Homogenization of the inviscid incompressible fluid flow through a 2D porous medium

We consider the non-stationary incompressible Euler equations in a 2D porous medium. We suppose a periodic porous medium, with the period proportional to the characteristic pore size and with connected fluid part. The flow is subject to an external force, corresponding to an inflow. We start from an initial irrotational velocity and prove that the effective filtration velocity satisfies a transient filtration law. It has similarities with Darcy's law, but it now connects the time derivative of the filtration velocity with the pressure gradient. The viscosity does not appear in the filtration law any more and the permeability tensor is determined through auxiliary problems of decomposition type. Using the limit problem, we construct the correction for the fluid velocity and prove that -norm of the error is of order . Similarly, we estimate the difference between the fluid pressure and its correction in as .

     

Some experimental results on flow in a diverging 2D channel

The effect of the free stream turbulence (FST) on the essential flow characteristics was investigated in the diverging 2D channel. The diverging channel was modelled in the closed type working section by fastening a displacement body above the flat plate that is parallel with the wind-tunnel axis. The angle of the channel expansion 11 degree induced the pressure gradient parameter with the flow velocity U₀ 30 m/s at the start of expansion, x = 0. The height of the channel is 0.15 m at x = 0. FST was either natural 0.3 percent or amplified by turbulence generating grids/screens with turbulence levels up to 5 percent. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

An inviscid flow with compact support in space-time

There exists a nonzero weak solution to the Euler equations of time-dependent incompressible fluid flow in the plane such that this solution has compact support in space-time.     

Global well-posedness for the 2D dispersive SQG equation and inviscid Boussinesq equations

In this paper, we obtain global well-posedness for the 2D dispersive SQG equation and inviscid Boussinesq equations. Our works are consistent with the corresponding works by Elgindi–Widmayer (SIAM J Math Anal 47:4672–4684, 2015) in the special case \({A=\kappa=1}\). In addition, our result concerning the SQG equation can be regarded as the borderline case of the work by Cannone et al. (Proc Lond Math Soc 106:650–674, 2013).     

High-resolution large time-step schemes for inviscid fluid flow

We present a set of sufficient conditions for conservative, consistent and total variation diminishing (TVD) Large Time Step (LTS) schemes. We generalize the modified flux approach of Harten (1986) to achieve second-order accuracy away from discontinuities. In particular, we derive a supplementary condition, providing a full set of sufficient conditions for high-resolution LTS.In this framework, a second-order extension (LTS-Roe2) of the LTS-Roe scheme is proposed and compared to the second order LTS scheme proposed by Harten. Tests show that LTS-Harten consistently gives good results with less oscillations than LTS-Roe2, but has a tendency to smear out discontinuities when the Courant number is increased.     

Concentrated force acting in an inviscid and incompressible parallel flow

The paper discusses the steady disturbance velocity field induced by a concentrated force acting at a fixed point of space in a parallel flow of an inviscid and incompressible fluid. Force and parallel flow have the same direction. The induced velocity field, its pressure field and its acceleration field are described by means of generalized functions (distributions). Copyright © 2000 John Wiley & Sons, Ltd.     

On the development of a wake vortex in inviscid flow

The evolution of an initial perturbation in an axisymmetric subsonic normal inviscid gas flow through a pipe is directly simulated. The basic (unperturbed) flow has a zero radial velocity component, while its axial velocity component (along the axis of symmetry) increases or decreases linearly with the radius. The perturbation is specified as a swirl (rotation about the axis) with a positive or negative velocity vanishing on the central axis and the lateral surface. Irrespective of its direction, the swirl gives rise to a steady-state vortex carried by the flow. It shape is spherical (contiguous to the rotation axis) or circular (sliding along the impermeable lateral surface).     

Transient inviscid flow in a passive linear model of the cochlea

Detailed knowledge of the fluid dynamics within the cochlea is important for a complete understanding of the physiology of this complex organ. We present a numerical simulation in the time domain for the inviscid two-dimensional flow in a passive cochlea. Numerical results are presented for a click stimulation and an impulsively started tone. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The asymptotics of blow-up in inviscid Boussinesq flow and related systems

We consider the approach to blow-up in two-dimensional inviscidflows with stagnation-point similitude, in particular a buoyancy-drivenflow resulting from a horizontally quadratic density variationin a horizontally unbounded slab. The blow-up, which is onlypossible because the flow has infinite energy, proceeds by intensificationof the vorticity and density gradient in a layer adjacent tothe upper boundary, while the remainder of the flow tends towardsirrotationality. The governing Boussinesq flow equations arefirst solved numerically, and the results suggest scalings whichare then used in an asymptotic analysis as 0, where is thetime remaining until blow-up. The structure of the asymptoticsolution, involving exponential orders as well as powers andlogarithms of the small parameter, is suggested by the analysisof a simpler related problem for which an exact solution isavailable. The expansion is uniformly valid across the upperboundary layer and the outer region, but there is a layer adjacentto the lower boundary where the flow remains dependent on theinitial conditions and is undetermined by the asymptotics.     

A simple 1D model of inviscid fluid-solid interaction

We analyze a one-dimensional fluid-particle interaction model, composed by the Burgers equation for the fluid velocity and an ordinary differential equation which governs the particle movement. The coupling is achieved through a friction term. One of the novelties is to consider entropy weak solutions involving shock waves. The difficulty is the interaction between these shock waves and the particle. We prove that the Riemann problem with arbitrary data always admits a solution, which is explicitly constructed. Besides, two asymptotic behaviors are described: the long-time behavior and the behavior for large friction coefficients.     

