2.1. Governing equations for streamwise-unsteady wake dynamics

To derive the governing equations for a turbine wake with streamwise unsteadiness, we define a control volume with a variable wake radius $R(x,t)$, as shown in figure 1. The flow inside the control volume is treated as incompressible and quasi-one-dimensional (quasi-1-D), such that all flow occurs in the streamwise direction and the streamwise velocity $U(x,t)$ does not vary in the radial direction. The flow is additionally assumed to be inviscid in the near wake, where tip vortices dominate and viscous contributions to the kinetic-energy budget are comparatively small. The turbine itself is modelled as an actuator disc, which generates a thrust force on the flow and defines the inlet boundary condition to the control volume at $x=0$. Variations in the thrust force due to unsteady inflow conditions or streamwise surge motions will create an oscillatory inlet condition $U_i(x=0,t)$ that will dictate the dynamics in the near wake. We do not seek to model the coupling between the turbine dynamics and this near-wake inflow condition here, assuming for simplicity a periodic inflow condition of the form $U_i(x=0,t) = \bar {U} + \hat {U}\sin (ft)$.

Figure 1.

Sketch of the quasi-1-D axisymmetric problem formulation for an unsteady turbine wake.

Given these assumptions, the equations for conservation of mass and momentum within the wake control volume can be derived (with details provided in Appendix A). The relation for conservation of mass is given as a first-order nonlinear partial differential equation in terms of $R$ and $U$:

(2.3)\begin{equation} \frac{\partial R}{\partial t} + \frac{1}{2} R \frac{\partial U}{\partial x} + U \frac{\partial R}{\partial x} = 0. \end{equation}

If $R$ is held constant, (2.3) reduces to the more familiar form ${\partial U}/{\partial x} = 0$. The equation therefore captures the effects of changes in the flow velocity on the local size of the wake. For a quasi-1-D incompressible flow, a decrease in the flow velocity across a given distance $\Delta x$ must result in a corresponding increase in the cross-sectional area of the control volume so that the mass flux remains constant. Similarly, an increase in the flow velocity will result in a decrease in the cross-sectional area. Hence, if $U(x,t)$ takes the form of a travelling wave, (2.3) dictates that the wake radius will also form travelling waves so that the mass flux at every streamwise location in the control volume is conserved.

The relation for conservation of momentum is the 1-D Euler equation,

(2.4)\begin{equation} \frac{\partial U}{\partial t} + U \frac{\partial U}{\partial x} ={-}\frac{1}{\rho} \frac{\partial p}{\partial x}. \end{equation}

Here, we have assumed that the pressure is constant in the radial direction, both inside and outside of the wake. The pressure term represents a heterogeneous forcing on a Burgers-type partial differential equation. In keeping with the assumptions of many turbine wake models (e.g. Bastankhah & Porté-Agel Reference Bastankhah and Porté-Agel2014), we choose to neglect this term, though in reality the pressure recovery in the near-wake region will be non-negligible.

For the remaining terms in the momentum equation, we decompose the flow velocity into phase-averaged and fluctuating components, in the style of a Reynolds decomposition:

(2.5)\begin{equation} U(x,t) = \tilde{U}(x,t) + U'(x,t). \end{equation}

The phase-averaged equation, (2.4), using this decomposition yields

(2.6)\begin{equation} \frac{\partial \tilde{U}}{\partial t} + \tilde{U} \frac{\partial \tilde{U}}{\partial x} ={-} \frac{1}{2} \frac{\partial}{\partial x} \widetilde{{U'}^2}. \end{equation}

To model the normal-stress term on the right-hand side of the equation, we combine two scaling relationships for wind-turbine wakes in steady inflow conditions. Quarton & Ainslie (Reference Quarton and Ainslie1990) show that the turbulence intensity $u'/U_\infty$ decays monotonically towards zero with increasing streamwise distance. It is also well known that the velocity in a turbine wake recovers monotonically towards the free stream wind speed in a power-law relationship with streamwise distance (cf. Porté-Agel, Bastankhah & Shamsoddin Reference Porté-Agel, Bastankhah and Shamsoddin2020). Thus, one can typically assume that the magnitude of the streamwise velocity gradient ${\partial u}/{\partial x}$ will decay monotonically towards zero as well. Combining these two observations, we argue that

