Research of Wind–Wave–Ship Coupled Effects on Ship Airwake and Helicopter Aerodynamic Characteristics

Abstract

The oceanic wind and waves, as well as the resultant ship motions, significantly impact the ship airwake and the operation of shipborne helicopters. A numerical method coupling wind, wave, ship and helicopter is developed using multiphase flow, in which the ship motions are simulated in real time by dynamic fluid body interaction module and the helicopter rotor is modeled using the momentum source approach. By integrating the ONRT ship with the UH-60A helicopter, the unsteady aerodynamic characteristics of the ship airwake and the helicopter rotor while the ship is pitching and heaving at sea state 36 that cover moderate to extreme marine environments are studied, and the time history of rotor thrust and pitch moment at four different sea states and different hovering heights are calculated. It is shown that ship motions and deck displacements in relative sea states are highly nonlinear, making the conditions faced by helicopter landing and take-off operations vary greatly from one sea state to another. The effects of each sea state when coupling waves and ship motions varies greatly. The fluctuation of velocity components and rotor air loads in sea state 6 is up to twice that of in sea state 5, while there are less differences between the velocity fluctuation and the corresponding helicopter airloads among common sea state 3~5. The dynamic aerodynamic interference resulting from the wind–wave–ship–helicopter coupling exhibits pronounced unsteady characteristics, as the hovering rotor continuously traverses areas with varying velocities and vorticities. At the most severe sea state 6, rotor thrust fluctuations can reach up to 20%, and strong perturbations of 5~10 Hz with an amplitude of 1/3 of the total range occur due to oscillating separated shear layers, which endanger the shipborne helicopter operation and needs to be eluded.

1. Introduction

Shipborne helicopters face greater and more complex challenges than land-based helicopters during takeoff and land on ships. The 6 degrees of freedom (6-DOF) motions of ships under the influence of wind and waves, the geometric size limitations of the ship superstructure and deck, and the unsteady airflow field generated by the coupling of the ship, wind, and waves significantly increase the difficulty of launching and recovering shipborne helicopters [1]. This difficulty is mainly reflected in the fact that when the sea state is higher, the ship will have a correspondingly large and long period of pitch and heave motions under the inducing of waves, which will cause the flight deck for helicopter launch and recovery to generate a great motion correspondingly. The resulting unsteady fluctuations of the ship airwake have a stronger effect on the flight safety of the helicopter.

Although numerous scholars worldwide have investigated the coupling between ship airwake fields and helicopter aerodynamic characteristics, existing research has predominantly focused on numerical simulations and experimental validations under static ship conditions or low sea states. Studies investigating the effects of ship motion on airwake fields and helicopter aerodynamic loads under high sea state conditions remain relatively scarce. For example, Reddy et al. [2] conducted steady-state numerical simulations using a structured grid and the k-ε turbulence model on a simplified frigate ship (SFS) model, but failed to adequately consider the effects of ship motion. While Polsky et al. [3] performed unsteady simulations using the Cobalt solver, their research primarily focused on the steady-state characteristics of ship wakes, with insufficient investigation into the unsteady characteristics of ship motion under high sea conditions. Studies comparing numerical methods including Reynolds-averaged Navier–Stokes (RANS), Detached Eddy Simulation (DES), and hybrid RANS-LES (Large Eddy Simulation) have been conducted by Forrest and Owen [4], Thornber et al. [5], Lawson et al. [6] and Muijden et al. [7]. It was illustrated that the RANS method is capable of capturing the vortex structure, velocity distribution and turbulence intensity of the flow field, although there are some deficiencies of high frequency vortex components.

Regarding the ship–helicopter dynamic interface, numerous studies have been conducted using one-way coupled simulation [8,9] and two-way coupled simulation, which takes into account the rotor-on-ship effect. Shi et al. [10] studied the ship–helicopter coupling flow field where helicopter rotor simulated by steady momentum source approach or unsteady moving overset mesh method, and found that either method can effectively simulate the dynamic interference between ship airwake and helicopter, while the momentum source method saves more calculation resources. Su, Shi et al. [11] and Su, Xu et al. [12] studied the aerodynamic loads of helicopters landing on ships with different rotational directions of single rotor and with coaxial rotors, respectively, and discussed the helicopter landing safety under different landing modes.

Recently, there have been studies focusing on the dynamic changes in airwake with ship motions. Sydney et al. [13] used PIV method to study the airwake of SFS2 when pitching, focusing on the effects of pitch angle and pitch frequency on the turbulent distribution and period of the flow field, and found that the pitch motion and attitude of the ship will significantly affect the formation and development of the airwake. Li et al. [14] numerically simulated the flow field of SFS2 under different pitch states, set the conditions of sudden start of pitch or halving the pitching period, and analyzed the changes in vortex structure and vertical velocities. The results show that the flow field in static state is quite different from the pitching state. Dooley et al. [15,16,17] numerically simulated the airwake of the multi-scale Office of Naval Research Tumblehome (ONRT) ship considering atmospheric turbulence, ship motions and wave pumping effects. These studies analyzed the effects of these factors by decomposing the velocity field and compared with their previous scaled wind tunnel test results under uniform wind conditions [18], which proved that the effect of ship motions is the most significant factor. In the above research, there are still few modellings and analyses on coupling ship motions, wind, current and waves. However, these factors on ship airwake and helicopter launch and recovery, especially at high sea states, cannot be ignored. At the same time, there is a lack of research on the aerodynamic characteristics of helicopters under the coupled effect of these factors.

