on the unsteady motion and stability of a heaving airfoil in ground effect

Acta Mech. Sin. (2011) 27(2):164–178DOI 10.1007/s10409-011-0445-9

RESEARCH PAPER

On the unsteady motion and stability of a heaving airfoil in ground effect

Juan Molina · Xin Zhang · David Angland

Received: 7 September 2010 / Revised: 8 November 2010 / Accepted: 26 January 2011©The Chinese Society of Theoretical and Applied Mechanics and Springer-Verlag Berlin Heidelberg 2011

Abstract This study explores the fluid mechanics and forcegeneration capabilities of an inverted heaving airfoil placedclose to a moving ground using a URANS solver with theSpalart-Allmaras turbulence model. By varying the meanground clearance and motion frequency of the airfoil, it waspossible to construct a frequency-height diagram of the var-ious forces acting on the airfoil. The ground was found toenhance the downforce and reduce the drag with respect tofreestream. The unsteady motion induces hysteresis in theforces’ behaviour. At moderate ground clearance, the hys-teresis increases with frequency and the airfoil loses energyto the flow, resulting in a stabilizing motion. By analogy witha pitching motion, the airfoil stalls in close proximity to theground. At low frequencies, the motion is unstable and couldlead to stall flutter. A stall flutter analysis was undertaken. Athigher frequencies, inviscid effects overcome the large sepa-ration and the motion becomes stable. Forced trailing edgevortex shedding appears at high frequencies. The sheddingmechanism seems to be independent of ground proximity.However, the wake is altered at low heights as a result of aninteraction between the vortices and the ground.

Keywords Oscillating airfoil · Flutter ·Wing in ground ef-fect · Vortex shedding

J. Molina · Xin Zhang (�) · D. AnglandSchool of Engineering Sciences,University of Southampton,Southampton, SO17 1BJ, UKe-mail: [email protected]

Nomenclature

a heaving amplitudec airfoil chordCD drag coefficientCL downforce coefficienth airfoil ride heighth non-dimensional airfoil displacement, 2(h − h0)/ah0 mean ride heightf heaving frequencyk reduced frequency, π f c/U∞L downforceMa Mach numberRe Reynolds number based on wing chord c, ρU∞c/μT heaving period, f −1

t timeU∞ freestream velocityW work done by the fluid on the airfoilx, y cartesian coordinates, x positive downstream,

y positive upy+ non-dimensional normal wall distanceα airfoil incidenceν kinematic viscosityνT turbulent viscosityρ densityτ non-dimensional time, t/Tω vorticity, ∂v/∂x − ∂u/∂yξ aerodynamic damping

1 Introduction

There appears to be a general understanding of the influenceof the most important parameters that define the oscillatingmotion of an airfoil on the forces that it sustains, namely theamplitude and frequency and combinations of these two, likethe Strouhal number or the effective incidence. It has been

On the unsteady motion and stability of a heaving airfoil in ground effect 165

possible to not only explain the flying motion of birds andinsects but also to emulate those mechanisms. On a differentscale to modern aircrafts, micro-air vehicles (MAVs) applythe basics of insect flight to low Reynolds man-made flyingvehicles [1–3]. By flapping the wings, MAVs are able to gen-erate additional amounts of thrust and enhance the lift withrespect to a fixed wing configuration.

In this work, we undertake the study of a heaving air-foil in close proximity to a ground plane at high Reynoldsnumbers (Re = 3.9 × 105). The effect of the ground on theaerodynamic performance of a stationary inverted airfoil hasalready been investigated in detail [4–6]. The heaving mo-tion represents the suspension movement of a racing car. Thedynamic motion might lead to a change in lift curve with rideheight in a similar way that a dynamically pitching airfoilwith a variable angle of attack results in delayed stall andhigher peak lift values (see Carr et al. [7], Sheng et al. [8]).In fact, Zerihan [9] found an analogy between the effects ofground proximity and angle of attack on an inverted airfoil,with increasing lift up to a point of stall due to separation.A proper understanding of the unsteady aerodynamic forcescould have implications on high Reynolds number applica-tions. Indeed, the use of an inverted airfoil makes this studymore appealing to race cars. Nevertheless, it is interesting toanalyse if the results for low Reynolds number applicationscan be extrapolated to larger Reynolds. As was shown in aprevious paper by Molina and Zhang [10], the behaviour athigh reduced frequencies is mainly inviscid, so low Reynoldsnumber studies might be relevant for this case. On a similarnote, Ohmi et al. [11] found that the general wake structureof an oscillating airfoil is insensitive to the Reynolds num-ber, while McCroskey [12] concluded that, the freestreamvelocity being included in the non-dimensional frequenciesand time parameters, Reynolds effects are apparent only inthe stationary results.

Another approach to this problem might be taken onthe basis of the aerodynamics of flapping wings. The per-formance of flapping airfoils is well documented in the lit-erature, but there is still a gap in this area which we hope tocover in this paper. Flying close to the ground can be ben-eficial to MAVs, since some insects and birds seem to takeadvantage of the ground effect by flying low. The conse-quences of this behaviour need to be quantified in order topredict the performance of MAVs in ground proximity.

Previous attempts to analyze the effect of the ground onan oscillating wing have proposed analytical models underrather restrictive assumptions and which fall short of cap-turing the real physics [13,14]. Moryossef and Levy [15]conducted a numerical study on an inverted airfoil in groundeffect. However, their work did not include ride heights be-low static stall.

One of the features of oscillating bodies which is per-haps of most interest to researchers is vortex shedding in thewake. There are mainly two types of vortex shedding thatcan be present in unsteady flows, both of them prompted by

the same underlying physical process: forced shedding andnatural shedding.

The physical mechanisms that triggers forced vortexshedding is related to the weak starting vortex that is gen-erated during the transient initial period when an airfoil ac-celerates from zero to a certain velocity. From basic aero-dynamics, it is well known that the effect of viscosity is tomove the rear stagnation point to the trailing edge, gener-ating an anticlockwise circulation around the airfoil (for adownforce-producing wing). This circulation accelerates theflow along the suction surface. Since the total circulation inthe whole domain must remain constant, a vortex of oppositesign is shed from the trailing edge. As well as a change in ve-locity, an abrupt change in the effective angle of attack resultsin a starting vortex being formed. In the case of a heavingairfoil, the abrupt change occurs when the airfoil reaches themean position and the angle of attack is at its maximum (up-stroke) or minimum (downstroke). The change can only beconsidered abrupt at high frequencies. Consequently, start-ing from the mean position and plunging down, the angle ofattack is negative and changes to positive as the airfoil re-verses. When the mean position is approached from below,the airfoil sees the maximum angle of attack. During thistime, clockwise vorticity has been accumulating into a vor-tex as the incidence increased. The vortex is shed around themean position and anticlockwise vorticity starts to accumu-late in the same manner. Two vortices of opposite sign areshed per period.

