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Aerodynamic admittance of a 5:1 rectangular cylinder in turbulent flow
Journal of Wind Engineering & Industrial Aerodynamics 189 (2019) 125–134
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Aerodynamic admittance of a 5:1 rectangular cylinder in turbulent flow Yang Yang a, Mingshui Li a, b, *, Yi Su a, Yanguo Sun a, b a b
Research Centre for Wind Engineering, Southwest Jiaotong University, Chengdu, 610031, China Key Laboratory for Wind Engineering of Sichuan Province, Chengdu, 610031, China
A R T I C L E I N F O
A B S T R A C T
Keywords: Aerodynamic admittance Turbulent flow BARC benchmark Wind tunnel test
It is known for some time that the buffeting force acting on a structure induced by turbulent flow is intrinsically three-dimensional. To understand the influence of the three-dimensional effects fundamentally, clarifications are made regarding the relations between the one-wavenumber aerodynamic admittance, the two-wavenumber aerodynamic admittance and the generalized aerodynamic admittance. Then via wind tunnel tests, these three lift aerodynamic admittances of a 5:1 rectangular cylinder in grid turbulence are identified. An empirical formula for determining the one-wavenumber aerodynamic admittance of a 5:1 rectangular cylinder is proposed based on the experimental data and analysis. In this process, an empirical model for the spanwise influence term and an adapted form of the lift coherence function are used, which enable the use of the measured lift coherence for the estimation of the spanwise influence term. Based on these results, this work indicates that the generalized aerodynamic admittance, which is usually used in the practical engineering, is the combined results of the onewavenumber aerodynamic admittance and the three-dimensionality effects. The influence of the threedimensionality effects can be integrally considered by a 3D effects factor, the generalized aerodynamic admittance can be therefore expressed as the product of the one-wavenumber aerodynamic admittance and the 3D effects factor. It is then further revealed that the 3D effects factor is less than 1 and depends on the ratio of the turbulence integral scale to the chord, which makes the generalized aerodynamic admittance being smaller than the one-wavenumber aerodynamic admittance and being flow field-dependent.
1. Introduction The flow in atmospheric boundary layer is usually turbulent, one of the important issues in wind engineering is therefore the estimation of buffeting forces on structures induced by turbulent flow. Due to the statistic features of this problem, a quantity very useful to the designer is the aerodynamic admittance function (AAF), which is the transfer function of turbulent fluctuations transmitting to buffeting forces. If the input flow is given, the aerodynamic admittance permits the calculation of the buffeting force at any frequency. The first attempt to calculate the lift aerodynamic admittance of a thin airfoil in turbulent flow was made by Liepmann (1952) under two assumptions: first, the turbulent flow was fully correlated in the spanwise direction and second, each chordwise strip responded according to Sears's analysis (Sears, 1938). As a result, the lift aerodynamic admittance was equal to the Sears function, which was only the function of the chordwise wavenumber. However, the lift measured in turbulent flow was found to be much less than the theoretical values calculated with the Sears function (Hakkinen and Richardson, 1957; Lamson, 1957). The
underlying cause for this discrepancy is the influence of the three-dimensional effects. The turbulent flow is three-dimensional, thus has the spanwise variation, leading to the aerodynamic admittance being the function of chordwise and spanwise wavenumbers. Graham (1970, 1971) calculated the exact numerical solution of the two-wavenumber lift aerodynamic admittance of a thin airfoil in a two-dimensional sinusoidal vertical gust. Other aerodynamicists (Filotas, 1969; Mugridge, 1971; Blake, 1986) derived the closed-form approximations for Graham's exact numerical solution. The validity of Graham's solution has been verified by lots of wind tunnel tests (Jackson et al., 1973; McKeough, 1976; Stapountzis and Graham, 1982). The concept of the aerodynamic admittance had been introduced to calculate the buffeting forces acting on bluff bodies (Davenport, 1962; Vickery, 1965). However, for bluff bodies, the aerodynamic admittance can not be obtained through the theoretical derivation due to the flow separation, which has to be determined by wind tunnel tests. At present, the experimental identifications of the aerodynamic admittance are mainly divided into two common strategies. One is the measurement of the one-wavenumber aerodynamic admittance in a one-dimensional
* Corresponding author. Research Centre for Wind Engineering, Southwest Jiaotong University, Chengdu, 610031, China. E-mail address: [email protected] (M. Li). https://doi.org/10.1016/j.jweia.2019.03.023 Received 2 September 2018; Received in revised form 23 January 2019; Accepted 23 March 2019 0167-6105/© 2019 Elsevier Ltd. All rights reserved.
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Journal of Wind Engineering & Industrial Aerodynamics 189 (2019) 125–134
sinusoidal gust, this special flow field can be generated by the vibrating vane or the active fan (Jancauskas and Melbourne, 1986; Diana et al., 2002; Ma et al., 2013; Yang et al., 2017). From these measurements, the one-wavenumber aerodynamic admittances of bluff bodies are between 1 and the Sears function at low frequencies. The other is the measurement of the generalized aerodynamic admittance in turbulent flow generated by the grid or the spire, which is directly the ratio of the one-dimensional force spectrum to the one-dimensional turbulence spectrum (Larose and Mann, 1998; Larose, 1999; Zhu and Xu, 2014; Yan et al., 2017; Ma et al., 2019). However, it is shown by these experiments that the generalized aerodynamic admittances of bluff bodies are not between 1 and the Sears functionand vary with the flow field parameters and the characteristic dimensions of the structure. For this, researchers (Larose and Mann, 1998; Larose, 1999; Ma et al., 2019) proposed some empirical models for determining the generalized aerodynamic admittances of bluff bodies. In fact, the generalized aerodynamic admittance implicitly integrates the influence of the three-dimensional effects, thus it is naturally different from the one-wavenumber aerodynamic admittance and varies with the flow field. To understand the influence of the three-dimensional effects as well as the resulting flow field-dependence of the generalized admittance fundamentally, it is necessary to investigate the relationships between the one-wavenumber aerodynamic admittance, the two-wavenumber aerodynamic admittance and the generalized aerodynamic admittance. This could not be achieved with the conventional identification approaches. The two-wavenumber approach proposed by Li et al. (2015) is a method used to determine the two-wavenumber aerodynamic admittance of a body in turbulent flow. The theoretical basis of this approach is the three-dimensional theory. Using this approach, they have obtained experimental values of the two-wavenumber lift aerodynamic admittance of a NACA 0015 airfoil. In this work, clarifications are made regarding the concepts of aerodynamic admittances, including the one-wavenumber aerodynamic admittance, the two-wavenumber aerodynamic admittance and the generalized aerodynamic admittance. Then via wind tunnel tests, these three kinds of aerodynamic admittances of a 5:1 rectangular cylinder in turbulent flow are identified based on the improved two-wavenumber approach. The study on the aerodynamics of a rectangular cylinder with the chord-to-depth ratio being equal to 5 is the bench mark study in bluff body aerodynamics (a bench mark on the aerodynamics of a 5:1 rectangular cylinder, BARC) (Bartoli et al., 2009; Bruno et al., 2014). For now, relatively few experimental works have been performed on three kinds of aerodynamic admittances of a 5:1 rectangular cylinder. The present work may be helpful to understand the three-dimensional effects better and provide more experimental results of the aerodynamic admittance for BARC study. The paper is organized as follows: in Section 2, three concepts of aerodynamic admittances, including the one-wavenumber aerodynamic admittance, the two-wavenumber aerodynamic admittance and the generalized aerodynamic admittance are clarified, a 3D effects factor is defined to reflect the influence of the three-dimensional effects; in Section 3, the pressure measurement test on a 5:1 rectangular cylinder in grid turbulence is carried out, the related experimental arrangements are described; in Section 4, the one-wavenumber lift aerodynamic admittance and the two-wavenumber lift aerodynamic admittance of the 5:1 rectangular cylinder are presented, the influence of the turbulence integral scale to the chord ratio on the generalized aerodynamic admittance is discussed.
