Unsteady PTV velocity field past an airfoil solved with DNS: Part 1. Algorithm of hybrid simulation and hybrid velocity field at <Emphasis Type="Italic">Re</Emphasis> ≈ 10<Superscript>3</Superscript>

We develop a hybrid unsteady-flow simulation technique combining direct numerical simulation (DNS) and particle tracking velocimetry (PTV) and demonstrate its capabilities by investigating flows past an airfoil. We rectify instantaneous PTV velocity fields in a least-squares sense so that they satisfy the equation of continuity, and feed them to the DNS by equating the computational time step with the frame rate of the time-resolved PTV system. As a result, we can reconstruct unsteady velocity fields that satisfy the governing equations based on experimental data, with the resolution comparable to numerical simulation. In addition, unsteady pressure distribution can be solved simultaneously. In this study, particle velocities are acquired on a laser-light sheet in a water tunnel, and unsteady flow fields are reconstructed with the hybrid algorithm solving the incompressible Navier–Stokes equations in two dimensions. By performing the hybrid simulation, we investigate nominally two-dimensional flows past the NACA0012 airfoil at low Reynolds numbers. In part 1, we introduce the algorithm of the proposed technique and discuss the characteristics of hybrid velocity fields. In particular, we focus on a vortex shedding phenomenon under a deep stall condition (α = 15°) at Reynolds numbers of Re = 1000 and 1300, and compare the hybrid velocity fields with those computed with two-dimensional DNS. In part 2, the extension to higher Reynolds numbers is considered. The accuracy of the hybrid simulation is evaluated by comparing with independent experimental results at various angles of attack and Reynolds numbers up to Re = 10⁴. The capabilities of the hybrid simulation are also compared with two-dimensional unsteady Reynolds-Averaged Navier–Stokes (URANS) solutions in part 2. In the first part of these twin papers, we demonstrate that the hybrid velocity field approaches the PTV velocity field over time. We find that intensive alternate vortex shedding past the airfoil, which is predicted by the two-dimensional DNS, is substantially suppressed in the hybrid simulation and the resultant flow field is similar to the PTV velocity field, which is projection of the three-dimensional velocity field on the streamwise plane. We attempt to identify the motion that originates three-dimensional flow patterns by highlighting the deviation of the PTV velocity field from the two-dimensional governing equations at each snapshot. The results indicate that the intensive spots of the deviation appear in the regions in which three-dimensional instabilities are induced in the shear layer separated from the pressure side.     

To formulating a nonlinear initial-boundary problem of an unsteady separated flow around an airfoil

A general formulation of a nonlinear initial-boundary problem of an unsteady separated flow around an airfoil by an ideal incompressible fluid is considered. The problem is formulated for a complex velocity. Conditions of shedding of vortex wakes from the airfoil are analyzed in detail. The proposed system of functional relations allows constructing algorithms for solving a wide class of problems of the wing theory. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 2, pp. 48–56, March–April, 2007.     

A study on the mechanism of high-lift generation by an airfoil in unsteady motion at low reynolds number

The aerodynamic force and flow structure of NACA 0012 airfoil performing an unsteady motion at low Reynolds number (Re=100) are calculated by solving Navier-Stokes equations. The motion consists of three parts: the first translation, rotation and the second translation in the direction opposite to the first. The rotation and the second translation in this motion are expected to represent the rotation and translation of the wing-section of a hovering insect. The flow structure is used in combination with the theory of vorticity dynamics to explain the generation of unsteady aerodynamic force in the motion. During the rotation, due to the creation of strong vortices in short time, large aerodynamic force is produced and the force is almost normal to the airfoil chord. During the second translation, large lift coefficient can be maintained for certain time period and , the lift coefficient averaged over four chord lengths of travel, is larger than 2 (the corresponding steady-state lift coefficient is only 0.9). The large lift coefficient is due to two effects. The first is the delayed shedding of the stall vortex. The second is that the vortices created during the airfoil rotation and in the near wake left by previous translation form a short “vortex street” in front of the airfoil and the “vortex street” induces a “wind”; against this “wind” the airfoil translates, increasing its relative speed. The above results provide insights to the understanding of the mechanism of high-lift generation by a hovering insect. The project supported by the National Natural Science Foundation of China (19725210)     

