Experimental Characterization of High-Amplitude Fluid–Structure Interaction of a Flexible Hydrofoil at High Reynolds Number

When a fluid flows over an object, it imparts stresses on the structure, which if sufficiently large can deform the structure and/or move it such that it modifies the flow pattern and consequently alters the stresses on the structure. Fluid–structure interaction (FSI) modeling aims to capture the interaction between the structure and the flow. Aerospace and marine applications are particularly interested in FSI modeling since lightweight and composite materials offer long life, high strength, and reduced weight while also producing significant FSI in unsteady flows. One of the most extreme operating conditions experienced by a ship or submarine propeller is a crashback maneuver [1–5], which occurs when a forward moving vessel reverses the propeller rotation to quickly stop. Development of computational tools to accurately predict the maximum stresses and deflections during crashback would significantly improve the propeller design process. Modeling of crashback events for traditional propulsors has been an open research topic for years [6–9]. This problem will become even more important as the use of composite materials [10–13] and new propulsion schemes [14–16] are explored. Improving crashback modeling capabilities was the focus of a companion paper [17] to the current study, which provides experimental results from a fin subjected to high-amplitude, flow-induced oscillations to mimic crashback conditions.

Ideally, the model would be two-way coupled such that the fluid and structure can affect one another. Interest in two-way coupled models has recently increased due to computational advances, maturing of flow/structural modeling, and realization of potential applications. While there is a large body of work related to small deformation, dynamic aeroelastic/hydroelastic modeling (e.g., flutter [18,19], vortex-induced vibrations [20–23]), there is a dearth of FSI literature when the structure experiences large deformations as observed with crashback. Campbell and Paterson [24] reviews the available literature and presents a method for computing pump performance with highly flexible impellers. Flexible turbomachinery has gained interest among researchers [25–28] due to the deformable structure aiding implantation for biomedical applications. In this application, the structural response and the fluid flow are strongly coupled due to the large flow-induced deformations, which require that the fluid flow and structural response be solved simultaneously. See Heil and Hazel [29] for a review of FSI-related internal physiological flows, and Dowell and Hall [30] for a review of FSI modeling approaches.

FSI models advancement has been limited, in part, due to the challenge of performing unsteady flow experiments that are strongly coupled with structural motions/deformation with sufficient accuracy needed for FSI model validation (e.g., Ref. [31]). One approach is to use a rigid fin oscillating in a controlled manner but mounted on a flexible base such that the fin can shift from the flow-induced loading [32–34]. An alternative approach is to mount a flexible membrane within a cylinder wake, which tip deflections and velocities from this approach have been reported up to a Reynolds number of 500 [35]. A similar study at higher Reynolds number (3.0 × 10⁴) [36] is one of the few turbulent FSI test cases.

The current study performs a similar experiment, but instead of a flexible membrane, a fin was mounted in the cylinder wake to mimic crashback. The cylinder was sized to induce fin angle-of-attack variations at a frequency such that a quasi-steady lift varied by nominally 40%. In addition, the flow was turbulent with a cylinder Reynolds number up to 2.8 × 10⁵. Measurements included two-component velocity in the wake, surface deflections, and constraint forces and moments on the fin. This study fills a void in the literature for high-fidelity, fully coupled FSI experiments with large structural deformations. A comparison between this study and a tightly coupled FSI simulation is available [17].

Testing was done in the Applied Research Lab 12-in. water tunnel [37,38]. The test section was 760 mm long with a circular cross section. To minimize acceleration from boundary layer growth, the diameter gradually increased from 305 mm (12-in.) at the inlet to 307 mm at the outlet. The empty test section freestream turbulence intensity was <0.3%. The test configuration was a backward-facing fin (to mimic crashback) mounted in the wake of a 60 mm diameter circular cylinder as shown in Fig. 1. The contraction upstream of the test section had an 8.9:1 area ratio, 1.5 m long and a fifth-order polynomial shape [39]. The coordinate system has the x-axis increasing in the streamwise direction with its origin at the cylinder center. The z-axis increases vertically upward with the origin at the fin base, which was 11.4 mm above inlet floor due a ramp used to make a flat surface for fin mounting (see Fig. 1). The elliptical shaped ramp spanned x = 70 mm to x = 119 mm. The y-axis completes a right-handed coordinate system with the origin at the tunnel centerline. Technical drawings and IGES files for the test section are in the Supplemental Material on the ASME Digital Collection.