unsteady lift on an airfoil is 1 ΔLðtÞ ¼ ρUBC'L w0 eiωt χ w ~k1 ; 2
(1)
ρ is the fluid density, U is the mean flow velocity, B is the chord, C'L is the slope of the lift coefficient curve, w0 is the gust amplitude, ω is the circular frequency, k1 ð¼ ω=UÞ is the chordwise wavenumber, χ ð~k1 Þ is w
the one-wavenumber lift aerodynamic admittance, ~k1 ð¼ k1 B=2Þ is the dimensionless chordwise wavenumber (the reduced frequency). Assuming that the turbulent flow is fully correlated in the spanwise direction, Liepmann (1952) gave the one-dimensional lift spectrum of a thin airfoil in turbulent flow as follow SL ðk1 Þ ¼
2 2 1 ρUBC'L Sw ðk1 Þχ w ~k 1 ; 2
(2)
Sw ðk1 Þ is the one-dimensional vertical turbulence spectrum, j2 is the symbol for the squared modulus. When there is no case of confusion, the squared modulus of the aerodynamic admittance will be also called the aerodynamic admittance for brevity. In reality, the turbulent flow is three-dimensional, a consideration of the spanwise variation of the turbulent flow would be closer to the physical nature of the problem. The Fourier component of the fluctuating velocity of the turbulent flow is therefore expressed as a two-dimensional sinusoidal gust of the form wðtÞ ¼ w0 eiðωtk1 xk2 yÞ , as is shown in Fig. 1. According to the three-dimensional theory, the lift on a thin airfoil in such a two-dimensional sinusoidal gust is 1 ΔLðtÞ ¼ ρUBC'L w0 eiðωtk2 yÞ χ w ~k1 ; ~k 2 ; 2
(3)
χ w ð~k1 ; ~k2 Þ is the two-wavenumber lift aerodynamic admittance, k2 is the
spanwise wavenumber, ~k2 ð¼ k2 B=2Þ is the dimensionless spanwise wavenumber. Hence, the lift per unit span carried by an infinitesimal strip (s → 0, s is the spanwise length) is 1 s→0 s
LðtÞ ¼ lim
Z
s=2
s=2
1 ΔLðtÞdy ¼ ρUBC'L w0 eiωt χ w ~k 1 ; ~k 2 : 2
(4)
The lift induced by turbulent flow may be composed of the lift forces due to a sum of such gusts (Ribner, 1956; Etkin, 1959). Thus the two-dimensional spectrum of L is 2 1 SL ðk1 ; k2 Þ ¼ ρUBC'L Sw ðk1 ; k2 Þχ w ~k 1 ; ~k 2 j2 ; 2
(5)
Sw ðk1 ; k2 Þ is the two-dimensional vertical turbulence spectrum. 2.2. Lift on a rectangular cylinder in turbulent flow Based on the quasi-steady buffeting force model of the bluff body, it is not hard to extend the above analysis to a rectangular cylinder. Taking
2. Theoretical considerations 2.1. Lift on a thin airfoil in turbulent flow In this section, the lift on a thin airfoil in turbulent flow is briefly reviewed based on the three-dimensional theory for the sake of clarity. In a one-dimensional sinusoidal gust of the form wðtÞ ¼ w0 eiðωtk1 xÞ , the
Fig. 1. Two-dimensional wise variations. 126
sinusoidal
gust
with
chordwise
and
span-
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Journal of Wind Engineering & Industrial Aerodynamics 189 (2019) 125–134
the one-wavenumber aerodynamic admittance and the threedimensional effects, leading to the significant difference between the one-wavenumber aerodynamic admittance and the generalized aerodynamic admittance. Here, the influence of the three-dimensional effects is reflected by the 3D effects factor g3D . Hence, the generalized lift aerodynamic admittance can be expressed as the product of the onewavenumber aerodynamic admittance and the 3D effects factor. To further investigate the characteristics of the 3D effects factor, the presentation of g3D is best given in terms of dimensionless parameters, γ ¼ Lxw =B, Lw is the vertical turbulence integral scale. If the turbulence is homogeneous and isotropic, the turbulence spectrum can be described by the von Karman spectral model (see Appendix B), g3D can be then given as follow,
the lift for instance, the two-dimensional lift spectrum of a rectangular cylinder in turbulent flow can be expressed as SL ðk1 ; k2 Þ ¼
2 1 ρUB 4C2L Su ðk1 ; k2 Þχ u ~k 1 ; ~k 2 j2 2 þ ðC'L þ CD Þ2 Sw ðk1 ; k2 Þχ ~k 1 ; ~k 2 j2 ; w
(6)
CL and CD are the lift and drag coefficients, Su ðk1 ; k2 Þ is the twodimensional longitudinal turbulence spectrum. However, it is not easy ~ 2 ~ 2 to identify χ u ðk 1 Þj and χ w ðk1 Þj respectively in turbulent flow. The corresponding aerodynamic admittances of the longitudinal and vertical turbulent fluctuations are generally assumed to be the same in practical application, which is the so-called auto-spectrum approach (Larose and Mann, 1998; Larose, 1999; Zhu and Xu, 2014; Yan et al., 2017; Ma et al., 2019). Eq. (6) is therefore simplified as SL ðk1 ; k2 Þ ¼
g3D ~k 1 ; γ ¼
2
2 0 1 ρUB Þχ ~k1 ; ~k 2 j2 4C2L Su ðk1 ; k2 Þ þ C L þ CD Sw ðk1 ; k2 Þ 2 (7)
(12) with
η ~k1 ; ~k 2 ¼
The two-wavenumber lift aerodynamic admittance can be expressed as the product of the one-wavenumber lift aerodynamic admittance and a spanwise influence term (Mugridge, 1971; Blake, 1986), which is χ ~k1 ; ~k 2 j2 ¼ χ ~k1 j2 η ~k 1 ; ~k 2 ;
R þ∞ η ~k1 ; ~k 2 4C2L ξu ~k1 ; ~k 2 ; γ þ ðC'L þ CD Þ2 ξw ~k1 ; ~k 2 ; γ ~k 2 ∞ ; R þ∞ 2 4CL ξu ~k 1 ; ~k 2 ; γ þ ðC'L þ CD Þ2 ξw ~k 1 ; ~k 2 ; γ d~k 2 ∞
(8)
ηð~k1 ; ~k2 Þ is the spanwise influence term (see Appendix A). Then by integrating out the spanwise wavenumber from Eq. (7), the onedimensional lift spectrum can be written as
c a~k 1 þ 1 ; 2 þ b~k þ 1
c a~k 1
4:78 ξu ~k 1 ; ~k2 ; γ ¼ h 2 i4=3 ; 2 1 þ 28:688γ 2 ~k1 þ ~k 1
(14)
2 2 182:872γ 2 ~k 1 þ ~k 2 2 i7=3 : 2 1 þ 28:688γ 2 ~k 1 þ ~k2
(15)
ξw ~k 1 ; ~k2 ; γ ¼ h
3 2 2 2 Z 2 þ∞ 0 1 2 ~ ~ ~ 4 SL ðk1 Þ ¼ ρUB Þχ k1 j η k 1 ; k2 4CL Su ðk1 ; k2 Þ þ C L þ CD Sw ðk1 ; k2 Þ5dk2 : 2 ∞
2.3. Generalized aerodynamic admittance
(9)
Empirical parameters a, b and c can be experimentally determined by the use of the measured lift coherence (see Appendix C). The generalized aerodynamic admittance can be now written as
By analogy with Liepmann's original expression Eq. (2), the onedimensional lift spectrum Eq. (9) can be written as the common form as follow, 2 1 SL ðk1 Þ ¼ ρUB A ~k 1 j2 4C2L Su ðk1 Þ þ ðC'L þ CD Þ2 Sw ðk1 Þ ; 2
(13)
2
2 A ~k 1 j ¼ χ ~k 1 j2 g3D ~k 1 ; γ ;
(16)
Apparently, the 3D effects factor g3D determines the degree of the deviation between the generalized aerodynamic admittance Að~k1 Þj2 and the one-wavenumber aerodynamic admittance χ ð~k1 Þj2 , which depends
(10)
with
on the value of the turbulence integral scale to the chord ratio γ.
Z þ∞ η ~k 1 ; ~k 2 4C2L Su ðk1 ; k2 Þ þ ðC'L þ CD Þ2 Sw ðk1 ; k2 Þ dk2 ~ 2 2 A k 1 j ¼ χ ~k1 j ∞ Z þ∞ 2 4C L Su ðk1 ; k2 Þ þ ðC'L þ CD Þ2 Sw ðk1 ; k2 Þ dk2 ∞ ¼ χ ~k1 j2 g3D :
3. Experimental arrangements The experiments are conducted in a closed-loop low-speed wind tunnel (XNJD-1) at Southwest Jiaotong University, which has a test section of 2.4 m width, 2 m height and 16 m length. A square passive grid with a mesh size of 0.33 m and a bar width of 0.07 m is installed at the entrance of the test section for generating the turbulent flow. Grid turbulence is considered as the simplest three-dimensional turbulent flow and the closest practical approximation to homogenous and isotropic turbulence, which can be conveniently implemented in wind tunnel tests. To ensure that grid turbulence has enough time to develop to a approximately homogeneous and isotropic state, the testing model is mounted horizontally (zero angle of attack) on a steel frame approximately 4.2 m downstream of the grid, as is shown in Fig. 2. The flow velocities are
(11) ~ 2 Aðk1 Þj is the generalized aerodynamic admittance (also called the three-dimensional aerodynamic admittance). Because the generalized aerodynamic admittance can be directly obtained with the measured one-dimensional lift spectrum and the measured one-dimensional turbulence spectrum, this kind of aerodynamic admittance is commonly used in the practical engineering. From Eq. (11), it can be seen that the generalized aerodynamic admittance is actually the combined results of 127
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4. Results and discussions 4.1. Characteristics of the turbulence In the present work, it is necessary to check the homogeneity and isotropy of the generated turbulent flow. For homogeneous turbulence, the r.m.s fluctuating velocity should be identical in the same planes parallel to the grid. Fig. 4a shows the spanwise distributions of the normalized r.m.s fluctuating velocities, σ u =U and σ w =U. It can be seen that the r.m.s fluctuating velocities at different spanwise positions are very close, good homogeneity can be confirmed in the measurement region. The turbulence integral scale, which reflects information of the mean size of the energy-containing eddies, is defined as the integration of the correlation function. For isotropic turbulence, the turbulence integral scale can be determined by comparison between the theoretical onedimensional turbulence spectrum and the measured one-dimensional turbulence spectrum. In this matching, the one-dimensional von Karman spectral models, Eqs. (B.5) and (B.6), can be used. The measured one-dimensional turbulence spectra of the longitudinal and vertical fluctuating velocities are shown in Fig. 4b and c. It can be seen that the measured one-dimensional turbulence spectra match with the von Karman spectral models. To further assess the isotropy of the turbulence, the isotropic relationships should be also checked. The normalized r.m.s longitudinal and vertical fluctuating velocities are σ u =U ¼ 0:074 and σ w = U ¼ 0:063, respectively, the ratio σ u =σ w 1:17, the longitudinal and vertical integral scales obtained from the von Karman spectral model are Lxu ¼ 0.112 m and Lxw ¼ 0.052 m, respectively, the ratio Lxu =Lxw 2:15. It seems that the strictly isotropic condition is difficult to be achieved in grid turbulence, as is concluded in previous grid turbulence experiments (Lavoie et al., 2007; Djenidi and Tardu, 2012; Li et al., 2018). Nevertheless, these results satisfy roughly the isotropic relationships, the generated turbulent flow can be still regarded to be approximately isotropic. Then substituting values of σ u , σ w , Lxu and Lxw into Eq. (B.3) and (B.4), the two-dimensional turbulence spectra are obtained, as is shown in Fig. 5.
