2.1 Unsteady lift response of a rectangular-section body in turbulent flow

The turbulent flow is three-dimensional. The phase of the fluctuating velocity varies not only in the chordwise direction but also in the spanwise direction. The Fourier component of the fluctuating velocity of the turbulent flow can therefore be expressed as an oblique harmonic gust convected with the velocity $U$ , which has the form $w=w_{0}\text{e}^{-\text{i}(\unicode[STIX]{x1D714}t-k_{1}x-k_{2}y)}$ , where $w_{0}$ is the gust amplitude, $\unicode[STIX]{x1D714}$ is the circular frequency, $k_{1}$ and $k_{2}$ are chordwise and spanwise wavenumbers, respectively. Let $L$ be the lift per unit span of a section of spanwise length $b$ , e.g. the finite span section shown in figure 1(a). According to the three-dimensional theory, the lift $L$ acting on such a section (centred on $y$ ) of an infinite rectangular-section body subjected to an oblique harmonic gust can be expressed as

where ${\hat{C}}_{s}$ is the static force term. For rectangular-section bodies, the static force term is equal to $\unicode[STIX]{x1D70C}Uc(C_{L}^{\prime }+C_{D})/2$ , where $\unicode[STIX]{x1D70C}$ is the fluid density, $c$ is the chord length, $C_{L}^{\prime }$ is the slope of the lift coefficient curve and $C_{D}$ is the drag coefficient. Also, $\text{sin}\,c(k_{2}b/2)$ is the sinc function, $\unicode[STIX]{x1D712}(\tilde{k}_{1},\tilde{k}_{2})$ is the two-wavenumber transfer function, $\tilde{k}_{1}~(=k_{1}c/2)$ and $\tilde{k}_{2}$ $(=k_{2}c/2)$ are the dimensionless chordwise and spanwise wavenumbers, respectively. The lift induced by turbulent flow may be composed of the lift forces generated by a sum of such harmonic gust components (Ribner Reference Ribner1956; Etkin Reference Etkin1959). The two-dimensional spectrum of $L$ is therefore written as

where $S_{w}(k_{1},k_{2})$ is the two-dimensional vertical turbulence spectrum and $|\cdot |^{2}$ is the symbol for the squared modulus. Note that in cases where it does not result in confusion, the squared modulus of the transfer function is also referred to as the transfer function for the sake of brevity. The two-wavenumber transfer function can be expressed as the product of the one-wavenumber transfer function and the spanwise influence term (Mugridge Reference Mugridge1971; Blake Reference Blake1986), which is

where $\unicode[STIX]{x1D712}(\tilde{k}_{1})$ is the one-wavenumber transfer function and $\unicode[STIX]{x1D702}(\tilde{k}_{1},\tilde{k}_{2})$ is the spanwise influence term. By integrating out the spanwise wavenumber, the one-dimensional lift spectrum can then be written as

Figure 1. Sketch of an infinite rectangular-section body in turbulent flow, in which the Fourier component of the fluctuating velocity is represented by an oblique harmonic gust: (a) a finite span section with a spanwise length, (b) two strips separated in the spanwise direction.

2.2 Influence of three-dimensional effects on lift force and transfer function

In order to bring in the one-dimensional vertical turbulence spectrum $S_{w}(k_{1})$ , the following 3-D effects factor is defined:

If the turbulence is homogeneous and isotropic, the turbulence spectrum can be described by the von Kármán spectral model (see appendix A). $F_{3D}$ can then be given in terms of dimensionless parameters: the ration of the vertical turbulence integral scale to the chord $\unicode[STIX]{x1D6FE}=L_{w}^{x}/c$ and the ratio of the spanwise length to the chord (the aspect ratio) $\unicode[STIX]{x1D703}=b/c$ ,

The one-dimensional lift spectrum then becomes

From (2.7), the normalized one-dimensional lift spectrum $S_{L}/({\hat{C}}_{s}^{2}S_{w})$ is the combination of the unsteady effects (reflected by the one-wavenumber transfer function) and the three-dimensional effects (reflected by the 3-D effects factor). This normalized one-dimensional lift spectrum can be directly obtained with the measured one-dimensional lift spectrum and the measured one-dimensional turbulence spectrum in practical measurements.

