- Email: [email protected]
Geometry effects on autorotation of rectangular prisms
J. Wind Eng. Ind. Aerodyn. 132 (2014) 92–100
Contents lists available at ScienceDirect
Journal of Wind Engineering and Industrial Aerodynamics journal homepage: www.elsevier.com/locate/jweia
Geometry effects on autorotation of rectangular prisms D.I. Greenwell Aircraft Research Association Ltd., Bedford, United Kingdom
art ic l e i nf o
a b s t r a c t
Article history: Received 2 February 2014 Received in revised form 5 June 2014 Accepted 8 June 2014
This paper presents an analysis of published data on the autorotation of rectangular prisms, supplemented by additional wind tunnel tests of container-like shapes typical of helicopter slung loads. Previous widely used correlations for effects of geometry on autorotation rate are shown to perform rather poorly, particularly for higher aspect ratios and thickness/chord ratios. A smooth and well-behaved 3D surface has been fitted to the data, permitting the likely autorotation rate of any rectangular prism to be determined graphically. Of particular significance to the behaviour of slung loads is a well-defined region where autorotation does not occur. Two new parameters for presenting autorotation data have been proposed. A normalised aspect ratio has been used which allows 2D and 3D data to be plotted on a single figure, and gives a more nearly linear variation of autorotation rate with aspect ratio. The geometric mean aspect ratio is a simple single parameter for the autorotation rate, which gives an approximate collapse of trends in the data. This parameter can be interpreted as a ratio between the strengths of driving vortices shed from spanwise and streamwise edges respectively, suggesting that the 3D structure of the flows driving autorotation would repay further investigation. & 2014 Elsevier Ltd. All rights reserved.
Keywords: Autorotation Slung load Rectangular prism
1. Introduction Military and civil helicopters are often used to carry external cargo as a slung load. For many loads, maximum operational flight speeds are limited not by the available power, but by the onset of divergent or limit-cycle oscillations of the load. Rectangular box loads are particularly susceptible to aerodynamic instabilities, especially when lightly loaded (Sheldon and Pryor, 1973b). One load instability that has been the subject of considerable recent interest is the continuous rotation in yaw of the 6 ft 6 ft 8 ft CONEX container when slung from a single point with a swivel joint (Fig. 1, Raz et al., 2008). This is an example of autorotation (Lugt, 1983), a well-known but still rather poorly understood aerodynamic phenomenon. However, flight and wind tunnel tests for other cuboid slung loads show a range of behaviour from steady autorotation to oscillatory yawing motion to a stable broadside-on posture (Raz et al., 2011; Sheldon and Pryor, 1973a), which raises the question of what aspect(s) of the load geometry govern autorotation rate, in particular the conditions for onset of steady rotation. There is an extensive body of experimental and computational work on the autorotation of rectangular prisms, but this is largely concerned with either quasi-two-dimensional (2D) geometries or with low aspect ratio ‘flat plate’ models.
E-mail address: [email protected] http://dx.doi.org/10.1016/j.jweia.2014.06.014 0167-6105/& 2014 Elsevier Ltd. All rights reserved.
For three-dimensional (3D) geometries representative of cargo containers, the only data available is from flight test and scaled dynamic wind tunnel tests of slung loads (Cicolani et al., 2010; Raz et al., 2008, 2011; Sheldon and Pryor, 1973b), which introduces the added complexity of large-amplitude coupled lateral and longitudinal pendulum-like load motions. Some recent work has been done on CFD simulation of rotating CONEX loads (Cicolani et al., 2009; Prosser and Smith, 2013), but validation against flight or wind tunnel data has as yet been limited in extent. Existing semi-empirical correlations for the effect of geometry on autorotation rate (Bustamante and Stone, 1969; Iversen, 1979) are based primarily on data for low-aspect ratio thin wings, and fail to predict behaviour for higher thickness/chord ratios (Hirata et al., 2009, Skews, 1990). In order to separate load sling dynamics from aerodynamics some simple exploratory studies were undertaken at City University London on a fixed-axis autorotation rig. The objectives were to (a) examine initial spin-up and limiting autorotation rate behaviour, and (b) assess the effect of geometry changes on autorotation of 3D bodies. The experimental procedure and some new aspects of autorotation dynamics are described in more detail in Greenwell (2014). This paper reports on the relationship between geometry and autorotation rate for rectangular prisms, presenting a synthesis of new and previously published data.
D.I. Greenwell / J. Wind Eng. Ind. Aerodyn. 132 (2014) 92–100
Nomenclature AR AR ARmean AR0 b c la
aspect ratio, ¼b/c normalised aspect ratio, ¼ 1/(1 þ1/AR) geometric mean aspect ratio, ¼b/√(tc) critical aspect ratio span (m) chord, reference length (m) rig offset from tunnel floor (mm)
2. Autorotation rates for rectangular prisms 2.1. Basic features of autorotation A comprehensive review of autorotation is given by Lugt (1983), covering motions about axes parallel and perpendicular to the flow. The detailed flow physics are rather complex, but it is generally accepted that autorotation of a body with a rectangular cross-section is driven by periodic vortex shedding from the spanwise edges, with spin rates found to coincide approximately with shedding frequencies measured in the wake (Lugt, 1983; Zaki et al., 1994). Links between autorotation and vortex shedding were developed in more depth by Lugt (1982). The specific component of the flow providing the net driving torque required to overcome bearing friction is the vortex shed downstream of the retreating edge (Izutsu, 1986; Zaki et al., 1994), as shown in Fig. 2a. Flow visualisation in Zaki et al. (1994) indicates that in comparison to a stationary prism the vortices rotating in the same sense as the prism are augmented, whilst those rotating in the opposite sense are inhibited. At the orientation shown in Fig. 2b the augmented retreating-edge vortex lies near to the aft face of the body, generating a suction peak close to the upper edge and hence giving a positive torque contribution. The net result is a steady time-averaged spin rate which varies linearly with freestream velocity. The moment of inertia can have a significant impact on autorotation rates for light models, but for the higher relative densities typical of cargo container shapes and construction (Cicolani et al., 2009), autorotation rate is essentially independent of inertia (Smith, 1971). Relatively little attention has been given in the literature to three-dimensional flow effects. Skews (1990) suggested that for very low aspect ratios the main vortices are shed from the radial
r t U V 2D 3D
ω Ωn
93
radius (m), ¼ 1/2√(t2 þ c2) thickness (m) freestream velocity (m/s) tip speed (m/s) two-dimensional three-dimensional angular velocity (rad/s) non-dimensional autorotation rate,¼ ωc/2U
edges rather than the spanwise edge, as shown schematically in Fig. 2b. The aerodynamic driving moment would then be smaller because of the combination of reduced radial vortex strength and reduced moment arm. In contrast, Iversen (1979) correlated the effect of aspect ratio on autorotation of flat plate wings by using the static pitch stability derived from thin-wing attached-flow theory. The underlying assumption was that the time-averaged strength of the driving vortex is related to the strength of the bound spanwise vortex for a stationary wing of the same aspect ratio at low incidence, therefore implying that autorotation is being driven by the spanwise edge vortex. Some 3D unsteady computational fluid dynamics simulations have been done for the CONEX container (Theron et al., 2006), but no details of the flow topology appear to have been published. CFD results in a very recent paper on autorotating square flat plates (Hargreaves et al., 2014) do show a very similar wake structure to that suggested by Skews (1990). 2.2. Autorotation similarity parameters For rectangular prisms,1 geometry is defined as shown in Fig. 3. Conventionally, autorotation rate is taken to be a function of the non-dimensional parameters planform aspect ratio AR ¼b/c and thickness to chord ratio t/c, where b is the span and chord cZthickness t. An additional parameter used in this work is the normalised aspect ratio AR, AR ¼
AR 1 ¼ AR þ 1 1 þ 1=AR
ð1Þ
which is a simplified version of a semi-empirical aspect ratio correlation from Iversen (1979),2. The normalised aspect ratio asymptotes to AR as AR-0 and to 1 as AR-1, making it a convenient means of presenting low and high aspect ratio behaviour on a single plot. Autorotation rate data is usually presented as non-dimensional spin rate plotted against freestream velocity or Reynolds Number. Conventions vary considerably, but the most common form is
Ωn ¼
ωc ¼ f nðUÞ 2U
ð2Þ
where c is a reference length (typically the chord for a thin wing model, or the longest streamwise edge for a rectangular crosssection) and U is the freestream velocity. Comparisons of wind tunnel and full-size flight test data for autorotating slung container loads (Raz et al., 2011) show very good agreement of rotation rates scaled using Eq. (2), even for an 11:1 variation in length scale.
Fig. 1. Blackhawk with CONEX slung load (Raz et al., 2008).
1 The terminology in the fluid dynamics and aerodynamics communities for container-like geometries is rather inconsistent. Alternative terms that have been used are ‘rectangular prism’, ‘rectangular cylinder’, ‘rectangular parallelepiped’, ‘cuboid’, or just ‘box’. 2 see Eq. (6b)
94
D.I. Greenwell / J. Wind Eng. Ind. Aerodyn. 132 (2014) 92–100
2.3. Semi-empirical correlations for autorotation rate retreating edge
An early attempt at relating autorotation rate to 3D wing geometry was made by Bustamante and Stone (1969), who gave a correlation for the steady autorotation rate for thin flat-plate wings as
U driving vortices
ω
Ωn ðARÞ ¼ 1 e 0:2AR
advancing edge
Curiously, the supporting figure in Bustamante and Stone shows no direct comparison with experimental results; detailed examination of data in this reference suggests that Eq. (5) was in fact a rather poor fit, particularly for lower aspect ratios. The most widely used correlation for aspect ratio and thickness/chord ratio effects was developed by Iversen (1979), based on the premise that pitch stability might be a valid correlation parameter for autorotation. Iversen's correlation is
ω
U
spanwise vortex
radial vortex
Ωn ðAR; t=cÞ ¼ f 1 ðARÞf 2 ðt=cÞ
Fig. 2. (a) 2D flow topology for an autorotating cylinder (redrawn from Zaki et al., 1994), (b) 3D driving vortex structure (Skews, 1990).
thickness, t
f 1 ðARÞ ¼
wind speed, U aspect ratio AR = b/c chord, c Fig. 3. Model geometry definitions.
ð3Þ
where t and c are the model thickness and chord respectively, and r is the maximum model radius (¼ 1/2√(t2 þ c2)). However, as demonstrated by Iversen (1979), these simple non-dimensional forms can give rise to spurious Reynolds Number effects. This is due to a combination of a zero shift in the linear variation of spin rate with freestream velocity due to bearing friction (Iversen, 1979), non-linear behaviour at low spin rates (Cheng, 1966), and coupling with structural modes at high spin rates (Greenwell, 2014). On the basis of a rather ad-hoc analysis of the equations of motion of an autorotating body, Iversen concluded that it is justifiable to account for the effects of bearing friction by taking the slope of the linear section, giving c ∂ω 2 ∂U
2 þ ð4 þ AR2 Þ1=2
( f 2 ðt=cÞ ¼
Ωn ¼
AR
#" 2
AR AR þ 0:595
0:76 #)2=3 ð6bÞ
The first term in f1 is an approximation to the lift-curve slope based on thin-wing theory, and the second an empirical expression for the aerodynamic centre. The overall function f1 can be thought of as corresponding to the static “aerodynamic overturning moment”, and varies from 0 for AR ¼0–1 for 2D configurations (AR ¼1). The function f2 was fitted to the effect of varying thickness/chord ratio from 0.0054 to 0.5, for low aspect ratio wings with AR ¼0.25–4. Aspect ratio effects were accounted for using f1.
span, b
An alternative parameter is the tip speed ratio sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 V rω t n ¼ Ω 1þ U c U
ð6aÞ
where f1 and f2 are functions of aspect ratio and thickness/chord ratio respectively, derived from curve-fits to experimental data from Glaser and Northup (1971).3 The function f1 was fitted to the effect of varying aspect ratio from 0.25 to 4 for thin wings, with t/c 0.03. ("
spin rate,
ð5Þ
ð4Þ
Autorotation rates presented in this work are non-dimensionalised using Eq. (4).