Fig. 2. 5:1 rectangular cylinder sectional model installed downstream of a square passive grid.
measured by the TFI Cobra Probe located at the model's leading edge position. The TFI Cobra Probe is a multi-hole pressure probe able to resolve 3 components of the flow velocity and has a frequency response of 0 Hz up to more than 2 kHz. To avoid the disturbance due to the model, the flow velocities are measured in the empty wind tunnel, which is implemented prior to model pressure measurements. The testing model has a constant 5:1 rectangular cross-section with a breadth (chord) B ¼ 0.3 m and a depth D ¼ 0.06 m. The model is made of ABS plastics, several transverse glass fibre ribs are added to enhance the stability of the model. To simulate the infinite span condition and prevent the air stream from being able to flow around the tips, the end plates are installed at two sides of the model. The pressures on the strip of the model are measured by the Scanivalve ZOC33 miniature pressure scanner. Multiple pressure scanners are connected to a same Scanivalve DSM3400 digital service module, achieving the synchronous acquisition of pressures on different strips. The ZOC33 miniature pressure scanner can sample pressure data from 64 pressure channels simultaneously, which has a measurement uncertainty of 0:08% full scale. The pressure scanners are placed inside the model to make the tubing length between the taps on the model surface and the scanner within 0.2 m, ensuring a good frequency response. The strips are located at the middle portion of the model, as shown in Fig. 3a. Each strip is equipped with 56 pressure taps, the positioning of the pressure taps is shown in Fig. 3b. The lift on each strip can be obtained by integrating the pressure around the strip, and the lift coherence between two strips can be then computed from the obtained lift on two strips. The flow velocity and the pressure are recorded at a sampling frequency of 256 Hz, the durations of measurements are 60 s. The mean flow velocity is set to U ¼ 11.5 ms1, the Reynolds number based on the depth of the model is Re ¼ 46000.
4.2. Lift force and its spanwise coherence In the theoretical analysis, the lift force is carried by a strip of a infinite span rectangular cylinder, and this strip can be taken anywhere along the span due to the assumed homogeneity of the turbulent flow. In the experiment, this is represented by a sectional model between two end plates to approximate the infinite span condition. Thus, it is necessary to check the two-dimensionality of the mean flow and the consistency of the measurements on the model to demonstrate that there are no or negligible effects of external disturbances on the measurements. Fig. 6 shows the distributions of the mean pressure coefficients and the r.m.s pressure coefficients on four strips. Fig. 7 shows the time-histories of the lift forces and the one-dimensional lift spectra on four strips. From these results, it
Fig. 3. (a) Arrangement of the strips on the sectional model. (b) Positioning of the pressure taps. 128
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Fig. 4. (a) Spanwise distributions of the normalized r.m.s fluctuating velocities. (b) One-dimensional longitudinal turbulence spectrum. (c) One-dimensional vertical turbulence spectrum.
Fig. 5. Two-dimensional turbulence spectra of (a) longitudinal fluctuations and (b) vertical fluctuations.
Fig. 6. Distributions of (a) the mean pressure coefficients CP and (b) the r.m.s pressure coefficients CPrms .
turbulent flow condition is larger than the one under the smooth flow condition. The spanwise coherence of the lift force on the model is shown in Fig. 8. The theoretical spanwise coherence of the vertical fluctuating velocity calculated with the von Karman coherence model (Roberts and Surry, 1973) is included for comparison. A small peak also appears in the lift coherence due to the vortex-shedding, similarly with the lift spectrum, thus this part of data would be not considered in the curve fitting. Then by the use of Eq. (15) to fit the measured lift coherence, the empirical parameters in the spanwise influence term are determined: a ¼ 7:0, b ¼ 2:2 and c ¼ 2:4. In fact, for bluff bodies, the formation mechanism of the force coherence is very complex, it may relate not only to the
can be seen that the pressure distributions and the lift forces on four strips are very close, the two-dimensionality of the mean flow and the consistency of the measurements on the model are confirmed. A small peak around the wavenumber k1 17 radm-1 (the frequency is about 31 Hz) appears in the lift spectrum (the highlighted region with a grey background). The corresponding Strouhal number is about St ¼ 0:162, which is in close agreement with the result provided in the work of Mannini et al. (2017) under the similar turbulence intensity conditions (in their work, the Strouhal number of a 5:1 rectangular cylinder is about 0.17 when the longitudinal turbulence intensity is about 7 %). Hence, it could be inferred that this peak may be caused by the vortex-shedding. Note that the Strouhal number of a 5:1 rectangular cylinder under the 129
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Fig. 7. (a) Time-histories of the lift forces on the model. (b) One-dimensional lift spectra on the model.
Fig. 8. Comparison between the coherence of the lift with those of the turbulent fluctuation.
fluctuating velocity, but also to the structure of the vortices formed in the separation bubble, thus only the approximations can be obtained. Nevertheless, it is believed that the fundamental properties of the results would not be lost because of this. The results show that the trends of the measured lift coherence can be approximately described by the proposed adapted form of the lift coherence function. From these results, it can be seen that the lift coherence is larger than the coherence of the vertical fluctuating velocity, as observed in many previous experiments (Jakobsen, 1997; Kimura et al., 1997; Larose and Mann, 1998; Ma et al., 2019). This suggests that for the lift measured on a
strip, the three-dimensional effects can not be neglected due to the small ratio of the turbulence integral scale to the chord, which inhibits the use of the strip assumption. To evaluate the influence of the three-dimensional effects, the 3D effects factor g3D is introduced, as mentioned in the theoretical analysis. Calculation of g3D requires information of the spanwise influence term. Substituting values of a, b and c into Eq. (A.3), the spanwise influence term of the 5:1 rectangular cylinder is then obtained, as shown in Fig. 9a. For the purposes of comparison, the spanwise influence term of a thin airfoil is also shown in Fig. 9b. By contrast, the spanwise influence term of the 5:1 rectangular cylinder has
Fig. 9. Spanwise influence terms of (a) the 5:1 rectangular cylinder and (b) the thin airfoil. 130
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Fig. 10. (a) One-wavenumber lift aerodynamic admittance and generalized lift aerodynamic admittance of the 5:1 rectangular cylinder. (b) 3D effects factors of the 5:1 rectangular cylinder and the thin airfoil.
Sears function as the frequency increases due to the thickness effects. This is because that, as the frequency increases, the incident turbulent fluctuations would be obviously distorted by the perturbation of the mean flow velocity caused by the thickness, leading to the flat plate assumption being invalid. Certainly, for a rectangular cylinder, besides the leading edge separation and the thickness effects, the trailing edge of the cross-section is also blunt (compared to a thin airfoil). However, the work of (Stapountzis and Graham, 1982) showed that an equivalent Kutta-Joukowski condition could be applicable when the trailing edge of the cross-section is blunt. Hence, the deviations between the one-wavenumber lift transfer function of the rectangular cylinder and the Sears function may be mainly attributed to the distortion of the incident turbulence caused by the thickness effects and the flow separation caused by the blunt leading edge. Based on the above experimental data and analysis, an empirical formula for determining the one-wavenumber aerodynamic admittance of a rectangular cylinder is proposed, which is
a similar characteristic as the spanwise influence term of the thin airfoil, which decreases as the spanwise wavenumber increases. However, as the chordwise wavenumber increases, the spanwise influence term of the 5:1 rectangular cylinder are larger than the spanwise influence term of the thin airfoil under the same spanwise wavenumber, which may result in larger values of the 3D effects factor. 4.3. Identification of the aerodynamic admittance The measured values of the generalized lift aerodynamic admittance are directly obtained with the measured one-dimensional lift spectrum and the measured one-dimensional turbulence spectrum, as shown in Fig. 10a. The spanwise influence term has been determined in the above section, so the 3D effects factor is calculated by the use of Eq. (12), as shown in Fig. 10b. Then the experimental values of the one-wavenumber lift aerodynamic admittance are obtained by the use of Eq. (16), as shown in Fig. 10a. The ratio of the turbulence integral scale to the chord used here is 0.17. From these results, it can be seen that the generalized lift aerodynamic admittance are smaller than the one-wavenumber lift aerodynamic admittance as well as the Sears function (especially at the low frequency range) because the 3D effect factor is less than 1. The 3D effect factor is relatively small at low frequencies, while increases as the frequency increases. This indicates the influence of the 3D effects mainly embodies at low frequencies, the increase of the frequency tends to reduce some of this influence. In addition, it is shown that the 3D effects factors of the 5:1 rectangular cylinder is larger than the 3D effects factors of the thin airfoil due to the differences of the spanwise influence terms, which is consistent with the analysis mentioned in the above section. The results also show that the measured values of the onewavenumber lift aerodynamic admittance of the 5:1 rectangular cylinder are larger than the Sears function at lower frequencies, while fall off faster than the Sears function gradually as the frequency increases. The reason for the deviations between the one-wavenumber lift aerodynamic admittance of the rectangular cylinder and the Sears function are still not fully understood. A possible explanation for the deviations at low frequencies was given by Jancauskas and Melbourne (1983, 1986). In their analysis, a proportion of the lift of a rectangular cylinder is considered to result from the nett effect of the pressures formed under the reattaching shear layers shed from the leading edge. The pressure under the reattaching shear layer results from the acceleration of the flow immediately outside the separation bubble, and is therefore not as dependent on the overall flow pattern as the pressures resulting from the fully attached flow. Consequently, at low frequencies, the lift generated by this mechanism may be far less sensitive to the wavelength of the turbulent fluctuations, leading to the lift aerodynamic admittance being closer to the quasi-steady value. On the other hand, previous studies (Atassi, 1984; Lysak et al., 2013, 2016; Li et al., 2018) have indicated that the lift aerodynamic admittance of a thick airfoil would fall off faster than the
1 ~ 2 ~ χ k 1 j ¼ c3 exp c2 k 1 : 1 þ c1 ~k 1
(17)
By fitting experimental data, the parameters in this empirical formula for the 5:1 rectangular cylinder are determined, which are c1 ¼ 4:3, c2 ¼ 1, c3 ¼ 3. To verify Eq. (16) and the proposed empirical formula better, calculated values of the generalized lift aerodynamic admittance is given in Fig. 10a. The results show that calculated values agree with the measured values (ignore the peak caused by the vortex-shedding). With the spanwise influence term and the one-wavenumber lift aerodynamic
Fig. 11. Two-wavenumber lift aerodynamic admittance of the 5:1 rectangular cylinder. 131
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Fig. 12. Numerical results of (a) the 3D effects factors g3D and (b) the generalized lift aerodynamic admittance Að~k1 Þj2 at different turbulence integral scale to chord ratios.
