Force and flow structures of a wing performing flapping motion at low Reynolds number

Summary Aerodynamic force and flow structures of a wing performing a simplified flapping motion that emulates the wing motion of small insects in normal hovering flight are studied, using the method of numerically solving the Navier-Stokes equations. For a typical case (wing rotation-axis is at 0.25 chord position and wing rotation is symmetrical with respect to stroke reversal), large peaks inC _L andC _D are produced near the end of a stroke by the wing-rotation, but the wing-rotation also generates a vortical structure which induces strong downwash velocity, reducing the lift production in the early part of the following stroke; averaging over one flapping cycle,C _L is 28% larger than the steady-state value and is about that needed to support the weight of a small insect. The timing of the wing-rotation at stroke reversal can change the size of the peaks inC _L andC _D and their averages; e.g. in the case of shifting the wing-rotation forward in time by 7.5% of a stroke period, the averageC _L becomes 63% larger than the steady-state value, which is larger than that needed for weight supporting (and might provide extra force for control or maneuvers). When the rotation-axis is moved rearward to the middlechord position, the lift and drag peaks due to the wing-rotation become smaller, and in this case, the wing-rotation generates a vortical structure that tends to prevent the relative motion between positive and negative vorticities, also reducing the lift production in the early part of the following stroke, resulting in a smaller averageC _L .     

Aerodynamic forces and flow structures of an airfoil in some unsteady motions at small Reynolds number

Summary The aerodynamic forces and flow structures of a NACA 0012 airfoil in some unsteady motions at small Reynolds number (Re=100) are studied by numerically solving the Navier-Stokes equations. These motions include airfoil acceleration and deceleration from one translational speed to another and rapidly pitching up in constant freestream (equivalent to pitching up during translational motion at constant speed). It is shown that at small Reynolds number (Re=100), when the airfoil is performing fast acceleration or deceleration from one speed to another, a large aerodynamic force can be generated during and for a time period after the acceleration or deceleration; a large aerodynamic force can also be generated when the airfoil is performing a fast pitching motion in a constant freestream. In these fast unsteady motions, an airfoil in low Re flow can produce a large aerodynamic force as effective as in large Re flow, or the effect of unsteady motion dominates the Reynolds number effect. During the fast unsteady motion of the airfoil, new layers of strong vorticity are formed near the upper and lower surfaces of the airfoil under the previously existing thick vorticity layers, and it is the generation and motion of the new vorticity layers that is mainly responsible for the generation of the large aerodynamic force; the large-scale structure and movement of the newly produced vorticity layers are similar to that of high Re flow.     

Optimization of micro Venturi diode in steady flow at low Reynolds number

Micro Venturi diodes have been broadly used in microfluidics to rectify liquid flow. A key feature of these diodes is that they have no moving parts. This article treats the numerical optimization of such a micro Venturi diode for the case of steady Newtonian flow, with Reynolds numbers less than 100, which typically applies to microfluidic devices. Key design factors are discussed, such as the ratio of length to width of the design domain, the values of lower and upper bounds for the constraint of the volume of the diodes, and the Reynolds number, all of which influence the outcome of the diode effect. Curved micro Venturi diodes are found by the optimization. The method presented provides a systematic way to design Venturi diodes for steady Newtonian flows.     

Asymptotics beyond all orders for a low Reynolds number flow

The solution for slow incompressible flow past a circular cylinder involves terms in powers of 1/log , times powers of 1/log , etc., where is the Reynolds number. Previously we showed how to determine the sum of all terms in powers of 1/log . Now we show how to go beyond all those terms to find the sum of all terms containing times a power of 1/log . The first sum gives the drag coefficient and represents a symmetric flow in the Stokes region near the cylinder. The second term reveals the asymmetry of the flow near the body. This problem is studied using a hybrid method which combines numerical computation and asymptotic analysis.     

低雷诺数下两类串列圆柱的涡激振动

为了研究上游圆柱的运动状况对下游圆柱涡激振动的影响,针对两类串列双圆柱(上游圆柱固定、下游圆柱可作两自由度振动,上下游圆柱均可作两自由度振动),在低雷诺数下(Re=100),采用数值模拟方法,研究了下游圆柱在不同尾流干扰下的振幅、振动频率、相位差等振动特性随折减速度的变化规律,从能量输入和尾流模态角度探讨了上游圆柱的振...     

Asymptotic analysis of the viscous micro/nano pump at low Reynolds number

The steady viscous parabolic flow past an eccentrically placed rotating cylinder is studied in the asymptotic limit of small Reynolds number. It is assumed that the flow around the rotating cylinder undergoes boundary slip described by the Navier boundary condition. This involves a single parameter to account for the slip, referred to as the slip length ?, and replaces the standard no-slip boundary condition at solid boundaries. The streamlines for ? > 0 are closer to the body than for ? = 0, and it is discovered that the loss of symmetry due to the rotation of the cylinder is significantly reduced by the inclusion of slip. This arises as a result of a balance between the rotation velocity and the slip velocity on that portion of the cylinder which rotates opposite to the free-stream flow. Streamline patterns for nonzero eccentricity partially agree with Navier–Stokes simulations of the viscous pump; the small discrepancy is primarily due to the fact that here wall effects are not explicitly considered. Expressions for the frictional drag and the torque on the cylinder are obtained. The expression for the torque agrees well with the lubrication solution for the flow past a rotating cylinder placed symmetrically in a fully developed channel flow. The results presented here may be used to validate numerical schemes developed to study the viscous pump.     