The anodized aluminum cylinder spanned the test section height with its axis of rotation (centerline) at (x,y) = (0,0). The cylinder leading edge was roughened between θ = ± 50 deg with 254 µm mean diameter silicon carbide grit, where θ is the angular position from the leading edge (see Fig. 2). This roughness was selected based on past work [40] that showed that the base coefficient and Strouhal number were Reynolds number independent. The cylinder was instrumented with four dynamic pressure (dp) transducers (105C02, PCB) and two static pressure (sp) ports. The dp transducers were located at θ = ±130 deg at the fin tip (z = 230 mm) and mid-span (z = 115 mm) heights (see Fig. 2). Static ports were at the same heights and θ = 180 deg. An accelerometer (303M231, PCB) was mounted in the cylinder.

The fin base must move freely to measure constraint forces. Typically, a load cell (LC) uses a stiff model with a small gap between the model and the surrounding structure (e.g., Ref. [41]), but the large flow-induced motion required a larger gap. In addition, the load cell flexure was weakened to improve sensitivity, which required the fin and load cell assembly to be considered as the FSI test article. The fin, including a 23 mm thick base, was machined from a single piece of Inconel 718. It had a 49.5 mm chord, 230 mm span, average surface roughness <1.6 µm, and was swept forward (to mimic crashback) 75 mm. The angle-of-attack (α) was fixed at −9.6 deg, which produced ∼3% solid blockage. The fin cross section was a NACA 4408 profile. From the base to mid-span, the fin was straight and necked down to a minimum chord of 35 mm at 25% span. Table 1 provides leading and trailing edge locations, chord length, and α along the span. The load cell was fabricated from beryllium copper with dimensions shown in Fig. 2. The technical drawings and IGES files for the fin (and load cell) are provided in the Supplemental Material on the ASME Digital Collection.

Static testing was done by applying point loads to the fin at 75% span and 62% chord (denoted by “×” in Fig. 2) with and without the load cell. The fin tip deflection (δ_t) versus applied force (gradual ramp or static) is shown in Fig. 3. The slopes for the fin only and fin with LC are 0.104 mm/N and 0.190 mm/N, respectively. The mode shapes and loss factors were determined from in-air and in-water modal impact tests with a roving hammer and accelerometers. Mode shapes, resonance frequencies, and loss factors were identified using a singular value decomposition approach combined with rational fraction polynomial curve fitting [42]. The frequency and loss factors of the first seven modes in-air and in-water are provided in Table 2.

Two-component (x and y) particle image velocimetry (PIV) was acquired at z = 188 mm. The image plane was illuminated with a laser sheet formed with an Nd:YLF diode pumped laser (DM50-527, Photonics Industries). Scattered light from 18 µm diameter tracer particles (iM30 K, 3 M) were imaged with a 1280 × 800 pixel CMOS camera (v1610, Phantom). A minimum of 3000 image pairs were acquired per condition at 1 kHz. The field-of-view spanned x = 100–340 mm and y = −80 to 85 mm. Vectors were computed using standard multiple pass, cross-correlation (DaVis 8, LaVision) with a final interrogation window of 24 × 24 pixels with 75% overlap (1.4 mm vector spacing). A subset of conditions used laser Doppler velocimetry (LDV), which used an argon ion laser (Innova 70C-5, Coherent) coupled to a fiber optic probe head (9832, TSI) fitted with a 350 mm lens (9253-350, TSI Incorporated). The nominal measurement volume was 1.5 mm (length) by 150 µm (diameter) [43]. Most conditions acquired 20,000 samples per component.

The fin motion and orientation were monitored with a rotary encoder and an optical switch [44]. For a subset of conditions, high-speed imaging provided fin tip deflection (translational motion) and twist (rotational motion). The motion was identified via cross-correlation between an image and a reference (no-flow) image, which was confirmed with 400 manually inspected images. This also showed that the unloaded fin tip position varied <10 µm (1 pixel) throughout testing. The fin surface velocity was measured with a laser vibrometer (PSV-400-3D, Polytec) at 21 locations (7 × 3 grid; 40, 50, 60, 70, 80, 90, 95% span; 10, 50, 90% chord) and integrated to get displacements. The beam was perpendicular to the fin surface at 90% chord with no-flow (α = −9.6 deg) and corrections for relative surface angle at each location were applied. A reference vibrometer (OFV-505/5000, Polytec) was fixed at 80% span, 90% chord. Following Bendat and Piersol [45] and assuming a 0.9 coherence, the normalized random errors for the autospectrum, cross-spectrum, coherence, coherent output power, and transfer function were 0.250, 0.264, 0.037, 0.276, and 0.0589, respectively.

