Spatially-averaged flows over mobile rough beds: equations for the second-order velocity moments

Konstantinos Papadopoulos* (Corresponding Author), Vladimir Nikora, Stuart Cameron, Mark Stewart, Christopher Gibbins

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

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The double-averaging methodology is used in this paper for deriving equations for the second-order velocity moments (i.e. turbulent and dispersive stresses) that emerge in the double-averaged momentum equation for incompressible Newtonian flows over mobile boundaries. The starting point in the derivation is the mass and momentum conservation equations for local (at a point) instantaneous variables that are up-scaled by employing temporal and spatial averaging. First, time-averaged conservation equations for mass, momentum, and turbulent stresses for mobile bed conditions are derived. Then, the double-averaged hydrodynamic equations obtained by spatial averaging the time-averaged equations are proposed. The derived second-order equations can serve as a basis for the construction of simplified mathematical and numerical models and for interpretation of experimental and simulation data when bed mobility is present. Potential applications include complex flow situations such as free-surface flows over vegetated or mobile sedimentary beds and flows through tidal and wind turbine arrays.

Original languageEnglish
Pages (from-to)133-151
Number of pages19
JournalJournal of Hydraulic Research
Issue number1
Early online date18 Jan 2019
Publication statusPublished - 31 Jan 2020

Bibliographical note

This study was part of the research project ‘Hydrodynamic Transport in Ecologically Critical Heterogeneous interfaces’ (HYTECH), the support of which, under the European Union’s Seventh Framework Programme (Marie Curie Actions FP7PEOPLE-2012-ITN, European Commission [grant agreement number 316546]),is gratefully acknowledged. Financial support was also provided by the Engineering and Physical Sciences Research Council (EPSRC)/UK grant “Bed friction in roughbed free-surface flows: a theoretical framework, roughness regimes, and quantification” [grant EP/K041088/1].


  • double averaging methodology
  • form-induced stress
  • mobile-boundary flows
  • second-order hydrodynamic equations
  • spatially-averaged turbulent stress
  • turbulence


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