Langmuir circulation due to shear flow over wavy topography
- Norwegian University of Science and Technology, Department of Energy & Process Engineering, Department of Energy & Process Engineering, Trondheim, Norway (simen.a.ellingsen@ntnu.no)
Langmuir circulations (LC) in their traditional form are large rolling fluid flow pattern created by the interplay of surface waves and a near-surface shear current, typically both created by the wind. A celebrated theory by Craik and Leibovich (1976) describes two kinematic mechanisms which cause instabilities which grow into Langmuir rolls, both involving only the shear of the flow and the kinematic driving of flow undulations by a wavy surface, but containing no direct reference to the wind as a driving force. The same kinematic processes are present also in boundary layer flow over a wavy bottom topography in almost perfect analogy.
We present a theory of Langmuir-like circulations created by boundary layer flow over a topography in the form of a regular pattern of two monochromatic waves crossing at an oblique angle. Thus, the Craik-Leibovich instability sometimes referred to as CL1 is triggered and the close analogy with surface waves allows us to follow the general procedure of Craik (1970).
A flow of arbitrary shear profile is assumed over the bottom topography. In the opposite limits of transient inviscid flow and steady-state viscous flow simple equations for the stream function in cross-current plane can be derived and easily solved numerically. For the special case of a power-law velocity profile, explicit leading-order solutions are available. This allows us to quickly map out the circulation response to different parameters: wavelength, crossing angle and wave amplitude. The study is supplemented with direct numerical simulations which verify the manifestation of Langmuir-like circulations over wavy geometries with a no-slip boundary condition.
References
Craik, A.D.D., A wave-interaction model for the generation of windrows. J. Fluid Mech. (1970) 41, 801-821.
Craik, A.D.D. & Leibovich, S. A rational model for Langmuir circulations. J. Fluid Mech. (1976) 73, 401-426.
How to cite: Akselsen, A. H., Brostrøm, A., and Ellingsen, S. Å.: Langmuir circulation due to shear flow over wavy topography, EGU General Assembly 2020, Online, 4–8 May 2020, EGU2020-20406, https://doi.org/10.5194/egusphere-egu2020-20406, 2020
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Hi. Very interesting presentation.
I am interested in Langmuir turbulence-like vortices in flow over wavy terrrain. In your work you focus on the CL1 mechanism. Have you thought about addressing the CL2 mechanism (with exponential growth rate, and that does not require crossed waves) as well? As currently treated by various authors (e.g. A. Craik and later W. R. C. Phillips), this situation appears to be more complicated than the one originally envisaged by Craik and Leibovich, since the strong shear in the wind profile renders the wave perturbations due to the wavy terrain non-irrotational. This appears only to be treatable rigorously using the GLM equations of motion. Are you aware of other possible approaches?
Thanks in advance.
Miguel Teixeira (University of Reading)
dear dr Teixeira,
thank you for your interest! Very good question. CL2 for the same type of set-up is indeed something we have given some thought to. There are some preliminary considerations in chapter 5 of our preprint: https://arxiv.org/abs/2005.00317 . As you say, GLM is the standard method for treating this, and when the shear is strong, it could be the only sensible (possible?) way to include the wave distortion properly. Straightforward mode-coupling like we do is very nice and simple, and attractive when possible.
CL2 for wind over hilly terrain has been noted by others as you know (Gong et al 1996). I'm not aware that it has been discussed for water currents over bathymetry, but I could be wrong. I think there is a lot of interesting material there, potentially.
Our motivation was partly to show this possibility to the more general fluid mechanics community. There is much interest in generating vortices in boundary layers (for mixing & drag reduction) and if you can make the wall the way you like, then CL1 is nice. I agree crossing waves are less common in natural flows.
On another note: our lab in Trondheim is now testing a brand new water channel (15m length) where turbulence and mean flow can be tailored with an active grid. We will be putting a wave-maker in there too. Testing some of your impressive theory work on wave/free surface/turbulence interactions will be ideal. This is something we could discuss.