Numerical Simulations of Lateral Dispersion by the Relaxation of Diapycnal Mixing Events

Abstract In this second of two companion papers, numerical simulations of lateral dispersion by small-scale geostrophic motions, or vortical modes, generated by the adjustment of mixed patches following diapycnal mixing events are examined. A three-dimensional model was used to solve the Navier–Stokes equations and an advection/diffusion equation for a passive tracer. Model results were compared with theoretical predictions for vortical mode stirring with results from dye release experiments conducted over the New England continental shelf. For “weakly nonlinear” cases in which adjustment events were isolated in space and time, lateral dispersion in the model was consistent to within a constant scale factor with the parameter dependencepredicted by Sundermeyer et al., where h and L are the vertical and horizontal scales of the mixed patches, ΔN 2 is the change in stratification associated with the mixed patches, f is the Coriolis parameter, ϕ is the frequency of diapycnal mixing events, and νB is the background viscosity. The associated scale factor, assumed to be of order 1, had an actual value of about 7, although this value will depend, in an unknown way, on the assumed horizontal scale of the mixed patches, which was here held constant at close to the deformation radius. A second more energetic parameter regime was also identified in which vortical mode stirring became strongly nonlinear and the effective lateral dispersion was larger. Estimates of the relevant parameters over the New England shelf suggest that this strongly nonlinear regime is more relevant to the real ocean than the weakly nonlinear regime, at least under late summer conditions. This suggests that stirring by small-scale geostrophic motion may, under certain conditions, contribute significantly to lateral dispersion on scales of 1–10 km in the ocean.

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Precession of a rapidly rotating cylinder flow: traverse through resonance

Journal of Fluid Mechanics ◽

10.1017/jfm.2015.524 ◽

2015 ◽

Vol 782 ◽

pp. 63-98 ◽

Cited By ~ 13

Author(s):

Francisco Marques ◽

Juan M. Lopez

Keyword(s):

Aspect Ratio ◽

Stokes Equations ◽

Complex Dynamics ◽

Nonlinear Models ◽

Transition To Turbulence ◽

Navier Stokes ◽

Small Scale ◽

Viscous Effects ◽

Navier Stokes Equations ◽

Weakly Nonlinear

Recent experiments using a rapidly rotating and precessing cylinder have shown that for specific values of the precession rate, aspect ratio and tilt angle, sudden catastrophic transitions to turbulence occur. Even if the precessional forcing is not too strong, there can be intermittent recurrences between a laminar state and small-scale chaotic flow. The inviscid linearized Navier–Stokes equations have inertial-wave solutions called Kelvin eigenmodes. The precession forces the flow to have azimuthal wavenumber $m=1$ (spin-over mode). Depending on the cylinder aspect ratio and on the ratio of the rotating and precessing frequencies, additional Kelvin modes can be in resonance with the spin-over mode. This resonant flow would grow unbounded if not for the presence of viscous and nonlinear effects. In practice, one observes a rapid transition to turbulence, and the precise nature of the transition is not entirely clear. When both the precessional forcing and viscous effects are small, weakly nonlinear models and experimental observations suggest that triadic resonance is at play. Here, we used direct numerical simulations of the full Navier–Stokes equations in a narrow region of parameter space where triadic resonance has been previously predicted from a weakly nonlinear model and observed experimentally. The detailed parametric studies enabled by the numerics reveal the complex dynamics associated with weak precessional forcing, involving symmetry-breaking, hysteresis and heteroclinic cycles between states that are quasiperiodic, with two or three independent frequencies. The detailed analysis of these states leads to associations of physical mechanisms with the various time scales involved.

