Aerodynamic interaction of two plane cascades of thin lightly loaded blades in relative motion in a subsonic gas flow

The problem of subsonic ideal-gas flow over two plane cascades of thin lightly loaded blades in relative motion is solved within the framework of the linear theory of small perturbations. By means of the method of integral equations [1] the problem is reduced to an infinite system of singular integral equations for the harmonic components of the oscillations in the distribution of the unknown aerodynamic load on the blades. The regularized system of integral equations for a finite number of harmonics is solved numerically by a collocation method.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 168–175, May–June, 1987.     

Unsteady aerodynamic interaction of two annular blade rows of thin lightly loaded blades rotating one relative to the other in a subsonic flow

The problem of subsonic unsteady ideal-gas flow over two annular blade rows of thin lightly loaded blades rotating one relative to the other is solved within the framework of linear small perturbation theory. As in the case of the interaction of two-dimensional cascades [1], the problem reduces to an infinite system of singular integral equations for the harmonic components of the oscillations in the distribution of the unknown aerodynamic load on one blade of each row. The system of integral equations for a finite number of harmonics is solved numerically by the collocation method. The kernels of the integral equations are regularized on the basis of the method proposed in [2].Translated from Izvestiya Akademii Nauk.SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 165–174, May–June, 1991.The author is grateful to A. A. Osipov and K. K. Butenko for their considerable assistance in the preparation of this paper.     

Cascaded thin lightly loaded airfoils vibrating in a subsonic separated flow

The problem of calculating the nonstationary aerodynamic characteristics of a cascade of thin lightly loaded airfoils in a subsonic flow with the formation of thin separation zones of finite extent is solved approximately. As in [1–5], an approach based on a linear small-perturbation analysis, within which the flow is assumed to be inviscid, is employed and the boundaries of the unsteady separation zones are simulated by oscillating lines of contact discontinuity. However, instead of the requirement of a given fixed pressure at the boundary of the separation zone, used in [1–5], this study proposes a more general condition according to which on each element of length of the thin separation layer the pressure oscillates with an amplitude proportional to the local value of the amplitude of its thickness oscillations. The problem is reduced to a system of two singular integral equations which can be solved numerically.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 181–191, January–February, 1995.     

Unsteady flow past a cascade of staggered thin profiles

Many studies have been made of plane flow of an incompressible inviscid fluid past a cascade of profiles with arbitrary stagger angle 0. For example, in the particular case of the motion of a cascade with the stagger angle at zero-oscillation phase-shift angle =0 Khaskind [1] determined the unsteady lift force theoretically by isolating the singularities with the Sedov method [2], applying a conformal mapping to the cascade of unstaggered flat plates. Belotserkovskii et al. [3] calculated the over-all unsteady aerodynamic characteristics of a cascade in the particular case =0 and for any on a computer by the method of discrete vortices, and for the more general case (0) Whitehead [4] has done the same using a vortex method. Gorelov and Dominas [5] calculated the over-all unsteady force and moment coefficients of a profile in a cascade with stagger angle 0 and phase shift 0.The calculation method was based on unsteady theory for a slender isolated profile whose flow pattern is known, with subsequent account for the interference of the profiles and the vortex wakes behind them.In the present study the singularity isolation method [2] is extended to slender profiles with arbitrary stagger angle 0 and arbitrary phase shift 0 of the oscillations between neighboring profiles. It is shown that the solution reduces to the solution of a Fredholm integral equation of the first kind in terms of the sum of the tangential velocity components along the profile. It is found that the relative effect of the unsteady flow due to the system of vortex trails behind the cascade with stagger may be determined without solving this integral equation. However, this solution must be found to calculate the added masses of the cascade and the total magnitudes of the unsteady forces. It is found that regularization transforms the integral equation of the first kind to an integral equation of the second kind, for which solution methods are known.Thus the expressions for the unsteady forces are determined in the form of separate terms, each of which has a physical significance: as a result we obtain finite formulas (improper integrals) for calculating the variable forces; from these formulas are derived the asymptotic expressions for the forces in the limiting cases of high and low solidities and Strouhal numbers, which as a rule are lost in numerical calculations. The proposed method may be considered as one of the techniques for improving the convergence of the numerical methods (elimination of singularities). Moreover, this method may be used to solve the problems of unsteady flow past cascades of arbitrary systems of slender profiles for various profile incidence angles relative to the x-axis and in the presence of a finite cavitation zone on the profiles.The limited practical application of this method is explained by the extreme theoretical difficulties in its applications to cascades with stagger angle. In the present studies these difficulties are examined using the example of a cascade with stagger angle.     

