Navier-Stokes Equations

In the study of local regularity of weak solutions to systems related to incompressible viscous fluids local energy estimates serve as important ingredients. However, this requires certain informations on the pressure. This fact has been used by V. Scheffer in the notion of a suitable weak to the Navier-Stokes equation, and in the proof of the partial regularity due to Caffarelli. Kohn and Nirenberg. In general domains, or in case of complex viscous fluid models a global pressure doesn't necessarily exist. To overcome this problem, in the present paper we construct a local pressure distribution by showing that every distribution ∂ t u + F , which vanishs on the set of smooth solenoidal vector fields can be represented by a distribution ∂ t ∇p h + ∇p 0 , where ∇p h ∼ u and ∇p 0 ∼ F .

This paper studies and contrasts the performances of three iterative methods for computing the solution of large sparse linear systems arising in the numerical computations of incompressible Navier-Stokes equations. The emphasis is on the traditional Gauss-Seidel (GS) and Point Successive Over-relaxation (PSOR) algorithms as well as Krylov projection techniques such as Generalized Minimal Residual (GMRES). The performances of these three solvers for the second-order finite difference algebraic equations are comparatively studied by their application to solve a benchmark problem in Computational Fluid Dynamics (CFD). It is found that as the mesh size increases, GMRES gives the fastest convergence rate in terms of cpu time and number of iterations.

In this work we present final solving Millennium Prize Problems formulated Clay Math. Inst., Cambridge. A new uniform time estimation of the Cauchy problem solution for the Navier-Stokes equations is provided.Describes the loss of smoothness of classical solutions for the Navier-Stokes equations.

The viability and accuracy of large-eddy simulation (LES) with wall modeling for high Reynolds number complex turbulent flows is investigated by considering the flow around a circular cylinder in the supercritical regime. A simple wall stress model is employed to provide approximate boundary conditions to the LES. The results are compared with those obtained from steady and unsteady Reynolds-averaged Navier–Stokes (RANS) solutions and the available experimental data. The LES solutions are shown to be considerably ...

One important area of Maritime Simulations is the Wave effects of the ocean. Whilst there is significant work done in modeling deep ocean waves, the area of shallow water wave modeling has taken precedence in recent times. The objective of this thesis is, to study the possibilities of overcoming the barrier of high computation shallow water wave models that cannot be used for real time applications. Quoting from literature there exists three main approaches to model ocean waves: 1) Geometrical Description Models(constructed using periodic functions), 2) Spectral Models (using empirical data from Oceanic researches) & 3) Physical based models (from Computational Fluid Dynamics(CFD) based on numerical models). One key interest of this research is towards models that are able to achieve high accuracy in terms of the wave properties; thus guiding the user in making critical decisions and to predict close to real wave energy effects. It is noteworthy that only the approach of numerically solved physical based models (from the 3 approaches mentioned above) can provide wave parameters with high accuracy. Hence, this experiment is constrained to such physically based numerical models. All such models available for shallow-water simulations are inherently limited to not considering the depth of the water volume. Thus, the main focus of this thesis is towards a wave model that considers "depth" as an integral parameter in its calculations. The challenge imposed by such accurate models is the computation complexity that results in long processing time. Hence, they are not the most suitable choice for real-time simulators. This thesis experiments the possibility of a solution to this problem by restricting the simulation area to only that which has an impact on the vessel and by introducing cell reductions. The results obtained within the duration of this study, reveal that maintaining an optimal accuracy with such mesh-restrictions is not feasible and further efforts needs to be put in terms of parallel/gpu processing, profiling & etc.

Apresente as equações de Navier-Stokes. Discuta as conseqüências para fluido incompressível, escoamento irrotacional, escoamento não viscoso. Identifique a hipótese de Stokes. Escreva esta equação na sua forma integral (para volume de controle).

Computational fluid dynamics provide an efficient way to solve complex flow problems. Here, 2-D incompressible Navier Stokes equation for flow over a rectangular cylinder is solved using the Gauss-Seidel method with relaxation as an iterative method for Re = 200. Stream function-vorticity formulation of Navier - Stokes equation has been used.