The fin constraint forces/moments were measured with a custom four-component load cell (Fig. 2). The center shaft had four 7.62 mm square pockets (P1 and P2 in Fig. 2) to form two thin membranes (rotated 90 deg relative to each other) for lift and drag measurements. A center hole created a 0.71 mm thin-walled section to facilitate torque and drag-moment measurements. Each cell had a Wheatstone bridge of strain gauges (lift, drag, torque: sk-06-062TW-350, Vishay; y -moment: TK-06-092P-10C/DP, Vishay). Bridges were excited with 10 volts direct current (VDC) and the output sampled at 1 kHz. The load cell measured lift force $(L)$⁠, drag force $(D)$⁠, y-axis moment (drag-moment; M_D), and the z-axis moment (torque; T_q) when at α = 0 deg, which were corrected for α = −9.6 deg. A benchtop calibration matrix was formed from 356 samples using a range of loads, locations, and fin orientations, which was confirmed with an in situ calibration. The random and bias errors were quantified from the mean residual loads, which are in Table 3. Given the nominal test range, the uncertainty of the lift and drag-moment were about 5%, while the drag and torque were 7–10%.

Water temperature (resistance temperature detector sensor; 911PL, Stow Laboratories), total tunnel pressure (Kiel probe), local static pressure, and tunnel impeller frequency (f_imp) were recorded at 1 kHz. Static pressure ports were located at x = 0, 75, 175, 280, 385, and 490 mm as well as by the Kiel probe. The local freestream speed was determined from the difference between total and static pressure, which was confirmed with PIV and LDV.

This study focuses on the fin in the rough cylinder wake, but three other configurations were tested: (i) empty test section, (ii) smooth cylinder only, and (iii) rough cylinder only. These results as well as additional setup details are in Elbing et al. [44], and some of these results are included here to assist in interpretation and provide model validation data. During forced fin testing, f_imp varied between 50 and 84 rpm and α was −9.6 deg. The average water temperature was 20.5 ± 0.5 °C, which sets the water density (ρ) and kinematic viscosity (ν) at 998 kg/m³ and 9.9 × 10⁻⁷ m²/s, respectively.

A direct measure of the inlet velocity was not possible due to the close proximity of the cylinder. Thus, the volumetric flowrate was determined from integration of the velocity across the test section cross-sectional area since detailed surveys of the wake, freestream, vertical variation, and wall boundary layers were performed. The mean wake velocity profiles in Fig. 4 scale like a self-similar plane wake [46], where U is the local mean streamwise velocity, U_S(=U_∞ − U(x, y = 0)) is the velocity deficit, and y_0.5 is the wake half-width defined as U(x, ±y_0.5) = U_∞ − 0.5U_S(x). The current results approach a self-similar constant turbulent viscosity plane wake, (U_∞ − U)/U_S = exp(−0.693(y/y_0.5(x))²), with increasing downstream distance. Table 4 provides the average velocity at the test section inlet (U_in) at each f_imp, which is the volumetric flowrate divided by the inlet area. Table 4 also includes the wake scaling parameters, U_sp the average freestream speed spanning the fin location (x = 175–490 mm), U_C the centerline (y = 0) streamwise velocity one chord length upstream of the fin tip (x = 185 mm), and U_∞ a PIV freestream speed measurement. The freestream speeds from PIV and static pressure were within ∼2%. The forced fin analysis primarily uses U_sp since it is the freestream at the fin.

The flow around the three cylinder configurations (smooth, rough, and rough with fin) were characterized with the base static and dynamic pressures. The base pressure coefficient $(Cpb=ΔPb/0.5ρU∞2)$ is commonly used to characterize cylinder flow because it is directly related to the drag, where ΔP_b is the difference between base (θ = 180 deg) and local static pressures. Figure 5 provides C_pb versus Re_D(=U_∞D/ν) at the fin mid-span (z = 115 mm) and tip (z = 230 mm) heights for all three configurations. For reference, a representative smooth cylinder curve [47] is included. The smooth results follow the reference curve with scatter that is consistent with historical data [40,47,48]. This shows that the smooth cylinder was in the transitional range, which motivated roughening the leading edge. The rough results closely follow the data used to size and position the roughness [40], which has a weak Re_D dependence. Minimal spanwise variation is observed with and without the fin installed, but there is a measurable shift in the cylinder performance with the fin installed. This shift was likely due to the oscillating fin impacting the freestream speed measurement, not the actual cylinder flow field. Regardless, it is apparent that with the fin installed there was minimal spanwise variation and over the operation range the behavior is nearly independent of Re_D.

