Leading-edge flow criticality as a governing factor in leading-edge vortex initiation in unsteady airfoil flows

A leading-edge suction parameter (LESP) that is derived from potential flow theory as a measure of suction at the airfoil leading edge is used to study initiation of leading-edge vortex (LEV) formation in this article. The LESP hypothesis is presented, which states that LEV formation in unsteady flows for specified airfoil shape and Reynolds number occurs at a critical constant value of LESP, regardless of motion kinematics. This hypothesis is tested and validated against a large set of data from CFD and experimental studies of flows with LEV formation. The hypothesis is seen to hold except in cases with slow-rate kinematics which evince significant trailing-edge separation (which refers here to separation leading to reversed flow on the aft portion of the upper surface), thereby establishing the envelope of validity. The implication is that the critical LESP value for an airfoil–Reynolds number combination may be calibrated using CFD or experiment for just one motion and then employed to predict LEV initiation for any other (fast-rate) motion. It is also shown that the LESP concept may be used in an inverse mode to generate motion kinematics that would either prevent LEV formation or trigger the same as per aerodynamic requirements.     

Shallow and deep dynamic stall for flapping low Reynolds number airfoils

We consider a combined experimental (based on flow visualization, direct force measurement and phase-averaged 2D particle image velocimetry in a water tunnel), computational (2D Reynolds-averaged Navier–Stokes) and theoretical (Theodorsen’s formula) approach to study the fluid physics of rigid-airfoil pitch–plunge in nominally two-dimensional conditions. Shallow-stall (combined pitch–plunge) and deep-stall (pure-plunge) are compared at a reduced frequency commensurate with flapping-flight in cruise in nature. Objectives include assessment of how well attached-flow theory can predict lift coefficient even in the presence of significant separation, and how well 2D velocimetry and 2D computation can mutually validate one another. The shallow-stall case shows promising agreement between computation and experiment, while in the deep-stall case, the computation’s prediction of flow separation lags that of the experiment, but eventually evinces qualitatively similar leading edge vortex size. Dye injection was found to give good qualitative match with particle image velocimetry in describing leading edge vortex formation and return to flow reattachment, and also gave evidence of strong spanwise growth of flow separation after leading-edge vortex formation. Reynolds number effects, in the range of 10,000–60,000, were found to influence the size of laminar separation in those phases of motion where instantaneous angle of attack was well below stall, but have limited effect on post-stall flowfield behavior. Discrepancy in lift coefficient time history between experiment, theory and computation was mutually comparable, with no clear failure of Theodorsen’s formula. This is surprising and encouraging, especially for the deep-stall case, because the theory’s assumptions are clearly violated, while its prediction of lift coefficient remains useful for capturing general trends.     

An inviscid/viscous coupling approach for vortex flow field calculations

A new computational approach is developed for the analysis of vortex-dominated flow fields around highly swept wings at high angles of attack. In this approach an inviscid Euler technology is coupled with viscous models, similar to inviscid/boundary layer coupling. The viscous nature of the vortex core is represented by an algebraic model derived from the Navier–Stokes equations. The approach also accounts for the effects of the viscous shear layer near a wing surface through a modified surface boundary condition. The inviscid/viscous coupling consistently provides improved predictions of leading edge separation, vortex bursting and secondary vortex formation at relatively low computational cost. Results for several cases are compared with wind tunnel tests and other Euler and Navier-Stokes solutions.     

Vortex shedding from edges including viscous effects

This paper describes results obtained by using the inviscid Cloud-in-Cell vortex method to model the vortex sheet which is shed and rolls up from a single sharp edge. There is good agreement between these results and previous (Pullin 1978) computations of the development of the sheet in impulsively started incompressible inviscid flow. The Cloud-in-Cell method has been modified to include viscous diffusion calculated by finite differences on the mesh to give a mixed Eulerian-Lagrangian Navier-Stokes solver. This method has been shown to model the diffusing free vortex and the Stokes boundary layer quite accurately. It is used to compute impulsively started flow past sharp right-angled edges and edges with small rounding. The effect of viscous diffusion on the development of the shed vortex is discussed.