Simulation of internal heat transfer in a porous permeable envelope at low values of Reynolds number

A slotted porous layer is used as an example for performing direct numerical simulation of internal heat transfer in a porous envelope under conditions of low values of Reynolds number and significant impact of longitudinal thermal conductivity. Comparison is made with the results of calculation using the porous-disperse model. Analysis is performed of the pattern of transition from convective to convective-conductive heat transfer and further to the mode of thermal conductivity from a porous body to surrounding continuous medium.     

低雷诺数下串联双圆柱涡激振动机理的数值研究

利用有限元数值方法求解不可压缩粘性流体的Navier-Stokes方程,结合任意拉格朗日-欧拉（ALE）动网格方法,对低雷诺数下（Re = 150）等直径串联圆柱的涡激振动问题进行了数值模拟研究。其中上游圆柱固定,下游圆柱在弹簧和阻尼作用下允许同时发生顺流向和横流向的运动。在约化速度Ur = 3.0 ~ 12.0的范围内（阻尼比ξ = 0.007）研究了两圆柱圆心间距比（LX / D = 3.0、5.0、8.0）及圆柱质量比（m* = 5.0、10.0、20.0）对下游圆柱的运动响应及受力的影响。数值结果表明,圆柱间距比的变化会导致锁定区间的变化,进而影响到圆柱涡激振动的位移响应和受力特性。这些方面都与尾流区涡旋脱落模式密切相关,体现了双圆柱间干涉作用对涡激振动的影响。进一步的研究表明,圆柱质量比的变化对以约化速度表征的锁定区间、运动响应和尾流模式等都有一定的影响作用。     

Multi-domain dual reciprocity for the solution of inelastic non-Newtonian flow problems at low Reynolds number

An efficient boundary element solution of the motion of inelastic non-Newtonian fluids at low Reynolds number is presented in this paper. For the numerical solution all the domain integrals of the boundary element formulation have been transformed into equivalent boundary integrals by means of the dual reciprocity method (DRM). To achieve an accurate approximation of the non-linear and non-Newtonian terms two major improvements have been made to the DRM, namely the use of augmented thin plate splines as interpolation functions, and the partition of the entire domain into smaller subregions or domain decomposition. In each subregion or domain element the DRM was applied together with some additional equations that ensure continuity on the interfaces between adjacent subdomains. After applying the boundary conditions the final systems of equations will be sparse and the approximation of the nonlinear terms will be more localised than in the traditional DRM. This new method known as multidomain dual reciprocity (MD-DRM) has been used to solve several non-Newtonian problems including the pressure driven flow of a power law fluid, the Couette flow and two simulations of industrial polymer mixers. Received 7 February 2001     

Structure and dynamics of low Reynolds number turbulent pipe flow

Using large-scale numerical calculations, we explore the proper orthogonal decomposition of low Reynolds number turbulent pipe flow, using both the translational invariant (Fourier) method and the method of snapshots. Each method has benefits and drawbacks, making the 'best' choice dependent on the purpose of the analysis. Owing to its construction, the Fourier method includes all the flow fields that are translational invariants of the simulated flow fields. Thus, the Fourier method converges to an estimate of the dimension of the chaotic attractor in less total simulation time than the method of snapshots. The converse is that for a given simulation, the method of snapshots yields a basis set that is more optimal because it does not include all of the translational invariants that were not a part of the simulation. Using the Fourier method yields smooth structures with definable subclasses based upon Fourier wavenumber pairs, and results in a new dynamical systems insight into turbulent pipe flow. These subclasses include a set of modes that propagate with a nearly constant phase speed, act together as a wave packet and transfer energy from streamwise rolls. It is these interactions that are responsible for bursting events and Reynolds stress generation. These structures and dynamics are similar to those found in turbulent channel flow. A comparison of structures and dynamics in turbulent pipe and channel flows is reported to emphasize the similarities and differences.     

Numerical simulations of Coriolis flow meters for low Reynolds number flows

In process industries Coriolis mass flow meters (CMFs) are widely employed for measuring mass flows. Quite often, especially in the oil and gas (O&G) industry, owing to fluids with high viscosities, flow measurements may lie in low Reynolds number regions. At low Reynolds numbers (Re), a CMF reading may deviate under the influence of fluid-dynamic forces. With the help of extensive Fluid-Structure-Interaction simulations (FSI), a detailed insight into physical mechanisms leading to this deviation is provided. The main finding is that this deviation is a function of the Reynolds number and the effect can be explained by a periodic shear mechanism which interacts with the oscillatory Coriolis force and reduces the tube deflection. Experimental results with and without a correction for this effect are shown and compared with corresponding numerical results. If the low Reynolds number effect were ignored, it would lead to errors as large as 0.5% to 1% at Re = 800, however by measuring the Re and making corrections, the effect is reduced to < 0.2%.     

