Numerical investigation of hydrodynamic damping of a pitching hydrofoil at different flow regimes

Due to the development of intermittent renewable energy resources, hydropower plants are mostly operated under off-design conditions. This may lead to natural frequency excitation shortening the turbine life-span. To accurately estimate the fatigue life, it is necessary to evaluate the hydrodynamic damping parameters. In the present study, different flow regimes and their relationship with hydrodynamic damping are analyzed numerically using the γ - Reθt transition SST ĸ - ω turbulence model. The test case is a NACA0009 hydrofoil pitching around its center of mass. A good agreement between the present and previous numerical results is obtained. Consistent with the literature, hydrodynamic damping coefficient demonstrate consistently two different regions. The phase shift between the displacement and moment increases with the rise of the pitching frequency. After reaching a peak value at a reduced frequency of around κ = 5, the phase shift starts to decrease, and eventually approaches zero again. The damping behavior demonstrates an opposite trend. First, it reduces in spite of the phase shift increase, and after the inflection point, where the flow field changes from the drag mode to the thrust mode, it rises due to the torque development. The maximum of the damping occurs at the low frequencies.


Introduction
The operation of hydraulic machinery at off-design results in fluid induced vibration (FIV) phenomenon.Owing to a large frequency content of the flow, FIV can excites natural frequencies of the system, leading to early fatigue failure [1].To accurately estimate the fatigue life, it is necessary to evaluate the hydrodynamic damping parameter.It is a key factor that influences the vibration amplitudes.However, its physical characteristics are not completely investigated yet [2].
Several authors studied key parameters affecting the hydrodynamic damping [3].Soltani et al. [4] investigated the fluid added parameters of a rigid Kaplan turbine where results indicated that the damping was constant at low disturbance frequencies, and then increased with the frequency.For an oscillating hydrofoil, Zeng et al. [4] showed that increasing the angle of attack before lock in, has no influence on the fluid damping.However, it decreases the damping after lock in [5].Two different behaviors were observed for the damping of a vibrating hydrofoil function of the free-stream velocity [6][7][8][9].The damping was constant at low velocity and before lock-in, whereas it increased linearly with the rise of the free-stream velocity after lock-in.
Bergan et al. [6] attributed this behaviour to the change of phase difference between the vortex shedding and the imposed oscillation velocity.However, they later observed that the phase of the vortex shedding had an adverse effect on the damping with the increase of the velocity [7].It was claimed that the change of the phase shift between displacement and force is the source of the altered behaviour.Zeng et al. [10] ascribed the behaviour to the wake width decrease with the rise in free-stream velocity and as a result interaction of the upper and lower vortices.The same trend has been observed for a pitching hydrofoil function of the reduced frequency � =  2 � by Munch et al. [11] where ,  and  are, the angular frequency, characteristics length and free-stream velocity, respectively.
In nature, animals such as birds and fishes generate lift and thrust by flapping their wings and fins, respectively.Several devices have been introduced to extract energy from wave or wind [12] utilizing similar mechanism.The dynamics of oscillating foil is widely studied in the literature [13].Several geometrical and kinematic parameters characterize the flow structure and performance of an oscillating foil.Although, the Reynolds number is a key parameter determining the flow regime for a stationary foil, apparently it has no effect on the flow structure in the wake and hydrodynamic coefficients of an oscillating foil [14][15][16][17].However, the reduced frequency alters the structure of the flow significantly.Miao and Ho [18] performed the numerical simulation of a flexible foil, and found that thrust and power coefficients were almost unchanged with the Reynolds number, while they increased with the reduced frequency.Liu et al. [19] investigated the hydrodynamic performance of a pitching hydrofoil where it was revealed that the Reynolds number may increase the propulsion efficiency.At low reduced frequency, a drag force is generated due to the formation of two rows of vortices in the wake of the foil having the same sign with the same side shear layer, i.e., Karman Vortex (KV) street.However, as the reduced frequency increases, the vortices in the wake switch their positions having opposite sign with the same side shear layer called reverse Karman vortex (RKV) street [20][21][22][23].In this state, a thrust force is exerted on the foil.RKV is necessary but not sufficient for thrust generation [24].
In this study, the hydrodynamic damping of a pitching NACA0009 hydrofoil with a rounded trailing edge is studied numerically using the Ansys CFX commercial software.The relationship between the flow regime and key parameters affecting the damping are investigated.In addition to a rounded trailing edge, the NACA0009 hydrofoil with a truncated trailing edge shape is employed to further validate the results with the available experimental data.