Named after Claude-Louis Navier and George Gabriel Stokes, the Navier Stokes Equations are the fundamental governing equations to describe the motion of a viscous, heat conducting fluid substances. These equations are obtained by applying Newton’s Law of motion to a fluid element and are also called the momentum equation. Supplemented by the mass conservation equation, these equations are also referred to as energy equation or continuity equation. Navier Stokes equations assume that the stress tensor in the fluid element is the sum of a diffusing viscous term that is proportional to the gradient of velocity, plus a pressure term (Batchelor 2000).
Navier Stokes equations have wide range of applications in both academic and economical benefits. They are used as the basic algorithms in computational tools to simulate ocean currents, fluid flow in pipes, airflow around a foil and model the weather (Dean 2012). A Beam-warming algorithm coupled with Euler/Navier-Stokes equations can be applied for simulation of a transonic viscous flow over wings and the design of aircrafts. They can also help with the design of cars, mathematical modelling of the arterial blood flow in human body, and the design of power stations (Batchelor 2000). Last but not least, Maxwell’s equations in conjunction with Navier-Stokes equations can be used to design and study magnetohydrodynamics. (Dean 2012)
The aim of this essay is to initially describe the properties of the Navier-Stokes equations and then create a simple way to understand the main components of the equation by describing the mass conservation, momentum conservation and heat equations. The paper only focuses on the motion of incompressible fluids.

Final Degree Dissertation for my undergraduate in Mathematics at the University of the Basque Country. The dissertation is intended as an introduction to Sobolev spaces, with the objective of applying abstract results of Functional Analysis and Sobolev Spaces results to the study of Partial Differential Equations (PDEs).

Uno de los campos de la física más complicados de estudiar son los fluidos, el comportamiento de gases y líquidos en movimiento. Comprender, por ejemplo, los flujos de aire turbulento o los remolinos que se forman cuando el agua escurre por una tubería o la sangre por una arteria son de suma importancia, tanto para la ingeniería como para la medicina.

Las ecuaciones que rigen la dinámica de fluidos son las conocidas como ecuaciones de Navier-Stokes, producto del francés constructor de puentes Claude-Louis Navier y del matemá-tico irlandés George Stokes. El primero en obtener estas ecuaciones fue el francés en una época (1822) en que no se comprendía muy bien cuál era la física de la situación que estaba matematizando. De hecho, lo único que hizo fue modificar unas ecuaciones ya existentes y obtenidas por el famoso matemático Euler, de modo que incluyesen las fuerzas existentes entre las moléculas del fluido. Aproximadamente 20 años después, Stokes justificó las ecuaciones del ingeniero francés deduciéndolas adecuadamente. A pesar de que las ecuaciones de Navier-Stokes son sólo una aproximación del comportamiento real de los fluidos, se utilizan para estudiar cualquier aspecto que tenga que ver con éstos; el problema es que si uno estudia el movimiento de un fluido con estas ecuaciones, es incapaz de prever si ese movimiento se va a mantener siempre o se va a complicar.

This PhD thesis focuses on numerical and analytical methods for simulating the dynamics of volcanic ash plumes.
The study starts from the fundamental balance laws for a multiphase gas– particle mixture, reviewing the existing models and developing a new set of Partial Differential Equations (PDEs), well suited for modeling multiphase dispersed turbulence. In particular, a new model generalizing the equilibrium–Eulerian model to two-way coupled compressible flows is developed.
The PDEs associated to the four-way Eulerian-Eulerian model is studied, in- vestigating the existence of weak solutions fulfilling the energy inequalities of the PDEs. In particular, the convergence of sequences of smooth solutions to such a set of weak solutions is showed.
Having explored the well-posedness of multiphase systems, the three-dimensional compressible equilibrium–Eulerian model is discretized and numerically solved by using the OpenFOAM® numerical infrastructure. The new solver is called ASHEE, and it is verified and validated against a number of well understood benchmarks and experiments. It demonstrates to be capable to capture the key phenomena involved in the dynamics of volcanic ash plumes. Those are: turbulence, mixing, heat transfer, compressibility, preferential concentration of particles, plume entrainment.
The numerical solver is tested by taking advantage of the newest High Perfor- mance Computing infrastructure currently available.
Thus, ASHEE is used to simulate two volcanic plumes in realistic volcanological conditions. The influence of model configuration on the numerical solution is analyzed. In particular, a parametric analysis is performed, based on: 1) the kinematic decoupling model; 2) the subgrid scale model for turbulence; 3) the discretization resolution.
In a one-dimensional and steady-state approximation, the multiphase flow model is used to derive a model for volcanic plumes in a calm, stratified atmosphere. The corresponding Ordinary Differential Equations (ODEs) are written in a compact, dimensionless formulation. The six non-dimensional parameters characterizing a multiphase plume are then written. The ODEs is studied both numerically and analytically. Different regimes are analyzed, extracting the first integral of motion and asymptotic solutions. An asymptotic analytical solution approximating the model in the general regime is derived and compared with numerical results. Such a solution is coupled with an electromagnetic model providing the infrared intensity emitted by a volcanic ash plume. Key vent parameters are then retrieved by means of inversion techniques applied to infrared images measured during a real volcanic eruption.

