Numerical and Experimental Study of Flow Field between the Main Hull and Demi-Hull of a Trimaran

The interactions between the main hull and demi-hull of trimarans have been arousing increasing attention, and detailed circumferential flow fields greatly influence trimaran research. In this research, the unsteady wake flow field of a trimaran was obtained by Reynolds-Averaged Navier-Stokes (RANS) equations on the basis of the viscous flow principles with consideration of the heaving and pitching of the trimaran. Then, we designed an experimental method based on particle-image velocimetry (PIV) and obtained a detailed flow field between the main hull and demi-hull of the trimaran. A trimaran model with one demi-hull made of polycarbonate material with 90% light transmission rate and a refractive index 1.58 (close to that of water 1.33) was manufactured as the experiment sample. Using polycarbonate material, the laser-sheet light-source transmission and high-speed camera recording problems were effectively rectified. Moreover, a nonstandard calibration was added into the PIV flow field measurement system. Then, we established an inverse three-dimensional (3D) distortion coordinate system and obtained the corresponding coordinates by using optics calculations. Further, the PIV system spatial mapping was corrected, and the real flow field was obtained. The simulation results were highly consistent with the experimental data, which showed the methods established in this study provided a strong reference for obtaining the detailed flow field information between the main hull and demi-hull of trimarans.


Introduction
As researchers seek to discover the fundamental properties associated with object motion, the focus of research on basic fluid mechanics problems in shipbuilding and ocean engineering has shifted from determining macroscopic forces and moments to measuring detailed flow fields. Investigations into the complete circumferential flow field are very important to research on multihull ships. The International Towing Tank Conference (ITTC) has established a specialist committee on detailed flow measurement since the 25th conference to enable coordination between scientists and researchers worldwide who are devoted to basic research on fluid mechanics to conduct in-depth studies in this area [1]. However, for high-performance ships with complicated structures, such as trimarans, there exists a complex flow field caused by the wave-making disturbance between the main hull and demi-hull. Consequently, obtaining accurate measurements of the detailed flow field between the main hull and demi-hull of trimarans is difficult because of the limitations caused by various long-standing issues such as the special structure of trimaran ships, fully transparent material of the demi-hull, and fluid measurement techniques.
In computational fluid dynamics (CFD)-related work on ship wakes, Zhang [2] simulated the viscous circumferential flow field of the KRISO container ship (KCS), assuming a free surface with/without propellers and using Reynolds-Averaged Navier-Stokes (RANS) equations, and then compared the numerical results with the experimental data. Sadat-Hosseini et al. [3] simulated the ship motion and added resistance of KVLCC2 in long and short head waves, and compared the results obtained to experimental data. They compared the wake flow fields at various time instances with the measured values obtained using particle-image velocimetry (PIV) by analyzing waveforms. Ahmed and Soares [4] analyzed the free surface flow around a very large crude carrier (VLCC) ship using viscous and potential flow methods. Choi et al. [5] conducted numerical analyses on the resistance, propulsion, and surrounding flow of multiple wide ships based on CFD results. He et al. [6] studied the disturbance effects of a high-speed catamaran on its resistance, heaving, and pitching. Ma et al. [7] simulated the bubbly flow around a ship using a subgrid air entrainment model. Wan et al. [8] implemented a numerical simulation on the viscous flow field around a Wigley ship using the level set method. Wang et al. [9] simulated the wake flow field behind a low-speed wide ship, where the "hook-like" effect was found to be prominent. In addition, Larsson et al. [10] collected, analyzed, and compiled 18 test cases pertaining to the KCS, KVLCC, and DTMB5414 ships, which were presented by 33 research groups at the Gothenburg 2010 Workshop on Numerical Hydrodynamics. These previous studies fully demonstrated the reliability of CFD tools in researching a ship's hydrodynamic performance, and also provide references for obtaining detailed flow fields between the main body and the demi-body of the trimaran considered in this work.