(2.7)\begin{equation} \widetilde{{U'}^2} \sim \left|\frac{{\rm d} \tilde{U}}{{\rm d}\kern0.7pt x}\right|, \end{equation}

or in the form of a turbulent-viscosity hypothesis,

(2.8)\begin{equation} -\widetilde{U'U'} = \nu_u \left|\frac{\partial \tilde{U}}{\partial x}\right|. \end{equation}

The proportionality parameter $\nu _u$ may vary as a function of $x$; we will obtain a rudimentary model from experimental measurements in § 4 of the form

(2.9)\begin{equation} \nu_u = \kappa x,\end{equation}

(cf. figure 13). It is important to note that $\nu _u$ is fundamentally different from a typical eddy viscosity based on the Reynolds shear stress, since it represents the interactions of streamwise quantities and has nothing to do with shear. Equation (2.8) thus suggests that stronger streamwise gradients are associated with stronger velocity fluctuations, and a linear model enforces a stronger coupling between the two quantities with increasing downstream distance.

Applying this model to (2.6) yields

(2.10)\begin{equation} \frac{\partial \tilde{U}}{\partial t} + \tilde{U} \frac{\partial \tilde{U}}{\partial x} + \nu_u(x) \frac{\partial^2 \tilde{U}}{\partial x^2} = 0. \end{equation}

This first-order nonlinear partial differential equation is a viscous Burgers equation, which describes the growth, propagation and steepening of nonlinear travelling waves. The proportionality parameter $\nu _u(x)$ therefore represents a damping term that inhibits the steepening of the waves and the formation of unphysical shock discontinuities, which would occur in the inviscid form of the Burgers equation ($\nu _u=0$). This is physically consistent with the underlying $\widetilde {U'U'}$ quantity that the damping term represents, which is related to turbulent convection and transport in the turbulent kinetic energy budget for axisymmetric wakes (Uberoi & Freymuth Reference Uberoi and Freymuth1970). This term thus models the transfer of energy from the phase-averaged base flow to turbulence via streamwise velocity gradients, which are created in this system by the time-varying inflow condition $U_i$, and the nonlinear wave steepening generated by the first two terms in (2.10) (i.e. the inviscid Burgers equation).

In summary, the dynamics of the unsteady wake can be written as a system of two first-order nonlinear partial differential equations,

(2.11a)$$\begin{gather} \frac{\partial R}{\partial t} + \frac{1}{2} R \frac{\partial U}{\partial x} + U \frac{\partial R}{\partial x} = 0\quad {\rm and} \end{gather}$$

(2.11b)$$\begin{gather}\frac{\partial U}{\partial t} + U \frac{\partial U}{\partial x} + \nu_u(x) \frac{\partial^2 U}{\partial x^2} = 0, \end{gather}$$

where the tildes denoting phase-averaging have been removed for clarity and will not be used for the remainder of this work. The system is subject to the boundary conditions

(2.12a)$$\begin{gather} \left. \frac{\partial R_i}{\partial t}\right|_{x=0,t} = 0\quad {\rm and} \end{gather}$$

(2.12b)$$\begin{gather}U_i|_{x=0,t} =\bar{U}+\hat{U}\sin(ft). \end{gather}$$

The equations exhibit one-way coupling, as the momentum equation only depends on $U$ and serves as a forcing on $R$ through the continuity equation. The time-varying boundary condition $U_i$ and damping term in (2.11b) preclude the straightforward derivation of explicit analytical solutions, but it is well known that the viscous Burgers equation generates damped nonlinear travelling waves and can be solved numerically. For (2.11a), the method of characteristics can be applied by defining a characteristic variable $\xi = x - Ut$, such that along lines of constant $\xi$, the partial differential equation reduces to a pair of ordinary differential equations,

(2.13a)$$\begin{gather} \left. \frac{{\rm d}\kern0.7pt x}{{\rm d} t}\right|_{\xi} = U \quad {\rm and} \end{gather}$$