An inviscid model for vortex shedding from a deforming body

An inviscid vortex sheet model is developed in order to study the unsteady separated flow past a two-dimensional deforming body which moves with a prescribed motion in an otherwise quiescent fluid. Following Jones (J Fluid Mech 496, 405–441, 2003) the flow is assumed to comprise of a bound vortex sheet attached to the body and two separate vortex sheets originating at the edges. The complex conjugate velocity potential is expressed explicitly in terms of the bound vortex sheet strength and the edge circulations through a boundary integral representation. It is shown that Kelvin’s circulation theorem, along with the conditions of continuity of the normal velocity across the body and the boundedness of the velocity field, yields a coupled system of equations for the unknown bound vortex sheet strength and the edge circulations. A general numerical treatment is developed for the singular principal value integrals arising in the solution procedure. The model is validated against the results of Jones (J Fluid Mech 496, 405–441, 2003) for computations involving a rigid flat plate and is subsequently applied to the flapping foil experiments of Heathcote et al. (AIAA J, 42, 2196–2204, 2004) in order to predict the thrust coefficient. The utility of the model in simulating aquatic locomotion is also demonstrated, with vortex shedding suppressed at the leading edge of the swimming body.      

Time resolved PIV analysis of flow over a NACA 0015 airfoil with Gurney flap

A NACA 0015 airfoil with and without a Gurney flap was studied in a wind tunnel with Re _c = 2.0 × 10⁵ in order to examine the evolving flow structure of the wake through time-resolved PIV and to correlate this structure with time-averaged measurements of the lift coefficient. The Gurney flap, a tab of small length (1–4% of the airfoil chord) that protrudes perpendicular to the chord at the trailing edge, yields a significant and relatively constant lift increment through the linear range of the C _L versus α curve. Two distinct vortex shedding modes were found to exist and interact in the wake downstream of flapped airfoils. The dominant mode resembles a Kàrmàn vortex street shedding behind an asymmetric bluff body. The second mode, which was caused by the intermittent shedding of fluid recirculating in the cavity upstream of the flap, becomes more coherent with increasing angle of attack. For a 4% Gurney flap at α = 8°, the first and second modes corresponded with Strouhal numbers based on flap height of 0.18 and 0.13. Comparison of flow around ‘filled’ and ‘open’ flap configurations suggested that the second shedding mode was responsible for a significant portion of the overall lift increment.     

On the stability and extension of reduced-order Galerkin models in incompressible flows

Proper orthogonal decomposition (POD) has been used to develop a reduced-order model of the hydrodynamic forces acting on a circular cylinder. Direct numerical simulations of the incompressible Navier–Stokes equations have been performed using a parallel computational fluid dynamics (CFD) code to simulate the flow past a circular cylinder. Snapshots of the velocity and pressure fields are used to calculate the divergence-free velocity and pressure modes, respectively. We use the dominant of these velocity POD modes (a small number of eigenfunctions or modes) in a Galerkin procedure to project the Navier–Stokes equations onto a low-dimensional space, thereby reducing the distributed-parameter problem into a finite-dimensional nonlinear dynamical system in time. The solution of the reduced dynamical system is a limit cycle corresponding to vortex shedding. We investigate the stability of the limit cycle by using long-time integration and propose to use a shooting technique to home on the system limit cycle. We obtain the pressure-Poisson equation by taking the divergence of the Navier–Stokes equation and then projecting it onto the pressure POD modes. The pressure is then decomposed into lift and drag components and compared with the CFD results.     

