613 research outputs found
The Inviscid Limit and Boundary Layers for Navier-Stokes Flows
The validity of the vanishing viscosity limit, that is, whether solutions of
the Navier-Stokes equations modeling viscous incompressible flows converge to
solutions of the Euler equations modeling inviscid incompressible flows as
viscosity approaches zero, is one of the most fundamental issues in
mathematical fluid mechanics. The problem is classified into two categories:
the case when the physical boundary is absent, and the case when the physical
boundary is present and the effect of the boundary layer becomes significant.
The aim of this article is to review recent progress on the mathematical
analysis of this problem in each category.Comment: To appear in "Handbook of Mathematical Analysis in Mechanics of
Viscous Fluids", Y. Giga and A. Novotn\'y Ed., Springer. The final
publication is available at http://www.springerlink.co
Boundary Layers Associated with Incompressible Navier-Stokes Equations: The Noncharacteristic Boundary Case
The goal of this article is to study the boundary layer of wall bounded flows in a channel at small viscosity when the boundaries are uniformly non-characteristic, i.e., there is injection and/or suction everywhere at the boundary. Following earlier work on the boundary layer for linearized Navier-Stokes equations in the case where the boundaries are characteristic (non-slip at the boundary and non-permeable), we consider here the case where the boundary is permeable and thus non-characteristic. the form of the boundary layer and convergence results are derived in two cases: linearized equation and full nonlinear equations. We prove that there exists a boundary layer at the outlet (downwind) of the form e-Uz/ε where U is the speed of injection/suction at the boundary, z is the distance to the outlet of the channel, and ε is the kinematic viscosity. We improve an earlier result of S. N. Alekseenko (1994, Siberian Math. J. 35, No. 2, 209-230) where the convergence in L2 of the solutions of the Navier-Stokes equations to that of the Euler equations at vanishing viscosity was established. in the two-dimensional case we are able to derive the physically relevant uniform in space (L∞ norm) estimates of the boundary layer. the uniform in space estimate is derived by properly developing our previous idea of better control on the tangential derivative and the use of an anisotropic Sobolev imbedding. to the best of our knowledge this is the first rigorously proved result concerning boundary layers for the full (nonlinear) Navier-Stokes equations for incompressible fluids. © 2002 Elsevier Science (USA)
A viscous inverse method for aerodynamic design
A numerical technique to solve two-dimensional inverse problems that arise in aerodynamic design is presented. The approach, which is well-established for inviscid, rotational flows, is here extended to the viscous case. Two-dimensional and axisymmetric configurations are here considered. The solution of the inverse problem is given as the steady state of an ideal transient during which the flowfield assesses itself to the boundary conditions by changing the boundary contour. Comparisons with theoretical and experimental results are used to validate the numerical procedure
Recommended from our members
Turbulent Drag Reduction Using Anisotropic Permeable Substrates.
The behaviour of turbulent flow over anisotropic permeable substrates is studied using linear stability analysis and direct numerical simulations (DNS). The flow within the permeable substrate is modelled using the Brinkman equation, which is solved analytically to obtain the boundary conditions at the substrate-channel interface for both the DNS and the stability analysis. The DNS results show that the drag-reducing effect of the permeable substrate, caused by preferential streamwise slip, can be offset by the wall-normal permeability of the substrate. The latter is associated with the presence of large spanwise structures, typically associated to a Kelvin-Helmholtz-like instability. Linear stability analysis is used as a predictive tool to capture the onset of these drag-increasing Kelvin-Helmholtz rollers. It is shown that the appearance of these rollers is essentially driven by the wall-normal permeability Ky+ . When realistic permeable substrates are considered, the transpiration at the substrate-channel interface is wavelength-dependent. For substrates with low Ky+ , the wavelength-dependent transpiration inhibits the formation of large spanwise structures at the characteristic scales of the Kelvin-Helmholtz-like instability, thereby reducing the negative impact of wall-normal permeability
Injection-suction control for Navier-Stokes equations with slippage
We consider a velocity tracking problem for the Navier-Stokes equations in a
2D-bounded domain. The control acts on the boundary through a injection-suction
device and the flow is allowed to slip against the surface wall. We study the
well-posedness of the state equations, linearized state equations and adjoint
equations. In addition, we show the existence of an optimal solution and
establish the first order optimality condition.Comment: 23 page
Adhesion and detachment fluxes of micro-particles from a permeable wall under turbulent flow conditions
We report a numerical investigation of the deposition and re-entrainment of Brownian particles from a permeable plane wall. The tangential flow was turbulent. The suspension dynamics were obtained through direct numerical simulation of the Navier–Stokes equations coupled to the Lagrangian tracking of individual particles. Physical phenomena acting on the particles such as flow transport, adhesion, detachment and re-entrainment were considered. Brownian diffusion was accounted for in the trajectory computations by a stochastic model specifically adapted for use in the vicinity of the wall. Interactions between the particles and the wall such as adhesion forces and detachment were modeled. Validations of analytical solutions for simplified cases and comparisons with theoretical predictions are presented as well. Results are discussed focusing on the interplay between the distinct mechanisms occurring in the fouling of filtration devices. Particulate fluxes towards and away from the permeable wall are analyzed under different adhesion strengths
Instability driven by boundary inflow across shear: a way to circumvent Rayleigh's stability criterion in accretion disks?
We investigate the 2D instability recently discussed by Gallet et al. (2010)
and Ilin \& Morgulis (2013) which arises when a radial crossflow is imposed on
a centrifugally-stable swirling flow. By finding a simpler rectilinear example
of the instability - a sheared half plane, the minimal ingredients for the
instability are identified and the destabilizing/stabilizing effect of
inflow/outflow boundaries clarified. The instability - christened `boundary
inflow instability' here - is of critical layer type where this layer is either
at the inflow wall and the growth rate is (as found by Ilin \&
Morgulis 2013), or in the interior of the flow and the growth rate is where measures the (small) inflow-to-tangential-flow
ratio. The instability is robust to changes in the rotation profile even to
those which are very Rayleigh-stable and the addition of further physics such
as viscosity, 3-dimensionality and compressibility but is sensitive to the
boundary condition imposed on the tangential velocity field at the inflow
boundary. Providing the vorticity is not fixed at the inflow boundary, the
instability seems generic and operates by the inflow advecting vorticity
present at the boundary across the interior shear. Both the primary bifurcation
to 2D states and secondary bifurcations to 3D states are found to be
supercritical. Assuming an accretion flow driven by molecular viscosity only so
, the instability is not immediately relevant for accretion
disks since the critical threshold is and the inflow boundary
conditions are more likely to be stress-free than non-slip. However, the
analysis presented here does highlight the potential for mass entering a disk
to disrupt the orbiting flow if this mass flux possesses vorticity.Comment: 44 pages, 14 figure
- …