(2.13b)$$\begin{gather}\left. \frac{{\rm d} R}{{\rm d} t}\right|_{\xi} ={-}\frac{1}{2}\frac{\partial U}{\partial x} R. \end{gather}$$

The first equation describes the advection of solutions for $R$ according to the travelling-wave velocity $U$, while the second governs the growth or decay of the wake-radius amplitude independent of advection. Given the wave steepening that is inherent to solutions of Burgers-type equations for $U$, we expect that the magnitude of ${\partial U}/{\partial x}$ will be greater on the downstream side of the wave (where ${\partial U}/{\partial x}<0$) than on the upstream side of the wave. Thus, from (2.13b), we anticipate that the period-averaged amplitude of $R$ will grow under the forcing provided by (2.11b) until streamwise velocity gradients are damped out, at which point the amplitude of $R$ will saturate and waves in the wake radius will simply advect downstream.

To demonstrate the dynamics of the system, results from numerical integrations are shown in figure 2. The upstream boundary conditions were set based on measured values from experiments (cf. figure 12 in § 4.2). A third-order upwind scheme was used to discretize the ${\partial U}/{\partial x}$ term, while a second-order central-difference scheme was used for the ${\partial ^2 U}/{\partial x^2}$ term. As we have anticipated in our analysis, the model produces damped nonlinear travelling waves in the velocity, which in turn result in waves in the wake radius that grow and saturate in amplitude as they propagate downstream.

Figure 2.

Numerical solutions to the coupled partial differential equation modelling framework for (a,c,e) the wake radius and (b,d,f) streamwise velocity, for three example cases. Here (a,b) $u^{*}= 0.1, k = 1.89, \lambda _0 = 4.50$; (c,d) $u^{*}= 0.1, k = 2.36, \lambda _0 = 4.50$; (e,f) $u^{*}= 0.2, k = 1.89, \lambda _0 = 4.50$.

2.2. Vortex dynamics

The kinematics of the control volume described by the system of equations derived in the previous section also allow predictions regarding the vortex dynamics in the wake to be made. Tip vortices shed by the turbine blades typically bound the wake of the turbine until they break down in the intermediate wake. This is a three-dimensional (3-D) helical structure, but given the axisymmetric nature of the turbine wake, the successive appearances of the helical vortex line at a single azimuthal orientation are often treated as a discrete series of two-dimensional (2-D) point vortices (e.g. de Vaal, Hansen & Moan Reference de Vaal, Hansen and Moan2014b; van den Broek et al. Reference van den Broek, Van den Berg, Sanderse and van Wingerden2023). In steady conditions, these vortex elements are generally arranged in relatively straight lines extending downstream from the blade tips. For an unsteady wake with a time-varying wake radius, the tip vortices will experience radial displacements along with the wake radius. We can therefore consider the dynamics of a series of discrete point vortices arranged in non-collinear patterns, as a representation of a helical tip vortex undergoing spatial and temporal changes in its radius. For more detailed stability analyses of helical tip vortices under dynamic conditions, we refer the reader to the work of Kleine et al. (Reference Kleine, Franceschini, Carmo, Hanifi and Henningson2022) on the wakes of moving FOWTs and Rodriguez, Jaworski & Michopoulos (Reference Rodriguez, Jaworski and Michopoulos2021) on the wakes of turbines with flexible blades.

Consider an infinitely long line of point vortices with equal circulation $\varGamma$ and equal spacing $\Delta x$. The induced velocity from the $i$th vortex on a given vortex $j$ is given by

(2.14)\begin{equation} \begin{bmatrix}u\\v\end{bmatrix}_{i,j} = \frac{\varGamma}{2{\rm \pi}((x_j-x_i)^2+(y_j-y_i)^2)}\begin{bmatrix} -(y_j - y_i) \\ x_j - x_i \end{bmatrix},\end{equation}

where we assume that radial perturbations ($\Delta r \equiv \Delta y$) are small such that $\varGamma$ is approximately constant. Due to the symmetries of the interactions, the induced velocity on a given vortex by its left-hand neighbour will be cancelled out by the induced velocity by its right-hand neighbour, and the line will not deform over time.

