scholarly journals Existence of Incompressible Vortex-Class Phenomena and Variational Formulation of Raleigh–Plesset Cavitation Dynamics

2021 ◽  
Vol 2 (3) ◽  
pp. 613-630
Author(s):  
Terry Moschandreou ◽  
Keith Afas
Keyword(s):  
Fluid Mechanics ◽  
Density Functional ◽  
Lagrangian Density ◽  
Stokes Equations ◽  
Closed Surface ◽  
Navier Stokes ◽  
The Novel ◽  
Navier Stokes Equations ◽  
Variational Framework ◽  
First Time

The following article extends a decomposition to the Navier–Stokes Equations (NSEs) demonstrated in earlier studies by corresponding author, in order to now demonstrate the existence of a vortex elliptical set inherent to the NSEs. These vortice elliptical sets are used to comment on the existence of solutions relative to the NSEs and to identify a potential manner of investigation into the classical Millennial Problem encompassed in Fefferman’s presentation. The article also presents the utilization of a recently developed versatile variational framework by both authors in order to study a related fluid-mechanics phenomena, namely the Raleigh–Plesset equations, which are ultimately obtained from the NSEs. The article develops, for the first time, a Lagrangian density functional for a closed surface which when minimized produced the Raleigh–Plesset equations. The article then proceeds with the demonstration that the Raleigh–Plesset equations may be obtained from thisenergy functional and identifies the energy dissipation predicted by the proposed Lagrangian density. The importance of the novel Raleigh–Plesset functional in the greater scheme of fluid mechanics is commented upon.

scholarly journals The inf-sup stability of the lowest order Taylor–Hood pair on affine anisotropic meshes

2019 ◽  
Vol 40 (4) ◽  
pp. 2377-2398
Author(s):  
Gabriel R Barrenechea ◽  
Andreas Wachtel
Keyword(s):  
Finite Element ◽  
Fluid Mechanics ◽  
Incompressible Fluid ◽  
Stokes Equations ◽  
Sufficient Conditions ◽  
Navier Stokes ◽  
Element Solution ◽  
Navier Stokes Equations ◽  
Anisotropic Meshes ◽  
Theoretical Results

Abstract Uniform inf-sup conditions are of fundamental importance for the finite element solution of problems in incompressible fluid mechanics, such as the Stokes and Navier–Stokes equations. In this work we prove a uniform inf-sup condition for the lowest-order Taylor–Hood pairs $\mathbb{Q}_2\times \mathbb{Q}_1$ and $\mathbb{P}_2\times \mathbb{P}_1$ on a family of affine anisotropic meshes. These meshes may contain refined edge and corner patches. We identify necessary hypotheses for edge patches to allow uniform stability and sufficient conditions for corner patches. For the proof, we generalize Verfürth’s trick and recent results by some of the authors. Numerical evidence confirms the theoretical results.

scholarly journals Fast solvers for finite difference approximations for the stokes and navier-stokes equations

1996 ◽  
Vol 38 (2) ◽  
pp. 274-290 ◽  
Author(s):  
Dongho Shin ◽  
John C. Strikwerda
Keyword(s):  
Finite Difference ◽  
Stokes Equations ◽  
Linear Equations ◽  
Computational Effort ◽  
Navier Stokes ◽  
Fast Solvers ◽  
Navier Stokes Equations ◽  
Finite Difference Approximations ◽  
Grid Points ◽  
First Time

AbstractWe consider several methods for solving the linear equations arising from finite difference discretizations of the Stokes equations. The two best methods, one presented here for the first time, apparently, and a second, presented by Bramble and Pasciak, are shown to have computational effort that grows slowly with the number of grid points. The methods work with second-order accurate discretizations. Computational results are shown for both the Stokes equations and incompressible Navier-Stokes equations at low Reynolds number.

