CIVE 530 - OPEN-CHANNEL HYDRAULICS

LECTURE 8A: GRADUALLY VARIED FLOW I

8.1  BASIC ASSUMPTIONS

  • The basic assumptions are:

    • The flow is steady (not unsteady)

    • The streamlines are essentially parallel (not with curvature); therefore, hydrostatic pressure distribution prevails over the channel section.

    • The head loss is the same as that corresponding to uniform flow; therefore, the uniform flow formula can be used to evaluate the energy slope.

    • The Manning's n roughness is the same as that of uniform flow (errors are believed to be small).

  • Other assumptions are:

    • The slope of the channel is small.

    • The pressure correction factor cos θ ≅ 1.

    • There is no air entrainment.

    • The channel is prismatic, i.e., of constant shape and cross section.

    • The velocity distribution is fixed; the velocity-distribution coefficients are constant.

    • The conveyance is an exponential function of the flow depth (not applicable in circular culverts).

    • The roughness is independent of the flow depth (only an approximation) and constant throughout the reach being considered.
8.2  DYNAMIC EQUATION FOR GRADUALLY VARIED FLOW

  • The gradient of hydraulic head in an open channel is:

    dH/dx = d [z + y + V2/(2g)] /dx = - Sf

  • The negative sign in front of the friction slope is because the flow is from left to right, and the derivative is taken from right to left by convention.

  • By definition, the friction slope is:

    Sf = hf / L

    where hf is the head loss due to friction and L is the length of the channel reach.

  • The gradient of specific energy is:

    d [y + V2/(2g)] /dx = - (dz/dx) - Sf

  • The gradient of the bed, or bed slope, is:

    dz/dx = (z2 - z1) / L

    - dz/dx = (z1 - z2) / L = So

  • Therefore, the gradient of specific energy is:

    d [y + V2/(2g)] /dx = So - Sf

  • Under steady flow, Q = VA = constant. Therefore:

    d [y + Q2/(2gA2)] /dx = So - Sf

  • Operating:

    dy/dx + d [Q2/(2gA2)] /dx = So - Sf

    dy/dx - [Q2/(gA3)] (dA/dx)= So - Sf

    dy/dx - [Q2/(gA3)] (dA/dy) (dy/dx)= So - Sf

  • But:

    dA/dy = T

  • Therefore, the flow depth gradient is:

    dy/dx { 1 - [(Q2T) / (gA3)]} = So - Sf

    dy/dx = (So - Sf) / { 1 - [(Q2T) / (gA3)]}

  • The friction slope following Chezy is:

    Sf = V2 / (C2R)

    Sf = Q2 / (C2A2R)

    R = A / P

    Sf = (Q2P) / (C2A3)

  • Therefore, the flow depth gradient is:

    dy/dx = {So - [(Q2P) / (C2A3)]} / { 1 - [(Q2T) / (gA3)]}

  • Multiplying and dividing by g and T, and rearranging C:

    dy/dx = {So - [(g/C2) (P/T) [(Q2T) / (gA3)]} / { 1 - [(Q2T) / (gA3)]}

  • The square of the Froude number is:

    F2 = V2/(gD) = Q2/(gA2D) = (Q2T) / (gA3)

  • Therefore, the flow depth gradient is:

    dy/dx = [So - (g/C2) (P/T) F2] / (1 - F2)

  • Therefore, the depth gradient dy/dx will be a function of:

    • So (channel slope),

    • g/C2 (friction coefficient),

    • P/T (shape of the cross section: narrow or hydraulically wide), and

    • F (Froude number).

  • For uniform flow, dy/dx = 0; then:

    So = (g/C2) (P/T) F2

  • The quantity g/C2 is a dimensionless Chezy friction factor, herein referred to as f (f is 1/8 of the Darcy-Weisbach fD):

    So = f (P/T) F2

  • For critical flow, F = 1; then:

    So = f (Pc/Tc) = Sc

  • Therefore, f (Pc/Tc) is the channel slope for which the flow is critical, or the critical slope Sc.

  • In terms of Sc, the flow depth gradient is:

    dy/dx = [So - (P/T) (Tc/Pc) Sc F2] / (1 - F2)

  • For a hydraulically wide channel, P ≅ T, and, Tc ≅ Pc; therefore, the flow depth gradient is:

    dy/dx = (So - Sc F2) / (1 - F2)

  • Or, better yet, the following relation is approximately satisfied:

    (P/T) (Tc/Pc) ≅ 1

    (P/T) ≅ (Pc/Tc)

  • Therefore, the flow depth gradient for a hydraulically wide channel reduces to:

    dy/dx = (So - Sc F2) / (1 - F2)

  • Now call the flow depth gradient:

    dy/dx = Sy

  • Then, the gradually varied flow equation [for hydraulically wide channels] is:

    Sy/Sc = [(So/Sc) - F2] / (1 - F2)

Go to Chapter 8B.

 
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