2 ) 1 1 0:0246 ln 0:329 ln t=c t=c
ð6cÞ
This function is an entirely empirical relationship for the variation of autorotation rate with the inverse of the thickness/ chord ratio, and varies from 1 for thin wings (t/c ¼0) to 0 for square cross-section (t/c ¼1). There are obvious problems with the limiting behaviour of this correlation: an infinite rotation rate for a thin wing is clearly non-physical, whilst square prisms in general autorotate quite readily. Fig. 4 presents the performance of this correlation for the data originally fitted by Eq. (6), shown as filled circles, and for more recent 2D and 3D autorotation measurements (Table 1). Iversen (1979) described the fit as ‘fairly good’, which is a reasonable assessment for the original data, but when applied to more recent data for higher aspect ratios and thickness/chord ratios the correlation clearly performs poorly. The deviation is particularly large for nominally 2D testing, where there is also a very large scatter in the data; for example reported non-dimensional autorotation rates for wings with thickness/chord ratios of 0.1–0.15 vary from 0.45 (Skews, 1990) to 1.1 (Smith, 1971). 3 The original reference for this data has been unobtainable – the data in Fig. 4 has been extracted from Figs. 6 and 7 in Iversen (1979)
D.I. Greenwell / J. Wind Eng. Ind. Aerodyn. 132 (2014) 92–100
2.4. Effect of endplates One likely major contributor to the scatter for 2D configurations is the difficulty in achieving ‘true’ 2D conditions for an unsteady separated flow. For example, previous work on drag and vortex shedding for 2D bluff bodies has shown in many cases a very significant sensitivity to the end conditions (Kubo and Kato, 1986; Gerich and Eckelmann, 1982; Ramberg, 1983). Even with careful endplate design, aspect ratios of greater than 7 can be required to achieve 2D flow conditions on model centrelines (Fox and West, 1990). Typical nominally 2D wind or water tunnel test configurations for autorotation experiments include: models spanning a closed test
95
section (Skews, 1990; Zaki et al., 1994), models spanning an open jet test section with endplates (Hirata et al, 2009), models with ‘fixed’ endplates (Smith, 1971, Cheng, 1966) and models with ‘rotating’ endplates (Smith, 1971, Cheng, 1966). ‘Fixed’ endplates are mounted to the tunnel, and are therefore stationary. These tend to be large and non-circular in planform, often based on optimum configurations determined for 2D aerofoil testing. ‘Rotating’ endplates are attached to the model and rotate with it. These are circular in planform and tend to be smaller. In three cases (Skews, 1990; Hirata et al. 2009 and Smith, 1971) the effect of varying thickness/chord ratio was also investigated, giving the trends shown in Fig. 5. Open symbols denote models with endplates: for rotating endplates and ○ for fixed endplates. Filled symbols denote 2D test sections of one form or
1.2
1.2 Glaser & Northup
Iversen correlation
0.8
0.4
0.0 0.0
0.2
rotating endplates 2D test section, high AR fixed endplates 2D test section, low AR 2D open jet, low AR 2D flume, low AR effect of endplates, AR = 3
rotating endplate
later 2D data
autorotation rate, Ω* = c/2U
autorotaion rate, Ω* = ωc/2U
later 3D data
0.4
0.6
0.8
1.0
0.8 fixed endplate no endplate
0.4
0.0 0.0
0.2
f1 (AR) × f2 (t/c)
0.4 0.6 thickness/chord ratio t/c
0.8
Fig. 5. Thickness effects for nominal ‘2D’ tests.
Fig. 4. Iversen's correlation for autorotation rate.
Table 1 Published data on autorotating rectangular prisms. Reference
AR
t/c
Slung load /fixed axis
End condition
Wing section
Bustamante and Stone (1967, 1969) Cheng (1966) Cicolani et al. (2009) Fletcher et al (2002) Glaser and Northup (1971)
0.167–6
Fixed axis
–
Flat plate
3, 2D 0.76 2D? 0.25–4
0.008– 0.045 0.047 0.72 0.025 0.005–0.5
Fixed axis Slung load Fixed axis Fixed axis
Fixed endplate – Spanning open jet –
chamfered flat plate Rectangular Flat plate ?
Hirata et al (2009)
2D
0.3–0.8
Fixed axis
Hirata et al. (2011) Martinez-Vasquez et al. (2011) Izutsu (1986), Oshima and Oshima (1983) Poreh and Wray (1979)
2–20 1
0.3 Free fall 0.025,0.083 Fixed axis
Large fixed endplates in Rectangular open jet – Flat plate – Flat plate
2D
0.15
Fixed axis
Rotating endplate
Elliptical
Geometric AR ¼ 3
2D
0.3
Fixed axis
Flat plate
Geometric AR ¼ 6.9
Raz et al. (2011) Riabouchinsky (1935)
0.4–1.2 4
1 0.133,1
Slung load Fixed axis
Spanning closed test Section – –
TRIO container
Roger and Aubert (2006) Sefat (2011)
2D? 2D
0.067 0.008
Fixed axis Fixed axis
Rectangular Chamfered flat plate, Cruciform Elliptical Flat plate
Geometric AR ¼ 4.7 Geometric AR ¼ 0.83
Sheldon and Pryor (1973a) Sheldon and Pryor (1973b) Sheldon and Pryor (1973c) Skews (1990)
0.333,1 0.125–0.5 0.044 2D
0.333,1 0.0625–0.5 0.077 0.1–1
Slung load Slung load Slung load Fixed axis
Rectangular Rectangular Flat plate Rectangular
Model of blade box Geometric AR ¼ 1.7–3.3
Smith (1970, 1971)
2D
Fixed axis
Tachikawa (1983a, 1983b) Tanabe and Ito (1996) Zaki et al. (1994)
1,2 3 2D
0.02–0.04, 0.15 0.029–1 0.038 1
Fixed axis Fixed axis Fixed axis
Spanning open jet Spanning free-surface flume
Spanning closed test section Rotating endplate – – Spanning closed test section
Flat plate, elliptical Rectangular Flat plate Rectangular
Comments
Also tested without endplates CONEX container Geometric AR ¼ 4.2 Original report unobtainable, Data taken from Iversen (1979) Geometric AR ¼ 10
Also tested with a fixed endplate Geometric AR ¼ 3, 6
Geometric AR ¼ 20
1.0
96
D.I. Greenwell / J. Wind Eng. Ind. Aerodyn. 132 (2014) 92–100
ω
II U
driving vortex I
U III
b
I 250mm
driving vortex III Fig. 6. Autorotating wing with asymmetric chamfer (after Riabouchinsky, 1935).
another: for high aspect ratio models and , , . This function is an entirely empirical relationship for low aspect ratio models. Eq. (6c) predicts autorotation rate falling rapidly as thickness/ chord is increased, becoming zero at t/c ¼1 (square cross-section). Instead, we see an initial reduction in rate which then levels out to a nearly constant value for t/c40.6. Autorotation rates fall in a band which appears to narrow as t/c increases. The lower boundary of the band corresponds to test configurations where the model is moving relative to the flow boundary and end effects may be significant – that is, low aspect ratio models spanning open and closed test sections, and wings with fixed endplates. The upper boundary corresponds to tests where end effects are less likely to be significant – that is, very high aspect ratio ( Z 10) models and wings with rotating endplates. Extrapolation from 3D test results tends to give autorotation rates towards the upper boundary, although the data coverage at intermediate aspect ratios is too sparse to be confident of the limiting values. An indication of the impact of model motion relative to the flow boundary is given by two datasets where endplate configuration was varied for low aspect ratio thin wings, indicated on Fig. 5 by the vertical dash-dot line at the left hand end. Cheng (1966) tested a flat plate wing (t/c¼ 0.047, AR ¼3) with and without a fixed (non-rotating) endplate. Smith (1970, 1971) tested a similar flat plate wing (t/c¼ 0.04, AR ¼2.6) with rotating and non-rotating endplates. For Cheng's baseline ‘no endplate’ model Ωn E0.52; adding a large fixed endplate gave a small increase to Ωn E0.54. For Smith's model, a small fixed endplate with diameter 1.3c gave Ωn E0.60; allowing the endplate to rotate with the model increased Ωn to 0.84. Reducing the rotating endplate diameter to 1.0c gave a further increase to Ωn E 1.0. Clearly, achieving 2D flow conditions is not straightforward, particularly for lower thickness/chord ratios. Configurations with the model rotating relative to an endplate or sidewall have the worst performance, but there is insufficient data to be clear as to whether this is a gap effect, or due to the interaction between the moving model and stationary boundary layer. The autorotation motion should at least constrain the driving vortices to be shed parallel to the model, as occurs with vortex shedding lock-in (Sarpakaya, 1979), so that spanwise asymmetry is probably not a contributing factor. 2.5. Effect of leading-edge geometry Continuing with the analogy to vortex shedding proposed by Lugt (1982), one would expect the strength of the driving vortex (and hence the autorotation rate) to depend strongly on the geometry of the spanwise edges. Fig. 5, and 3D results presented later in Section 4, gives some support to this hypothesis;
la
Fig. 7. Model mounting arrangement in T7 wind tunnel.
autorotation rate is most sensitive to wing thickness for thin wings (t/c o0.2). A similar sensitivity to small variations in edge geometry can be seen for the leading-edge vortices generated by swept delta wings (e.g. Kirby and Kirkpatrick, 1969). However, no parametric study has been undertaken on the effect of edge shape. Most published autorotation datasets are for models with simple rectangular cross-sections. The exceptions are a handful of tests of 15% thick elliptical cross-section wings (Smith, 1970, Izutsu, 1986, Oshima 1983), and two studies of thin wings with chamfered edges. Cheng (1966) tested a 4.7% thick flat plate wing with the leading and trailing-edge chamfered at 121 on one face only. The combination of small chamfer angle and longitudinal symmetry gave an autorotation rate close to a similar squareedged wing tested by Smith (1970). Riabouchinsky (1935) tested a 13% thick 3D wing (AR¼4) with square edges and with an asymmetric 301 chamfer (Fig. 6). Autorotation rate for the square-edged wing (case II in Fig. 6) is consistent with other published results at similar aspect ratios and thicknesses. For the chamfered wing the asymmetry resulted in an autorotation rate that depended on direction of rotation: 18% slower than the square-edged wing with the chamfered face to windward on the retreating side (case I in Fig. 6) and 20% faster for rotation in the opposite direction with the flat face to windward (case III). This is consistent with the effect of leading-edge chamfer on the strength and position of delta wing vortices (e.g. Kegelman and Roos, 1989) – the driving vortex on the retreating side is stronger with the flat face to windward (case III).