In order to more realistically simulate the operation of ships and helicopters in the marine environment, this paper uses the numerical simulation method of wind–wave–ship–helicopter coupling air flow field, simulates the dynamic induced motions of ships advancing against the wind and waves at sea state 3~6, and analyses the unsteady characteristics of the airwake generated by the induced pitch and heave motions of the ship and the effects of nonlinear deck displacements on helicopter landing and takeoff operations in different sea states. The rotor of helicopter is simulated by momentum source approach to simulate the ship–helicopter dynamic interface with wind, waves and ship motions, and study the effects of all these factors on the airloads of the helicopter.

2. Methods and Models

2.1. Numerical Method

The numerical simulation method for the wind–wave–ship–helicopter dynamic interface established in this paper encompasses air-water two-phase fluid media, the free surface wave model, the ship motion model simulated using the Dynamic Fluid Body Interaction (DFBI) module, which enables the ship to produce motions closer to real situations under the dynamic action of the fluid, thereby better capturing the airwake field under the combined action of wind, waves, and ship motion, and the helicopter rotor model simulated using the momentum source approach.

2.1.1. CFD Method

In this paper, the Reynolds-averaged Navier–Stokes (RANS) equations are used as the main governing equation for the calculation, and the helicopter rotor is simulated by the momentum source approach. The expression in the integral form of the RANS model is as follows:

$${\displaystyle \frac{\partial }{\partial \mathrm{t}}}\underset{\mathrm{V}}{\iiint }\mathrm{W} \mathrm{d}\mathrm{V}+\underset{\partial \mathrm{V}}{∯}\left({\mathrm{F}}_{\mathrm{c}}\left(\mathrm{W}\right)-{\mathrm{F}}_{\mathrm{v}}\left(\mathrm{W}\right)\right)\mathrm{d}\mathrm{S}=\iiint \mathrm{R} \mathrm{d}\mathrm{V}$$

(1)

$$W=\left[\begin{array}{c}\rho \\ \rho u\\ \rho v\\ \rho w\\ \rho E\end{array}\right] F_c=\left[\begin{array}{c}\rho U\\ \rho uU+n_{x}p\\ \rho vU+n_{y}p\\ \rho wU+n_{z}p\\ \rho HU\end{array}\right] F_v=\left[\begin{array}{c}0\\ n_x{\tau }_{xx}+n_y{\tau }_{xy}+n_z{\tau }_{xz}\\ n_x{\tau }_{yx}+n_y{\tau }_{yy}+n_z{\tau }_{yz}\\ n_x{\tau }_{zx}+n_y{\tau }_{zy}+n_z{\tau }_{xzz}\\ n_x{\Phi }_x+n_y{\Phi }_y+n_z{\Phi }_z\end{array}\right]$$

(2)

where $W$ is the conserved variable, $F_c$ and $F_v$ are the convective and viscous fluxes, respectively, and $\mathrm{R}$ is the momentum source term.

Under high sea conditions, we ensured the reliability of the results through grid convergence and time step sensitivity analysis, yet we must still be mindful of the potential errors arising from numerical uncertainty. Future research may consider using higher-precision numerical methods, such as LES or DNS.

2.1.2. Momentum Source Model

To enhance calculation efficiency, the momentum source model is employed to replace the real helicopter rotor in the numerical simulation of the ship–helicopter coupling flow field. This method simplifies the periodically rotating blades into a time average virtual disk. Although at the cost of sacrificing part of the computational accuracy, the research in this paper focuses on the flow field characteristics of the rotor as a whole, and the method can accurately compute the rotor downwash flow field by utilizing only a small amount of computational resources, and at the same time, effectively simulate the aerodynamic load of the rotor as well as the rotor’s effect on the surrounding flow field [10]. The specific expression of momentum source term R is as follows:

$$R={\left[0\hspace{0.33em}\hspace{0.33em}{S}_x\hspace{0.33em}\hspace{0.33em}{S}_y\hspace{0.33em}\hspace{0.33em}{S}_z\hspace{0.33em}\hspace{0.33em}0\right]}^T$$

(3)

The components of the momentum source term in the three directions in the above equation are determined using the blade element theory. In specific numerical calculations, the virtual actuator disk is represented by a two-dimensional interpolation grid within the rotor disk plane, where each interpolation grid cell corresponds to a micro-segment of the blade. Figure 1 shows a schematic diagram of the interpolation grid for the momentum source model. It can be seen from the figure that each cell of the interpolation grid is a micro-element of the rotor disk, obtained by differentiating the rotor disk in the circumferential and radial directions, respectively.

Figure 2 shows a schematic diagram of the spatial relative positions of the paddle disk rectangular coordinate system $\left(X_s,Y_s,Z_s\right)$ and the global coordinate system $\left(X_D,Y_D,Z_D\right)$. Let the rear chamfer and right tilt angle of the paddle disk in the global coordinate system be A and B, respectively, and the center coordinate Os of the paddle disk be $\left(x_c,y_c,z_c\right)$, then the coordinate transformation matrix from the global coordinate system to the paddle disk rectangular coordinate system is as follows:

$$M=\left[\begin{array}{ccc}\mathit{cos}A& 0& -\mathit{sin}A\\ \mathit{sin}A\mathit{sin}B& \mathit{cos}B& \mathit{cos}A\mathit{sin}B\\ \mathit{sin}A\mathit{cos}B& -\mathit{sin}B& \mathit{cos}A\mathit{cos}B\end{array}\right]$$

(4)

In the paddle disk rectangular coordinate system, the linear velocity of the paddle blade micro-segment caused by the rotational movement of the rotor $V_b$ is as follows:

$$V_b={\omega }^{\prime }\times R^{\prime }$$

(5)

where ${\omega }^{\prime }={\omega }_{x^{\prime }}i+{\omega }_{y^{\prime }}j+{\omega }_{z^{\prime }}k$, $R^{\prime }=R_{x^{\prime }}i+R_{y^{\prime }}j+R_{z^{\prime }}k$.