The 3-D effects factor includes the spanwise influence term due to the influence of three-dimensional effects on the transfer function. Following the work of Li et al. (Reference Li, Yang, Li and Liao2018), the dimensionless factor $\unicode[STIX]{x1D713}$ , which is the ratio of the one-dimensional lift spectrum calculated from the three-dimensional theory and that calculated from the strip theory, can be used to evaluate the degree of this influence. According to the strip theory, the one-dimensional lift spectrum is

where $F_{3DST}(\tilde{k}_{1},\unicode[STIX]{x1D6FE},\unicode[STIX]{x1D703})$ is the 3-D effects factor derived from the strip theory, in which the spanwise influence term is not included (or say, the spanwise influence term is equal to 1). It can be expressed as

The dimensionless factor $\unicode[STIX]{x1D713}$ is then expressed as

It can be seen that when the dimensionless factor is close to 1, $F_{3D}$ can be approximated by $F_{3DST}$ , which means that the influence of three-dimensional effects on the transfer function becomes negligible and the strip theory is applicable.

Actually, nothing has been gained so far because both the one-wavenumber transfer function and the spanwise influence term are unknown for the rectangular-section body. However, the lift coherence function is the ratio of the lift cross-spectrum to the lift auto-spectrum and consequently does not include the one-wavenumber transfer function. The possibility of using the measured lift coherence to determine the spanwise influence term becomes apparent.

2.3 Coherence function of lift force

The lift coherence is generally measured on two separate strips in the spanwise direction, e.g. strip 1 and strip 2 shown in figure 1(b). In this case, the lift of an infinitesimal span length $b\rightarrow 0$ , and therefore the sinc function $\text{sin}\,c(k_{2}b/2)\rightarrow 1$ . The lift coherence function is defined as

where $\unicode[STIX]{x0394}y$ is the separation between two strips and $S_{L}(k_{1},\unicode[STIX]{x0394}y)$ is the lift cross-spectrum, which can be expressed as the inverse Fourier transform of $S_{L}(k_{1},k_{2})$ as follows:

The lift coherence function is then written as

The spanwise influence term of the rectangular-section body cannot be theoretically derived since the flow over the rectangular cross-section is partially separated. However, a previous study (Li & Li Reference Li and Li2017) indicated that an empirical solution may still be applicable in this case. For this, an empirical model of the spanwise influence term is given as follows:

where $\unicode[STIX]{x1D6FC}$ , $\unicode[STIX]{x1D6FD}$ and $a$ are empirical parameters. In (2.13), a relation between the lift coherence and the spanwise influence term is established which enables the use of the measured lift coherence for the estimation of the empirical parameters in the spanwise influence term. Despite the empirical nature of this solution, it is believed that the fundamental properties of the results would not be lost. For comparison, the coherence function of the vertical fluctuating velocity is also given according to the preceding concepts as

Comparing (2.13) and (2.15), it can be seen that the lift coherence is naturally not equal to the coherence of the fluctuating velocity due to the influence of three-dimensional effects on the transfer function.

4.1 Characteristics of turbulent flow

Grid turbulence is considered to be the closest practical approximation to homogeneous and isotropic turbulence. For homogeneous turbulence, the root mean square (r.m.s) of the fluctuating velocity should be identical everywhere in the same plane parallel to the grid. As shown in figure 3(a), the normalized r.m.s fluctuating velocities at different spanwise positions are very close. The homogeneity of the generated turbulence can be confirmed. The longitudinal and vertical normalized r.m.s fluctuating velocities are $\unicode[STIX]{x1D70E}_{u}/U=0.095$ and $\unicode[STIX]{x1D70E}_{w}/U=0.083$ , respectively. The ratio $\unicode[STIX]{x1D70E}_{u}/\unicode[STIX]{x1D70E}_{w}=1.14$ . The turbulence integral scale, which reflects the mean size of the energy-containing eddies, can be determined by a comparison between the measured one-dimensional turbulence spectrum and the von Kármán spectral model (see appendix A). The measured one-dimensional spectra of the longitudinal and vertical fluctuating velocities are shown in figure 3(b). It can be seen that the measured turbulence spectra match well with the von Kármán spectral models, as observed in previous grid turbulence experiments (Roberts & Surry Reference Roberts and Surry1973; Nagata et al. Reference Nagata, Hunt, Sakai and Wong2011; Li et al. Reference Li, Li and Liao2015). The longitudinal and vertical turbulence integral scales (in the chordwise direction) determined from the one-dimensional turbulence spectra are $L_{u}^{x}=0.133~\text{m}$ and $L_{w}^{x}=0.062~\text{m}$ , respectively. The ratio $L_{u}^{x}/L_{w}^{x}=2.15$ . To further assess the isotropy of the turbulence, the spanwise correlation coefficients of fluctuating velocities are measured, as shown in figure 4. The vertical turbulence integral scale in the spanwise direction can then be determined from the definition: $L_{w}^{y}=0.06~\text{m}$ and the ratio $L_{u}^{x}/L_{w}^{y}=2.21$ . Though it seems that the strictly isotropic condition is difficult to achieve, these results roughly satisfy the isotropic relationships and the generated turbulence can be regarded as approximately isotropic.

