Shock-vortex shear-layer interaction in the transonic flow around a supercritical airfoil at high Reynolds number in buffet conditions

This paper provides a conceptual analysis and a computational model for how the unsteady ‘buffeting’ phenomenon develops in transonic, low incidence flow around a supercritical aerofoil, the OAT15A, at Reynolds number of 3.3 million. It is shown how a low-frequency buffet mode is amplified in the shock-wave region and then develops upstream and downstream interaction with the alternating von Kármán eddies in the wake past the trailing-edge as well as with the shear-layer, Kelvin–Helmholtz vortices. These interactions are tracked by wavelet analysis, autoregressive (AR) modelling and by Proper Orthogonal Decomposition. The frequency modulation of the trailing-edge instability modes is shown in the spectra and in the wall-pressure fluctuations. The amplitude modulation of the buffet and von Kármán modes has been also quantified by POD analysis. The thinning of the shear layers, both at the outer edge of the turbulent boundary layers and the wake, caused by an ‘eddy-blocking’ mechanism is modelled by stochastic forcing of the turbulent kinetic energy and dissipation, by small-scale straining of the higher-order POD modes. The benefits from thinning the shear-layers by taking into account the interfacial dynamics are clearly shown in the velocity profiles, and wall pressure distribution in comparison with the experimental data.     

Transonic flow past a convex corner with free streamline

An asymptotic analysis in the limit of large Reynolds numbers Re → ∞ is made of the system of Navier—Stokes equations in the neighborhood of a corner in a profile past which there is a transonic gas flow. The flow with free streamline from the corner point at which the velocity of sound is reached is taken as a limiting case. In the first approximation, it is described by a self-similar solution to the Kármán—Fal'kovich equation with self-similarity exponent n = 6/5 [1]. In such a flow, the favorable pressure gradient becomes infinite as the corner is approached from the side of the oncoming flow.     

Control of flow past a circular cylinder via a spanwise surface wire: effect of the wire scale

Flow phenomena induced by a single spanwise wire on the surface of a circular cylinder are investigated via a cinema technique of particle image velocimetry (PIV). The primary aim of this investigation is to assess the effect of the wire scale. To this end, consideration is given to wires with different diameters that are 0.5, 1.2, and 2.9% of the cylinder diameter. The Reynolds number has a subcritical value of 10,000. Compared to the thickness of the unperturbed boundary layer developing around the cylinder between 5° and 75° from the forward stagnation point, the former two wires have smaller scales and the latter has a larger scale. Two angular locations of the wire, defined with respect to the forward stagnation point of the cylinder, are found to be critical. When the wire is located at these critical angles, either the most significant extension or the contraction of the time-mean separation bubble occurs in the near wake. These critical angles depend on the wire scale: the smaller the wire, the larger the critical angle. The small-scale and large-scale wires that have diameters of 1.2 and 2.9% of the cylinder diameter induce bistable shear-layer oscillations between different separation modes when placed at their respective critical angles corresponding to maximum extension of the near-wake bubble. These oscillations have irregular time intervals that are much longer than the time scale associated with the classical Kármán instability. Moreover, the large-scale wire can either significantly attenuate or intensify the Kármán mode of vortex shedding at the critical states; in contrast, the small-scale wires do not notably alter the strength of the Kármán instability.     

Secondary wake instabilities of a blunt trailing edge profiled body as a basis for flow control

Flow in the wake of a blunt trailing edge profiled body, comprised of an elliptical leading edge and a rectangular trailing edge, has been investigated experimentally, to identify and characterize the secondary instabilities accompanying the von Kármán vortices. The experiments, which involve laser-induced fluorescence for visualization and particle image velocimetry for quantitative measurement of the wake instabilities, cover Reynolds numbers ranging from 250 to 2,150 based on thickness of the body, to include the wake transition regime. The dominant secondary instability appears as spanwise undulations in von Kármán vortices, which evolve into pairs of counter-rotating vortices, with features resembling the instability mechanism predicted by Ryan et al. (J Fluid Mech 538:1–29, 2005). Feasibility of a flow control approach based on interaction with the secondary instability using a series of discrete trailing edge injectors has also been investigated. The control approach mitigates the adverse effects of vortex shedding in certain conditions, where it is able to amplify the secondary instability effectively.     

Exotic vortex wakes—point vortex solutions

Von Kármán was the first to present a quantitative model of the “vortex street” wake as a double row of point vortices, to determine which configurations propagate in the direction of the rows, and to consider the linear stability theory for such states. In the early literature one works with infinite rows of vortices. The vortex street is assumed to continue to infinity both upstream and downstream. Another analytical approach is to use periodic boundary conditions in the direction of the wake. This representation was used by Domm in his analysis of the instability of the Kármán vortex street. Birkhoff and Fisher in 1959 were the first to treat vortices in a periodic strip as a dynamical system in its own right. We have used the periodic system to address problems of vortex wake patterns, in particular vortex wakes that are more complicated than the traditional two-vortices-per-strip configurations. We use the term “exotic” for such wakes. We submit that this approach can yield a number of insights, including results of direct relevance to experiments, in the same sense that von Kármán's analysis has been helpful to the understanding of the regular vortex street wake, and we present the results obtained to date following this program.     

