Nonlinear aeroelastic analysis of a two-dimensional wing with control surface in supersonic flow

Based on the piston theory of supersonic flow and the energy method, a two dimensional wing with a control surface in supersonic flow is theoretically modeled, in which the cubic stiffness in the torsional direction of the control surface is considered. An approximate method of the cha- otic response analysis of the nonlinear aeroelastic system is studied, the main idea of which is that under the condi- tion of stable limit cycle flutter of the aeroelastic system, the vibrations in the plunging and pitching of the wing can approximately be considered to be simple harmonic excita- tion to the control surface. The motion of the control surface can approximately be modeled by a nonlinear oscillation of one-degree-of-freedom. The range of the chaotic response of the aeroelastic system is approximately determined by means of the chaotic response of the nonlinear oscillator. The rich dynamic behaviors of the control surface are represented as bifurcation diagrams, phase-plane portraits and PS diagrams. The theoretical analysis is verified by the numerical results.     

Soret and Dufour effects on mixed convection unsteady MHD boundary layer flow over stretching sheet in porous medium with chemically reactive species

This paper studies the thermal-diffusion and diffusion thermo-effects in the hydro-magnetic unsteady flow by a mixed convection boundary layer past an imperme- able vertical stretching sheet in a porous medium in the presence of chemical reaction. The velocity of t~he stretching surface, the surface temperature, and the concentration are assumed to vary linearly with the distance along the surface. The governing partial differential equations are transformed into self-similar unsteady equations using similarity transformations .and solved numerically by the Runge-Kutta fourth order scheme in as- sociation with the shooting method for the whole transient domain from the initial state to the final steady state flow. Numerical results for the velocity, the temperature, the concentration, the skin friction, and the Nusselt and Sherwood numbers are shown graph- ically for various flow parameters. The results reveal that there is a smooth transition of flow from unsteady state to the final steady state. A special case of our results is in good agreement with an earlier published work.     

Numerical simulation of motion and deformation of ring bubble along body surface

Numerical simulation for fluid flow over an attached rigid body with a deformable ring bubble is analyzed based on the velocity potential theory together with the boundary element method （BEM）. The analysis is focused on the axisymmetric case. The bubble surface is treated as a well defined air-liquid interface and is tracked by a mixed Eulerian-Lagrangian method. The points of intersection between the bubble and body are treated, specially in the numerical procedure. The auxiliary function method is adopted to calculate the pressure on the body surface and in the flow field. The convergence study is undertaken to assess the developed numerical method and the computation code. Some case studies are undertaken in which the interactions between the bubble/body and the incoming flow field are simulated. The effects of various physical parameters on the interactions are investigated.     

Integrated numerical model for vegetated surface and saturated subsurface flow interaction

The construction of an integrated numerical model is presented in this paper to deal with the interactions between vegetated surface and saturated subsurface flows. A numerical model is built by integrating the previously developed quasi-three-dimensional (Q3D) vegetated surface flow model with a two-dimensional (2D) saturated groundwater flow model. The vegetated surface flow model is constructed by coupling the explicit finite volume solution of 2D shallow water equations (SWEs) with the implicit finite difference solution of Navier-Stokes equations (NSEs) for vertical velocity distribution. The subsurface model is based on the explicit finite volume solution of 2D saturated groundwater flow equations (SGFEs). The ground and vegetated surface water interaction is achieved by introducing source-sink terms into the continuity equations. Two solutions are tightly coupled in a single code. The integrated model is applied to four test cases, and the results are satisfactory.     

MHD flow and heat transfer of micropolar fluid between two porous disks

A numerical study is carried out for the axisymmetric steady laminar incompressible flow of an electrically conducting micropolar fluid between two infinite parallel porous disks with the constant uniform injection through the surface of the disks. The fluid is subjected to an external transverse magnetic field. The governing nonlinear equations of motion are transformed into a dimensionless form through von Karman’s similarity transformation. An algorithm based on a finite difference scheme is used to solve the reduced coupled ordinary differential equations under associated boundary conditions. The effects of the Reynolds number, the magnetic parameter, the micropolar parameter, and the Prandtl number on the flow velocity and temperature distributions are discussed. The results agree well with those of the previously published work for special cases. The investigation predicts that the heat transfer rate at the surfaces of the disks increases with the increases in the Reynolds number, the magnetic parameter, and the Prandtl number. The shear stresses decrease with the increase in the injection while increase with the increase in the applied magnetic field. The shear stress factor is lower for micropolar fluids than for Newtonian fluids, which may be beneficial in the flow and thermal control in the polymeric processing.     