Figure 4. Spanwise correlation coefficients of longitudinal and vertical fluctuating velocities, $\unicode[STIX]{x0394}y$ is the spanwise separation.

4.2 Lift coherence and spanwise influence term

The measured lift coherence and the measured coherence of the vertical fluctuating velocity are shown in figure 5. The theoretical velocity coherence is calculated by (2.15). Using (2.13) to fit the measured lift coherence, the empirical parameters of the spanwise influence term are obtained, as listed in table 1. The theoretical lift coherence of an ideal flat plate is also included for comparison. It is shown that the measured velocity coherence matches well with the theoretical velocity coherence, while the measured lift coherence is much stronger than the velocity coherence. This phenomenon has been shown experimentally on many occasions (Etkin Reference Etkin1971; Jakobsen Reference Jakobsen1997; Kimura et al. Reference Kimura, Fujino, Nakato and Tamura1997; Larose & Mann Reference Larose and Mann1998; Ma Reference Ma2007). Etkin (Reference Etkin1971) considered that the stronger correlation is an intrinsic property of the lift induced by the three-dimensional turbulence due to the influence of three-dimensional effects on the transfer function.

Figure 5. Coherence of lift forces and vertical fluctuating velocities.

It can be observed that there are different degrees of deviation between the measured lift coherence of the model and the theoretical lift coherence of the ideal flat plate. Considering that the lift coherence reflects the structure of the flow around the body, the distortion of the velocity fluctuation and the flow separation may be the reasons for these deviations. Compared to the flow around an ideal flat plate, the mean velocity field of the flow around a real body is disturbed, which distorts the velocity fluctuation (or eddies) of the turbulent flow. The distortion of the velocity fluctuation reduces its spatial correlation (Santana et al. Reference Santana, Christophe, Schram and Desmet2016) and consequently leads to the decrease of the coherence of the induced lift (compared to the theoretical lift coherence of the ideal flat plate). It should be noted that the deviation between the theoretical lift coherence and the measurement is frequency-dependent. As mentioned above, the turbulent flow is distorted by the disturbance of the mean velocity field; however, the degree of the distortion is different for eddies with different scales. The high-frequency velocity fluctuation (small-scale eddy) is more easily distorted by the disturbance than the low-frequency velocity fluctuation (large-scale eddy). Hence, the measured lift coherence agrees better with the theoretical value for lower frequencies, while for higher frequencies the measured lift coherence deviates significantly from the theoretical value due to the serious distortion of the velocity fluctuation. On the other hand, the flow over the rectangular cross-section is partially separated. The results indicate that the spatial correlation of the velocity fluctuation under separated flow conditions may be weaker than that under attached flow conditions, further reducing the lift coherence of the rectangular-section body compared to the lift coherence of the airfoil. It can be seen that the decrease of the lift coherence becomes more significant as the chord-to-depth ratio of the rectangular cross-section becomes smaller, accompanied by more serious flow separation.

It can be seen from these results that the turbulence distortion and flow separation have a significant influence on the lift coherence. Due to this influence, the lift coherence of a real body is lower than the theoretical lift coherence of an ideal flat plate, and the spanwise influence term of a real body is also different from the theoretical spanwise influence term. Compared to a rectangular-section body with larger chord-to-depth ratio, one with smaller chord-to-depth ratio experiences more serious turbulence distortion and flow separation. Correspondingly, the deviation between its lift coherence and the theoretical lift coherence is more obvious, and the difference between its spanwise influence term and the theoretical value is also more significant.

Figure 6. Normalized one-dimensional lift spectrum and transfer function of model A.