A rotating elliptic airfoil in fluid at rest and in a parallel freestream

This paper reports results of DPIV measurements on a two-dimensional elliptic airfoil rotating about its own axis of symmetry in a fluid at rest and in a parallel freestream. In the former case, we examined three rotating speeds (Re _c,Ω = 400, 1,000 and 2,000), and in the later case, four rotating speeds (Ro _c,Ω = 2.4, 1.2, 0.6 and 0.4), together with two freestream velocities (Re _c,u  = 200 and 1,000) and two starting configurations of the airfoil (i.e., chord parallel to (α ₀ = 0°) or normal (α ₀ = 90°) to the freestream). Results show that a rotating airfoil in a stationary fluid produces two distinct types of vortex structures depending on the Reynolds number. The first type occurs at the lowest Reynolds number (Re _c,Ω = 400), where vortices shed from the two edges or tips of the airfoil dissipated quickly, resulting in the airfoil rotating in a layer of diffused vorticity. The second type occurs at higher Reynolds numbers (i.e., Re _c,Ω = 1,000 and 2,000), where the corresponding vortices rotated together with the airfoil. Due to the vortex suction effect, the torque characteristics are likely to be heavily damped for the first type because of the rapidly subsiding vortex shedding, and more oscillatory for the second type due to persistent presence of tip vortices. In a parallel freestream, increasing the tip-speed ratio (V/U) of the airfoil (i.e., decreasing the Rossby number, Ro _c,Ω) transformed the flow topology from periodic vortex shedding at Ro _c,Ω = 2.4 to the generation of a “hovering vortex” at Ro _c,Ω = 0.6 and 0.4. The presence of the hovering vortex, which has not been reported in literature before, is likely to enhance the lift characteristics of the airfoil. Freestream Reynolds number is found to have minimal effect on the vortex formation and shedding process, although it enhances shear layer instability and produces more small-scale flow structures that affect the dynamics of the hovering vortex. Likewise, initial starting configuration of the airfoil, while affecting the flow transient during the initial phase of rotation, has insignificant effect on the overall flow topology. Unfortunately, technical constraint of our apparatus prevented us from carrying out complimentary force measurements; nevertheless, the results presented herein, which are more extensive than those computed by Lugt and Ohring (1977), will provide useful benchmark data, from which more advanced numerical calculations can be carried out to ascertain the corresponding force characteristics, particularly for those conditions with the presence of hovering vortex.     

Effect of one-sided unsteady energy supply on aerodynamic characteristics of airfoils in a transonic flow

The possibility of controlling the aerodynamic characteristics of airfoils in transonic flight regimes by means of one-sided pulsed-periodic energy supply is studied. Based on the numerical solution of two-dimensional unsteady gas-dynamic equations, the change in the flow structure in the vicinity of a symmetric airfoil at different angles of attack and the aerodynamic characteristics of the airfoil as functions of the amount of energy supplied asymmetrically (with respect to the airfoil) are determined. The results obtained are compared with the data calculated for the flow past the airfoil at different angles of attack without energy supply. It is found that a given lift force can be obtained with the use of energy supply at a much better lift-to-drag ratio of the airfoil, as compared to the case of the flow past the airfoil at an angle of attack. The moment characteristics of the airfoil are found. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 49, No. 6, pp. 82–87, November–December, 2008.     

An improved method for calculating flow past flapping and hovering airfoils

A method is reported here for calculating unsteady aerodynamics of hovering and flapping airfoil for two-dimensional flow via the following improved methodologies: (a) a correct formulation of the problem using stream function (ψ) and vorticity (ω) as dependent variables; (b) calculating loads and moment by a new method to solve the governing pressure Poisson equation (PPE) in a truncated part of the computational domain on a nonstaggered grid; (c) accurate solution using high accuracy compact difference scheme for the vorticity transport equation (VTE) and (d) accelerating the computations by using a high-order filter after each time step of integration. These have been used to solve Navier–Stokes equation for flow past flapping and hovering NACA 0014 and 0015 airfoils at typical Reynolds numbers relevant to the study of unsteady aerodynamics of micro air vehicle (MAV) and insect/bird flight.     

Scanning PIV measurements of a laminar separation bubble

Scanning PIV is applied to a laminar separation bubble to investigate the spanwise structure and dynamics of the roll-up of vortices within the bubble. The laminar flow separation with turbulent reattachment is studied on the suction side of an airfoil SD7003 at Reynolds numbers of 20,000–60,000. The flow is recorded with a CMOS high-speed camera in successive light-sheet planes over a time span of 1–2 s to resolve the temporal evolution of the flow in the different planes. The results show the quasi-periodic development of large vortex-rolls at the downstream end of the separation bubble, which have a convex structure and an extension of 10–20% chord length in the spanwise direction. These vortices possess an irregular spanwise pattern. The evolution process of an exemplary vortex structure is shown in detail starting from small disturbances within the separation bubble transforming into a compact vortex at the downstream end of the separation bubble. As the vortex grows in size and strength it reaches a critical state that leads to an abrupt burst of the vortex with a large ejection of fluid into the mean flow.     