Supercritical as well as subcritical Hopf bifurcation in nonlinear flutter systems

The Hopf bifurcations of an airfoil flutter system with a cubic nonlinearity are investigated, with the flow speed as the bifurcation parameter. The center manifold theory and complex normal form method are Used to obtain the bifurcation equation. Interestingly, for a certain linear pitching stiffness the Hopf bifurcation is both supercritical and subcritical. It is found, mathematically, this is caused by the fact that one coefficient in the bifurcation equation does not contain the first power of the bifurcation parameter. The solutions of the bifurcation equation are validated by the equivalent linearization method and incremental harmonic balance method.     

BIFURCATIONS OF AIRFOIL IN INCOMPRESSIBLE FLOW

Bifurcations of an airfoil with nonlinear pitching stiffness in incompressible flow are investigated. The pitching spring is regarded as a spring with cubic stiffness. The motion equations of the airfoil are written as the four dimensional one order differential equations. Taking air speed and the linear part of pitching stiffness as the parameters, the analytic solutions of the critical boundaries of pitchfork bifurcations and Hopf bifurcations are obtained in 2 dimensional parameter plane. The stabilities of the equilibrium points and the limit cycles in different regions of 2 dimensional parameter plane are analyzed. By means of harmonic balance method, the approximate critical boundaries of 2-multiple semi-stable limit cycle bifurcations are obtained, and the bifurcation points of supercritical or subcritical Hopf bifurcation are found. Some numerical simulation results are given.     

An explicit finite volume element method for solving characteristic level set equation on triangular grids

Level set methods are widely used for predicting evolutions of complex free surface topologies,such as the crystal and crack growth,bubbles and droplets deformation,spilling and breaking waves,and two-phase flow phenomena.This paper presents a characteristic level set equation which is derived from the two-dimensional level set equation by using the characteristic-based scheme.An explicit finite volume element method is developed to discretize the equation on triangular grids.Several examples are presented to demonstrate the performance of the proposed method for calculating interface evolutions in time.The proposed level set method is also coupled with the Navier-Stokes equations for two-phase immiscible incompressible flow analysis with surface tension.The Rayleigh-Taylor instability problem is used to test and evaluate the effectiveness of the proposed scheme.     

The performance of proper orthogonal decomposition in discontinuous flows

In this paper,flow reconstruction accuracy and flow prediction capability of discontinuous transonic flow field by means of proper orthogonal decomposition(POD) method is studied.Although linear superposition of ‘‘high frequency waves' ' in different POD modes can achieve the reconstruction of the shock wave,the smoothness of the solution near the shock wave cannot be guaranteed.The modal coefficients are interpolated or extrapolated and different modal components are superposed to realize the prediction of the flow field beyond the snapshot sets.Results show that compared with the subsonic flow,the transonic flow with shock wave requires more POD modes to reach a comparative reconstruction accuracy.When a shock wave exists,the interpolation prediction ability is acceptable.However,large errors exist in extrapolation,and increasing the number of POD modes cannot effectively improve the prediction accuracy of the flow field.     

Study of modeling unsteady blade row interaction in a transonic compressor stage part 1: code development and deterministic correlation analysis

The average-passage equation system (APES) provides a rigorous mathematical framework for accounting for the unsteady blade row interaction through multistage compressors in steady state environment by introducing deterministic correlations (DC) that need to be modeled to close the equation system.The primary purpose of this study is to provide insight into the DC characteristics and the influence of DC on the time-averaged flow field of the APES.In Part 1 of this two-part paper,firstly a 3D viscous unsteady and time-averaging flow CFD solver is developed to investigate the APES technique.Then steady and unsteady simulations are conducted in a transonic compressor stage.The results from both simulations are compared to highlight the significance of the unsteady interactions.Furthermore,the distribution characteristics of DC are studied and the DC at the rotor/stator interface are compared with their spatial correlations (SC).Lastly,steady and time-averaging (employing APES with DC) simulations for the downstream stator alone are conducted employing DC derived from the unsteady results.The results from steady and time-averaging simulations are compared with the time-averaged unsteady results.The comparisons demonstrate that the simulation employing APES with DC can reproduce the time-averaged field and the 3D viscous time-averaging flow solver is validated.     

Limit cycle oscillation behavior of transonic control surface buzz considering free-play nonlinearity

The limit cycle oscillation (LCO) behaviors of control surface buzz in transonic flow are studied. Euler equations are employed to obtain the unsteady aerodynamic forces for Type B and Type C buzz analyses, and an all-movable control surface model, a wing/control surface model and a three-dimensional wing with a full-span control surface are adopted in the study. Aerodynamic and structural describing functions are used to deal with aerodynamic and structural nonlinearities, respectively. Then the buzz speed and buzz frequency are obtained by V-g method. The LCO behavior of the transonic control surface buzz system with linear structure exhibits subcritical or supercritical bifurcation at different Mach numbers. For nonlinear structural model with a free-play nonlinearity in the control surface deflection stiffness, the double LCO phenomenon is observed in certain range of flutter speed. The free-play nonlinearity changes the stability of LCOs at small amplitudes and turns the unstable LCO into a stable one. The LCO behavior is dominated by the aerodynamic nonlinearity for the case with large control surface oscillation amplitude but by the structural nonlinearity for the case with small amplitude. Good agreements between LCO behaviors obtained by the present method and available experimental data show that our study may help to explain the experimental observation in wind tunnel tests and to understand the physical mechanism of transonic control surface buzz.     