The method is also used to study the effect of rounding on the vortex shedding from a right-angled edge in oscillatory flow. This problem is particularly important in determining the roll damping and hence response of certain types of ship hull in waves. It is shown that the strength and effect of the shed vortices rapidly decrease as the ratio of the edge radius to the oscillation amplitude increases, and that at larger values of this ratio the mode of shedding changes from two vortices per cycle from one edge to a more complicated mode. The computed results are compared with flow visualisation using dye and neutrally buoyant particles in water flow around an oscillating edge.     

旋转振动圆柱绕流周期解和Floquet稳定性

对低雷诺数旋转振动圆柱绕流问题运用低维Galerkin方法将N－S方程约化为一组非线性常微分方程组。运用打靶法数值求解了这组方程的周期解,并用Tloquet理论对周期解的稳定性进行了分析,确定了流动失稳的机制。     

Detection and tracking of vortex phenomena using Lagrangian coherent structures

The formation and shedding of vortices in two vortex-dominated flows around an actuated flat plate are studied to develop a better method of identifying and tracking coherent structures in unsteady flows. The work automatically processes data from the 2D simulation of a flat plate undergoing a \(45^{\circ }\) pitch-up maneuver, and from experimental particle image velocimetry data in the wake of a continuously pitching trapezoidal panel. The Eulerian \(\varGamma _1\), \(\varGamma _2\), and Q functions, as well as the Lagrangian finite-time Lyapunov exponent are applied to identify both the centers and boundaries of the vortices. The multiple vortices forming and shedding from the plates are visualized well by these techniques. Tracking of identifiable features, such as the Lagrangian saddle points, is shown to have potential to identify the timing and location of vortex formation, shedding, and destruction more precisely than by only studying the vortex cores as identified by the Eulerian techniques.     

Topology of the vorticity field in three-dimensional shear layers and wakes

An experimental and numerical study of the three-dimensional transition of plane wakes and shear layers behind a flat plate is presented. Flow visualization techniques are used to monitor the response of laminar flows at moderate Reynolds numbers (≈100) to perturbations periodically distributed along the span. In this way, the formation and evolution of streamwise vortex tubes and their interaction with the spanwise vortices are analyzed. The flow was studied numerically by means of three-dimensional inviscid vortex dynamics. Assuming periodicity in the spanwise and the streamwise direction, we discretize the vorticity field into two layers of vortex filaments with finite core diameter. Comparison between experiment and visualization indicates that important features of the three-dimensional evolution can be reproduced by inviscid vortex dynamics. Vortex stretching in the strain field of the spanwise rollers appears to be the primary mechanism for the three-dimensional transition in this type of flows.     

Fast numerical evaluation of flow fields with vortex cells

A vortex cell (in this paper) is an aerodynamically shaped cavity in the surface of a body, for example a wing, designed specially to trap the separated vortex within it, thus preventing large-scale unsteady vortex shedding from the wing. Vortex stabilisation can be achieved either by the special geometry, as has already been done experimentally, or by a system of active control. In realistic conditions the boundary and mixing layers in the vortex cell are always turbulent. In the present study a model for calculating the flow in a vortex cell was obtained by replacing the laminar viscosity with the turbulent viscosity in the known high-Reynolds-number asymptotic theory of steady laminar flows in vortex cells. The model was implemented numerically and was shown to be faster than solving the Reynolds-averaged Navier–Stokes equations. An experimental facility with a vortex cell was built and experiments performed. Comparisons of the experimental results with the predictions of the model are reasonably satisfactory. The results also indicate that at least for flows in near-circular vortex cells it is sufficient to have accurate turbulence models only in thin viscous layers, while outside the viscosity should only be small enough to make the flow effectively inviscid.     

