The optimal conical twist of a delta wing

The problem of reducing the drag of a wing at a specified lift in a supersonic flow is investigated. A solution for a delta wing is obtained in a simplified formulation of the optimization problem and a theoretical analysis. It is shown that the optimal conical wing is formed by elements of elliptical cones and planes. Numerical modelling of the flow of a non-viscous non-heat-conducting gas past the wing is performed, and the results of the theoretical analysis and direct optimization are compared. ©2012     

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...     

The vibration and stability of non-homogeneous orthotropic conical shells with clamped edges subjected to uniform external pressures

In this paper an analytical procedure is given to study the free vibration and stability characteristics of homogeneous and non-homogeneous orthotropic truncated and complete conical shells with clamped edges under uniform external pressures. The non-homogeneous orthotropic material properties of conical shells vary continuously in the thickness direction. The governing equations according to the Donnell’s theory are solved by Galerkin’s method and critical hydrostatic and lateral pressures and fundamental natural frequencies have been found analytically. The appropriate formulas for homogeneous orthotropic and isotropic conical shells and for cylindrical shells made of homogeneous and non-homogeneous, orthotropic and isotropic materials are found as a special case. Several examples are presented to show the accuracy and efficiency of the formulation. The closed-form solutions are verified by accurate different solutions. Finally, the influences of the non-homogeneity, orthotropy and the variations of conical shells characteristics on the critical lateral and hydrostatic pressures and natural frequencies are investigated, when Young’s moduli and density vary together and separately. The results obtained for homogeneous cases are compared with their counterparts in the literature.     

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.     

The acoustic impedance of a semi-infinite tube fitted with a conical flange

The acoustic characteristics of a semi-infinite tube of constant cross sectionσ, fitted with a semi-infinite conical flange of solid angleω, are investigated. Let the acoustic potential in the tube be given by $$\varphi = e^{ - ikx} - Ae^{ikx}$$ where exp(?ikx) represents the incident wave travelling along the tube towards the open endx=0 and ?A exp(ikx) represents the reflected wave. It is shown that for long waves,A can be written in the form $$A = \exp \{ - 2ik\alpha (\sigma /\omega )^{1/2} - 2k^2 \sigma /\omega \}$$ whereα is a parameter which depends only on the solid angleω. From existing work the value ofα is known forω=2π (the infinite plane flange) andω=4π (the unflanged tube). In this paper, the variation ofα for small values ofω is calculated. Because 1≦α≦1.2266 over the whole range 0≦ω≦4π, it is suggested that general values ofα can be interpolated from the existing results. An explicit interpolation formula is proposed.     

Effect of wing deformation on highlift systems of transport aircraft

An elastic multibody model is developed that combines highlift system models consisting of flap and slat mechanisms with an elastic wing model. It describes the motions and deformations of the highlift systems with airloads applied and, in particular, the effect of wing deformations on the highlift systems. In addition the model contains a flexible wing with the engine and the middle section of the fuselage that is fixed at the symmetry plane of the aircraft. While previously wing deformation was taken into account in static FE simulations only the developed multibody model provides the investigation of the motions and inner loads of the highlift mechanisms with consideration of the wing deformations with dynamical effects. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The vertical plate in laminar free convection: Effects of leading and trailing edges and discontinuous temperature

Summary For laminar free-convection flow past a heated vertical plate of finite length, the local asymptotic flow structure is studied in regions where the boundary-layer equations do not provide a correct approximation at large Grashof numbers. The leading-edge region is shown to contribute a secondorder term to the integrated heat transfer. An integral form of the energy equation permits calculation of this correction in terms of the second-order boundary-layer solution away from the edge, without knowledge of the flow details near the edge, which can be obtained only by solution of the full Navier-Stokes equations. Near the trailing edge and near a jump in the prescribed plate temperature the longitudinal pressure gradient is found to be important in a thin sublayer adjacent to the plate, and the transverse pressure gradient is important in the remainder of the boundary layer, each for a distance along the plate which is slightly larger in order of magnitude than the boundary-layer thickness. At the trailing edge the sublayer problem is nonlinear and cannot be solved analytically, but it can be shown that the local correction to the total heat transfer is of slightly larger order of magnitude than the leading-edge correction. It is pointed out that the trailing-edge flow is identical in form to the flow near the edge of a rotating disc in a stationary fluid. The temperature-jump problem is linear and a solution is given which shows how the singularity in streamline slope predicted by the boundary-layer solution is removed.

