Fluid forces on a very low Reynolds number airfoil and their prediction

This paper presents the measurements of mean and fluctuating forces on an NACA0012 airfoil over a large range of angle (α) of attack (0-90°) and low to small chord Reynolds numbers (Re_c), 5.3 × 10³-5.1 × 10⁴, which is of both fundamental and practical importance. The forces, measured using a load cell, display good agreement with the estimate from the LDA-measured cross-flow distributions of velocities in the wake based on the momentum conservation. The dependence of the forces on both α and Re_c is determined and discussed in detail. It has been found that the stall of an airfoil, characterized by a drop in the lift force and a jump in the drag force, occurs at Re_c ? 1.05 × 10⁴ but is absent at Re_c = 5.3 × 10³. A theoretical analysis is developed to predict and explain the observed dependence of the mean lift and drag on α.     

Analysis of the aerodynamic interaction between two plunging plates in tandem at low Reynolds number for maximum propulsive efficiency

The thrust generated by two heaving plates in tandem is analyzed computationally by solving the Navier–Stokes equations for an incompressible and two-dimensional flow at low Reynolds numbers. We consider with detail two particular sets of configurations of interest in forward flight in a wide range of heaving amplitudes and frequencies: a plunging leading plate with the trailing plate at rest, and the two plates heaving with the same frequency and amplitude, but varying the phase difference. In almost all cases the thrust efficiency of the leading plate is augmented in relation to a single plate heaving with the same frequency and amplitude. In the first configuration with a trailing plate at rest, we characterize the range of nondimensional heaving frequencies and amplitudes of the leading plate for which the stationary trailing plate contributes positively to the global thrust. The maximum global thrust efficiency of this configuration, reached for an advance ratio slightly less than unity and a reduced frequency close to 5, is about the same as the maximum efficiency for an isolated plate, reached for slightly smaller frequencies. But for low frequencies the tandem configuration with the trailing plate at rest is more thrust efficient than the isolated plate. We also characterize the nondimensional frequency and amplitude regions for which the flow becomes chaotic. In the second configuration, the maximum of the total thrust efficiency is reached for a phase lag of 180° (counterstroking), particularly for an advance ratio unity and a reduced frequency 4.4. It is almost the same as the maximum thrust efficiency in the other configuration with the trailing plate at rest and that of a single plate. We discuss the flow structures and the aerodynamic interaction between plates responsible for the optimal thrust configuration in both cases.     

A low Reynolds number variant of partially-averaged Navier-Stokes model for turbulence

A low Reynolds number (LRN) formulation based on the Partially Averaged Navier-Stokes (PANS) modelling method is presented, which incorporates improved asymptotic representation in near-wall turbulence modelling. The effect of near-wall viscous damping can thus be better accounted for in simulations of wall-bounded turbulent flows. The proposed LRN PANS model uses an LRN k-ε model as the base model and introduces directly its model functions into the PANS formulation. As a result, the inappropriate wall-limiting behavior inherent in the original PANS model is corrected. An interesting feature of the PANS model is that the turbulent Prandtl numbers in the k and ε equations are modified compared to the base model. It is found that this modification has a significant effect on the modelled turbulence. The proposed LRN PANS model is scrutinized in computations of decaying grid turbulence, turbulent channel flow and periodic hill flow, of which the latter has been computed at two different Reynolds numbers of Re = 10,600 and 37,000. In comparison with available DNS, LES or experimental data, the LRN PANS model produces improved predictions over the standard PANS model, particularly in the near-wall region and for resolved turbulence statistics. Furthermore, the LRN PANS model gives similar or better results - at a reduced CPU time - as compared to the Dynamic Smagorinsky model.     