Without considering the angular velocity of the pitch and roll directions of the paddle disk, it can be written as follows:

$$V_b=-{\omega }_{z^{\prime }}{R}_{y^{\prime }}i+{\omega }_{z^{\prime }}{R}_{x^{\prime }}j$$

(6)

The velocity of the downstream flow relative to the blade in the paddle disk rectangular coordinate system $V^{rel}$ is as follows:

$$\begin{array}{cc}\hfill V^{rel}& =V^{\prime }-V_b\hfill \\ & =\left(V_{x^{\prime }}+{\omega }_{z^{\prime }}{R}_{y^{\prime }}\right)i+\left(V_{y^{\prime }}-{\omega }_{z^{\prime }}{R}_{x^{\prime }}\right)j+V_{z^{\prime }}k\hfill \end{array}$$

(7)

For any blade profile, the angle relative to the inflow $\beta$ is as follows:

$$\beta =\mathit{arctan}\left[{\displaystyle \frac{-V_{z^{\prime }}^{rel}}{\sqrt{{\left({V_{x^{\prime }}}^{rel}\right)}^2+{\left({V_{y^{\prime }}}^{rel}\right)}^2}}}\right]$$

(8)

Finally, the angle of attack of the blade profile $\alpha$ can be expressed as follows:

$$\begin{array}{cc}\hfill \alpha & =\theta -\beta \hfill \\ & ={\theta }_0+{\displaystyle \frac{r}{R}}{\theta }_{tw}-A_1\mathit{cos}\psi -B_1\mathit{sin}\psi -\beta \hfill \end{array}$$

(9)

where $\theta$ is the pitch angle, ${\theta }_0$ is the root installation angle, ${\theta }_{tw}$ is the blade twist, $A_1$ is the lateral periodic variable pitch, $B_1$ is the longitudinal periodic variable pitch, and $\psi$ represents the direction angle.

Based on wind tunnel test data, interpolate the angle of attack and Mach number of the inflow for the blade profile to obtain the lift coefficient, $C_l$, and drag coefficient of the blade profile, $C_d$, under this condition. Then, the lift and drag of the blade element are, respectively, as follows:

$$\begin{array}{l}{L}^{\prime }={\displaystyle \frac{1}{2}}\times \rho \times {\left|V^{rel}\right|}^2\times C_l\times c\times dr\\ D^{\prime }={\displaystyle \frac{1}{2}}\times \rho \times {\left|V^{rel}\right|}^2\times C_d\times c\times dr\end{array}$$

(10)

where $c$ is the chord length of the blade, and $\rho$ is the air density.

Converting the above leaf aerodynamic force to the paddle disk rectangular coordinate system, we obtain the following:

$$\begin{array}{l}{F}_x^{\prime }=\left(L^{\prime }\mathit{sin}\beta -D^{\prime }\mathit{cos}\beta \right)\times \mathit{sin}\psi \\ F_y^{\prime }=-\left(L^{\prime }\mathit{sin}\beta -D^{\prime }\mathit{cos}\beta \right)\times \mathit{cos}\psi \\ F_z^{\prime }=L^{\prime }\mathit{cos}\beta -D^{\prime }\mathit{sin}\beta \end{array}$$

(11)

Convert it to the global coordinate system to obtain the following:

$$\left[\begin{array}{c}{F}_x\\ F_y\\ F_z\end{array}\right]=M^T\left[\begin{array}{c}{F}_x^{\prime }\\ F_y^{\prime }\\ F_z^{\prime }\end{array}\right]$$

(12)

Finally, the force exerted by the micro-segment of the paddle blade on the flow field is obtained as $-F$, and the momentum source term $R=\left(S_x,S_y,S_z\right)$ on the paddle disk can be expressed as follows:

$$S=N{\displaystyle \frac{\Delta \varphi }{2\pi }}\times \left(-F\right)$$

(13)

where $N$ is the number of rotor blades, and ${\displaystyle \frac{\Delta \varphi }{2\pi }}$ is the ratio of the time required for the two-dimensional interpolation grid cell corresponding to the momentum source term to pass through the blade to the rotation period.

The k-ε turbulence model, which was used by Reddy et al. [2] and exhibits good adaptability for air–water two-phase calculations at high Reynolds numbers, is adopted, where k is the turbulent kinetic energy, and ε is the turbulent dissipation rate. The implicit unsteady solution method is adopted for time progress, including the calculation time step of the without helicopter is 0.01 s, and the calculation time step of adding the momentum source rotor to the ship airwake is relatively short, which is 0.00323 s for 5 degrees of true blade rotation.

2.1.3. Wave and Ship Motion Model

Waves refer to the periodic movement of the water surface under gravity after being disturbed by wind pressure. Give the liquid surface a velocity potential, φ, caused by the initial wind pressure impulse. Then, the velocity potential $\phi$ of the linearized infinite water depth wave can be expressed as follows:

$$\phi ={\displaystyle \frac{Ag}{\omega }}{e}^{kz}\mathit{cos}\left(\omega t-kx\right)$$

(14)

where A is the wave amplitude, and ω Is the circular frequency.

The regular waves acting on the ship adopt the fifth-order Volume of Fluid (VOF) wave model [19]. Based on the fifth-order approximation of Stokes theory, the wave is closer to the reality than that generated by the first-order method. The waveform and wave phase velocity depend on the water depth, wave height and water flow velocity.

The DFBI module is added in the simulation to let the ship move with the advance of waves. The DFBI module can simulate the motion of a rigid body in response to forces exerted by physics continuum, or to any additional forces defined. In this paper, the origin of the global coordinate system is located at the intersection of the bowsprit and the waterline plane. The origin of the ship’s coordinate system is located at the center of gravity. Axis X pointing to the stern is positive, axis Y pointing to the starboard side is positive, and axis Z pointing vertically up is positive. Taking the ship’s center of gravity as the origin, rotation around the axis X is defined as rolling motion, and tilting to port is positive; rotation around the axis Y is defined as pitching motion, and lifting the ship’s bow is positive; translation along the axis Z is defined as heaving, and upward is positive. As shown in Figure 3. The forces on the ship as a rigid body includes gravity as well as the dynamic fluid forces from both water and air. Only the displacement of axis Z and the rotation of axis Y is activated to make the ship pitch and heave.

