arXiv subjects

A. I. Dyachenko

Publications and source records attributed to A. I. Dyachenko.

14 recordsLinked to original sources

arXiv2022

Free Surface in 2D Potential Flow: Singularities, Invariants and Virtual Fluid

We study a 2D potential flow of an ideal fluid with a free surface with decaying conditions at infinity. By using the conformal variables approach, we study a particular solution of Euler equations having a pair of square-root branch points in the conformal plane, and find that the analytic continuation of the fluid complex potential and conformal map define a flow in the entire complex plane, excluding a vertical cut between the branch points. The expanded domain is called the "virtual" fluid, and it contains a vortex sheet whose dynamics is equivalent to the equations of motion posed at the free surface. The equations of fluid motion are analytically continued to both sides of the vertical branch cut (the vortex sheet), and additional time-invariants associated with the topology of conformal plane and Kelvin's theorem for virtual fluid are explored. We called them "winding" and virtual circulation. This result can be generalized to a system of many cuts connecting many branch points, and resulting in a pair of invariants for each pair of branch points. We develop an asymptotic theory that shows how a solution originating from a single vertical cut forms a singularity at the free surface in infinite time, the rate of singularity approach is double-exponential, and supercedes the previous result of the short branch cut theory with finite time singularity formation. The present work offers a new look at fluid dynamics with free surface by unifying the problem of motion of vortex sheets, and the problem of 2D water waves. A particularly interesting question that arises in this context is whether instabilities of the virtual vortex sheet are related to breaking of steep ocean waves when gravity effects are included.

physics.flu-dyn↗

arXiv2021

Short branch cut approximation in $2$D Hydrodynamics with Free Surface

A potential motion of ideal incompressible fluid with a free surface and infinite depth is considered in two-dimensional geometry. A time-dependent conformal mapping of the lower complex half-plane of the auxiliary complex variable $w$ into the area filled with fluid is performed with the real line of $w$ mapped into the free fluid's surface. The fluid dynamics can be fully characterized by the motion of the complex singularities in the analytical continuation of both the conformal mapping and the complex velocity. We consider the short branch cut approximation of the dynamics with the small parameter being the ratio of the length of the branch cut to the distance between its center and the real line of $w$. We found that the fluid dynamics in that approximation is reduced to the complex Hopf equation for the complex velocity coupled with the complex transport equation for the conformal mapping. These equations are fully integrable by characteristics producing the infinite family of solutions, including the pairs of moving square root branch points. The solutions are compared with the simulations of the fully nonlinear Eulerian dynamics giving excellent agreement even when the small parameter approaches about one.

physics.flu-dyn↗

arXiv2019

Dynamics of Poles in 2D Hydrodynamics with Free Surface: New Constants of Motion

We address a problem of potential motion of ideal incompressible fluid with a free surface and infinite depth in two dimensional geometry with gravity forces and surface tension. A time-dependent conformal mapping z(w,t) of the lower complex half-plane of the variable w into the area filled with fluid is performed. We study the dynamics of singularities of both z(w,t) and the complex fluid potential Pi(w,t) in the upper complex half-plane of w. We show the existence of solutions with an arbitrary finite number N of complex poles in z_w(w,t) and Pi_w(w,t) which are the derivatives of z(w,t) and Pi(w,t) over w. The orders of poles can be arbitrary for zero surface tension while all orders are even for nonzero surface tension. We find that the residues of z_w(w,t) at these N points are new, previously unknown constants of motion, see also Ref. V.E. Zakharov and A. I. Dyachenko, arXiv:1206.2046 (2012) for the preliminary results. All these constants of motion commute with each other in the sense of underlying Hamiltonian dynamics. In absence of both gravity and surface tension, the residues of Pi_w(w,t) are also the constants of motion while nonzero gravity g ensures a trivial linear dependence of these residues on time. A Laurent series expansion of both z_w(w,t) and Pi_w(w,t) at each poles position reveals an existence of additional integrals of motion for poles of the second order. If all poles are simple then the number of independent real integrals of motion is 4N for zero gravity and 4N-1 for nonzero gravity. For the second order poles we found 6N motion integral for zero gravity and 6N-1 for nonzero gravity. We suggest that the existence of these nontrivial constants of motion provides an argument in support of the conjecture of complete integrability of free surface hydrodynamics in deep water.

nlin.SI↗

arXiv2019

Non-Canonical Hamiltonian Structure and Poisson Bracket for 2D Hydrodynamics with Free Surface

We consider Euler equations for potential flow of ideal incompressible fluid with a free surface and infinite depth in two dimensional geometry. Both gravity forces and surface tension are taken int account. A time-dependent conformal mapping is used which maps a lower complex half plane of the auxiliary complex variable $w$ into a fluid's area with the real line of $w$ mapped into the free fluid's surface. We reformulate the exact Eulerian dynamics through a non-canonical nonlocal Hamiltonian structure for a pair of the Hamiltonian variables. These two variables are the imaginary part of the conformal map and the fluid's velocity potential both evaluated of fluid's free surface. The corresponding Poisson bracket is non-degenerate, i.e. it does not have any Casimir invariant. Any two functionals of the conformal mapping commute with respect to the Poisson bracket. New Hamiltonian structure is a generalization of the canonical Hamiltonian structure of Ref. V.E. Zakharov, J. Appl. Mech. Tech. Phys. 9, 190 (1968) which is valid only for solutions for which the natural surface parametrization is single valued, i.e. each value of the horizontal coordinate corresponds only to a single point on the free surface. In contrast, new non-canonical Hamiltonian equations are valid for arbitrary nonlinear solutions (including multiple-valued natural surface parametrization) and are equivalent to Euler equations. We also consider a generalized hydrodynamics with the additional physical terms in the Hamiltonian beyond the Euler equations. In that case we identified powerful reductions which allowed to find general classes of particular solutions.

