## 1 Introduction  
A deep understanding of the hydrodynamic loads on vertical piles induced by waves and currents is very important for designing marine structures such as sea-crossing bridges, offshore platforms, and wind farms [1; 2; 3; 4; 5]. As marine engineering develops gradually into deeper water and its scale expands continuously, pile foundations must withstand the huge forces associated with strong wave and current actions, which are the key control factors in the design and construction of such engineering structures. Therefore, a crucial issue for marine engineering design is accurate consideration of the wave-current loads on piles. In practical engineering design, the influence of piles on wave propagation is usually ignored, with consideration given mainly to the viscous and added-mass effects of waves on structures. The classic Morison formula [6] is used to estimate the wave forces on piles, but the factors that influence the drag force coefficient \(C_{D}\) and inertia force coefficient \(C_{M}\) in the formula are very complicated and difficult to derive directly via theory, so instead these hydrodynamic coefficients are determined mainly via model experiments [7; 8; 9; 10; 11]. [PERSON] and [PERSON] [7] used oscillating flow in a U-shaped water tunnel to experiment on the wave force acting on a cylinder; they considered how \(C_{D}\) and \(C_{M}\) varied with the Reynolds number (\(Re\)) and Keulegan-Carpenter number (\(KC=u_{m}T/D\)), finding that the hydrodynamic coefficients of the cylinder differed under wave-only and wave-current conditions. Subsequently, [PERSON] and [PERSON] [8], [PERSON] [9], and [PERSON] et al. [11; 12] conducted hydrodynamic tests on piles under wave-only and wave-current actions; they calculated the hydrodynamic coefficients using the time-domain least-squares method and identified where in parameter space the drag and inertia forces were greatest. [PERSON] et al. [13] used numerical methods to consider the conditions of fluid viscosity and low Reynolds number, and they obtained the stable hydrodynamic coefficients of a pile model under the combined actions of linear waves and currents. Wave-current interactions can cause the forces acting on piles to change significantly [14]. Via flume model experiments, [PERSON] et al. [15] discussed how wave-current interactions affect the loads on piles, finding that the force coefficients under wave-current conditions are always lower than those under wave-only conditions. Via large-eddy simulations, [PERSON] and [PERSON] [16] discussed the critical \(Re\) for a square pile under wave-current actions for \(KC=0.6\) and \(Re=1\times 10^{3}\) to \(6\times 10^{5}\), where the drag coefficient and turbulence characteristics become more obvious when the critical \(Re\) is exceeded. [PERSON] et al. [17] reported how the drag and inertia coefficients varied with \(KC\) for an inclined pile, and [PERSON] et al. [18] analyzed the wave force components in the frequency domain for a pile with different inclination angles. Via model experiments and numerical simulations, [PERSON] et al. [19; 20] analyzed the pressure distribution and total wave force on a pile under wave actions, as well as the generation and evolution of vortices around the pile; the results indicated that using the Morison formula with the finite-amplitude linear wave hypothesis to estimate the wave force on a pile under nonlinear wave action may result in significant errors, and also that the values of the drag and inertia coefficients are very uncertain at high Reynolds number (\(Re=10^{6}\)-\(10^{7}\)). Furthermore, [PERSON] et al. [21] conducted a quantitative model test on the turbulence distribution around a cylinder under a combination of waves and currents. However, most of the above research was for piles with small diameters under either wave-only conditions or a combination of waves and low-speed currents, with more attention given to the inline force. In addition, there were few cases to reveal how the transverse force develops under wave-current action.