The Fundamental theorem of vector calculus is based on the Helmholtz decomposition (sometimes called Helmholtz-Hodge decomposition) of any vector field into an irrotational part and a solenoidal part. In this paper we prove that Helmholtz decomposition is opened and require major revision. For Fundamental theorem of vector calculus we establish a new formula which completely corresponds to the Navier–Stokes and Lame (also called Navier) equations (equations of linear elasticity). This paper written in a way that gives insight to mathematicians, physicists, engineers who may not be experts in this topic (it is only comparison of information in different textbooks and its improving by clear counterexamples).

In this work, we present final solving Millennium Prize Problems formulated by Clay Math. Inst., Cambridge. A new uniform time estimation of the Cauchy problem solution for the Navier-Stokes equations is provided. We also describe the loss of smoothness of classical solutions for the Navier-Stokes equations.

Selecting compute nodes and solution grid generation are the first steps of numerical solutions. The most distinct manner is storing the values of dependent variables in the same set of nodes and using the identical control volumes for all variables. Such a grid is called Collocated. Collocated grid arrangement has many positive results in problems with complex solving range, especially with discontinuous boundary conditions. But this arrangement was not used for a long time for incompressible flow due to pressure and velocity isolation problems and creation of fluctuations in pressure. So the researchers in the mid-60s, have developed a new arrangement to reduce this isolation and increase the coupling between pressure and velocity. This new arrangement called staggered grid, provided the field of a new method for solving fluid flow problems called SIMPLE (Semi-Implicit Method for Pressure-Linked Equation) algorithm [1]. This report presents the solution to the continuity, Navier-Stokes equations. Standard fundamental methods like SIMPLER and primary variable formulation have been utilized. The results were analyzed for standard CFD test case-cavity flow. Different Reynold number (1000, 3000) and grid sizes with the finest meshes ie. (100×100), (1000×1000) have been studied.

A class of similarity solutions for two-dimensional unsteady flow in the neighbourhood of a front or rear stagnation point on a plane boundary is considered, and a wide range of possible behaviour is revealed, depending on whether the flow in the far field is accelerating or decelerating. The solutions, when they exist, are exact solutions of the Navier–Stokes equations, having a boundary-layer character analogous to that of the classical steady front stagnation point flow. The velocity profiles are obtained by numerical integration of a nonlinear ordinary differential equation. For the front-flow situation, the solution is unique for the accelerating case, but bifurcates for modest deceleration, while for sufficient rapid deceleration there exists a one-parameter family of solutions. For the rear-flow situation, a unique solution exists (remarkably!) for sufficiently strong acceleration, and a one-parameter family again exists for sufficient strong deceleration. Analytic results, which are consistent with the numerical results, are obtained in the limits of strong acceleration or deceleration, and for the asymptotic behaviour far from the boundary.

We present a high-order Implicit Large-Eddy Simulation (ILES) approach for transitional aerodynamic flows. The approach encompasses a hybridized Discontinuous Galerkin (DG) method for the discretization of the Navier–Stokes (NS) equations, and a parallel preconditioned Newton-GMRES solver for the resulting nonlinear system of equations. The combination of hybridized DG methods with an efficient solution procedure leads to a high-order accurate NS solver that is competitive to alternative approaches, such as finite volume and finite difference codes, in terms of computational cost. The proposed approach is applied to transitional flows over the NACA 65-(18)10 compressor cascade and the Eppler 387 wing at Reynolds numbers up to 460,000. Grid convergence studies are presented and the required resolution to capture transition at different Reynolds numbers is investigated. Numerical results show rapid convergence and excellent agreement with experimental data. In short, this work aims to demonstrate the potential of high-order ILES for simulating transitional aerodynamic flows. This is illustrated through numerical results and supported by theoretical considerations.

We derive an exact formula for solutions to the Stokes equations in the half-space with an external forcing term. This formula is used to establish local and global existence and uniqueness in a suitable Besov space for solutions to the Navier-Stokes equations. In particular, wellposedness is proved for initial data in L3(@+ ).