3. Additional 3D data 3.1. Wind tunnel and models Testing was undertaken in the T7 industrial wind tunnel at City University London. This is a closed return wind tunnel, with working section dimensions of 3 m 1.5 m 8 m and a speed range of 0–25 m/s. Turbulence levels are of the order of 3% (Sykes, 1977), high by aeronautical standards but not high enough to have a significant effect on vortex shedding frequencies for rectangular cylinders (Noda and Nakayama, 2003). Reynolds Numbers based on chord were 16,000 to 45,000, well above the critical range for vortex shedding (Okajima, 1982). Autorotation tests were carried out using models mounted on a modified Vector Instruments A101M anemometer body (Fig. 7) which provided a
D.I. Greenwell / J. Wind Eng. Ind. Aerodyn. 132 (2014) 92–100
Table 2 LegoTM model geometries tested. AR
t/c
Ωn
AR
t/c
Ωn
0.2 0.3 0.4 0.5 0.6 0.7 0.24 0.36 0.3 0.45 0.6 0.75 0.3 0.4 0.5 0.6 0.7
0.167 0.167 0.167 0.167 0.167 0.167 0.2 0.2 0.25 0.25 0.25 0.25 0.333 0.333 0.333 0.333 0.333
0.06 0.107 0.13 0.168 0.176 0.189 0.05 0.1 0 0.1 0.123 0.163 0 0 0.055 0.090 0.120
0.36 0.48 0.6 0.72 0.84 1.08 0.45 0.6 0.75 1.05 1.35 0.4 0.8 0.8 1 1.2 1.4 1.6 1.8 2 3
0.4 0.4 0.4 0.4 0.4 0.4 0.5 0.5 0.5 0.5 0.5 0.667 0.667 1 1 1 1 1 1 1 1
0 0 0.088 0.127 0.147 0.228 0 0 0.077 0.218 0.321 0 0.080 0 0.17 0.213 0.278 0.285 0.289 0.341 0.374
small-amplitude oscillations to fully-developed autorotation in less than 30 s. Because of the small size of the model relative to the tunnel working section no blockage corrections were applied. 3.2. Data validation Fig. 8 shows typical experimental results for the variation of autorotation rate with freestream velocity, for four models with a thickness/chord ratio of 0.5 and aspect ratios varying from 0.6 to 1.08. The mid-range behaviour is consistent with Iversen's discussion, with a repeatable linear trend and a small zero offset due to bearing friction. For the lowest aspect ratio there is a minimum critical velocity below which autorotation does not occur, as reported by Cheng (1966). For the higher aspect ratios shown in Fig. 8 the critical velocity is below the lowest achievable tunnel speed. For some configurations the autorotation characteristics at the extremes of the speed range become highly non-linear locking-in to structural natural frequencies at high speeds, and displaying static hysteresis at low speeds. These aspects are reported in more detail in Greenwell (2014); this paper focuses on the effect of geometry on autorotation rate in the linear regime. Fig. 9 compares autorotation rates determined using Eq. (4) with the limited amount of ‘overlapping’ data available (Table 1). Fig. 9a shows the effect of aspect ratio for cross-sections with thickness/chord ratios of 0.3 (Hirata et al., 2011) and 1.0 (Tachikawa, 1983b; Zaki et al., 1994; Riabouchinsky, 1935; Sheldon and Pryor, 1973a; Raz et al., 2011),
0.6 present work, t/c = 1 fixed axis models, t/c = 1
autorotation rate, Ω* = ωc/2U
very low-friction bearing and a 13 pulse/rev output from an optical sensor. Pulse frequency was measured using a Racal Instruments 1990 Universal Counter set to a 2 s averaging time and hence giving spin rate averaged over typically 10–60 rotations. Freestream velocity was measured using a standard NPL pitot-static probe mounted on the sidewall at the start of the working section, and a Furness Controls FCO318 digital pressure indicator. Tunnel speed and autorotation rate were recorded manually, with an estimated uncertainty of the order of 7 2% in speed and 71% in angular velocity. This is consistent with repeatability tests for a single configuration, which gave a standard deviation of 2% for the non-dimensional autorotation rate Ωn. Models were made of standard Lego™ bricks, permitting thickness, chord and span to be varied quickly. Aspect ratio was varied between 0.2 and 3, thickness/chord ratio was varied between 0.167 and 1 (Table 2). Tests reported here were carried out with the studs on the upper face of the models ground off; however, some exploratory tests showed no significant difference in autorotation behaviour between modified and unmodified bricks. Average density of the bricks used was approximately 500 kg/m3, giving moments of inertia well above the critical range identified by Smith (1971). For these geometries rotation started spontaneously, building up rapidly from
97
underslung loads, t/c = 1 present work, t/c = 0.33 free-fall, t/c = 0.3
0.4
0.2
t/c = 0.33
0.0 0.0
t/c = 1.0 (square section)
0.2
0.4
0.6
0
0.25
0.5
1
2
4
2D
0.4
autorotation rate, Ω* = ωc/2U
t/c = 0.4
rotation rate, ω (rad/s)
1.0
aspect ratio, AR
40
30
20
10
0
0.8
normalised aspect ratio, 1/(1+1/AR)
AR = 0.6 AR = 0.72 AR = 0.84 AR = 1.08
increasing AR
present work, AR = 0.5 Glaser & Northup, AR = 0.5
0.3
0.2
0.1
0.0 0
5
1 10
15
20
freestream velocity, U (m/s) Fig. 8. Typical variation of rotation rate with freestream velocity and aspect ratio for LegoTM models.
0.0
0.2
0.4
0.6
0.8
1.0
thickness/chord ratio , t/c Fig. 9. Comparison with published autorotation data for prisms: (a) effect of AR for t/c ¼0.33 and 1.0, (b) effect of t/c for AR ¼0.5.
98
D.I. Greenwell / J. Wind Eng. Ind. Aerodyn. 132 (2014) 92–100
the data, giving the mesh in Fig. 10a and the corresponding contours in Fig. 10b. Data for 2D tests (B) was not used to generate the fitted surface, because of the issues discussed in Section 2.4 above. The variation of autorotation rate with aspect ratio is nearly linear, although there are signs of a discontinuity in slope at aspect ratios of around 0.8–1. For higher aspect ratios the variation with thickness/chord ratio is similar to the 2D case (Fig. 5): autorotation rate initially drops rapidly as thickness is increased before levelling out to a nearly constant value. As noted previously, extrapolating 3D trends in Fig. 10 to the 2D case (AR ¼ 1) gives values towards the upper boundary in Fig. 5. For lower aspect ratios (o 0.8) autorotation rate does not level out, but falls to zero as thickness is increased. The resulting low aspect ratio autorotation boundary is shown in more detail in Fig. 11. Crosses ( ) and open symbols (◯, ☐) indicate published data (Table 1) for configurations that do, and do not, rotate respectively. Filled symbols indicate extrapolated ‘zero rotation’ cases from parametric variations of aspect ratio ( , Table 2) and thickness/chord ratio ( , Glaser and Northup, 1971), see for example Fig. 9. The autorotation boundary is reasonably well defined for higher thicknesses; however, at low thicknesses there is some uncertainty as to whether the boundary extends to the origin, or intercepts the x-axis at a non-zero thickness.
4.2. Autorotation rate correlations Correlations for the effects of thickness/chord ratio on autorotation are constrained by symmetry requirements. For a rectangular prism the choice of which edge length to take as thickness and which as chord is to some extent arbitrary, so for a general correlating function f(t/c) should equal f(c/t) and hence df/d(t/c)¼0 at t/c ¼1. Fig. 10b shows that a surface fitted to experimental data for autorotation rate comes close to meeting this requirement, with the contour lines having near-zero slope for t/c 40.5. Fig. 11 indicates that there is a critical aspect ratio AR0 below which a rectangular prism will not autorotate. An empirical relationship for the variation of AR0 with t/c that approximately meets the symmetry requirement AR0 (t/c)¼AR0 (c/t) is t=c 2 1:05 1 1:0026e 0:89 AR0 ¼ ð7Þ t=c Eq. (7) is shown as a dashed line in Fig. 11: the boundary crosses the x-axis at t/c ¼0.045 and has zero slope at t/c ¼1. Attempts to find a general function that would correlate autorotation rate were not particularly successful. More complex exponential relationships of the general form of Eq. (7) can be fitted to the data shown in Fig. 10a, but these are rather unwieldy and lack any Fig. 10. Effect of aspect ratio and thickness/chord ratio on autorotation rate – fit to experimental data (J, - Table 1, 3DB- Table 1, 2D☐- Table 2).
4. Geometry effects revisited 4.1. Effect of aspect ratio and thickness/chord ratio Fig. 10a shows autorotation rates from published data (Table 1: and results from the present work (Table 2: ☐) plotted against normalised aspect ratio and thickness/chord ratio. Cases with no rotation are denoted by . A smoothed 3D surface was fitted to J, ,B)
1.00
aspect ratio, AR
while Fig. 9b shows the effect of thickness/chord ratio for planforms with aspect ratio AR¼ 0.5 (Glaser and Northup, 1971). Both magnitudes and trends with aspect ratio and thickness compare very well with published data. Fig. 9a also demonstrates that the normalised aspect ratio AR is a suitable parameter for presenting 3D geometry effects, giving near-linear trends for t/c¼0.3 and 1.0.
1.25
0.75
0.50 rotation - fixed axis rotation - underslung loads no rotation - underslung loads Lego models extrapolated Glaser & Northup extrapolated
0.25
0.00
0
0.2
0.4
0.6
thickness/chord ratio,t/c Fig. 11. Autorotation limits.
0.8
1
D.I. Greenwell / J. Wind Eng. Ind. Aerodyn. 132 (2014) 92–100
physical significance. An alternative is shown in Fig. 12, which relates autorotation rate to a more general form of the aspect ratio. As discussed in Section 2, autorotation rate is governed by the strength of the driving vortex shed by the retreating spanwise edge, which in turn should be a function of the aspect ratio. However, for a rectangular cross-section the appropriate aspect ratio to consider would depend on which edge was shedding the vortex. All four spanwise edges contribute to the time-averaged driving moment; therefore one might expect the average moment to depend on the average aspect ratio of the top/bottom and front/ rear faces, where ARtop
¼
b ; c
ARf ront
b t
¼
ð8aÞ
The geometric mean was found to give a much better collapse of the data than the arithmetic mean, hence qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b AR ARmean ¼ ARtop ARf ront ¼ pffiffiffiffiffi ¼ pffiffiffiffiffiffiffi ð8bÞ tc t=c which also meets the symmetry constraint discussed above. To be entirely consistent, one might also expect that a similar argument should apply to the choice of reference length in the nondimensional autorotation rate Ωn; however, it was found that the best results were obtained using the chord c (e.g. Eq. (2)) rather than an average value such as the radius (e.g. Eq. (3)). This may indicate that although the average strength of the driving vortices is related to the average aspect ratio, the corresponding average moment arm is fixed by the length of the deepest face. Fig. 12 shows the variation of non-dimensional autorotation rate for 3D prisms, as defined in Eq. (4), plotted against the mean aspect ratio, as defined in Eq. (8b). There is still a significant degree of scatter, but when compared with Fig. 3 there is a much better collapse of the general trends from different datasets. There is a well-defined and consistent break in the general slope at a mean aspect ratio of around 1.4; above this break point a rough approximation for the autorotation rate is
Ωn 0:21 AR0:33 mean
ð9Þ
while below it the autorotation rate falls off rapidly with reducing aspect ratio and/or increasing thickness. 4.3. Suggested relationship to flow physics Referring back to Fig. 2b and Eq. (8b), the mean aspect ratio ARmean could also be interpreted as a ratio between the strength of
1
autorotation rate, Ω* = ω c/2U
transition region ?
0.1 fully developed autorotation
Glaser & Northup later 3D data Lego models
0.01 0.1
1
10
ARmean = b/√(tc) Fig. 12. General effect of ‘aspect ratio’ on autorotation rate.