4.3 Roles of 3-D effects factor and transfer function in lift force

For a body in turbulent flow, factoring the lift only by the static force term would fail as the frequency increases since the generation of the lift is simultaneously affected by both the unsteady effects and the three-dimensional effects. As shown in figures 6–9, the one-dimensional lift spectrum, which is normalized by the one-dimensional turbulence spectrum and the static force term, decreases as the frequency increases. In this case, the 3-D effects factor $F_{3D}$ , which is defined by (2.6a,b ), can be used to reflect the influence of three-dimensional effects on the lift. Using the spanwise influence terms determined in § 4.2, the 3-D effects factors of the airfoil and rectangular-section bodies are calculated, as shown in the insets. The static force terms and ratios of turbulence integral scale to chord used in calculations are listed in table 2 in appendix C. Note that the aspect ratios are all equal to zero in the current experiment. The results show that the 3-D effects factor is a function with the value of less than 1, which is one of the reasons for the decreasing of the normalized lift spectrum compared to the quasi-steady value. The 3-D effect factor is relatively small at lower frequencies, but increases as the frequency increases. This indicates that the three-dimensional effects mainly affect the lift at lower frequencies, and the increase of the frequency tends to reduce this influence. In fact, we will show in the next section that the influence of three-dimensional effects on the lift reaches a peak at a certain frequency and then decreases as the frequency increases. Certainly, the 3-D effects factors of these aerodynamic configurations are different due to their different spanwise influence terms as well as the ratios turbulence integral scale to chord.

Figure 7. Normalized one-dimensional lift spectrum and transfer function of model B.

The flow being unsteady is another reason for the decreasing of the normalized lift spectrum compared to the quasi-steady value. These unsteady effects can be reflected by the one-wavenumber transfer function, which would apply unchanged to the spanwise correlated unsteady incident flow. After separating the influence of three-dimensional effects on the lift with the 3-D effects factor, the one-wavenumber transfer functions of these aerodynamic configurations can then be determined based on experiment data. Due to the empirical nature of the determination, an empirical model for the one-wavenumber transfer function is introduced (see appendix C). For the NACA 0015 airfoil, the one-wavenumber transfer function agrees well with the Sears function at lower frequencies, while falling at a faster rate as the frequency increases, as shown in figure 6. According to previous theoretical studies (Goldstein & Atassi Reference Goldstein and Atassi1976; Atassi Reference Atassi1984; Lysak, Capone & Jonson Reference Lysak, Capone and Jonson2013), this may be caused by the thickness effects. The Sears function is derived based on the flat plate assumption. This may be a reasonable assumption at lower frequencies. However, at higher frequencies, the turbulence is significantly distorted by the disturbance of the mean velocity caused by the thickness. In this case, the flat plate assumption is invalid and leads to the overestimation of the transfer function. Relatively little research has investigated the one-wavenumber transfer function of the rectangular-section body. From the results of figures 7–9, the one-wavenumber transfer functions of the rectangular-section bodies are larger than the Sears function at lower frequencies, while also falling more quickly than the Sears function as the frequency increases. The high-frequency deviation may be attributed to the distortion of the turbulence, similarly to the case involving a thick airfoil. However, the physics that causes the low-frequency deviation is still not fully understood, to the extent that the role of the flow separation in the generation mechanism of the lift is not clear.

Figure 8. Normalized one-dimensional lift spectrum and transfer function of model C.

Figure 9. Normalized one-dimensional lift spectrum and transfer function of model D.

From the current results, it can be seen that the low-frequency deviation becomes more significant as the chord-to-depth ratio becomes smaller. This is accompanied by more serious flow separation. It may be considered that the flow separation reduces the influence of the unsteady effects on the lift at lower frequencies. The more serious the flow separation, the lower the influence of the unsteady effects on the low-frequency lift, and consequently the closer the one-wavenumber transfer function of the rectangular-section body to the quasi-steady value. To better illustrate these differences, prediction 1 is calculated with the Sears function and the 3-D effects factor, while prediction 2 is calculated with the modified transfer function and the same 3-D effects factor. It can be seen that, although both predictions consider the three-dimensional effects, prediction 1 cannot capture the experimental values because the Sears function does not consider the effects of the flow separation and the distortion. Prediction 2, wherein the transfer function accounts for the effects of the flow separation and the distortion by modifying the Sears function, agrees well with experimental values. Hence, for rectangular-section bodies, the use of the Sears function would underestimate the lift at lower frequencies while overestimating the lift at higher frequencies. The modified transfer function proposed here may be an appropriate alternative in this case.

4.4 Influence of turbulence integral scale to chord ratio and aspect ratio

In § 4.3, we have shown that the three-dimensional effects have a significant influence on the lift induced by turbulent flow. This influence can be calculated by the 3-D effects factor $F_{3D}$ derived from the three-dimensional theory. However, it requires the spanwise influence term to be known. If the strip theory is applicable, the simpler factor $F_{3DST}$ derived from the strip theory can be used to approximate $F_{3D}$ , avoiding the measurement of the spanwise influence term. The key is to what extent the strip theory is applicable. For this, $F_{3D}$ and $F_{3DST}$ are numerically calculated under different turbulence integral scale to chord ratios $\unicode[STIX]{x1D6FE}$ and aspect ratios $\unicode[STIX]{x1D703}$ , as shown in figure 10. The relative deviation between $F_{3D}$ and $F_{3DST}$ (the accuracy of the strip theory) is quantitatively evaluated by the dimensionless factor $\unicode[STIX]{x1D713}$ under the same parameters, as shown in figure 11. Note that $F_{3DST}$ does not contain the spanwise influence term, and is therefore independent of the aerodynamic configuration.

