Search results for "Prandtl equation"
showing 4 items of 4 documents
Well-posedness of the boundary layer equations
2004
We consider the mild solutions of the Prandtl equations on the half space. Requiring analyticity only with respect to the tangential variable, we prove the short time existence and the uniqueness of the solution in the proper function space. Theproof is achieved applying the abstract Cauchy--Kowalewski theorem to the boundary layer equations once the convection-diffusion operator is explicitly inverted. This improves the result of [M. Sammartino and R. E. Caflisch, Comm. Math. Phys., 192 (1998), pp. 433--461], as we do not require analyticity of the data with respect to the normal variable.
High Reynolds number Navier-Stokes solutions and boundary layer separation induced by a rectilinear vortex array
2008
Numerical solutions of Prandtl’s equation and Navier Stokes equations are considered for the two dimensional flow induced by an array of periodic rec- tilinear vortices interacting with an infinite plane. We show how this initial datum develops a separation singularity for Prandtl equation. We investigate the asymptotic validity of boundary layer theory considering numerical solu- tions for the full Navier Stokes equations at high Reynolds numbers.
The inviscid limit and Prandtl's asymptotic expansion for incompressible flows in the half space
2022
The validity of the inviscid limit for the incompressible Navier-Stokes equations is one of the most important and challenging problems in the mathematical theory of fluid dynamics: the motion of inviscid fluids is described by the Euler equations, so, when the viscosity goes to zero, one would expect the convergence of NS solutions to the Euler solutions. However, NS equations are a singular perturbation of the Euler equations: the change of order of the equation implies that fewer boundary conditions can be imposed on the inviscid flows. Therefore, the no-slip boundary conditions, imposed on the NS solutions, are not satisfied by the Euler flow, for which a tangential slip is allowed. Thi…
Unsteady Separation for High Reynolds Numbers Navier-Stokes Solutions
2010
In this paper we compute the numerical solutions of Navier-Stokes equations in the case of the two dimensional disk impulsively started in a uniform back- ground flow. We shall solve the Navier-Stokes equations (for different Reynolds numbers ranging from 1.5 · 10^3 up to 10^5 ) with a fully spectral numerical scheme. We shall give a description of unsteady separation process in terms of large and small scale interactions acting over the flow. The beginning of these interactions will be linked to the topological change of the streamwise pressure gradient on the disk. Moreover we shall see how these stages of separation are related to the complex singularities of the solution. Infact the ana…