Shallow water wave turbulence

• The dynamics of irrotational shallow water wave turbulence forced in large scales and dissipated at small scales is investigated. First, we derive the shallow water analogue of the `four-fifths law' of Kolmogorov turbulence for a third order structure function involving velocity and displacement increments. Using this relation and assuming that the flow is dominated by shocks we develop a simple model predicting that the shock amplitude scales as (ϵd)1/3, where ϵ is the mean dissipation rate and d the mean distance between the shocks, and that the pth order displacement and velocity structure functions scale as (ϵd)p/3r/d, where r is the separation. Then we carry out a series of forced simulations with resolutions up to 76802, varying the Froude number, Ff=ϵ1/3/ckf1/3, where kf is the forcing wave number and c is the wave speed. In all simulations a stationary state is reached in which there is a constant spectral energy flux and equipartition between kinetic and potential energy in the constant flux range. The third order structure function relation is satisfied with a high degree of accuracy. Mean energy is found to scale as E∼√(ϵc/kf), and is also dependent on resolution, indicating that shallow water wave turbulence does not fit into the paradigm of a Richardson-Kolmogorov cascade. In all simulations shocks develop, displayed as long thin bands of negative divergence in flow visualisations. The mean distance between the shocks is found to scale as d∼Ff1/2/kf. Structure functions of second and higher order are found to scale in good agreement with the model. We conclude that in the weak limit, Ff→0, shocks will become denser and weaker and finally disappear for a finite Reynolds number. On the other hand, for a given Ff, no matter how small, shocks will prevail if the Reynolds number is sufficiently large.

Ämnesord

TEKNIK OCH TEKNOLOGIER  -- Maskinteknik -- Strömningsmekanik och akustik (hsv//swe)

ENGINEERING AND TECHNOLOGY  -- Mechanical Engineering -- Fluid Mechanics and Acoustics (hsv//eng)

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Physics

Engineering Mechanics

Teknisk mekanik

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