Figure 10. 3-D effects factors $F_{3D}$ under different ratios of turbulence integral scale to chord $\unicode[STIX]{x1D6FE}$ and aspect ratios $\unicode[STIX]{x1D703}$ .

Figure 11. Dimensionless factors $\unicode[STIX]{x1D713}$ under different of ratios turbulence integral scale to chord $\unicode[STIX]{x1D6FE}$ and aspect ratios $\unicode[STIX]{x1D703}$ .

It is shown that the 3-D effects factors $F_{3D}$ of all configurations are much smaller than $F_{3DST}$ when the values of $\unicode[STIX]{x1D6FE}$ and $\unicode[STIX]{x1D703}$ are small. The use of the strip theory would thus lead to the significant overestimation of the lift in this case. As $\unicode[STIX]{x1D6FE}$ and $\unicode[STIX]{x1D703}$ increase, $F_{3D}$ gradually approaches $F_{3DST}$ . When the values of $\unicode[STIX]{x1D6FE}$ and $\unicode[STIX]{x1D703}$ are large, the deviation between $F_{3D}$ and $F_{3DST}$ becomes small. The same trend can be observed in the results of the dimensionless factor. The dimensionless factor $\unicode[STIX]{x1D713}$ is much less than 1 when $\unicode[STIX]{x1D6FE}$ and $\unicode[STIX]{x1D703}$ have small values, and increases with the increase of $\unicode[STIX]{x1D6FE}$ and $\unicode[STIX]{x1D703}$ . When $\unicode[STIX]{x1D6FE}$ and $\unicode[STIX]{x1D703}$ have large values, the dimensionless factor $\unicode[STIX]{x1D713}$ is close to 1. The deviation between $F_{3D}$ and $F_{3DST}$ (or say, the dimensionless factor $\unicode[STIX]{x1D713}$ ) also depends on the frequency. The deviation at lower frequencies is significantly larger than the deviation at higher frequencies, and therefore the dimensionless factor $\unicode[STIX]{x1D713}$ is closer to 1 at higher frequencies. In addition, both $F_{3D}$ and $F_{3DST}$ reach a peak at a certain frequency then decrease as the frequency increases. The occurrence of the peak may be related to the shape of the turbulence spectrum. As is known, the turbulence spectrum also has a peak at the energy-containing range. From the correlation point of view, at the peak of the spectrum, the correlation of the turbulence is the strongest, and the three-dimensional effect is the weakest. $F_{3D}$ and $F_{3DST}$ reflect the influence of the 3-D effects on the lift, and therefore reach a peak at the frequency where the three-dimensional effect is the weakest. When the values of $\unicode[STIX]{x1D6FE}$ and $\unicode[STIX]{x1D703}$ are small, the peak position of $F_{3D}$ is not identical with the peak position of $F_{3DST}$ due to the influence of three-dimensional effects on the transfer function. With the increase of $\unicode[STIX]{x1D6FE}$ and $\unicode[STIX]{x1D703}$ , the influence of three-dimensional effects on the transfer function gradually weakens, and the positions of these peaks tend to be identical. It can be concluded from these results that for rectangular-section bodies, the accuracy of the strip theory depends on the ratio of the turbulence integral scale to the chord as well as the aspect ratio, and is proportional to these two parameters. If these two parameters have suitable values, the strip theory can be applicable to the rectangular-section body, and $F_{3DST}$ can be used to estimate the lift response of the rectangular-section body in turbulent flow.

The results also show that the dimensionless factors of the rectangular-section bodies are larger than the dimensionless factor of the airfoil under the same parameters. The dimensionless factor of the rectangular-section body with smaller chord-to-depth ratio is larger than that of the rectangular-section body with larger chord-to-depth ratio. As mentioned earlier, compared to the attached flow over the airfoil, the flow over the rectangular-section body is partially separated, and the flow separation experienced by the rectangular-section body with smaller chord-to-depth ratio is more serious. These trends imply that the accuracy of the strip theory is improved under separated flow conditions. The flow separation is one of the salient aerodynamic features of bluff bodies. Hence, compared to the airfoil, the lift response of the bluff body calculated by the strip theory will have higher accuracy, and the more serious the flow separation, the higher the accuracy.