Importance of turbulent diffusion in transverse mixing in rivers

ABSTRACT

An expression for the transverse mixing coefficient in a rectangular channel is derived to evaluate the importance of transverse turbulent diffusion. Although some formulas for the transverse mixing coefficient account for both turbulent diffusion and shear dispersion, most calculations of the transverse dispersion coefficient do not include the effect of transverse turbulent diffusion on shear dispersion. Both vertical and transverse turbulent diffusion contribute appreciably to transverse shear dispersion, and the direct contribution of transverse turbulent diffusion to transverse mixing is important for a range of conditions commonly observed in natural channels. A comparison of predicted and measured transverse mixing coefficients supports including turbulent diffusion in analyses of transverse mixing – both in its effect on shear dispersion and as a direct mechanism of transport.

Keywords:

• Dispersion processes and models
• secondary currents/flows
• streams and rivers
• turbulent diffusion
• wall-bounded shear flow turbulence

Disclosure statement

The authors report no potential conflict of interest.

Supplemental material

The supplemental material includes full details of the derivation of Eq. (6) and the solution for the channel of rectangular cross section. It also includes a solution for a channel of semicircular cross-section.

Notation

a	=	regression coefficient for Eq. (4), 77.88 (–)
a00	=	constant arising from Eq. (8) (–)
a0n	=	constant arising from Eq. (8) (–)
am0	=	constant arising from Eq. (8) (–)
amn	=	constant arising from Eq. (8) (–)
B	=	channel width (m)
C	=	concentration (kg m−3)
C′	=	concentration fluctuation (kg m−3)
$\overline{C}$	=	average concentration (kg m−3)
Cmn	=	coefficients in the eigenfunction expansion of the concentration fluctuation (kg m−3)
Deff	=	effective dispersion coefficient (m2 s−1)
Dy	=	transverse mixing coefficient (m2 s−1)
Dym	=	measured transverse mixing coefficient (m2 s−1)
Dyp	=	predicted transverse mixing coefficient (m2 s−1)
Dyx	=	transverse dispersion coefficient due to fluctuations in streamwise velocity (m2 s−1)
Dyy	=	transverse dispersion coefficient due to fluctuations in transverse velocity (m2 s−1)
f1(η)	=	normalized transverse profile of streamwise velocity (–)
f2(η)	=	normalized transverse profile of transverse velocity (–)
Fy	=	total transverse flux (kg m−1 s−1)
g1(ζ)	=	normalized vertical profile of streamwise velocity (–)
g2(ζ)	=	normalized vertical profile of transverse velocity (–)
H	=	channel depth (m)
Iyx	=	normalized dispersion coefficient – see Eqs (12) and (13) (–)
Iyy	=	normalized dispersion coefficient – see Eqs (14) and (15) (–)
$I_{yy}^{max}$	=	maximum value of Iyy (–)
L	=	length of reach (m)
m	=	mode number for eigenfunction (–)
${\dot{M}}_y$	=	transverse dispersive flux (kg m−1 s−1)
M0	=	zeroth moment of concentration (kg m−2)
M1	=	first moment of concentration (kg m−1)
M2	=	second moment of concentration (kg)
n	=	mode number for eigenfunction (–)
P	=	$UH/u_{\ast }{R}_C$ (–)
p	=	power law exponent (–)
Rc	=	radius of curvature (m)
RD	=	discrepancy ratio (–)
rh	=	ratio of the depths of the main channel and floodplain (–)
t	=	time (s)
T95	=	time to reach 95% complete mixing (s)
U	=	streamwise velocity scale, which equals the cross-sectional average value $\overline{u}$ (m s−1)
U	=	streamwise velocity (m s−1)
u′	=	streamwise velocity fluctuation (m s−1)
u*	=	shear velocity (m s−1)
$\overline{u}$	=	cross-sectional average streamwise velocity (m s−1)
$\tilde{u}$	=	depth-averaged streamwise velocity (m s−1)
V	=	transverse velocity scale (m s−1)
v	=	transverse velocity (m s−1)
vs	=	surface value of the transverse velocity (m s−1)
v′	=	transverse velocity fluctuation (m s−1)
$\overline{v}$	=	cross-sectional average transverse velocity (m s−1)
X	=	streamwise coordinate moving at mean velocity, $x-\overline{u}t$ (m)
x	=	streamwise coordinate (m)
y	=	transverse coordinate (m)
z	=	vertical coordinate (m)
α	=	shape parameter for velocity profile (–)
β	=	shape parameter for velocity profile (–)
Γ	=	gamma function (–)
γv	=	$\frac{10}{3}\left(1-u_{\ast }/3\kappa U\right)f_1^2\left(\frac{1}{2}\right)$, coefficient in normalizing transverse velocity (–)
εx	=	streamwise eddy diffusivity (m2 s−1)
εy	=	transverse eddy diffusivity (m2 s−1)
εz	=	vertical eddy diffusivity (m2 s−1)
ζ	=	normalized vertical coordinate (–)
η	=	normalized transverse coordinate (–)
η*	=	location of the peak in $f_1\left(\eta \right)$ (–)
θy	=	proportionality coefficient, 0.13 (–)
θz	=	proportionality coefficient, 0.067 (–)
κ	=	von Kármán constant (–)
λmn	=	eigenvalue (s−1)
σ2	=	spatial variance of concentration (m2)
τ	=	ratio of mixing times, $\left(H^2/{ϵ}_z\right)/\left(B^2/{ϵ}_y\right)$ (–)
ϕmn	=	eigenfunction (–)

Notes

1 The case of a semi-circular channel is solved in the Supplemental Material.