An unsteady point vortex method for coupled fluid–solid problems

A method is proposed for the study of the two-dimensional coupled motion of a general sharp-edged solid body and a surrounding inviscid flow. The formation of vorticity at the body’s edges is accounted for by the shedding at each corner of point vortices whose intensity is adjusted at each time step to satisfy the regularity condition on the flow at the generating corner. The irreversible nature of vortex shedding is included in the model by requiring the vortices’ intensity to vary monotonically in time. A conservation of linear momentum argument is provided for the equation of motion of these point vortices (Brown–Michael equation). The forces and torques applied on the solid body are computed as explicit functions of the solid body velocity and the vortices’ position and intensity, thereby providing an explicit formulation of the vortex–solid coupled problem as a set of non-linear ordinary differential equations. The example of a falling card in a fluid initially at rest is then studied using this method. The stability of broadside-on fall is analysed and the shedding of vorticity from both plate edges is shown to destabilize this position, consistent with experimental studies and numerical simulations of this problem. The reduced-order representation of the fluid motion in terms of point vortices is used to understand the physical origin of this destabilization.      

Instantaneous structure and statistical feature of unsteady flow in a channel obstructed by a square rod

A two-dimensional numerical computation has been made for an unsteady flow in a channel obstructed by an inserted square rod. The results of the computation made for the flow with a parabolic inlet velocity profile at a specific value of channel Reynolds number are analyzed in detail. The obtained results reveal that momentum transfer is enhanced due to the apparent shear stress resulting from the nonzero value of cross-correlation between the streamwise and normal components of fluctuating velocity, , just as in turbulent shear flows, although the studied flow is quite different from turbulent flows in the sense that it is highly periodical and therefore free from randomness. This periodicity leads to a quick recovery of the velocity defect in some region of the wake of the rod. Special attention is paid to the time variation of flow structure. The crisscross motion of the Karman vortex previously found to occur is discussed again, and how it appears is explained in terms of the interaction between the Karman vortex and the disturbed wall shear layer. In the discussion of this relationship, wavering motion of the separation vorticity layers formed on both sides of the rod and the periodic formation of an isolated vortex island from the lifted tip of the wall vorticity layer are analyzed. The vortex island is found to play an important role not only for the occurrence of the crisscross motion of Karman vortex but also for the generation of the nonzero value of .     

Numerical study of an inviscid incompressible flow through a channel of finite length

A two‐dimensional inviscid incompressible flow in a rectilinear channel of finite length is studied numerically. Both the normal velocity and the vorticity are given at the inlet, and only the normal velocity is specified at the outlet. The flow is described in terms of the stream function and vorticity. To solve the unsteady problem numerically, we propose a version of the vortex particle method. The vorticity field is approximated using its values at a set of fluid particles. A pseudo‐symplectic integrator is employed to solve the system of ordinary differential equations governing the motion of fluid particles. The stream function is computed using the Galerkin method. Unsteady flows developing from an initial perturbation in the form of an elliptical patch of vorticity are calculated for various values of the volume flux of fluid through the channel. It is shown that if the flux of fluid is large, the initial vortex patch is washed out of the channel, and when the flux is reduced, the initial perturbation evolves to a steady flow with stagnation regions. Copyright © 2008 John Wiley & Sons, Ltd.     

Limit-cycle oscillations in unsteady flows dominated by intermittent leading-edge vortex shedding

High-frequency limit-cycle oscillations of an airfoil at low Reynolds number are studied numerically. This regime is characterized by large apparent-mass effects and intermittent shedding of leading-edge vortices. Under these conditions, leading-edge vortex shedding has been shown to result in favorable consequences such as high lift and efficiencies in propulsion/power extraction, thus motivating this study. The aerodynamic model used in the aeroelastic framework is a potential-flow-based discrete-vortex method, augmented with intermittent leading-edge vortex shedding based on a leading-edge suction parameter reaching a critical value. This model has been validated extensively in the regime under consideration and is computationally cheap in comparison with Navier–Stokes solvers. The structural model used has degrees of freedom in pitch and plunge, and allows for large amplitudes and cubic stiffening. The aeroelastic framework developed in this paper is employed to undertake parametric studies which evaluate the impact of different types of nonlinearity. Structural configurations with pitch-to-plunge frequency ratios close to unity are considered, where the flutter speeds are lowest (ideal for power generation) and reduced frequencies are highest. The range of reduced frequencies studied is two to three times higher than most airfoil studies, a virtually unexplored regime. Aerodynamic nonlinearity resulting from intermittent leading-edge vortex shedding always causes a supercritical Hopf bifurcation, where limit-cycle oscillations occur at freestream velocities greater than the linear flutter speed. The variations in amplitude and frequency of limit-cycle oscillations as functions of aerodynamic and structural parameters are presented through the parametric studies. The excellent accuracy/cost balance offered by the methodology presented in this paper suggests that it could be successfully employed to investigate optimum setups for power harvesting in the low-Reynolds-number regime.     