On steady long waves on a viscous liquid at small Reynolds number

Summary The generation of steady surface waves on a viscous liquid flowing down an irregular inclined plane is investigated in the shallow-liquid approximation. A non-linear differential equation gives the surface elevation and a numerical solution is presented for a periodic two-dimensional flow. Linearisation of this equation enables three-dimensional small-amplitude disturbances to be considered.     

Confined laminar wakes at small and large Reynolds number

Summary The development of the wake flowfield behind a symmetric cascade of finite-thickness flat plates in steady two-dimensional laminar incompressible flow is investigated for a wide range of Reynolds number. A spectral method is used to obtain the solution to a low-Reynolds-number expansion of the Navier-Stokes equations as well as a second approximation to the Oseen equations. Comparisons to the results of the second Oseen approximation are made with previously obtained solutions to the slender-channel equations for large Reynolds number as well as with solutions to the low-Reynolds-number expansion.     

Nonstationary diffusion flow to a moving drop at low Reynolds numbers

The process of nonstationary convective diffusion to a moving drop at low Re is examined. Equations for the dimensionless diffusion flow Nu to the surface of the drop and for the length of the nonstationary region and the time required for the establishment of steady-state diffusion conditions are obtained.     

MHD flow past a sphere at low and moderate Reynolds numbers

The steady incompressible magnetohydrodynamic (MHD) flow past a sphere with an applied magnetic field parallel to the main flow is studied for the Reynolds number, Re, up to 100 and interaction parameter, N, up to 0.7. The magnetic Reynolds number is assumed to be small. It is observed that drag coefficient increases as N increases. The Finite difference method is used to solve the vorticity-stream function form of the non-linear Navier-Stokes equations. The multigrid method is used to solve the finite difference equations. The fifth decimal place convergent solutions are obtained upto the finest grid of 1024×512. Graphs of the streamlines, vorticity lines, surface vorticity and drag coefficient are presented.The authors are grateful to the Principal, Pondicherry Engineering College and Head, Department of Mathematics, Pondicherry Engineering College for providing the necessary facilities. The authors are thankful to Dr. Umamaheswara Rao, Department of Applied Mathematics, Andhra University, Visakhapatnam and Dr. R. Sivakumar, Department of Physics, Pondicherry Engineering College, for their encouragement.     

Relaminarization in supersonic microjets at low Reynolds numbers

Results of measuring the length of the supersonic portion of the air jets that flow out of axisymmetric sonic nozzles 10.4 μm-1 mm in diameter are presented. The measurements are carried out in a range of degree of jet noncalculation of 1–30 and in a wide Reynolds number range, including the laminar and turbulent flow modes. It is shown that the Reynolds number calculated from the nozzle diameter and the outlet parameters of gas is the parameter that governs jet flow. It is found that, for a laminar jet mixing layer, the length of the supersonic portion sharply increases. When the jet mixing layer becomes turbulent, the length of the supersonic portion decreases. The effect of increasing the length of the supersonic portion after its decrease due to the turbulization of flow in a jet and a growth in the Reynolds number is first discovered.     

An improved low Reynolds number k-ɛ model for heat transfer calculations

The paper presents numerical calculations of turbulent exchange of momentum and heat in the flow of air and water respectively at the inlet region of a circular pipe. The calculations are based on the low Reynolds number k-? turbulence model, taking into account the changes of the turbulent Prandtl number Pr_t across the pipe. Basing on results of experimental work available in the literature, three variants of changes of the Prandtl number Pr_t were developed. The obtained results of numerical calculations were compared with corresponding results of experimental work and other analytical ones available in the literature. The comparative analysis of the numerical results and other, both experimental and theoretical ones, evidenced a high degree of their conformity, especially in the case of calculations based on case I or III of the change pattern of the Pr_t number.     

The critical layer in pipe flow at high Reynolds number

We report the computation of a family of travelling wave solutions of pipe flow up to Re=75000. As in all lower branch solutions, streaks and rolls feature prominently in these solutions. For large Re, these solutions develop a critical layer away from the wall. Although the solutions are linearly unstable, the two unstable eigenvalues approach 0 as Re-->infinity at rates given by Re-0.41 and Re-0.87; surprisingly, the solutions become more stable as the flow becomes less viscous. The formation of the critical layer and other aspects of the Re-->infinity limit could be universal to lower branch solutions of shear flows. We give implementation details of the GMRES-hookstep and Arnoldi iterations used for computing these solutions and their spectra, while pointing out the new aspects of our method.