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.     

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.     

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.     

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%.     

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.     

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.     

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.     

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.     

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.     

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.     

Aerodynamic interactions of two airfoils in unsteady motion

Summary Aerodynamic interactions of two airfoils in tandem configuration moving parallelly forward and down at large angle of attack after an initial acceleration from rest are studied, using the method of solving the Navier-Stokes equations in moving overset grids. In the early time of the motion, force coefficients on the fore- and hind-airfoils are almost the same and are both enhanced in comparison with that of the single airfoil. The mechanism for the enhancement is that each airfoil sees a faster incoming flow because of the blockage effect caused by the presence of the other airfoil. After the early time, for the cases having only horizontal spacing,C _L on the fore-airfoil keeps to be larger than that of the single airfoil, and the smaller the spacing is, the larger theC _L is;C _L on the hind-airfoil rapidly decreases to a lower level because of the interaction between the starting vortex of the fore-airfoil and the dynamic stall vortex of the hind-airfoil, and this detrimental effect becomes more severe as the spacing becomes smaller.C _d behaves similarly. When the hind-airfoil is lower in vertical position than the fore-airfoil and their horizontal spacing is small (e.g. the vertical and horizontal spacings are 0.25c), large enhancement of the forces on the fore-airfoil (e.g.C _L=3.0) and a noticeable enhancement on the hind-airfoil (e.g.C _L2.2) can be obtained.     

Motion of a solid sphere in a general flow near a plane boundary at zero Reynolds number

It is demonstrated how the hydrodynamic force and moment of force acting on a solid sphere may be calculated when it is placed at rest at an arbitrary position in a two dimensional flow at zero Reynolds number in which the region of flow is bounded by either an undeformable planar free surface or by a plane solid wall. The results so obtained are used to calculate the motion of a freely moving solid sphere in an asymmetric vortex in the presence of an underformable free surface. It is seen that the sphere, depending on the direction of the undisturbed flow, will either spiral into or out of the vortex. This implies that when a dilute suspension of such spherical particles undergoes such a vortex motion in the presence of the free surface, the vortex will either fill up with particles from the surrounding flow or become devoid of particles.Deceased, July 31, 1995     

Flow of a viscous fluid at small Reynolds number past a porous sphere with a solid core

Summary Uniform flow of an incompressible viscous fluid at small Reynolds number past a porous sphere of radius a with a solid concentric spherical core of radius b has been discussed. The region of the porous shell is called zone I which is fully saturated with the viscous fluid, and the flow in this zone is governed by the Brinkman equation. The space outside the shell where clear fluid flows is divided into two zones (II and III). In these zones the flow is discussed following Proudman and Pearson's method of expanding Stokes' stream function in powers of Reynolds number and then matching Stokes' solution with Oseen's solution. The stream function of zone II is matched with that of zone I at the surface of the shell by the condition suggested by Ochoa – Tapia and Whitaker. It is found that the drag on the spherical shell increases with the increase of the λ (=b/a) and also with the increase of the Darcy number. The graph of dimensionless drag against λ for various values of Reynolds number shows that the drag increases with the increase of the Reynolds number for all values of λ.     

Flow past an impulsively started circular cylinder at high Reynolds number

The unsteady viscous incompressible flow around a circular cylinder is studied numerically for high Reynolds number. The two dimensional Navier-Stokes equation in stream function-vorticity formulation is solved. A fractional step method is used to solve the stream function equation while the vorticity transport equation is solved using a third order upwinding scheme. The computed solutions by the finite difference method agree reasonably well with the available experimental and other computational results at a Reynolds number of 9500 and 4500. These comparisons are for the initial stages of flow evolution when the wake bubble is symmetric.     

Convective heat transfer in air flow around a cylinder at low Reynolds numbers

The principal objective of the present work is to conduct investigations leading to a more complete explanation of heat-transfer processes on the external wall of a heated cylinder in laminar axial flow around it under high pressures. Investigations are aimed at determination of the limits of existence of mixed convection, explanation of the influence of free convection on the disturbances of heat transfer during laminar flow of a medium, and final explanation of intensification of heat-transfer processes occurring in a flow at high pressures. Published in Inzhenerno-Fizicheskii Zhurnal, Vol. 78, No. 6, pp. 163–169, November–December, 2005.