Problem Description
The test case is a NACA0009 hydrofoil [11], see figure 1, pitching around its center of mass with a sinusoidal movement such as () =  0 sin (), where  is the pitching angle,  0 is the pitching angle amplitude,  is the angular frequency, and  is the time.The hydrofoil has a chord length (c), maximum thickness, and maximum angle of attack ( 0 ) of 100 mm, 9.27 mm and 2 °, respectively.Figure 2 shows a schematic of the computational domain and mesh.The no-slip boundary condition is imposed on the hydrofoil solid boundary, while the symmetry condition is assumed at the upper and the lower boundaries.At the inlet boundary, a fixed velocity with 1% turbulent intensity and eddy viscosity ratio of 10 are set whilst a constant average static pressure is used for the outlet boundary.The reduced frequency () and Reynolds number (Re) are varied in the ranges 0.03 <  < 20 and 10 4 < Re < 10 6 , respectively.The working conditions are given in Table 1.

Numerical methods
The unsteady and two-dimensional fluid flow of the pitching hydrofoil is simulated using the finite volume method in Ansys CFX by solving the incompressible continuity and momentum equations written as: where   and  are the mean velocity and pressure, respectively. is the fluid density,  is the kinematic viscosity of the fluid and −  ′   ′ is the unknown Reynolds stress.The above equations are closed by the shear stress transport (SST) turbulence model [25], and the  −   transition SST model [26].The SST turbulence model yields remarkably precise predictions for both the initiation and extent of flow separation in conditions involving adverse pressure gradients, while the  −   transition model improves the prediction of onset of transition from laminar to turbulent regime.
The governing equations are discretized by the second order backward Euler scheme in time and high resolution scheme with specified blend factor of one in space leading to a second order method.
The dynamic equation of the submerged structure can be modelled as follows: where  is the angular displacement,   is the external torque acting on the structure and  is the total hydrodynamic torque exerted on the structure by the fluid flow.  ,   and   are the moment of inertia, damping and stiffness, coefficients of the structure, respectively.Assuming   ,   and   are the added moment of inertia, damping and stiffness coefficients, respectively, the hydrodynamic added torque on the structure can be modelled as: Based on linear approximation and complex transfer function method [11], the added damping coefficient (  ) is obtained as: where,  is the phase shift between the angular displacement and the torque,  is the angular frequency, and , is the magnitude of the complex transfer function.The non-dimensional form of the fluid damping is as follows: Performance parameters characterizing the flow field of an oscillating foil are torque coefficient (  ), lift coefficient (  ) and drag or thrust coefficient (  ) which can be described as: where   ,   , c and s are lift, drag, chord and span length, respectively.