Spectral analysis of the dynamic pressure on the aft side of the cylinder was used to identify the vortex shedding frequency. The smooth cylinder showed increasing shedding frequency and root-mean-squared (RMS) levels with increasing speed until ∼3 m/s (Re_D = 1.8 × 10⁵). At higher speeds, the peak weakens and eventually disappears (Re_D = 2.4 × 10⁵) due to the laminar-to-turbulent transition of the boundary layer, which makes the wake incoherent. The rough cylinder has increasing frequency and RMS levels up to the max speed of 6 m/s (Re_D = 3.6 × 10⁵), but it also had three additional spectral peaks. For example, the power spectra (Φ_dp) at U_sp = 2.99 m/s are shown in Fig. 6 and spectral peaks are observed at 13 (shedding frequency), 16, 17.1, and 18.6 Hz. Estimating the fluctuating lift and drag forces on the cylinder showed that the three higher frequency peaks are only in the drag force spectra. Since LDV confirmed that the tunnel speed was stable and repeatable, the most likely cause for these extra peaks in the drag spectra were either noise contamination or spanwise variations. Fortunately, the fin loading was primarily dependent on the lift, which did not have these extra peaks, but it does have the 13 Hz shedding frequency. The spanwise coherence at the shedding frequency was high on both sides of the cylinder. Comparison of rough cylinder spectra with and without the fin shows that there was no significant variation in peak RMS levels at the shedding frequency, in spite of the slight variation in C_pb. This finding supports the previous conjecture that the C_pb variation was due to the fin’s influence on the velocity measurement and not an alteration to the flow field.

Figure 7 shows the Strouhal number (St = f_sD/U_∞) versus Re_D for all configurations, where f_s is the cylinder vortex shedding frequency. The current smooth data are in agreement with a composite curve for smooth cylinders [49], though no shedding was observed for Re_D ≥ 3.3 × 10⁵. Conversely, the rough cylinders had a nearly constant St (0.26) over the range of Re_D tested, which is slightly lower than that reported in the literature [40] over the same Re_D range. Bias error was quantified by comparing independent measurements (dynamic pressure, accelerometer, and LDV), and the random error was determined from comparisons between repeated test conditions. The bias and random errors were ±1.2% and ±2% with a 95% confidence, which sets the accuracy of f_s at ±2.3%.

The streamwise (u) and cross-stream (v) velocities were measured upstream of the fin at 82% span (z = 188 mm). Figure 8 provides the average streamwise (U), fluctuating streamwise (u′), average cross-stream (V), and fluctuating cross-stream (v′) velocity profiles scaled with U_sp and plotted versus the y-coordinate scaled with D at three streamwise locations (x/D = 2.25, 2.67, 3.08). The upstream cylinder blockage causes the mean streamwise velocity outside of the wake to exceed U_sp in the near-wake region, with this effect decreasing with increasing downstream distance. The mean and fluctuating streamwise velocity profiles show the wake width spreading and wake deficit decreasing with increasing downstream distance. The u′ profiles have a bimodal distribution with peaks at nominally ±0.4D, which also decreases with increasing downstream distance. The asymmetry of the peaks is likely due to the fin oscillations causing a periodic blockage downstream, which was biased toward the negative y side due to the angle-of-attack. However, due to the spreading wake, the fluctuating velocity increases with increasing downstream distance for |y/D| > 0.6. The V profiles are approximately zero at the centerline and symmetric in magnitude about the centerline with peaks near y/D = ±0.5. Since these positions are beyond the recirculation region (x/D ∼ 2.2), the peaks decrease with increasing downstream distance. The centerline v′ increases with increasing downstream distance, but this trend ceases beyond x = 185 mm (data not shown since the goal is to characterize the cylinder wake upstream of the fin).