100
99
the spanwise driving vortices (proportional to spanwise edge length b) and the strength of the radial or edge vortices (proportional to mean streamwise edge length √(tc)). Fig. 12 then suggests a possible explanation for the behaviour seen in Figs. 10 and 11. For higher mean aspect ratios autorotation rate is driven by the flow separating from the spanwise edges, with 3D flow effects confined to a spanwise variation in driving vortex strength. At lower mean aspect ratios the flow around the end faces begins to have an effect, with the flow separating from the streamwise edges further reducing the driving vortex strength. This hypothesis would have implications for the prediction, modelling and control of autorotation. For example, there has been considerable work on predicting the trajectories of windborne projectiles (e.g. Baker, 2007; Tachikawa, 1983a, 1983b), where the flight path of sheet debris is dominated by the effect of autorotation on lift and drag. Sheet debris such as roof tiles tends to be thin, giving a high mean aspect ratio and therefore aerodynamic characteristics likely to be dominated by the shape of the spanwise edges. Conversely, standard containers used as slung loads have low mean aspect ratios, suggesting that autorotation could be controlled by modifications to the streamwise edge geometry. It therefore seems likely that two aspects of autorotation for low mean aspect ratio prisms would repay further investigation. Firstly, the 3D structure of the flowfield that is driving autorotation. Very little had been published on this, until the recent work of Hargreaves et al. (2014) on square flat plates. Secondly, the effect of changes such as local chamfering to the streamwise and spanwise edges.
5. Conclusions Analysis of published data on the autorotation of rectangular prisms has demonstrated that: (a) The very widely used correlation due to Iversen (1979) for autorotation rate performs rather poorly, particularly for higher aspect ratios and thickness/chord ratios. (b) Achieving two-dimensional test conditions is difficult. Test setups with low aspect ratio models rotating relative to the flow boundary (tunnel sidewall or endplate) give consistently lower autorotation rates than either very high aspect ratio models, or models with rotating endplates. (c) The effects of size and freestream velocity on rotation rate can be accounted for using the non-dimensional autorotation rate Ωn. Additional parametric wind tunnel tests of container-like geometries have filled in a number of gaps in the available data on the effects of aspect ratio and thickness/chord ratio on autorotation rate. A reasonably smooth and well-behaved 3D surface has been fitted to the data, permitting the likely autorotation rate of any rectangular prism to be determined. Of particular significance to slung load behaviour is a well-defined region at low aspect ratio and high thickness/chord ratio where autorotation does not occur. Two new parameters for presenting geometry effects on autorotation rate have been proposed: (a) In place of the usual aeronautical definition of aspect ratio AR which varies from 0 to 1, a normalised aspect ratio AR ¼ 1/(1 þ1/AR) has been used which varies from 0 to 1. This allows 2D and 3D data to be conveniently plotted on a single figure, and gives a more nearly linear variation of autorotation rate with aspect ratio. (b) A simple single parameter for the autorotation rate is the geometric mean aspect ratio ARmean ¼b/√(tc), which gives an approximate collapse of the trends in the available data.
100
D.I. Greenwell / J. Wind Eng. Ind. Aerodyn. 132 (2014) 92–100
The mean aspect ratio can be interpreted as a ratio between the strengths of driving vortices shed from the spanwise and streamwise edges respectively. There is a clear change in behaviour for ARmean o1.4, suggesting perhaps a switch from autorotation dominated by the spanwise edge flows to a strong interaction with the streamwise edge flows. Further investigation of (i) the structure of the 3D flows driving autorotation for lower aspect ratio prisms, and (ii) of the effects of local changes to edge geometry, is recommended.
References Baker, C.J., 2007. The debris flight equations. J. Wind Eng. Ind. Aerodyn. 95 (5), 329–353. Bustamante, A.C. and Stone, G.W., 1967. Autorotational Characteristics of Flat Plates and Right Circular Cylinders at Subsonic Speeds, Sandia Laboratory report SCRR-67-778. Bustamante, A.C. and Stone, G.W., 1969. The Autorotation Characteristics of Various Shapes for Subsonic and Hypersonic Flows, AIAA paper AIAA-69-0132, In: Proceedings of the 7th AIAA Aerospace Sciences Meeting, New York. Cheng, S., 1966. An Experimental Investigation of the Autorotation of a Flat Plate (MSc. thesis). University of British Columbia. Cicolani, L.S., Cone, A., Theron, J.N., Robinson, D., Lusardi, J., Tischler, M.B., Rosen, A., Raz, R., 2009. Flight test and simulation of a cargo container slung load in forward flight. J.Am. Helicopter Soc. 54 (3). Cicolani, L.S., Lusardi, J., Greaves, L.D., Robinson, D., Rosen, A., Raz, R., 2010. Flight Test Results for the Motions and Aerodynamics of a Cargo Container and a Cylindrical Slung Load. NASA (Report No. TP-2010-216380). Fletcher, N.H., Tarnopolsky, A.Z., Lai, J.C.S., 2002. Rotational aerophones. J. Acoust. Soc. Am. 111 (3), 1189–1196. Fox, T.A., West, G.S., 1990. On the use of end plates with circular cylinders. Exp. Fluids 9, 237–239. Gerich, D., Eckelmann, H., 1982. Influence of end plates and free ends on the shedding frequency of circular cylinders. J. Fluid Mech. 122, 109–121. Glaser, J.C., Northup, L.L., 1971. Aerodynamic Study of Autorotating Flat Plates. Engineering Research Institute, Iowa State University, Iowa (Report No. ISUERL-Ames 71037). Greenwell, D.I., 2014. Autorotation dynamics of a low aspect-ratio rectangular prism. J. Fluids Struct.. Hargreaves, D.M., Kakimpa, B., Owen, J.S., 2014. The computational fluid dynamics modelling of the autorotation of square, flat plates. J. Fluids Struct. 46, 111–133. Hirata, K., Kondo, M., Nishikawa, T., Shimuzu, K. and Funaki, J., 2009. Basic characteristics of autorotating flat plate in flow. In: Proceedings of the 10th Asian International Conference on Fluid Machinery, Kuala Lumpur. Hirata, K., Hayakawa, M., Funaki, J., 2011. On tumbling of a two-dimensional plate under free flight. J. Fluid Sci. Technol. 6 (2), 177–191. Iversen, J.D., 1979. Autorotating flat-plate wings: the effect of the moment of inertia, geometry and Reynolds number. J. Fluid Mech. 92, 327–348. Izutsu, N., 1986. Numerical and Experimental Study of Dynamics of TwoDimensional Body in Uniform Flow. The Institute of Space and Astronautical Science, Japan (Report No. 619). Kegelman, J. and Roos, F., 1989. Effects of Leading-Edge Shape and Vortex Burst on the Flowfield of a 70 Degree Sweep Delta Wing, AIAA paper AIAA-89-0086. Kirby, D.A., Kirkpatrick, D.L.I., 1969. An Experimental Investigation of the Effect of Thickness on the Subsonic Longitudinal Stability Characteristics of Delta Wings of 701 Sweep-Back. Aeronautical Research Council, UK. (Report No. 3673). Kubo, Y., Kato, K., 1986. The role of end plates in two dimensional wind tunnel tests. Proc. Jpn Soc. Civil Eng. 368, 179–186.
Lugt, H.J., 1982. Analogies Between Oscillation and Rotation of Bodies Induced or Influenced by Vortex Shedding, In: Vortex Motion; Proceedings of a Colloquium Held at Gottingen, pp. 82–96. Lugt, H.J., 1983. Autorotation. Annu. Rev. Fluid Mech. 15, 123–147. Martinez-Vazquez, P., Sterling, M., Baker, C.J., Quinn, A.D., Richards, P.J., 2011. Autorotation of square plates, with application to windborne debris. Wind and Structures 14 (2), 167–186. Noda, H., Nakayama, A., 2003. Free-stream turbulence effects on the instantaneous pressure and forces on cylinders of rectangular cross section. Exp. Fluids 34 (3), 332–344. Okajima, A., 1982. Strouhal numbers of rectangular cylinders. J. Fluid Mech. 123, 379–398. Oshima, K., Oshima, Y., 1983. Lift generation due to vortex shedding, Japan Aerospace Exploration Agency. Institute of Space and Astronautical Science Report 1, 65–72. Poreh, M., Wray, R.N., 1979. On the motion of rectangular prismatic bodies. J. Fluids Eng. 101, 193–199. Prosser, D. and Smith, M., 2013. Navier-Stokes-Based Dynamic Simulations of Sling Loads, In: Proceedings of the 54th AIAA/ASME/ASCE/AHS/ASC Structures, Dynamics, and Materials Conference, Boston, Massachusetts; AIAA 2013-1922. Ramberg, S.E., 1983. “The effects of yaw and finite length upon the vortex wakes of stationary and vibrating circular cylinders”. J. Fluid Mech. 128, 81–107. Raz, R., Rosen, A., Carmeli, A., Lusardi, J., Cicolani, L.S., and Robinson, D., 2008. “Wind tunnel and flight evaluation of stability and passive stabilization of cargo container slung load”, In: Proceedings of the American Helicopter Society 64th Annual Forum, Montreal, Canada. Raz, R., Rosen, A., Cicolani, L.S, Lusardi, J., Gassaway, B. and Thompson, T., 2011. Using wind tunnel tests for slung load clearance, In: Proceedings of the American Helicopter Society 67th Annual Forum, Virginia Beach. Riabouchinsky, D.P., 1935. Thirty years of theoretical and experimental research in fluid mechanics. J. R. Aeronaut. Soc. 39, 377–444 (pp. 282–348). Roger, M., Aubert, S., 2006. Aeroacoustics of the Bullroarer. Acta Acust. 92, 826–841. Sarpakaya, T., 1979. Vortex-induced oscillations: a selective review. J. Appl. Mech. 46, 241–258. Sefat, S.M., 2011, Fluttering and autorotation of a Hinged Vertical Flat Plate Induced by Uniform Current (Ph.D. thesis). Universidade Federal do Rio de Janeiro. Sheldon, D.F. and Pryor, J., 1973a. An Appreciation of the problems in stabilising underslung loads beneath a helicopter, Technical Note AM/37, Royal Military College of Science Sheldon, D.F. and Pryor, J., 1973b. Study in Depth of a single point and two point lateral and tandem suspension of rectangular box loads, Technical Note AM/38, Royal Military College of Science Sheldon, D.F. and Pryor, J., 1973c. A study on the stability and aerodynamic characteristics of particular military loads underslung from a helicopter, Technical Note AM/40, Royal Military College of Science. Skews, B.W., 1990. “Autorotation of rectangular plates”. J. Fluid Mech. 217, 33–40. Smith, E.H., 1970. Autorotating Wings: An Experimental Investigation, technical report for the US Army Research Office project ORA 01954. University of Michigan, Michigan. Smith, E.H., 1971. Autorotating wings: an experimental investigation. J. Fluid Mech. 50 (3), 513–534. Sykes, D.M, 1977. “A new wind tunnel for industrial aerodynamics”. J. Wind Eng. Ind. Aerodyn. 2 (1), 65–78. Tachikawa, M., 1983a. Trajectories of flat plates in uniform flow with application to wind-generated missiles. J. Wind Eng. Ind. Aerodyn. 14 (1-3), 443–453. Tachikawa, M., 1983b. Trajectories of typhoon-generated missiles. J. Wind Eng. 17, 21–30. Tanabe, A., Ito, A., 1996. Flow visualization on flow around autorotating flat plates. J. Vis. Soc. Jpn. 16, 99–102. Theron, J.N., Gordon, R., Rosen, A., Cicolani, L., Duque, E.P.N. and Halsey, R.H., 2006. Simulation of helicopter slung load aerodynamics: wind tunnel validation of two computational fluid dynamics codes, AIAA paper 2006-3374, In: Proceedings of the 36th AIAA Fluid Dynamics Conference, San Francisco, California. Zaki, T.G., Sen, M., Gad-el-Hak, M., 1994. “Numerical and experimental investigation of flow past a freely rotatable square cylinder”. J. Fluids Struct. Vol. 8 (No. 7), 555–582.
