# Turbulent channel flow perturbed by triangular ripples

## Abstract

This paper presents an experimental investigation of the perturbation of a turbulent closed-conduit flow by two-dimensional triangular ripples. Two ripple configurations were employed: one single asymmetric triangular ripple, and two consecutive asymmetric triangular ripples, all of them with the same geometry. Different water flows were imposed over either one or two ripples fixed to the bottom wall of a closed conduit, and the flow field was measured by particle image velocimetry (PIV). Reynolds numbers based on the channel height were moderate, varying between 27500 and 34700. The regime was hydraulically smooth, and the blockage ratio was significant. Experimental data for this specific case remain scarce, and the physics involved has yet to be fully elucidated. Using the instantaneous flow fields, the mean velocities and fluctuations were computed, and the shear stresses over the ripples were determined. The velocities and stresses obtained in this way for the single ripple and for the pair of ripples are compared, and the surface shear stress is discussed in terms of bed stability. Our results show that for the single and upstream ripples, the shear stress increases at the leading edge and decreases toward the crest, while for the downstream ripple it decreases monotonically between the reattachment point and the crest. The stress distribution over the downstream ripple, which is different from both the upstream and single ripples, is shown to sustain existing ripples over loose beds.

## Keywords

Closed-conduit flow Turbulent boundary layer Perturbation Ripples## Nomenclature

## Roman symbols

*A*Constant

*B*Constant

*d*Diameter (m)

*f*Darcy friction factor

*h*Local height of the ripple (m)

*H*Channel height (m)

- \(H_\text{ eff }\)
Distance between the PVC plates and the top wall of the channel (m)

*i*Imaginary number

*k*Wave number (m\(^{-1}\))

*L*Ripple length (m)

- \(l_v\)
Viscous length (m)

*P*Turbulence production term (m\(^2\)/s\(^3\))

- \(P_{\text {max}}\)
Maximum value of turbulence production (m\(^2\)/s\(^3\))

*Q*Volumetric flow rate (m\(^3\)/h)

*Re*Reynolds number based on the channel height

- \(Re_{\text {dh}}\)
Reynolds number based on the hydraulic diameter

*t*Time (s)

*u*Longitudinal component of the velocity (m/s)

- \(\overline{U}\)
Cross-sectional mean velocity (m/s)

- \(U_0\)
Velocity at the channel centerline (m/s)

- \(u_*\)
Shear velocity (m/s)

*v*Vertical component of the velocity (m/s)

- \(\mathbf {V}\)
Velocity vector (m/s)

*x*Horizontal coordinate (m)

*y*Vertical coordinate (m)

## Greek symbols

- \(\Delta\)
Perturbation field

- \(\kappa\)
von Kármán constant

- \(\lambda\)
Wavelength (m)

- \(\mu\)
Dynamic viscosity (Pa s)

- \(\nu\)
Kinematic viscosity (m\(^2\)/s)

- \(\rho\)
Density (kg/m\(^3\))

- \(\tau\)
Shear stress (N/m\(^2\))

- \(\theta\)
Angle between the horizontal and the ripple surface (\(^o\))

## Subscripts

- 0
Relative to the flat surface (unperturbed flow)

- bla
Relative to Blasius correlation

*d*Relative to the displaced coordinate system

- dh
Relative to the hydraulic diameter

*k*Relative to Fourier space

- max
Maximum value

*v*Relative to the viscous layer

- \(\theta\)
Aligned with the ripple surface

## Superscripts

- \(+\)
Normalized by the viscous length \(l_v\) or by the shear velocity \(u_{*,0}\)

- \(\overline{\quad }\)
Averaged in time

- \('\)
Fluctuation

- \(\hat{\quad }\)
Perturbation

## Notes

### Acknowledgements

Fernando David Cúñez Benalcázar is grateful to SENESCYT (Programa de Becas Convocatoria Abierta 2014 Segunda Fase) and to FAPESP (grant no. 2016/18189-0), and Gabriel Victor Gomes de Oliveira is grateful to FAPESP (grant no. 2015/15001-8) and to CAPES for providing financial support. Erick de Moraes Franklin would like to express his gratitude to FAPESP (grant nos. 2012/19562-6 and 2016/13474-9) and CNPq (grant no. 400284/2016-2) for the financial support they provided.

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