Low relative-submergence effects in a rough-bed open-channel flow

ABSTRACT

Multi-plane stereoscopic PIV measurements were performed in an open-channel flume fitted with cubes to investigate very low submergence ratios, $h/k=\{1.5,2,3\}$, where h is the water depth and k the roughness height. The spatial standard deviation of the mean flow components reveals that the extent of the roughness sublayer increases drastically with the decrease in h/k to span the entire water column for the lowest h/k investigated. Despite this, the logarithmic law is still observed on the double-averaged velocity profiles for all h/k, first with a fixed von Kármán constant κ and, second, via the indicator function where κ is a free parameter. Also, the longitudinal and vertical normal stresses indicate a universal boundary layer behaviour independent of h/k. The results suggest that the logarithmic and wake-defect laws can still be applied at such low h/k. However, the lateral normal stress depends on h/k in the range investigated as well as on the geometry of the roughness pattern.

Keywords:

• Double-averaging
• logarithmic law
• rough-wall boundary-layers
• roughness sublayer
• stereoscopic PIV
• turbulent open-channel flows
• relative submergence

Acknowledgements

The authors thank S. Cazin, M. Marchal and S. Font for their valuable support and help with the experiments.

ORCID

Maxime Rouzes http://orcid.org/0000-0002-3329-7790

Frédéric Yann Moulin http://orcid.org/0000-0003-4682-7348

Olivier Eiff https://orcid.org/0000-0002-6451-6378

Notation
$A_f$	=	frontal area of the periodic roughness pattern (m)
$A_p$	=	planar area of the periodic roughness pattern (m)
$A_w$	=	wave amplitude (m)
$B_r$	=	constant of the logarithmic law for rough beds (–)
d	=	displacement height (m)
$D_s$	=	non-dimensional spatial standard deviation coefficient (–)
$D_{84}$	=	84th percentile of grain size distribution (m)
$Fr$	=	Froude number (–)
g	=	gravitational acceleration ( m s−2)
h	=	water depth (m)
$h_n$	=	normal water depth (m)
$h_{rs}$	=	top of the roughness sublayer (m)
k	=	roughness height (m)
$k_s$	=	equivalent-sand-roughness scale (m)
$k_s^{+}$	=	equivalent-sand-roughness Reynolds number (–)
$k_w$	=	wavenumber (m−1)
${\ell }_m$	=	mixing length (m)
L	=	roughness pattern length (m)
Q	=	water discharge (m3 s−1)
$R_h$	=	hydraulic radius (m)
u	=	streamwise component of the velocity ( m s−1)
$U_d$	=	free-stream bulk velocity ( m s−1)
$U_r$	=	velocity for wave resonance ( m s−1)
$u_{\ast }$	=	friction velocity at top of roughness elements (m s−1)
v	=	lateral component of the velocity ( m s−1)
w	=	vertical component of the velocity ( m s−1)
W	=	wake function ( m s−1)
x	=	streamwise coordinate (m)
$x_M$	=	streamwise position of the measurement area in the flume (m)
$x_0$	=	streamwise origin for resonant waves (m)
y	=	lateral coordinate (m)
z	=	vertical coordinate (m)
$z_0$	=	roughness length (m)
$z_0^{+}$	=	roughness-length Reynolds number (–)
$z_{fs}$	=	free-surface z-location
$z_m$	=	lower bound of the linear regression for the logarithmic law (m)
$z_m^{ϵ}$	=	extended lower bound of the linear regression for the logarithmic law (m)
$z_M$	=	upper bound of the linear regression for the logarithmic law (m)
$z_M^{ϵ}$	=	extended upper bound of the linear regression for the logarithmic law (m)
β	=	camera angle (${}^{\circ }$)
δ	=	boundary-layer thickness (m)
${\delta }^{+}$	=	boundary-layer thickness Reynolds number (–)
$\mathrm{\Delta }{U}^{+}$	=	roughness function (–)
${\mathrm{\Delta }}_x$	=	streamwise grid step for PIV (m)
${\mathrm{\Delta }}_z$	=	vertical grid step for PIV (m)
${ϵ}_{⟨\overline{\phi }⟩}$	=	spatial convergence error with 95% confidence for the double-averaged quantity $⟨\overline{\phi }⟩$ (–)
η	=	external variable (–)
${\eta }_{\max }$	=	relative height of the upper bound for the logarithmic law (–)
${\eta }_{\max }^{ϵ}$	=	relative height of the extended upper bound for the logarithmic law (–)
κ	=	von Kármán constant (–)
${\kappa }_{h/k}$	=	von Kármán constant found with the indicator function (–)
${\lambda }_f$	=	frontal density (–)
ω	=	wave frequency (s−1)
Φ	=	canopy porosity (–)
Π	=	Coles' wake parameter (–)
${\tau }_0$	=	bed shear stress ( N m−2)
${\tau }_{ij}$	=	total shear stress tensor ( N m−2)
${\tau }_k$	=	shear stress at z=k equal to $\rho u_{\ast }^2$ ( N m−2)
$\overline{\cdot \cdot \cdot }$	=	time-averaging operator
$⟨\cdot \cdot \cdot {⟩}_x$	=	single-plane spatial-averaging operator (in x-direction)
$⟨\cdot \cdot \cdot ⟩$	=	spatial-averaging operator (in both y- and x-directions)
${\cdot \cdot \cdot }^{\prime }$	=	turbulent fluctuation component
$\widetilde{\cdot \cdot \cdot }$	=	dispersive component

Additional information

Funding

This work was supported by FP7 Research infrastructures Hydralab IV/PISCES project [grant number 261520] and by the Agence National de la Recherche (ANR) FlowRes project [grant number 14-CE03-0010]. M. Rouzès benefited during his PhD from a financial support by the Direction Générale de l'Armement (DGA).