The fluid forces and torques acting on the surface of a ship are as follows:

$$f_{\mathit{fluid}}=\sum _f{p}_f{a}_f-\sum _f{\tau }_f{a}_f$$

(15)

$$n_{\mathit{fluid}}=\sum _f[r_f\times (p_f{a}_f)]-\sum _f[r_f\times ({\tau }_f{a}_f)]$$

(16)

The translation and rotation equations relative to the center of mass are as follows:

$$m{\displaystyle \frac{dv}{dt}}=F$$

(17)

where $m$ is the mass of the rigid body, F is the resultant force acting on the body, and $v$ is the velocity of center of mass.

$$M{\displaystyle \frac{d\omega }{dt}}+\omega \times M\omega =N$$

(18)

where $M$ is the moment of inertia tensor, $\omega$ is the angular velocity of rigid body, and $N$ is the resultant moment acting on the body.

$$M=\left(\begin{array}{ccc}{M}_{xx}& M_{xy}& M_{xz}\\ M_{xy}& M_{yy}& M_{yz}\\ M_{xz}& M_{yz}& M_{zz}\end{array}\right)$$

(19)

2.2. Models and Meshes

2.2.1. Model and Operating Conditions

This paper focuses on the full-scale ONR Tumblehome Ship (ONRT) model provided by the International Towing Tank Conference (ITTC). The ship is an experimental ship model of the U.S. Navy Zumwalt-class destroyer. It includes high-precision hull and superstructure shape under and above waterline, and has complete hydrodynamic parameters and experimental and simulation data for reference [20,21], so that it is capable of the calculation of ship airwake under ship–wave coupling motions. As shown in Figure 3, the rudder, tail fin, bilge keel and other components that have a great impact on the ship motions are retained, while the superstructure does not include small components such as mast. Specific parameters of ONRT ship are shown in

Table 1. ONRT ship dimensions.

Variable	Values
Length of waterline (LWL, m)	154
Beam (B, m)	18.78
Draft (D, m)	5.494
Displacement (Δ, tons)	8507
Longitudinal center of gravity (xCG, m after FP)	79.625
Vertical center of gravity (zCG, m above BL)	7.65
Pitch moment of inertia coefficient (Kyy/L, kg·m)	0.25

. The specific parameters of helicopter rotor simulated by momentum source approach are shown in

Table 2. Rotor parameters.

Variable	Values
Number of blades	4
Rotor radius (m)	8.18
Blade heel cutting (m)	1.167
Blade chord (m)	0.5273
Blade negative twist (°)	−18
Rotor speed (rpm)	258
Rotor airfoil	SC1095

, the main dimensions of which refer to the UH-60A helicopter. For calculational efficiency, the SC1095 airfoil is employed across the full span of the blades.

2.2.2. Grid System

The computational domain is defined as 5 L × 4 L × 6 L, where L denotes the Design Waterline Length (DWL) of the ONRT ship. The front, up and down of the domain are defined as velocity inlets, and the rear is defined as pressure outlet. The boundary conditions of the inlet and outlet are defined by the field function of the corresponding fifth-order VOF wave model, and the left and right are symmetrical boundaries. In this way, although the distance from the bow to the inlet boundary is 1 L, the use of field function definition can ensure the stability of the flow field and accelerate the speed of waves acting on the ship; the depth below the water surface is 4 L, more than 600 m, which can simulate the ocean environment of infinite depth waves.

In this paper, the trimmed cell mesh with the commercial software Simcenter STAR-CCM+2206.0001 is used for mesh generation, and the prism layer mesh is added to the ship surface to capture the viscous boundary layer. In order to improve the calculation accuracy of ship–wind–wave coupling motions and airwake flow field, several mesh encryptions is set near the ship and the helicopter deck. Several free surface encryption areas are set for different wave heights and ship motion amplitudes under different sea states. For the cases with momentum source rotor, cylindrical local dense mesh is adopted to ensure the accurate application of momentum source term and the calculation accuracy of rotor downwash flow. A typical mesh generation result with the momentum source rotor at sea state 6 is shown in Figure 4. In this case, the total number of mesh is about 22 million, while the number of mesh is less for other cases at lower sea states or without momentum source rotor, but they are all more than 15 million, which can fully ensure the calculation accuracy and take into account the efficiency.

2.3. Validation

2.3.1. Velocity Distribution Validation of Ship’s Flow Field

The effectiveness of the numerical method used in this paper is validated by comparing the results of fixed ship without waves with the those of scaled model wind tunnel experiments [18]. The Reynolds number independence of ONRT ship airwake has also been verified by Dooley et al. [17]. The verification examples retain the multiphase flow model. Regular head waves are not imposed, while there are still Kelvin free surface waves, the body of ship is set to be fixed, the speed of current relative to the ship is 15 knots, and the wind speed is 35 knots (the wind speed relative to the ship is 50 knots), with the ship advancing against the wind and current. Six measuring lines are set at the stern along the longitudinal center plane, which are, respectively, located at x/L = 0.683, 0.708 above the hangar at the rear of the bridge (the experimental ship model hangar is lower than the standard model) and at x/L = 0.833, 0.883, 0.958, 0.983 on the stern flight deck as shown in Figure 5a, in order to capture the distribution of streamwise velocity, u, and vertical velocity, w, after the calculation converges, and then compare them with the experimental values.