Next, consider a similar system of point vortices with circulations $-\varGamma$, now arranged on a parabola of constant concavity given by $y=\alpha x^2$. For the vortex initially at $(x_j,y_j) = (0,0)$, the total induced velocity from its left and right neighbours at $j-1$ and $j+1$ is

(2.15)\begin{equation} \begin{bmatrix}u\\v\end{bmatrix}_{j\pm1,j} ={-}\frac{\varGamma}{{\rm \pi}(1+(\alpha \Delta x)^2)}\begin{bmatrix} \alpha \\ 0 \end{bmatrix}. \end{equation}

A point-vortex distribution with positive concavity will thus induce a negative tangential velocity in a given vortex, and the magnitude of this induced motion scales with the concavity $\alpha$. Conversely, a distribution with negative concavity will induce a positive tangential velocity in a vortex on the curve.

This principle can be demonstrated for a system of point vortices initialized along a sine wave with no background flow and numerically integrated forward in time using (2.14), shown in figure 3. Vortices initially located on the wave crest move in the positive direction, while vortices initially located on the wave trough move in the negative direction. In both cases, vortices travelling in the same direction are pushed into closer proximity with each other as they move, rolling up into a vortex aggregate.

Figure 3.

The 2-D point-vortex simulation. Vortices with equal and negative (clockwise) circulation are initialized at the open circles on the grey dotted line. Their trajectories are shown in solid lines, and their final positions in the simulation are given by closed circles. Vortex locations and trajectories are coloured by the concavity of the curve at the vortex's initial position.

Applying this analysis to the sample model solutions for the wake radius shown in figure 2, we can predict the evolution of tip vortices in the turbine wake as they are advected downstream. Assuming the initial distribution of the tip vortices follows the model solutions for $r(x,t)$, we would expect to see a similar aggregation behaviour as that observed in figure 3. Furthermore, since the nonlinear steepening of the wake-radius waves leads to higher concavities at the wave crests relative to the wave troughs, the tip vortices should tend to aggregate most strongly downstream of the wave crests and ‘surf’ on these waves as they travel downstream.

A complementary mechanism for tip-vortex aggregation can be found by considering the effect of streamwise-velocity gradients. Consider again a line of point vortices with initial spacing $\Delta x$, advected by a flow with a positive streamwise gradient ${\Delta U}/{\Delta x}$. If we follow one point vortex at its advection velocity $U$, after some time $\Delta t$, the distance between it and its neighbours will have increased to $\Delta x + \Delta U \Delta t$. By (2.14), this increase in separation will result in weaker interactions between neighbouring vortices. Conversely, in flows with a negative streamwise gradient, the separation between vortices will decrease as a function of time, thereby increasing the magnitude of vortex interactions by mutual induction.

Returning to the simulated traces of $R$ and $U$ in figure 2, we observe that regions where ${\partial U}/{\partial x} < 0$ coincide with peaks in the wake radius where the concavity of $R$ is most negative. The negative streamwise velocity gradients should therefore supplement the effect of negative concavity described previously and augment the tendency of tip-vortex elements to aggregate just downstream of peaks in the wake-radius waveform. By contrast, positive streamwise velocity gradients align with troughs in $R$, which will discourage the roll-up of individual vortices in these regions. These analyses suggest that tip vortices will primarily aggregate in a single structure just downstream of peaks in the wake-radius waves. The strength of this behaviour will depend on the amplitude and frequency of the unsteady wake forcing $U_i(x=0,t)$, which dictate both the magnitude of the streamwise velocity gradients in the wake and the sharpness of the peaks in the wake-radius waveform.