A new bionic MAV＇s flapping motion based on fruit fly hovering at low Reynolds number

On the basis of the studies on the high unsteady aerodynamic mechanisms of the fruit fly hovering the aerodynamic advantages and disadvantages of the fruit fly flapping motion were analyzed. A new bionic flapping motion was proposed to weaken the disadvantages and maintain the advantages, it may be used in the designing and manufacturing of the micro air vehicles (MAV’s). The translation of the new bionic flapping motion is the same as that of fruit fly flapping motion. However, the rotation of the new bionic flapping motion is different. It is not a pitching-up rotation as the fruit fly flapping motion, but a pitching-down rotation at the beginning and the end of a stroke. The numerical method of 3rd-order Roe scheme developed by Rogers was used to study these questions. The correctness of the numerical method and the computational program was justified by comparing the present CFD results of the fruit fly flapping motion in three modes, i.e., the advanced mode, the symmetrical mode and the delayed mode, with Dickinson’s experimental results. They agreed with each other very well. Subsequently, the aerodynamic characteristics of the new bionic flapping motion in three modes were also numerically simulated, and were compared with those of the fruit fly flapping. The conclusions could be drawn that the high unsteady lift mechanism of the fruit fly hovering is also effectively utilized by this new bionic flapping. Compared with the fruit fly flapping, the unsteady drag of the new flapping decreases very much and the ratio of lift to drag increases greatly. And the great discrepancies among the mean lifts of three flapping modes of the fruit fly hovering are effectively smoothed in the new flapping. On the other hand, this new bionic flapping motion should be realized more easily. Finally, it must be pointed out that the above conclusions were just drawn for the hovering flapping motion. And the aerodynamic characteristics of the new bionic flapping motion in forward flight are going to be studied in the next step. The project supported by the National Natural Science Foundation of China (10232010, 10032060, 90605005).     

Treadmilling swimmers near a no-slip wall at low Reynolds number

The motion of a circular treadmilling low Reynolds number swimmer near a no-slip wall is studied analytically. First, the exact solution of Jeffrey and Onishi [Q. J. Mech. Appl. Math., 34 (1981)] for a translating and rotating solid cylinder near a no-slip wall is rederived using a novel conformal mapping approach that differs from the original derivation which employed bipolar coordinates. Then it is shown that this solution can be combined with the reciprocal theorem, and the calculus of residues, to produce an explicit non-linear dynamical system for the treadmilling swimmer's velocity and angular velocity. The resulting non-linear dynamical system governing the swimmer motion is used to corroborate the qualitative results obtained by an approximate model of the same swimmer recently presented in Crowdy and Or [Phys. Rev. E., 81 (2010)].     

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.     

Thrust augmentation of flapping airfoils in low Reynolds number flow using a flexible membrane

The unsteady aerodynamic thrust and aeroelastic response of a two-dimensional membrane airfoil under prescribed harmonic motion are investigated computationally with a high-order Navier–Stokes solver coupled to a nonlinear membrane structural model. The effects of membrane prestress and elasticity are examined parametrically for selected plunge and pitch–plunge motions at a chord-based Reynolds number of 2500. The importance of inertial membrane loads resulting from the prescribed flapping is also assessed for pure plunging motions. This study compares the period-averaged aerodynamic loads of flexible versus rigid membrane airfoils and highlights the vortex structures and salient fluid–membrane interactions that enable more efficient flapping thrust production in low Reynolds number flows.     

Suppression of vortex shedding from a rectangular cylinder at low Reynolds numbers

Small elements of circular, square, triangular and thin-strip cross-sections are used to suppress vortex shedding from a rectangular cylinder of stream-wise to transverse scale ratio L/B=3.0 at Reynolds numbers in the range of Re=V_∞B/ν=75–130, where V_∞ is the on-coming velocity of the stream, and ν is the kinematic viscosity. The relative transverse dimension of the small element b/B is fixed at 0.2. The results of numerical simulation and visualization experiment show that, vortex shedding from both sides of the cylinder can be suppressed and the fluctuating drag and lift of the cylinder can be greatly reduced, if the element is placed in a certain region referred to as the effective zone. Comparisons at a specific Reynolds number indicate that the square element produces the largest size of the effective zone, whereas the triangular element yields the smallest. Results also show that the effective zone for the square element shrinks with increasing Re and disappears at Re>130. Independent of element cross-section shape and Reynolds number, the center of the effective zone is always at X/B=2.5–3.0 and Y/B≈1.0. The mechanism of the suppression is discussed from the view points of velocity profile stability and stress distribution.     

