scholarly journals Vectorial variational problems in L ∞ constrained by the Navier–Stokes equations*

Nonlinearity ◽  
2021 ◽  
Vol 35 (1) ◽  
pp. 470-491
Author(s):  
Ed Clark ◽  
Nikos Katzourakis ◽  
Boris Muha
Keyword(s):  
Stokes Equations ◽  
Weather Forecasting ◽  
Variational Problems ◽  
Euler System ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Atmospheric Flows ◽  
Lagrange System ◽  
Minimisation Problem ◽  
Cost Functionals

Abstract We study a minimisation problem in L p and L ∞ for certain cost functionals, where the class of admissible mappings is constrained by the Navier–Stokes equations. Problems of this type are motivated by variational data assimilation for atmospheric flows arising in weather forecasting. Herein we establish the existence of PDE-constrained minimisers for all p, and also that L p minimisers converge to L ∞ minimisers as p → ∞. We further show that L p minimisers solve an Euler–Lagrange system. Finally, all special L ∞ minimisers constructed via approximation by L p minimisers are shown to solve a divergence PDE system involving measure coefficients, which is a divergence-form counterpart of the corresponding non-divergence Aronsson–Euler system.

A Godunov-Type Scheme for Atmospheric Flows on Unstructured Grids: Euler and Navier-Stokes Equations

2007 ◽  
Vol 164 (1) ◽  
pp. 217-244 ◽  
Author(s):  
Nash'at Ahmad ◽  
Zafer Boybeyi ◽  
Rainald Löhner ◽  
Ananthakrishna Sarma
Keyword(s):  
Stokes Equations ◽  
Unstructured Grids ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Atmospheric Flows

scholarly journals Partial differential equations for oceanic artificial intelligence

2021 ◽  
Vol 70 ◽  
pp. 137-146
Author(s):  
Jules Guillot ◽  
Guillaume Koenig ◽  
Kadi Minbashian ◽  
Emmanuel Frénod ◽  
Héléne Flourent ◽  
...  
Keyword(s):  
Data Model ◽  
Stokes Equations ◽  
Weather Forecasting ◽  
Navier Stokes ◽  
Model Coupling ◽  
Navier Stokes Equations ◽  
Coupling Approach ◽  
Prediction Systems ◽  
Applications Of Machine Learning ◽  
The One

The Sea Surface Temperature (SST) plays a significant role in analyzing and assessing the dynamics of weather and also biological systems. It has various applications such as weather forecasting or planning of coastal activities. On the one hand, standard physical methods for forecasting SST use coupled ocean- atmosphere prediction systems, based on the Navier-Stokes equations. These models rely on multiple physical hypotheses and do not optimally exploit the information available in the data. On the other hand, despite the availability of large amounts of data, direct applications of machine learning methods do not always lead to competitive state of the art results. Another approach is to combine these two methods: this is data-model coupling. The aim of this paper is to use a model in another domain. This model is based on a data-model coupling approach to simulate and predict SST. We first introduce the original model. Then, the modified model is described, to finish with some numerical results.

Bounds for turbulent shear flow

1970 ◽  
Vol 41 (1) ◽  
pp. 219-240 ◽  
Author(s):  
F. H. Busse
Keyword(s):  
Shear Flow ◽  
Turbulent Flow ◽  
Stokes Equations ◽  
Equations Of Motion ◽  
Variational Problems ◽  
Navier Stokes ◽  
Turbulent Shear ◽  
Turbulent Shear Flow ◽  
Navier Stokes Equations ◽  
Experimental Values

Bounds on the transport of momentum in turbulent shear flow are derived by variational methods. In particular, variational problems for the turbulent regimes of plane Couette flow, channel flow, and pipe flow are considered. The Euler equations resemble the basic Navier–Stokes equations of motion in many respects and may serve as model equations for turbulence. Moreover, the comparison of the upper bound with the experimental values of turbulent momentum transport shows a rather close similarity. The same fact holds with respect to other properties when the observed turbulent flow is compared with the structure of the extremalizing solution of the variational problem. It is suggested that the instability of the sublayer adjacent to the walls is responsible for the tendency of the physically realized turbulent flow to approach the properties of the extremalizing vector field.