Effects of a dynamic trailing-edge flap on the aerodynamic performance and flow structures in hovering flight

To examine the effects of wing morphing on unsteady aerodynamics, deformable flapping plates are numerically studied in a low-Reynolds-number flow. Simulations are carried out using an in-house immersed-boundary-method-based direct numerical simulation (DNS) solver. In current work, chord-wise camber is modeled by a hinge connecting two rigid components. The leading portion is driven by a biological hovering motion along a horizontal stroke plane. The hinged trailing-edge flap (TEF) is controlled by a prescribed harmonic deflection motion. The effects of TEF deflection amplitude, deflection phase difference, hinge location, and Reynolds number on the aerodynamic performance and flow structures are investigated. The results show that the unsteady aerodynamic performance of deformable flapping plates is dominated by the TEF deflection phase difference, which directly affects the strength of the leading-edge vortex (LEV) and thus influences the entire vortex shedding process. The overall lift enhancement can reach up to 26% by tailoring the deflection amplitude and deflection phase difference. It is also found that the role of the dynamic TEF played in the flapping flight is consistent over a range of hinge locations and Reynolds numbers. Results from a low aspect-ratio (AR=2) deformable plate show the same trend as those of 2-D cases despite the effect of the three-dimensionality.     

Dynamics of large-scale eddies in turbulent flows

A model equation based on the equipartition of the turbulent dissipation is proposed for describing the dynamics of large-scale eddies in turbulent flows. The equation is reducible to the equation of motion of an inviscid fluid, so that the motion of the large-scale eddies can be described in terms of inviscid fluid dynamics. It is found that the large-scale eddies are always weakened by the background turbulence and their evolution is slowed down compared with the corresponding inviscid motion. In the case of turbulent mixing layer, its linear growth in downstream direction is accounted for by the exponential growth in time of the perturbation in an inviscid plane vortex sheet.     

Point vortex model of the unsteady separated flow past a semi-infinite plate with transverse motion

Two-dimensional unsteady separated flow past a semi-infinite plate with transverse motion is considered. The rolling-up of the separated shear-layer is modelled by a point vortex whose time-dependent circulation is predicted by an unsteady Kutta condition. A power-law starting flow is assumed along with a power law for the transverse motion. The effects of the motion of the plate on the starting vortex circulation and trajectory are presented. A suitable vortex shedding mechanism is introduced and a class of flows involving several vortices is presented. Finally, some possibilities for actively controlling the production of circulation by moving the plate are discussed.     

On the origins of vortex shedding in two-dimensional incompressible flows

An exegesis of a novel mechanism leading to vortex splitting and subsequent shedding that is valid for two-dimensional incompressible, inviscid or viscous, and external or internal or wall-bounded flows, is detailed in this research. The mechanism, termed the vortex shedding mechanism (VSM) is simple and intuitive, requiring only two coincident conditions in the flow: (1) the existence of a location with zero momentum and (2) the presence of a net force having a positive divergence. Numerical solutions of several model problems illustrate causality of the VSM. Moreover, the VSM criteria is proved to be a necessary and sufficient condition for a vortex splitting event in any two-dimensional, incompressible flow. The VSM is shown to exist in several canonical problems including the external flow past a circular cylinder. Suppression of the von Kármán vortex street is demonstrated for Reynolds numbers of 100 and 400 by mitigating the VSM.     

