A numerical methods for 2-D flow of flapping airfoil

To study the physics of the flow of flapping flight, a numerical model for the two-dimensional flow around an airfoil undergoing prescribed oscillatory motions in a viscous flow is described. The model is used to examine the flow characteristics and power coefficients of a symmetric airfoil heaving sinusoidally over a range of frequencies and amplitudes. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Effect of the wing shape on the thrust of flapping wing

Simulations of harmonically oscillating wings were performed using two-dimensional and three-dimensional boundary element method computer programs: the corresponding hydrodynamic forces were obtained by assuming potential flow. The maximum thickness of symmetric 4-digit NACA airfoils was varied in order to assess the effect of changing the foil shape on the generated thrust. It was found that the thrust coefficient per unit of wing mass decreases in magnitude when the thickness increases. The result indicates that, if the wing mass is fixed, its thickness has to be minimised in order to maximise the generated thrust. Another important finding is the dependence on the motion frequency, i.e. for a fixed foil thickness the thrust coefficient per unit of wing mass increases with the motion frequency. However, when the foil thickness becomes larger, the motion frequency effect on the generated thrust becomes less pertinent, i.e. the thrust range for a slender foil is larger than that of a thicker one over the same motion frequency range. The three-dimensional effect due to the wing spanwise shape was also investigated by changing the wing sweep angle and the influence on the generated thrust was insignificant within the range of investigated parameters. The outcome reinforces the idea that the function of swept wings for low-speed flyers, such as birds, is mainly structural as the sweep angle can be related to a desired aeroelastic response of the oscillating wing.     

A further note on the force discrepancy for wing theory in Euler flow

Uniform steady potential flow past a wing aligned at a small angle to the flow direction is considered. The standard approach is to model this by a vortex sheet, approximated by a finite distribution of horseshoe vortices. In the limit as the span of the horseshoe vortices tends to zero, an integral distribution of infinitesimal horseshoe vortices over the vortex sheet is obtained. The contribution to the force on the wing due to the presence of one of the infinitesimal horseshoe vortices in the distribution is focused upon. Most of the algebra in the force calculation is evaluated using Maple software and is given in the appendices. As in the two previous papers by the authors on wing theory in Euler flow [E Chadwick, A slender-wing theory in potential flow, Proc. R. Soc. A461 (2005) 415–432, and E Chadwick and A Hatam, The physical interpretation of the lift discrepancy in Lanchester-Prandtl lifting wing theory for Euler flow, leading to the proposal of an alternative model in Oseen flow, Proc. R. Soc. A463 (2007) 2257–2275], it is shown that the normal force is half that expected. In this further note, in addition it is demonstrated that the axial force is infinite. The implications and reasons for these results are discussed.     

Supersonic flow of a Chaplygin gas past a delta wing

Science China Mathematics - We consider the problem of supersonic flow of a Chaplygin gas past a delta wing with a shock or a rarefaction wave attached to the leading edges. The flow under study is...     

Numerical solution of steady and unsteady flow over a vibrating airfoil in a channel

The work deals with a numerical solution of 2D steady and unsteady inviscid incompressible flow over the profile NACA 0012 in a channel. The finite volume method (FVM) in a form of cell-centered explicit schemes at quadrilateral C-mesh is used. Governing system of equations is the system of incompressible Euler equations. The method of artificial compressibility and time dependent method is applied to steady computations. The small disturbance theory (SDT) applied to a numerical solution of flow over a rotated profile by a small angle only is mentioned. Brief introduction is given to the Arbitrary (Semi) Lagrangian-Eulerian (ALE) method used for unsteady computations. Some numerical results of unsteady flow over a vibrating profile achieved by both SDT and ALE method are presented. Unsteady flow is caused by prescribed oscillations of the profile (one degree of freedom) fixed in an elastic axis. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A taxonomy of flexible flow line scheduling procedures

We present a taxonomy for flexible flow line scheduling procedures. Flexible flow lines are flow lines with parallel machines on some or all production stages. The taxonomy groups procedures according to their general solution approach. It distinguishes optimal and heuristic procedures. Heuristic procedures are split into holistic and decomposition approaches. While holistic approaches consider the complete scheduling problem in an integrated way, decomposition approaches divide the problem with respect to the production stages, the individual jobs, or the sub-problems to be solved (batching, loading, and sequencing). We explicitly mention the inclusion of setups in a study and list the respective objective criteria.     

Numerical study of the viscous hypersonic flow over a blunt delta wing

A previously developed two-dimensional Navier-Stokes solver is extended to three dimensions. The hypersonic steady flow of an perfect gas (with a constant adiabatic index) over a delta wing with a spherical nose and a cylindrical leading edge is computed as an example. The characteristic features of the flow are analyzed. Pressure, heat flux, and friction coefficient distributions are computed in the shock layer and on the wing surface. The results are compared with available numerical and experimental data.     

Instabilities of a flexible surface in supersonic flow

The stability of a supersonic boundary layer above a flexiblesurface is considered in the limit of large Reynolds numberand for Mach numbers O(1). Asymptotic theory of viscous–inviscidinteraction has been used for this purpose. We found that fora simple elastic surface, for which deflections are proportionalto local pressure differences, the boundary-layer flow remainsstable as it is for a rigid wall. However, when either dampingor surface inertia is included the flow becomes unstable. Moreover,in a certain range of wave numbers the boundary layer developsmore then one unstable mode. It is interesting that these modesare connected to one another via saddle points in the complex-frequencyplane. A more complex Kramer-type surface is also analysed andin some parameter ranges is found to permit the evolution ofunstable Tollmien–Schlichting waves. The neutral curvesare found for a variety of situations related to the parametersassociated with the flexible surface.     