Navier-Stokes Simulation of the Flow Around an Airfoil in Darrieus Motion

1994 ◽  
Vol 116 (4) ◽  
pp. 870-876 ◽  
Author(s):  
Ko-Foa Tchon ◽  
Ion Paraschivoiu
Keyword(s):  
Stokes Equations ◽  
Navier Stokes ◽  
External Boundary ◽  
Time Step ◽  
Navier Stokes Equations ◽  
Infinite Elements ◽  
Quadrilateral Elements ◽  
Minimum Residual ◽  
Naca 0015 ◽  
First Time

In order to study the dynamic stall phenomenon on a Darrieus wind turbine, the incompressible flow field around a moving airfoil is simulated using a noninertial stream function-vorticity formulation of the two-dimensional unsteady Navier-Stokes equations. Spatial discretization is achieved by the streamline upwind Petrov-Galerkin finite element method on a hybrid mesh composed of a structured region of quadrilateral elements in the vicinity of solid boundaries, an unstructured region of triangular elements elsewhere, and a layer of infinite elements surrounding the domain and projecting the external boundary to infinity. Temporal discretization is achieved by an implicit second order finite difference scheme. At each time step, a nonlinear algebraic system is solved by a Newton method. To accelerate computations, the generalized minimum residual method with an incomplete triangular factorization preconditioning is used to solve the linearized Newton systems. The solver is applied to simulate the flow around a NACA 0015 airfoil in Darrieus motion and the results are compared to experimental observations. To the authors’ knowledge, it is the first time that the simulation of such a motion has been performed using the Navier-Stokes equations.

scholarly journals Comparing fluid mechanics models with experimental data

2003 ◽  
Vol 358 (1437) ◽  
pp. 1567-1576 ◽  
Author(s):  
G. R. Spedding
Keyword(s):  
Fluid Mechanics ◽  
Stokes Equations ◽  
Physical World ◽  
Experimental Models ◽  
Experimental Results ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Dimensionless Groups ◽  
Model Predictions ◽  
Provided Examples

The art of modelling the physical world lies in the appropriate simplification and abstraction of the complete problem. In fluid mechanics, the Navier–Stokes equations provide a model that is valid under most circumstances germane to animal locomotion, but the complexity of solutions provides strong incentive for the development of further, more simplified practical models. When the flow organizes itself so that all shearing motions are collected into localized patches, then various mathematical vortex models have been very successful in predicting and furthering the physical understanding of many flows, particularly in aerodynamics. Experimental models have the significant added convenience that the fluid mechanics can be generated by a real fluid, not a model, provided the appropriate dimensionless groups have similar values. Then, analogous problems can be encountered in making intelligible but independent descriptions of the experimental results. Finally, model predictions and experimental results may be compared if, and only if, numerical estimates of the likely variations in the tested quantities are provided. Examples from recent experimental measurements of wakes behind a fixed wing and behind a bird in free flight are used to illustrate these principles.

Numerical Analysis of the Sulcus Vocalis Disorder on the Function of the Vocal Folds

2016 ◽  
Vol 33 (4) ◽  
pp. 513-520 ◽  
Author(s):  
A. Vazifehdoostsaleh ◽  
N. Fatouraee ◽  
M. Navidbakhsh ◽  
F. Izadi
Keyword(s):  
Stokes Equations ◽  
Vocal Folds ◽  
Element Model ◽  
Navier Stokes ◽  
Arbitrary Lagrangian Eulerian ◽  
Navier Stokes Equations ◽  
Flow Parameters ◽  
Complex Process ◽  
Sulcus Vocalis ◽  
First Time

AbstractSpeaking is a very complex process resulting from the interaction between the air flow along the larynx and the vibrating structure of the vocal folds. Sulcus is a disease missing layers in the vocal folds result in cracks resulting in some disorders in producing sounds. Sulcus and its effects on the vocal cord vibrations are numerically studied for the first time in this paper. An ideal model of healthy vocal folds and Sulcus vocalis has been two-dimensionally defined and the finite element model of vocal folds is solved in a fully coupled form. The proposed calculative model was used in a fluid range of the computational fluid dynamics, arbitrary Lagrangian-Eulerian (ALE), incompressible continuity and Navier-Stokes equations and in a structure range of a three-layer elastic linear model. Self-excited oscillations were presented for vocal folds among type II patients and compared with healthy models. Responses were qualitatively and quantitatively studied. The healthy model was compared with numerical and empirical results. In addition, the effects of the disease on the flow parameters and the vibration frequency of the vocal folds were studied. According to the simulated model, the oscillation frequency decreased 25% and the average and instantaneous volume flux significantly increased compared to healthy samples. Results may help present a guideline for surgery and subsequently evaluate patients’ improvement.