Validation
In this section, the numerical model is validated and compared to the results of Munch et al. [11] as well as the experimental data from Ausoni [27].Three different grid densities, each with different first-cell height and cells number, were used for a mesh independence analysis whose details are given in Table 2. Instantaneous moment variations for four oscillation cycles are shown in Figure 3 for G1-G3.The peak value difference between G1 and G2, and between G2 and G3 are 1.7% and 0.2%, respectively.The uncertainty of the grid 2, which should be reported as 0.6%, is in an acceptable range.To save computational time, the grid G2 is selected for the subsequent analyses.To analyze the influence of the time step, three different time steps were used and compared based on the maximum torque results, as presented in Table 3.The difference between a time step of 2 × 10 −5 s and 1 × 10 −5 s is less than 0.6 %.Thus, time step 2 × 10 −5 s is employed for further investigations.To investigate the influence of turbulence model, the SST and the  −   transition SST turbulence models are employed and compared to the experimental results of Ausoni [27].Figure 5 shows the frequency of the vortex shedding () and Strouhal number � = ℎ  � function of free-stream velocity, where ℎ is the maximum thickness of the hydrofoil.It is observed that the frequency of the vortex shedding increases linearly with the free-stream velocity for both trailing edges, while the Strouhal number is almost constant.The vortex shedding frequency and Strouhal number of the rounded trailed edge is higher than the truncated trailing edge.It is noted that the SST turbulence model underestimates the frequency, whereas the results of  −   transition model is more consistent with the experimental results.�for a Reynolds number of   = 2 × 10 6 .The SST turbulence model underpredicts the velocity profiles within the boundary layer; however, there is a good agreement between the predictions of the  −   transition SST model and the measured data from Ausoni [27].The boundary layer thickness changes significantly in the region between   = 0.6 and   = 0.8, a sign of laminar to turbulence transition.This is confirmed by the profile of normalized wall shear stress (  * = 2   2 ⁄ ) along the hydrofoil, as shown in Figure 5 (b), where the wall shear stress starts to increase from   = 0.6 .However, the SST turbulence model fails to reproduce the change in the wall shear stress at the precise location.Consequently, the  −   transition SST turbulence model provides a more faithful prediction of the fluid flow.Therefore, the  −   transition SST turbulence model is used for further investigation.Figure 6 shows hydrodynamic torque and incidence angle at  = 0.21.A good agreement is observed between the current numerical results and those obtained by Munch et al. [11].

Results
Figure 7 shows the phase shift () between the angular displacement and the torque, magnitude of the transfer function (||) and added damping coefficient (c  * ) at different reduced frequencies.Two different behaviors can be observed for all of the plots.The phase shift resembles a reverse parabola.At low reduced frequency, the phase shift increases with the rise of , but decreases at high reduced frequency after reaching a maximum.The trend for the added damping is the opposite.The fluid damping, first, decreases, then, after a minimum, it increases as the reduced frequency rises.The trend is almost independent of the Reynolds number.Small differences with Munch et al. [11] results are observed for the phase shift and the damping.However, the trend matches well.These differences may be attributed to the different turbulence models used, SST versus  −   transition SST turbulence model.Figure 8 depicts the instantaneous non-dimensional vorticity (Ω * = Ω/ ) for four distinct reduced frequencies, where Ω represents vorticity around z-axis and is given as: The color bar used for case (a) and (c) is equivalently applicable to case (b) and case (d), respectively.The wake structure undergoes significant changes with the increase in reduced frequency.In cases (a) and (b), the vortices exhibit a KV configuration, while in cases (c) and (d), they take on an RKV pattern.The magnitude of vorticity in the wake augments as κ increases.It is notable that the torque coefficients (  ) increase as the reduced frequency rises.However, the curve's slope is nearly zero before  = 1; then, it progressively increases with the frequency.A similar trend is observed for the lift coefficient (  ), remaining almost constant for  < 1 and subsequently experiencing a steady ascent.On the other hand, the drag coefficient (  ) decreases at low reduced frequencies and converges to nearly zero around  = 9.Subsequently, the sign shifts from negative to positive, indicative of a thrust force.Worth noting is that the lift force considerably surpasses the drag force in magnitude, making it the principal contributor to the torque.This correlation between torque and lift force behavior can be inferred from their alignment.

