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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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