Asymptotics of small spacings for subsonic nonstationary streamlined flow of a thin airfoil close to a solid boundary

In an asymptotic approximation of small spacings an analytic solution to the problem on harmonic oscillations of a thin airfoil which is moving with a subsonic velocity near a solid plane boundary is given. Results of a computation of the lifting force are given.Translated from Dinamicheskie Sistemy, No. 7, pp. 48–53, 1988.     

A computational comparison of flow formulations for the capacitated location-routing problem

In this paper we present a computational comparison of four different flow formulations for the capacitated location-routing problem. We introduce three new flow formulations for the problem, namely a two-index two-commodity flow formulation, a three-index vehicle-flow formulation and a three-index two-commodity flow formulation. We also consider an existing two-index vehicle-flow formulation and extend it by considering new families of valid inequalities and separation algorithms. We introduce new branch-and-cut algorithms for each of the formulations and compare them on a wide number of instances. Our results show that compact formulations can produce tight gaps and solve many instances quickly, whereas three-index formulations scale better in terms of computing time.     

The Zhukovskii problem of the flow around a sheet pile

The solution of the Zhukovskii problem of the flow around a sheet pile is given using the principles of two-dimensional steady-state seepage in the case when, accompanying the motion of the seeping water, there is a layer of saline ground waters at a certain depth under the sheet pile and this layer is located above an impermeable thickness of rock salt. The mixed boundary-value problem of the theory of analytic functions which arises is solved using Polubarinova-Kochina's method, which is based on the application of the analytical theory of linear differential equations and, also, the method, developed by us, of the conformal mappings of circular polygons in polar meshes, which are extremely typical for the velocity hodograph domains of such flows. While reflecting the specific details and individual properties of such flows, the solution constructed below turns out to be expressed in closed form in terms of elementary functions and, consequently, it is the simplest and most convenient solution. In addition, it is the most general solution for the class of problems being considered. The well known results Zhukovskii, Vedernikov and others are obtained from it as special and limiting çases A detailed hydrodynamic analysis and the specific features of the seepage process being considered, as well as the effects of all the physical parameters of the model on the pattern of the phenomenon, are presented using this solution and by numerical calculations.     

A computational comparison of the dinic and network simplex methods for maximum flow

We study the implementation of two fundamentally different algorithms for solving the maximum flow problem: Dinic's method and the network simplex method. For the former, we present the design of a storage-efficient implementation. For the latter, we develop a "steepest-edge" pivot selection criterion that is easy to include in an existing network simplex implementation. We compare the computational efficiency of these two methods on a personal computer with a set of generated problems of up to 4 600 nodes and 27 000 arcs.This research was supported in part by the National Science Foundation under Grant Nos. MCS-8113503 and DMS-8512277.     

POD and CVT Galerkin reduced-order modeling of the flow around a cylinder

Reduced-order models are applied to the laminar vortex-shedding flow around a circular cylinder. The models rely on approximations of the solutions in the space spanned by a set of POD or CVT reduced basis vectors, which are computed from snapshots of a numerical solution. To enable the computation of drag and lift forces, the modeling of the velocity and pressure is realized via a one-way coupling. A comparison of the results of the CVT and POD reduced-order models is presented. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the plane problem of the flow around a submerged beam

We consider the problem of the steady flow of an ideal heavy fluid around a submerged beam. The problem is obtained from the free-boundary problem of the flow past a submerged obstacle in the limit of bodies of vanishing thickness. We introduce a special Sobolev space formulation of the problem in term of a perturbed stream function and prove its unique solvability for every value of the unperturbed flow velocity, with the possible exception of a discrete set depending on the geometry of the domain. The asymptotic properties of the solutions are discussed.     

Lift maximization in the flow around a contour over a screen

The problem of lift maximization for a smooth contour of given length placed in a flow near a screen is analyzed. The distance between the contour and the screen is assumed to be given. Optimal contours are constructed, and the lift coefficient is derived as a function of the contour-screen separation. The results can be useful as accurate upper bounds for the lift coefficient of actual ekranoplan airfoils.     

A weighted L q -approach to Stokes flow around a rotating body

Considering time-periodic Stokes flow around a rotating body in or we prove weighted a priori estimates in L ^q -spaces for the whole space problem. After a time-dependent change of coordinates the problem is reduced to a stationary Stokes equation with the additional term in the equation of momentum, where ω denotes the angular velocity. In cylindrical coordinates attached to the rotating body we allow for Muckenhoupt weights which may be anisotropic or even depend on the angular variable and prove weighted L ^q -estimates using the weighted theory of Littlewood-Paley decomposition and of maximal operators. The research was supported by the Academy of Sciences of the Czech Republic, Institutional Research Plan no. AV0Z10190503, by the Grant Agency of the Academy of Sciences No. IAA100190505, and by the joint research project of DAAD (D/04/25763) and the Academy of Sciences of the Czech Republic (D-CZ 3/05-06).