rectangular cylinder. The generalized aerodynamic admittance, which is the product of the one-wavenumber aerodynamic admittance and the 3D effects factor, is directly obtained with the measured one-dimensional lift spectrum and the measured one-dimensional turbulence spectrum. The one-wavenumber lift aerodynamic admittance of a 5:1 rectangular cylinder is then obtained with the generalized lift aerodynamic admittance and the 3D effects factor. In this process, an adapted form of the lift coherence function is derived, then the spanwise influence term, which requires to be known in the calculation of the 3D effects factor, is estimated by the use of the measured lift coherence. The results show that the one-wavenumber aerodynamic admittance of a 5:1 rectangular plates is obviously different from the Sears function. Based on the experimental data and analysis, an empirical formula is proposed to determine the lift aerodynamic admittance function of a 5:1 rectangular cylinder. The results also show that the 3D effects factor determines the deviations between the generalized aerodynamic admittance and the onewavenumber aerodynamic admittance. This factor depends on the ratio of the turbulence integral scale to the chord length. When the turbulence integral scale to the chord ratio is smaller, the 3D effects factor is smaller, the generalized aerodynamic admittance is much less than the onewavenumber aerodynamic admittance. With the increase of the ratio of the turbulence integral scale to the chord, the value of the 3D effects factor increases, the generalized lift aerodynamic admittance approaches to the one-wavenumber aerodynamic admittance gradually. Hence, the generalized lift aerodynamic admittances in different flow fields are different due to different turbulence integral scale to the chord ratios. With the measured one-wavenumber aerodynamic admittance and the calculated 3D effects factor, the generalized lift aerodynamic admittance in different flow fields may be predicted. It should be emphasized that the approach presented in this paper is not restricted to the simple rectangular cross-section considered herein. It can be extended to bluff structures with more complicated cross-sections, such as bridges and buildings.
admittance, the two-wavenumber lift aerodynamic admittance of the 5:1 rectangular cylinder is then obtained by the use of Eq. (8), as shown in Fig. 11. Note that the spanwise influence term ηð~k1 ; ~k2 Þ → 1 when the
spanwise wavenumber k2 → 0. Thus the one-wavenumber aerodynamic admittance can be regarded as a specific solution of the two-wavenumber aerodynamic admittance. 4.4. Discussion on the influence of the turbulence integral scale to the chord ratio In the above section, the 3D effects factor is calculated with the experimental value of the turbulence integral scale to the chord ratio. In fact, according to the theoretical analysis, the 3D effects factor as well as the generalized aerodynamic admittance would vary with the turbulence integral scale to the chord ratio. Hence, to further investigate the influence of the turbulence integral scale to the chord ratio, the 3D effects factor can be numerically varied over a larger parameter range, which would result in different generalized aerodynamic admittances. The 3D effects factors calculated with different turbulence integral scale to chord ratios are shown in Fig. 12a, the corresponding generalized aerodynamic admittances are shown in Fig. 12b. It can be seen that when the ratio of the turbulence integral scale to the chord is relatively small, the 3D effects factor is small, which means that the influence of the 3D effects is significant. In this case, the generalized lift aerodynamic admittance is much less than the one-wavenumber aerodynamic admittance. With the increase of the ratio of the turbulence integral scale to the chord, the influence of the 3D effects weakens, the value of the 3D effects factor increases, and thus the generalized lift aerodynamic admittance approaches to the one-wavenumber aerodynamic admittance gradually. Apparently, if the ratio of the turbulence integral scale to the chord is known, a reasonable prediction of the generalized lift aerodynamic admittance in different flow fields could be given with the measured onewavenumber aerodynamic admittance and the 3D effects factor. Nevertheless, to extend the approach to the practical application better, more extensive and systematic measurements involving other cross-sections in different flow fields would be worthwhile.
Acknowledgements This work is supported by National Natural Science Foundation of China under Grant Nos. 51478402, 51608074.
5. Conclusions This paper has investigated the lift aerodynamic admittances of a 5:1
Appendix D. Supplementary data Supplementary data to this article can be found online at https://doi.org/10.1016/j.jweia.2019.03.023.
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Appendix A This appendix gives the spanwise influence term used in the calculation of the 3D effects factor g3D . The theoretical models for the spanwise influence term of the thin airfoil are derived by Mugridge (1971) and Blake (1986), which are
ηMugridge ~k1 ; ~k 2 ¼
2 4:9~k1 þ 1 ; 2 2 4:9~k þ 4:9~k þ 1 1
(A.1)
2
1=2 4:5~k1 þ 1 : 1=2 2 4:5~k þ 9:6~k þ 1
ηBlake ~k1 ; ~k 2 ¼
1
(A.2)
2
For rectangular cylinders, modelling the spanwise influence term with empirical parameters becomes necessary, since the flow over the rectangular cylinder is partially separated, which inhibits the use of existing models. Hence, an empirical model for the spanwise influence term of the rectangular cylinder is given as follow
c a~k1 þ 1 : 2 þ b~k þ 1
η ~k 1 ; ~k 2 ¼
c a~k 1
(A.3)
2
a, b and c are empirical parameters, which can be experimentally determined by the use of the measured lift coherence (see Appendix C). Appendix B For homogeneous and isotropic turbulence, the spectral tensor can be described by the total kinetic energy as follow (Batchelor, 1953; Panofsky and Dutton, 1984), Φij ðkÞ ¼
x 2 Lu EðkÞ 2 k δij ki kj ; 4π k4
(B.1)
i and j are from 1 to 3, k is the wavenumber vector fk1 ; k2 ; k3 g. The number corresponds to fluctuating velocity components u, v and w in Φij ðkÞ, respectively, and to x (chordwise), y (spanwise), and z (vertical) in ki , respectively. δij is the Kronecker delta. EðkÞ is the energy spectrum. Combining theories of Kolmogorov and Loitsyansky, von Karman (1948) proposed an interpolation formula for the energy spectrum. It is 8 ðLkÞ4 EðkÞ ¼ σ 2 Lxu 17=6 : π 1 þ ðLkÞ2
(B.2)
This interpolation formula represents EðkÞ for small and large values of k and has the advantage that calculations can be carried out analytically. σ is pffiffiffi the root mean square of the fluctuating velocity, L is the turbulence length scale, L ¼ ½Γð1=3Þ= π Γð5=6ÞLxu 1:339Lxu , Γ is the Gamma function. Substituting Eq. (B.2) into Eq. (B.1), the two-dimensional spectra of the u-component and the w-component are Z Su ðk1 ; k2 Þ ¼
þ∞
∞
2 1:195σ 2u Lxu 2 i4=3 ; π 1 þ 1:793 Lxu k 21 þ k22
Φuu ðkÞdk3 ¼ h
(B.3)
4 45:718σ 2w Lxw k 21 þ k 22 Φww ðkÞdk3 ¼ h 2 i7=3 : π 1 þ 7:172 Lxw k 21 þ k22
(B.4)
and Z Sw ðk1 ; k2 Þ ¼
þ∞
∞
The corresponding one-dimensional turbulence spectra are Z Su ðk1 Þ ¼
þ∞
∞
Z
þ∞
∞
2σ 2u Lxu Φuu ðkÞdk2 dk3 ¼ 2 5=6 ; π 1 þ 1:793 Lxu k21
(B.5)
h 2 i 2σ 2w Lxw 1 þ 19:125 Lxw k 21 Φww ðkÞdk2 dk3 ¼ 2 11=6 : π 1 þ 7:172 Lxw k 21
(B.6)
and Z Sw ðk1 Þ ¼
þ∞
∞
Z
þ∞
∞
Lxu and Lxw are the longitudinal and vertical turbulence integral scales. For isotropic turbulence, there are isotropic relationships Lxu ¼ 2Lxw and σ u ¼ σ w (Mann, 1994). Theoretically, these turbulence parameters can be interchangeable in the analysis according to above isotropic relationships.
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Appendix C The lift coherence function is defined as CohL ðk1 ; ΔyÞ ¼
SL ðk1 ; ΔyÞ ; SL ðk1 Þ
(C.1)
Δy is the separation between two strips. SL ðk1 ; ΔyÞ is the cross-spectrum of the lift, it is the inverse Fourier transform of SL ðk1 ; k2 Þ, which is Z SL ðk1 ; ΔyÞ ¼
þ∞
∞
SL ðk1 ; k2 Þexpðik2 ΔyÞdk2 :
(C.2)
Combining Eqs. (7), (9), (C.1) and (C.2), the lift coherence function can be then written as CohL ðk1 ; ΔyÞ ¼
R þ∞ 2 4C L Su ðk1 ; k2 Þ þ ðC'L þ CD Þ2 Sw ðk1 ; k2 Þ η ~k1 ; ~k 2 expðik2 ΔyÞdk2 ∞ : R þ∞ 2 4CL Su ðk1 ; k2 Þ þ ðC'L þ CD Þ2 Sw ðk1 ; k2 Þ η ~k 1 ; ~k 2 dk2 ∞
(C.3)
From Eq. (C.3), the relation between the lift coherence and the spanwise influence term is established. Empirical parameters a, b and c in the spanwise influence term can be then determined by fitting the measured lift coherence.