The proposed mechanisms for vortex aggregation in unsteady wakes are similar in principle to that of tip-vortex pairing, a phenomenon that is related to the breakdown of turbine wakes in steady inflow conditions (Okulov & Sørensen Reference Okulov and Sørensen2007; Felli, Camussi & Di Felice Reference Felli, Camussi and Di Felice2011; Sarmast et al. Reference Sarmast, Dadfar, Mikkelsen, Schlatter, Ivanell, Sørensen and Henningson2014; Lignarolo et al. Reference Lignarolo, Ragni, Scarano, Simão Ferreira and Van Bussel2015; Quaranta, Bolnot & Leweke Reference Quaranta, Bolnot and Leweke2015). However, while the mutual-induction instabilities considered in canonical turbine wakes typically involve two or three vortex elements, the present analysis involves larger systems of vortices. The number of tip vortices shed into the wake over a single unsteady forcing period can be estimated as

(2.16)\begin{equation} N_v = 2 N_b \frac{\lambda}{k}, \end{equation}

where $N_b$ is the number of turbine blades, $\lambda$ is the tip-speed ratio ($\lambda = \omega D/2U_\infty$) and $k$ is the reduced frequency. For the experiments that will be detailed in this work (cf. table 1 in § 3), $11 < N_v < 33$. The unsteady vortex-interaction mechanisms described above occur across length scales of the order of half the wavelength of the wake-radius perturbations, thus involving approximately $N_v/2$ vortices per period. Therefore, while vortex-pairing instabilities may be present in periodically forced unsteady wakes, it is expected that the additional mechanisms described in this section will play a non-negligible role in the vortex dynamics of the wake, due to the number and distribution of the vortices present in the interactions.

Table 1.

Operational parameters for the turbine used in these experiments, including steady-flow characteristics ($\lambda _0$ and $C_{p,0}$), surge-motion parameters ($T$, $u^*$, and $k$) and the resulting time-averaged performance ($\bar {\lambda }$ and $\overline {C_p}$). For all cases, $U_\infty = 1\ {\rm ms}^{-1}$.

In summary, the modelling approach outlined in this section provides an interpretable theoretical description of the dynamics of the wake radius, streamwise velocity and tip-vortex aggregations in a periodically unsteady turbine wake. The framework offers insights into the dominant physics of the problem and can be used for qualitative predictions of trends in the dynamics, including that

1. (1) nonlinear travelling waves in the streamwise velocity $U$ are generated by the oscillatory boundary condition, undergo steepening and are damped in amplitude as they advect downstream;

2. (2) travelling wakes in the wake radius $R$ are generated by the streamwise-velocity waves, grow with increasingly prominent crests and broad troughs and saturate as velocity gradients dissipate;

3. (3) tip-vortex elements are encouraged by the unsteady wake dynamics to aggregate ahead of crests in the wake radius and advect downstream with these crests; and

4. (4) increasing the strength of the forcing in the boundary condition $U_i(x=0,t)$ increases the amplitudes of $U$ and $R$ in the wake, as well as the tendency towards vortex aggregation.

The modelling approach relies on several assumptions that limit its quantitative accuracy and restrict its applicability to the prediction of perturbations in the near wake. However, in the following sections we will demonstrate that it still captures many of the key dynamical features of unsteady turbine wakes.

3.1. Experimental apparatus

Wake measurements downstream of a periodically surging rotor were conducted in an optically accessible towing-tank facility at Queen's University. The apparatus is described in detail by El Makdah et al. (Reference El Makdah, Ruzzante, Zhang and Rival2019); a brief overview is given here, and a schematic is provided in figure 4.

Figure 4.

Schematic of the experimental apparatus in the optical towing tank, including the turbine, turbine sensors, traverse and the fields of view of the four high-speed cameras. A sample velocity waveform $\mathcal {U}(t)$ for the traverse is shown in an inset.

A three-dimensionally printed turbine with a diameter of $D=0.3$ m was used in the experiments. The turbine blades were designed with SD7003 airfoil profiles of constant chord and a spanwise twist that targeted a constant angle of attack of $10^\circ$ along the blade at a tip-speed ratio (TSR) of $\lambda =4$. The turbine had a maximum measured coefficient of power of $C_p=0.29$ at a TSR of $\lambda = 3.89$. The turbine was mounted on a sting at the centre of the towing-tank test section, and the turbine shaft was connected to a rotational encoder (Baumer ITD69H00), torque sensor (HBM T22) and frictional brake by means of a chain drive. The turbine system had a low rotational inertia relative to the hydrodynamic torques on the blades (El Makdah, Zhang & Rival Reference El Makdah, Zhang and Rival2021). The estimated blockage of the turbine based on swept area was 7.1 %.