Natural shedding is typical of bluff bodies like cylin-ders, but it can also occurs on airfoils with sharp edges, inwhich the flow separates upstream of the trailing edge andan effective blunt trailing edge is created. This shedding isalso related to the starting vortex but, as opposed to forcedshedding, a vertical motion is not necessary. Since there isno longer a sharp trailing edge, the stagnation point keepsmoving around the trailing edge until the flow stalls and thestagnation point moves in the other direction. This process isknown as stall-and-reversal and produces two alternate rowsof shed vortices. The body will have a natural frequency as-sociated to this shedding process. While forced shedding isan inviscid phenomenon, natural shedding is caused by vis-cous effects. The transition to turbulence may lead to thesuppression of laminar separation bubbles towards the trail-ing edge and to the cessation of natural shedding [16].

Depending on the configuration taken by the vorticesin the wake, the airfoil will generate either drag, thrust or abalance of both [17,18]. Natural shedding induces drag, asis common in stationary bodies. Usually only natural shed-ding is present at low frequencies and small amplitudes ofoscillation. At higher frequencies or large amplitudes, Youngand Lai [16] showed that an interaction between natural andforced shedding can occur and the wake can take a compli-cated form with more than two vortices shed per cycle, asobserved by Lai and Platzer [19]. Further increasing thefrequency, a phenomenon known as vortex lock-in appears

166 J. Molina, et al.

and the wake oscillates at the flapping frequency [16]. Theswitch from drag to thrust production is not related to the pre-vious interaction, so it is possible to still generate drag at thelock-in state. In fact, it was von Karman and Burgess [20]who first showed that the vortices could be arranged into areverse vortex sheet as to generate thrust, yet prior to thatKnoller [21] and Betz [22] showed in separate studies thatan oscillating airfoil generates thrust due to the lift vectortilting forward.

The effect of the ground on these wake structures is un-clear. In stationary bluff bodies, there is a critical gap be-tween the body and the ground below which vortex sheddingis suppressed [23]. While a similar effect can be expected ofairfoils, the change in ride height with time will undoubtedlyintroduce an additional constraint to the flow.

Finally, the subject of stability and flutter should alsobe considered. Recalling the similarity between incidenceand ground proximity, the concept of dynamic stall couldalso be applicable to a heaving airfoil in ground effect. Pre-vious works in this area have focused on the hysteresis dueto high incidence angles [7,12,24–26]. Since static stall oc-curs at low ride heights, we surmise that dynamic stall andstall flutter could occur for an airfoil in ground effect due tothe plunge degree of freedom. A similar approach to flutteron a pitching airfoil is undertaken in this study for a heavingairfoil in ground effect.

2 Case definition

The airfoil used in this study and sketched in Fig. 1 is an in-verted single element also used by Molina and Zhang [10],with a finite trailing edge of 0.01c. The incidence α wasfixed to 5◦, such that the stationary behaviour in ground ef-fect, including separation onset and downforce magnitude,is consistent with previous studies on wings in ground ef-fect [5,27]. Positive incidence is taken as the airfoil pitchingdown. The ride height h is defined as the distance from thelowest point on the suction surface to the ground plane. Theground plane moved at the same velocity as the freestreamflow.

Fig. 1 Sketch of the computational domain and details of the air-foil near the ground plane, showing the definition of airfoil chord c,amplitude a, ride height h, angle of attack α and freestream velocityU∞

The axes were orientated such that x is aligned withthe incoming flow and positive downstream, and y is nor-mal to the ground and positive upwards. For nomenclaturepurposes downforce, or negative lift, is considered positivewhen pointing to the ground.

3 Numerical method

The flow around the airfoil was resolved by means of atwo-dimensional (2D), unsteady Reynolds-averaged Navier–Stokes (URANS), incompressible solver. The equationswere discretized using second-order accuracy in space andfirst-order implicit time discretization, with 30 sub-iterationsper time step. The use of first-order for the unsteady termswas justified by Molina and Zhang [10]. More details on thenumerical model can also be found in that reference. Thenon-dimensional time step ranged from Δτ = 1 × 10−3 fork = 4.37 to Δτ = 1 × 10−4 for k = 0.005.

The Mach number was Ma = 0.12 and the Reynoldsnumber based on the chord was Re = 3.9 × 105. The flowwas forced to be turbulent in the entire domain and the turbu-lence was modelled with the one-equation Spalart-Allmaras(S-A) turbulence model [28]. A fully turbulent field repro-duces real conditions on race cars and removes the need toknow the transition point a priori.

The 2D domain consists of four boundaries as shownin Fig. 1. The inlet was located on the left hand side of thedomain, upstream of the airfoil. A freestream velocity in thex direction and a turbulent viscosity ratio of νT/ν = 68 wereimposed on the inlet boundary. A pressure outlet was placeddownstream of the airfoil with the same turbulent viscosityratio as the inlet. The ground plane was placed below theairfoil and extended upstream to the inlet and downstream tothe outlet. To recreate the real racing conditions, a horizontalvelocity equal to freestream was prescribed on the ground.The top wall did not influence the flow around the airfoil soa symmetry condition was enforced. Finally, the airfoil wasmodeled as a wall with a no-slip condition. The flow fieldinside the domain was initiated with uniform freestream con-ditions, equal to the velocity inlet, to allow the unsteadinessto develop from the onset of the computation.

The suitability of the method to model ground effectwas extensively proven. For instance, Zerihan [9] success-fully compared URANS simulations of an inverted airfoilin ground effect using the S-A model to real experimentsat Re = 4.6 × 105, while Nishino et al. [29] showed thatS-A is able to qualitatively capture the real flow physicsof a cylinder in ground effect, including the wake and theflow between airfoil and ground. While the previous sim-ulations were only validated for stationary bodies, the suc-cessful comparison to experiments can still hold at low fre-quencies under the quasi-stationary hypothesis, as will beshown during the analysis of the results. In the high fre-


quency range, the characteristic flow is mainly inviscid andthe turbulence model is of secondary importance. This effectwill also be shown in more detail later.