One-dimensional von Kármán models for elastic ribbons

By means of a variational approach we rigorously deduce three one-dimensional models for elastic ribbons from the theory of von Kármán plates, passing to the limit as the width of the plate goes to zero. The one-dimensional model found starting from the “linearized” von Kármán energy corresponds to that of a linearly elastic beam that can twist but can deform in just one plane; while the model found from the von Kármán energy is a non-linear model that comprises stretching, bendings, and twisting. The “constrained” von Kármán energy, instead, leads to a new Sadowsky type of model.     

Buckling of a pre-compressed or pre-stretched membrane in shear flow

Membranes enclosing capsules and biological cells undergo periodic compression and stretching due to an imparted hydrodynamic traction as they rotate in a shear flow. Compression may cause transient or permanent buckling manifested by the onset of wrinkled shapes. To study the effect of pre-compression and pre-stretching on the critical conditions for buckling, the response of an elastic circular plate flush mounted on a plane wall and deforming under the action of a uniform tangential load due to an over-passing simple shear flow is considered. Working under the auspices of the theory of elastic instability of plates governed by the linear von Kármán equation, an eigenvalue problem is formulated resulting in a fourth-order partial differential equation with position-dependent coefficients parametrized by the Poisson ratio. Solutions are computed by applying Fourier series expansions to derive an infinite system of coupled ordinary differential equations, and then implementing orthogonal collocation. The solution space is illustrated, critical values for buckling are identified, the associated eigenfunctions representing possible modes of deformation are displayed, and the effect of the Poisson ratio is discussed.     

Buckling of a flush-mounted plate in simple shear flow

The buckling of an elastic plate with arbitrary shape flush-mounted on a rigid wall and deforming under the action of a uniform tangential load due to an overpassing simple shear flow is considered. Working under the auspices of the theory of elastic instability of plates governed by the linear von Kármán equation, an eigenvalue problem is formulated for the buckled state resulting in a fourth-order partial differential equation with position-dependent coefficients parameterized by the Poisson ratio. The governing equation also describes the deformation of a plate clamped around the edges on a vertical wall and buckling under the action of its own weight. Solutions are computed analytically for a circular plate by applying a Fourier series expansion to derive an infinite system of coupled ordinary differential equations and then implementing orthogonal collocation, and numerically for elliptical and rectangular plates by using a finite-element method. The eigenvalues of the resulting generalized algebraic eigenvalue problem are bifurcation points in the solution space, physically representing critical thresholds of the uniform tangential load above which the plate buckles and wrinkles due to the partially compressive developing stresses. The associated eigenfunctions representing possible modes of deformation are illustrated, and the effect of the Poisson ratio and plate shape is discussed.     

Direct Numerical Simulation of the Three-Dimensional Transition to Turbulence in the Transonic Flow around a Wing

The three-dimensional transition to turbulence in the transonic flow around a NACA0012 wing of constant spanwise section has been analysed at zero incidence and within the Reynolds number range [3000, 10000], by performing the direct numerical simulation. The successive stages of the 2D and 3D transition beyond the first bifurcation have been identified. A 2D study has been carried out near the threshold concerning the appearance of the first bifurcation that is the von-Kármán instability. The critical Mach number associated with this flow transition has been evaluated. Three other successive stages have been detected as the Mach number further increases in the range [0.3, 0.99]. Concerning the 3D transition of this nominally 2D flow configuration, the amplification of the secondary instability has been studied within the Reynolds number range of [3000, 5000]. The formation of counter-rotating longitudinal vorticity cells and the consequent appearance of a large-scale spanwise wavelength have been obtained downstream of the trailing-edge shock. A vortex dislocation pattern is developed as a consequence of the shock-vortex interaction near the trailing edge. The subcritical nature of the present 3D transition to turbulence has been proven by means of the DNS amplification signals and the Landau global oscillator model.     

Thickness effect of NACA foils on hydrodynamic global parameters,boundary layer states and stall establishment

The present study experimentally investigates the hydrodynamic behaviour of 2-D NACA (15%, 25% and 35%) symmetric hydrofoils at Reynolds number 0.5×10⁶. Particular attention was paid to the hysteretic behaviour at the static stall angle, and a detailed cartography of the boundary layer structures (integral quantities and velocity profiles) is given to support the detachment mechanism and the onset of von Kármán instability for thick hydrofoils.     