Contents lists available at ScienceDirect
Journal of Wind Engineering and Industrial Aerodynamics journal homepage: www.elsevier.com/locate/jweia
Geometry effects on autorotation of rectangular prisms D.I. Greenwell Aircraft Research Association Ltd., Bedford, United Kingdom
art ic l e i nf o
a b s t r a c t
Article history: Received 2 February 2014 Received in revised form 5 June 2014 Accepted 8 June 2014
This paper presents an analysis of published data on the autorotation of rectangular prisms, supplemented by additional wind tunnel tests of container-like shapes typical of helicopter slung loads. Previous widely used correlations for effects of geometry on autorotation rate are shown to perform rather poorly, particularly for higher aspect ratios and thickness/chord ratios. A smooth and well-behaved 3D surface has been fitted to the data, permitting the likely autorotation rate of any rectangular prism to be determined graphically. Of particular significance to the behaviour of slung loads is a well-defined region where autorotation does not occur. Two new parameters for presenting autorotation data have been proposed. A normalised aspect ratio has been used which allows 2D and 3D data to be plotted on a single figure, and gives a more nearly linear variation of autorotation rate with aspect ratio. The geometric mean aspect ratio is a simple single parameter for the autorotation rate, which gives an approximate collapse of trends in the data. This parameter can be interpreted as a ratio between the strengths of driving vortices shed from spanwise and streamwise edges respectively, suggesting that the 3D structure of the flows driving autorotation would repay further investigation. & 2014 Elsevier Ltd. All rights reserved.
Keywords: Autorotation Slung load Rectangular prism
1. Introduction Military and civil helicopters are often used to carry external cargo as a slung load. For many loads, maximum operational flight speeds are limited not by the available power, but by the onset of divergent or limit-cycle oscillations of the load. Rectangular box loads are particularly susceptible to aerodynamic instabilities, especially when lightly loaded (Sheldon and Pryor, 1973b). One load instability that has been the subject of considerable recent interest is the continuous rotation in yaw of the 6 ft 6 ft 8 ft CONEX container when slung from a single point with a swivel joint (Fig. 1, Raz et al., 2008). This is an example of autorotation (Lugt, 1983), a well-known but still rather poorly understood aerodynamic phenomenon. However, flight and wind tunnel tests for other cuboid slung loads show a range of behaviour from steady autorotation to oscillatory yawing motion to a stable broadside-on posture (Raz et al., 2011; Sheldon and Pryor, 1973a), which raises the question of what aspect(s) of the load geometry govern autorotation rate, in particular the conditions for onset of steady rotation. There is an extensive body of experimental and computational work on the autorotation of rectangular prisms, but this is largely concerned with either quasi-two-dimensional (2D) geometries or with low aspect ratio ‘flat plate’ models.
E-mail address: [email protected] http://dx.doi.org/10.1016/j.jweia.2014.06.014 0167-6105/& 2014 Elsevier Ltd. All rights reserved.
For three-dimensional (3D) geometries representative of cargo containers, the only data available is from flight test and scaled dynamic wind tunnel tests of slung loads (Cicolani et al., 2010; Raz et al., 2008, 2011; Sheldon and Pryor, 1973b), which introduces the added complexity of large-amplitude coupled lateral and longitudinal pendulum-like load motions. Some recent work has been done on CFD simulation of rotating CONEX loads (Cicolani et al., 2009; Prosser and Smith, 2013), but validation against flight or wind tunnel data has as yet been limited in extent. Existing semi-empirical correlations for the effect of geometry on autorotation rate (Bustamante and Stone, 1969; Iversen, 1979) are based primarily on data for low-aspect ratio thin wings, and fail to predict behaviour for higher thickness/chord ratios (Hirata et al., 2009, Skews, 1990). In order to separate load sling dynamics from aerodynamics some simple exploratory studies were undertaken at City University London on a fixed-axis autorotation rig. The objectives were to (a) examine initial spin-up and limiting autorotation rate behaviour, and (b) assess the effect of geometry changes on autorotation of 3D bodies. The experimental procedure and some new aspects of autorotation dynamics are described in more detail in Greenwell (2014). This paper reports on the relationship between geometry and autorotation rate for rectangular prisms, presenting a synthesis of new and previously published data.
D.I. Greenwell / J. Wind Eng. Ind. Aerodyn. 132 (2014) 92–100
Nomenclature AR AR ARmean AR0 b c la
aspect ratio, ¼b/c normalised aspect ratio, ¼ 1/(1 þ1/AR) geometric mean aspect ratio, ¼b/√(tc) critical aspect ratio span (m) chord, reference length (m) rig offset from tunnel floor (mm)
2. Autorotation rates for rectangular prisms 2.1. Basic features of autorotation A comprehensive review of autorotation is given by Lugt (1983), covering motions about axes parallel and perpendicular to the flow. The detailed flow physics are rather complex, but it is generally accepted that autorotation of a body with a rectangular cross-section is driven by periodic vortex shedding from the spanwise edges, with spin rates found to coincide approximately with shedding frequencies measured in the wake (Lugt, 1983; Zaki et al., 1994). Links between autorotation and vortex shedding were developed in more depth by Lugt (1982). The specific component of the flow providing the net driving torque required to overcome bearing friction is the vortex shed downstream of the retreating edge (Izutsu, 1986; Zaki et al., 1994), as shown in Fig. 2a. Flow visualisation in Zaki et al. (1994) indicates that in comparison to a stationary prism the vortices rotating in the same sense as the prism are augmented, whilst those rotating in the opposite sense are inhibited. At the orientation shown in Fig. 2b the augmented retreating-edge vortex lies near to the aft face of the body, generating a suction peak close to the upper edge and hence giving a positive torque contribution. The net result is a steady time-averaged spin rate which varies linearly with freestream velocity. The moment of inertia can have a significant impact on autorotation rates for light models, but for the higher relative densities typical of cargo container shapes and construction (Cicolani et al., 2009), autorotation rate is essentially independent of inertia (Smith, 1971). Relatively little attention has been given in the literature to three-dimensional flow effects. Skews (1990) suggested that for very low aspect ratios the main vortices are shed from the radial
r t U V 2D 3D
ω Ωn
93
radius (m), ¼ 1/2√(t2 þ c2) thickness (m) freestream velocity (m/s) tip speed (m/s) two-dimensional three-dimensional angular velocity (rad/s) non-dimensional autorotation rate,¼ ωc/2U
edges rather than the spanwise edge, as shown schematically in Fig. 2b. The aerodynamic driving moment would then be smaller because of the combination of reduced radial vortex strength and reduced moment arm. In contrast, Iversen (1979) correlated the effect of aspect ratio on autorotation of flat plate wings by using the static pitch stability derived from thin-wing attached-flow theory. The underlying assumption was that the time-averaged strength of the driving vortex is related to the strength of the bound spanwise vortex for a stationary wing of the same aspect ratio at low incidence, therefore implying that autorotation is being driven by the spanwise edge vortex. Some 3D unsteady computational fluid dynamics simulations have been done for the CONEX container (Theron et al., 2006), but no details of the flow topology appear to have been published. CFD results in a very recent paper on autorotating square flat plates (Hargreaves et al., 2014) do show a very similar wake structure to that suggested by Skews (1990). 2.2. Autorotation similarity parameters For rectangular prisms,1 geometry is defined as shown in Fig. 3. Conventionally, autorotation rate is taken to be a function of the non-dimensional parameters planform aspect ratio AR ¼b/c and thickness to chord ratio t/c, where b is the span and chord cZthickness t. An additional parameter used in this work is the normalised aspect ratio AR, AR ¼
AR 1 ¼ AR þ 1 1 þ 1=AR
ð1Þ
which is a simplified version of a semi-empirical aspect ratio correlation from Iversen (1979),2. The normalised aspect ratio asymptotes to AR as AR-0 and to 1 as AR-1, making it a convenient means of presenting low and high aspect ratio behaviour on a single plot. Autorotation rate data is usually presented as non-dimensional spin rate plotted against freestream velocity or Reynolds Number. Conventions vary considerably, but the most common form is
Ωn ¼
ωc ¼ f nðUÞ 2U
ð2Þ
where c is a reference length (typically the chord for a thin wing model, or the longest streamwise edge for a rectangular crosssection) and U is the freestream velocity. Comparisons of wind tunnel and full-size flight test data for autorotating slung container loads (Raz et al., 2011) show very good agreement of rotation rates scaled using Eq. (2), even for an 11:1 variation in length scale.
Fig. 1. Blackhawk with CONEX slung load (Raz et al., 2008).
1 The terminology in the fluid dynamics and aerodynamics communities for container-like geometries is rather inconsistent. Alternative terms that have been used are ‘rectangular prism’, ‘rectangular cylinder’, ‘rectangular parallelepiped’, ‘cuboid’, or just ‘box’. 2 see Eq. (6b)
94
D.I. Greenwell / J. Wind Eng. Ind. Aerodyn. 132 (2014) 92–100
2.3. Semi-empirical correlations for autorotation rate retreating edge
An early attempt at relating autorotation rate to 3D wing geometry was made by Bustamante and Stone (1969), who gave a correlation for the steady autorotation rate for thin flat-plate wings as
U driving vortices
ω
Ωn ðARÞ ¼ 1 e 0:2AR
advancing edge
Curiously, the supporting figure in Bustamante and Stone shows no direct comparison with experimental results; detailed examination of data in this reference suggests that Eq. (5) was in fact a rather poor fit, particularly for lower aspect ratios. The most widely used correlation for aspect ratio and thickness/chord ratio effects was developed by Iversen (1979), based on the premise that pitch stability might be a valid correlation parameter for autorotation. Iversen's correlation is
ω
U
spanwise vortex
radial vortex
Ωn ðAR; t=cÞ ¼ f 1 ðARÞf 2 ðt=cÞ
Fig. 2. (a) 2D flow topology for an autorotating cylinder (redrawn from Zaki et al., 1994), (b) 3D driving vortex structure (Skews, 1990).
thickness, t
f 1 ðARÞ ¼
wind speed, U aspect ratio AR = b/c chord, c Fig. 3. Model geometry definitions.
ð3Þ
where t and c are the model thickness and chord respectively, and r is the maximum model radius (¼ 1/2√(t2 þ c2)). However, as demonstrated by Iversen (1979), these simple non-dimensional forms can give rise to spurious Reynolds Number effects. This is due to a combination of a zero shift in the linear variation of spin rate with freestream velocity due to bearing friction (Iversen, 1979), non-linear behaviour at low spin rates (Cheng, 1966), and coupling with structural modes at high spin rates (Greenwell, 2014). On the basis of a rather ad-hoc analysis of the equations of motion of an autorotating body, Iversen concluded that it is justifiable to account for the effects of bearing friction by taking the slope of the linear section, giving c ∂ω 2 ∂U
2 þ ð4 þ AR2 Þ1=2
( f 2 ðt=cÞ ¼
Ωn ¼
AR
#" 2
AR AR þ 0:595
0:76 #)2=3 ð6bÞ
The first term in f1 is an approximation to the lift-curve slope based on thin-wing theory, and the second an empirical expression for the aerodynamic centre. The overall function f1 can be thought of as corresponding to the static “aerodynamic overturning moment”, and varies from 0 for AR ¼0–1 for 2D configurations (AR ¼1). The function f2 was fitted to the effect of varying thickness/chord ratio from 0.0054 to 0.5, for low aspect ratio wings with AR ¼0.25–4. Aspect ratio effects were accounted for using f1.
span, b
An alternative parameter is the tip speed ratio sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 V rω t n ¼ Ω 1þ U c U
ð6aÞ
where f1 and f2 are functions of aspect ratio and thickness/chord ratio respectively, derived from curve-fits to experimental data from Glaser and Northup (1971).3 The function f1 was fitted to the effect of varying aspect ratio from 0.25 to 4 for thin wings, with t/c 0.03. ("
spin rate,
ð5Þ
ð4Þ
Autorotation rates presented in this work are non-dimensionalised using Eq. (4).