The spatial distribution of the fluctuating velocity was examined via spatial correlations and wavenumber spectra. The spatial autocorrelation for a homogeneous velocity component u is defined as R_uu(ξ, η) = E[u(x, y)u(x + ξ, y + η)], where E[–] is the expected value and ξ and η are relative distances in the x and y directions, respectively. The spatial autocorrelation is even (symmetric), thus all spatial autocorrelations are shown for positive separation distances only. Contour plots of the spatial autocorrelation distribution for u and v are in Elbing et al. [44] with a general shape similar between speeds and the peak correlations increasing with velocity. The autocorrelation for u and v with streamwise (Δx) and cross-stream (Δy) separation distances is shown in Fig. 9. The autocorrelations are normalized by the variance (⁠$σu2$ or $σv2$⁠), which was 0.02 m²/s² for the v and u was 0.16, 0.2, and 0.35 m²/s² for U_sp = 2.51, 2.99, and 3.62 m/s, respectively. The streamwise variance was proportional to tunnel speed squared, and all the profiles have similar trends for a given speed and direction. The minimum at approximately 70 mm followed by a local maximum at approximately 140 mm in the streamwise direction (x) corresponds to the half wavelength (i.e., out-of-phase) and full wavelength of the shed vortices.

Since the spatial autocorrelation is even along both axes, the corresponding autospectrum is real and even (symmetric). Thus, only the positive wavenumbers (k_x, k_y) are shown. These wavenumber spectra were computed using the indirect fast Fourier transform (FFT) approach [50]. The range was extended by padding the velocity with zeroes and a Hanning window was applied to the spatial data. The FFT of the extended velocity data was used to estimate the autospectral density, and then the inverse FFT of the autospectral density gives the autocorrelation. Assuming a coherence of 1.00, the normalized random error for the autospectrum was 0.154 or 0.6 dB. Contour plots of the spatial distribution of S_uu and S_vv are in Elbing et al. [44], and Fig. 10 shows autospectra curves extracted from the contour plots versus k_x for various constant k_y values. Comparison of these wavenumber spectra with those acquired without the fin installed (not shown) show negligible variation, which shows that there were no feedback mechanisms between the upstream flow-field and the fin at this location.

High-speed imaging measured the maximum deflections (translational motion) and twist (rotation) at the fin tip. The fin tip twist (ϕ_t) results reveal a linear relationship with the tunnel speed, ϕ_t = 0.20U_sp − 0.372, between 2.5 ≤ U_sp ≤ 3.4. In general, there was minimal twist with the maximum mean and RMS twist being 0.31 deg and 0.33 deg, respectively. Fin tip deflection mean (δ_t) and RMS (δ_tʹ) values scaled by the fin span (b = 230 mm) as a function of the chord (c = 49.5 mm) based Reynolds number (Re_c = U_spc/ν) are in Fig. 11. The mean and RMS fin tip deflections were linearly fitted resulting in δ_t/b = −1.13 × 10⁻⁷ Re_c + 0.011 and $δt′/b=9.85×10−8Rec−0.0083$⁠, respectively. Note that a power-law fit was equally valid, but with a narrow Re_c range the power could not be accurately estimated. Also in Fig. 11 are dashed lines marking the nominal range of the fin tip deflection extrema, which were approximately ±2.9 times δ_t. The mean deflection became more negative with increasing speed due to the negative angle-of-attack. The uncertainty in the mean and RMS tip deflections are ±7% (≤ ± 0.15 mm) and ±9% (≤ ± 0.2 mm) of the measurement, respectively. This uncertainty was primarily due to the combination of a relatively short sample period (3 s) and broadband frequency content.

While the broadband frequency was apparent in the fin tip displacement time traces, spectral analysis was not performed due to the limited period. However, the peak in the autocorrelation was at f_s. Consequently, spectral density of the fin motion was examined with the surface laser vibrometer. A typical deflection RMS amplitude spectrum Φ_δ (square root of the autospectrum level) from a single point (90% span, 50% chord) is shown in Fig. 12. Two peaks were observed below 50 Hz, with the first and second corresponding to f_s and first fin bending mode, respectively. The peak at about 100 Hz corresponds to the in-water second cross-stream bending mode with no apparent peak at the first streamwise bending mode (∼44 Hz). This is consistent since surface vibrometers only measure the normal surface motion (i.e., more sensitive to cross-stream motion than streamwise in the current setup). As expected given the deflection spectrum, the predominant mode near the shedding frequency is the first bending mode with the largest deflections occurring near the fin tip (see Ref. [44]; for example, contour plots). The surface deflection results are provided in Table 5, including the peak frequency (f_pk), the corresponding RMS level, and the quality factor. The quality factor (Q = f_pk/(f_U − f_L)) is a measure of the bandwidth of the peak, where f_U and f_L are the frequency 3 dB below the peak on the upper and lower side of the peak, respectively. In addition, the average and standard deviation of f_pk and Q are provided. These results show that f_pk increases with speed but remains relatively stable independent of speed. The width of the peak broadens (i.e., Q increases) with increasing speed and experiences larger fluctuations.