As shown in Figure 5b,c, the numerical simulation results used in this paper are basically consistent with the experimental results in the Buchholz et al. [18] in the overall distribution trend. The small deviations may be a normal phenomenon caused by the difference in Reynolds number. In Figure 5, there is a large reverse flow area at the position of x/L = 0.683, 0.708 above the hangar, which is caused by the strong step separate flow generated by the air flow bypassing the superstructure. The airflow at x/L = 0.833, 0.883 in front of the flight deck is strongly disturbed and the affected range is larger, which is in line with the common distribution of ship airwake flow field [2]. Therefore, the numerical calculation method used in this paper has the effectiveness needed for research.

2.3.2. Isolated Rotor Aerodynamic Validation

The C-T rotor test results [22] are used for the arithmetic validation of the aerodynamic characteristics of the wash flow field under the rotor with isolated momentum source. The specific parameters of the rotor are shown in

Table 3. C-T rotor parameters.

Variable	Values
Number of rotor blades N	2
Rotor radius R (m)	0.9144
Blade root cut r0 (m)	0.2286
Blade chord length c (m)	0.1
Blade negative twist $△\phi$ (°)	0
Total pitch $\phi$ (°)	11
Rotor height above ground Hhub (m)	3.6 R
Rotor speed n (rad/s)	122.2
Rotor airfoil	NACA0012

.

Figure 6 shows the dynamic pressure distributions at 0.104 R, 0.215 R, 0.326 R and 0.993 R below the C-T rotor, which are compared with the experimental values. Figure 7 shows the dynamic pressure distribution cloud map of the flow field after convergence of the C-T rotor calculation. It can be seen that in the three sections within 0.326 R below the paddle disk, the dynamic pressure distribution of the downwash flow field corresponds well with the experimental values, and there is less than 10% error only at the paddle tip; in the position 1 R farther away, there is a certain degree of error with the experimental values, but the dynamic pressure at the tip of the paddle is high and decreases gradually to the inside and outside of the paddle with the same trend. In general, the rotor under-wash flow field conforms to the time-averaged characteristics of the momentum source method, and the distribution is in high consistency with the test, which can prove the effectiveness of the momentum source rotor simulation method adopted in this paper.

3. Effects of Waves and Ship Motions on Ship Airwakes

This section will use full-size ONRT ships to keep the bow facing the wind and waves and explore the differences between the four sea states from 3 to 6, as well as the effects of wave pumping and ship motions on the ships airwake flow field. Firstly, it introduces the physical model setting of every sea state in the numerical simulation and the corresponding ship motions. Then cases with and without waves or motions are compared with the baseline cases at the same sea states. Finally, the unsteady aerodynamic characteristics of the ship airwake at different sea states are analyzed and displayed.

3.1. Ship Motions at Four Different Sea States

Calculations were performed using typical wind and wave data for sea states 3–6 in the North Atlantic region [23]. The specific parameters of regular waves and uniform wind used in the physical model are shown in

Table 4. Wave and wind parameters at sea state 3~6.

Sea State	Wind Speed (Uwind, kn)	Wave Period (T, s)	Wave Height (H, m)	Wavelength (λ, m)
3	15	7.480	1.25	184.8
4	20	7.911	2	201.4
5	27	8.365	3.2	219.6
6	35	8.853	5	240.1

. The ship advances against the wind and waves, with its speed relative to the current fixed at 15 knots. As shown in Table 4, the wave period, T, ranges from 7 to 9 s, and the wavelength, λ, and wave height, H, increases with increase in sea states, where the wavelength, λ, is proportional to the square of wave period, T. To synchronize the ship’s pitch and heave periods with the wave, the wavelength λ of each sea state is slightly larger than the ship length L. Although these wave lengths are relatively rare for lower sea state 3 and 4, they are set like these to control the variables.

The results of the ship’s pitch and heave motions induced over one period, as simulated by the DFBI module, are shown in Figure 8. Using the 0° and positive pitch rates as the start time of a period, a phase lag of 1/4 T is observed for heave relative to pitch, which is quite different from the imposed ship motions simulations. The physical time of each calculation case in this paper is 120 s. The results when the ship moves regularly after 30 s are taken after. To synchronize the ship’s longitudinal rocking and pitching periods with the waves, the wavelength, λ, is slightly larger than the length of the ship for each sea state L. The average results of pitch and heave amplitudes and periods calculated are shown in

Table 5. Mean results of pitch and heave motions.

Sea State	Pitch Amplitude (°)	Heave Amplitude (m)	Period (s)
3	−1.05, 0.95	−0.44, 0.35	7.483
4	−1.66, 1.56	−0.84, 0.51	7.907
5	−2.47, 2.35	−1.42, 0.89	8.363
6	−3.47, 3.32	−1.65, 2.07	8.859

. It can be seen from the table that the pitch and heaving periods of ONRT ships are almost consistent with the wave period, and the amplitudes of motions increase with the increase in sea state. Since the wavelength is slightly larger than the ship length, i.e., λ/L > 1, the ship is in the condition of harmonic pitch, and pitch and heave occur with the advancement of wave phase with large amplitudes. In Figure 8a, for sea state 6, the maximum pitch angle of the ship reaches about ±3.5°, and the bow down amplitude is slightly greater than the bow up amplitude at all sea states, which is due to the sharp and thin shape of bow. In Figure 8b, the descending displacements at sea state 3~5 are larger than ascending, which is the common characteristic of ONRT, a freeboard inward inclined ship. However, the ascending displacement of sea state 6 is larger, which may be caused by its displacement exceeding the relatively gentle part of the profile near the design waterline.