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Head-on collision of internal waves with trapped cores

10.5194/npg-2017-31 ◽

2017 ◽

Author(s):

Vladimir Maderich ◽

Kyung Tae Jung ◽

Kateryna Terletska ◽

Kyeong Ok Kim

Keyword(s):

Nonlinear Waves ◽

Wave Amplitude ◽

Stokes Equations ◽

Three Dimensional ◽

Navier Stokes ◽

Class Iii ◽

Navier Stokes Equations ◽

Weakly Nonlinear ◽

Strongly Nonlinear ◽

Conservation Of Momentum

Abstract. The dynamics and energetics of a head-on collision of internal solitary waves (ISWs) with trapped cores propagating in thin pycnocline were studied numerically within the framework of the Navier-Stokes equations for a stratified fluid. The peculiarity of this collision is that it involves the trapped masses of a fluid. The interaction of ISWs differs for three classes of ISWs: (i) weakly nonlinear waves without trapped cores, (ii) stable strongly nonlinear waves with trapped cores, and (iii) shear unstable strongly nonlinear waves. The wave phase shift grows as the amplitudes of the interacting waves increase for colliding waves of classes (i) and (ii) and remains almost constant for those of class (iii). The excess of the maximum runup amplitude over the sum of the amplitudes of colliding waves almost linearly increases with increasing amplitude of the interacting waves belonging to classes (i) and (ii); however, it decreases somewhat for those of class (iii). The waves of class (ii) with a normalized on thickness of pycnocline amplitude lose fluid trapped by the wave cores in the range approximately between 1 and 1.75. The interacting stable waves of higher amplitude capture cores and carry trapped fluid in opposite directions with little mass loss. The collision of locally shear unstable waves of class (iii) is accompanied by the development of three-dimensional instability and turbulence. The dependence of loss of energy on the wave amplitude is not monotonous. Initially, the energy loss due to the interaction increases as the wave amplitude increases. Then, the energy losses reach a maximum due to the loss of potential energy of the cores upon collision and then start to decrease. With further amplitude growth, collision is accompanied by the development of instability and an increase in the loss of energy. The collision process is modified for waves of different amplitudes because of the exchange of trapped fluid between colliding waves due to the conservation of momentum.

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Numerical simulations of two-fluid turbulent mixing at large density ratios and applications to the Rayleigh–Taylor instability

Philosophical Transactions of The Royal Society A Mathematical Physical and Engineering Sciences ◽

10.1098/rsta.2012.0185 ◽

2013 ◽

Vol 371 (2003) ◽

pp. 20120185 ◽

Cited By ~ 55

Author(s):

D. Livescu

Keyword(s):

Numerical Simulations ◽

Stokes Equations ◽

Taylor Instability ◽

Continuum Approximation ◽

Small Scale ◽

Gaseous Mixtures ◽

Rayleigh Taylor Instability ◽

Density Ratios ◽

Mixing Problem ◽

Two Fluid

A tentative review is presented of various approaches for numerical simulations of two-fluid gaseous mixtures at high density ratios, as they have been applied to the Rayleigh–Taylor instability (RTI). Systems exhibiting such RTI behaviour extend from atomistic sizes to scales where the continuum approximation becomes valid. Each level of description can fit into a hierarchy of theoretical models and the governing equations appropriate for each model, with their assumptions, are presented. In particular, because the compressible to incompressible limit of the Navier–Stokes equations is not unique and understanding compressibility effects in the RTI critically depends on having the appropriate basis for comparison, two relevant incompressible limits are presented. One of these limits has not been considered before. Recent results from RTI simulations, spanning the levels of description presented, are reviewed in connection to the material mixing problem. Owing to the computational limitations, most in-depth RTI results have been obtained for the incompressible case. Two such results, concerning the asymmetry of the mixing and small-scale anisotropy anomaly, as well as the possibility of a mixing transition in the RTI, are surveyed. New lines for further investigation are suggested and it is hoped that bringing together such diverse levels of description may provide new ideas and increased motivation for studying such flows.

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Diffusion Experiments with a Global Discontinuous Galerkin Shallow-Water Model

Monthly Weather Review ◽

10.1175/2009mwr2843.1 ◽

2009 ◽

Vol 137 (10) ◽

pp. 3339-3350 ◽

Cited By ~ 25

Author(s):

Ramachandran D. Nair

Keyword(s):