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References Atassi, H.M., 1984. The Sears problem for a lifting airfoil revisited - new results. J. Fluid Mech. 141, 109–122. Bartoli, G., Bruno, L., Buresti, G., Ricciardelli, F., Salvetti, M.V., Zasso, A., 2009. BARC: a benchmark on the aerodynamics of a rectangular 5:1 cylinder. J. Fluids Struct. 25, 586. Batchelor, G.K., 1953. The Theory of Homogeneous Turbulence. Cambridge University Press. Blake, W.K., 1986. Mechanics of Flow-Induced Sound and Vibration. Academic Press. Bruno, L., Salvetti, M.V., Ricciardelli, F., 2014. Benchmark on the aerodynamics of a rectangular 5:1 cylinder: an overview after the first four years of activity. J. Wind Eng. Ind. Aerod. 126, 87–106. Davenport, A.G., 1962. Buffeting of suspension bridge by storm winds. J. Struct. Div. 88, 233–270. Diana, G., Bruni, S., Cigada, A., Zappa, E., 2002. Complex aerodynamic admittance function role in buffeting response of a bridge deck. J. Wind Eng. Ind. Aerod. 90, 2057–2072. Djenidi, L., Tardu, S.F., 2012. On the anisotropy of a low–Reynolds–number grid turbulence. J. Fluid Mech. 702, 332–353. Etkin, B., 1959. A theory of the response of airplanes to random atmospheric turbulence. J. Aerosp. Sci. 26, 409–420. Filotas, L.T., 1969. Theory of airfoil response in a gusty atmosphere. part i - aerodynamic transfer function. Tech. Rep. 139 (UTIAS). Graham, J.M.R., 1970. Lifting-surface theory for the problem of an arbitrary yawed sinusoidal gust incident on a thin aerofoil in incompressible flow. Aeronaut. Q. 21, 182–198. Graham, J.M.R., 1971. A lift-surface theory for the rectangular wing in non-stationary flow. Aeronaut. Q. 22, 83–100. Hakkinen, R.J., Richardson, A.S., 1957. Theoretical and experimental investigation of random gust loads. part I – aerodynamic transfer function of a simple wing configuration in incompressible flow. Tech. Rep. 3878 (NACA). Jackson, R., Graham, J.M.R., Maull, D.J., 1973. The lift on a wing in a turbulent flow. Aeronaut. Q. 24, 155–166. Jakobsen, J.B., 1997. Span-wise structure of lift and overturning moment on a motionless bridge girder. J. Wind Eng. Ind. Aerod. 69, 795–805. Jancauskas, E.D., Melbourne, W.H., 1983. The aerodynamic admittance of a slender box girder bridge section. In: Eighth Australasian Fluid Mechanics Conference. Newcastle, Australia. Jancauskas, E.D., Melbourne, W.H., 1986. The aerodynamic admittance of twodimensional rectangular section cylinders in smooth flow. J. Wind Eng. Ind. Aerod. 23, 395–408. Kimura, K., Fujino, Y., Nakato, S., Tamura, H., 1997. Characteristics of buffeting forces on flat cylinders. J. Wind Eng. Ind. Aerod. 69, 365–374. Lamson, P., 1957. Measurements of lift fluctuations due to turbulence. Tech. Rep. 3880 (NACA). Larose, G.L., 1999. Experimental determination of the aerodynamic admittance of a bridge deck segment. J. Fluids Struct. 13, 1029–1040. Larose, G.L., Mann, J., 1998. Gust loading on streamlined bridge decks. J. Fluids Struct. 12, 511–536.
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Journal of Wind Engineering & Industrial Aerodynamics journal homepage: www.elsevier.com/locate/jweia
Aerodynamic admittance of a 5:1 rectangular cylinder in turbulent flow Yang Yang a, Mingshui Li a, b, *, Yi Su a, Yanguo Sun a, b a b
Research Centre for Wind Engineering, Southwest Jiaotong University, Chengdu, 610031, China Key Laboratory for Wind Engineering of Sichuan Province, Chengdu, 610031, China
A R T I C L E I N F O
A B S T R A C T
Keywords: Aerodynamic admittance Turbulent flow BARC benchmark Wind tunnel test
It is known for some time that the buffeting force acting on a structure induced by turbulent flow is intrinsically three-dimensional. To understand the influence of the three-dimensional effects fundamentally, clarifications are made regarding the relations between the one-wavenumber aerodynamic admittance, the two-wavenumber aerodynamic admittance and the generalized aerodynamic admittance. Then via wind tunnel tests, these three lift aerodynamic admittances of a 5:1 rectangular cylinder in grid turbulence are identified. An empirical formula for determining the one-wavenumber aerodynamic admittance of a 5:1 rectangular cylinder is proposed based on the experimental data and analysis. In this process, an empirical model for the spanwise influence term and an adapted form of the lift coherence function are used, which enable the use of the measured lift coherence for the estimation of the spanwise influence term. Based on these results, this work indicates that the generalized aerodynamic admittance, which is usually used in the practical engineering, is the combined results of the onewavenumber aerodynamic admittance and the three-dimensionality effects. The influence of the threedimensionality effects can be integrally considered by a 3D effects factor, the generalized aerodynamic admittance can be therefore expressed as the product of the one-wavenumber aerodynamic admittance and the 3D effects factor. It is then further revealed that the 3D effects factor is less than 1 and depends on the ratio of the turbulence integral scale to the chord, which makes the generalized aerodynamic admittance being smaller than the one-wavenumber aerodynamic admittance and being flow field-dependent.
1. Introduction The flow in atmospheric boundary layer is usually turbulent, one of the important issues in wind engineering is therefore the estimation of buffeting forces on structures induced by turbulent flow. Due to the statistic features of this problem, a quantity very useful to the designer is the aerodynamic admittance function (AAF), which is the transfer function of turbulent fluctuations transmitting to buffeting forces. If the input flow is given, the aerodynamic admittance permits the calculation of the buffeting force at any frequency. The first attempt to calculate the lift aerodynamic admittance of a thin airfoil in turbulent flow was made by Liepmann (1952) under two assumptions: first, the turbulent flow was fully correlated in the spanwise direction and second, each chordwise strip responded according to Sears's analysis (Sears, 1938). As a result, the lift aerodynamic admittance was equal to the Sears function, which was only the function of the chordwise wavenumber. However, the lift measured in turbulent flow was found to be much less than the theoretical values calculated with the Sears function (Hakkinen and Richardson, 1957; Lamson, 1957). The
underlying cause for this discrepancy is the influence of the three-dimensional effects. The turbulent flow is three-dimensional, thus has the spanwise variation, leading to the aerodynamic admittance being the function of chordwise and spanwise wavenumbers. Graham (1970, 1971) calculated the exact numerical solution of the two-wavenumber lift aerodynamic admittance of a thin airfoil in a two-dimensional sinusoidal vertical gust. Other aerodynamicists (Filotas, 1969; Mugridge, 1971; Blake, 1986) derived the closed-form approximations for Graham's exact numerical solution. The validity of Graham's solution has been verified by lots of wind tunnel tests (Jackson et al., 1973; McKeough, 1976; Stapountzis and Graham, 1982). The concept of the aerodynamic admittance had been introduced to calculate the buffeting forces acting on bluff bodies (Davenport, 1962; Vickery, 1965). However, for bluff bodies, the aerodynamic admittance can not be obtained through the theoretical derivation due to the flow separation, which has to be determined by wind tunnel tests. At present, the experimental identifications of the aerodynamic admittance are mainly divided into two common strategies. One is the measurement of the one-wavenumber aerodynamic admittance in a one-dimensional
* Corresponding author. Research Centre for Wind Engineering, Southwest Jiaotong University, Chengdu, 610031, China. E-mail address: [email protected] (M. Li). https://doi.org/10.1016/j.jweia.2019.03.023 Received 2 September 2018; Received in revised form 23 January 2019; Accepted 23 March 2019 0167-6105/© 2019 Elsevier Ltd. All rights reserved.
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Journal of Wind Engineering & Industrial Aerodynamics 189 (2019) 125–134
sinusoidal gust, this special flow field can be generated by the vibrating vane or the active fan (Jancauskas and Melbourne, 1986; Diana et al., 2002; Ma et al., 2013; Yang et al., 2017). From these measurements, the one-wavenumber aerodynamic admittances of bluff bodies are between 1 and the Sears function at low frequencies. The other is the measurement of the generalized aerodynamic admittance in turbulent flow generated by the grid or the spire, which is directly the ratio of the one-dimensional force spectrum to the one-dimensional turbulence spectrum (Larose and Mann, 1998; Larose, 1999; Zhu and Xu, 2014; Yan et al., 2017; Ma et al., 2019). However, it is shown by these experiments that the generalized aerodynamic admittances of bluff bodies are not between 1 and the Sears functionand vary with the flow field parameters and the characteristic dimensions of the structure. For this, researchers (Larose and Mann, 1998; Larose, 1999; Ma et al., 2019) proposed some empirical models for determining the generalized aerodynamic admittances of bluff bodies. In fact, the generalized aerodynamic admittance implicitly integrates the influence of the three-dimensional effects, thus it is naturally different from the one-wavenumber aerodynamic admittance and varies with the flow field. To understand the influence of the three-dimensional effects as well as the resulting flow field-dependence of the generalized admittance fundamentally, it is necessary to investigate the relationships between the one-wavenumber aerodynamic admittance, the two-wavenumber aerodynamic admittance and the generalized aerodynamic admittance. This could not be achieved with the conventional identification approaches. The two-wavenumber approach proposed by Li et al. (2015) is a method used to determine the two-wavenumber aerodynamic admittance of a body in turbulent flow. The theoretical basis of this approach is the three-dimensional theory. Using this approach, they have obtained experimental values of the two-wavenumber lift aerodynamic admittance of a NACA 0015 airfoil. In this work, clarifications are made regarding the concepts of aerodynamic admittances, including the one-wavenumber aerodynamic admittance, the two-wavenumber aerodynamic admittance and the generalized aerodynamic admittance. Then via wind tunnel tests, these three kinds of aerodynamic admittances of a 5:1 rectangular cylinder in turbulent flow are identified based on the improved two-wavenumber approach. The study on the aerodynamics of a rectangular cylinder with the chord-to-depth ratio being equal to 5 is the bench mark study in bluff body aerodynamics (a bench mark on the aerodynamics of a 5:1 rectangular cylinder, BARC) (Bartoli et al., 2009; Bruno et al., 2014). For now, relatively few experimental works have been performed on three kinds of aerodynamic admittances of a 5:1 rectangular cylinder. The present work may be helpful to understand the three-dimensional effects better and provide more experimental results of the aerodynamic admittance for BARC study. The paper is organized as follows: in Section 2, three concepts of aerodynamic admittances, including the one-wavenumber aerodynamic admittance, the two-wavenumber aerodynamic admittance and the generalized aerodynamic admittance are clarified, a 3D effects factor is defined to reflect the influence of the three-dimensional effects; in Section 3, the pressure measurement test on a 5:1 rectangular cylinder in grid turbulence is carried out, the related experimental arrangements are described; in Section 4, the one-wavenumber lift aerodynamic admittance and the two-wavenumber lift aerodynamic admittance of the 5:1 rectangular cylinder are presented, the influence of the turbulence integral scale to the chord ratio on the generalized aerodynamic admittance is discussed.