On the other hand, PIV is a newly developed flow visualization-based flow measurement technique that takes advantage of the research results of modern computer science, optics, and image analysis techniques. It can not only demonstrate the physical flow pattern of a flow field, but can also provide quantitative information regarding the instantaneous flow, which enables the leap from qualitative research to quantitative research in the field of flow visualization. Arslan et al. [11] measured the flow field of the gap between two interacting ship-like sections, from which various flow phenomena, such as vortex formation, flow separation, and reattachment mechanism were observed. Wen et al. [12] studied the flow structures of underwater jets with free surfaces using time-resolved PIV. Gim [13] conducted measurements of the circumferential flow field for a twin-rudder model in a circulating water channel under various angles of attack and separation distances using stereoscopic PIV (SPIV). Longo et al. [14] measured the nominal wake flow field of a DTMB5512 warship cruising over waves using phase-averaged PIV.
Trimarans have been attracting increasing attention in recent years as a result of advantages they possess that cannot be matched by monohull ships. For example, the wider deck possessed by a trimaran offers a better overall configuration, the long and thin shape with appropriate demi-hull positions can achieve smaller wave-making resistance at high cruise speeds, and the wave-resistance performance is generally superior to that of monohull ships. The excellent hydrodynamic performance guarantees a promising future for the applications of trimarans. There has been progress regarding the theoretical calculations, model tests, and numerical simulations of the wave-making resistance theories of trimarans worldwide. Several Chinese researchers, including Han and Huang [15], Huang et al. [16], and Wang et al. [17][18][19], have conducted research on multihull ships. Pavkov and Morabito [20] experimentally investigated the resistance, heaving, and pitching properties of two trimaran ship models in shallow water. Fang and Chen [21] studied the hydrodynamic performance of a trimaran in waves using the spectral analysis method. Hafez and El-Kot [22] studied the effects of the stagger variation of the outriggers on the disturbance hydrodynamic performance of a high-speed trimaran.
In this study, we first solved the RANS equations based on six degrees of freedom (DoFs) for the unsteady circumferential flow field of a trimaran with two unconstrained DoFs-specifically, heaving and pitching. Then, we performed nonstandard calibration for PIV, and calculated the corresponding spatial distortion coordinate system. Subsequently, we manufactured a demi-hull in accordance with the calculated refractive index, and measured the detailed flow field between the main hull and the demi-hull of the trimaran in the Ship Model Towing Tank Laboratory at Harbin Engineering University. The results of this research are expected to provide better details pertaining to fluid flow. Both the numerical and experimental methods employed in this study can also serve as important reference material for further research into the flow fields of catamaran and trimaran ships.

Governing Equations
The motion of incompressible Newtonian fluid satisfies the continuity and momentum conservation equations [23].
where u i and u j are the time-averaged velocity components (i,j = 1,2,3), p is the time-averaged pressure, ρ is the fluid density, µ is the dynamic viscosity coefficient, ρu i u j is the Reynolds stress term, and S j is the generalized source term of the momentum equation.

Turbulence Model and Free-Surface Treatment
In this work, pressure-based coupled solver was used for the control equations of the numerical computation in which spatial discretization using a second-order upwind scheme was performed on the convection term and discretization using a second-order central difference scheme was performed on the dissipative term [24]. By taking into account the effect of model wall shear and to simulate a strong adverse pressure gradient flow field, the shear stress transport k − ω model was employed [25]. The model combines the advantages of the k − ω and k − ε models and is able to compute the flow separation region. It is one of the most advanced models among two-equation turbulence models, which is superior in its calculation of viscous ambient flow field. Free surface was captured using the volume-of-fluid (VOF) method [26].

Wave Damping
A numerical wave beach was embedded at the inlet, outlet, and side boundaries within the computational domain to prevent the reflection of free-surface waveforms, and vertical damping was introduced to balance the vertical motion using the method introduced by Choi and Sung [27]. The addition of velocity w in the damping equation is shown below. where x sd is the starting point for the wave damping (propagation in the x-direction). x ed is the end point or boundary for the wave damping. f 1 , f 2 , and n d are the damping model parameters. w is the vertical velocity component.