Figure 9 is the curve of vertical coordinates and velocities of flight deck center at x = 135 m (x/L = 0.877) from front perpendicular (7 m above the design waterline) at four sea states. The superposition of pitch and heaving motions generated by the high sea states inevitably causes large unsteady vertical displacement of the flight deck at the stern. In addition to the effect of the unsteadiness of air flow field caused by the ship motions, the vertical displacements and velocities of the deck themselves also cause great interference to the helicopter pilot’s take-off and landing operation. It can be seen from Figure 9 that the vertical displacement of the deck center in sea state 6 ranges up to 6 m. In comparison, the vertical displacement range at sea state 5 is about 3.5 m. Meanwhile, the maximum vertical velocity of deck center at sea state 6 is 2 m/s, which is quite large relative to a landing helicopter with descent speed of 0.5~1 m/s. Although the periods T are slightly shortened, the overall launch and recovery difficulties are greatly reduced at lower sea states. There are critical nonlinear characteristics of the ship motions and flight deck displacement according to the sea states, which provide quite different conditions for helicopter operation, so coupling the waves and induced ship motions makes the simulation close to reality.

3.2. Respective Effects of Waves and Ship Motions on Ship Airwakes

In this subsection, the effects of waves and ship motions on ship airwakes are examined, respectively. In addition to the baseline cases of fixed ship without waves and the cases including both waves and ship motions mentioned above, the cases of ship with only waves but no motions are added to study the separate effects of waves or ship motions on airwake and their respective effects proportion.

Figure 10 and Figure 11 are the curves of the streamwise velocity, u, and vertical velocity, w, of the measuring point 9 m above the flight deck center in about three periods at sea state 3 and 6, respectively. The baseline cases with no waves and no motions are in quasi steady state. Comparing the velocity changes between the two sea states, the amplitudes of streamwise velocity, u, and vertical velocity, w, are larger at sea state 6. The velocity change curves for the two results with 5 m waves height, H, show more obvious synchronicity, and the velocity changes between the case only with waves and the baseline case are also more obvious. The time synchronicity between the two results with waves at sea state 3 is slightly weak, indicating that the pumping effect of waves on the air flow field above the deck is limited under lower 1.25 m wave height. Furthermore, the effects of waves on the vertical velocity, w, are not as obvious as that of the streamwise velocity, u, at both two sea states, but it still plays a certain role.

Comparing the two results with waves in Figure 11, the ship motions and waves together effects on the vortices structure and velocity components of ship airwake have strong nonlinearity. The averaged period range of streamwise velocity, u, is 11% of the baseline at sea state 3 and 16% at sea state 6. The averaged period ranges of vertical velocity, w, are 37% of the baseline at sea state 3 while 62% at sea state 6. The change in velocity caused by waves alone accounts for about 25% of the cases with waves and motions, indicating that the periodic pumping effect of waves with large wave height, H on, the air flow field at high sea states cannot be ignored. However, the waves have much less effects at lower sea states, which account for about 10% of the cases with all factors. Due to the nonlinearity of wave pumping action and ship motions, based on the water/air multiphase flow and DFBI module, the wind–wave–ship coupled method developed in this paper has the superiority over previous methods presented in chapter 1 when simulating the shipborne helicopter operation in real condition.

3.3. Wind–Wave–Ship Coupling Flow Field

The previous subsection demonstrated that the effects of both wave and ship motion on the airwake field and helicopter landing and takeoff in higher sea states are not negligible, so the above subsection proves that the effects of waves and ship motions on the airwake and helicopter launch and recovery at high sea states cannot be ignored. Therefore, the calculation results including both waves and ship motions are used to further analyze the unsteady aerodynamic characteristics of the ship airwake flow field at four sea states from 3~6.

Figure 12 illustrates the wave form when the wave crest is in the middle of the ship and the iso−surface of ship airwake vorticity diagram at sea state 6. The vorticity is defined using Q criterion, and the colors of the free surface represent the wave height. When the coupled simulation of waves and ship motions is added, the turbulent vortex structure of the airwake flow field of ONRT ship is basically consistent with the wind tunnel test and single-phase simulation of air flow field of other destroyers and frigates [5,24], which mainly includes bow separation vortices, side vortices, superstructure shedding vortices and stern shedding vortices.

Figure 13 shows the Streamlines on longitudinal center planes when the ship is pitching and heaving for four equal phases during a period T at sea state 6, and the comparison with the baseline result without waves and motions. The background of the sections in these figures are contours with vorticity ranging from 0 to 10. The lower part of the ship is red for the wet surface, and the wave form is displayed in different colors. The pitch angle reaches the extreme value at 0.25 T and 0.75 T, and the pitch angular velocity reaches the extreme value at 0.5 T and 1 T.

It can be seen from Figure 13 that in one period, with the periodic motions of ship and waves, the ship airwake presents the corresponding periodic contraction-expansion trend. At the moment of 0.25 T, the ship stern inclines greatly. At this time, the angles of the superstructure separated shear layer and the stern shedding vortices are basically consistent with the reference situation, but the recirculation zone behind the superstructure is significantly expanded. At the moment of 0.75 T, the ship bow inclines greatly, and the recirculation zone behind the superstructure is greatly reduced. At the moments of 0.5 T and 1 T, the vertical motion velocity of the stern flight deck caused by pitching reaches positive and negative maximums, respectively, and with the wave crest or trough reaching the middle of the ship, the heave displacement also reaches the extreme value. At this time, the vorticity behind the superstructure and the stern increases significantly, which enhances the unsteady characteristics of the flow field.