unsteady lift on an airfoil is 1 ΔLðtÞ ¼ ρUBC'L w0 eiωt χ w ~k1 ; 2
(1)
ρ is the fluid density, U is the mean flow velocity, B is the chord, C'L is the slope of the lift coefficient curve, w0 is the gust amplitude, ω is the circular frequency, k1 ð¼ ω=UÞ is the chordwise wavenumber, χ ð~k1 Þ is w
the one-wavenumber lift aerodynamic admittance, ~k1 ð¼ k1 B=2Þ is the dimensionless chordwise wavenumber (the reduced frequency). Assuming that the turbulent flow is fully correlated in the spanwise direction, Liepmann (1952) gave the one-dimensional lift spectrum of a thin airfoil in turbulent flow as follow SL ðk1 Þ ¼
2 2 1 ρUBC'L Sw ðk1 Þχ w ~k 1 ; 2
(2)
Sw ðk1 Þ is the one-dimensional vertical turbulence spectrum, j2 is the symbol for the squared modulus. When there is no case of confusion, the squared modulus of the aerodynamic admittance will be also called the aerodynamic admittance for brevity. In reality, the turbulent flow is three-dimensional, a consideration of the spanwise variation of the turbulent flow would be closer to the physical nature of the problem. The Fourier component of the fluctuating velocity of the turbulent flow is therefore expressed as a two-dimensional sinusoidal gust of the form wðtÞ ¼ w0 eiðωtk1 xk2 yÞ , as is shown in Fig. 1. According to the three-dimensional theory, the lift on a thin airfoil in such a two-dimensional sinusoidal gust is 1 ΔLðtÞ ¼ ρUBC'L w0 eiðωtk2 yÞ χ w ~k1 ; ~k 2 ; 2
(3)
χ w ð~k1 ; ~k2 Þ is the two-wavenumber lift aerodynamic admittance, k2 is the
spanwise wavenumber, ~k2 ð¼ k2 B=2Þ is the dimensionless spanwise wavenumber. Hence, the lift per unit span carried by an infinitesimal strip (s → 0, s is the spanwise length) is 1 s→0 s
LðtÞ ¼ lim
Z
s=2
s=2
1 ΔLðtÞdy ¼ ρUBC'L w0 eiωt χ w ~k 1 ; ~k 2 : 2
(4)
The lift induced by turbulent flow may be composed of the lift forces due to a sum of such gusts (Ribner, 1956; Etkin, 1959). Thus the two-dimensional spectrum of L is 2 1 SL ðk1 ; k2 Þ ¼ ρUBC'L Sw ðk1 ; k2 Þχ w ~k 1 ; ~k 2 j2 ; 2
(5)
Sw ðk1 ; k2 Þ is the two-dimensional vertical turbulence spectrum. 2.2. Lift on a rectangular cylinder in turbulent flow Based on the quasi-steady buffeting force model of the bluff body, it is not hard to extend the above analysis to a rectangular cylinder. Taking
2. Theoretical considerations 2.1. Lift on a thin airfoil in turbulent flow In this section, the lift on a thin airfoil in turbulent flow is briefly reviewed based on the three-dimensional theory for the sake of clarity. In a one-dimensional sinusoidal gust of the form wðtÞ ¼ w0 eiðωtk1 xÞ , the
Fig. 1. Two-dimensional wise variations. 126
sinusoidal
gust
with
chordwise
and
span-
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Journal of Wind Engineering & Industrial Aerodynamics 189 (2019) 125–134
the one-wavenumber aerodynamic admittance and the threedimensional effects, leading to the significant difference between the one-wavenumber aerodynamic admittance and the generalized aerodynamic admittance. Here, the influence of the three-dimensional effects is reflected by the 3D effects factor g3D . Hence, the generalized lift aerodynamic admittance can be expressed as the product of the onewavenumber aerodynamic admittance and the 3D effects factor. To further investigate the characteristics of the 3D effects factor, the presentation of g3D is best given in terms of dimensionless parameters, γ ¼ Lxw =B, Lw is the vertical turbulence integral scale. If the turbulence is homogeneous and isotropic, the turbulence spectrum can be described by the von Karman spectral model (see Appendix B), g3D can be then given as follow,
the lift for instance, the two-dimensional lift spectrum of a rectangular cylinder in turbulent flow can be expressed as SL ðk1 ; k2 Þ ¼
2 1 ρUB 4C2L Su ðk1 ; k2 Þχ u ~k 1 ; ~k 2 j2 2 þ ðC'L þ CD Þ2 Sw ðk1 ; k2 Þχ ~k 1 ; ~k 2 j2 ; w
(6)
CL and CD are the lift and drag coefficients, Su ðk1 ; k2 Þ is the twodimensional longitudinal turbulence spectrum. However, it is not easy ~ 2 ~ 2 to identify χ u ðk 1 Þj and χ w ðk1 Þj respectively in turbulent flow. The corresponding aerodynamic admittances of the longitudinal and vertical turbulent fluctuations are generally assumed to be the same in practical application, which is the so-called auto-spectrum approach (Larose and Mann, 1998; Larose, 1999; Zhu and Xu, 2014; Yan et al., 2017; Ma et al., 2019). Eq. (6) is therefore simplified as SL ðk1 ; k2 Þ ¼
g3D ~k 1 ; γ ¼
2
2 0 1 ρUB Þχ ~k1 ; ~k 2 j2 4C2L Su ðk1 ; k2 Þ þ C L þ CD Sw ðk1 ; k2 Þ 2 (7)
(12) with
η ~k1 ; ~k 2 ¼
The two-wavenumber lift aerodynamic admittance can be expressed as the product of the one-wavenumber lift aerodynamic admittance and a spanwise influence term (Mugridge, 1971; Blake, 1986), which is χ ~k1 ; ~k 2 j2 ¼ χ ~k1 j2 η ~k 1 ; ~k 2 ;
R þ∞ η ~k1 ; ~k 2 4C2L ξu ~k1 ; ~k 2 ; γ þ ðC'L þ CD Þ2 ξw ~k1 ; ~k 2 ; γ ~k 2 ∞ ; R þ∞ 2 4CL ξu ~k 1 ; ~k 2 ; γ þ ðC'L þ CD Þ2 ξw ~k 1 ; ~k 2 ; γ d~k 2 ∞
(8)
ηð~k1 ; ~k2 Þ is the spanwise influence term (see Appendix A). Then by integrating out the spanwise wavenumber from Eq. (7), the onedimensional lift spectrum can be written as
c a~k 1 þ 1 ; 2 þ b~k þ 1
c a~k 1
4:78 ξu ~k 1 ; ~k2 ; γ ¼ h 2 i4=3 ; 2 1 þ 28:688γ 2 ~k1 þ ~k 1
(14)
2 2 182:872γ 2 ~k 1 þ ~k 2 2 i7=3 : 2 1 þ 28:688γ 2 ~k 1 þ ~k2
(15)
ξw ~k 1 ; ~k2 ; γ ¼ h
3 2 2 2 Z 2 þ∞ 0 1 2 ~ ~ ~ 4 SL ðk1 Þ ¼ ρUB Þχ k1 j η k 1 ; k2 4CL Su ðk1 ; k2 Þ þ C L þ CD Sw ðk1 ; k2 Þ5dk2 : 2 ∞
2.3. Generalized aerodynamic admittance
(9)
Empirical parameters a, b and c can be experimentally determined by the use of the measured lift coherence (see Appendix C). The generalized aerodynamic admittance can be now written as
By analogy with Liepmann's original expression Eq. (2), the onedimensional lift spectrum Eq. (9) can be written as the common form as follow, 2 1 SL ðk1 Þ ¼ ρUB A ~k 1 j2 4C2L Su ðk1 Þ þ ðC'L þ CD Þ2 Sw ðk1 Þ ; 2
(13)
2
2 A ~k 1 j ¼ χ ~k 1 j2 g3D ~k 1 ; γ ;
(16)
Apparently, the 3D effects factor g3D determines the degree of the deviation between the generalized aerodynamic admittance Að~k1 Þj2 and the one-wavenumber aerodynamic admittance χ ð~k1 Þj2 , which depends
(10)
with
on the value of the turbulence integral scale to the chord ratio γ.
Z þ∞ η ~k 1 ; ~k 2 4C2L Su ðk1 ; k2 Þ þ ðC'L þ CD Þ2 Sw ðk1 ; k2 Þ dk2 ~ 2 2 A k 1 j ¼ χ ~k1 j ∞ Z þ∞ 2 4C L Su ðk1 ; k2 Þ þ ðC'L þ CD Þ2 Sw ðk1 ; k2 Þ dk2 ∞ ¼ χ ~k1 j2 g3D :
3. Experimental arrangements The experiments are conducted in a closed-loop low-speed wind tunnel (XNJD-1) at Southwest Jiaotong University, which has a test section of 2.4 m width, 2 m height and 16 m length. A square passive grid with a mesh size of 0.33 m and a bar width of 0.07 m is installed at the entrance of the test section for generating the turbulent flow. Grid turbulence is considered as the simplest three-dimensional turbulent flow and the closest practical approximation to homogenous and isotropic turbulence, which can be conveniently implemented in wind tunnel tests. To ensure that grid turbulence has enough time to develop to a approximately homogeneous and isotropic state, the testing model is mounted horizontally (zero angle of attack) on a steel frame approximately 4.2 m downstream of the grid, as is shown in Fig. 2. The flow velocities are
(11) ~ 2 Aðk1 Þj is the generalized aerodynamic admittance (also called the three-dimensional aerodynamic admittance). Because the generalized aerodynamic admittance can be directly obtained with the measured one-dimensional lift spectrum and the measured one-dimensional turbulence spectrum, this kind of aerodynamic admittance is commonly used in the practical engineering. From Eq. (11), it can be seen that the generalized aerodynamic admittance is actually the combined results of 127
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4. Results and discussions 4.1. Characteristics of the turbulence In the present work, it is necessary to check the homogeneity and isotropy of the generated turbulent flow. For homogeneous turbulence, the r.m.s fluctuating velocity should be identical in the same planes parallel to the grid. Fig. 4a shows the spanwise distributions of the normalized r.m.s fluctuating velocities, σ u =U and σ w =U. It can be seen that the r.m.s fluctuating velocities at different spanwise positions are very close, good homogeneity can be confirmed in the measurement region. The turbulence integral scale, which reflects information of the mean size of the energy-containing eddies, is defined as the integration of the correlation function. For isotropic turbulence, the turbulence integral scale can be determined by comparison between the theoretical onedimensional turbulence spectrum and the measured one-dimensional turbulence spectrum. In this matching, the one-dimensional von Karman spectral models, Eqs. (B.5) and (B.6), can be used. The measured one-dimensional turbulence spectra of the longitudinal and vertical fluctuating velocities are shown in Fig. 4b and c. It can be seen that the measured one-dimensional turbulence spectra match with the von Karman spectral models. To further assess the isotropy of the turbulence, the isotropic relationships should be also checked. The normalized r.m.s longitudinal and vertical fluctuating velocities are σ u =U ¼ 0:074 and σ w = U ¼ 0:063, respectively, the ratio σ u =σ w 1:17, the longitudinal and vertical integral scales obtained from the von Karman spectral model are Lxu ¼ 0.112 m and Lxw ¼ 0.052 m, respectively, the ratio Lxu =Lxw 2:15. It seems that the strictly isotropic condition is difficult to be achieved in grid turbulence, as is concluded in previous grid turbulence experiments (Lavoie et al., 2007; Djenidi and Tardu, 2012; Li et al., 2018). Nevertheless, these results satisfy roughly the isotropic relationships, the generated turbulent flow can be still regarded to be approximately isotropic. Then substituting values of σ u , σ w , Lxu and Lxw into Eq. (B.3) and (B.4), the two-dimensional turbulence spectra are obtained, as is shown in Fig. 5.
