A numerical method for flexible airfoils in incompressible inviscid flow

Summary The paper deals with the aerodynamic analysis of flexible airfoils, based on a quasi-lattice vortex method (QVLM). The analysis is formulated in matrix form and leads, as in other similar studies, to a linear algebraic system when the angle of attack is nonzero, and to an eigenvalue problem when the incidence angle is zero. The aerodynamic characteristic curvesC _L -,C _m - are presented. Finally, the airfoil shapes for several values of the tension coefficient and angles of attack are drawn. The results obtained with the present method are in good agreement with those reported in previous studies and evidentiate the flexibility of the QVLM as applied to flexible airfoils.Notation A aerodynamic matrix, defined in QVL method, (8) - B matrix, see Eq. (18) - c chord of airfoil - C matrix defined asAB - C _L lift coefficient, 2L/V ² c) - C _p moment coefficient, 2M/(V ² c ²) - C _p pressure coefficient,C _p =2p/(V ² ) - C _T tension coefficient, 2T/(V ² c) - D matrix, see Eq. (11) - I unit matrix - l curvilinear length of the flexible airfoil - N number of collocation points on the airfoil shape - q dynamic pressure, V ² /2 - T tension force in the sail - V freestream velocity - w downwash - x nondimensional coordinate,x/c - X _i control points, Eq. (9) - X _max dimensionless position of the maximum camber - Y _k source points, Eq. (9) - z coordinate normal tox axis - Z nondimensional coordinate,z/c - Z _s camber equation in dimensionless form,z _s /c - incidence with respect to the upstream flow velocity - column vector of the local curvatures {₁, ₂,..., _N } ^T - nondimensional membrane excess ratio - eigenvalue of the problem (23) - _k zeroes of the Chebyshev polynomia of the first kind, 1kN - column vector of the local slopes, {₀, ₁, ₂,..., _N } ^T - column vector, {₁, ₂,..., _N } ^T - ₀ slope at airfoil leading edge     

Forced Korteweg-de Vries equation in an elastic tube filled with an inviscid fluid

In the present work, treating the arteries as a prestressed thin walled elastic tube with a stenosis and the blood as an inviscid fluid, we have studied the propagation of weakly nonlinear waves in such a composite medium, in the long wave approximation, by use of the reductive perturbation method [C.S. Gardner, G.K. Morikawa, Similarity in the asymptotic behavior of collision-free hydromagnetic waves and water waves, Courant Institute Math. Sci. Report, NYO-9082 (1960) 1-30, T. Taniuti, C.C. Wei, Reductive perturbation method in non-linear wave propagation I, J. Phys. Soc. Jpn., 24 (1968) 941-946]. We obtained the forced Korteweg-de Vries (FKdV) equation with variable coefficients as the evolution equation. By use of the coordinate transformation, it is shown that this type of evolution equation admits a progressive wave solution with variable wave speed. As might be expected from physical consideration, the wave speed reaches its maximum value at the center of stenosis and gets smaller and smaller as we go away from the center of the stenosis. The variations of radial displacement and the fluid pressure with the distance parameter are also examined numerically. The results seem to be consistent with Bernoulli’s law for inviscid fluid.     

Small oscillations of inviscid incompressible stratified parallel and swirling flows

Existence theorems, minimax principles and comparison theorems are obtained for the real eigenfrequencies of inviscid, incompressible, stratified plane parallel flow and swirling flow.     

Rotary oscillations of a spheroid in an incompressible micropolar fluid

The paper discusses the flow generated by rotary oscillations of a spheroid (prolate and oblate) in incompressible micropolar fluid. The velocity and microrotation components are determined explicitly in terms of spheroidal wave functions and are expressed in infinite series form. The couple on the oscillating spheroid is evaluated and numerical studies are undertaken to examine the effects of the geometric parameter and material constant parameters of the fluid.     

Non-symmetrical waves in a prestressed elastic tube filled with an inviscid fluid

In order for a better understanding of the effect of initial stress on flow in elastic tubes, the propagation of time harmonic waves in a prestressed elastic tube filled with an inviscid fluid is studied. Although the blood is known to be a non-Newtonian fluid, for simplicity in the mathematical analysis, it is assumed to be non-viscous while the tube material is considered to be incompressible, isotropic and elastic. Utilizing the theory of small deformations superimposed on large initial static deformation, for a non-symmetrical perturbed motion, the governing differential equations are obtained in the cylindrical polar coordinates. Due to variability of the coefficients of the resulting differential equations of the solid body, the field equations are solved by a truncated power series method. Applying the boundary conditions, the dispersion relation is obtained as a function of inner pressure, axial stretch and the thickness ratio. It is observed that the wave speed of the non-symmetrical wave is large as compared to the symmetrical case. Various special cases as well as some numerical results are also discussed in the paper.     

Axially symmetrical jet mixing of an incompressible dusty fluid

Summary Incompressible laminar jet mixing of a dustry fluid issuing from a circular jet has been considered by using Saffman's model for dusty fluid. Assuming the velocity and temperature in the jet to differ only slightly from that of the surrounding stream, a perturbation method has been employed to linearize the basic equations. The linearized equations have been solved by using Hankel transform, technique. Numerical computations have been made to discuss the veoocity and temperature profiles. It is observed that the particles move faster than the fluid in the down stream direction and the particles temperature is lower than the fluid temperature.With 6 Figures     

Numerical analysis of the inviscid incompressible flow in two-dimensional radial-flow pump impellers

The flow through a two-dimensional centrifugal impeller fitted with equiangular blades of arbitrary geometry is investigated using a combination of conformal mapping with a boundary element technique. The blades can be thin or thick of arbitrary cross-section. A theoretical analysis and a numerical procedure are developed to determine the pressure distributions along the blade.     