Boundary Conditions and Meshing
In this study, the model used for the numerical simulations and experiments was a trimaran ship model, with the demi-hulls symmetrically distributed on the left and right sides of the main hull. In terms of the longitudinal position of the demi-hull, its stern transform plate was flush with the stern transform plate of the main hull; the two demi-hulls were transversely separated by a distance of 0.6 m. The main parameters of this trimaran are given in Table 1. Using the intersecting point of the hull bottom surface and the intersecting line between the central longitudinal section and the central transverse section as the origin, the calculation domain (x is the hull length direction, y is the hull width direction, and z is the hull height direction) has the following ranges: Figure 1. The boundary conditions are listed in Table 2.  Meshing is the basic process used to realize the numerical discretization of the governing flow equations. Therefore, a good mesh-generating technique is one of the prerequisites for a successful numerical simulation in CFD, as the mesh quality directly determines the convergence and accuracy of the calculation. In this study, we used the Star-CCM+ software to conduct surface remeshing of the ship model. We processed the surface mesh to generate a high-quality triangular mesh, and then generated the volume mesh with a prism layer mesh and a trimmed mesh based on the face mesh. The boundary layer was captured using the prism boundary layer mesh, and was processed based on the two-layer all y + wall processing, with y + ranging between 30 and 100. The total number of cells was 2.46 million. A schematic of the meshing is shown in Figure 2. The criterion employed to evaluate the quality of the trimmed mesh was the volume change rate between neighboring volume cells, where an excessive volume change could lead to potential errors, and thus numerical instability. Table 3 lists the statistics of the mesh quality obtained for the numerical model. It can be seen that we achieved a fairly good meshing quality, which does not negatively affect the simulation accuracy or numerical stability, in this study.

Experimental Model
Considering the calculations for the test model and the designed experimental scheme, we used polycarbonate material to manufacture the demi-hull of the trimaran. Polycarbonate material has excellent light transmission characteristics (90%) and a refractive index that is close to that of water (polycarbonate: 1.58, water: 1.33), which can adequately address the problems that affect laser-sheet light-source transmission and high-speed camera recording. However, to directly satisfy the experimental requirements, the entire demi-hull would need to be made of polycarbonate, which would have resulted in a significant weight problem in addition to high cost; polycarbonate has a material density of 1.20 g/cm 3 (at 17 • C). Thus, compared to other hollow demi-hull materials, polycarbonate offers no buoyancy, and is heavier than an equivalent volume of water. This makes it difficult for a ship model to properly float on water. Moreover, because the demi-hull is located on one side of the hull, the addition of a weight to the other regular demi-hull in order to balance the overall weight would further increase the total weight of the ship model. Traditional model manufacturing approaches cannot leave such a large margin of displacement for model balance. Therefore, in this study, we employed a novel manufacturing method to construct the model and indirectly satisfy the requirements. Specifically, we carved the keel of the entire hull using a polyvinyl chloride (PVC) sheet instead of the original plywood. Then, for overall stability and structural support, we used hollow aluminum tubes. After the hull was manufactured, we removed both the internal and external surfaces, which did not contribute to the structural strength, and refabricated all of the necessary connecting parts for the guide system (e.g., connecting plate, rail, and guide vane) using aluminum alloy. The resulting light trimaran ship model constructed is shown in Figure 3.