3.1 Heaving motion

The oscillating motion is achieved by means of moving theairfoil vertically. The heaving motion is sinusoidal, with am-plitude a and frequency f . At the start of each period, theairfoil is at the mean position h = h0. Thus, the ride height isgiven by

h(t) = h0 − a2

sin(2π f t). (1)

A non-dimensional form of the vertical displacement is

h(τ) = − sin(2πτ), (2)

with τ representing the non-dimensional time. h varies fromhmin = −1 to hmax = 1.

The amplitude of the heaving motion is a/c = 0.08,leaving only two possible variables to define the motion. Bysweeping through different mean positions and frequencies,a map of results was elaborated. Instead of using the fre-quency, the results are presented as a function of the reducedfrequency, defined as

k =π f cU∞. (3)

The reduced frequency represents the ratio between thetime a particle takes to traverse the chord of the airfoil, c/U∞,and the period of oscillation, 1/ f .

3.2 Grid structure

A 2D hybrid grid was created around the airfoil (Fig. 2). Byusing a hybrid grid, the advantages of both structured and un-structured elements could be exploited. Structured elementswere used around the airfoil and ground, since accuracy isimproved with respect to the unstructured elements. Themesh was wrapped around the airfoil, forming a C-shapedstructure around the airfoil and its boundary layer block.Common features like gradients, separation, vortex shed-ding and wake are better captured with a structured mesh.However, unstructured meshes are suitable for deforming do-mains, so at a large enough distance from the airfoil as tonot make a difference on the results, unstructured elementswere used. With these considerations in mind, the domainwas split into three horizontal blocks extending from inlet tooutlet: an upper unstructured block far away from the air-foil; a middle block split into a structured region containingthe airfoil and extending 0.72c above it plus an unstructuredregion above it; and a lower structured block that enclosesthe boundary layer built up on the ground. The higher-than-freestream velocity in the region between airfoil and groundresults in a boundary layer forming on the ground, so it isimportant to refine the mesh to capture the boundary layerin this region. To avoid distortion of the cells around the

airfoil, the middle block moves as a rigid structure with theairfoil. The elements in the upper block deform like springs,although the deformation is relatively small compared to thesize of the elements. In the interface between the middleand lower blocks, a method known as dynamic layering wasused. The idea is to create or destroy the layer of elements onthe boundary as the middle block moves upwards or down-wards respectively. To create a new layer, the layer of cellsadjacent to the moving boundary, which is expanding, splitsinto two new layers once it has reached a certain maximumheight. To remove a layer of cells which is compressing, thatone is merged with the following layer once the original layerhas reached a certain minimum height. Cells are expandedduring the upstroke and compressed during the downstroke.

Fig. 2 Illustration of the hybrid mesh, with details of the blockstructure

An independence study was conducted to determine theappropriate dimensions for the domain. The symmetry planewas placed 10.5c above the airfoil. The grid extended 10cupstream of the leading edge of the airfoil onto the veloc-ity inlet and 10c downstream of the trailing edge onto thepressure outlet. The initial distance from the ground to theairfoil depended on the mean ride height and varied throughthe cycle.

A mesh refinement study was undertaken under bothstationary and heaving conditions. Results presented in thisstudy are based on a 1.51×105-point grid, for which conver-gence of the force coefficients was achieved and the wakeregion was sufficiently refined to capture the unsteady phe-nomena. The airfoil surface contains 505 points and theboundary layer block 30 points, the initial cell spacing nor-mal to the surface being 1×10−4c. This gives a y+ ≈ 1. Max-imum skewness was limited to an equi-angle (Computed asmax[(Qmax − 90)/90, (90 − Qmin)/90], where Qmax and Qmin

are the largest and smallest angle (in degrees) in the cell, re-spectively. A value lower than 0.8 is generally consideredacceptable.) of 0.55, while the average was 0.08.

Small changes were required to simulate the airfoil infreestream. The domain was extended vertically by addingunstructured elements under the lower block down to 10.5cbelow the airfoil. The new wall was set as a symmetry plane.


4 Results and analysis

4.1 Stationary airfoil

Before engaging in the oscillatory case analysis, the resultsfor a stationary airfoil at Re = 3.9 × 105 are discussed. Thevariation of downforce coefficient with ride height is pre-sented in Fig. 3a. These results are in qualitative agreementwith Zerihan and Zhang’s experiments for an inverted wingin ground effect [5]. Downforce in freestream is CL = 0.98.As the ground starts to modify the flowfield around the air-foil at high ride heights, downforce increases with respect tofreestream. If the ride height decreases, downforce initiallyincreases linearly. However, the airfoil is still at a relativelylarge distance from the ground and the forces are relativelysmall. The airfoil is in the force enhancement region. Theincrease of downforce in this region is due to the flow beingaccelerated in the channel formed between the airfoil and theground. This results in greater suction on the lower surface.For even lower ride heights, the rate of increase of downforceis larger. However, as the suction peak becomes larger, the

adverse pressure gradient is also stronger downstream of itsince the pressure has to recover before reaching the trailingedge. The onset of separation on the suction surface resultsin the slope of the curve decreasing in the force enhance-ment slowdown region. When the gain in downforce due toincreased suction is overcome by the large separation, down-force reaches a maximum of CL = 1.68 at h/c = 0.17, theairfoil stalls and downforce starts to decrease. The maximumrepresents an increase of 71% with respect to freestream. Forh/c < 0.17, the airfoil is in the force reduction region. Thesimilarity between decreasing ride height and increasing an-gle of attack becomes obvious: in both cases the force in-creases until boundary layer separation leads to stall.

In the case of Fig. 3b, the drag coefficient increases ex-ponentially with reduced ground clearance. As lift increases,drag also rises, from CD = 0.022 in freestream to CD = 0.053at the ride height of maximum downforce. Once the flowseparates from the suction surface, the component of pres-sure drag escalates to large values. For the lowest ride heightof h/c = 0.12 tested in this study, CD = 0.073, which is 232%higher than in freestream.

Fig. 3 Variation of force coefficients with ride height for a stationary airfoil. a Lift coefficient; b Drag coefficient

4.2 Forces on an oscillating airfoil

4.2.1 Methodology

The study of the heaving airfoil was performed at four dif-ferent mean ride heights. These cover the main phenomenaobserved on a stationary airfoil, namely: out of ground ef-fect, weak ground effect, strong ground effect with force en-hancement and separation and eventual stall. It is believedthat the analysis of these ride heights is enough to charac-terize the whole range of unsteady forces. The rest is inter-polated based on the justified premise that the variation issmooth. Ride heights lower than those studied for the sta-tionary airfoil were not addressed so the interpolation is cutat h0/c = 0.16.