A parametric investigation of the propulsion of 2D chordwise-flexible flapping wings at low Reynolds number using numerical simulations

This paper presents a numerical investigation of the effects of chordwise flexibility on flapping wings at low Reynolds number. The numerical simulations are performed with a partitioned fluid–structure interaction algorithm using artificial compressibility stabilization. The choice of the structural dimensionless parameters is based on scaling arguments and is compared against parameters used by other authors. The different regimes, namely inertia-driven and pressure-driven wing deformations, are presented along with their effects on the topology of the flow and on the performance of a heaving and pitching flapping wing in propulsion regime. It is found that pressure-driven deformations can significantly increase the thrust efficiency if a suitable amount of flexibility is used. Significant thrust increases are also observed in zero pitching amplitude cases. The effects of the second and third deformation modes on the performances of pressure-driven deformation cases are discussed. On the other hand, inertia-driven deformations generally deteriorate aerodynamic performances of flapping wings unless the behavior of the wing deformation is modified by the presence of sustainable superharmonics in a way that produces slight improvements. It is also shown that wing flexibility can act as an efficient passive pitching mechanism that allows fair thrust and better efficiency to be achieved when compared to a rigid pitching–heaving wing.     

Velocity measurement of compressible air flows utilizing a high-speed video camera

PIV-measurements of the flow above a pitching airfoil were conducted in a transonic wind tunnel. An ultra high-speed video camera was used for separate recording of two exposures. The data was analysed using the cross-correlation method. The results show the applicability of the technique in high speed flows.     

On the role of thickness ratio and location of axis of rotation in the flat plate motions

Wind tunnel experiments were performed to characterize the flow-induced rotations and pitching of various flat plates as a function of the thickness ratio and the location of the axis of rotation. High-resolution telemetry, laser tachometer, and hotwire were used to get time series of the plates motions and the signature of the wake flow at a specific location. Results show that small axis offset can induce high-order modes in the plate rotation due to torque unbalance, and can trigger self-initiated pitching. The spectral decomposition of the flow velocity in the plate wake reveals the existence of a dominating high-frequency mode that corresponds to a static-like vortex shedding occurring at the maximum plate pitch. The associated characteristic length scale is the projected width at maximum pitching angle. The increase of the plate thickness ratio implies lower angular velocity in rotation cases. A simple model based on aerodynamic forces is used to explain the linear relation between pitching frequency and wind speed, the pitching frequency increase with axis offset, and the onset of pitching.     

AERODYNAMIC FORCES ACTING ON AN ALBATROSS FLYING ABOVE SEA-WAVES

IntroductionIt was observed long time ago that the large sea bird such as albatross has excellent flightskills.The albatross can keep its soaring for a long time even in rough sea without flappingwings.This problemwas discussed byWood C.J.[1].He supposed …     

跨音速翼型上的激波／边界层干扰自适应控制计算

在激波区使用自适应壁对跨音速翼型的激波／边界层的相互作用（干扰）进行控制，可改变机翼的气动性能，这种被动控制可通过在翼型的激波区开一凹腔，其上覆盖一弹性橡胶膜柔壁来，本文给出用Ｎａｖｉｅｒ－Ｓｔｏｋｅｒ方程数值模拟这一自适应控制翼型的跨音速粘性绕流，提出了一个适应于本特殊情况（物面边界局部地区在求解过程中有变化）的处理办法。并探讨了自适应柔壁对当代跨音速翼绕流的影响。     

An efficient approach for calculation of pitching moment in nonlinear reduced frequency method at low Mach number transonic flows

In this research, an efficient methodology for calculation of pitching moment coefficient at low Mach number transonic flows by using the perturbed nonlinear reduced frequency approach is presented. The proposed approach uses the perturbation technique in the nonlinear frequency domain (NLFD) method to estimate the solution at high harmonics. In this approach, the density and velocity fields at high harmonics are perturbed about those at low harmonics. Perturbing the density and velocity fields, the semi‐linear form of the governing equations is obtained. The resulting solution vector and spatial operator are then approximated by discrete form of Fourier transformation and governing equations are solved by using the pseudo‐spectral approach. Numerical results show that the proposed approach predicts good pitching moment coefficient at low Mach number transonic flows with up to 50% savings in computational time. Copyright © 2011 John Wiley & Sons, Ltd.     

