Numerical Simulation of Turbidity Current in Approach Channels with a Closed End

Acknowledgments

Notation

C

sediment concentration of the turbidity current (kg · m⁻³);

C₀

sediment concentration at the position of density current formed (kg · m⁻³);

C_in

sediment concentration at the inlet (kg · m⁻³);

$\overline{C}$

layer-averaged concentration of the turbidity current (kg · m⁻³);

d

sediment particle size (m);

Fr_ini

densimetric Froude number at the initial stage;

g′

reduced gravity acceleration = g(ρ − ρ_w)/ρ_w;

g

gravity acceleration (m · s⁻²);

H

total depth of model (m);

h

thickness of density current (m);

k

turbulence energy;

k_s

parameter applied to describe the hindered settling of activated sludge;

L

length of the channel (m);

l

dimensionless length can be defined as l = (x − 1)/(L − 1);

p

static pressure (Pa);

q

inlet unit discharge (m² · s⁻¹);

q_d

inlet unit discharge of the lower layer (m² · s⁻¹);

q_s

sediment transport volume of single width (m² · s⁻¹);

q_sd

sediment transport volume of the lower layer (m² · s⁻¹);

q_su

sediment transport volume of the upper layer (m² · s⁻¹);

q_u

inlet unit discharge of the upper layer (m² · s⁻¹);

t

time (s);

U

average flow velocity along water depth (m · s⁻¹);

U₀

velocity at the position of density current formed (m · s⁻¹);

U_ini

velocity of turbidity current at the inlet (m · s⁻¹);

u

velocity in the x-direction (m · s⁻¹);

v

velocity in the y-direction (m · s⁻¹);

v_s

settling velocity (m · s⁻¹);

W

dimensionless deposition rate = (q_sd − q_su)/q_sd;

x

coordinate in the horizontal direction (m);

y

coordinate in the vertical direction (m);

γ

unit weight of turbid water (N · m⁻³);

γ_s

unit weight of sediment (N · m⁻³);

δ

distance from the bottom to the position where u becomes to be 0 (m);

ɛ

dissipation rate of turbulence energy;

λ₁, λ₂, λ₃

parameters in settling equations, which related to the particle size, surface properties, iron characteristics;

μ

viscosity of clean water (kg · m⁻¹ · s⁻¹);

μ_p

plastic viscosity of turbid water (kg · m⁻¹ · s⁻¹);

ρ

mixture density of fluid (kg · m⁻³);

ρ_p

dry density of sediment (kg · m⁻³);

ρ_w

clean water density (kg · m⁻³);

σ_c

turbulent Schmidt number (–);

τ_b

yield stress in the Bingham method (kg · m⁻¹ · s⁻²);

τ_ij

shearing stress of turbid fluid (kg · m⁻¹ · s⁻²);

υ

viscosity kinematic coefficient (m² · s⁻¹); and

υ_t

turbulent kinematic viscosity (m² · s⁻¹).

References