Computation of three-dimensional flow using the Euler equations and a multiple-grid scheme

The unsteady Euler equations are numerically solved using the finite volume one-step scheme recently developed by Ron-Ho Ni. The multiple-grid procedure of Ni is also implemented. The flows are assumed to be homo-enthalpic; the energy equation is eliminated and the static pressure is determined by the steady Bernoulli equation; a local time-step technique is used. Inflow and outflow boundaries are treated with the compatibility relations method of ONERA. The efficiency of the multiple-grid scheme is demonstrated by a two-dimensional calculation (transonic flow past the NACA 12 aerofoil) and also by a three-dimensional one (transonic lifting flow past the M6 wing). The third application presented shows the ability of the method to compute the vortical flow around a delta wing with leading-edge separation. No condition is applied at the leading-edge; the vortex sheets are captured in the same sense as shock waves. Results indicate that the Euler equations method is well suited for the prediction of flows with shock waves and contact discontinuities, the multiple-grid procedure allowing a substantial reduction of the computational time.     

A novel slightly compressible model for low Mach number perfect gas flow calculation

By analyzing the characteristics of low Mach number perfect gas flows, a novel Slightly Compressible Model (SCM) for low Mach number perect gas flows is derived. In view of numerical calculations, this model is proved very efficient, for it is kept within thep-v frame but does not have to satisfy the time consuming divergence-free condition in order to get the incompressible Navier-Stokes equation solution. Writing the equations in the form of conservation laws, we have derived the characteristic systems which are necessary for numerical calculations. A cell-centered finite-volume method with flux difference upwind-biased schemes is used for the equation solutions and a new Exact Newton Relaxation (ENR) implicit method is developed. Various computed results are presented to validate the present model. Laminar flow solutions over a circular cylinder with wake developing and vortex shedding are presented. Results for inviscid flow over a sphere are compared in excellent agreement with the exact analytic incompressible solution. Three-dimensional viscous flow solutions over sphere and prolate spheroid are also calculated and compared well with experiments and other incompressible solutions. Finally, good convergent performances are shown for sphere viscous flows. The project supported by the Basic Research on Frontier Problems in Fluid and Aerodynamics in China and the National Natural Science Foundation of China (19772069)     

A parametric study of LES on laminar-turbulent transitional flows past an airfoil

Low-Reynolds-number aerodynamic performance of small-sized air vehicles is an area of increasing interest. In this study, low-Reynolds-number flows past an SD7003 airfoil are investigated to understand important viscous features of laminar separation and transitional flow followed by the complicated behavior of the flow reattachment process. In order to satisfy the three-dimensional (3D) requirement of the code, a simple “3D wing” is constructed from a two-dimensional (2D) airfoil. A parametric study of large eddy simulation (LES) on the airfoil flows at Re = 60,000 is performed. Effects of grid resolution and sub-grid scale (SGS) models are investigated. Although 3D effects cannot be accurately captured owing to the limitation of the grid resolution in the spanwise direction, the preliminary LES calculations do reveal some important flow characteristics such as leading-edge laminar separation and vortex shedding from the primary laminar separation bubble on the low-Reynolds-number airfoil.     

三角翼的双襟翼控涡作用的数值模拟研究

对装有“前端襟翼”和“前缘襟翼”的７４°后掠三角翼的不可压缩流场作了数值模拟,以研究襟翼的旋涡控制作用．数值模拟是用拟压缩性方法求解一般曲线坐标系下的三维不可压缩Ｎａｖｉｅｒ－Ｓｔｏｋｅｓ方程,时间离散用向后Ｅｕｌｅｒ差分,空间无粘项的离散用二阶迎风ＴＶＤ格式,所得的离散方程用对角化形式的近似隐式因子分解格式求解．湍流模型用Ｂａｌｄｗｉｎ－Ｌｏｍａｘ代数模式．计算了三种平面形状的机翼在迎角范围为１０°～５０°的绕流和气动特性．计算和实验的比较表明,襟翼向下偏转可以推迟旋涡破裂,且对提高机翼的减阻能力、升阻比和改善失速前后的气动特性有明显效果,双襟翼具有更佳的控涡效果．