For each region, a total of 17 frequencies were tested,

ranging from k = 0.005 to k = 4.37. The distribution wasnot uniform but rather clustered towards the low frequencieswhere the regime changes occur [10]. That gave a total of68 points, which was considered adequate for the purpose ofthis study. The results for each variable can be plotted in ak–h0 contour map as a function of the frequency k and themean ride height h0/c. We opted to use the inverse of themean ride height, c/h0, since this allows the inclusion of thefreestream case under the assumption that h0 → ∞. Hence,c/h0 = 0 when out of ground effect.

Each simulation was run until a periodic solution wasobtained. While the transient lasted for less than a cycle,the stabilization of the forces by means of the characteristicvalues (i.e. the maximum, the minimum and the average) re-quired a larger number of oscillating periods. The stabiliza-tion depends on the reduced frequency, such that a greater


number of cycles is required as the frequency increases. Theexplanation for this follows from the definition of reducedfrequency: it is the ratio between the time a particle takes totravel the length of the airfoil and the period of oscillation.The higher this ratio, the larger the number of cycles for theflow to adapt.

Once the simulation becomes fully periodic, the prob-lem reduces to studying one cycle of oscillation. The fol-lowing section deals with the most relevant forces extractedfrom the data, specifically the average force and the peakforces in a period. In addition to a breakdown of the forcesat each point, the force distribution along the cycle is alsoconsidered.

4.2.2 Average forces

Figure 4a presents a contour map of the average downforcecoefficient against the two independent parameters. It canbe observed that the average downforce is consistent withthe stationary case, in that it increases with ground proxim-ity at all frequencies. However, that variation depends onthe frequency range. At low frequencies the variation withground proximity is small with respect to high frequencies,in which the downforce can be almost tripled by placing theairfoil very close to the ground. So, while being close to theground is beneficial at high frequencies, it does not seem tobe the optimal strategy at low frequencies. A larger gain isobtained at high ride heights, c/h0 < 4, than at c/h0 > 4, inwhich downforce barely increases. This plateau is a directconsequence of the quasi-stationary behaviour of the airfoilat low frequencies. In quasi-stationary motion (Not to beconfused with quasi-steady motion, which assumes that theairfoil moves with constant heave velocity at each instant.Quasi-stationary motion further assumes that the vertical ve-locity is zero.), the forces at each ride height are analogousto those of a stationary airfoil at the same ride height. There-fore, once the airfoil enters the force reduction region for asufficiently large amount of time during the cycle, the aver-age downforce will not increase as in the force enhancementregion. On the other hand, at high frequencies the quasi-stationary hypothesis is not valid anymore and the force re-duction region seems to disappear, resulting in a continuousforce enhancement region with no stall. The airfoil behavesas if the flow were inviscid [10]. This does not mean thatthere is no separation, but that viscous effects do not play arole in the aerodynamic performance. The forces are dom-inated by inviscid phenomena, particularly added mass ef-fects [10]. Added mass forces appear when a body is ac-celerating and are usually negligible in fluids of low rela-tive density, except for large accelerations as is the case athigh frequencies of oscillation. The almost-diagonal iso-lines across the lower right side of the map show that in-creasing either c/h0 or k is beneficial, whereas the horizontaliso-lines along the left side establish that frequency variationfor k ≤ 1.09 does not have a significant effect. The quasi-stationary assumption holds true. Moreover, in freestream

downforce barely changes with frequency. This is arguablythe most noticeable aspect of ground effect: forces are mag-nified. Remarkably, the effect of the ground is much largerthan the effect of geometry and incidence. The airfoil is notsymmetric and the angle of attack is set to α = 5◦, so it is op-timized to generate positive downforce in freestream, but theadvantage of using this setup is negligible in comparison tothe benefit of heaving close to the ground at any frequency.

Likewise, the effect of frequency and mean ride heighton the average drag is plotted in Fig. 4b. A decrease in dragwith frequency through all the ride height range is observed.This phenomenon is well-known in freestream conditionsand is the primary advantage of flapping flight. If the wingsflaps strongly enough, by sufficiently increasing frequencyor amplitude, negative drag is produced, which is equivalentto generating thrust. Positive thrust will push the object for-ward without the need to use a power unit. In this case thrustis generated for k ≥ 2.31 in freestream. Interestingly, thisproperty is retained in ground effect in spite of the onset ofseparation close to the ground. However, the switch fromdrag to thrust generation moves to higher frequencies as themean distance to the ground decreases. Zero drag occursalong the line

k = 0.028(c/h)2 − 0.025(c/h) + 2.313. (4)

Looking at the low frequencies, it is possible to discernthe quasi-stationary behaviour again. As in fully stationarysimulations, drag increases indefinitely with ground proxim-ity. This pattern remains until high frequencies. Even whenthe regime changes from ground-effect-dominated at low fre-quencies to incidence-effect-dominated at medium frequen-cies [10], the adverse pressure gradient caused by the ef-fective angle of attack is reinforced by the presence of theground, promoting separation and increasing drag. It is af-ter the switch to thrust production that we start to noticea change in trend. This is manifested by a change in theslope of the iso-contours. Now the airfoil tends to gener-ate more thrust (less drag) at moderate ground distances thanin freestream. Moryossef and Levy [15] observed the samephenomenon but no mention was made as to why it hap-pened. As the frequency is increased even further the max-imum average thrust moves to lower mean ride heights, asillustrated by the white line in Fig. 4b. In the range of fre-quencies covered in this study, this maximum does not reachthe lowest ride height, but it is expected that for even higherfrequencies the maximum average thrust will be obtained atminimum h/c. The Kriging interpolation used in processingthe data does not seem to capture this pattern very well closeto the bottom right boundaries, but it is clear from the sourcedata that the rate of increase of thrust with frequency is largeras the ground is approached (not shown here). The immedi-ate conclusion is that the proximity of the ground is detri-mental at low frequencies due to viscous effects, but it be-comes advantageous once thrust is produced. Furthermore,the higher the frequency the closer to the ground the airfoilcan be placed for an improved performance.


Fig. 4 Contours of the average downforce and drag as a function of mean ride height h/c and reduced frequency k. a CL; b CD

4.2.3 Peak forces

From the periodicity of the force history it is possible to ex-tract a dominant frequency which, for each case, correspondsto the frequency of oscillation. This implies that there willbe a maximum and a minimum force in each period, whichgives an idea of the amplitude and safety limits in terms ofstructural requirements.