A hydrodynamic model for the interaction of Cucker–Smale particles and incompressible fluid

2014 ◽  
Vol 24 (11) ◽  
pp. 2311-2359 ◽  
Author(s):  
Seung-Yeal Ha ◽  
Moon-Jin Kang ◽  
Bongsuk Kwon
Keyword(s):  
Incompressible Fluid ◽  
Hydrodynamic Model ◽  
Stokes Equations ◽  
Euler System ◽  
Navier Stokes ◽  
Classical Solutions ◽  
Hydrodynamic Models ◽  
Navier Stokes Equations ◽  
Proposed Model ◽  
Non Local

We present a new hydrodynamic model for the interactions between collision-free Cucker–Smale flocking particles and a viscous incompressible fluid. Our proposed model consists of two hydrodynamic models. For the Cucker–Smale flocking particles, we employ the pressureless Euler system with a non-local flocking dissipation, whereas for the fluid, we use the incompressible Navier–Stokes equations. These two hydrodynamic models are coupled through a drag force, which is the main flocking mechanism between the particles and the fluid. The flocking mechanism between particles is regulated by the Cucker–Smale model, which accelerates global flocking between the particles and the fluid. We show that this model admits the global-in-time classical solutions, and exhibits time-asymptotic flocking, provided that the initial data is appropriately small. In the course of our analysis for the proposed system, we first consider the hydrodynamic Cucker–Smale equations (the pressureless Euler system with a non-local flocking dissipation), for which the global existence and the time-asymptotic behavior of the classical solutions are also investigated.

Modelling atmospheric flows

Acta Numerica ◽  
2007 ◽  
Vol 16 ◽  
pp. 67-154 ◽  
Author(s):  
Mike Cullen
Keyword(s):  
Current Practice ◽  
Stokes Equations ◽  
Numerical Algorithms ◽  
Navier Stokes ◽  
Atmospheric Models ◽  
Navier Stokes Equations ◽  
Atmospheric Flows ◽  
Production Models ◽  
Weather And Climate ◽  
Limit Solutions

This article demonstrates how numerical methods for atmospheric models can be validated by showing that they give the theoretically predicted rate of convergence to relevant asymptotic limit solutions. This procedure is necessary because the exact solution of the Navier–Stokes equations cannot be resolved by production models. The limit solutions chosen are those most important for weather and climate prediction. While the best numerical algorithms for this purpose largely reflect current practice, some important limit solutions cannot be captured by existing methods. The use of Lagrangian rather than Eulerian averaging may be required in these cases.

scholarly journals Numerical investigations of the compressible Navier-Stokes system

2021 ◽  
Vol 70 ◽  
pp. 1-13
Author(s):  
Bilal Al-Taki ◽  
Kevin Atsou ◽  
Jean-Jéróme Casanova ◽  
Thierry Goudon ◽  
Pauline Lafitte ◽  
...  
Keyword(s):  
Stokes Equations ◽  
Euler System ◽  
Navier Stokes ◽  
List Type ◽  
Navier Stokes Equations ◽  
One Dimensional ◽  
Staggered Scheme ◽  
The One ◽  
Lagrangian Scheme ◽  
Good Agreement

In this paper we write, analyze and experimentally compare three different numerical schemes dedicated to the one dimensional barotropic Navier-Stokes equations: a staggered scheme based on the Rusanov one for the inviscid (Euler) system,a staggered pseudo-Lagrangian scheme in which the mesh “follows” the fluid,the Eulerian projection (on a fixed mesh) of the preceding scheme. All these schemes only involve the resolution of linear systems (all the nonlinear terms are solved in an explicit way). We propose numerical illustrations of their behaviors on particular solutions in which the density has discontinuities (hereafter called Hoff solutions). We show that the three schemes seem to converge to the same solutions, and we compare the evolution of the amplitude of the discontinuity of the numerical solution (with the pseudo-Lagrangian scheme) with the one predicted by Hoff and observe a good agreement.

Effects of stator splitter blades on aerodynamic performance of a single-stage transonic axial compressor

2020 ◽  
Vol 14 (4) ◽  
pp. 7369-7378
Author(s):  
Ky-Quang Pham ◽  
Xuan-Truong Le ◽  
Cong-Truong Dinh
Keyword(s):  
Stokes Equations ◽  
Three Dimensional ◽  
Axial Compressor ◽  
Aerodynamic Performance ◽  
Numerical Results ◽  
Single Stage ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Operating Stability ◽  
Length Coverage

Splitter blades located between stator blades in a single-stage axial compressor were proposed and investigated in this work to find their effects on aerodynamic performance and operating stability. Aerodynamic performance of the compressor was evaluated using three-dimensional Reynolds-averaged Navier-Stokes equations using the k-e turbulence model with a scalable wall function. The numerical results for the typical performance parameters without stator splitter blades were validated in comparison with experimental data. The numerical results of a parametric study using four geometric parameters (chord length, coverage angle, height and position) of the stator splitter blades showed that the operational stability of the single-stage axial compressor enhances remarkably using the stator splitter blades. The splitters were effective in suppressing flow separation in the stator domain of the compressor at near-stall condition which affects considerably the aerodynamic performance of the compressor.

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