Bifurcations in the transonic flow past a symmetric airfoil

Transonic flow past an airfoil with a small curvature in its midchord region is numerically investigated. The branching of the stationary solutions of the Euler equations is established and attributed to flow instability at certain angles of attack and freestream Mach numbers. The dependence of the lift coefficient on these parameters is studied.     

Nonuniqueness of the Transonic Flow past an Airfoil

The inviscid transonic flow past a symmetric airfoil having a curvature minimum in the middle is numerically investigated. It is shown that at zero angle of attack both symmetric and asymmetric steady-state flow patterns can exist on a certain freestream Mach number range M_min < M_max. On this range, the asymmetric flows are stable against small perturbations, whereas the symmetric flows are stable only if M does not coincide with a singular Mach number at which small variations in M or can result in flow restructuring.     

Bifurcations of transonic flow past simple airfoils with elliptic and wedge-shaped noses

A turbulent transonic flow past two symmetric airfoils with flat midparts is studied numerically. Using the Reynolds-averaged Navier-Stokes equations, we analyze the flow past a 9% thick airfoil with an elliptic nose. A range of the free-stream Mach number M_∞, in which flow bifurcations occur, is determined. Values of M_∞ that give rise to significant changes in the lift coefficient with variations of the angle of attack are specified. Flow bifurcations are also revealed for a thin double wedge, i.e., a sort of a hexagon.     

Structure and bifurcations of plane shock waves in a vibrationally excited gas with an external pumping source

The structure of the density profiles in stationary plane shock waves in a vibrationally excited gas is investigated. For self-similar solutions a bifurcation diagram is plotted in the parametric “traveling wave velocity—degree of nonequilibrium” plane. The bifurcation boundaries of the domains of existence of the structures of different types are analytically derived. It is shown that weak plane shock waves are unstable, accelerate, and break down into a sequence of pulses or-at a fairly high pumping rate-waves with nonzero asymptotics, whose amplitude and propagation velocity are independent of the initial disturbance and are determined by the parameters of the medium itself.     

Using the perturbation method to solve the problem of separated incompressible flow past thin airfoils

A model for separated incompressible flow past thin airfoils in the neighborhood of the “shockless entrance” condition is constructed based on the averaging of the vortex shedding flow past the airfoil edges. By approximation of the vortex shedding by two vortex curves, determination of the average hydrodynamic parameters is reduced to a twofold solution of an integral singular equation equivalent to the equation describing steady-state nonseparated airfoil flow. In this case, the calculation time is two orders of magnitude smaller than the time required for the solution of the corresponding evolution problem. The results of a test calculation using the proposed method are in fair agreement with available results of calculations and experiments. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 47, No. 3, pp. 49–63, May–June, 2006.     

Self-sustained oscillations of shear flow past a slotted plate coupled with cavity resonance

Shear flow past a slotted plate configuration can give rise to highly coherent, self-sustained oscillations when coupling occurs with a resonant mode of an adjacent cavity. The distinctive feature of these oscillations is that the wavelength of the coherent instability along the plate is of the order of the plate length. This observation is in contrast to previous investigations of flow past perforated or slotted surfaces, where the instability scales on the diameter of the perforation or the gap length of a slot. The present oscillations occur even when the inflow boundary layer is turbulent and an inflectional form of the shear flow cannot develop along the cavity opening, due to the presence of the slotted plate. Instigation of a resonant mode of the cavity, in conjunction with an inherent instability of the shear flow along the plate, gives rise to ordered clusters of instantaneous vorticity and instantaneous velocity correlation. During the oscillation, ejection of flow occurs from the cavity to the region of the shear flow; this ejection is in accord with the convection of the large-scale cluster of vorticity along the slotted plate. This oscillation can be effectively detuned by adjusting the inflow velocity, such that the inherent instability of the shear flow past the slotted plate is no longer coincident with the resonant frequency of the cavity. Certain features of this self-sustained oscillation are directly analogous to recent findings of oscillations due to shear flow past a perforated plate bounded by a cavity, but in the absence of cavity resonance effects.     

Distinctive features of the transonic flow past a cone-cylinder body at large angles of the bend in the body generator at the leading corner edge

The results of an experimental investigation of the flow past cone-cylinder bodies with small angles of the bend in the body generator at the leading corner edge (θ_s ≤ 20°) and the available data on the flow past cone-cylinder and segment-cylinder bodies with large angles of the bend in the generator (θ_{s, c} ≥ 30°) are generalized. On the basis of these data, the distinctive features of the transonic flow past these bodies and, primarily, the flow restructuring downstream of the leading corner edge at θ_s ≥ 30° are established and the characteristic gasdynamic parameters are determined. Emphasis is placed on the critical stage of flow restructuring, the nature of the stationary aerodynamic hysteresis, and the effect of the Reynolds number and wave disturbances.     