As shown in Fig. 5a, the maximum downforce gener-ated by the airfoil during a period increasingly grows withreduced frequency. The increase in maximum downforce ismainly due to an increase in peak effective incidence andto larger added mass forces at high frequencies. As the peakvertical velocity increases with frequency, the airfoil sees theincoming flow at a larger angle and this induces larger circu-lation around the airfoil. Just as the average downforce, themaximum downforce increases with ground proximity fora constant frequency. This is however not completely truefor the low, quasi-stationary frequencies. In those cases, themaximum downforce, roughly given by the stationary equiv-alent, stops increasing with ground proximity once the airfoilpasses through the ride height of maximum stationary down-force (h/c = 0.17). Thus, for h0/c ≤ 0.21 (c/h0 ≥ 4.76)and in the range of ride heights studied, the maximum down-force is fairly constant and located at h/c = 0.17 (see Molinaand Zhang [10] for a more in-depth explanation of where themaximum downforce is located). At higher frequencies, thecombined effect of oscillation and ground vicinity is to am-plify the maximum downforce, first by an increased effec-tive incidence and later by added mass effects. Notably, atmedium frequencies the effect of separation due to incidencenear the ground (which reduces the incidence at which max-

imum downforce is achieved with respect to the high-ride-heights, no-separation regions where maximum downforceoccurs at the location of maximum incidence) does not seemto cause a drop in the contour map. It is therefore possibleto obtain more downforce at a low effective incidence closeto the ground than at a high effective incidence farther awayfrom the ground. At very high frequencies, added mass ef-fects take over. It is remarkable that, although the averagedownforce does not reach values higher than CL = 3.24 inthe range of frequencies studied (k ≤ 4.37), the maximumdownforce can reach values up to CL,max = 10.35. Even infreestream, CL,max = 5.94. The peak forces are also highin freestream, but it is the lack of ground proximity thatcauses the average forces to be very low. The ground in-duces an additional asymmetry besides the geometrical onewhich is beneficial for an inverted airfoil. Indeed, the pres-ence of a nearby solid boundary has been shown to increasethe added mass due to the narrow gap between the airfoiland the wall [30]. Although it is not expected that a race carwing could reach these high frequencies of oscillation, thehigh amount of force could lead to serious structural prob-lems.

Regarding the minimum downforce, Fig. 5b presentsthe corresponding contour map. In reciprocating manner tothe maximum downforce, the minimum downforce decayswith increasing frequency, resulting in a larger peak-to-peakamplitude. This is a direct consequence of the symmetricmotion. However, absolute values are smaller than for themaximum downforce, basically due to the aforementionedasymmetry (i.e. geometry and ground presence where ap-plicable). At low and high frequencies the same principleapplies: more downforce is generated close to the ground. In


the range of medium frequencies (0.34 ≤ k ≤ 1.71), thereis a drop in downforce if the airfoil enters the force reduc-tion region at some point during the oscillation, when com-pared to higher ride heights. This sudden drop could resultin the car losing excessive grip if an instability causes thewing to enter the force reduction region. At high frequen-cies, the minimum downforce becomes negative, pushing theairfoil upwards. This effect does not seem to be related tothe ground proximity as the isolines are markedly vertical,as opposed to the maximum downforce, revealing that the

minimum downforce is reasonably constant with ride heightchanges at high frequencies. The capability of the airfoil togenerate lift due to added mass (which occurs near the topposition) is given by the pressure region created on the bot-tom surface, which is relatively independent of ground prox-imity, and by the suction region created on top of the airfoil,which is not influenced by any boundaries above the airfoil.Obviously this is not the case when downforce is generated,since the development of the suction region is determined bythe confinement of the flow between airfoil and ground.

Fig. 5 Contours of the maximum and minimum downforce as a function of mean ride height h/c and reduced frequency k.a CL,max; b CL,min

On a last note, it is easy to understand how birds canstay in the air, as their wings can generate huge amounts oflift. Properly adjusting the incidence by pitching with theright phase, they can also reduce the negative lift appearingduring part of the stroke.

In accordance with the stationary case, the maximumdrag increases with ground proximity, as well as with re-duced frequency, as depicted in Fig. 6a. The rate of in-crease of maximum drag increases with decreasing meanride height. Furthermore, the maximum drag grows in a pro-gressive manner with frequency, increasing by an order ofmagnitude from low to high frequencies.

As seen before, the average drag decreases with fre-quency, so this increase in the maximum drag has to be bal-anced by a reduction in the minimum drag, as shown in Fig.6b. In the quasi-stationary regime, the minimum drag in-creases as the ground is approached. This is once again con-sistent with the stationary results. But, in a similar man-

ner to the average drag, the minimum drag becomes higherwith increasing frequency away from the ground. As soon asthrust is produced at some point in the cycle, the minimumdrag is no longer lower in freestream than in ground effect(shown by the white line, which traces the minimum value ateach frequency). The switch from drag to thrust occurs veryearly, at 0.34 ≤ k ≤ 0.60 depending on the mean ride height.As the frequency is increased inside the thrust regime, theminimum moves to lower ride heights, until thrust increasesindefinitely with ground proximity for k ≥ 2.51. Also, it canbe noticed that the rate of increase of thrust with frequencyincreases not only with frequency but also with ground prox-imity (i.e. the isolines get closer). Hence, the effect of theground on drag production is two-fold: on the one hand itincreases the maximum drag (Fig. 6a), and on the other handit enhances the rate of decrease of minimum drag (Fig. 6b),increasing thrust above a certain frequency.


Fig. 6 Contours of the maximum and minimum drag as a function of mean ride height h/c and reduced frequency k.a CD,max; b CD,min

4.3 Stability: downforce hysteresis diagramsThe variation of downforce with ride height depends

on the direction of motion. This hysteresis effect can alsoappear in static wind tunnel tests in which the flow is notstopped between ride height changes [27]. Since the ver-tical velocity, and consequently the position, is imposed be-forehand, aeroelastic responses (e.g. flutter) are not possible.Therefore, the downforce curves are essentially identical attwo consecutive periods. However, it is viable to define theenergy transfer between the airfoil and the fluid in terms ofthe work done by the aerodynamic forces.

The work required to move the airfoil a small displace-ment dh is

dW = −Ldh, (5)

where the negative sign accounts for the fact that positivedownforce and positive displacement act in opposite direc-tions. The total work done in one period is computed byintegrating Eq. (5) over a complete period

W = −∮

Ldh. (6)

If the work is positive, then the fluid does work onthe airfoil, and the airfoil is gaining energy. In aeroelasticterms, this would mean that the amplitude of the oscillationincreases (negative damping), unless that energy were lostby other means (e.g. friction). If it is negative, work is doneby the airfoil on the fluid, with the former losing energy. Ifthe motion of the wing were not forced, then the amplitude ofmotion would tend to decay (positive damping). In this studywe attempt to estimate the work done to gain a rough predic-tion of the aeroelastic behaviour of an airfoil in ground effect.

