela Wash, Lambayeque, Peru

    • Use theoretical velocity distributions.

    • Measure roughness directly (expensive, and stage-dependent).


      Streamgaging station at Campo Creek, San Diego County.

5.8  FACTORS AFFECTING MANNING'S ROUGHNESS COEFFICIENT


  • The value of n is highly variable and depends on a number of factors.

    • A. Surface rougness

        Fine grains result in a generally low value of n; coarse grains in a high value of n.

    • B. Vegetation

        Vegetation retards the flow.

        Maintenance of the channel determines the value of n.

        Small depths require higher n.

        Minimum n = 0.04 for drainage ditches cleared annually.

        Use n = 0.05 if clearance is every two years.

        Values greater than n = 0.1 may be warranted when channels are not cleared for several years.

        Values up to n = 1 may be warranted for flow over vegetation.

          

      Santa Cruz river, near Tucson, Arizona, 1942 and 1989.

    • C. Channel irregularities

        Presence of sand bars, sand waves, ridges, depressions, and holes and humps in the channel bed.

        The increase in n may be 0.005 or more.


      Tocmoche Canyon, Cajamarca, Peru.

    • D. Channel alignment

        Smooth curvature with large radius will give a relatively low value of n.

        Meandering increases n by about 30%.


      Meta river, Meta, Colombia.

    • E. Silting and scouring

        Silting decreases n; scouring increases n.

        Uneven deposits such as sand bars and sand waves are channel irregularities and will increase the roughness.


      Pirai river, Santa Cruz de la Sierra, Bolivia.

    • F. Obstructions

        The presence of log jams and bridge piers tend to increase n.


      Santo Domingo river, Pernambuco,Brazil.

    • G. Size and shape of channel

        An increase in hydraulic radius may increase or decrease n, depending on the condition of the channel. There is no correlation between n and size and shape of channel.


      Paraguay river at Porto Murtinho, Mato Grosso do Sul, Brazil.

    • H. Stage and discharge

        The value of n varies with stage and discharge (See Ponce, page 274); first it decreases, then it increases, then it decreases again, as stages go from low flow to bankfull flow to overbank flow to very high flood stage.

        n is a minimum at or somewhat above bankfull stage (This depends on the cross section).

    • I. Seasonal change

        n increases in the growing season and decreases in the dormant season.

    • J. Suspended material and bedload

        Suspended material and bedload consume energy and cause an increase in the apparent channel roughness.


      Debris flow, foot of the Wasatch Mountains, Utah.

  • Procedure for estimating the value of n:

    n = (n0 + n1 + n2 + n3 + n4) m5

  • n0 = basic n value for a straight, uniform, smooth channel

  • n1 = value added to account for surface irregularities

  • n2 = value added to account for variations in the shape and size of the cross section

  • n3 = value added to account for obstructions

  • n4 = value added to account for vegetation and flow conditions

  • m5 = correction factor for channel meandering.


    Table 5-5 (Chow)

 

Go to Chapter 6.

 
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