Nonstandard Calibration for PIV System
The detailed configuration of PIV equipment is shown in Figure 4. The Scheimpflug angle, focus, and aperture can be adjusted through the controller box using dynamic studio software [28,29]. In this study, we utilized the PIV technique as follows. We evenly seeded a sufficient number of tracer particles with good scattering properties into the flow field, in which a luminous laser sheet from a pulse laser was to be formed at the cross section. Next, by capturing two or more exposure particle images using image-recording systems, we obtained images containing particle movements. Then, we used the cross-correlation method to analyze the PIV images in order to determine the average particle displacement in each small interrogation window. With this displacement information and the separation time between two exposures, we acquired the velocity vector at each point of the flow field. We were subsequently able to calculate other movement parameters (such as the velocity component, streamline, and vorticity). This procedure was utilized because PIV uses the velocities of the laser tracer particles to represent the real fluid velocity at each corresponding point within the flow field. Thus, based on the basic definition of velocity, the PIV technique in essence acquires the particle movement velocity measurement by calculating the displacements of tracer particles within a given time interval. The working principle of PIV is demonstrated in Figure 5. By tracking and measuring multiple tracer particles in a two-dimensional (2D) plane, we obtained the measurement of the entire 2D flow field. Figure 6 shows a schematic of the tracer particles in a measurement section, with the solid circles representing the position of the particle family at time instance t 1 , dashed circles representing the position of the particle family at time instance t 2 (t 2 = t 1 + ∆t), and arrows representing the displacement of each particle within the time interval ∆t. When using PIV to measure flow fields, we used natural light or a laser as a light source to form a luminous plane, and image-recording equipment such as CCD cameras to capture the images of the tracer particle movement. Then, we analyzed the image sequence to obtain the velocity-vector distribution of the 2D flow field. Polyamide seeding particles with a diameter of 50 µm (Dantec Dynamics Inc.) were used as the tracer particles for this PIV test. Approximately 150 g/m 3 of particles were mixed with the water using a motorized stirrer in the particle-seeding device. To ensure uniformity of particles in the test region, a customized rake-shaped pipe was used during the particle seeding process.  In general, a PIV system consists of two parts: hardware and software. The hardware part comprises the control circuit, CCD camera, light-source, image-collection circuit, a PC used for image collection, and a PC used for capturing the synchronization signal. The software part comprises both control and analysis software. The control software primarily includes image collection and signal synchronization, as well as scale-calibration software. The analysis software comprises software responsible for the processing and analysis of particle images, and software responsible for the display and correction of the flow-field velocity vectors. Image processing requires that the velocity field be obtained from two successive particle frames captured from two exposures. The particle image is divided into multiple interrogation windows, and all the particles within the same interrogation window are assumed to have the same displacement velocity with linear movement. Further, the maximum particle displacement within the interrogation window is usually less than one-quarter of the window size; in the out-of-plane direction, the particle displacement should be less than one-quarter of the laser sheet thickness, and the particle in-plane displacement should be more than twice the particle size on the image. The basic equations for calculating the instantaneous velocity can be written as where ∆t is the separation time of the CCD camera between two successive exposures, which is equal to 400 µs. When this separation time is sufficiently small, the average velocity should accurately represent the instantaneous velocity of the tracer particle. The commonly used algorithm for the PIV velocity calculation is the fast Fourier cross-correlation method. This algorithm is usually used for signal analysis. By introducing the fast Fourier transform, we were able to obtain the average displacement and velocity within a small window in the particle image by calculating the cross-correlation function from two successive images at the same point in the window. One of the advantages of this algorithm is its ability to automatically identify the velocity directions, and thus the measuring range of the velocity is significantly greater than that of self-correlation. When conducting correlation calculations, the small interpretation box on the subsequent image with the largest similarity to the one on the previous image can be automatically identified. Consequently, any background noise in correlation calculations is reduced, and the effective particles for correlation increase. As the signal-to-noise ratio increases, the accuracy of the interpretation and identification is significantly enhanced. The size of the interrogation region is 2.45 × 2.45 mm. The 16 × 16-pixel interrogation window was employed to calculate the velocity vectors based on adaptive correlation method [30][31][32]. And the average number of particles in each interrogation region is about 22. Thus far, the theories discussed above apply only to the fundamental assumption of homogeneous liquid with a constant refractive index. However, because the laser beam passes through the transparent demi-hull, it will deflect as a result of the difference in the refractive indexes of polycarbonate and water. Further, this deflection is a 3D deflection, instead of a unidirectional one. Consequently, if the original coordinate system is still used to conduct tests and flow-field analysis, the final results will be distorted, and would, therefore, not describe the real flow field. Figure 7 shows the optical path distortion in a real-life application. The calibration process of the PIV measurement system used in the experiment was completed with a 400 × 310 mm multilevel dot matrix Dantec Dynamics calibration target in the towing tank. The multi-level dot matrix calibration target is divided into two parts. Half of the dot matrix is at the position of the measurement section, and the other half of the dot matrix is at the plane level far away or close to the shooting system. The multi-level dot matrix calibration target avoids the influence of the movement of the single-level dot calibration target during the calibration process. Before calibrating the measurement area, it is necessary to calibrate the position of the measurement section-that is, the multi-level dot matrix calibration target needs to be installed at the position of the laser sheet light section where the measurement area is located. The calibration tool with mirror function is used to display the laser sheet light section and calibration. The alignment of the center line of the target can be used to fine-tune the position. The schematic diagram of the multi-level calibration target and the measurement section calibration tool is shown in Figure 8. During the calibration process of the plane measurement area, the green LED light is used to evenly illuminate the calibration target and at the same time, the PIV image acquisition system uses a single frame model for calibration image acquisition. Therefore, the spatial coordinate system needed to be recalibrated and an inverse distortion coordinate system established. We obtained the real flow-field information by analyzing the experimental data using the new coordinate system. Considering the complexity of the calculations, we present a 2D measurement example here.