Unsteady supersonic flow past a sphere moving through a thermal inhomogeneity

Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 186–189, March–April, 1989.     

Unsteady characteristics of low-Re flow past two tandem cylinders

This study investigated the two-dimensional flow past two tandem circular or square cylinders at Re = 100 and D / d = 4–10, where D is the center-to-center distance and d is the cylinder diameter. Numerical simulation was performed to comparably study the effect of cylinder geometry and spacing on the aerodynamic characteristics, unsteady flow patterns, time-averaged flow characteristics and flow unsteadiness. We also provided the first global linear stability analysis and sensitivity analysis on the physical problem for the potential application of flow control. The objective of this work is to quantitatively identify the effect of the cylinder geometry and spacing on the characteristic quantities. Numerical results reveal that there is wake flow transition for both geometries depending on the spacing. The characteristic quantities, including the time-averaged and fluctuating streamwise velocity and pressure coefficient, are quite similar to that of the single cylinder case for the upstream cylinder, while an entirely different variation pattern is observed for the downstream cylinder. The global linear stability analysis shows that the spatial structure of perturbation is mainly observed in the wake of the downstream cylinder for small spacing, while moves upstream with reduced size and is also observed after the upstream cylinder for large spacing. The sensitivity analysis reflects that the temporal growth rate of perturbation is the most sensitive to the near-wake flow of downstream cylinder for small spacing and upstream cylinder for large spacing.     

Unsteady incompressible fluid flow past a cascade of thin curved plates

A large number of papers have been devoted to the study of unsteady flow past airfoil cascades. The majority of authors solve the problem for slightly curved profiles oscillating at low angles of attack.Among other work, we note that of Söhngen [1] on the flow past a dense cascade of plates oscillating synchronously and in phase in a potential fluid flow at a high angle of attack. Samoilovich [2] studied the flow past a cascade of plates of arbitrary shape oscillating with an arbitrary phase shift between neighboring plates. He presents the solution for the case of variable circulation in the quasisteady formulation. Stepanov [3] studied the same question with a linear approach to the flow behind the cascade. Musatov [4] examined the problem of the flow past a cascade of plates oscillating with an arbitrary phase shift between neighboring plates in a fluid flow, again at a high angle of attack, and considered the variation of the relative position of the plates durilng the oscillation process.The present paper considers the flow of a perfect incompressible fluid past a cascade of thin curved oscillating plates with account for the relative displacements of the plates during oscillation. To determine the intensity of the bound vortices per unit length, a linear integral equation is obtained. This represents a generalization of the Birnbaum equation to the case considered (see [5]). Equations are presented for calculating the unsteady aerodynamic forces and moments acting on the plates. As an example, the aerodynamic forces and moments are calculated for the quasistationary formulation of the problem.     