2 ) 1 1 0:0246 ln 0:329 ln t=c t=c
ð6cÞ
This function is an entirely empirical relationship for the variation of autorotation rate with the inverse of the thickness/ chord ratio, and varies from 1 for thin wings (t/c ¼0) to 0 for square cross-section (t/c ¼1). There are obvious problems with the limiting behaviour of this correlation: an infinite rotation rate for a thin wing is clearly non-physical, whilst square prisms in general autorotate quite readily. Fig. 4 presents the performance of this correlation for the data originally fitted by Eq. (6), shown as filled circles, and for more recent 2D and 3D autorotation measurements (Table 1). Iversen (1979) described the fit as ‘fairly good’, which is a reasonable assessment for the original data, but when applied to more recent data for higher aspect ratios and thickness/chord ratios the correlation clearly performs poorly. The deviation is particularly large for nominally 2D testing, where there is also a very large scatter in the data; for example reported non-dimensional autorotation rates for wings with thickness/chord ratios of 0.1–0.15 vary from 0.45 (Skews, 1990) to 1.1 (Smith, 1971). 3 The original reference for this data has been unobtainable – the data in Fig. 4 has been extracted from Figs. 6 and 7 in Iversen (1979)
D.I. Greenwell / J. Wind Eng. Ind. Aerodyn. 132 (2014) 92–100
2.4. Effect of endplates One likely major contributor to the scatter for 2D configurations is the difficulty in achieving ‘true’ 2D conditions for an unsteady separated flow. For example, previous work on drag and vortex shedding for 2D bluff bodies has shown in many cases a very significant sensitivity to the end conditions (Kubo and Kato, 1986; Gerich and Eckelmann, 1982; Ramberg, 1983). Even with careful endplate design, aspect ratios of greater than 7 can be required to achieve 2D flow conditions on model centrelines (Fox and West, 1990). Typical nominally 2D wind or water tunnel test configurations for autorotation experiments include: models spanning a closed test
95
section (Skews, 1990; Zaki et al., 1994), models spanning an open jet test section with endplates (Hirata et al, 2009), models with ‘fixed’ endplates (Smith, 1971, Cheng, 1966) and models with ‘rotating’ endplates (Smith, 1971, Cheng, 1966). ‘Fixed’ endplates are mounted to the tunnel, and are therefore stationary. These tend to be large and non-circular in planform, often based on optimum configurations determined for 2D aerofoil testing. ‘Rotating’ endplates are attached to the model and rotate with it. These are circular in planform and tend to be smaller. In three cases (Skews, 1990; Hirata et al. 2009 and Smith, 1971) the effect of varying thickness/chord ratio was also investigated, giving the trends shown in Fig. 5. Open symbols denote models with endplates: for rotating endplates and ○ for fixed endplates. Filled symbols denote 2D test sections of one form or
1.2
1.2 Glaser & Northup
Iversen correlation
0.8
0.4
0.0 0.0
0.2
rotating endplates 2D test section, high AR fixed endplates 2D test section, low AR 2D open jet, low AR 2D flume, low AR effect of endplates, AR = 3
rotating endplate
later 2D data
autorotation rate, Ω* = c/2U
autorotaion rate, Ω* = ωc/2U
later 3D data
0.4
0.6
0.8
1.0
0.8 fixed endplate no endplate
0.4
0.0 0.0
0.2
f1 (AR) × f2 (t/c)
0.4 0.6 thickness/chord ratio t/c
0.8
Fig. 5. Thickness effects for nominal ‘2D’ tests.
Fig. 4. Iversen's correlation for autorotation rate.
Table 1 Published data on autorotating rectangular prisms. Reference
AR
t/c
Slung load /fixed axis
End condition
Wing section
Bustamante and Stone (1967, 1969) Cheng (1966) Cicolani et al. (2009) Fletcher et al (2002) Glaser and Northup (1971)
0.167–6
Fixed axis
–
Flat plate
3, 2D 0.76 2D? 0.25–4
0.008– 0.045 0.047 0.72 0.025 0.005–0.5
Fixed axis Slung load Fixed axis Fixed axis
Fixed endplate – Spanning open jet –
chamfered flat plate Rectangular Flat plate ?
Hirata et al (2009)
2D
0.3–0.8
Fixed axis
Hirata et al. (2011) Martinez-Vasquez et al. (2011) Izutsu (1986), Oshima and Oshima (1983) Poreh and Wray (1979)
2–20 1
0.3 Free fall 0.025,0.083 Fixed axis
Large fixed endplates in Rectangular open jet – Flat plate – Flat plate
2D
0.15
Fixed axis
Rotating endplate
Elliptical
Geometric AR ¼ 3
2D
0.3
Fixed axis
Flat plate
Geometric AR ¼ 6.9
Raz et al. (2011) Riabouchinsky (1935)
0.4–1.2 4
1 0.133,1
Slung load Fixed axis
Spanning closed test Section – –
TRIO container
Roger and Aubert (2006) Sefat (2011)
2D? 2D
0.067 0.008
Fixed axis Fixed axis
Rectangular Chamfered flat plate, Cruciform Elliptical Flat plate
Geometric AR ¼ 4.7 Geometric AR ¼ 0.83
Sheldon and Pryor (1973a) Sheldon and Pryor (1973b) Sheldon and Pryor (1973c) Skews (1990)
0.333,1 0.125–0.5 0.044 2D
0.333,1 0.0625–0.5 0.077 0.1–1
Slung load Slung load Slung load Fixed axis
Rectangular Rectangular Flat plate Rectangular
Model of blade box Geometric AR ¼ 1.7–3.3
Smith (1970, 1971)
2D
Fixed axis
Tachikawa (1983a, 1983b) Tanabe and Ito (1996) Zaki et al. (1994)
1,2 3 2D
0.02–0.04, 0.15 0.029–1 0.038 1
Fixed axis Fixed axis Fixed axis
Spanning open jet Spanning free-surface flume
Spanning closed test section Rotating endplate – – Spanning closed test section
Flat plate, elliptical Rectangular Flat plate Rectangular
Comments
Also tested without endplates CONEX container Geometric AR ¼ 4.2 Original report unobtainable, Data taken from Iversen (1979) Geometric AR ¼ 10
Also tested with a fixed endplate Geometric AR ¼ 3, 6
Geometric AR ¼ 20
1.0
96
D.I. Greenwell / J. Wind Eng. Ind. Aerodyn. 132 (2014) 92–100
ω
II U
driving vortex I
U III
b
I 250mm
driving vortex III Fig. 6. Autorotating wing with asymmetric chamfer (after Riabouchinsky, 1935).
another: for high aspect ratio models and , , . This function is an entirely empirical relationship for low aspect ratio models. Eq. (6c) predicts autorotation rate falling rapidly as thickness/ chord is increased, becoming zero at t/c ¼1 (square cross-section). Instead, we see an initial reduction in rate which then levels out to a nearly constant value for t/c40.6. Autorotation rates fall in a band which appears to narrow as t/c increases. The lower boundary of the band corresponds to test configurations where the model is moving relative to the flow boundary and end effects may be significant – that is, low aspect ratio models spanning open and closed test sections, and wings with fixed endplates. The upper boundary corresponds to tests where end effects are less likely to be significant – that is, very high aspect ratio ( Z 10) models and wings with rotating endplates. Extrapolation from 3D test results tends to give autorotation rates towards the upper boundary, although the data coverage at intermediate aspect ratios is too sparse to be confident of the limiting values. An indication of the impact of model motion relative to the flow boundary is given by two datasets where endplate configuration was varied for low aspect ratio thin wings, indicated on Fig. 5 by the vertical dash-dot line at the left hand end. Cheng (1966) tested a flat plate wing (t/c¼ 0.047, AR ¼3) with and without a fixed (non-rotating) endplate. Smith (1970, 1971) tested a similar flat plate wing (t/c¼ 0.04, AR ¼2.6) with rotating and non-rotating endplates. For Cheng's baseline ‘no endplate’ model Ωn E0.52; adding a large fixed endplate gave a small increase to Ωn E0.54. For Smith's model, a small fixed endplate with diameter 1.3c gave Ωn E0.60; allowing the endplate to rotate with the model increased Ωn to 0.84. Reducing the rotating endplate diameter to 1.0c gave a further increase to Ωn E 1.0. Clearly, achieving 2D flow conditions is not straightforward, particularly for lower thickness/chord ratios. Configurations with the model rotating relative to an endplate or sidewall have the worst performance, but there is insufficient data to be clear as to whether this is a gap effect, or due to the interaction between the moving model and stationary boundary layer. The autorotation motion should at least constrain the driving vortices to be shed parallel to the model, as occurs with vortex shedding lock-in (Sarpakaya, 1979), so that spanwise asymmetry is probably not a contributing factor. 2.5. Effect of leading-edge geometry Continuing with the analogy to vortex shedding proposed by Lugt (1982), one would expect the strength of the driving vortex (and hence the autorotation rate) to depend strongly on the geometry of the spanwise edges. Fig. 5, and 3D results presented later in Section 4, gives some support to this hypothesis;
la
Fig. 7. Model mounting arrangement in T7 wind tunnel.
autorotation rate is most sensitive to wing thickness for thin wings (t/c o0.2). A similar sensitivity to small variations in edge geometry can be seen for the leading-edge vortices generated by swept delta wings (e.g. Kirby and Kirkpatrick, 1969). However, no parametric study has been undertaken on the effect of edge shape. Most published autorotation datasets are for models with simple rectangular cross-sections. The exceptions are a handful of tests of 15% thick elliptical cross-section wings (Smith, 1970, Izutsu, 1986, Oshima 1983), and two studies of thin wings with chamfered edges. Cheng (1966) tested a 4.7% thick flat plate wing with the leading and trailing-edge chamfered at 121 on one face only. The combination of small chamfer angle and longitudinal symmetry gave an autorotation rate close to a similar squareedged wing tested by Smith (1970). Riabouchinsky (1935) tested a 13% thick 3D wing (AR¼4) with square edges and with an asymmetric 301 chamfer (Fig. 6). Autorotation rate for the square-edged wing (case II in Fig. 6) is consistent with other published results at similar aspect ratios and thicknesses. For the chamfered wing the asymmetry resulted in an autorotation rate that depended on direction of rotation: 18% slower than the square-edged wing with the chamfered face to windward on the retreating side (case I in Fig. 6) and 20% faster for rotation in the opposite direction with the flat face to windward (case III). This is consistent with the effect of leading-edge chamfer on the strength and position of delta wing vortices (e.g. Kegelman and Roos, 1989) – the driving vortex on the retreating side is stronger with the flat face to windward (case III).