Figure 14 shows the time history of streamwise velocity, u, and vertical velocity, w, of 5 measuring points and compares them with the ship pitch and haeve in one period. The other 4 points are set 6.135 m (0.75 R of the UH-60A rotor) around the deck center point all at 9 m height. In one period u and w reach the minimum and maximum during 0.5 T to 0.75 T, when the superposition and 1/4 T phase lag of ship pitch and heave causes the stern flight deck height to reach the maximum. Both velocity components encounter small perturbations at 5~10 Hz frequency, which is much longer than the time step 0.01 s. These perturbations are caused by the superstructure separated shear layer with small vortices Furthermore, the velocity components at front point are smaller than those at the center and back because it is closer to the recirculation zone. The measuring points at left and right encounter less fluctuations and the velocities are almost the same due to the symmetry of ship airwake at 0° wind and wave angle, so the roll moment is neglected.

Figure 15 shows the time history of streamwise velocity, u, and vertical velocity, w, of the measuring point 9 m above the flight deck center during the physical time of 60~120 s at sea state 3~6. The streamwise velocity, u, of each sea state is different, so u is de-averaged, Δu = u−U. It can be seen from Figure 15a that generally the fluctuation period of velocity components in each sea state are consistent with that of waves and ship motions, and the amplitudes in every period are not exactly the same, especially at higher sea states. The amplitude of de-averaged streamwise velocity Δu at sea state 6 is much larger than that at the other three lower sea states, and the amplitude of velocity reduction is more obvious, which is more than 2 times of the reduction in sea state 5. There is less difference in the amplitude of streamwise velocity Δu at sea state 3~5. It can be seen from Figure 15b that the flow field at the measuring point at each sea state presents a downward velocity component due to the recirculation zone, and the downward velocity component increases with the increase in the sea state nonlinearly. The higher sea states also correspond to a larger fluctuation amplitude of vertical velocity, w, and the amplitude at sea state 6 is 1.5 times that at sea state 5.

4. Wind–Wave–Ship Coupled Effects on Airloads of Helicopter

4.1. Ship–Helicopter Coupling Flow Field

Based on the wind–wave–ship coupling flow field shown above, the rotor of UH-60A helicopter simulated using momentum source approach hovers 9 m above the flight deck center (x/L = 0.877). The specific parameters of the rotor are shown in Table 2 in Section 2.2. During the calculation, the main rotor collective pitch is kept at 8 degrees without cyclical pitch controls. The main objectives are to observe the dynamic interference in the ship–helicopter coupled flow field during helicopter hovering and to monitor the changes in helicopter thrust and pitch moment over time.

Figure 16 shows the wave form when the wave crest is in the middle of the ship and the iso-surface of ship–helicopter flow field vorticity diagram at the sea state 6. The presence of momentum source rotor significantly increases the vorticity of the downstream flow field, and there is a vortex ring around the rotor and two obvious blade tips shedding vortices downstream. The momentum source approach can well simulate the interaction between the helicopter rotor and ship airwake.

Figure 17 shows the Streamlines on longitudinal centerlane with a helicopter rotor when the ship is pitching and heaving for four equal phases during a period T at sea state 6. Except for adding momentum source disk and local grid encryption around it, other settings are consistent with Figure 13 of ship airwake in Section 3.3. In Figure 17, the location with large vorticity above the stern flight deck is the downwash flow generated by the momentum source disk. Comparing Figure 17 with Figure 13, due to the contraction-expansion phenomenon of the recirculation zone caused by waves and ship motions, the helicopter rotor periodically crosses over the area with strong shedding vortices after the superstructure, and then strong interferences occur at 0.5 T and 1 T, with the flow state of the rotor downwash also changing greatly.

Figure 18 shows the time history and power spectral density of rotor thrust and pitch moment when hovering 9 m above the flight deck center during the physical time of 60~120 s at sea state 3~6. These cases with helicopter rotor use shorter time steps of 0.00323 s rather than 0.01 s.

As observed in Figure 18a, since wind speed increases with increasing sea states, the helicopter thrust increases with increasing horizontal inflow velocity. At each sea state, where the collective pitch is maintained at 8 degrees, the thrust is more than 90 kN, which can ensure that the UH-60A helicopter with over 9 tons of maximum takeoff weight can complete the station-keeping, taking off and landing process, leaving sufficient collective pitch control margin as well. It is mentioned in Section 3.3 that the higher the sea state, the greater the fluctuation amplitude of the vertical velocity is, resulting in the corresponding nonlinear increase in the amplitude of the helicopter thrust fluctuation, especially around 0.11~0.14 Hz which is the waves and ship motions frequency. The fluctuation range of thrust at sea state 6 is 20% of the average, which is a huge ratio for a hovering helicopter, while this ratio is only 5% at sea state 3 and 8%, 11% at sea state 4 and 5, respectively.

In Figure 18b, the values themselves of pitch moment have no practical significance. However, it can still be seen that with the increase in sea state, the amplitude of pitch moment fluctuation becomes larger. The fluctuation range at sea state 6 is 1.5 times that at sea state 3. There are relatively stronger high-frequency perturbations at 5~10 Hz, especially when the stern deck height moves fast. The amplitudes of these perturbations are less than 5% of the long period fluctuation, which may have very limited effects. In addition, the helicopter has a larger nose down moment at higher sea state 5 and 6, because the recirculation zone area of the superstructure is larger and plays a greater role. In these conditions, the longitudinal stability and maneuverability of helicopters are facing greater difficulties.

However, the periods of these coupled motions are long, all exceeding 7 s, which is much larger than the pilot’s shortest closed-loop response time of 0.5 s [25,26], so that the pilot can maintain the stability of the helicopter to a certain extent by applying control, and it is possible to take off and land by using the quiescent period of ship airwake [14].

4.2. Airloads of Helicopter Hovering at Different Heights

Aiming at the ship motions and airwake rest period in the helicopter recovery process, this subsection explores the optimization of helicopter recovery process at high sea states by setting the helicopter hovering height as the variable Due to the most serious and representative unsteady interference phenomenon of ship airwake especially at sea state 6, at such sea state the helicopter rotor hub is located 7 m, 9 m, 11 m and 13 m above the flight deck center. The collective pitch still maintains at 8 degrees, and cyclical pitch control is not applied. The changes in helicopter thrust and pitch moment over time are calculated and monitored.

