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Water
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Stochastic Modeling in Fluid Dynamics
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Stochastic Modeling in Fluid Dynamics

A special issue of Water (ISSN 2073-4441). This special issue belongs to the section "Hydraulics and Hydrodynamics".

Deadline for manuscript submissions: closed (31 December 2020) | Viewed by 8202

Special Issue Editor

Prof. Dr. Ana Bela Cruzeiro

E-Mail Website
Guest Editor
GFMUL and Dep. Mathematics IST, Univ. Lisbon, Av. Rovisco Pais, 1049-001 Lisboa, Portugal
Interests: stochastic analysis; infinite dimensional analysis; probability in hydrodynamics; stochastic geometric mechanics

Special Issue Information

Dear Colleagues,

In recent years, stochastic modeling in fluid dynamics has witnessed significant progress. Weather forecast, ocean modeling, turbulence or gas dynamics are among the topics of application.

Probability has for a long time already played a role in modeling many physical phenomena, since uncertainty is always present in measurements and in every computational simulation. There are several ways to introduce randomness in a model: describing initial data by probability measures is one of them and corresponds to statistical approaches, while another consists in considering noise perturbations or random forces in the equations. Since fluid dynamics is described by partial differential equations, this leads to stochastic partial differential equations.

One can also interpret deterministic models such as the celebrated Navier–Stokes equation using stochastic formulations, revealing a more intrinsic probability nature of fluid dynamics. Numerical schemes for stochastic equations have also been largely developed recently.

The Special Issue will be devoted to contributions to stochastic models in fluid dynamical models, either theoretical or numerical.

Submitted contributions will go through a peer review process performed by independent reviewers. Original review papers are also invited for publication in this Special Issue.

Prof. Dr. Ana Bela Cruzeiro
Guest Editor

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Water is an international peer-reviewed open access semimonthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • Stochastic fluid dynamics
  • Stochastic Lagrangian flows
  • Stochastic partial differential equations
  • Stochastic numerical methods
  • Euler and Navier–Stokes equations

Published Papers (4 papers)

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Research

44 pages, 2193 KiB  
Article
Uncertainty Relations in Hydrodynamics
by Gyell Gonçalves de Matos, Takeshi Kodama and Tomoi Koide
Water 2020, 12(11), 3263; https://doi.org/10.3390/w12113263 - 21 Nov 2020
Cited by 9 | Viewed by 2287
Abstract
The qualitative behaviors of uncertainty relations in hydrodynamics are numerically studied for fluids with low Reynolds numbers in 1+1 dimensional system. We first give a review for the formulation of the generalized uncertainty relations in the stochastic variational method (SVM), following [...] Read more.
The qualitative behaviors of uncertainty relations in hydrodynamics are numerically studied for fluids with low Reynolds numbers in 1+1 dimensional system. We first give a review for the formulation of the generalized uncertainty relations in the stochastic variational method (SVM), following the work by two of the present authors [Phys. Lett. A 382, 1472 (2018)]. In this approach, the origin of the finite minimum value of uncertainty is attributed to the non-differentiable (virtual) trajectory of a quantum particle and then both of the Kennard and Robertson-Schrödinger inequalities in quantum mechanics are reproduced. The same non-differentiable trajectory is applied to the motion of fluid elements in the Navier-Stokes-Fourier equation or the Navier-Stokes-Korteweg equation. By introducing the standard deviations of position and momentum for fluid elements, the uncertainty relations in hydrodynamics are derived. These are applicable even to the Gross-Pitaevskii equation and then the field-theoretical uncertainty relation is reproduced. We further investigate numerically the derived relations and find that the behaviors of the uncertainty relations for liquid and gas are qualitatively different. This suggests that the uncertainty relations in hydrodynamics are used as a criterion to classify liquid and gas in fluid. Full article
(This article belongs to the Special Issue Stochastic Modeling in Fluid Dynamics)
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16 pages, 320 KiB  
Article
Weak Solution for 3D-Stochastic Third Grade Fluid Equations
by Adilson Almeida and Fernanda Cipriano
Water 2020, 12(11), 3211; https://doi.org/10.3390/w12113211 - 16 Nov 2020
Cited by 3 | Viewed by 1844
Abstract
This article studies the stochastic evolution of incompressible non-Newtonian fluids of differential type. More precisely, we consider the equations governing the dynamic of a third grade fluid filling a three-dimensional bounded domain O, perturbed by a multiplicative white noise. Taking the initial [...] Read more.
This article studies the stochastic evolution of incompressible non-Newtonian fluids of differential type. More precisely, we consider the equations governing the dynamic of a third grade fluid filling a three-dimensional bounded domain O, perturbed by a multiplicative white noise. Taking the initial condition in the Sobolev space H2(O), and supplementing the equations with a Navier slip boundary condition, we establish the existence of a global weak stochastic solution with sample paths in L∞(0,T;H2(O)). Full article
(This article belongs to the Special Issue Stochastic Modeling in Fluid Dynamics)
17 pages, 330 KiB  
Article
Stochastic Modelling of Small-Scale Perturbation
by Franco Flandoli and Umberto Pappalettera
Water 2020, 12(10), 2950; https://doi.org/10.3390/w12102950 - 21 Oct 2020
Cited by 5 | Viewed by 1810
Abstract
In this paper we propose a stochastic model reduction procedure for deterministic equations from geophysical fluid dynamics. Once large-scale and small-scale components of the dynamics have been identified, our method consists in modelling stochastically the small scales and, as a result, we obtain [...] Read more.
In this paper we propose a stochastic model reduction procedure for deterministic equations from geophysical fluid dynamics. Once large-scale and small-scale components of the dynamics have been identified, our method consists in modelling stochastically the small scales and, as a result, we obtain that a transport-type Stratonovich noise is sufficient to model the influence of the small scale structures on the large scales ones. This work aims to contribute to motivate the use of stochastic models in fluid mechanics and identifies examples of noise of interest for the reduction of complexity of the interaction between scales. The ideas are presented in full generality and applied to specific examples in the last section. Full article
(This article belongs to the Special Issue Stochastic Modeling in Fluid Dynamics)
16 pages, 339 KiB  
Article
A Hamiltonian Interacting Particle System for Compressible Flow
by Simon Hochgerner
Water 2020, 12(8), 2109; https://doi.org/10.3390/w12082109 - 25 Jul 2020
Cited by 1 | Viewed by 1635
Abstract
The decomposition of the energy of a compressible fluid parcel into slow (deterministic) and fast (stochastic) components is interpreted as a stochastic Hamiltonian interacting particle system (HIPS). It is shown that the McKean–Vlasov equation associated to the mean field limit yields the barotropic [...] Read more.
The decomposition of the energy of a compressible fluid parcel into slow (deterministic) and fast (stochastic) components is interpreted as a stochastic Hamiltonian interacting particle system (HIPS). It is shown that the McKean–Vlasov equation associated to the mean field limit yields the barotropic Navier–Stokes equation with density-dependent viscosity. Capillary forces can also be treated by this approach. Due to the Hamiltonian structure, the mean field system satisfies a Kelvin circulation theorem along stochastic Lagrangian paths. Full article
(This article belongs to the Special Issue Stochastic Modeling in Fluid Dynamics)
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