3. Additional 3D data 3.1. Wind tunnel and models Testing was undertaken in the T7 industrial wind tunnel at City University London. This is a closed return wind tunnel, with working section dimensions of 3 m 1.5 m 8 m and a speed range of 0–25 m/s. Turbulence levels are of the order of 3% (Sykes, 1977), high by aeronautical standards but not high enough to have a significant effect on vortex shedding frequencies for rectangular cylinders (Noda and Nakayama, 2003). Reynolds Numbers based on chord were 16,000 to 45,000, well above the critical range for vortex shedding (Okajima, 1982). Autorotation tests were carried out using models mounted on a modified Vector Instruments A101M anemometer body (Fig. 7) which provided a
D.I. Greenwell / J. Wind Eng. Ind. Aerodyn. 132 (2014) 92–100
Table 2 LegoTM model geometries tested. AR
t/c
Ωn
AR
t/c
Ωn
0.2 0.3 0.4 0.5 0.6 0.7 0.24 0.36 0.3 0.45 0.6 0.75 0.3 0.4 0.5 0.6 0.7
0.167 0.167 0.167 0.167 0.167 0.167 0.2 0.2 0.25 0.25 0.25 0.25 0.333 0.333 0.333 0.333 0.333
0.06 0.107 0.13 0.168 0.176 0.189 0.05 0.1 0 0.1 0.123 0.163 0 0 0.055 0.090 0.120
0.36 0.48 0.6 0.72 0.84 1.08 0.45 0.6 0.75 1.05 1.35 0.4 0.8 0.8 1 1.2 1.4 1.6 1.8 2 3
0.4 0.4 0.4 0.4 0.4 0.4 0.5 0.5 0.5 0.5 0.5 0.667 0.667 1 1 1 1 1 1 1 1
0 0 0.088 0.127 0.147 0.228 0 0 0.077 0.218 0.321 0 0.080 0 0.17 0.213 0.278 0.285 0.289 0.341 0.374
small-amplitude oscillations to fully-developed autorotation in less than 30 s. Because of the small size of the model relative to the tunnel working section no blockage corrections were applied. 3.2. Data validation Fig. 8 shows typical experimental results for the variation of autorotation rate with freestream velocity, for four models with a thickness/chord ratio of 0.5 and aspect ratios varying from 0.6 to 1.08. The mid-range behaviour is consistent with Iversen's discussion, with a repeatable linear trend and a small zero offset due to bearing friction. For the lowest aspect ratio there is a minimum critical velocity below which autorotation does not occur, as reported by Cheng (1966). For the higher aspect ratios shown in Fig. 8 the critical velocity is below the lowest achievable tunnel speed. For some configurations the autorotation characteristics at the extremes of the speed range become highly non-linear locking-in to structural natural frequencies at high speeds, and displaying static hysteresis at low speeds. These aspects are reported in more detail in Greenwell (2014); this paper focuses on the effect of geometry on autorotation rate in the linear regime. Fig. 9 compares autorotation rates determined using Eq. (4) with the limited amount of ‘overlapping’ data available (Table 1). Fig. 9a shows the effect of aspect ratio for cross-sections with thickness/chord ratios of 0.3 (Hirata et al., 2011) and 1.0 (Tachikawa, 1983b; Zaki et al., 1994; Riabouchinsky, 1935; Sheldon and Pryor, 1973a; Raz et al., 2011),
0.6 present work, t/c = 1 fixed axis models, t/c = 1
autorotation rate, Ω* = ωc/2U
very low-friction bearing and a 13 pulse/rev output from an optical sensor. Pulse frequency was measured using a Racal Instruments 1990 Universal Counter set to a 2 s averaging time and hence giving spin rate averaged over typically 10–60 rotations. Freestream velocity was measured using a standard NPL pitot-static probe mounted on the sidewall at the start of the working section, and a Furness Controls FCO318 digital pressure indicator. Tunnel speed and autorotation rate were recorded manually, with an estimated uncertainty of the order of 7 2% in speed and 71% in angular velocity. This is consistent with repeatability tests for a single configuration, which gave a standard deviation of 2% for the non-dimensional autorotation rate Ωn. Models were made of standard Lego™ bricks, permitting thickness, chord and span to be varied quickly. Aspect ratio was varied between 0.2 and 3, thickness/chord ratio was varied between 0.167 and 1 (Table 2). Tests reported here were carried out with the studs on the upper face of the models ground off; however, some exploratory tests showed no significant difference in autorotation behaviour between modified and unmodified bricks. Average density of the bricks used was approximately 500 kg/m3, giving moments of inertia well above the critical range identified by Smith (1971). For these geometries rotation started spontaneously, building up rapidly from
97
underslung loads, t/c = 1 present work, t/c = 0.33 free-fall, t/c = 0.3
0.4
0.2
t/c = 0.33
0.0 0.0
t/c = 1.0 (square section)
0.2
0.4
0.6
0
0.25
0.5
1
2
4
2D
0.4
autorotation rate, Ω* = ωc/2U
t/c = 0.4
rotation rate, ω (rad/s)
1.0
aspect ratio, AR
40
30
20
10
0
0.8
normalised aspect ratio, 1/(1+1/AR)
AR = 0.6 AR = 0.72 AR = 0.84 AR = 1.08
increasing AR
present work, AR = 0.5 Glaser & Northup, AR = 0.5
0.3
0.2
0.1
0.0 0
5
1 10
15
20
freestream velocity, U (m/s) Fig. 8. Typical variation of rotation rate with freestream velocity and aspect ratio for LegoTM models.
0.0
0.2
0.4
0.6
0.8
1.0
thickness/chord ratio , t/c Fig. 9. Comparison with published autorotation data for prisms: (a) effect of AR for t/c ¼0.33 and 1.0, (b) effect of t/c for AR ¼0.5.
98
D.I. Greenwell / J. Wind Eng. Ind. Aerodyn. 132 (2014) 92–100
the data, giving the mesh in Fig. 10a and the corresponding contours in Fig. 10b. Data for 2D tests (B) was not used to generate the fitted surface, because of the issues discussed in Section 2.4 above. The variation of autorotation rate with aspect ratio is nearly linear, although there are signs of a discontinuity in slope at aspect ratios of around 0.8–1. For higher aspect ratios the variation with thickness/chord ratio is similar to the 2D case (Fig. 5): autorotation rate initially drops rapidly as thickness is increased before levelling out to a nearly constant value. As noted previously, extrapolating 3D trends in Fig. 10 to the 2D case (AR ¼ 1) gives values towards the upper boundary in Fig. 5. For lower aspect ratios (o 0.8) autorotation rate does not level out, but falls to zero as thickness is increased. The resulting low aspect ratio autorotation boundary is shown in more detail in Fig. 11. Crosses ( ) and open symbols (◯, ☐) indicate published data (Table 1) for configurations that do, and do not, rotate respectively. Filled symbols indicate extrapolated ‘zero rotation’ cases from parametric variations of aspect ratio ( , Table 2) and thickness/chord ratio ( , Glaser and Northup, 1971), see for example Fig. 9. The autorotation boundary is reasonably well defined for higher thicknesses; however, at low thicknesses there is some uncertainty as to whether the boundary extends to the origin, or intercepts the x-axis at a non-zero thickness.
4.2. Autorotation rate correlations Correlations for the effects of thickness/chord ratio on autorotation are constrained by symmetry requirements. For a rectangular prism the choice of which edge length to take as thickness and which as chord is to some extent arbitrary, so for a general correlating function f(t/c) should equal f(c/t) and hence df/d(t/c)¼0 at t/c ¼1. Fig. 10b shows that a surface fitted to experimental data for autorotation rate comes close to meeting this requirement, with the contour lines having near-zero slope for t/c 40.5. Fig. 11 indicates that there is a critical aspect ratio AR0 below which a rectangular prism will not autorotate. An empirical relationship for the variation of AR0 with t/c that approximately meets the symmetry requirement AR0 (t/c)¼AR0 (c/t) is t=c 2 1:05 1 1:0026e 0:89 AR0 ¼ ð7Þ t=c Eq. (7) is shown as a dashed line in Fig. 11: the boundary crosses the x-axis at t/c ¼0.045 and has zero slope at t/c ¼1. Attempts to find a general function that would correlate autorotation rate were not particularly successful. More complex exponential relationships of the general form of Eq. (7) can be fitted to the data shown in Fig. 10a, but these are rather unwieldy and lack any Fig. 10. Effect of aspect ratio and thickness/chord ratio on autorotation rate – fit to experimental data (J, - Table 1, 3DB- Table 1, 2D☐- Table 2).
4. Geometry effects revisited 4.1. Effect of aspect ratio and thickness/chord ratio Fig. 10a shows autorotation rates from published data (Table 1: and results from the present work (Table 2: ☐) plotted against normalised aspect ratio and thickness/chord ratio. Cases with no rotation are denoted by . A smoothed 3D surface was fitted to J, ,B)
1.00
aspect ratio, AR
while Fig. 9b shows the effect of thickness/chord ratio for planforms with aspect ratio AR¼ 0.5 (Glaser and Northup, 1971). Both magnitudes and trends with aspect ratio and thickness compare very well with published data. Fig. 9a also demonstrates that the normalised aspect ratio AR is a suitable parameter for presenting 3D geometry effects, giving near-linear trends for t/c¼0.3 and 1.0.
1.25
0.75
0.50 rotation - fixed axis rotation - underslung loads no rotation - underslung loads Lego models extrapolated Glaser & Northup extrapolated
0.25
0.00
0
0.2
0.4
0.6
thickness/chord ratio,t/c Fig. 11. Autorotation limits.
0.8
1
D.I. Greenwell / J. Wind Eng. Ind. Aerodyn. 132 (2014) 92–100
physical significance. An alternative is shown in Fig. 12, which relates autorotation rate to a more general form of the aspect ratio. As discussed in Section 2, autorotation rate is governed by the strength of the driving vortex shed by the retreating spanwise edge, which in turn should be a function of the aspect ratio. However, for a rectangular cross-section the appropriate aspect ratio to consider would depend on which edge was shedding the vortex. All four spanwise edges contribute to the time-averaged driving moment; therefore one might expect the average moment to depend on the average aspect ratio of the top/bottom and front/ rear faces, where ARtop
¼
b ; c
ARf ront
b t
¼
ð8aÞ
The geometric mean was found to give a much better collapse of the data than the arithmetic mean, hence qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b AR ARmean ¼ ARtop ARf ront ¼ pffiffiffiffiffi ¼ pffiffiffiffiffiffiffi ð8bÞ tc t=c which also meets the symmetry constraint discussed above. To be entirely consistent, one might also expect that a similar argument should apply to the choice of reference length in the nondimensional autorotation rate Ωn; however, it was found that the best results were obtained using the chord c (e.g. Eq. (2)) rather than an average value such as the radius (e.g. Eq. (3)). This may indicate that although the average strength of the driving vortices is related to the average aspect ratio, the corresponding average moment arm is fixed by the length of the deepest face. Fig. 12 shows the variation of non-dimensional autorotation rate for 3D prisms, as defined in Eq. (4), plotted against the mean aspect ratio, as defined in Eq. (8b). There is still a significant degree of scatter, but when compared with Fig. 3 there is a much better collapse of the general trends from different datasets. There is a well-defined and consistent break in the general slope at a mean aspect ratio of around 1.4; above this break point a rough approximation for the autorotation rate is
Ωn 0:21 AR0:33 mean
ð9Þ
while below it the autorotation rate falls off rapidly with reducing aspect ratio and/or increasing thickness. 4.3. Suggested relationship to flow physics Referring back to Fig. 2b and Eq. (8b), the mean aspect ratio ARmean could also be interpreted as a ratio between the strength of
1
autorotation rate, Ω* = ω c/2U
transition region ?
0.1 fully developed autorotation
Glaser & Northup later 3D data Lego models
0.01 0.1
1
10
ARmean = b/√(tc) Fig. 12. General effect of ‘aspect ratio’ on autorotation rate.
