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• Enhanced Abutment Scour Studies for Compound Channels

Enhanced Abutment Scour Studies for Compound Channels

LIST OF SYMBOLS

a = blocked flow area by abutment

A = unobstructed flow area in approach channel

A_f1 = blocked flow area in approach-channel floodplain

b = bridge pier width

B_f = width of floodplain in compound channel

B_m = width of main channel in compound channel

B_m1 = width of approach main channel

B_m2 = width of main channel in contracted section

C_n = units conversion coefficient used in equation 23

C₀ = best-fit constant in proposed scour formula

C_r = best-fit coefficient in proposed scour formula

C_LB = live-bed scour coefficient

d₅₀ = median diameter of sediment

d_s = local scour depth

d_sc = theoretical long contraction scour depth

d_st = unsteady scour depth at any time t

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

F₁ = approach-flow Froude number

F_ab = Froude number adjacent to abutment face

k- = 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

k_f = spiral-flow adjustment factor in Maryland scour formula

k_s = equivalent sand-grain roughness

k_v = 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

K_s^* = abutment shape factor in Melville scour formula

K_G = channel geometric factor in Melville scour formula

K_I = flow-intensity factor in Melville scour formula

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

K^* = abutment alignment factor in Melville scour formula

L_a = 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

n_fp = Manning's n in floodplain of compound channel

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

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

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

N_sc = 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

q_bv = volumetric sediment discharge per unit width

q_f0 = flow rate per unit width in floodplain at normal depth (unconstricted)

q_f0c = critical flow rate per unit width in floodplain at normal depth (unconstricted)

q_f1 = flow rate per unit width in the approach floodplain (constricted)

q_f2 = flow rate per unit width in the contracted floodplain

q_m1 = flow rate per unit width in the approach main channel at the beginning of scour

q_m2 = flow rate per unit width in the contracted section in the main channel

Q = total discharge in compound channel

Q_f1 = discharge in the approach floodplain

Q_m0 = discharge in main channel for uniform flow in a compound channel

Q_m1 = discharge in the approach main channel

Q_m2 = discharge in the contracted main channel

Q_obst1 = obstructed discharge in the approach section

R = channel hydraulic radius

SG = specific gravity of sediment

t = time since beginning of scour

t_e = time to reach equilibrium scour depth

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

u*_c = critical value of shear velocity = (₁/)^0.5

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

V_c = critical velocity for initiation of motion

V_f0 = floodplain velocity at normal depth

V_f0c = critical velocity for unconstricted floodplain depth of uniform flow

V_f1 = bridge approach velocity in floodplain of compound channel

V_f2 = floodplain velocity in the contracted floodplain after scour

V_m0c = critical velocity for the unconstricted depth in the main channel

V_m1 = bridge approach velocity in main channel of compound channel

V_m2c = critical velocity in the main channel at the contracted section at equilibrium scour

V_x = mean contraction velocity in GKY scour formula

V_R = 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

y_ab = depth near the upstream corner of the abutment face

y_f0 = normal depth in floodplain

y_f1 = bridge approach depth in floodplain of compound channel, including backwater

y_ftw = tailwater depth in floodplain

ym0 = normal depth in main channel

y_m1 = bridge approach depth in main channel of compound channel, including backwater

y_m2 = depth of flow in main channel in contracted section at equilibrium scour

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

= specific weight of fluid

s = specific weight of sediment

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

= ratio of boundary shear force to streamwise component of weight

= angle of repose of sediment

= fluid density

s = sediment density

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

g = geometric standard deviation of sediment grain size distribution

c = critical shear stress for initiation of sediment motion

0 = average bed shear stress in uniform flow

1 = bed shear stress in approach flow

* = Shields' parameter, /[(s - ) d₅₀]

*_c = critical value of Shields' parameter, c/[(s - ) d₅₀]

*1 = Shields' parameter in approach flow section

*2 = Shields' parameter in contracted flow section

= independent dimensionless ratio in proposed scour formula = q_f1/Mq_f0c

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