Figure 19 shows the curves of dimensionless helicopter thrust and pitch moment generated by the rotor during a ship motion period when the helicopter hovers at different heights, as well as the curve of the vertical coordinate of the flight deck center. Among them, the left axis and colored solid line are the aerodynamic loads of the rotor, and the right axis and black dotted line are the vertical coordinate of deck center, the measuring height of which is 7 m above the waterline when fixed.

It can be seen from Figure 19a that the phase of the rotor thrust fluctuation at each height is basically the same, and the change in the thrust lags about 1/4 T behind the height of the deck center, the reason of which is that the action of the deck displacement on the rotor through the air flow field is indirect and has a delay. At the same time, the incompressibility of the low-speed air makes the phases of the thrust at each height mostly consistent. Within 1 to 3 s after the deck displacement reaches the highest or lowest position, there is a platform period (horizontal black line shown) for the rotor thrust. At this time, the helicopter is relatively stable. When the vertical motion speed of the deck reaches the maximum of ±2 m/s at 3.4 s and 7.8 s, the rotor crosses over the separated shear layer of the superstructure as shown in Figure 17, and the thrust experiences large-scale perturbation above 1/3 of the total fluctuation range, and large-scale disturbances in thrust exceeding one-third of the total fluctuation range increase pilot workload, endangering helicopter launch and recovery safety.

As can be seen from Figure 19b, the trends of pitching moment at the height of 9~13 m above the deck are consistent. The absolute value of pitch moment at the 7 m height is small, but the fluctuation range is larger, because this lower height is closer to the recirculation zone. The pitch moment at each height is prone to high-frequency perturbation at the same time as the thrust, but the range is more than half of total range. This phenomenon is caused by the fore-and-aft non-uniformity of the shear layer when the rotor passes through. This will cause great damage to the pitch stability of the helicopter because of large-scale high-frequency oscillations of the pitching moment, while the change in pitch moment is relatively gentle in other time periods.

Since the vertical motion amplitude of the deck reaches 3 m at sea states 6 [23], and the rotor hub height of the UH-60A helicopter is about 3.8 m, the landing gears are close to the deck when the helicopter hovers at 7 m height and the deck reaches the highest vertical displacement 3 m. For the ONRT ship and UH-60A helicopter used in this paper, it is much appropriate for the helicopter during landing to hover at about 9 m, then reduce the collective pitch to descend, and complete the ship–helicopter contact when there is a slight downward trend right after the flight deck reaches the highest position and with the least vertical velocity. In this process, the rotor thrust is at a high level and the pitch moment changes slightly. It is conducive to maintaining the stability of the helicopter and ensures sufficient control margin. The relative vertical motion speed of the helicopter and the ship is low, which is also more favorable to meet the structural dynamics requirements of the fuselage, rotor and landing gears. The above is the primary attempt of helicopter recovery in the rest period at high sea states, according to both the flow field fluctuation and deck movement.

5. Conclusions

In this paper, a coupling wind–wave–ship–helicopter numerical simulation method for helicopter launch and recovery on ships is established. The unsteady aerodynamic characteristics of the ONRT ship airwake flow field when the ship pitches and heaves at sea states 3~6 are studied, and the changes in UH-60A rotor thrust and pitch moment over time at different sea states are calculated. The specific conclusions are as follows:

• The numerical simulation results of ship airwakes in this paper are in excellent agreement with the wind tunnel experimental data. The simulation of ship pitch and heave motions are in line with the actual situation. The ship motions and flight deck displacement exhibit critical nonlinear characteristics according to the sea states, which provide significantly different conditions for helicopter operation. Therefore, coupling the waves and induced ship motions makes the simulation more accurate and closer to reality.
• Either of the waves and ship motions have non negligible effects on ship airwake. In sea state 6, the wave pumping effect on the airflow accounts for up to 25% of the fluctuation in velocity components compared to the cases with both wave and motion, while at sea state 3, it accounts for only 10% with much less time synchronicity. The ship motions have more obvious effects, especially when the pumping effect of low height waves on the air flow field is relatively weak at lower sea states. The maximum vertical velocity of deck center at sea state 6 is 2 m/s, which is quite large relative to a landing helicopter with descent speed of 0.5~1 m/s.
• The effects of each sea state under the coupling of waves and ship motions vary significantly. At sea state 6, for the ship airwake without a helicopter, the fluctuation amplitude of the streamwise velocity and vertical velocity are 2 times and 1.5 times that of the level 5 sea state, respectively, while there are less differences between the velocity fluctuation and the corresponding helicopter airloads among sea state 3~5. For the ship–helicopter coupling flow field, the rotor thrust fluctuation ranges are 5%~20%, which is almost unacceptable at highest sea state 6. In addition, at sea state 5 and 6, the nose-down trend in the rotor pitch moment is more obvious due to the more effective recirculation of ship airwake.
• The dynamic aerodynamic interference of the wind–wave–ship–helicopter coupling system exhibits pronounced unsteady characteristics. The angle and position of the separated shear layer of the superstructure change with the ship pitch. The recirculation area presents a contraction-expansion trend with the ship heave. The ship pitch and heave with 1/4 T phase lag makes the helicopter rotor hover at a certain height constantly cross over areas with different velocities and vorticities. In addition to the large-scale air loads fluctuation in the same period of 7~9 s as the wave–ship coupling motions, the rotor thrust, and pitch moment produce high-frequency perturbations of 5~10 Hz when encountering the swinging separated shear layer. The amplitude of these disturbances can reach more than 1/3 of the total fluctuation at sea state 6, which seriously endangers the launch and recovery of shipborne helicopters and needs to be eluded.