Mathematical Modeling of Time-Dependent Aerodynamic Loads on an Airfoil with Dynamic Deflection of a Control on the Trailing Edge

The linear problem of the time-dependent inviscid flow past a thin symmetric airfoil with a control on its trailing edge deflected in accordance with an arbitrary law is considered. The aerodynamic loads on the airfoil are calculated. The intensity of the vortex wake shed from the airfoil is determined by numerically solving a Volterra integral equation of the first kind. Questions of the mathematical modeling of the time-dependent aerodynamic loads in a form convenient for the joint solution of the problems of aerodynamics and flight dynamics are also considered. The results of the modeling are compared with the numerical solutions obtained.__________Translated from Izvestiya Rossiiskoi Academii Nauk, Mekhanika Zhidkosti i Gaza, No. 3, 2005, pp. 157–169.Original Russian Text Copyright © 2005 by Khrabrov.     

Numerical study of self-sustained oscillatory flows and heat transfer in channels with a tandem of transverse vortex generators

A numerical investigation was conducted into channel flows with a tandem of transverse vortex generators in the form of rectangular cylinders. The oscillatory behavior of the flow is studied. Data for heat transfer and flow losses are presented for 100≤Re≤400 and cylinder separation distances 1≤S/H≤4. The results are obtained by numerical solution of the full Navier-Stokes equations and the energy equation. Self-sustained flow oscillations are found for Re>100. Alternate and dynamic shedding of large vortex structures from the cylinders is observed by visualization of the numerically determined flow field. A heat transfer enhancement up to a factor 1.78 compared to plane channel flow is observed. Received on 16 July 1997     

Experimental estimation of a D-shaped cylinder wake using body-mounted sensors

The effectiveness of a small array of body-mounted sensors, for estimation and eventually feedback flow control of a D-shaped cylinder wake is investigated experimentally. The research is aimed at suppressing unsteady loads resulting from the von-Kármán vortex shedding in the wake of bluff-bodies at a Reynolds number range of 100–1,000. A low-dimensional proper orthogonal decomposition (POD) procedure was applied to the stream-wise and cross-stream velocities in the near wake flow field, with steady-state vortex shedding, obtained using particle image velocimetry (PIV). Data were collected in the unforced condition, which served as a baseline, as well as during influence of forcing within the “lock-in” region. The design of sensor number and placement was based on data from a laminar direct numerical simulation of the Navier-Stokes equations. A linear stochastic estimator (LSE) was employed to map the surface-mounted hot-film sensor signals to the temporal coefficients of the reduced order model of the wake flow field in order to provide accurate yet compact estimates of the low-dimensional states. For a three-sensor configuration, results show that the estimation error of the first two cross-stream modes is within 20–40% of the PIV-generated POD time coefficients. Based on previous investigations, this level of error is acceptable for a moderately robust controller required for feedback flow control.     

Design of a separation-free airfoil with outer-flow suction over a certain range of angles of attack

The problem of the design of an airfoil with slot air suction from the outer flow for a prescribed velocity distribution over the airfoil contour that ensures the absence of flow separation over a given range of angles of attack is formulated and solved. The proposed combined numerical and analytical method of airfoil design within the framework of the inviscid incompressible fluid model is based on the theory of inverse problems of aerohydrodynamics. Separationless flow past the airfoil is achieved by eliminating the falling velocity intervals from the specified velocity distribution in two given flow regimes. The flow past an airfoil with outer-flow suction is determined not only by the angle of attack as for an impermeable airfoil but also by the value of the suction mass flow. The slot is modeled by an annular channel with constant velocities on the walls. To satisfy the problem solvability conditions, free parameters are introduced into the initial velocity distribution. Examples of airfoil design are given. Kazan, Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 4, pp. 185–191, July–August, 2000.     

Reduced‐order controllers for control of flow past an airfoil

Reduced‐order controller design by means of reduced‐order model for control of a wake flow is presented. Reduced‐order model is derived by combining the Galerkin projection with proper orthogonal decomposition (POD) or with other related reduced‐order approaches such as singular value decomposition or reduced‐basis method. In the present investigation, we discuss the applicability of the reduced‐order approaches for fast computation of the optimal control for control of vortex shedding behind a thin airfoil through unsteady blowing on the airfoil surface. Accuracy of the reduced‐order model is quantified by comparing flow fields obtained from the reduced‐order models with those from the full‐order simulations under the same free‐stream conditions. A control of vortex shedding is demonstrated for Reynolds number 100. It is found that downstream directed blowing on the upper surface of the airfoil near the leading edge is more efficient in mitigating flow separation and suppressing the vortex shedding. Copyright © 2005 John Wiley & Sons, Ltd.     