Comparative study of 1D, 2D and 3D simplified gas kinetic schemes for simulation of inviscid compressible flows

With assumption that all the particles in the phase velocity space are concentrated on a circle and on a sphere, the circular function-based gas kinetic scheme and sphere function-based gas kinetic scheme have been developed by Shu and his coworkers [21], [22], [23]. These schemes are simpler than the Maxwellian function-based gas kinetic schemes. The simplicity is due to the fact that the integral domain of phase velocity of circular function and sphere function is a finite region while the integral domain of Maxwellian distribution function is infinite. In this work, the 1D delta function-based gas kinetic scheme is also developed to form a complete set of the simplified gas kinetic schemes. The 1D, 2D and 3D simplified gas kinetic schemes can be viewed as the truly 1D, 2D and 3D flux solvers since they are based on the multi-dimensional Boltzmann equation. On the other hand, to solve the 3D flow problem, the tangential velocities are needed to be approximated by some ways for the 1D and 2D simplified gas kinetic schemes, and to solve the 1D flow problem, the tangential velocities should be taken as zero for the 2D and 3D simplified gas kinetic schemes. The performances of these three schemes for simulation of inviscid compressible flows are investigated in this work by their application to solve the test problems from 1D to 3D cases. Numerical results showed that the efficiency of the delta function-based gas kinetic scheme is slightly superior to that of the circular function- and sphere function-based gas kinetic schemes, while its stability is inferior significantly to the latter. For simulation of the 3D hypersonic flows, the sphere function-based gas kinetic scheme could be the best choice.     

Yudovich type solution for the 2D inviscid Boussinesq system with critical and supercritical dissipation

In this paper we consider the Yudovich type solution of the 2D inviscid Boussinesq system with critical and supercritical dissipation. For the critical case, we show that the system admits a global and unique Yudovich type solution; for the supercritical case, we prove the local and unique existence of Yudovich type solution, and the global result under a smallness condition of _θ0${\theta }_0$. We also give a refined blowup criterion in the supercritical case.     

A two-phase flow with a viscous and an inviscid fluid

We study the free boundary between a viscous and an inviscid fluid satisfying the Navier-Stokes and Euler equations respectively. Surface tension is incorporated. We read the equations as a free boundary problem for one viscous fluid with a nonlocal boundary force. We decompose the pressure distribution in the inviscid fluid into two contributions. A positivity result for the first, and a compactness property for the second contribution are dervied. We prove a short time existence theorem for the two-phase problem     

Analysis of a splitting method for incompressible inviscid rotational flow problems

This paper is devoted to analyze a splitting method for solving incompressible inviscid rotational flows. The problem is first recast into the velocity–vorticity–pressure formulation by introducing the additional vorticity variable, and then split into three consecutive subsystems. For each subsystem, the L²$L^2$ least-squares finite element approach is applied to attain accurate numerical solutions. We show that for each time step this splitting least-squares approach exhibits an optimal rate of convergence in the H¹$H^1$ norm for velocity and pressure, and a suboptimal rate in the L²$L^2$ norm for vorticity. A numerical example in two dimensions is presented, which confirms the theoretical error estimates.     

Magneto-gasdynamic channel flow

Zusammenfassung Die Gleichungen für die ?quasi-eindimensionale? Str?mung eines elektrisch leitenden Gases werden im Falle eines Stromfadens mit ver?nderlichem Querschnitt hergeleitet, wobei die Str?mung unter dem Einfluss eines elektrischen und magnetischen Feldes steht. Obwohl kein allgemeines Integral angegeben werden konnte, ergeben sich interessante Folgerungen für die Beschleunigung der Str?mung und für die ?nderung der Mach-Zahl. Zum Beispiel kann in gewissen Geschwindigkeitsbereichen eine überschallstr?mung ?magneto-gasdynamisch? auch bei konstantem Str?mungsquerschnitt beschleunigt werden. Bei variablem Querschnitt sind die Gleichungen für einen besonderen Fall integriert worden; das Ergebnis dieser Integration kann als weiteres Beispiel für die Veranschaulichung der Wirkung einer elektromagnetischen Energiezufuhr dienen. Die auch bei konstantem Querschnitt bestehenden mannigfaltigen M?glichkeiten werden für verschiedene Anfangsbedingungen diskutiert, wobei F?lle sich zeigen, bei denen Beschleunigungen und Verz?gerungen mit Durchg?ngen durch die Schallgeschwindigkeit auftreten sowie auch neue F?lle von ?Blockierungen? sich offenbaren.      

A novel variational method for 3D viscous flow in flow channel of turbomachines based on differential geometry

ABSTRACT

This paper presents a novel variational method for treating three-dimensional rotational Navier-Stokes equations in flow channel of turbomachines. The proposed method establishes a new semi-geodesic coordinate system on the central surface of blades. From the perspective of differential geometry, the system under concern is split into a set of membrane operator equations on two-dimensional manifolds and bending operator equations along hub circle. The third variable of the new coordinate system is approximated by the central difference scheme. We derive a new formulation of two-dimensional Navier-Stokes equations with three components on the manifolds in the variational sense. The well-posedness of the proposed variational formulation is rigorously justified.