The Oseen Equations

Slip with friction and penetration with resistance boundary conditions for the Navier–Stokes equations—numerical tests and aspects of the implementation

We prove that the steady state Navier–Stokes equations have a solution in an exterior Lipschitz domain of , vanishing at infinity, provided the boundary datum belongs to .

This thesis investigates the flexibility associated to Discontinuous Galerkin (DG) discretization on very general meshes obtained by means of agglomeration techniques. The work begins with a brief overview of the main tools that have been extended or specifically developed to deal with arbitrarily shaped elements in the DG context. Then two different implementations of the BRMPS scheme introduced by Bassi, Rebay, Mariotti, Pedinotti and Savini in [16] for the DG discretization of the Laplace operator on arbitrarily shaped elements have been presented. The validation of the scheme on a Poisson problem shows that the discrete polynomial space preserves optimal convergence properties. The discretization of the second order differential operator has been directly extended to the Navier-Stokes equations and the Reynolds Averaged Navier-Stokes (RANS) equations coupled with the k-w turbulence model of Wilcox [54]. In this regard, an implicit time integration strategy has been considered and assessed on classical validation test cases for the compressible f uid dynamics. Then a simple alternative approach to high-order mesh generation is presented. Indeed, once a standard fine grid able to provide an accurate domain discretization has been produced by means of standard low-order grid generation tools, a computational mesh suitable for the desired accuracy and computationally affordable can be obtained via agglomeration while keeping the boundary resolution of the fine grid. The effectiveness of this approach in representing the geometry of the domain is numerically assessed both on a Poisson model problem and on challenging inviscid and viscous test cases. Finally, the freedom in simply defining the topology of agglomerated meshes leads to a nonstandard approach to h-adaptivity that exploits adaptive agglomeration coarsening of a properly fine underlying grid. The effectiveness of this approach has been assessed on test cases involving both error-based and fl ow feature-based simple estimators.

Stationary or self similar solutions with suitable homogeneity often play a crucial role in the regularity theory of nonlinear problems, which are physically or geometrically interesting. This has been manifested in the regularity theory of harmonic maps and minimal surfaces. The local partial regularity theorem in [CKN] implies that there are no self-similar solutions with small local energy (also see [TX] for generalizations). Making use of some arguments in [NRS], Tsai has ruled out the existence of any self-similar solutions with a finite local energy. Yet it is unclear whether or not solutions of the incompressible Navier– Stokes equation in three space dimensions would develop singularities in finite time. Therefore, it may be still interesting to construct special solutions of the 3-dimensional Navier–Stokes equation. In this note, we construct a one-parameter family of explicit smooth solutions of the 3-dimensional impressible Navier–Stokes equation on R\p, where p is any given point. These solutions are axisymmetric, homogeneous of degree −1. They are steady solutions to the Navier–Stokes equations and also solve the self-similar

Edge noise is generated if turbulence interacts with solid edges. Reduction of trailing edge noise of airfoils can be achieved by replacing the solid material at the trailing edge by inlays of porous permeable material. The acoustic benefit of approximately 6 dB of such treatment is known from experiments. Enroute to numerically optimized porous properties, this paper presents a first principle based Computational Aeroacoustics (CAA) method for predicting the acoustic effect of a porous NACA0012 trailing edge. In a hybrid two-step CFD/CAA procedure the turbulence statistics from a solution of the Volume Averaged Navier-Stokes (VANS) equations is used as a basis for the prediction of turbulent-boundary-layer trailing-edge noise (TBL-TEN). For the acoustic part of the calculation, the Acoustic Perturbation Equations (APE) are solved in the flow field. Inside the porous regions, a different set of governing equations, referred to as Linear Perturbation Equations (LPE) will be solved. The LPE represent a modified form of the Linearized Euler Equations (LEE) with the APE vorticity source term shifted to the right-hand side. The new set of equations is derived by volume averaging the Navier-Stokes equations and decomposing the flow variables into a time-averaged mean part and a fluctuating part and isolating the vorticity source term to the right-hand side of the momentum equation. The LPE are verified by an analytical solution. The simulation results of a NACA0012 airfoil geometry with and without porous trailing edge treatment are compared to wind tunnel measurements. The noise reduction effect of such a trailing edge treatment is successfully demonstrated.

The solution of the Oseen's equations has been obtained by Lamb, Oseen, Bairstow and others only in the cases of a sphere, and a circular or elliptic cylinder. And their methods cannot be applied to other forms of boundary. A new method proposed by Prof. Isao Imai of Tokyo University is epoch-making, being applicable to various boundaries in the two-dimmensional problem. In this report the author, using Imai's method, integrates the Oseen's equations in the case of the thin aerofoil of the symmetrical Joukowski type placed in the uniform viscous flow. And the unique solution, without necessity of any assumption such as Joukowski's hypothesis in the perfect fluid, is obtained only from the boundary conditions.

DRBEM solutions of Stokes and Navier–Stokes equations in cavities under point source magnetic field

In this paper we analyze the theoretical properties of a stochastic representation of the incompressible Navier–Stokes equations defined in the framework of the modeling under location uncertainty (LU). This setup built from a stochastic version of the Reynolds transport theorem incorporates a so-called transport noise and involves several specific additional features such as a large scale diffusion term, akin to classical subgrid models, and a modified advection term arising from the spatial inhomogeneity of the small-scale velocity components. This formalism has been numerically evaluated in a series of studies with a particular interest on geophysical flows approximations and data assimilation. In this work we focus more specifically on its theoretical analysis. We demonstrate, through classical arguments, the existence of martingale solutions for the stochastic Navier–Stokes equations in LU form. We show they are pathwise and unique for 2D flows. We then prove that if the noise intensity goes to zero, these solutions converge, up to a subsequence in dimension 3, to a solution of the deterministic Navier–Stokes equation. similarly to the grid convergence property of well established large-eddies simulation strategies, this result allows us to give some guarantee on the interpretation of the LU Navier–Stokes equations as a consistent large-scale model of the deterministic Navier–Stokes equation.

A Chebyshev collocation method for solving the unsteady two‐dimensional Navier–Stokes equations in vorticity–streamfunction variables is presented and discussed. The discretization in time is obtained through a class of semi‐implicit finite difference schemes. Thus at each time cycle the problem reduces to a Stokes‐type problem which is solved by means of the influence matrix technique leading to the solution of Helmholtz‐type equations with Dirichlet boundary conditions. Theoretical results on the stability of the method are given. Then a matrix diagonalization procedure for solving the algebraic system resulting from the Chebyshev collocation approximation of the Helmholtz equation is developed and its accuracy is tested. Numerical results are given for the Stokes and the Navier–Stokes equations. Finally the method is applied to a double‐diffusive convection problem concerning the stability of a fluid stratified by salinity and heated from below.

In this paper, we study the existence and regularity of solutions to the Stokes and Oseen equations with a nonhomogeneous divergence condition. We also prove the existence of global weak solutions to the 3D Navier-Stokes equations when the divergence is not equal to zero. These equations intervene in control problems for the Navier-Stokes equations and in fluid-structure interaction problems.

Downscaling data assimilation algorithm with applications to statistical solutions of the Navier–Stokes equations

Unsteady axisymmetric stagnation-point flow of a viscous fluid on a cylinder

Analyticity estimates for the Navier–Stokes equations

Comparison between continuous and discontinuous boundary elements in the multidomain dual reciprocity method for the solution of the two-dimensional Navier–Stokes equations

A Direct Discrete-time IDA-PBC Design Method for a Class of Underactuated Hamiltonian Systems