nlin.SI↗

arXiv2016

Evolution of One-Dimensional Wind-Driven Sea Spectra

We analyze modern operational models of wind wave prediction on the subject for compliance dissipation. Our numerical simulations from the "first principle" demonstrate that heuristic formulas for damping rate of free wind sea due to "white capping" (or wave breaking) dramatically exaggerates the role of this effect in these models.

physics.flu-dyn↗

arXiv2015

Super compact equation for water waves

We derive very simple compact equation for gravity water waves which includes nonlinear wave term (`a la NLSE) and advection term (may results in wave breaking).

physics.flu-dyn↗

arXiv2015

Numerical simulation of surface waves instability on a discrete grid

We perform full-scale numerical simulation of instability of weakly nonlinear waves on the surface of deep fluid. We show that the instability development leads to chaotization and formation of wave turbulence. We study instability both of propagating and standing waves. We studied separately pure capillary wave unstable due to three-wave interactions and pure gravity waves unstable due to four-wave interactions. The theoretical description of instabilities in all cases is included into the article. The numerical algorithm used in these and many other previous simulations performed by authors is described in details.

physics.comp-ph↗

arXiv2012

Free-Surface Hydrodynamics in the conformal variables

The potential flow of two-dimensional ideal incompressible fluid with a free surface is studied. Using the theory of conformal mappings and Hamiltonian formalism allows us to derive exact equations of surface evolution. Simple form of the equations helped to discover new integrals of motion. These integrals are connected with the analytical properties of conformal mapping and complex velocity. Simple form of the equations also makes the numerical simulations of the free surface evolution very straightforward. In the limit of almost flat surface the equations can be reduced to the Hopf equation.

physics.flu-dyn↗

arXiv2012

Collision of two breathers at surface of deep water

We applied canonical transformation to water wave equation not only to remove cubic nonlinear terms but to simplify drastically fourth order terms in Hamiltonian. This transformation explicitly uses the fact of vanishing exact four waves interaction for water gravity waves for 2D potential fluid. After the transformation well-known but cumbersome Zakharov equation is drastically simplified and can be written in X-space in compact way. This new equation is very suitable as for analytic study as for numerical simulation. Localized in space breather-type solution was found. Numerical simulation of collision of two such breathers strongly supports hypothesis of integrability of 2-D free surface hydrodynamics.

physics.flu-dyn↗

arXiv2007

Theory of weakly damped free-surface flows: a new formulation based on potential flow solutions

Several theories for weakly damped free-surface flows have been formulated. In this paper we use the linear approximation to the Navier-Stokes equations to derive a new set of equations for potential flow which include dissipation due to viscosity. A viscous correction is added not only to the irrotational pressure (Bernoulli's equation), but also to the kinematic boundary condition. The nonlinear Schrödinger (NLS) equation that one can derive from the new set of equations to describe the modulations of weakly nonlinear, weakly damped deep-water gravity waves turns out to be the classical damped version of the NLS equation that has been used by many authors without rigorous justification.

physics.ao-ph↗

arXiv2005

Mesoscopic wave turbulence

We report results of sumulation of wave turbulence. Both inverse and direct cascades are observed. The definition of "mesoscopic turbulence" is given. This is a regime when the number of modes in a system involved in turbulence is high enough to qualitatively simulate most of the processes but significantly smaller then the threshold which gives us quantitative agreement with the statistical description, such as kinetic equation. Such a regime takes place in numerical simulation, in essentially finite systems, etc.

physics.flu-dyn↗

arXiv2004

Weak Turbulent Kolmogorov Spectrum for Surface Gravity Waves

We study the long-time evolution of gravity waves on deep water exited by the stochastic external force concentrated in moderately small wave numbers. We numerically implement the primitive Euler equations for the potential flow of an ideal fluid with free surface written in canonical variables, using expansion of the Hamiltonian in powers of nonlinearity of up to fourth order terms. We show that due to nonlinear interaction processes a stationary energy spectrum close to $|k| \sim k^{-7/2}$ is formed. The observed spectrum can be interpreted as a weak-turbulent Kolmogorov spectrum for a direct cascade of energy.

physics.flu-dyn↗

arXiv2003

Decay of the monochromatic capillary wave

It was demonstrated by direct numerical simulation that, in the case of weakly nonlinear capillary waves, one can get resonant waves interaction on the discrete grid when resonant conditions are never fulfilled exactly. The waves's decay pattern was obtained. The influence of the mismatch of resonant condition was studied as well.

physics.flu-dyn↗

arXiv2003

Weak turbulence of gravity waves

For the first time weak turbulent theory was demonstrated for the surface gravity waves. Direct numerical simulation of the dynamical equations shows Kolmogorov turbulent spectra as predicted by analytical analysis from kinetic equation.

physics.flu-dyn↗