Fig. 2. 5:1 rectangular cylinder sectional model installed downstream of a square passive grid.
measured by the TFI Cobra Probe located at the model's leading edge position. The TFI Cobra Probe is a multi-hole pressure probe able to resolve 3 components of the flow velocity and has a frequency response of 0 Hz up to more than 2 kHz. To avoid the disturbance due to the model, the flow velocities are measured in the empty wind tunnel, which is implemented prior to model pressure measurements. The testing model has a constant 5:1 rectangular cross-section with a breadth (chord) B ¼ 0.3 m and a depth D ¼ 0.06 m. The model is made of ABS plastics, several transverse glass fibre ribs are added to enhance the stability of the model. To simulate the infinite span condition and prevent the air stream from being able to flow around the tips, the end plates are installed at two sides of the model. The pressures on the strip of the model are measured by the Scanivalve ZOC33 miniature pressure scanner. Multiple pressure scanners are connected to a same Scanivalve DSM3400 digital service module, achieving the synchronous acquisition of pressures on different strips. The ZOC33 miniature pressure scanner can sample pressure data from 64 pressure channels simultaneously, which has a measurement uncertainty of 0:08% full scale. The pressure scanners are placed inside the model to make the tubing length between the taps on the model surface and the scanner within 0.2 m, ensuring a good frequency response. The strips are located at the middle portion of the model, as shown in Fig. 3a. Each strip is equipped with 56 pressure taps, the positioning of the pressure taps is shown in Fig. 3b. The lift on each strip can be obtained by integrating the pressure around the strip, and the lift coherence between two strips can be then computed from the obtained lift on two strips. The flow velocity and the pressure are recorded at a sampling frequency of 256 Hz, the durations of measurements are 60 s. The mean flow velocity is set to U ¼ 11.5 ms1, the Reynolds number based on the depth of the model is Re ¼ 46000.
4.2. Lift force and its spanwise coherence In the theoretical analysis, the lift force is carried by a strip of a infinite span rectangular cylinder, and this strip can be taken anywhere along the span due to the assumed homogeneity of the turbulent flow. In the experiment, this is represented by a sectional model between two end plates to approximate the infinite span condition. Thus, it is necessary to check the two-dimensionality of the mean flow and the consistency of the measurements on the model to demonstrate that there are no or negligible effects of external disturbances on the measurements. Fig. 6 shows the distributions of the mean pressure coefficients and the r.m.s pressure coefficients on four strips. Fig. 7 shows the time-histories of the lift forces and the one-dimensional lift spectra on four strips. From these results, it
Fig. 3. (a) Arrangement of the strips on the sectional model. (b) Positioning of the pressure taps. 128
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Fig. 4. (a) Spanwise distributions of the normalized r.m.s fluctuating velocities. (b) One-dimensional longitudinal turbulence spectrum. (c) One-dimensional vertical turbulence spectrum.
Fig. 5. Two-dimensional turbulence spectra of (a) longitudinal fluctuations and (b) vertical fluctuations.
Fig. 6. Distributions of (a) the mean pressure coefficients CP and (b) the r.m.s pressure coefficients CPrms .
turbulent flow condition is larger than the one under the smooth flow condition. The spanwise coherence of the lift force on the model is shown in Fig. 8. The theoretical spanwise coherence of the vertical fluctuating velocity calculated with the von Karman coherence model (Roberts and Surry, 1973) is included for comparison. A small peak also appears in the lift coherence due to the vortex-shedding, similarly with the lift spectrum, thus this part of data would be not considered in the curve fitting. Then by the use of Eq. (15) to fit the measured lift coherence, the empirical parameters in the spanwise influence term are determined: a ¼ 7:0, b ¼ 2:2 and c ¼ 2:4. In fact, for bluff bodies, the formation mechanism of the force coherence is very complex, it may relate not only to the
can be seen that the pressure distributions and the lift forces on four strips are very close, the two-dimensionality of the mean flow and the consistency of the measurements on the model are confirmed. A small peak around the wavenumber k1 17 radm-1 (the frequency is about 31 Hz) appears in the lift spectrum (the highlighted region with a grey background). The corresponding Strouhal number is about St ¼ 0:162, which is in close agreement with the result provided in the work of Mannini et al. (2017) under the similar turbulence intensity conditions (in their work, the Strouhal number of a 5:1 rectangular cylinder is about 0.17 when the longitudinal turbulence intensity is about 7 %). Hence, it could be inferred that this peak may be caused by the vortex-shedding. Note that the Strouhal number of a 5:1 rectangular cylinder under the 129
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Fig. 7. (a) Time-histories of the lift forces on the model. (b) One-dimensional lift spectra on the model.
Fig. 8. Comparison between the coherence of the lift with those of the turbulent fluctuation.
fluctuating velocity, but also to the structure of the vortices formed in the separation bubble, thus only the approximations can be obtained. Nevertheless, it is believed that the fundamental properties of the results would not be lost because of this. The results show that the trends of the measured lift coherence can be approximately described by the proposed adapted form of the lift coherence function. From these results, it can be seen that the lift coherence is larger than the coherence of the vertical fluctuating velocity, as observed in many previous experiments (Jakobsen, 1997; Kimura et al., 1997; Larose and Mann, 1998; Ma et al., 2019). This suggests that for the lift measured on a
strip, the three-dimensional effects can not be neglected due to the small ratio of the turbulence integral scale to the chord, which inhibits the use of the strip assumption. To evaluate the influence of the three-dimensional effects, the 3D effects factor g3D is introduced, as mentioned in the theoretical analysis. Calculation of g3D requires information of the spanwise influence term. Substituting values of a, b and c into Eq. (A.3), the spanwise influence term of the 5:1 rectangular cylinder is then obtained, as shown in Fig. 9a. For the purposes of comparison, the spanwise influence term of a thin airfoil is also shown in Fig. 9b. By contrast, the spanwise influence term of the 5:1 rectangular cylinder has
Fig. 9. Spanwise influence terms of (a) the 5:1 rectangular cylinder and (b) the thin airfoil. 130
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Fig. 10. (a) One-wavenumber lift aerodynamic admittance and generalized lift aerodynamic admittance of the 5:1 rectangular cylinder. (b) 3D effects factors of the 5:1 rectangular cylinder and the thin airfoil.
Sears function as the frequency increases due to the thickness effects. This is because that, as the frequency increases, the incident turbulent fluctuations would be obviously distorted by the perturbation of the mean flow velocity caused by the thickness, leading to the flat plate assumption being invalid. Certainly, for a rectangular cylinder, besides the leading edge separation and the thickness effects, the trailing edge of the cross-section is also blunt (compared to a thin airfoil). However, the work of (Stapountzis and Graham, 1982) showed that an equivalent Kutta-Joukowski condition could be applicable when the trailing edge of the cross-section is blunt. Hence, the deviations between the one-wavenumber lift transfer function of the rectangular cylinder and the Sears function may be mainly attributed to the distortion of the incident turbulence caused by the thickness effects and the flow separation caused by the blunt leading edge. Based on the above experimental data and analysis, an empirical formula for determining the one-wavenumber aerodynamic admittance of a rectangular cylinder is proposed, which is
a similar characteristic as the spanwise influence term of the thin airfoil, which decreases as the spanwise wavenumber increases. However, as the chordwise wavenumber increases, the spanwise influence term of the 5:1 rectangular cylinder are larger than the spanwise influence term of the thin airfoil under the same spanwise wavenumber, which may result in larger values of the 3D effects factor. 4.3. Identification of the aerodynamic admittance The measured values of the generalized lift aerodynamic admittance are directly obtained with the measured one-dimensional lift spectrum and the measured one-dimensional turbulence spectrum, as shown in Fig. 10a. The spanwise influence term has been determined in the above section, so the 3D effects factor is calculated by the use of Eq. (12), as shown in Fig. 10b. Then the experimental values of the one-wavenumber lift aerodynamic admittance are obtained by the use of Eq. (16), as shown in Fig. 10a. The ratio of the turbulence integral scale to the chord used here is 0.17. From these results, it can be seen that the generalized lift aerodynamic admittance are smaller than the one-wavenumber lift aerodynamic admittance as well as the Sears function (especially at the low frequency range) because the 3D effect factor is less than 1. The 3D effect factor is relatively small at low frequencies, while increases as the frequency increases. This indicates the influence of the 3D effects mainly embodies at low frequencies, the increase of the frequency tends to reduce some of this influence. In addition, it is shown that the 3D effects factors of the 5:1 rectangular cylinder is larger than the 3D effects factors of the thin airfoil due to the differences of the spanwise influence terms, which is consistent with the analysis mentioned in the above section. The results also show that the measured values of the onewavenumber lift aerodynamic admittance of the 5:1 rectangular cylinder are larger than the Sears function at lower frequencies, while fall off faster than the Sears function gradually as the frequency increases. The reason for the deviations between the one-wavenumber lift aerodynamic admittance of the rectangular cylinder and the Sears function are still not fully understood. A possible explanation for the deviations at low frequencies was given by Jancauskas and Melbourne (1983, 1986). In their analysis, a proportion of the lift of a rectangular cylinder is considered to result from the nett effect of the pressures formed under the reattaching shear layers shed from the leading edge. The pressure under the reattaching shear layer results from the acceleration of the flow immediately outside the separation bubble, and is therefore not as dependent on the overall flow pattern as the pressures resulting from the fully attached flow. Consequently, at low frequencies, the lift generated by this mechanism may be far less sensitive to the wavelength of the turbulent fluctuations, leading to the lift aerodynamic admittance being closer to the quasi-steady value. On the other hand, previous studies (Atassi, 1984; Lysak et al., 2013, 2016; Li et al., 2018) have indicated that the lift aerodynamic admittance of a thick airfoil would fall off faster than the
1 ~ 2 ~ χ k 1 j ¼ c3 exp c2 k 1 : 1 þ c1 ~k 1
(17)
By fitting experimental data, the parameters in this empirical formula for the 5:1 rectangular cylinder are determined, which are c1 ¼ 4:3, c2 ¼ 1, c3 ¼ 3. To verify Eq. (16) and the proposed empirical formula better, calculated values of the generalized lift aerodynamic admittance is given in Fig. 10a. The results show that calculated values agree with the measured values (ignore the peak caused by the vortex-shedding). With the spanwise influence term and the one-wavenumber lift aerodynamic
Fig. 11. Two-wavenumber lift aerodynamic admittance of the 5:1 rectangular cylinder. 131
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Fig. 12. Numerical results of (a) the 3D effects factors g3D and (b) the generalized lift aerodynamic admittance Að~k1 Þj2 at different turbulence integral scale to chord ratios.