scholarly journals Application of Local Gauge Theories to Fluid Mechanics

2019 ◽  
Author(s):  
Thomas Merz
Keyword(s):  
Fluid Dynamics ◽  
Fluid Mechanics ◽  
Strain Rate ◽  
Gauge Theories ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Gauge Fields ◽  
Strain Rate Tensor ◽  
Hidden Symmetries ◽  
Navier Stokes Equations

The problem of fluid dynamics can be greatly simplified if, for every point in space, the strain-rate tensor is diagonalized. This tensor is introduced into the Navier-Stokes equations via material law and divergence of the stress tensor. This article shows that local SO(3)xU(1) gauge fields can be used to locally diagonalize the diffusion components of the strain-rate tensor. The gauge fields resulting from the connection can be interpreted as convection components of the flow, they show properties of quasiparticles and can be interpreted as elementary vortices. Thus, the proposed approach not only offers new insights for the solution and situative simplification of the Navier-Stokes equations, it also uncovers hidden symmetries within the flow convection, allowing - depending on boundary conditions - further interpretation.

Introductory Incompressible Fluid Mechanics

2021 ◽  
Author(s):  
Frank H. Berkshire ◽  
Simon J. A. Malham ◽  
J. Trevor Stuart
Keyword(s):  
Fluid Mechanics ◽  
Incompressible Fluid ◽  
Stokes Equations ◽  
Fluid Flows ◽  
Teaching Experience ◽  
Navier Stokes ◽  
Inviscid Fluid ◽  
Science And Engineering ◽  
Navier Stokes Equations ◽  
Global Existence Of Solutions

This introduction to the mathematics of incompressible fluid mechanics and its applications keeps prerequisites to a minimum – only a background knowledge in multivariable calculus and differential equations is required. Part One covers inviscid fluid mechanics, guiding readers from the very basics of how to represent fluid flows through to the incompressible Euler equations and many real-world applications. Part Two covers viscous fluid mechanics, from the stress/rate of strain relation to deriving the incompressible Navier-Stokes equations, through to Beltrami flows, the Reynolds number, Stokes flows, lubrication theory and boundary layers. Also included is a self-contained guide on the global existence of solutions to the incompressible Navier-Stokes equations. Students can test their understanding on 100 progressively structured exercises and look beyond the scope of the text with carefully selected mini-projects. Based on the authors' extensive teaching experience, this is a valuable resource for undergraduate and graduate students across mathematics, science, and engineering.

scholarly journals Hidden fluid mechanics: Learning velocity and pressure fields from flow visualizations

Science ◽  
2020 ◽  
Vol 367 (6481) ◽  
pp. 1026-1030 ◽  
Author(s):  
Maziar Raissi ◽  
Alireza Yazdani ◽  
George Em Karniadakis
Keyword(s):  
Fluid Mechanics ◽  
Stokes Equations ◽  
Fluid Motion ◽  
Quantitative Information ◽  
Navier Stokes ◽  
Observation Data ◽  
Navier Stokes Equations ◽  
Learning Framework ◽  
Pressure Fields ◽  
Potential Applications

For centuries, flow visualization has been the art of making fluid motion visible in physical and biological systems. Although such flow patterns can be, in principle, described by the Navier-Stokes equations, extracting the velocity and pressure fields directly from the images is challenging. We addressed this problem by developing hidden fluid mechanics (HFM), a physics-informed deep-learning framework capable of encoding the Navier-Stokes equations into the neural networks while being agnostic to the geometry or the initial and boundary conditions. We demonstrate HFM for several physical and biomedical problems by extracting quantitative information for which direct measurements may not be possible. HFM is robust to low resolution and substantial noise in the observation data, which is important for potential applications.

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