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Federal Highway Administration Research and Technology
Coordinating, Developing, and Delivering Highway Transportation Innovations

Report
This report is an archived publication and may contain dated technical, contact, and link information
Publication Number: FHWA-RD-99-156
Date: August 2004

Enhanced Abutment Scour Studies for Compound Channels

LIST OF SYMBOLS

    a = blocked flow area by abutment

    A = unobstructed flow area in approach channel

    Af1 = blocked flow area in approach-channel floodplain

    b = bridge pier width

    Bf = width of floodplain in compound channel

    Bm = width of main channel in compound channel

    Bm1 = width of approach main channel

    Bm2 = width of main channel in contracted section

    Cn = units conversion coefficient used in equation 23

    C0 = best-fit constant in proposed scour formula

    Cr = best-fit coefficient in proposed scour formula

    CLB = live-bed scour coefficient

    d50 = median diameter of sediment

    ds = local scour depth

    dsc = theoretical long contraction scour depth

    dst = unsteady scour depth at any time t

    F0 = approach value of densimetric grain Froude number in Lim scour formula

    F1 = approach-flow Froude number

    Fab = Froude number adjacent to abutment face

    k-Epsilon = k-epsilon refers to a numerical modeling technique that uses the ratio of turbulent kinetic energy squared to turbulent energy dissipation rate to define the eddy viscosity term for computations

    kf = spiral-flow adjustment factor in Maryland scour formula

    ks = equivalent sand-grain roughness

    kv = velocity adjustment factor in Maryland scour formula

    K1 = geometric shape factor for abutment in Froehlich scour formula

    K2 = embankment skewness factor in Froehlich scour formula

    Ks* = abutment shape factor in Melville scour formula

    KG = channel geometric factor in Melville scour formula

    KI = flow-intensity factor in Melville scour formula

    KST = spill-through abutment shape factor for scour formula in present study

    KTheta* = abutment alignment factor in Melville scour formula

    La = abutment/embankment length

    M = discharge distribution factor in approach section = unblocked Q over total Q
        = discharge contraction ratio for a bankline abutment

    n = Manning's n

    nfp = Manning's n in floodplain of compound channel

    nmc = Manning's n in main channel of compound channel

    nK = Manning's n calculated from Keulegan's equation

    Ns = sediment number (equivalent to densimetric grain Froude number)

    Nsc = critical value of sediment number

    q1 = flow rate per unit width in approach section in Maryland scour formula

    q2 = flow rate per unit width in contracted section in Maryland scour formula

    qbv = volumetric sediment discharge per unit width

    qf0 = flow rate per unit width in floodplain at normal depth (unconstricted)

    qf0c = critical flow rate per unit width in floodplain at normal depth (unconstricted)

    qf1 = flow rate per unit width in the approach floodplain (constricted)

    qf2 = flow rate per unit width in the contracted floodplain

    qm1 = flow rate per unit width in the approach main channel at the beginning of scour

    qm2 = flow rate per unit width in the contracted section in the main channel

    Q = total discharge in compound channel

    Qf1 = discharge in the approach floodplain

    Qm0 = discharge in main channel for uniform flow in a compound channel

    Qm1 = discharge in the approach main channel

    Qm2 = discharge in the contracted main channel

    Qobst1 = obstructed discharge in the approach section

    R = channel hydraulic radius

    SG = specific gravity of sediment

    t = time since beginning of scour

    te = time to reach equilibrium scour depth

    u*1 = approach value of shear velocity = (Tau1/Rho)0.5

    u*c = critical value of shear velocity = (Tau1/Rho)0.5

    Vab = maximum resultant velocity near the upstream corner of the abutment face

    Vc = critical velocity for initiation of motion

    Vf0 = floodplain velocity at normal depth

    Vf0c = critical velocity for unconstricted floodplain depth of uniform flow

    Vf1 = bridge approach velocity in floodplain of compound channel

    Vf2 = floodplain velocity in the contracted floodplain after scour

    Vm0c = critical velocity for the unconstricted depth in the main channel

    Vm1 = bridge approach velocity in main channel of compound channel

    Vm2c = critical velocity in the main channel at the contracted section at equilibrium scour

    Vx = mean contraction velocity in GKY scour formula

    VR = resultant velocity adjacent to tip of abutment in GKY scour formula

    V1 = bridge approach velocity in rectangular channel

    y = generic symbol for depth of flow in main channel or floodplain for calculating critical velocity

    yab = depth near the upstream corner of the abutment face

    yf0 = normal depth in floodplain

    yf1 = bridge approach depth in floodplain of compound channel, including backwater

    yftw = tailwater depth in floodplain

    ym0 = normal depth in main channel

    ym1 = bridge approach depth in main channel of compound channel, including backwater

    ym2 = depth of flow in main channel in contracted section at equilibrium scour

    ysc = total depth of flow after scour, including the contraction scour depth only (Maryland formula)

    y1 = bridge approach depth in rectangular channel

    y2 = total flow depth in contracted section after scour (Maryland formula)

    y' = vertical coordinate measuring distance above the main-channel bed

    y'/ym= ratio of distance above the main-channel bed to depth of the flow in the main channel

    Gamma = specific weight of fluid

    Gammas = specific weight of sediment

    Muf = general discharge per unit width contraction ratio for the floodplain

    Phi = ratio of boundary shear force to streamwise component of weight

    Small Phi = angle of repose of sediment

    Rho = fluid density

    Rhos = sediment density

    Sigma = ratio of Manning's n in the main channel with compound-channel flow to the value for the flow in the main channel alone

    Sigmag = geometric standard deviation of sediment grain size distribution

    Tauc = critical shear stress for initiation of sediment motion

    Tau0 = average bed shear stress in uniform flow

    Tau1 = bed shear stress in approach flow

    Tau* = Shields' parameter, Tau/[(Gammas - Gamma) d50]

    Tau*c = critical value of Shields' parameter, Tauc/[(Gammas - Gamma) d50]

    Tau*1 = Shields' parameter in approach flow section

    Tau*2 = Shields' parameter in contracted flow section

    Xi= independent dimensionless ratio in proposed scour formula = qf1/Mqf0c

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