The load cell measured the constraint lift $(L)$⁠, drag $(D)$⁠, torque (T_q), and drag-moment (M_D), which is the constraint moment about the axis perpendicular to the chord length. Time traces show that f_s dominates the oscillations with additional broadband content, and $L$ is 180 deg out-of-phase with the other constraint forces/moments [44]. The mean coefficient of lift $(CL≡L/(0.5ρUsp2bc))$⁠, drag $(CD≡D/(0.5ρUsp2bc))$⁠, drag-moment $(CMD≡MD/(0.5ρUsp2b2c))$⁠, and torque $(CTq≡Tq/(0.5ρUsp2b2c))$ were all independent of Reynolds number over the narrow test range (10⁵ to 1.8 × 10⁵). The average values were −0.200 ± 0.018, 0.114 ± 0.007, 0.040 ± 0.002, and 0.0287 ± 0.002 for C_L, C_D, C_MD, and C_Tq, respectively. Here, the uncertainties are twice the standard deviation from measurements at different Reynolds numbers. Note that the mean negative $L$ is consistent with non-zero mean fin tip deflections at a given speed. Torque and drag-moment data enable the calculation of the center of pressure in two directions. Torque on the fin was induced by the lift and drag forces, suggesting that the resultant load acts at a distance R_T from the center of the load cell. This distance was determined by dividing the torque by the magnitude of the lift and drag forces and should be a constant because the constraint force coefficients and the span were constant. The resultant torque moment arm was R_T = 29.2 ± 2.0 mm, with the uncertainty being twice the standard deviation from results at each speed. This shows that the center of pressure was located more than half a chord length from the load cell center due to the forward sweep of the fin. In addition, the drag-moment was used to determine the spanwise location of the center of pressure. Since the load cell was zeroed before testing, the weight was assumed negligible, and thus, the force parallel to the chord was the only loading considered. The spanwise center of pressure was at z = 119 ± 14 mm with the uncertainty being twice the standard deviation at each speed. The center of pressure was slightly above the center of the span and experienced larger variations than the torque moment arm.

The RMS constraint forces/moments were scaled with their respective mean values and plotted versus the chord-based Reynolds number (Re_c) in Fig. 14. Linear regression analysis of these curves shows that the slope of the fluctuating drag and torque is statistically significant (p-value < 0.05). The resulting linear fits for the fluctuating drag and torque curves are $D′/|D|=−1.07×10−6Rec+0.405$ and $Tq′/|Tq|=−3.92×10−6Rec+2.21$⁠, respectively. Conversely, the fluctuating lift and drag-moment curves were independent of Re_c with $L′/|L|=0.326±0.038$ and $MD′/|MD|=1.935±0.174$⁠, where the uncertainty is twice the standard deviation. In general, these data are more scattered than the mean due to the broadband nature of the cylinder vortex shedding coupled with the structural response of the fin. Consequently, it is more informative to investigate the spectral content.

The power spectral density of $L$ was examined to assess the repeatability and sensitivity to angle-of-attack. Figure 15(a) shows repeated (seven total) measurements of the power spectral density of the fin constraint lift force (Φ_$L$) at α = −0.8 deg and U_sp = 3.1 m/s. These results consistently show a peak RMS lift force at the shedding frequency of 16.66 ± 0.38 N, or approximately ±2.3% uncertainty. The peak (shedding) frequency varied by ±1.5% at 13.67 ± 0.02 Hz. Figure 15(b) compares the sensitivity of the lift power spectral density to angle-of-attack at the same speed (U_sp = 3.1 m/s). The angle-of-attack has minimal impact on the fluctuating fin lift forces at the shedding frequency, with the peak RMS lift force (16.69 ± 0.43 N) within the uncertainty range of the repeated −0.8 deg condition. While this implies that the angle-of-attack does not impact the fluctuating lift force, the angle-of-attack does impact the mean lift force. Note that due to concerns of overstraining the load cell and/or exceeding the designed fin deflections, a mechanical stop that bracketed the fin tip was installed for these initial measurements. Figure 15(b) includes a repeated condition with the mechanical stop removed that shows the stop had negligible impact on the fin loading. Overall these results provide confidence in the repeatability of the facility and the fluid loading.