The test section of the towing tank was 15 m long and had a $1\ {\rm m} \times 1\ {\rm m}$ cross-section. This was filled with water and enclosed by a ceiling that served to mitigate free-surface waves. The turbine was suspended from a traverse through a 50-mm-wide opening in the ceiling. A rotary encoder on the traverse enabled its linear position to be recorded. The traverse was driven at a mean speed of $U_\infty =1\ {\rm ms}^{-1}$ for steady-flow measurements. For unsteady surge cases, periodic oscillations in the traverse velocity of up to $\pm 0.4\ {\rm ms}^{-1}$ were superimposed on this mean speed. The average magnitude of the error between the desired velocity profile and the measured traverse velocity was 0.023 ${\rm ms}^{-1}$. Each traverse run was initiated with a constant-acceleration ramp-up profile that transitioned smoothly into the test motion waveform, and a similar ramp-down profile was used to bring the traverse to rest after the experiment. A light sensor mounted 3.84 m from the starting position of the traverse served as a trigger to start flow measurements. This position was chosen such that the traverse would reach steady-state operation before data recording was initiated. The towing tank was allowed to settle for at least 4 min between individual runs, as it was determined that longer settling durations did not further reduce the average particle displacements significantly. Each phase-averaged ensemble for a given case was compiled from 20 separate runs of the corresponding traverse-motion profile.

Four high-speed cameras (Photron SA4) with a resolution of $1024\times 1024$ pixels were arranged in a line outside of the towing tank to capture a combined field of view of approximately $1.15\ {\rm m} \times 0.39\ {\rm m}$. The individual fields of view overlapped by approximately 30 %. The cameras were triggered simultaneously by the aforementioned light sensor and recorded at a frame rate of 500 Hz. A high-speed laser (Photonics DM40-527) was used to illuminate the measurement domain along the rotational axis of the turbine. The flow was seeded by neutrally buoyant polyamide spherical particles (LaVision) with a diameter of 60 $\mathrm {\mu }{\rm m}$. The turbine and sting were spray-painted black to minimize reflections.

3.2. Data analysis

The 2-D particle image velocimetry (PIV) was performed using the raw image data recorded by the cameras. An automated masking routine was written to mask the turbine, sting and shadow cast by the turbine blades, based on the identified location of the nose of the turbine in the images. Only images recorded between the start trigger signal and the start of the traverse ramp-down motion were processed. Velocity vectors were computed using multipass cross-correlation with a final interrogation-window size of $32\times 32$ pixels with 50 % overlap, using the open-source MATLAB package PIVlab (Thielicke & Stamhuis Reference Thielicke and Stamhuis2014). A high-pass filter was applied to the images before correlation, and standard deviation and median thresholds were applied to the vector fields after processing.

The individual vector fields from each camera were stitched together by interpolating onto a common spatial grid and averaging the velocities in the overlap regions. These composite laboratory-fixed velocity fields were then transferred into a reference frame moving at the mean speed of the turbine by identifying their locations in space and time within a single turbine surge period, interpolating onto a common spatial grid in the new reference frame and stitching overlapping regions via linearly weighted blending. For the steady-flow reference cases, the representative period was set as $T=1$ s (for $U_\infty \leq 1$) or $T=0.5$ s (for $U_\infty > 1$). Vorticity fields were then calculated using Gaussian-filtered velocity fields to smooth out spurious results from numerical differentiation. Finally, the fields were phase-averaged across all 20 traverse runs, yielding time-resolved 2-D velocity fields over a single period that covered at least 12 turbine diameters of the streamwise extent of the wake and over 1 turbine diameter in the radial direction. In some cases, over $18D$ of the wake was measured.

To serve as quantities for comparison with the flow model described in § 2, the wake radius and streamwise velocity were also computed from the PIV data. The wake radius was defined at a given location $x$ as the radial location $r$ at which the streamwise velocity reached 95 % of the free stream velocity, i.e. $R(x,t)$ such that $u(x,R,t) = 0.95U_\infty$. These profiles of $R(x,t)$ were smoothed using a moving-average filter with a width of three samples. The radially averaged streamwise velocity $U(x,t)$ was computed by averaging the local velocity $u$ across a streamwise slice of the wake at $x$, from the centre of the wake to the wake radius $R(x,t)$.