Obviously this is far from a legitimate aeroelastic analysis,since in reality the structural and aerodynamic systems arecoupled and it would be too naive to try to solve the aerody-namic side alone. The change in downforce with ride heightresults in a change in acceleration which is omitted here, onthe grounds that the motion is fixed by Eq. (1).

In the case of a plunging airfoil, the aerodynamic damp-ing ξ can be defined as

ξ =

∮CL

dhc. (7)

And the stability depends on the sign of ξ. If ξ > 0 thenthe motion is stable, whereas if ξ < 0 the motion is unsta-ble. The aerodynamic damping is a non-dimensional formof the work, thus both are equivalent but with opposite sign.ξ represents the energy transfer between the heaving motionand the flow. When the airfoil gains energy, the motion isunstable, ξ is negative and the work is positive, as shown be-fore. When the airfoil loses energy, the motion is stable, ξ ispositive and the work is negative.

Based on the previous definitions, the stability can bededuced directly from the hysteresis curves according to thesense of motion (Fig. 7). If the sense of motion is clockwise,the work is negative and the motion is stable. However, if thesense of motion is anticlockwise, the work is positive and themotion is unstable. Furthermore, the amount of work done isproportional to the area inside the curve. Therefore, the de-gree of hysteresis is given by the magnitude of aerodynamicdamping.

Figure 7a presents the hysteresis curve in freestream atdifferent reduced frequencies. As the frequency increases,the peak forces diverge, as reported previously. The area


inside the curve becomes larger, increasing the hysteresisbetween upstroke and downstroke. The sense of motion isalways clockwise, so that at any single ride height, down-force is larger when the airfoil is moving up than down. For

each ride height, the effective incidence is greater during theupstroke than during the downstroke, and the difference be-tween both increases with frequency as the incidence is afunction of k.

Fig. 7 Variation of downforce coefficient with vertical position a in freestream; b at h0/c = 0.25; c at h0/c = 0.16 for various reducedfrequencies

Regarding the energy transfer, for all frequencies workis done by the airfoil on the fluid, so that the motion isstable and the transfer of energy from the airfoil to thesurrounding fluid will result in the oscillatory motion be-ing damped. The aerodynamic damping increases with fre-quency, as expected from the progressively increasing areainside the curve (Fig. 8). Intuitively, it can be seen that bothincidence and added mass effects play an important role onthe stability. Assuming the original angle of attack is zero,effective incidence is positive through the upstroke and neg-ative through the downstroke. If we further assume that theairfoil is symmetric, then the sign of downforce follows thatof the incidence (e.g. positive downforce on the upstroke).Thus, the motion can only be attained if the airfoil does workto move against the force that pushes it in the opposite di-rection. The inclusion of an initial angle of attack and ge-ometrical asymmetry does not seem to perturb the stability

according to the simulations. Added mass effects becomeimportant as the frequency increases. But the effect of addedmass is to increase the effective mass of the body and shouldnot alter the stability. However, the geometrical asymmetryand ground proximity work with the added mass to improvethe stability.

A single-degree-of-freedom unstalled motion does notgenerally produce flutter, since the motion is stable infreestream. An exception would be the existence of negativespring or damping forces. At least two degrees of freedomare required to create an instability. In fact, the work doneby the airfoil on the fluid increases with frequency and theaerodynamic damping increases, as shown in Fig. 8. Typi-cally, a combined plunging and pitching motion is requiredto produce flutter, the reason being that there must be an ex-change of energy between the modes. A pure heaving mo-tion consists of only one degree of freedom, thus flutter is


very unlikely to happen if the flow remains attached. How-ever, this is not necessarily the case in ground effect. Whileat medium and high frequencies the main effects are due toincidence and added mass, respectively, at low frequenciesthe motion is quasi-stationary and both the force enhance-ment region and force reduction region prevail accordinglywith the ride height.

Fig. 8 Contours of aerodynamic damping over a period as a func-tion of mean ride height h/c and reduced frequency k

When the airfoil is placed in the force enhancementregion (e.g. h0/c = 0.25), the hysteresis curves are quali-tatively similar to freestream (Fig. 7b). The hysteresis in-creases with frequency, resulting in larger forces than fora stationary airfoil. The sense of motion is also clock-wise, so that work is done by the airfoil on the fluid. Thistransfer of energy would lead to positive damping, decreas-ing the amplitude of the oscillations if the motion were un-forced. The main difference can be found at low-mediumfrequencies, where the ground effect and separation are rel-evant. Initially the curves are geometrically inclined to-wards the left since the quasi-stationary behaviour dictatesthat downforce increases with ground proximity (Fig. 7b1).The incidence effect is nonexistent in the limit k → 0, butit appears at even the lowest frequencies, so the curves be-come a combination of the stationary ground-effect curveand the freestream curve dominated by the effective inci-dence. Therefore, downforce is larger during the upstrokethan during the downstroke, resulting in positive damping.This motion is stable, not due to the effect of the ground(which is statically unstable) but due to the effect of inci-dence on the unsteady motion. As the frequency increases

and the incidence effect becomes preponderant, the curvesdo not adopt a symmetric shape like that shown in Fig. 7a1,the reason being the boundary layer separation at high inci-dences (i.e. near the mean position) boosted by the groundand the contribution of the ground effect at such low rideheights as well. At h0/c = 0.43, where the contribution ofthe incidence effect is the only important factor at mediumfrequencies [10], the curves tend towards a more symmetricconfiguration (not shown here).

Also, the amount of work done by the airfoil at highfrequencies is larger than in freestream, so the aerodynamicdamping as well (Fig. 8), essentially due to the larger peak-to-peak force amplitude. This is mainly a result of the largeincrease in maximum downforce when in ground effect.