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Zusammenfassung Bei laminarer natürlicher Strömung längs einer senkrecht stehenden endlichen geheizten Platte bei sehr grossen Grashof'schen Zahlen gibt es kleine Bereiche, bei denen die Grenzschichtgleichungen keine echte asymptotische Darstellung liefern. Im Vorderkantenbereich findet man ein Zusatzglied zweiter Ordnung für die gesamte Wärmeübertragung. Die Berechnung dieser Korrektur erfolgt durch eine Integralform der Energiegleichung, wobei nur die Lösung der Grenzschichtgleichung zweiter Ordnung fern von der Kante benützt wird, während die Kenntnis der Strömung in der Nähe der Kante nur durch eine Lösung der exakten Navier-Stokes'schen Gleichungen gefunden werden könnte. In der Nähe der Hinterkante und in der Nähe einer Diskontinuität der gegebenen Plattentemperatur wird das Längsdruckgefälle in einer dünnen Unterschicht wichtig; das Querdruckgefälle wird in dem restlichen Teil der Grenzschicht berücksichtigt, für eine Länge, deren Grössenordnung etwas grösser ist, als die der Grenzschichtdicke. Bei der Hinterkante ist das Unterschichtsproblem nicht linear, und es gibt keine analytische Lösung, aber man kann zeigen, dass die Grössenordnung der Korrektur der gesamten Wärmeübertragung etwas grösser ist als die Korrektur bei der Vorderkante. Die Strömung bei der Hinterkante ist dieselbe wie die Strömung bei der Kante einer rotierenden Scheibe in einer sonst ruhenden Flüssigkeit. Das Temperatur-Diskontinuitätsproblem ist linear, und eine Lösung ist hier gefunden worden, wodurch die von der Grenzschichttheorie gegebene Singularität (unendliche Quergeschwindigkeit) entfernt wird.

     

The vibrations of a conical shell filled with a variable volume of fluid

We consider the vibrations of a conical shell filled with a fluid. We obtain a system of differential equations that describes the interaction of a variable volume of fluid with the shell. We solve the problem of determining the natural frequencies for a completely full shell. We obtain a formula for computing the fundamental frequency as a function of the geometric parameters of the shell. One figure. Bibliography: 4 titles. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 22, pp. 85–88, 1991.     

The maxwell operator in domains with edges

In the problem of the oscillations of a hollow resonator with ideally conducting walls, one describes the domain of definition of the corresponding self-adjoint Maxwell operator. One discusses the case when the boundary has inward edges.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 147, pp. 3–9, 1985.     

Edge-bipancyclicity of a hypercube with faulty vertices and edges

A bipartite graph G=(V,E) is said to be bipancyclic if it contains a cycle of every even length from 4 to |V|. Furthermore, a bipancyclic G is said to be edge-bipancyclic if every edge of G lies on a cycle of every even length. Let F_v (respectively, F_e) be the set of faulty vertices (respectively, faulty edges) in an n-dimensional hypercube Q_n. In this paper, we show that every edge of Q_n-F_v-F_e lies on a cycle of every even length from 4 to 2ⁿ-2|F_v| even if |F_v|+|F_e|?n-2, where n?3. Since Q_n is bipartite of equal-size partite sets and is regular of vertex-degree n, both the number of faults tolerated and the length of a longest fault-free cycle obtained are worst-case optimal.     

Multigrid fictitious boundary and grid deformation methods for flow-induced rotation of wing

Numerical simulations of flow-induced rotation of wing by multigrid fictitious boundary and grid deformation methods are presented. The flow is computed by a special ALE formulation with a multigrid finite element solver. The solid wing is allowed to move freely through the computational mesh which is adaptively aligned by a special mesh deformation method. The advantage of this approach is that no expensive remeshing has to be performed. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Stability of a flexible orthotropic plate with free edges

An investigation is made into the postcritical behavior of rectangular flexible orthotropic plates under the action of distributed and concentrated loads. In particular, glass-reinforced plastic plates with different ratios of the moduli of elasticity are studied. The calculation is performed by the method of successive approximations, using finite differences, on a BESM-2M computer. The solutions obtained are compared with experimental data.Mekhanika Polimerov, Vol. 2, No. 4, pp. 585–592, 1966     

Covering a graph with cycles passing through given edges

We propose a conjecture: for each integer k ≥ 2, there exists N(k) such that if G is a graph of order n ≥ N(k) and d(x) + d(y) ≥ n + 2k - 2 for each pair of non-adjacent vertices x and y of G, then for any k independent edges e₁, …, e_k of G, there exist k vertex-disjoint cycles C₁, …, C_k in G such that e_i ∈ E(C_i) for all i ∈ {1, …, k} and V(C₁ ∪ ···∪ C_k) = V(G). If this conjecture is true, the condition on the degrees of G is sharp. We prove this conjecture for the case k = 2 in the paper. © 1997 John Wiley & Sons, Inc. J Graph Theory 26: 105–109, 1997     

The development of a laminar wing type turbine bucket

Zusammenfassung In Analogie zum Gedankengang, der zur Einführung des Laminarflügels führte, wurde eine Leitschaufel für eine Gleichdruckturbine konstruiert. Die Methode für die Originalkonstruktion beruhte auf einer Kombination von Singularit?ten in der Hodographenebene und einem Netzverfahren in der (x, y)-Ebene. Die Schaufel wurde im Gitterversuch, in einer einstufigen Luftturbine und schliesslich in einer dreistufigen Dampfturbine genau geprüft. Sie ergab einen Gewinn am Wirkungsgrad zwischen einem halben und einem Prozent. Sie wurde daraufhin im Turbinenbau eingeführt. Ein v?llig automatisches Hodographenverfahren zur Berechnung dieser Schaufeln ist in stetigem Gebrauch. Es wird bald ver?ffentlicht werden.