The effect of the Reynolds number on the Reynolds stresses and vorticity in a turbulent far-wake

Using two orthogonal arrays of 16 X-wires, eight in the (x,y)-plane and eight in the (x,z)-plane, the effect of the Reynolds number in a turbulent plane far-wake has been investigated for two values of Re_θ (based on the free stream velocity and the momentum thickness), i.e. 1350 and 4600. It is observed that as the Reynolds number increases the magnitudes of the measured Reynolds stresses increase, as does the size of two-point vorticity correlation iso-contours. Discernible differences are also observed in probability density function, spectra and three-dimensional topologies. The Reynolds number dependence seems to vanish when Re_θ5000.     

随机湍流工况低雷诺数风力机翼型优化研究

为改善小型风力机随机湍流工况适应性,以NACA0012翼型为研究对象,采用非嵌入式概率配置点法,获得随机湍流工况下小型风力机叶片翼型运行攻角分布规律;在气动优化中耦合层流分离预测,基于Transition SST模型、拉丁超立方试验设计、Kriging模型和带精英策略非支配排序遗传算法NSGA-II进行高湍流低雷诺数风力机翼型气动优化。结果表明,优化翼型叶片平均风能捕获效率分别提高3.01%和4.76%,标准差分别降低4.76%和14.93%,优化翼型湍流适应性增强。该方法将翼型设计与湍流风况相匹配,为湍流工况低雷诺数翼型及小型风力机设计提供参考。     

Reynolds number, thickness and camber effects on flapping airfoil propulsion

The effect of varying airfoil thickness and camber on plunging and combined pitching and plunging airfoil propulsion at Reynolds number Re=200, 2000, 20 000 and 2×10⁶ was studied by numerical simulations for fully laminar and fully turbulent flow regimes. The thickness study was performed on 2-D NACA symmetric airfoils with 6-50% thick sections undergoing pure plunging motion at reduced frequency k=2 and amplitudes h=0.25 and 0.5, and for combined pitching and plunging motion at k=2, h=0.5, phase ?=90°, pitch angle θ_o=15° and 30° and the pitch axis was located at 1/3 of chord from leading edge. At Re=200 for motions where positive thrust is generated, thin airfoils outperform thick airfoils. At higher Re significant gains could be achieved both in thrust generation and propulsive efficiency by using a thicker airfoil section for plunging and combined motion with low pitch amplitude. The camber study was performed on 2-D NACA airfoils with varying camber locations undergoing pure plunging motion at k=2, h=0.5 and Re=20 000. Little variation in thrust performance was found with camber. The underlying physics behind the alteration in propulsive performance between low and high Reynolds numbers has been explored by comparing viscous Navier-Stokes and inviscid panel method results. The role of leading edge vortices was found to be key to the observed performance variation.     

低雷诺数下附面层组合抽吸方案对压气机特性影响的研究

为了分析附面层抽吸流动控制对低雷诺数下压气机特性的影响,本文采用数值方法模拟了低雷诺数下附面层组合抽吸方案对NASA Rotor 37跨音速压气机性能和稳定性的影响特点及作用机理。通过在该压气机转子叶片吸力面和机匣上分别设计附面层抽吸槽,探讨了组合抽吸方案对低雷诺数下(H=20km)压气机性能和稳定性的影响。结果表明:采用组合抽吸方案后,压气机峰值效率提高约1.3%;压气机最大增压比提高约2.5%;压气机转子的近失速点流量减小约14.6%。进一步分析作用机理发现,组合抽吸槽有效抑制了附面层径向涡向叶顶的运动和聚集,使叶顶附面层分离区减少约70%从而有效改善了压气机的流场特性。     

Wake-induced vibrations of an elastically mounted cylinder located downstream of a stationary larger cylinder at low Reynolds numbers