Problems of incompressible flow past thin blades and correction of their shape

Solutions to the problem of the flow of an ideal fluid past a blade or cascade of blades with low blocking factor are found in the framework of the first approximation of the theory of perturbations of the flow past infinitely thin arcs. Problems of correction of the shape of the blades are also considered. Problems associated with the application of perturbation theory in problems of flow past bodies are discussed in Van Dyke's monograph [1]. The present paper includes an example of realization of this theory for the thin curved blades that are widely used in compressor construction. Searches for effective methods for calculating the shape of blades to ensure necessary gas-dynamic properties, for example, a given distribution on the blades of the velocity of separationless flow, led to the appearance of algorithms based on the theory of small perturbations for a thin wing of finite span [2] and a single airfoil in a gas flow [3]. In such an approach, the problem of constructing the required profile can be formulated as a sequence of corrections of the boundary of the flow region with respect to small variations of the boundary values of the flow velocity. The paper contains a general formulation of the linear problem of the correction of the flow boundary, an algorithm for its solution in the case of thin blades in an incompressible flow, and analysis of the obtained solutions. Examples of calculations are presented.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 130–137, January–February, 1992.We thank S.A. Smirnov for assistance in the calculations.     

Effects of viscous dissipation on heat transfer in an oscillating flow past a flat plate

Theoretical investigation has been carried out of laminar thermal boundary layer response to harmonic oscillations in velocity associated with a progressive wave imposed on a steady free stream velocity and convected in the free stream direction. Series solutions are derived both to velocity and temperature field and the resulting equations are solved numerically. The functions affecting the temperature field are shown graphically for different values of Prandtl number. It is observed that there is more reduction in the rate of heat transfer for P _r<1 and a rise in the rate of heat transfer for P _r>1 due to the presence of oscillatory free-stream.Nomenclature u, v velocity components in the x and y direction - x, y Cartesian coordinate axes - t time - U, U ₀ instantaneous value of and mean free stream velocity - density of fluid - kinematic viscosity - T, T _w, T temperature of the fluid, wall and free stream fluid - c _p specific heat at constant pressure - thermal diffusivity - amplitude of free stream velocity - frequency - _p non-dimensional temperature (T–T /T _w–T ) - P _r Prandtl number (c _p/K) - E _c Eckert number (U ₀ ² /c _p(T _w–T )) - a parameter ( ) - ₀ boundary layer thickness of the oscillation of a harmonic oscillation of frequency ( ) - ordinary boundary layer thickness ( ) - time-averaged, time-independent external velocity - A, B, C, D, E, K, L, M, N, P functions used in expansion for u and - Nu Nusselt number (hx/k) - T _w–% MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8V4rqqrFfpeea0Jc9yq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepGe9fr-xfr-x% frpeWZqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaacIcadaGcaa% qaaiaadAhacaWG4bGaai4laiqadwfagaqeaaWcbeaakiaacMcaaaa!3CA6!\[(\sqrt {vx/\bar U} )\] - k thermal conductivity     

Unsteady natural convective flow past a moving vertical cylinder with heat and mass transfer

Transient natural convection boundary layer flow of an incompressible viscous fluid past an impulsively moving semi- infinite vertical cylinder is considered. The temperature and concentration of the cylinder surface are taken to be uniform. The unsteady, nonlinear and coupled governing equations of the flow are solved using an implicit finite difference scheme. The finite difference scheme is unconditionally stable and accurate. Numerical results are presented with various sets of parameters for both air and water. Transient effects of velocity, temperature and concentration profiles are analyzed. Local and average skin friction, rates of heat and mass transfer are shown graphically. Received on 1 November 1999     

Numerical modeling of gas flow around oscillating compressor blades

The problem of gas flow around a plane cascade of oscillating blades is numerically solved using the ANSYS CFX package. The blade surface displacement is taken into account using a movable grid generated before the beginning of the calculations at each time step. The calculated and experimental data are compared. The calculated results are used for determining the blade stability against flutter.     

Heat transfer in flow past a continuously moving porous flat plate with heat flux

The analysis of the heat transfer in flow past a continuously moving semi-infinite plate in the presence of suction/ injection with heat flux has been presented. Similarity solutions have been derived and the resulting equations are integrated numerically. It has been observed that an increase in suction value leads to an increase in temperature whereas it is opposite in the case of injection.