ON THE REFINED FIRST-ORDER SHEAR DEFORMATION PLATE THEORY OF KARMAN TYPE

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Thermal-induced snap-through buckling of simply-supported functionally graded beams

The instability of functionally graded material (FGM) structures is one of the major threats to their service safety in engineering applications. This paper aims to clarify a long-standing controversy on the thermal instability type of simply-supported FGM beams. First, based on the Euler-Bernoulli beam theory and von Kármán geometric nonlinearity, a nonlinear governing equation of simply-supported FGM beams under uniform thermal loads by Zhang's two-variable method is formulated. Second, an approximate analytic solution to the nonlinear integro-differential boundary value problem under a thermal-induced inhomogeneous force boundary condition is obtained by using a semiinverse method when the coordinate axis is relocated to the bending axis (physical neutral plane), and then the analytical predictions are verified by the differential quadrature method (DQM). Finally, based on the free energy theorem, it is revealed that the symmetry breaking caused by the material inhomogeneity can make the simply-supported FGM beam under uniform thermal loads occur snap-through postbuckling only in odd modes; furthermore, the nonlinear critical load of thermal buckling varies non-monotonically with the functional gradient index due to the stretching-bending coupling effect. These results are expected to provide new ideas and references for the design and regulation of FGM structures.     

Global stability behaviour for the BEK family of rotating boundary layers

Numerical simulations were conducted to investigate the linear global stability behaviour of the Bödewadt, Ekman, von Kármán (BEK) family of flows, for cases where a disc rotates beneath an incompressible fluid that is also rotating. This extends the work reported in recent studies that only considered the rotating-disc boundary layer with a von Kármán configuration, where the fluid that lies above the boundary layer remains stationary. When a homogeneous flow approximation is made, neglecting the radial variation of the basic state, it can be shown that linearised disturbances are susceptible to absolute instability. We shall demonstrate that, despite this prediction of absolute instability, the disturbance development exhibits globally stable behaviour in the BEK boundary layers with a genuine radial inhomogeneity. For configurations where the disc rotation rate is greater than that of the overlying fluid, disturbances propagate radially outwards and there is only a convective form of instability. This replicates the behaviour that had previously been documented when the fluid did not rotate beyond the boundary layer. However, if the fluid rotation rate is taken to exceed that of the disc, then the propagation direction reverses and disturbances grow while convecting radially inwards. Eventually, as they approach regions of smaller radii, where stability is predicted according to the homogeneous flow approximation, the growth rates reduce until decay takes over. Given sufficient time, such disturbances can begin to diminish at every radial location, even those which are positioned outwards from the radius associated with the onset of absolute instability. This leads to the confinement of the disturbance development within a finitely bounded region of the spatial–temporal plane.     

ON EXACT SOLUTION OF KÁRMÁN’S EQUATIONS OF RIGID CLAMPED CIRCULAR PLATE AND SHALLOW SPHERICAL SHELL UNDER A CONCENTRATED LOAD

It is extremely difficult to obtain an exact solution of von Karman’s equations because the equations are nonlinear and coupled. So far many approximate methods have been used to solve the large deflection problems except that only a few exact solutions have been investigated but no strict proof on convergence is presented yet. In this paper, first of all, we reduce the von KÁrmÁn’s equations to equivalent integral equations which are nonlinear, coupled and singular. Secondly the sequences of continuous function with general form are constructed using iterative technique. Based on the sequences to be uniformly convergent, we obtain analytical formula of exact solutions to von Karman’s equations related to large deflection problems of circular plate and shallow spherical shell with clamped boundary subjected to a concentrated load at the centre.     

On the refined first-order shear deformation plate theory of Kármán type

A new refined first-order shear-deformation plate theory of the Kármán type is presented for engineering applications and a new version of the generalized Kármán large deflection equations with deflection and stress functions as two unknown variables is formulated for nonlinear analysis of shear-deformable plates of composite material and construction, based on the Mindlin/Reissner theory. In this refined plate theory two rotations that are constrained out in the formulation are imposed upon overall displacements of the plates in an implicit role. Linear and nonlinear investigations may be made by the engineering theory to a class of shear-deformation plates such as moderately thick composite plates, orthotropic sandwich plates, densely stiffened plates, and laminated shear-deformable plates. Reduced forms of the generalized Kármán equations are derived consequently, which are found identical to those existe in the literature. Foundation item: the National Natural Science Foundation of China (59675027) Biography: Zhang Jianwu (1954-)     

Power-Spectral density estimate of the Bloor-Gerrard instability in flows around circular cylinders

There have been differences in the literature concerning the power law relationship between the Bloor-Gerrard instability frequency of the separated shear layer from the circular cylinder, the Bénard-von Kármán vortex shedding frequency and the Reynolds number. Most previous experiments have shown a significant degree of scatter in the measurement of the development of the shear layer vortices. Shear layers are known to be sensitive to external influences, which can provide a by-pass transition to saturated growth, thereby camouflaging the fastest growing linear modes. Here, the spatial amplification rates of the shear layer instabilities are calculated using power-spectral density estimates, allowing the fastest growing modes rather than necessarily the largest structures to be determined. This method is found to be robust in determining the fastest growing modes, producing results consistent with the low scatter results of previous experiments.     