Shallow Water ◽

Discontinuous Galerkin ◽

Water Model ◽

Order System ◽

Small Scale ◽

Shallow Water Model ◽

First Order ◽

Advection Diffusion Equation ◽

Diffusion Scheme ◽

Cubed Sphere

Abstract A second-order diffusion scheme is developed for the discontinuous Galerkin (DG) global shallow-water model. The shallow-water equations are discretized on the cubed sphere tiled with quadrilateral elements relying on a nonorthogonal curvilinear coordinate system. In the viscous shallow-water model the diffusion terms (viscous fluxes) are approximated with two different approaches: 1) the element-wise localized discretization without considering the interelement contributions and 2) the discretization based on the local discontinuous Galerkin (LDG) method. In the LDG formulation the advection–diffusion equation is solved as a first-order system. All of the curvature terms resulting from the cubed-sphere geometry are incorporated into the first-order system. The effectiveness of each diffusion scheme is studied using the standard shallow-water test cases. The approach of element-wise localized discretization of the diffusion term is easy to implement but found to be less effective, and with relatively high diffusion coefficients, it can adversely affect the solution. The shallow-water tests show that the LDG scheme converges monotonically and that the rate of convergence is dependent on the coefficient of diffusion. Also the LDG scheme successfully eliminates small-scale noise, and the simulated results are smooth and comparable to the reference solution.

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Nonlinear cellular instabilities of planar premixed flames: numerical simulations of the Reactive Navier–Stokes equations

Combustion Theory and Modelling ◽

10.1080/13647830500472354 ◽

2006 ◽

Vol 10 (3) ◽

pp. 483-514 ◽

Cited By ~ 18

Author(s):

G. J. Sharpe ◽

S. A. E. G. Falle

Keyword(s):

Numerical Simulations ◽

Stokes Equations ◽

Premixed Flames ◽

Navier Stokes ◽

Navier Stokes Equations

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Nonlinear evolution of the elliptical instability: an example of inertial wave breakdown

Journal of Fluid Mechanics ◽

10.1017/s0022112099005959 ◽

1999 ◽

Vol 396 ◽

pp. 73-108 ◽

Cited By ~ 35

Author(s):

D. M. MASON ◽

R. R. KERSWELL

Keyword(s):

Finite Amplitude ◽

Complex Conjugate ◽

Small Scale ◽

Conjugate Pair ◽

Weakly Nonlinear ◽

Elliptical Instability ◽

Elliptical Flow ◽

Complex Conjugate Pair ◽

Generic Instability ◽

Secondary Instabilities

A direct numerical simulation is presented of an elliptical instability observed in the laboratory within an elliptically distorted, rapidly rotating, fluid-filled cylinder (Malkus 1989). Generically, the instability manifests itself as the pairwise resonance of two different inertial modes with the underlying elliptical flow. We study in detail the simplest ‘subharmonic’ form of the instability where the waves are a complex conjugate pair and which at weakly supercritical elliptical distortion should ultimately saturate at some finite amplitude (Waleffe 1989; Kerswell 1992). Such states have yet to be experimentally identified since the flow invariably breaks down to small-scale disorder. Evidence is presented here to support the argument that such weakly nonlinear states are never seen because they are either unstable to secondary instabilities at observable amplitudes or neighbouring competitor elliptical instabilities grow to ultimately disrupt them. The former scenario confirms earlier work (Kerswell 1999) which highlights the generic instability of inertial waves even at very small amplitudes. The latter represents a first numerical demonstration of two competing elliptical instabilities co-existing in a bounded system.

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Feedback Control of Turbulence

Applied Mechanics Reviews ◽

10.1115/1.3124438 ◽

1994 ◽

Vol 47 (6S) ◽

pp. S3-S13 ◽

Cited By ~ 92

Author(s):

Parviz Moin ◽

Thomas Bewley

Keyword(s):

Feedback Control ◽

Turbulent Flows ◽

Systems Approach ◽

Stokes Equations ◽

Navier Stokes ◽

Small Scale ◽

Flow Equations ◽

Active Feedback ◽

Active Feedback Control ◽

Mathematical Techniques

A brief review of current approaches to active feedback control of the fluctuations arising in turbulent flows is presented, emphasizing the mathematical techniques involved. Active feedback control schemes are categorized and compared by examining the extent to which they are based on the governing flow equations. These schemes are broken down into the following categories: adaptive schemes, schemes based on heuristic physical arguments, schemes based on a dynamical systems approach, and schemes based on optimal control theory applied directly to the Navier-Stokes equations. Recent advances in methods of implementing small scale flow control ideas are also reviewed.

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