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On Displacement Height, from Classical to Practical Formulation: Stress, Turbulent Transport and Vorticity Considerations

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Abstract

Displacement height (d) is an important parameter in the simple modelling of wind speed and vertical fluxes above vegetative canopies, such as forests. Here we show that, aside from implicit definition through a (displaced) logarithmic profile, accepted formulations for d do not consistently predict flow properties above a forest. Turbulent transport can affect the displacement height, and is an integral part of what is called the roughness sublayer. We develop a more general approach for estimation of d, through production of turbulent kinetic energy and turbulent transport, and show how previous stress-based formulations for displacement height can be seen as simplified cases of a more general definition including turbulent transport. Further, we also give a simplified and practical form for d that is in agreement with the general approach, exploiting the concept of vortex thickness scale from mixing-layer theory. We assess the new and previous displacement height formulations by using flow statistics derived from the atmospheric boundary-layer Reynolds-averaged Navier–Stokes model SCADIS as well as from wind-tunnel observations, for different vegetation types and flow regimes in neutral conditions. The new formulations tend to produce smaller d than stress-based forms, falling closer to the classic logarithmically-defined displacement height. The new, more generally defined, displacement height appears to be more compatible with profiles of components of the turbulent kinetic energy budget, accounting for the combined effects of turbulent transport and shear production. The Coriolis force also plays a role, introducing wind-speed dependence into the behaviour of the roughness sublayer; this affects the turbulent transport, shear production, stress, and wind speed, as well as the displacement height, depending on the character of the forest. We further show how our practical (‘mixing-layer’) form for d matches the new turbulence-based relation, as well as correspondence to previous (stress-based) formulations.

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Notes

  1. Some formulations for displacement height also attempt to describe mean scalar profiles (e.g. Thom et al. 1975).

  2. We introduce the friction velocity \(u_{*}\) via the mean momentum flux \(\left\langle {{u}^{\prime }{w}^{\prime }} \right\rangle =-u_*^{2}\), where turbulent fluctuations about the mean are denoted by primes.

  3. The Lambert-W function is sometimes labelled the ‘product-log’ function; see e.g. Warburton and Wang (2004).

  4. \(z_{0,sfc} \left\langle {{u}^{\prime }{w}^{\prime }} \right\rangle _{0}\ll h\left\langle {{u}^{\prime }{w}^{\prime }} \right\rangle _\mathrm{h}\) is valid in the mean, since Brunet et al. (1994) confirmed that \(\left\langle {{u}^{\prime }{w}^{\prime }} \right\rangle _{0}\ll \left\langle {{u}^{\prime }{w}^{\prime }} \right\rangle _\mathrm{h}\) experimentally, and the roughness length \(z_{0,sfc} \) is much smaller than the canopy height h.

  5. RSL theory uses a ‘vorticity thickness’ scale to modify the form of vertical profiles, but does not typically modify the displacement height d (e.g. Arnqvist and Bergstrom 2015). However, RSL theory does allow for a variation of d with stability, but uses a stress-based form such (as in Eqs. 4–7).

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Acknowledgments

Part of this work was supported by the Center for Computational Wind Turbine Aerodynamics and Atmospheric Turbulence, funded by the Danish Council for Strategic Research (‘DSF’) under grant number 09-067216. The authors would also like to thank the anonymous reviewers for constructive comments, as well as continued discussion on this subject.

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  1. Meteorology Group, Wind Energy Department, Risø Lab/Campus of Technical University of Denmark, Building 118, Box 49, 4000, Roskilde, Denmark

    Andrey Sogachev & Mark Kelly

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  1. Andrey Sogachev
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Correspondence to Mark Kelly.

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The authors (AS and MK) declare that they have no conflict of interest with regards to this work.

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Sogachev, A., Kelly, M. On Displacement Height, from Classical to Practical Formulation: Stress, Turbulent Transport and Vorticity Considerations. Boundary-Layer Meteorol 158, 361–381 (2016). https://doi.org/10.1007/s10546-015-0093-x

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