3.3. Experimental cases

The parameter space investigated in these experiments is summarized in table 1. Two loading conditions were applied to the turbine. In the first case, the frictional brake was manually tuned between runs to obtain a steady-flow TSR of $\lambda _0=4.50$. In the second case, the brake was released such that the only load on the turbine was due to shaft friction, yielding a steady-flow TSR of $\lambda _0=5.07$. Both of these scenarios result in higher TSRs than the power-maximizing loading condition. This was intended to mitigate the effects of flow separation and stall on the turbine blades, in accordance with the observations of Wei & Dabiri (Reference Wei and Dabiri2022).

At each loading condition, both steady-reference and unsteady surge-motion cases were investigated. Steady-flow reference cases were carried out for each loading condition at a constant inflow velocity of $U_\infty = 1\ {\rm ms}^{-1}$. Additional constant-velocity cases at $U_\infty = 0.8\ {\rm ms}^{-1}$ and $U_\infty = 1.2\ {\rm ms}^{-1}$ were collected to represent a quasisteady range of wake profiles for a surge-velocity amplitude of $u^* = 0.2$, which will be compared with unsteady cases with the same surge-velocity amplitude to highlight differences between quasisteady and unsteady wake dynamics.

Four unsteady cases for each loading condition spanned a range of surge-velocity amplitudes $u^*$ and reduced frequencies $k$: a baseline unsteady case with $u^*=0.1$ and $k=1.89$, a case with the same $u^*$ and higher $k$, a case with the same $k$ as the baseline case and higher $u^*$ and finally a case with as high of a value of $u^*$ as could be achieved with the apparatus, which required $k$ to be halved. For the case with $u^* = 0.4$, the displacement of the traverse over a single period exceeded the length of the measurement domain, since acceleration limitations of the traverse stipulated a lower frequency than that of the other cases. Therefore, 20 additional runs were carried out in which the turbine motion waveform was offset by 1 m from its usual starting location, so that the combined set of 40 runs covered the entire wake over a single surge period. The surge-velocity amplitudes and reduced frequencies investigated in this study are relatively high and represent significant motions when applied to full-scale FOWTs, which in certain conditions experience motions with $u^*>0.25$ (Wayman Reference Wayman2006; Larsen & Hanson Reference Larsen and Hanson2007; de Vaal, Hansen & Moan Reference de Vaal, Hansen and Moan2014a). The range of reduced frequencies is somewhat higher than the range expected for FOWTs as a function of wave and platform frequencies, which Messmer et al. (Reference Messmer, Hölling and Peinke2024) estimate to be $k\in [0,1.5]$. This was unavoidable due to limitations in the traverse velocity and length of the towing tank. Still, based on the findings of Messmer et al. (Reference Messmer, Hölling and Peinke2024), we expect the dynamics observed at higher reduced frequencies to apply to somewhat lower reduced frequencies ($k\gtrsim 0.5$) as well.

The phase-averaged TSRs for all unsteady cases, referenced to the apparent inflow velocity in the rotor frame $U_\infty - \mathcal {U}(t)$, are shown in figure 5. The fact that these signals are almost perfectly in phase with the surge-velocity waveform $\mathcal {U}(t)$ confirms that the inertia of the turbine is very low compared with the surge dynamics. We also note that, for most of the unsteady cases, the time-averaged TSR and coefficient of power (shown in the rightmost columns of table 1) remained relatively constant with changes in $u^*$ and $k$. For one case ($u^* = 0.4$ and $\lambda _0 = 4.50$), the time-averaged TSR and power dropped due to the onset of stall in the turbine at low instantaneous inflow velocities, as evidenced by the drop in rotation rate observed around $t/T=0.8$ for this case in figure 5(a).

Figure 5.

Phase-averaged TSR data, defined by the inflow velocity in the rotor frame, as a function of time for the two loading conditions covered in this study (a,b). The steady reference values are shown as black dashed lines.