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Wärmeübergang in einer Strömung hinter einer wärmeabgebenden, kontinuierlich bewegten, porösen Platte

Zusammenfassung Es wird eine Analyse des Wärmeübergangs in einer Strömung hinter einer bewegten, halbunendlichen Platte bei gleichzeitiger Absaugung bzw. Ausströmung präsentiert. Mit Hilfe von Ähnlichkeitsbetrachtungen wurden Lösungen erarbeitet und die daraus resultierenden Gleichungen wurden numerisch integriert. Als Ergebnis zeigte sich, daß zunehmende Absaugung zu einer Erhöhung der Temperatur führt, während bei der Ausströmung das Gegenteil der Fall ist.

     

Global stability of separated flows: subsonic flow past corners

The triple-deck equations for the steady subsonic flow past a convex corner are solved numerically using a novel technique based on Chebychev collocation in the direction normal to the body combined with finite differences in the direction along the flow. The resulting set of nonlinear algebraic equations are solved with Newton linearization and using the GMRES method for the solution of the linear system of equations. The stability of the computed steady flows is then examined using global stability analysis. It is found that for small corner angles, the Tollmien?CSchlichting modes are globally unstable and these persist to larger corner angles. Multiple steady state solutions also exist beyond a critical corner angle but these are globally unstable because of the presence of the Tollmien?CSchlichting modes.     

Method of calculating subsonic gas flow with separation past wings

The steady nonlinear problem of subsonic compressible gas flow past a wing of arbitrary shape in plan is considered. A numerical method was devized for solving the problem; this is a further development of the method of discrete vortices. The surface of the body and the vortex wake behind it are simulated by systems of discrete vortex sections, but, in contrast to the case of an incompressible medium, it is necessary in this case for the sources to be distributed outside the wing. The circulations of the attached vortices, the strengths of the sources, and the shape of the wake are determined by iterations.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 140–147, July–August, 1984.     

Unsteady flow of a fourth-grade fluid due to an oscillating plate

The governing non-linear high-order, sixth-order in space and third-order in time, differential equation is constructed for the unsteady flow of an incompressible conducting fourth-grade fluid in a semi-infinite domain. The unsteady flow is induced by a periodically oscillating two-dimensional infinite porous plate with suction/blowing, located in a uniform magnetic field. It is shown that by augmenting additional boundary conditions at infinity based on asymptotic structures and transforming the semi-infinite physical space to a bounded computational domain by means of a coordinate transformation, it is possible to obtain numerical solutions of the non-linear magnetohydrodynamic equation. In particular, due to the unsymmetry of the boundary conditions, in numerical simulations non-central difference schemes are constructed and employed to approximate the emerging higher-order spatial derivatives. Effects of material parameters, uniform suction or blowing past the porous plate, exerted magnetic field and oscillation frequency of the plate on the time-dependent flow, especially on the boundary layer structure near the plate, are numerically analysed and discussed. The flow behaviour of the fourth-grade non-Newtonian fluid is also compared with those of the Newtonian fluid.     

Unsteady aerodynamic characteristics of a three-dimensional plate cascade in a subsonic gas flow

M. A. Lavrent'ev Institute of Hydrodynamics of the Siberian Division of the Russian Academy of Sciences, 630090 Novosibirsk. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, No. 2, pp. 45–55, March–April, 1995.     

Acoustic resonance in turbomachinery with aerodynamic interaction of the cascades in subsonic gas flow

Sergiev Posad, Moscow Oblast. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, No. 4, pp. 78–85. July–August, 1994.     

Mathematical modeling of the sound induced in a subsonic flow around cascades in relative motion

A method of mathematical modeling of the tonal sound induced by the unsteady aerodynamic interaction of two plane airfoil cascades in a subsonic flow and in uniform relative motion in the direction of their fronts is developed. The method is based on the numerical integration of the unsteady flow equations using a simplified model for the periodic system of edge wakes shed from the airfoils of the first (leading) cascade in the viscous flow and acting on the second (trailing) cascade. An analysis of the distinctive features of the flow under consideration demonstrates the efficiency of the model proposed.