On vortex shedding behind a circular disk

Abtract  Experiments were performed for individual realizations of the vortex shedding process behind a circular disk at Reynolds numbers of 10³–10⁵, at which periodic vortex shedding prevails in the wake. The phase differences regarding the individual vortex shedding structures detected at multiple circumferential locations in the wake were obtained by analyzing the hot-wire signals with a conditional-sampling scheme. The phase differences of vortex shedding detected at circumferential positions 90° apart show a wide scatter, but the anti-phase character is largely preserved in the individual vortex shedding process as detected at circumferential locations 180° apart. The randomness of phase differences involved in the vortex shedding process is noted to be essential in order to satisfy the axisymmetric property of the global flow. Received: 4 April 19969/Accepted: 29 January 1997     

Kinematic problem of an ideal incompressible fluid flow around an arbitrarily moving airfoil

An unsteady kinematic problem for arbitrary two-dimensional motion of an airfoil in an ideal incompressible fluid with formation of one and two vortex wakes is solved. The problem is solved by the method of conformal mapping of the flow domain onto a circle exterior; solution singularities in the vicinity of a sharp edge are analyzed, and the initial asymptotics of the solution is taken into account. The calculated results are found to be in good agreement with available experimental data on visualization of the flow pattern. The necessity of correct modeling of the initial stage of vortex-wake formation is demonstrated. A regular flow pattern is found to form after three and more periods of oscillations. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 50, No. 2, pp. 120–128, March-April, 2009.     

Time-resolved and volumetric PIV measurements of a transitional separation bubble on an SD7003 airfoil

To comprehensively understand the effects of Kelvin–Helmholtz instabilities on a transitional separation bubble on the suction side of an airfoil regarding as to flapping of the bubble and its impact on the airfoil performance, the temporal and spatial structure of the vortices occurring at the downstream end of the separation bubble is investigated. Since the bubble variation leads to a change of the pressure distribution, the investigation of the instantaneous velocity field is essential to understand the details of the overall airfoil performance. This vortex formation in the reattachment region on the upper surface of an SD7003 airfoil is analyzed in detail at different angles of attack. At a Reynolds number Re _c < 100,000 the laminar boundary layer separates at angles of attack >4°. Due to transition processes, turbulent reattachment of the separated shear layer occurs enclosing a locally confined recirculation region. To identify the location of the separation bubble and to describe the dynamics of the reattachment, a time-resolved PIV measurement in a single light-sheet is performed. To elucidate the spatial structure of the flow patterns in the reattachment region in time and space, a stereo scanning PIV set-up is applied. The flow field is recorded in at least ten successive light-sheet planes with two high-speed cameras enclosing a viewing angle of 65° to detect all three velocity components within a light-sheet leading to a time-resolved volumetric measurement due to a high scanning speed. The measurements evidence the development of quasi-periodic vortex structures. The temporal dynamics of the vortex roll-up, initialized by the Kelvin–Helmholtz (KH) instability, is shown as well as the spatial development of the vortex roll-up process. Based on these measurements a model for the evolving vortex structure consisting of the formation of c-shape vortices and their transformation into screwdriver vortices is introduced.     

Scanning PIV investigation of the laminar separation bubble on a SD7003 airfoil

A laminar separation bubble occurs on the suction side of the SD7003 airfoil at an angle of attack α =  4–8° and a low Reynolds number less than 100,000, which brings about a significant adverse aerodynamic effect. The spatial and temporal structure of the laminar separation bubble was studied using the scanning PIV method at α =  4° and Re = 60,000 and 20,000. Of particular interest are the dynamic vortex behavior in transition process and the subsequent vortex evolution in the turbulent boundary layer. The flow was continuously sampled in a stack of parallel illuminated planes from two orthogonal views with a frequency of hundreds Hz, and PIV cross-correlation was performed to obtain the 2D velocity field in each plane. Results of both the single-sliced and the volumetric presentations of the laminar separation bubble reveal vortex shedding in transition near the reattachment region at Re = 60,000. In a relatively long distance vortices characterized by paired wall-normal vorticity packets retain their identities in the reattached turbulent boundary layer, though vortices interact through tearing, stretching and tilting. Compared with the restricted LSB at Re = 60,000, the flow at Re = 20,000 presents an earlier separation and a significantly increased reversed flow region followed by “huge” vortical structures.