Two-dimensional numerical simulations of flow past two unequal-sized circular cylinders in tandem arrangement are performed at low Reynolds numbers (Re). The upstream larger cylinder is stationary, while the downstream cylinder has both one (transverse-only) and two (transverse and in-line) degrees of freedom (1-dof and 2-dof, respectively). The Re, based on the free stream velocity U_∞ and the downstream cylinder diameter d, varies between 50 and 200 with a wide range of reduced velocities U_r. The diameter of the upstream cylinder is twice that of the downstream cylinder, and the center-to-center spacing is 5.5d. In general, for the 1-dof case, the calculations show that the wake-induced vibrations (WIV) of the downstream cylinder are greatly amplified when compared to the case of a single cylinder or two equal-sized cylinders. The transverse amplitudes build up to a significantly higher level within and beyond the lock-in region, and the U_r associated with the peak amplitude shifts toward a higher value. The dominant wake pattern is 2S mode for Re=50 and 100, while with the increase of Re to 150 and 200, the P+S mode can be clearly observed at some lower U_r. For the 2-dof vibrations, the transverse response characteristics are similar to those presented in the corresponding 1-dof case. The in-line responses are generally much smaller, except for several significant vibrations resulting from in-line resonance. The obvious in-line vibration may induce a C (chaotic) vortex shedding mode for higher Re (Re=200). With regard to the 2-dof motion trajectories, besides the typical figure-eight pattern, several odd patterns such as figure-double eight and single-looped trajectories are also obtained due to the wake interference effect.     

Comparative studies of numerical methods for evaluating aerodynamic characteristics of two-dimensional airfoil at low Reynolds numbers

This study investigates the predictability of the aerodynamic performance of some numerical methods at low Reynolds numbers and their dependency on the geometric shape of airfoil. We conducted three-dimensional large-eddy simulations (3-D LES), two-dimensional laminar simulations (2-D Lam), and Reynolds-averaged Navier–Stokes simulations with Baldwin–Lomax (2-D RANS(BL)) and Spalart–Allmaras (2-D RANS(SA)) turbulence models. Although there is little discrepancy between the 3-D LES, 2-D Lam, and 2-D RANS(SA) results in terms of the lift and drag characteristics, significant differences are observed in the predictability of the separation and reattachment points. The predicted lift, separation, and reattachment points of the 2-D Lam are qualitatively similar to those of the 3-D LES, except for high angles of attack at which a massive separation occurs. The 2-D RANS(SA) shows good predictability of the lift and separation points, but it does not estimate reattachment points accurately. The 2-D RANS(BL) fails to predict the precise separation points, which results in a poor lift predictability. These characteristics appear regardless of the airfoil geometry shapes. The results suggest that a 2-D Lam without any turbulence models can be used to estimate qualitative airfoil aerodynamic characteristics at the low Reynolds numbers.     

Analysis of a meshless solver for high Reynolds number flow

In this paper, the extension of an upwind least‐square based meshless solver to high Reynolds number flow is explored, and the properties of the meshless solver are analyzed both theoretically and numerically. Existing works have verified the meshless solver mostly with inviscid flows and low Reynolds number flows, and in this work, we are interested in the behavior of the meshless solver for high Reynolds number flow, especially in the near‐wall region. With both theoretical and numerical analysis, the effects of two parameters on the meshless solver are identified. The first one is the misalignment effect caused by the significantly skewed supporting points, and it is found that the meshless solver still yields accurate prediction. It is a very interesting property and is opposite to the median‐dual control volume based vertex‐centered finite volume method, which is known to give degraded result with stretched triangular/tetrahedral cells in the near‐wall region. The second parameter is the curvature, and according to theoretical analysis, it is found in the region with both large aspect ratio and curvature, and the streamwise residual is less affected; however, the wall‐normal counterpart suffers from accuracy degradation. In this paper, an improved method that uses a meshless solver for the streamwise residual and finite difference for wall‐normal residual is developed. This method is proved to be less sensitive to the curvature and provides improved accuracy. This work presents an understanding of the meshless solver for high Reynolds number flow computation, and the analysis in this paper is verified with a series of numerical experiments. Copyright © 2012 John Wiley & Sons, Ltd.