Adhesive interaction of elastically deformable spherical particles

Two spherical particles that attract each other by van der Waals volume forces and can undergo deformation as a result of the attraction are considered. Small deformations of such particles can be described by the solution of the Hertz problem. The deformation of particles, in turn, alters the force of attraction between them. It has been established that the relationship between the adhesion and elasticity of the indicated particles is determined by the degree to which these particles deform and that the adhesion force acting between the particles depends on their elasticity, size, and the Hamaker constants.     

Thermoconvective waves in a horizontal bounded layer of an incompressible fluid

The propagation of finite-amplitude thermoconvective waves in a horizontal fluid layer with rigid boundaries is investigated.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol.31, No.1, pp.79–85, July, 1976.     

Slow steady rotation of a spheroid in an incompressible micropolar fluid

The paper discusses the flow generated by the slow steady rotation of a spheroid (prolate and oblate) in incompressible micropolar fluid. The velocity and the microrotation components are determined explicitly in terms of Legendre, Associated Legendre and Spheroidal wave functions and are expressed in infinite series form. The couple on the rotating spheroid is evaluated and numerical studies are undertaken to examine the effects of the geometric parameter and material constant parameters of the fluid.     

The rectilinear oscillations of an elliptic cylinder in incompressible micropolar fluid

The paper is a study of the flow of incompressible micropolar fluid arising from the harmonic oscillation of an elliplic cylinder parallel to either of the principal axes of its cross section. The stream function as well as the velocity and microrotation vectors are obtained and expressed in infinite series form involving Mathieu functions and functions allied to them. The drag on the cylinder is evaluated and the effect of variations of the polarity and frequency parameters on the drag as revealed by numerical studies is shown through figures.     

Isothermal axisymmetric flow of an incompressible fluid in radial contact apparatus

A method is proposed for the calculation of the axisymmetric flow of an incompressible fluid in a radial apparatus with a nonmoving layer of granular material. An engineering method is developed to evaluate the degree of flow nonuniformity in equipment of this type, thus making it possible operationally to choose among structural solutions in the design of this equipment.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 56, No. 4, pp. 555–562, April, 1989.     

Explicit solution for the potential flow due to an assembly of stirrers in an inviscid fluid

The motion of an irrotational incompressible fluid driven by an assembly of stirrers, of arbitrary shape, moving at specified velocities in the fluid is considered. The problem is shown to be equivalent to a standard mathematical problem in potential theory known as the modified Schwarz problem. It turns out that the solution to this problem can be written down, in closed form, as an explicit integral depending on the conformal mapping to the fluid region from a canonical pre-image region and a kernel function expressed in terms of a transcendental function called the Schottky–Klein prime function. In this way, an explicit integral solution, up to conformal mapping, for the complex potential of the flow generated by an arbitrary assembly of stirrers can be written down.     

Statistical theory of the nonuniform turbulence of an incompressible fluid

The nonuniform turbulence problem is treated by a statistical approach based on the use of a finite number of equations for the higher-order single-point correlations. Additional differential equations are derived for the unknown moments in the single-point correlation equations. The equations are closed by means of approximate expressions for the anisotroplc two-point correlation tensors for near points. A closed system of seven tensorial differential equations is given, describing the variation of the fundamental characteristics of nonuniform turbulence.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 19, No. 5, pp. 789–802, November, 1970.     

Adhesive force between a spherical rigid particle and an incompressible elastic substrate

Adhesion between a spherical rigid particle and an incomprssible elastic substrate is studied on the basis of the Lennard–Jones (L–J) potential, and the aim is to explore limitations of the well-known Derjaguin approximation. A new expression of the adhesive force is derived, in which the contribution from the elastic deformation of the substrate is incorporated naturally. Numerical results show that the Derjaguin approximation is valid down to particle radii of the order of the interaction range.     

Flexural- and capillary-gravity waves due to fundamental singularities in an inviscid fluid of finite depth

Wave motion due to line, point and ring sources submerged in an inviscid fluid are analytically investigated. The initially quiescent fluid of finite depth, covered by a thin elastic plate or by an inertial surface with the capillary effect, is assumed to be incompressible and homogenous. The strengths of the sources are time-dependent. The linearized initial-boundary-value problem is formulated within the framework of potential flow. The perturbed flow is decomposed into the regular and the singular components. An image system is introduced for the singular part to meet the boundary condition at the flat bottom. The solutions in integral form for the velocity potentials and the surface deflexions due to various singularities are obtained by means of a joint Laplace-Fourier transform. To analyze the dynamic characteristics of the flexural- and capillary-gravity waves due to unsteady disturbances, the asymptotic representations of the wave motion are explicitly derived for large time with a fixed distance-to-time ratio by virtue of the Stokes and Scorer methods of stationary phase. It is found that the generated waves consist of three wave systems, namely the steady-state gravity waves, the transient gravity waves and the transient flexural/capillary waves. The transient wave system observed depends on the moving speed of the observer in relation to the minimal and maximal group velocities. There exists a minimal depth of fluid for the possibility of the propagation of capillary-gravity waves on an inertial surface. Furthermore, the results for the pure gravity and capillary-gravity waves in a clean surface can also be recovered as the flexural and inertial parameters tend to zero.