As shown in Figure 9, the pre-seeded tracer particles in the flow field between the main body and demi-hull are lighted by the incoming laser beam after it passes through the demi-hull. Then, the reflected light that is scattered from the particles again passes through the demi-hull, before being received by Cam1 and Cam2. Because the transparent demi-hull has a greater refractive index than water, the reflected light is deflected on passing through the demi-hull. However, the positions of the flow-field points used in PIV calculations remain the inverse focal points of the reflected light before deflection, which means that there is a large discrepancy between the calculation positions and the real positions. Therefore, the coordinate system needed to be redefined in order for them to be mapped. Figure 9 shows the schematic for the 2D measurement. We generalized this to a 3D space to obtain a new coordinate system to correct for spatial distortion. Then, we used this new coordinate system to conduct optical calculations in order to obtain the coordinates of the corresponding points and to display the real flow field from PIV data by correcting the spatial mapping. Figure 10 illustrates the calibration of the awaiting measurement area after spatial-mapping correction. Figure 11 shows arrangement of the measurement system.

Results and Discussion
We set the time step of the numerical calculation to 0.02 s, and the calculation time for each ship-speed condition to 80 s. We used the SIMPLE algorithm for pressure and velocity coupling, and adopted the finite-difference method with second-order accuracy for discretization of the momentum equation and pressure, and the first-order difference method for the turbulence model. The heaving and pitching of the ship model, i.e., the linear translation along the z axis and rotation about the y axis, respectively, were unconstrained in the experiments, thus they were also unconstrained in the numerical simulation. The hull can be freely translated along the z direction and rotated about the y direction in the free state. As a result, the velocity field and the pressure field around the hull will vary significantly, leading to a large-scale variation of the forces acting upon the hull. In the numerical simulation, after several time-step iterations, the loaded movement function can conduct real-time calculations at specific mesh points on the hull surface. Thus, we obtained the force acting upon the entire hull by calculating the total combined forces at each mesh point on the hull surface. Variations of the hull force (torque along the y-axis) with time, under two different ship speeds, are shown in Figure 12. In the figure, it can be seen that at the onset of the flow field, the hull has a large torque in the y direction, resulting in a pitching torque for the hull in addition to the heaving motion. Then, a larger draft leads to another balance between the buoyancy and gravity. The pitching angle then changes and a new balance of force is reached. After performing iterations for a specific period of time, the forces acting on the hull gradually reach a balance, and thus the circumferential flow field around the hull also reaches a dynamic equilibrium under the current ship speed.  Figure 13 shows the configuration of the model test setup. The resistance test data of the ship model were obtained by means of a force sensor mounted on the airworthiness instrument. A comparison of the resistance between the numerically calculated values and the experimentally determined values is given in Table 4. It is clear that the resistance errors are below 2%, which confirms the reliability of the numerical results obtained in this work.  The PIV system started collecting data when the ship model reached a stable speed, with Cam1 and Cam2 (installed on the underwater torpedo bodies) eventually capturing 90 double-frame particle images. We processed these double-frame particle images by performing the following steps. (1) Noise reduction: We reduced the noise by subtracting the background from the 90-pixel images using the image-subtraction arithmetic algorithm. This method enabled arithmetic operations to be performed on pixel values. We implemented the process for the images recorded by Cam1 and Cam2. (2) Analysis of the noise-reduced data: We employed the adaptive cross-correlation method to process the 90 noise-reduced double-frame images in which the pseudo vectors were removed. After this step, we generated a data set containing 90 vector fields, with each field representing the vector field at one of the 90 instances in time. (3) Data averaging: We averaged the data set containing 90 vector fields in order to obtain final results containing velocity and scalar information. The "good" vectors and "bad" vectors within the interrogation region were not evaluated, the local validation algorithm along with the adaptive correlation was employed so that viewed on all the calculation process [33][34][35], hence less "bad" vectors are generated. The sample image that shows the particle density is presented in Figure 14.    