100
99
the spanwise driving vortices (proportional to spanwise edge length b) and the strength of the radial or edge vortices (proportional to mean streamwise edge length √(tc)). Fig. 12 then suggests a possible explanation for the behaviour seen in Figs. 10 and 11. For higher mean aspect ratios autorotation rate is driven by the flow separating from the spanwise edges, with 3D flow effects confined to a spanwise variation in driving vortex strength. At lower mean aspect ratios the flow around the end faces begins to have an effect, with the flow separating from the streamwise edges further reducing the driving vortex strength. This hypothesis would have implications for the prediction, modelling and control of autorotation. For example, there has been considerable work on predicting the trajectories of windborne projectiles (e.g. Baker, 2007; Tachikawa, 1983a, 1983b), where the flight path of sheet debris is dominated by the effect of autorotation on lift and drag. Sheet debris such as roof tiles tends to be thin, giving a high mean aspect ratio and therefore aerodynamic characteristics likely to be dominated by the shape of the spanwise edges. Conversely, standard containers used as slung loads have low mean aspect ratios, suggesting that autorotation could be controlled by modifications to the streamwise edge geometry. It therefore seems likely that two aspects of autorotation for low mean aspect ratio prisms would repay further investigation. Firstly, the 3D structure of the flowfield that is driving autorotation. Very little had been published on this, until the recent work of Hargreaves et al. (2014) on square flat plates. Secondly, the effect of changes such as local chamfering to the streamwise and spanwise edges.
5. Conclusions Analysis of published data on the autorotation of rectangular prisms has demonstrated that: (a) The very widely used correlation due to Iversen (1979) for autorotation rate performs rather poorly, particularly for higher aspect ratios and thickness/chord ratios. (b) Achieving two-dimensional test conditions is difficult. Test setups with low aspect ratio models rotating relative to the flow boundary (tunnel sidewall or endplate) give consistently lower autorotation rates than either very high aspect ratio models, or models with rotating endplates. (c) The effects of size and freestream velocity on rotation rate can be accounted for using the non-dimensional autorotation rate Ωn. Additional parametric wind tunnel tests of container-like geometries have filled in a number of gaps in the available data on the effects of aspect ratio and thickness/chord ratio on autorotation rate. A reasonably smooth and well-behaved 3D surface has been fitted to the data, permitting the likely autorotation rate of any rectangular prism to be determined. Of particular significance to slung load behaviour is a well-defined region at low aspect ratio and high thickness/chord ratio where autorotation does not occur. Two new parameters for presenting geometry effects on autorotation rate have been proposed: (a) In place of the usual aeronautical definition of aspect ratio AR which varies from 0 to 1, a normalised aspect ratio AR ¼ 1/(1 þ1/AR) has been used which varies from 0 to 1. This allows 2D and 3D data to be conveniently plotted on a single figure, and gives a more nearly linear variation of autorotation rate with aspect ratio. (b) A simple single parameter for the autorotation rate is the geometric mean aspect ratio ARmean ¼b/√(tc), which gives an approximate collapse of the trends in the available data.
100
D.I. Greenwell / J. Wind Eng. Ind. Aerodyn. 132 (2014) 92–100
The mean aspect ratio can be interpreted as a ratio between the strengths of driving vortices shed from the spanwise and streamwise edges respectively. There is a clear change in behaviour for ARmean o1.4, suggesting perhaps a switch from autorotation dominated by the spanwise edge flows to a strong interaction with the streamwise edge flows. Further investigation of (i) the structure of the 3D flows driving autorotation for lower aspect ratio prisms, and (ii) of the effects of local changes to edge geometry, is recommended.
References Baker, C.J., 2007. The debris flight equations. J. Wind Eng. Ind. Aerodyn. 95 (5), 329–353. Bustamante, A.C. and Stone, G.W., 1967. Autorotational Characteristics of Flat Plates and Right Circular Cylinders at Subsonic Speeds, Sandia Laboratory report SCRR-67-778. Bustamante, A.C. and Stone, G.W., 1969. The Autorotation Characteristics of Various Shapes for Subsonic and Hypersonic Flows, AIAA paper AIAA-69-0132, In: Proceedings of the 7th AIAA Aerospace Sciences Meeting, New York. Cheng, S., 1966. An Experimental Investigation of the Autorotation of a Flat Plate (MSc. thesis). University of British Columbia. Cicolani, L.S., Cone, A., Theron, J.N., Robinson, D., Lusardi, J., Tischler, M.B., Rosen, A., Raz, R., 2009. Flight test and simulation of a cargo container slung load in forward flight. J.Am. Helicopter Soc. 54 (3). Cicolani, L.S., Lusardi, J., Greaves, L.D., Robinson, D., Rosen, A., Raz, R., 2010. Flight Test Results for the Motions and Aerodynamics of a Cargo Container and a Cylindrical Slung Load. NASA (Report No. TP-2010-216380). Fletcher, N.H., Tarnopolsky, A.Z., Lai, J.C.S., 2002. Rotational aerophones. J. Acoust. Soc. Am. 111 (3), 1189–1196. Fox, T.A., West, G.S., 1990. On the use of end plates with circular cylinders. Exp. Fluids 9, 237–239. Gerich, D., Eckelmann, H., 1982. Influence of end plates and free ends on the shedding frequency of circular cylinders. J. Fluid Mech. 122, 109–121. Glaser, J.C., Northup, L.L., 1971. Aerodynamic Study of Autorotating Flat Plates. Engineering Research Institute, Iowa State University, Iowa (Report No. ISUERL-Ames 71037). Greenwell, D.I., 2014. Autorotation dynamics of a low aspect-ratio rectangular prism. J. Fluids Struct.. Hargreaves, D.M., Kakimpa, B., Owen, J.S., 2014. The computational fluid dynamics modelling of the autorotation of square, flat plates. J. Fluids Struct. 46, 111–133. Hirata, K., Kondo, M., Nishikawa, T., Shimuzu, K. and Funaki, J., 2009. Basic characteristics of autorotating flat plate in flow. In: Proceedings of the 10th Asian International Conference on Fluid Machinery, Kuala Lumpur. Hirata, K., Hayakawa, M., Funaki, J., 2011. On tumbling of a two-dimensional plate under free flight. J. Fluid Sci. Technol. 6 (2), 177–191. Iversen, J.D., 1979. Autorotating flat-plate wings: the effect of the moment of inertia, geometry and Reynolds number. J. Fluid Mech. 92, 327–348. Izutsu, N., 1986. Numerical and Experimental Study of Dynamics of TwoDimensional Body in Uniform Flow. The Institute of Space and Astronautical Science, Japan (Report No. 619). Kegelman, J. and Roos, F., 1989. Effects of Leading-Edge Shape and Vortex Burst on the Flowfield of a 70 Degree Sweep Delta Wing, AIAA paper AIAA-89-0086. Kirby, D.A., Kirkpatrick, D.L.I., 1969. An Experimental Investigation of the Effect of Thickness on the Subsonic Longitudinal Stability Characteristics of Delta Wings of 701 Sweep-Back. Aeronautical Research Council, UK. (Report No. 3673). Kubo, Y., Kato, K., 1986. The role of end plates in two dimensional wind tunnel tests. Proc. Jpn Soc. Civil Eng. 368, 179–186.
Lugt, H.J., 1982. Analogies Between Oscillation and Rotation of Bodies Induced or Influenced by Vortex Shedding, In: Vortex Motion; Proceedings of a Colloquium Held at Gottingen, pp. 82–96. Lugt, H.J., 1983. Autorotation. Annu. Rev. Fluid Mech. 15, 123–147. Martinez-Vazquez, P., Sterling, M., Baker, C.J., Quinn, A.D., Richards, P.J., 2011. Autorotation of square plates, with application to windborne debris. Wind and Structures 14 (2), 167–186. Noda, H., Nakayama, A., 2003. Free-stream turbulence effects on the instantaneous pressure and forces on cylinders of rectangular cross section. Exp. Fluids 34 (3), 332–344. Okajima, A., 1982. Strouhal numbers of rectangular cylinders. J. Fluid Mech. 123, 379–398. Oshima, K., Oshima, Y., 1983. Lift generation due to vortex shedding, Japan Aerospace Exploration Agency. Institute of Space and Astronautical Science Report 1, 65–72. Poreh, M., Wray, R.N., 1979. On the motion of rectangular prismatic bodies. J. Fluids Eng. 101, 193–199. Prosser, D. and Smith, M., 2013. Navier-Stokes-Based Dynamic Simulations of Sling Loads, In: Proceedings of the 54th AIAA/ASME/ASCE/AHS/ASC Structures, Dynamics, and Materials Conference, Boston, Massachusetts; AIAA 2013-1922. Ramberg, S.E., 1983. “The effects of yaw and finite length upon the vortex wakes of stationary and vibrating circular cylinders”. J. Fluid Mech. 128, 81–107. Raz, R., Rosen, A., Carmeli, A., Lusardi, J., Cicolani, L.S., and Robinson, D., 2008. “Wind tunnel and flight evaluation of stability and passive stabilization of cargo container slung load”, In: Proceedings of the American Helicopter Society 64th Annual Forum, Montreal, Canada. Raz, R., Rosen, A., Cicolani, L.S, Lusardi, J., Gassaway, B. and Thompson, T., 2011. Using wind tunnel tests for slung load clearance, In: Proceedings of the American Helicopter Society 67th Annual Forum, Virginia Beach. Riabouchinsky, D.P., 1935. Thirty years of theoretical and experimental research in fluid mechanics. J. R. Aeronaut. Soc. 39, 377–444 (pp. 282–348). Roger, M., Aubert, S., 2006. Aeroacoustics of the Bullroarer. Acta Acust. 92, 826–841. Sarpakaya, T., 1979. Vortex-induced oscillations: a selective review. J. Appl. Mech. 46, 241–258. Sefat, S.M., 2011, Fluttering and autorotation of a Hinged Vertical Flat Plate Induced by Uniform Current (Ph.D. thesis). Universidade Federal do Rio de Janeiro. Sheldon, D.F. and Pryor, J., 1973a. An Appreciation of the problems in stabilising underslung loads beneath a helicopter, Technical Note AM/37, Royal Military College of Science Sheldon, D.F. and Pryor, J., 1973b. Study in Depth of a single point and two point lateral and tandem suspension of rectangular box loads, Technical Note AM/38, Royal Military College of Science Sheldon, D.F. and Pryor, J., 1973c. A study on the stability and aerodynamic characteristics of particular military loads underslung from a helicopter, Technical Note AM/40, Royal Military College of Science. Skews, B.W., 1990. “Autorotation of rectangular plates”. J. Fluid Mech. 217, 33–40. Smith, E.H., 1970. Autorotating Wings: An Experimental Investigation, technical report for the US Army Research Office project ORA 01954. University of Michigan, Michigan. Smith, E.H., 1971. Autorotating wings: an experimental investigation. J. Fluid Mech. 50 (3), 513–534. Sykes, D.M, 1977. “A new wind tunnel for industrial aerodynamics”. J. Wind Eng. Ind. Aerodyn. 2 (1), 65–78. Tachikawa, M., 1983a. Trajectories of flat plates in uniform flow with application to wind-generated missiles. J. Wind Eng. Ind. Aerodyn. 14 (1-3), 443–453. Tachikawa, M., 1983b. Trajectories of typhoon-generated missiles. J. Wind Eng. 17, 21–30. Tanabe, A., Ito, A., 1996. Flow visualization on flow around autorotating flat plates. J. Vis. Soc. Jpn. 16, 99–102. Theron, J.N., Gordon, R., Rosen, A., Cicolani, L., Duque, E.P.N. and Halsey, R.H., 2006. Simulation of helicopter slung load aerodynamics: wind tunnel validation of two computational fluid dynamics codes, AIAA paper 2006-3374, In: Proceedings of the 36th AIAA Fluid Dynamics Conference, San Francisco, California. Zaki, T.G., Sen, M., Gad-el-Hak, M., 1994. “Numerical and experimental investigation of flow past a freely rotatable square cylinder”. J. Fluids Struct. Vol. 8 (No. 7), 555–582.