Energy Scaling of Compressed Elastic Films –Three-Dimensional Elasticity¶and Reduced Theories

We derive an optimal scaling law for the energy of thin elastic films under isotropic compression, starting from three-dimensional nonlinear elasticity. As a consequence we show that any deformation with optimal energy scaling must exhibit fine-scale oscillations along the boundary, which coarsen in the interior. This agrees with experimental observations of folds which refine as they approach the boundary. We show that both for three-dimensional elasticity and for the geometrically nonlinear Föppl-von Kármán plate theory the energy of a compressed film scales quadratically in the film thickness. This is intermediate between the linear scaling of membrane theories which describe film stretching, and the cubic scaling of bending theories which describe unstretched plates, and indicates that the regime we are probing is characterized by the interplay of stretching and bending energies. Blistering of compressed thin films has previously been analyzed using the Föppl-von Kármán theory of plates linearized in the in-plane displacements, or with the scalar eikonal functional where in-plane displacements are completely neglected. The predictions of the linearized plate theory agree with our result, but the scalar approximation yields a different scaling.     

Existence Result for a Dynamical Equations of Generalized Marguerre-von Kármán Shallow Shells

In a recent work in the static case, Gratie (Appl. Anal. 81:1107–1126, 2002) has generalized the classical Marguerre-von Kármán equations studied by Ciarlet and Paumier in (Comput. Mech. 1:177–202, 1986), where only a portion of the lateral face of the shallow shell is subjected to boundary conditions of von Kármán type, while the remaining portion is subjected to boundary conditions of free edge. Then Ciarlet and Gratie (Math. Mech. Solids 11:83–100, 2006) have established an existence theorem for these equations. In Chacha et al. (Rev. ARIMA 13:63–76, 2010), we extended formally these studies to the dynamical case. More precisely, we considered a three-dimensional dynamical model for a nonlinearly elastic shallow shell with a specific class of boundary conditions of generalized Marguerre-von Kármán type. Using technics from formal asymptotic analysis, we showed that the scaled three-dimensional solution still leads to two-dimensional dynamical boundary value problem called the dynamical equations of generalized Marguerre-von Kármán shallow shells. In this paper, we establish the existence of solutions to these equations using a compactness method of Lions (Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires, Dunod, Paris, 1969).     

Effects of Compressibility and Shock-Wave Interactions on Turbulent Shear Flows

Compressibility effects are present in many practical turbulent flows, ranging from shock-wave/boundary-layer interactions on the wings of aircraft operating in the transonic flight regime to supersonic and hypersonic engine intake flows. Besides shock wave interactions, compressible flows have additional dilatational effects and, due to the finite sound speed, pressure fluctuations are localized and modified relative to incompressible turbulent flows. Such changes can be highly significant, for example the growth rates of mixing layers and turbulent spots are reduced by factors of more than three at high Mach number. The present contribution contains a combination of review and original material. We first review some of the basic effects of compressibility on canonical turbulent flows and attempt to rationalise the differing effects of Mach number in different flows using a flow instability concept. We then turn our attention to shock-wave/boundary-layer interactions, reviewing recent progress for cases where strong interactions lead to separated flow zones and where a simplified spanwise-homogeneous problem is amenable to numerical simulation. This has led to improved understanding, in particular of the origin of low-frequency behaviour of the shock wave and shown how this is coupled to the separation bubble. Finally, we consider a class of problems including side walls that is becoming amenable to simulation. Direct effects of shock waves, due to their penetration into the outer part of the boundary layer, are observed, as well as indirect effects due to the high convective Mach number of the shock-induced separation zone. It is noted in particular how shock-induced turning of the detached shear layer results in strong localized damping of turbulence kinetic energy.     

THE THREE-DIMENSIONAL UNSTEADY BOUNDARY LAYER OVER AN IMPULSIVELY STARTED ROTATING DISK

The unsteady boundary layer over an impulsively started rotating disk isstudied,a complete solution describing the smooth transition from vortex diffusionat ωt=0 to Kármán’s steady solution is obtained by series expansion and itsnumerical continuation. The angle of body streamlines,together withexperimental values, are given as the function of time t as well as the momentcoefficient C_M and on-coming velocity w(∞).