rectangular cylinder. The generalized aerodynamic admittance, which is the product of the one-wavenumber aerodynamic admittance and the 3D effects factor, is directly obtained with the measured one-dimensional lift spectrum and the measured one-dimensional turbulence spectrum. The one-wavenumber lift aerodynamic admittance of a 5:1 rectangular cylinder is then obtained with the generalized lift aerodynamic admittance and the 3D effects factor. In this process, an adapted form of the lift coherence function is derived, then the spanwise influence term, which requires to be known in the calculation of the 3D effects factor, is estimated by the use of the measured lift coherence. The results show that the one-wavenumber aerodynamic admittance of a 5:1 rectangular plates is obviously different from the Sears function. Based on the experimental data and analysis, an empirical formula is proposed to determine the lift aerodynamic admittance function of a 5:1 rectangular cylinder. The results also show that the 3D effects factor determines the deviations between the generalized aerodynamic admittance and the onewavenumber aerodynamic admittance. This factor depends on the ratio of the turbulence integral scale to the chord length. When the turbulence integral scale to the chord ratio is smaller, the 3D effects factor is smaller, the generalized aerodynamic admittance is much less than the onewavenumber aerodynamic admittance. With the increase of the ratio of the turbulence integral scale to the chord, the value of the 3D effects factor increases, the generalized lift aerodynamic admittance approaches to the one-wavenumber aerodynamic admittance gradually. Hence, the generalized lift aerodynamic admittances in different flow fields are different due to different turbulence integral scale to the chord ratios. With the measured one-wavenumber aerodynamic admittance and the calculated 3D effects factor, the generalized lift aerodynamic admittance in different flow fields may be predicted. It should be emphasized that the approach presented in this paper is not restricted to the simple rectangular cross-section considered herein. It can be extended to bluff structures with more complicated cross-sections, such as bridges and buildings.
admittance, the two-wavenumber lift aerodynamic admittance of the 5:1 rectangular cylinder is then obtained by the use of Eq. (8), as shown in Fig. 11. Note that the spanwise influence term ηð~k1 ; ~k2 Þ → 1 when the
spanwise wavenumber k2 → 0. Thus the one-wavenumber aerodynamic admittance can be regarded as a specific solution of the two-wavenumber aerodynamic admittance. 4.4. Discussion on the influence of the turbulence integral scale to the chord ratio In the above section, the 3D effects factor is calculated with the experimental value of the turbulence integral scale to the chord ratio. In fact, according to the theoretical analysis, the 3D effects factor as well as the generalized aerodynamic admittance would vary with the turbulence integral scale to the chord ratio. Hence, to further investigate the influence of the turbulence integral scale to the chord ratio, the 3D effects factor can be numerically varied over a larger parameter range, which would result in different generalized aerodynamic admittances. The 3D effects factors calculated with different turbulence integral scale to chord ratios are shown in Fig. 12a, the corresponding generalized aerodynamic admittances are shown in Fig. 12b. It can be seen that when the ratio of the turbulence integral scale to the chord is relatively small, the 3D effects factor is small, which means that the influence of the 3D effects is significant. In this case, the generalized lift aerodynamic admittance is much less than the one-wavenumber aerodynamic admittance. With the increase of the ratio of the turbulence integral scale to the chord, the influence of the 3D effects weakens, the value of the 3D effects factor increases, and thus the generalized lift aerodynamic admittance approaches to the one-wavenumber aerodynamic admittance gradually. Apparently, if the ratio of the turbulence integral scale to the chord is known, a reasonable prediction of the generalized lift aerodynamic admittance in different flow fields could be given with the measured onewavenumber aerodynamic admittance and the 3D effects factor. Nevertheless, to extend the approach to the practical application better, more extensive and systematic measurements involving other cross-sections in different flow fields would be worthwhile.
Acknowledgements This work is supported by National Natural Science Foundation of China under Grant Nos. 51478402, 51608074.
5. Conclusions This paper has investigated the lift aerodynamic admittances of a 5:1
Appendix D. Supplementary data Supplementary data to this article can be found online at https://doi.org/10.1016/j.jweia.2019.03.023.
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Appendix A This appendix gives the spanwise influence term used in the calculation of the 3D effects factor g3D . The theoretical models for the spanwise influence term of the thin airfoil are derived by Mugridge (1971) and Blake (1986), which are
ηMugridge ~k1 ; ~k 2 ¼
2 4:9~k1 þ 1 ; 2 2 4:9~k þ 4:9~k þ 1 1
(A.1)
2
1=2 4:5~k1 þ 1 : 1=2 2 4:5~k þ 9:6~k þ 1
ηBlake ~k1 ; ~k 2 ¼
1
(A.2)
2
For rectangular cylinders, modelling the spanwise influence term with empirical parameters becomes necessary, since the flow over the rectangular cylinder is partially separated, which inhibits the use of existing models. Hence, an empirical model for the spanwise influence term of the rectangular cylinder is given as follow
c a~k1 þ 1 : 2 þ b~k þ 1
η ~k 1 ; ~k 2 ¼
c a~k 1
(A.3)
2
a, b and c are empirical parameters, which can be experimentally determined by the use of the measured lift coherence (see Appendix C). Appendix B For homogeneous and isotropic turbulence, the spectral tensor can be described by the total kinetic energy as follow (Batchelor, 1953; Panofsky and Dutton, 1984), Φij ðkÞ ¼
x 2 Lu EðkÞ 2 k δij ki kj ; 4π k4
(B.1)
i and j are from 1 to 3, k is the wavenumber vector fk1 ; k2 ; k3 g. The number corresponds to fluctuating velocity components u, v and w in Φij ðkÞ, respectively, and to x (chordwise), y (spanwise), and z (vertical) in ki , respectively. δij is the Kronecker delta. EðkÞ is the energy spectrum. Combining theories of Kolmogorov and Loitsyansky, von Karman (1948) proposed an interpolation formula for the energy spectrum. It is 8 ðLkÞ4 EðkÞ ¼ σ 2 Lxu 17=6 : π 1 þ ðLkÞ2
(B.2)
This interpolation formula represents EðkÞ for small and large values of k and has the advantage that calculations can be carried out analytically. σ is pffiffiffi the root mean square of the fluctuating velocity, L is the turbulence length scale, L ¼ ½Γð1=3Þ= π Γð5=6ÞLxu 1:339Lxu , Γ is the Gamma function. Substituting Eq. (B.2) into Eq. (B.1), the two-dimensional spectra of the u-component and the w-component are Z Su ðk1 ; k2 Þ ¼
þ∞
∞
2 1:195σ 2u Lxu 2 i4=3 ; π 1 þ 1:793 Lxu k 21 þ k22
Φuu ðkÞdk3 ¼ h
(B.3)
4 45:718σ 2w Lxw k 21 þ k 22 Φww ðkÞdk3 ¼ h 2 i7=3 : π 1 þ 7:172 Lxw k 21 þ k22
(B.4)
and Z Sw ðk1 ; k2 Þ ¼
þ∞
∞
The corresponding one-dimensional turbulence spectra are Z Su ðk1 Þ ¼
þ∞
∞
Z
þ∞
∞
2σ 2u Lxu Φuu ðkÞdk2 dk3 ¼ 2 5=6 ; π 1 þ 1:793 Lxu k21
(B.5)
h 2 i 2σ 2w Lxw 1 þ 19:125 Lxw k 21 Φww ðkÞdk2 dk3 ¼ 2 11=6 : π 1 þ 7:172 Lxw k 21
(B.6)
and Z Sw ðk1 Þ ¼
þ∞
∞
Z
þ∞
∞
Lxu and Lxw are the longitudinal and vertical turbulence integral scales. For isotropic turbulence, there are isotropic relationships Lxu ¼ 2Lxw and σ u ¼ σ w (Mann, 1994). Theoretically, these turbulence parameters can be interchangeable in the analysis according to above isotropic relationships.
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Appendix C The lift coherence function is defined as CohL ðk1 ; ΔyÞ ¼
SL ðk1 ; ΔyÞ ; SL ðk1 Þ
(C.1)
Δy is the separation between two strips. SL ðk1 ; ΔyÞ is the cross-spectrum of the lift, it is the inverse Fourier transform of SL ðk1 ; k2 Þ, which is Z SL ðk1 ; ΔyÞ ¼
þ∞
∞
SL ðk1 ; k2 Þexpðik2 ΔyÞdk2 :
(C.2)
Combining Eqs. (7), (9), (C.1) and (C.2), the lift coherence function can be then written as CohL ðk1 ; ΔyÞ ¼
R þ∞ 2 4C L Su ðk1 ; k2 Þ þ ðC'L þ CD Þ2 Sw ðk1 ; k2 Þ η ~k1 ; ~k 2 expðik2 ΔyÞdk2 ∞ : R þ∞ 2 4CL Su ðk1 ; k2 Þ þ ðC'L þ CD Þ2 Sw ðk1 ; k2 Þ η ~k 1 ; ~k 2 dk2 ∞
(C.3)
From Eq. (C.3), the relation between the lift coherence and the spanwise influence term is established. Empirical parameters a, b and c in the spanwise influence term can be then determined by fitting the measured lift coherence.
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