Experimental study of the reverse flow in the forward separation region in a pulsating flow around a spiked body

The flow with a free-stream Mach number M _∞ = 6 around a cylindrical body with a sharp spike is studied. The existence of a supersonic reverse flow for one of the phases of the pulsating flow regime is experimentally validated. A range of spike lengths is determined, which ensures a region of a supersonic reverse flow near the side surface of the spike. The time of existence of the supersonic reverse flow region is shown to be 0.15 of the period of pulsations if the spike length equals the model diameter. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 4, pp. 30–39, July–August, 2007.     

Distinctive features of the transonic flow past a cone-cylinder body at small angles of the bend in the body generator at the leading corner edge

The distinctive features of the transonic flow past cone-cylinder bodies with small angles of the bend in the body generator at the leading corner edge are considered. The mechanism of transonic flow restructuring downstream of the edge and the associated variation of the gasdynamic parameters are studied, the reasons for the formation of aerodynamic hysteresis are established, and the effect of the length of the cylindrical part of the body is determined.     

The structural instability of transonic flow associated with amalgamation/splitting of supersonic regions

We study inviscid transonic flow in a channel with a bump modeling an airfoil at zero incidence. A few simple bump configurations are considered. Numerical simulations show the existence of singular free-stream Mach numbers, which trigger the splitting/amalgamation of local supersonic regions. An analysis of this phenomenon contributes to the understanding of the nonuniqueness of transonic solutions at certain free-stream conditions.     

Oblique shock/bow shock interference in rarefied flow past a cylinder

Direct statistical simulation is employed to study the flow of a rarefied diatomic gas past a cylinder in the presence of an incident oblique shock. The distinctive features of the formation of a high-pressure compressed-gas jet in the case of interference between the oblique shock and the bow shock are studied for different Reynolds numbers. The variation of the pressure and the heat transfer to the surface with the shock position relative to the center of the cylinder, the Reynolds number, and the surface temperature is analyzed. The results obtained are compared with the experimental data and the results of the numerical solutions of the Euler and boundary layer equations. Free-molecular-to-continuum flow transition is demonstrated with reference to the example of interference-free flow past a cylinder.Translated from Izvestiya Rossiiskoi Academii Nauk, Mekhanika Zhidkosti i Gaza, No. 5, 2004, pp. 171–180. Original Russian Text Copyright © 2004 by Gusev and Erofeev.     

Structure of Time-Dependent Transonic Flows in Plane Channels

The evolution and interaction of supersonic zones and the shock waves closing them are considered on the basis of the Lin-Reissner-Tsien equation for flow in plane channels with a local bottleneck.__________Translated from Izvestiya Rossiiskoi Academii Nauk, Mekhanika Zhidkosti i Gaza, No. 2, 2005, pp. 168–179.Original Russian Text Copyright © 2005 by Bibik, Duesperov, and Popov.     

弯度翼型大迎角时分离涡流场的多平衡态现象

基于kω的SST两方程湍流模型,在时间域求解雷诺平均Navier-Stokes方程,模拟弯度翼型大迎角时的分离流动。通过给翼型施加一定形式的扰动,重点关注了翼型弯度对大迎角分离涡流场平衡态转移的影响。研究结果表明:与相同厚度20%以上的对称翼型相比,2%弯度的翼型出现分离涡流场平衡态转移的起始迎角变小2°左右,迎角区间变宽约1°;在厚度相对较小的NACA2416翼型上也发现上述分离涡平衡态转移现象。由此说明翼型弯度在一定程度上促使了分离涡平衡态的转移。     

Local and global bifurcations of valve mechanism

In this paper we study in detail problems of nonlinear oscillations of valve mechanism at internal combustion engine. The practical measurement indicates that stiffness of valve mechanism is not constant but is a function of the rotational angle of the cam. For simplicity of analysis we replace valve mechanism of internal combustion engine with a nonlinear oscillator of single degree of freedom under combined parametric and forcing excitation. We use the method of multiple scales and normal form theory to study local and global bifurcations of valve mechanism at internal combustion engine.     

