Method of determining the aerodynamic coefficients of asymmetric bodies with allowance for nonlinear body shape influence factors

The numerical method proposed makes it possible to determine the aerodynamic coefficients of asymmetric bodies of fairly arbitrary shape (including those with discontinuities of the transverse contour) at small solid angles of attack. The method allows an aerodynamically sound transition from the three-dimensional system of equations of gas dynamics to a two-dimensional system, which considerably simplifies the problem and reduces by an order the machine time required. The method takes into account the nonlinear body shape influence factors, which substantially improves the accuracy of the calculations. The efficiency and accuracy of the method are demonstrated by comparing the results of the calculations with the results of a numerical solution of the three-dimensional problem.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.2, pp. 121–128, March–April, 1992.     

Instability of the Rossby–Haurwitz wave in the invariant sets of perturbations

Stability of the Rossby–Haurwitz (RH) wave of subspace H₁H_n in an ideal incompressible fluid on a rotating sphere is analytically studied (H_n is the subspace of homogeneous spherical polynomials of degree n). It is shown that any perturbation of the RH wave evolves in such a way that its energy K(t) and enstrophy η(t) decrease, remain constant or increase simultaneously. A geometric interpretation of variations in the perturbation energy is given. A conservation law for arbitrary perturbations is obtained and used to classify all the RH-wave perturbations in four invariant sets M₋ⁿ, M₊ⁿ, H_n and M₀ⁿ−H_n depending on the value of their mean spectral number χ(t)=η(t)/K(t). The energy cascade of growing (or decaying) perturbations has opposite directions in the sets M₋ⁿ and M₊ⁿ due to a hyperbolic dependence between K(t) and χ(t). A factor space with a factor norm of the perturbations is introduced using the invariant subspace H_n of neutral perturbations as the zero factor class. While the energy norm controls the perturbation part belonging to H_n, the factor norm controls the perturbation part orthogonal to H_n. It is shown that in the set M₋ⁿ (χ(t)<n(n+1)), any nonzonal RH wave of subspace H₁H_n (n2) is Liapunov unstable in the energy norm. This instability has nothing in common with the orbital (Poincaré) instability and is caused by asynchronous oscillations of two almost coinciding RH-wave solutions. It is also shown that the exponential instability is possible only in the invariant set M₀ⁿ−H_n. A necessary condition for this instability is given. The condition states that the spectral number χ(t) of the amplitude of each unstable mode must be equal to n(n+1), where n is the RH-wave degree. The growth rate is estimated and the orthogonality of the unstable normal modes to the RH wave is shown. The instability in the invariant set M₊ⁿ of small-scale perturbations (χ(t)>n(n+1)) is still open problem.     

Use of the aerodynamic equivalence method in the determination and analysis of the aerodynamic coefficients of asymmetric bodies

In [1] the validity of the linear method of aerodynamic equivalence (AE) was demonstrated and the results of calculating the aerodynamic characteristics (ADC) of certain asymmetric (nonaxisymmetric) bodies, differing only slightly from the axisymmetric, were presented. In this paper a nonlinear AE method is proposed. This method is based on the principle of the equivalence of two bodies, one of which has a cross section of arbitrary shape while the other has a cross section described by a smooth function. This function is the sun of the first N + 1 terms of the Fourier series of the initial (discontinuous) function describing the shape of the body. The effectiveness of the AE method is illustrated with reference to certain examples of star-shaped bodies. The accuracy of the results obtained is estimated and a comparison is made with the experimental data. It is also shown that the AE method makes it possible to give a simple explanation of certain results of aerodynamics from a new standpoint.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 98–105, January–February, 1986.     

Aerodynamics of an asymmetrically deformed body in nonstationary supersonic motion

A method for making numerical calculations of the stationary and nonstationary aerodynamic characteristics of bodies of variable shape is considered. The results of calculations of the aerodynamic coefficents are given [1]. The results of numerical calculations are compared with the results of Newton's theory.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 162–167, March–April, 1980.     

X-quasinormal subgroups

Considering two subgroups A and B of a group G and ? ≠ X ? G, we say that A is X-permutable with B if AB ^x = B ^x A for some element x ∈ X. We use this concept to give new characterizations of the classes of solvable, supersolvable, and nilpotent finite groups.     

Nonlinear and linear instability of the Rossby-Haurwitz wave

As is known, the large-scale dynamics of barotropic atmosphere can approximately be described by the nonlinear barotropic vorticity equation. It is also well known that the Rossby-Haurwitz (RH) waves, being exact solutions to this equation, represent one of the main features of meteorological fields. Therefore the stability properties of the RH wave are of considerable interest for deeper understanding of the low-frequency variability of the atmosphere. Many works has been devoted to the barotropic instability of flows on a beta-plane and a sphere. However, mathematically, the nonlinear stability problem of the RH wave is still far from its complete solution. Indeed, some of the stability results have been obtained numerically, and hence, contain calculation errors. Severe truncation of perturbations used in the spectral stability analysis, though leads to interesting and useful conclusions, does not allow obtaining comprehensive results. The weak point of some analytical nonlinear instability studies consists in using inappropriate norms for perturbations. It should also be noted that a necessary condition for the linear instability of the RH wave was obtained only recently (Skiba, 2000). In the present work, the nonlinear stability of the RH wave in an ideal incompressible fluid on a rotating sphere is analytically studied. Let H(n) be a subspace of homogeneous spherical polynomials of degree n. Mathematically, a RH wave of degree n is the sum of a super-rotating flow of subspace H(1) and a homogeneous spherical polynomial of subspace H(n). First, we derive a conservation law for arbitrary RH-wave perturbations which asserts that any perturbation evolves in such a way that its kinetic energy E(t) and enstrophy q(t) decrease, remain constant or increase simultaneously. The law is used to divide all the perturbations into three invariant sets depending on the value of their mean spectral number k(t)=q(t)/E(t) introduced by Fjortoft (1953). These sets are denoted as M where k(t)¡n(n+1) (large-scale perturbations), N where k(t)¿n(n+1) (small-scale perturbations), and Z where k(t)=n(n+1) (boundary surface between the sets M and N). Note that Z includes one more invariant set, namely, the subspace H(n). The existence of invariant sets of perturbations allows us to study the RH wave instability in each set separately. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Giant faults in deformed Gum Metal

We have experimentally characterized and theoretically described plastic flow localization in Gum Metal, a special titanium alloy with high strength, low Young’s modulus, excellent cold workability and low resistance to shear in certain crystallographic planes. The electron transmission microscopy experiments demonstrate that plastic flow is localized in giant faults – macroscopic planar defects carrying very large plastic strains (thousand percent or more) – in deformed Gum Metal. Also, regions with highly inhomogeneous elastic strains and varying crystal lattice orientation are experimentally observed in the vicinity of giant faults. A theoretical model is suggested describing the generation of giant faults as a process resulting from generation and evolution of nanodisturbances (nanoscopic planar areas of local shear) in Gum Metal. It is shown that giant faults can effectively nucleate and evolve in Gum Metal, and their intersection with grain boundaries produces both elastic strain accumulation and inhomogeneities of crystal lattice orientation. This behavior of giant faults is expected to be essential for excellent cold ductility of high-strength Gum Metal.