But, undoubtedly, the most interesting case is that ofan airfoil operating in both the force enhancement and forcereduction regions. It has already been noted that the forcereduction region disappears at medium-high frequencies andthe airfoil behaves as in the previous cases. However, a dif-ferent pattern can be discerned at low frequencies, as seenin Fig. 7c. First of all, at very low frequencies the curvesfollow the stationary curve, with an initial increase and asubsequent drop with ground proximity. The sense of mo-tion is now anticlockwise. Secondly, it is noticeable that theforce reduction region is pushed to lower ride heights as thefrequency increases, so that a larger part of the downstrokeoccurs inside the force enhancement region. As a direct con-sequence, there is no longer a maximum in the upstroke,but a new minimum appears shortly after initiating the up-stroke. Downforce then increases continuously throughoutthe upstroke. This phenomenon can be observed clearly fork = 0.11. By delaying the onset of the force reduction re-gion, the airfoil is under increasing downforce for most ofthe stroke. On the other hand, the drop in downforce whenin the force reduction region is more severe. For higher fre-quencies, the force reduction region disappears completelyand the trend resembles the previous case: increasing down-force throughout the downstroke and decreasing downforcethroughout the upstroke. Nevertheless, the sense of motionis still anticlockwise. At k = 0.27, the upstroke and down-stroke branches overlap at h = 0.2, such that the upstrokebranch generates more downforce for higher ride heights.The curve reverses completely at k = 0.34, the clockwisemotion restored as seen before.

This hysteresis effect was also apparent in stationarysimulations of a double-element wing in ground effect whengoing through the force reduction region [27]. The down-force for decreasing ride heights was larger than for increas-ing ride heights. This was attributed to a residual separationregion on the flap surface at increasing ride heights.

An anticlockwise sense of motion implies that energyis transferred from the fluid to the airfoil, making the motionunstable. It is therefore possible for a plunging airfoil to havenegative aerodynamic damping. Evidence suggests that thisdepends on the extent to which the airfoil is in the force re-


duction region (Fig. 8). An airfoil outside the force reductionregion will always be stable, but as soon as it breaks into it(i.e. h0/c ≤ 0.21), instabilities will occur at low frequencies.The range of frequencies at which the motion is unstable thenincreases with ground proximity. The unstable region is lo-cated at the bottom left corner of the damping map, at lowfrequencies and low mean ride heights. It remains to be seenhow the map would look at even lower ride heights, when theairfoil is in the force reduction region for the whole cycle.

This type of flutter belongs to the non-linear aeroelasticresponse family and, as such, it can be considered as stallflutter. Stall flutter can appear on one-degree-of-freedommotions due to the continuously separating and reattachingflow [24,31]. As its name suggests, it is closely related todynamic stall. Stall flutter applied to pitching airfoils showsthat flutter will occur once the airfoil goes in and out of stall,with large areas of separation and reattachment. In this case,the airfoil enters the region of static stall for most of the cy-cle, and it was shown in Fig. 7c that this region remains atlow frequencies.

Finally, it should be noted that, at very high frequen-cies, the amount of work required for the airfoil to move islarger as the mean ride height decreases. That makes the mo-tion more difficult to materialize from an aeroelastic point ofview (i.e. the oscillations would be damped by the structuralrestore forces more quickly).

4.4 Wake structure of an oscillating airfoil

The instantaneous vorticity magnitude contour plot can notalways provide a clear picture of the vortex shedding pro-cess. However, in this case the use of the Q-criterion [32] toidentify vortex cores yields similar results. The Q-criterionconsists in the identification of regions with positive secondinvariant of the velocity gradient. In 2D, this is equivalentto the more advanced λ2 method [33]. A positive Q valuemeans that the local rotational effects dominate over the de-formation, so it is safe to assume that the vortex cores com-prise the region of Q > 0.

This particular airfoil does not produce natural shed-ding. For more details refer to Molina and Zhang [10]. Atlow frequencies, the wake consists of two shear layers of op-posite sign. The positive vorticity layer grows on the lowersurface while the negative vorticity layer grows on the uppersurface of the airfoil. A few differences between freestreamand in ground effect can be seen in Fig. 9. Ground prox-imity results in an increase in thickness of the shear layers,particularly the lower one. Separation of the boundary layeron the lower surface enhances vorticity in that area. Also,the wake is deflected horizontally near the ground, as op-posed to a deflection parallel to the chord in freestream. Thiswas also shown for a stationary wing in ground effect [6].Note that the venturi channel created between the airfoil andthe ground also induces an adverse pressure gradient on theground surface, resulting in negative vorticity building up

along the ground plane.

Fig. 9 Contours of instantaneous non-dimensional vorticity atk = 0.22, τ = 0.25, h = −1.00. a Freestream; b h0/c = 0.16

This helps to explain why the rate of increase of thrustis higher in ground effect than in freestream once the switchfrom drag to thrust generation occurs (Figs. 4b and 6b). It isobvious that stronger vorticity enhances the wake, so when astationary airfoil is placed closer to the ground vorticity in-creases and the momentum deficit in the wake increases aswell, as a result of the rotational flow. This can be extrapo-lated to an oscillating wing, which can see periods of drag-generation and periods of thrust-generation during its oscil-lation. When the airfoil is generating thrust instead of drag,the momentum surplus will be higher in ground effect thanin freestream since the rotation of the flow is enhanced but inthe opposite direction than for a drag-generating wake. Forany frequency at which the airfoil in freestream generatesthrust at some ride height (Fig. 6b), the thrust generated bythe airfoil near the ground will be enhanced by the strongervorticity. It makes sense that the airfoil will tend to generatemore thrust than in freestream, but at the same time if dragis being produced at another ride height, then it will generatemore drag than in freestream. So the effect of the ground onthe horizontal force is two-fold.

The patterns in the wake change as the frequency in-creases, as observed in the instantaneous field in Fig. 10 forthree characteristic frequencies (freestream). By compari-son to different ride heights, it seems that the wake evolu-tion is independent of ground proximity, except for the afore-mentioned increase in vorticity at very low ride heights (i.e.h0/c = 0.16). The shear layers flap at the same frequency asthe motion (Fig. 10c). As the period decreases, the oscilla-


tory pattern in the wake repeats itself within a shorter length(U∞/ f ). Consequently, the vertical to horizontal span (Thewake amplitude relative to the length of the vortex sheet isf a/U∞ = ka/c, so this ratio is proportional to k, a beingconstant.) increases. This tends to break the shear layersinto discrete structures since each layer finds vorticity of op-posite sign blocking its translational path. At this stage, theQ-criterion identifies vortex structures stretched in the lon-gitudinal direction. As the switch from flapping wake tovortex shedding is gradual, it is difficult to pinpoint an ex-act switching frequency. The Q-criterion shows proper vor-tices for k ≥ 1.71, independently of the mean ride height.The arrangement of the vortices agrees with the average dragobtained in freestream. For instance, at k = 2.18, averagedrag is zero and the vortices form a neutral wake (Fig. 10b).At higher frequencies, the vortex structures are tilted down-stream, with the anticlockwise-rotating vortices accumulat-ing above the clockwise-rotating vortices (Fig. 10a). Thisconfiguration results in a jet-like time-averaged velocity pro-file in the wake. The momentum surplus translates intothrust. Note that whether the positive vorticity is shed aboveor below the negative vorticity depends on the vertical ve-locity of the airfoil relative to that of the shed vortices (andthese are assumed to convect with the freestream velocity).