Direct and adjoint global stability analysis of turbulent transonic flows over a NACA0012 profile

In this work, various turbulent solutions of the two‐dimensional (2D) and three‐dimensional compressible Reynolds averaged Navier–Stokes equations are analyzed using global stability theory. This analysis is motivated by the onset of flow unsteadiness (Hopf bifurcation) for transonic buffet conditions where moderately high Reynolds numbers and compressible effects must be considered. The buffet phenomenon involves a complex interaction between the separated flow and a shock wave. The efficient numerical methodology presented in this paper predicts the critical parameters, namely, the angle of attack and Mach and Reynolds numbers beyond which the onset of flow unsteadiness appears. The geometry, a NACA0012 profile, and flow parameters selected reproduce situations of practical interest for aeronautical applications. The numerical computation is performed in three steps. First, a steady baseflow solution is obtained; second, the Jacobian matrix for the RANS equations based on a finite volume discretization is computed; and finally, the generalized eigenvalue problem is derived when the baseflow is linearly perturbed. The methodology is validated predicting the 2D Hopf bifurcation for a circular cylinder under laminar flow condition. This benchmark shows good agreement with the previous published computations and experimental data. In the transonic buffet case, the baseflow is computed using the Spalart–Allmaras turbulence model and represents a mean flow where the high frequency content and length scales of the order of the shear‐layer thickness have been averaged. The lower frequency content is assumed to be decoupled from the high frequencies, thus allowing a stability analysis to be performed on the low frequency range. In addition, results of the corresponding adjoint problem and the sensitivity map are provided for the first time for the buffet problem. Finally, an extruded three‐dimensional geometry of the NACA0012 airfoil, where all velocity components are considered, was also analyzed as a Triglobal stability case, and the outcoming results were compared to the previous 2D limited model, confirming that the buffet onset is well detected. Copyright © 2014 John Wiley & Sons, Ltd.     

Shocks in local supersonic zones

The results of numerical integration of the Euler equations governing two-dimensional and axisymmetric flows of an ideal (inviscid and non-heat-conducting) gas with local supersonic zones are presented. The subject of the study is the formation of shocks closing local supersonic zones. The flow in the vicinity of the initial point of the closing shock is calculated on embedded, successively refined grids with an accuracy much greater than that previously achieved. The calculations performed, together with the analysis of certain controversial issues, leave no doubt that it is the intersection of C ^?-characteristics proceeding from the sonic line inside the supersonic zone that is responsible for the closing shock formation.     

Heteroclinic orbit and subharmonic bifurcations and chaos of nonlinear oscillator

Dynamical behavior of nonlinear oscillator under combined parametric and forcing excitation, which includes van der Pol damping, is very complex. In this paper, Melnikov'smethod is used to study the heteroclinic orbit bifurcations, subharmonic bifurcations and chaos in this system. Smale horseshoes and chaotic motions can occur from odd subharmonic bifurcation of infinite order in this system for various resonant cases.Finally the numerical computing method is used to study chaotic motions of this system. The results achieved reveal some new phenomena.     

Investigation of transonic and supersonic flow past a pair of bodies

Transonic and supersonic flows past a pair of bodies have been experimentally investigated. The leading bodies were spheres, cylinders, and cones, while the trailing bodies were flat-ended circular cylinders. The leading and trailing bodies were joined by cylindrical rods of various lengths, aligned with the axis of symmetry. For these models, the pattern of flow between the bodies and the Mach number dependence of the drag coefficientC _x were determined in the acceleration and deceleration flow regimes in a wind tunnel. The experimental results are used to analyze the properties of the flow between the bodies and the variation of the aerodynamic coefficients of the models. The reasons for the hysteresis in the behavior of the coefficients in the acceleration and deceleration stages are discussed. The influence of the shape and dimensions of the leading body on the modelC _x is evaluated. Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 2, pp. 158–164, March–April, 1998.     

Effect of one-sided unsteady energy supply on aerodynamic characteristics of airfoils in a transonic flow

The possibility of controlling the aerodynamic characteristics of airfoils in transonic flight regimes by means of one-sided pulsed-periodic energy supply is studied. Based on the numerical solution of two-dimensional unsteady gas-dynamic equations, the change in the flow structure in the vicinity of a symmetric airfoil at different angles of attack and the aerodynamic characteristics of the airfoil as functions of the amount of energy supplied asymmetrically (with respect to the airfoil) are determined. The results obtained are compared with the data calculated for the flow past the airfoil at different angles of attack without energy supply. It is found that a given lift force can be obtained with the use of energy supply at a much better lift-to-drag ratio of the airfoil, as compared to the case of the flow past the airfoil at an angle of attack. The moment characteristics of the airfoil are found. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 49, No. 6, pp. 82–87, November–December, 2008.     

Supersonic Flow Around a V-Shaped Wing at Incidence and Yaw

A solution of the problem of the flow around a V-wing with supersonic leading edges at low angles of attack and yaw is obtained within the framework of the linear theory. Possible patterns of nonsymmetric flow around the wing are analyzed as functions of the wing geometry and the freestream velocity direction, and the ranges of angles of attack and yaw on which these patterns are realized are established. Some previously undescribed shock wave configurations are found to exist in the wing-induced conical flows.