We explore interactions of elastic waves propagating in plates (with soil parameters) structured with concrete pillars buried in the soil. Pillars are 2 m in diameter, 30 m in depth and the plate is 50 m in thickness. We study the frequency range 5 to 10 Hz, for which Rayleigh wave wavelengths are smaller than the plate thickness. This frequency range is compatible with frequency ranges of particular interest in earthquake engineering. It is demonstrated in this paper that two seismic cloaks' configurations allow for an unprecedented flow of elastodynamic energy associated with Rayleigh surface waves. The first cloak design is inspired by some approximation of ideal cloaks' parameters within the framework of thin plate theory. The second, more accomplished but more involved, cloak design is deduced from a geometric transform in the full Navier equations that preserves the symmetry of the elasticity tensor but leads to Willis' equations, well approximated by a homogenization procedure, as corroborated by numerical simulations. The two cloaks's designs are strikingly different, and the superior efficiency of the second type of cloak emphasizes the necessity for rigor in transposition of existing cloaks's designs in thin plates to the geophysics setting. Importantly, we focus our attention on geometric transforms applied to thick plates, which is an intermediate case between thin plates and semi-infinite media, not studied previously. Cloaking efficiency (reduction of the disturbance of the wave wavefront and its amplitude behind an obstacle) and protection (reduction of the wave amplitude within the center of the cloak) are studied for ideal and approximated cloaks' parameters. These results represent a preliminary step towards designs of seismic cloaks for surface Rayleigh waves propagating in sedimentary soils structured with concrete pillars.

From the principle of least action the equation of motion for viscous compressible and charged fluid is derived. The viscosity effect is described by the 4-potential of the energy dissipation field, dissipation tensor and dissipation stress-energy tensor. In the weak field limit it is shown that the obtained equation is equivalent to the Navier-Stokes equation. The equation for the power of the kinetic energy loss is provided, the equation of motion is integrated, and the dependence of the velocity magnitude is determined. A complete set of equations is presented, which suffices to solve the problem of motion of viscous compressible and charged fluid in the gravitational and electromagnetic fields.

The motive of this paper is to put forward a general solution to Navier-stokes equation which describes the motion of viscous fluid substances, derived by applying Newton's second law to fluid motion. These equations are the set of coupled differential equations, which are too difficult to solve analytically.

We prove the global well-posedness for the 3D Navier-Stokes equations in critical Fourier-Herz spaces, by making use of the Fourier localization method and the Littlewood-Paley theory. The advantage of working in Fourier-Herz spaces lies in that they are more adapted than classical Besov spaces, for estimating the bilinear paraproduct of two distributions with the summation of their regularity indexes exactly zero. Our result is an improvement of a recent theorem by Lei and Lin (2011) [10].

We report the results of a study on the spectral properties of Laplace and Stokes operators modified with a volume penalization term designed to approximate Dirichlet conditions in the limit when a penalization parameter, η, tends to zero. The eigenvalues and eigenfunctions are determined either analytically or numerically as functions of η, both in the continuous case and after applying Fourier or finite difference discretization schemes. For fixed η, we find that only the part of the spectrum corresponding to eigenvalues λ≲η−1 approaches Dirichlet boundary conditions, while the remainder of the spectrum is made of uncontrolled, spurious wall modes. The penalization error for the controlled eigenfunctions is estimated as a function of η and λ. Surprisingly, in the Stokes case, we show that the eigenfunctions approximately satisfy, with a precision O(η), Navier slip boundary conditions with slip length equal to √η. Moreover, for a given discretization, we show that there exists a value of η, corresponding to a balance between penalization and discretization errors, below which no further gain in precision is achieved. These results shed light on the behavior of volume penalization schemes when solving the Navier–Stokes equations, outline the limitations of the method, and give indications on how to choose the penalization parameter in practical cases.

A genuinely two-dimensional discretization of general drift-diffusion (including incompressible
Navier-Stokes) equations is proposed. Its numerical fluxes are derived by computing
the radial derivatives of “bubbles” which are deduced from available discrete data by exploiting the
stationary Dirichlet-Green function of the convection-diffusion operator. These fluxes are reminiscent
of Scharfetter-Gummel’s in the sense that they contain modified Bessel functions which allow to
pass smoothly from diffusive to drift-dominating regimes. For certain flows, monotonicity properties
are established in the vanishing viscosity limit (“asymptotic monotony”) along with second-order
accuracy when the grid is refined. Practical benchmarks are displayed to assess the feasibility of the
scheme, including the “western currents” with a Navier-Stokes-Coriolis model of ocean circulation.