Fig. 10 Contours of instantaneous non-dimensional vorticity andboundaries of the vortex cores at k = 4.37, τ = 1.00, h = 0.00 fordifferent k. a k = 4.37; b k = 2.18; c k = 0.55

The shedding process is independent of mean ride

height and the timing of vortex shedding does not change.However, drag values depend on the proximity to the ground,so those differ among different mean ride heights. Then it ispossible to find wake structures which do not correspond tothe average drag produced. The answer to this apparent con-tradiction lies in the interaction between the vortices and theground. Although the upper part of the wake is not affectedby the presence of the ground, the lower part is and the aver-age velocity profile in the wake is modified. This interactioncan be clearly seen in Fig. 11. The shear layer building up onthe lower surface of the airfoil and which will eventually rollinto a discrete vortex is of positive vorticity. A shear layer ofthe opposite sign grows on the ground surface just beneaththe airfoil. The ground layer interacts with the shed vortices,such that there is a three-way interference (Fig. 12):

(1) Anticlockwise vortex: this vortex is pushed downstreamby the ground vorticity, so that it gets closer to the pre-viously shed clockwise vortex; that clockwise vortexpushes it up, to a higher position than in freestream.

(2) Clockwise vortex: this vortex is surrounded by a ring ofpositive vorticity, a result of the large positive vorticityformed on the suction surface and which is still linked tothe previously shed anticlockwise vortex while the clock-wise vortex rolls up and sheds.

(3) Ground vorticity: this vorticity is pushed back by theclockwise vortex, while the anticlockwise vortex en-hances and lifts the vorticity from the ground.

Fig. 11 Contours of instantaneous non-dimensional vorticity atk = 4.37, τ = 1.00, h = 0.00. a Freestream; b h0/c = 0.16

Figure 11a shows that the gap between vortices infreestream is not uniform, but that the anticlockwise vor-tex is always closer to the upstream than to the downstream


clockwise vortex (i.e. a < b). This is by no means relatedto the timing of shedding from the airfoil surface since theanticlockwise and clockwise vortices are shed at τ = 0.06and τ = 0.60, respectively, regardless of the ground distance.This lag is caused by the asymmetric geometry which resultsin unequal roll-up and shedding processes on each side of thetrailing edge. On the other hand, the three-way interferencein proximity to the ground leads to the gap between vorticesbeing reversed. Now the anticlockwise vortex is closer tothe downstream clockwise vortex (i.e. a′ > b′), as seen inFig. 11b.

Fig. 12 Box diagram of the interaction between the shed vorticesand the vorticity produced on the ground

This forced shedding is an inherently inviscid phe-nomenon, since inviscid simulations fully resolve the vor-tices shed in the wake. Furthermore, as there is no vortic-ity on the ground plane in inviscid flow, the three-way in-teraction becomes a dual interaction between clockwise andanticlockwise vortices. For instance, the vortices are moreuniformly spaced and the ring of vorticity is weaker.

Finally, another circumstance to consider is that the non-dimensional vorticity magnitude inside the vortex cores in-creases with ground proximity due to the larger rotationalflow in the boundary layer. Also, a larger amount of negativethan positive vorticity is shed, both factors enhancing the in-teraction between vortices illustrated before. The differencein shed vorticity between vortices is compensated by the pos-itive vorticity remaining in the boundary layer and the largercirculation around the airfoil due to increased downforce.

Although there does not seem to be a direct correlationbetween the timing of shedding and the drag history, at highfrequencies the perfectly sinusoidal curves are replaced byotherwise periodic curves in which the maximum and min-imum peaks are moved closer together. The gap betweenCD,min and CD,max is approximately τ = 0.50 up to k = 0.55,gradually decreasing for higher frequencies down to τ =0.40 and remaining relatively constant for k ≥ 1.71. Thegap is substantially similar in ground effect, albeit decreas-ing slightly with ground proximity, reaching τ = 0.38. The

shift of the drag curves at high frequencies was attributed tothe geometrical asymmetry being critical throughout a rathersymmetrical response to motion (i.e. added mass).

5 Conclusions

Building on the foundations of the literature on a stationaryairfoil in ground effect and a heaving airfoil in freestreamas separate entities, a study on an inverted heaving airfoil inground effect was performed. As a complement to a previouswork that focused on the aerodynamic mechanisms whichgenerate the forces on a heaving airfoil in ground effect [10],this work analyses the actual forces on the airfoil and theflow structure in the wake. The following conclusions weredrawn from the study:

(1) The resultant forces exhibit hysteresis between upstrokeand downstroke. The forces are periodic with the samefrequency as the heaving motion. As the frequency in-creases, the forces diverge from their stationary counter-parts.

(2) The force reduction region is apparent at low frequen-cies, but it disappears at medium and high frequenciesas viscous effects become less significant. As in thefreestream case, average downforce increases with fre-quency in proximity to the ground. Furthermore, the rel-ative increase with respect to a stationary airfoil is muchlarger in ground effect than in freestream.

(3) Thrust can be achieved in ground effect at high enoughfrequencies. Although at low frequencies boundary layerseparation results in higher drag close to the ground, oncethe airfoil produces thrust the larger vorticity means thatmore thrust is obtained progressively closer to the groundas the component of thrust produced overcomes the vis-cous drag inherent to low ride heights. However, the ef-fect of the ground is also to increase the maximum dragin the cycle.

(4) The motion was shown to be unstable for low ride heightsand low frequencies. Due to the existence of a force re-duction region at those frequencies, stall flutter occurs.In fact, the heaving motion in ground effect is compa-rable to a pitching motion in which dynamic stall canoccur at high enough amplitudes. At higher frequencies,the work required to move the airfoil increases with de-creasing ground clearance, such that in unforced motionthe oscillations would tend to be damped more quickly.

(5) A progressively increasing flapping motion of the shearlayers in the wake with frequency leads to forced vor-tex shedding at k ≥ 1.71. The shedding process in andout of ground effect is identical, although at mean rideheights as low as h0/c = 0.16 the vortices interact withthe ground, altering the wake configuration.

Acknowledgements The authors would like to thank


Mercedes GP for their support, in particular David Jeffrey,Jonathan Zerihan and Jonathan Eccles.

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on the unsteady motion and stability of a heaving airfoil in ground effect

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