Figure 16(a) shows the power spectral density of the constraint forces/moments at U_sp = 3.44 m/s and $α=−9.6deg$⁠. These data show that the fin lift force at the shedding frequency of 14.9 Hz was approximately 10 dB greater than the drag component. The drag force and drag-moment have an additional peak at ∼44 Hz, which was not observed in the torque or lift force. This is due to the first streamwise bending mode (Table 2), since modes in the streamwise direction would only be observed in the loading associated with the drag. The first cross-stream bending mode (23 Hz) was not apparent in these data, but a slight plateau in the spectral levels coming down from the shedding frequency appears to hold until ∼23 Hz before decreasing more rapidly. At the shedding frequency, the RMS fin torque (twist) was 0.275 N·m (−11.2 dB), which is nearly an order of magnitude less than the moment applied during the static load testing. This result is consistent with the RMS fin tip twist measurements that showed only a few tenths of a degree of movement.

The speed dependence was also examined with Fig. 16(b) showing Φ_$L$ spanning the range of test speeds. It is apparent that the peak frequency and amplitude increase with increasing speed. Note that the lowest speeds appear to have a weak secondary peak between 20 and 25 Hz that corresponds to the first cross-stream bending mode (Table 2). The peak frequencies match the cylinder shedding frequency with a constant St = 0.263 ± 0.007, where the uncertainty is twice the standard deviation. The peak amplitudes were 11.2, 13.4, 15.3, 16.7, 21.7, and 23.1 N for U_sp = 2.5, 2.7, 2.95, 3.05, 3.3, and 3.4 m/s, respectively. These amplitudes are well approximated as a power-law function with the amplitude being proportional to $Usp2.6$⁠.

A turbulent high-amplitude, low-frequency flow-induced FSI experiment was performed with a backward-facing, 49.5 mm chord swept-fin mounted on a load cell and located in the wake of 60 mm diameter cylinder to mimic crashback [17]. The fin angle-of-attack was fixed at −9.6 deg since the fluctuating loading was insensitive to angle-of-attack. The geometry (tunnel, cylinder, fin, and load cell), fluid properties, and material properties/response are provided in Sec. 2. The flow-field and fin response are characterized with wake flow-field measurements (PIV and LDV), fin motion (high-speed imaging and surface laser vibrometer), and constraint forces/moments (lift, drag, torque, and drag-moment).

Tip deflections increased linearly with Re_c with the peak-to-peak fin tip deflections being 5.8 times the mean deflections. The corresponding mean constraint force/moment coefficients were constant. The center of pressure for the resultant load is located at z = 119 mm with a torque moment arm of 29.2 mm from the center of the load cell. Wavenumber spectral analysis of the cylinder fluctuating wake profiles shows that there was no feedback mechanism between the fin and the upstream cylinder wake. The fluctuating lift force and drag-moment are constant, but the drag force and torque are proportional to Re_c. The fin surface deflections show distinct peaks at the cylinder shedding frequency (St = 0.26), the first cross-stream bending mode (23 Hz) and the second cross-stream bending mode (103 Hz). The constraint force/moment spectra capture the first two peaks, and the first streamwise bending mode (44 Hz) was also observed in the drag related spectra. The constraint force/moment amplitudes at the shedding frequency were proportional to $Usp2.6$⁠.

This paper provides a challenging real-world benchmark problem for FSI validation due to the large fin oscillations induced by a complex turbulent flow-field. Presented is the information required to setup the initial conditions for modelers as well as several independent measurements characterizing the response of the forced fin. This rich dataset may be used for FSI validation. Further, it will give modelers the ability to identify strengths and weaknesses of various modeling approaches for solving such complex FSI problems. This will become more important as composite materials are explored for propulsion schemes, especially for marine applications.

The authors would like to thank the technical staff at the ARL, especially Mr. Neil Kimerer for the load cell design and Dr. Arnie Fontaine with the experimental design. We would also like to thank Drs. Peter Chang and Scott Black for guidance throughout the experiment.

• This research was sponsored by DARPA under Contract No. N0002402-D-6604 (Dr. Christopher Warren, Program Manager). This document does not necessarily reflect the position or policy of the U.S. Government, and no official endorsement should be inferred.