From Figures 16 and 17, it is clear that the flow field between the main body and the demi-body of the trimaran can be captured by both numerical calculations and PIV tests, and there is fairly good agreement in terms of the microscopic and macroscopic structures of the flow field. Moreover, they both capture the interacting properties of the flow field between the main hull and the demi-hull. The most straightforward way to obtain the wake flow field behind the hull is to perform an experimental measurement. However, in PIV measurements, because a sufficient number of laser tracer particles need to be seeded in order to represent the real mass points of the flow field, the wake flow field obtained using numerical calculations is smoother and more homogeneous than that determined via PIV tests. From these results, we clearly observed the flow-field distributions around the main hull, demi-hull, and between the two. Stratification of the boundary layer was prominent. An increase in the ship speed resulted in a decrease in the wake homogeneity and an expansion of the boundary layer. Further, the water quality during the PIV tests had a direct impact on the data quality. In addition, the discrepancies between the experimental and numerical results also included errors caused by the inaccurate calibration of the coordinate system for PIV.
Specifically, when the wake velocity contour is around 1 m/s, the numerical simulation value and the PIV measurement value have a similar velocity gradient. Judging from the outline of the wake field, the value is more similar to the wake distribution on the main body in the experiment and simulation. In addition, the wake field around the demi-body at x/Lpp = −0.4833 also has a certain similarity, as shown in Figure 17. The main quantitative discrepancies between the numerical and experimental values come from the velocity distribution (the red contour) near the solid wall (hull boundary). The velocity boundary layers present a slight shift from the main and demi-hull especially at x/Lpp = −0.18 due to the wall reflection effect and calibration accuracy, suggesting that the non-standard calibration method still needs to be carefully designed in the near-wall field. While at x/Lpp = −0.4833, the difference of near field velocity contour between numerical simulation and experiments become small, which implies that the non-standard calibration process should be handled differently at different longitudinal position.

Conclusions
In this study, we numerically and experimentally investigated in detail the flow field between the main body and demi-body of a trimaran. First, we solved the unsteady circumferential flow field of the trimaran using RANS, in which we considered the heaving and pitching of the hull and analyzed the calculation domain, boundary conditions, and meshing. Then, in what to the best of our knowledge is a first, we applied the PIV technique to measure in detail the flow field between the main body and demi-hull of a trimaran. This process involved the design of an innovative experimental method. As the test object, we constructed a trimaran ship model with a polycarbonate demi-hull. Subsequently, we performed nonstandard calibration in the flow field for the PIV measurement system, and established an inverse 3D distortion coordinate system to calculate the corresponding flow-field points affected by light deflection. Then, we displayed and calculated the real flow field obtained from the PIV tests by making corrections for spatial mapping. Finally, we measured the wake flow fields between the main body and demi-body near the stern transform plate (x/Lpp = −0.4833), and near the bow (x/Lpp = −0.18) under the designed draft. A comparison of the numerical results and PIV data showed that there is good agreement in terms of the micro-and macro-structures in the flow field. Numerical simulations and PIV measurements are both effective techniques for studying complicated flows such as the flow field between the main hull and demi-hull of a trimaran. The research methods proposed in this study provide an important reference for obtaining the complete circumferential flow fields of multihull ships.