Discussion
The oscillation of the hydrofoil results in angular acceleration of the surrounding fluid, but not to the same extent as the foil's acceleration.As a result, the pressure and suction sides switch before the hydrofoil reaches its ultimate position.This leads to a phase shift between the angular displacement and the torque, as depicted in Figure 7. Alongside angular acceleration, Coriolis, Euler, and centrifugal accelerations also contribute to the pressure distribution.The power results from Coriolis and Euler acceleration, while the thrust is produced as a result of centrifugal acceleration [24].
Figure 7 indicates that the phase shift and damping are independent of the Reynolds number but vary with the reduced frequency.At low reduced frequencies, the phase shift increases while the magnitude of the transfer function remains constant.This increasing phase shift has a positive influence on damping.However, the rise in frequency outweighs the effect of the phase shift, ultimately leading to a reduction in damping.The inflection point for the phase shift and magnitude of the transfer function occurs at around  = 1, while for damping, it is located at  = 5.As anticipated, the change in behavior for the magnitude of the transfer function coincides with the torque and lift force.At  = 5, increasing torque and decreasing || overcome the negative effects of frequency and phase shift, causing damping to rise.
While the torque and force magnitudes increase with higher frequency, it is important to note that maximum damping occurs at lower frequencies.The trend for damping resembles that of drag force.Initially, it decreases and then, after reaching a minimum point, begins to increase.However, the minimum points do not align.The minimum point for drag occurs at  = 9, whereas for damping, it is located at κ=5.

Conclusion
This study investigates the hydrodynamic damping of a pitching hydrofoil across various flow parameters.The results revealed that hydrodynamic damping does not depend on the Reynolds number; but is strongly affected by the reduced frequency.At low reduced frequencies, the torque and lift force coefficients remain constant despite frequency changes; however, both coefficients increase at higher reduced frequencies.Consistent with literature, the current study identified two distinct fluid damping behaviors.At low reduced frequencies, hydrodynamic damping decreases with a rise in reduced frequency, despite an increase in the phase shift.On the contrary, at high reduced frequencies, damping increases after reaching a minimum value, attributable to torque development.A direct correlation was found between drag force and damping.

Figure 4 .
Figure 4. Frequency of the vortex shedding and Strouhal number for rounded and truncated trailing edges.Experiments form Ausoni[27].

Figure 6 .
Figure 6.Hydrodynamic torque and incidence angle for a one period of oscillation at  = 0.21 for grid G2.

Figure 7 .
Figure 7. Phase shift () between the angular displacement and the torque, magnitude of the transfer function () and hydrodynamic damping coefficient (  * ) at different reduced frequencies and Reynolds number.

Figure 9 (
Figure9(a) presents the torque and force coefficients at Re = 10 4 .It is notable that the torque coefficients (  ) increase as the reduced frequency rises.However, the curve's slope is nearly zero before  = 1; then, it progressively increases with the frequency.A similar trend is observed for the lift coefficient (  ), remaining almost constant for  < 1 and subsequently experiencing a steady ascent.On the other hand, the drag coefficient (  ) decreases at low reduced frequencies and converges to nearly zero around  = 9.Subsequently, the sign shifts from negative to positive, indicative of a thrust force.Worth noting is that the lift force considerably surpasses the drag force in magnitude, making it the principal contributor to the torque.This correlation between torque and lift force behavior can be inferred from their alignment.Figure9(b) displays the mean stream-wise velocity at Re = 10 4 for different reduced frequencies.Notably, the non-dimensional velocities ( * = /) are below one for  = 2.51 and  = 6.28 in the wake region, while surpassing for κ=12.56 and κ=18.86.This variation indicates the formation of a jet.

Figure 9 (
Figure9(a) presents the torque and force coefficients at Re = 10 4 .It is notable that the torque coefficients (  ) increase as the reduced frequency rises.However, the curve's slope is nearly zero before  = 1; then, it progressively increases with the frequency.A similar trend is observed for the lift coefficient (  ), remaining almost constant for  < 1 and subsequently experiencing a steady ascent.On the other hand, the drag coefficient (  ) decreases at low reduced frequencies and converges to nearly zero around  = 9.Subsequently, the sign shifts from negative to positive, indicative of a thrust force.Worth noting is that the lift force considerably surpasses the drag force in magnitude, making it the principal contributor to the torque.This correlation between torque and lift force behavior can be inferred from their alignment.Figure9(b) displays the mean stream-wise velocity at Re = 10 4 for different reduced frequencies.Notably, the non-dimensional velocities ( * = /) are below one for  = 2.51 and  = 6.28 in the wake region, while surpassing for κ=12.56 and κ=18.86.This variation indicates the formation of a jet.

Table 1 .
Flow and hydrofoil conditions.