Comparative Analysis of the Characteristics of the Vortex Wake behind a Flapping Wing Performing Oscillations of Different Types

The wake characteristics of a custom-designed airfoil performing pitching oscillations, heaving oscillations, and a combination of pitch and heave oscillations are compared in this study. The influence of flapping parameters are investigated at a constant Reynolds number Re\(_{c} = 2640\) and is presented for the Strouhal numbers based on the oscillation amplitude, St_A, varying in the \(0.1 \leqslant {\text{S}}{{{\text{t}}}_{A}} \leqslant 0.4\) range. The generation of vorticity above and below the airfoil depends on the airfoil’s initial direction of motion and remains the same for all types of flapping oscillations investigated. The evolution of the leading-edge and trailing-edge vortices is presented. The heaving oscillations of the airfoil are found to have a greater influence on the characteristics of the leading edge vortex. The wake behind the combined pitch-heave oscillations appears to be governed by pitching oscillations below \({\text{S}}{{{\text{t}}}_{A}} = 0.24\), whereas it is driven by heaving oscillations above \({\text{S}}{{{\text{t}}}_{A}} = 0.24\). The force computations indicate that the mere existence of the reverse von Kármán street is not sufficient to develop the thrust on the airfoil. The periodic component of velocity fluctuations significantly influences the wake characteristics. The anisotropic stress field developed around the airfoil due to the periodic fluctuations of the velocity is presented. The coherent structures developed in the wake are identified using the proper orthogonal decomposition and a qualitative comparison of the structures for different flapping oscillations is presented. The energy transfer from the flapping airfoil to the fluid for different flapping oscillations is highest for heaving oscillations followed by combined pitch-heave oscillations and pitching oscillations.

     

Effect of flow regime change from subsonic to transonic on the air loads of an oscillating airfoil

In this research, the effect of flow regime change from subsonic to transonic on the air loads of a pitching NACA0012 airfoil is investigated. To do this, the effect of change in flow regime on the lift and pitching moment coefficients hysteresis cycles is studied. The harmonic balance approach is utilized for numerical calculation due to its low computational time. Verifications are also made with previous works and good agreements are observed. The assessment of flow regime change on the aforementioned hysteresis cycles is accomplished in the Mach number range of M=0.65–0.755. The reduced frequency and pitch amplitude also vary from k=0.03 to 0.1 and α₀=1–2.51°, respectively. Results show that the effect of increase in Mach number is to increase and decrease the lift coefficient during downstroke and upstroke, respectively, whereas at low reduced frequencies, the effect of increase in Mach number may lead to a reverse manner when airfoil moves toward its extremum angle of attack. Results also reveal that as the pitch amplitude varies, the shape of lift coefficient hysteresis cycle depends more on the pitch amplitude than on the appearance of shock. It is shown that as the Mach number increases, the incidence angles correspond to the extremum pitching moment, and depending on the reduced frequency, lie between zero and extremum angle of attack. These incidence angles shift toward the extremum angle of attack as the reduced frequency decreases. Results also show that the increase in pitch amplitude at low Mach number, in such a way that leads to the formation of shock around the extremum angle of attack, causes the extremum pitching moment to appear around these angles and at high Mach number, depending on the reduced frequency, the extremum pitching moment incidence angles would be between zero and extremum incidence angle.     

Energy harvesting of a heaving and forward pitching wing with a passively actuated trailing edge

This study experimentally investigates the energy harvesting capabilities of an oscillating wing with a passively actuated trailing edge. The oscillation kinematics are composed of a combined heaving and forward pitching motions, where the pitching axis is well behind the wing center of mass. Passive actuation is attained by connecting the trailing edge with the wing body using a torsion rod. The degree of flexibility of the trailing edge is represented by the Strouhal number based on the trailing edge natural frequency. The trailing edge passive response is studied for oscillation Strouhal numbers of 0.017, 0.025 and 0.033. Instantaneous aerodynamic forces are measured in a closed loop wind tunnel at a Reynolds number of 40 000, based on the free stream velocity and the wing chord length. Measured results include the effective angle of attack induced by the trailing edge actuation as well as the lift and moment during the oscillation cycle. For the imposed kinematics in this study, the pitching motion has a positive contribution to the mean power output whereas the heaving motion has a relatively small but negative contribution. Additionally, by decreasing the natural frequency of the trailing edge closer to that of the imposed oscillation frequency, the magnitude of the lift and moment forces and hence the mean power output, increases. It is found that there exists a strong correlation between mean power output and the effective angle of attack, shown through the passive trailing edge response, resulting in an increase in energy harvesting potential.