Evaluation of Mock Quay Wall Dimensions for Tank Tests Based on Numerical Analysis of Wall Effects

1 Evaluation of Mock Quay Wall Dimensions for Tank Tests Based on Numerical Analysis of Wall Effects Masatoshi Hirota1*, Yasushi Kitagawa1, and Hiroshi Kobayashi1 1 National Maritime Research Institute, Japan Abstract. This study presents numerical analyses of wall effects during berthing and unberthing maneuvers. As a preliminary step toward future physical model experiments, it aims to provide insights into the design of a mock quay wall — a simplified vertical wall used to replicate wall effects in tank tests. Numerical analyses are conducted using a RANS-based numerical analysis applied to the DTC benchmark hull form under deep-water conditions. Two types of maneuvers are considered: (1) straight-ahead run parallel to the quay wall and (2) lateral berthing and unberthing with the bow and stern remaining parallel to the quay wall. This study focuses on the effect of mock quay wall geometry, particularly its depth. When the hull moves parallel to the quay wall, the hydrodynamic forces acting on the hull are affected by the quay wall depth. In contrast, under lateral motion, the hydrodynamic forces are found to be insensitive to the mock quay wall depth. During lateral berthing and unberthing, oscillatory time histories of the lateral forces on the hull are observed, attributed to circulating flows that develop around the bottom edge of the mock quay wall. Such oscillatory time histories are not observed in the deep full depth wall structure without open space. These results indicate that the mock quay wall is not capable of adequately reproducing the wall effects. Keywords: Numerical analysis, Wall effect, Port navigation. 1. Introduction When a ship navigates in the vicinity of a quay wall, suction forces toward the wall and yaw moments acting on the hull are known to occur [1]. These hydrodynamic interactions are collectively known as wall effects. Wall effects induce non-negligible hydrodynamic forces during berthing and unberthing operations and are directly related to ship maneuverability and safety. Therefore, accurately evaluating the associated hydrodynamic characteristics is essential for ensuring safe berthing and unberthing operations. To understand the influence of quay walls in restricted waters, numerous experimental and numerical studies have been conducted under conditions where both bank effects and shallow water effects are simultaneously present. Experimental investigations have addressed wall forces acting on the hull [2], hydrodynamic interactions [3], and the effects of wall geometry and water depth [4]. On the numerical analysis side, CFD simulations [5], [6], [7] and fast prediction methods using potential flow theory have been conducted [8]. This study aims to isolate and evaluate the wall effect, focusing specifically on its effects during berthing and unberthing maneuvers. Physical model tests conducted in experimental tanks with vertical wall structures (hereafter referred to as mock quay wall) are essential for evaluating wall effects. However, to appropriately reproduce wall-induced effects in such experiments, the dimensions and structural configurations of the mock quay wall must be carefully examined in advance. In our previous studies, we conducted numerical simulations under sufficiently deep-water conditions to isolate wall effects from shallow water effects. Two types of ship motion were considered: (1) parallel motion to the quay wall [9], and (2) lateral berthing and unberthing while maintaining a parallel heading to the wall [10]. Hydrodynamic forces acting on both the hull and the mock quay wall were analyzed, with a focus on how the depth and length of the quay affected the results. The simulations revealed that during parallel motion, the depth of the mock quay wall significantly influenced the hydrodynamic forces acting on the hull. In contrast, no significant influence of quay wall depth on the forces was observed during lateral motion. In this study, we further investigate these findings. In addition to analyzing hydrodynamic forces during parallel motions, such as the pressure distribution on the hull surface and bow-out moments, we emphasize the oscillation * Correspondence to: [email protected] 16th International Symposium on Practical Design of Ships and Other Floating Structures PRADS 2025 Ann Arbor, MI, USA, October 19th – 23rd 2025 2 of the forces during lateral motion. The analysis considers the two types of ship maneuvers: (1) parallel motion to the quay wall, and (2) lateral berthing and unberthing while maintaining a parallel heading to the quay wall. For each case, numerical analyses based on RANS CFD models are conducted under two conditions: a mock quay wall with an open space at the edge and a full depth wall configuration with no openings. The mock quays are also simulated at different depths. This study is a preliminary numerical investigation to support the design and interpretation of future physical model tests. Its goal is to clarify the effects of mock quay wall configurations on the hydrodynamic forces acting on the hull. 2. Methodology of CFD analysis 2.1. Numerical schemes 2.1.1. Overset grid approach The numerical analyses in this study are performed using an overset grid method. This method entails the utilization of multiple computational grids, including a hull, a quay, and a basin, which are then enclosed to represent the computational domain collectively and overlap each other without requiring face-to-face matching between grids. All computational grids are generated using the commercial grid generation software, PointwiseTM. An in-house overset assembling system called UP_GRID [11] computes the domain connectivity information (DCI) for the overset grid method. UP_GRID is based on a structured grid DCI system and developed by National Maritime Research Institute (NMRI) of Japan. 2.1.2. Navier-Stokes Solver All simulations are carried out by the in-house flow solver NAGISA [12], a 3D incompressible Navier-Stokes solver developed by NMRI. Incompressible Reynolds-averaged Navier–Stokes equations are solved to obtain a steady solution by introducing pseudo-compressibility for velocity-pressure coupling. Unsteady flow can also be simulated by using a dual time-stepping approach. Spatial discretization is based on a finite-volume method, and inviscid fluxes are evaluated by the third-order upwind scheme based on the flux-difference splitting. The evaluation of viscous fluxes is the second-order centered difference. The artificial compressibility approach is used for velocity-pressure coupling. The single-phase level-set method used in this study for capturing free surface. Various turbulence models such as one-equation, two-equation, and explicit algebraic stress model(EASM [13]) are implemented. The solver is capable of the overset grid method for complex geometry and can cope with overlapped grids with DCI generated by UP_GRID [11]. 2.2. Target Ship and Computed Cases The geometry of the target ship is the Duisburg Test Case (DTC) [14], which is a post-Panamax size container ship. Table 1 shows the principal particulars of DTC, in which the value for an actual ship and a model ship of 𝐿 = 3.740 m; the subscript 𝑀 is the model ship scale. The DTC without bilge keels, rudder, or other appendages is used in this study. Table 1. Principal particulars of the DTC. Actual Model Length between perpendiculars: 𝐿  [m] 355.0 3.740 Breadth: 𝐵 [m] 51.0 0.537 Draft: 𝑑 [m] 14.5 0.153 Block Coefficient: 𝐶  0.661 Design Speed: 𝑉  [knot] 25.0 - This study aims to investigate the specifications of the mock quay wall for tank tests through numerical analyses, with the objective of evaluating the wall effects in the Actual-sea Model Basin (AMB; 80 m (length) x 40 m (width) x 4.5 m (depth)) of the NMRI. A water depth of 7.48 m is equivalent to twice the 𝐿 of the model 3 ship, is selected as sufficiently deep. When a hull navigates near a quay wall, hydrodynamic interaction between the hull and the quay wall occurs, and this interference is expected to depend on the dimensions of the quay wall. Therefore, numerical analyses are conducted under the conditions shown in Table 2 to evaluate the effect of quay wall dimensions on the wall effect. In Table 2, the overall quay length indicates the total length of the mock quay, which is the same as the full longitudinal extent of the computational domain. Figure 1 shows two types of hull motions considered in this study. The left diagram represents straight-ahead run parallel to the quay wall, while the right diagram shows berthing and unberthing operations by lateral motion with the heading angle maintained parallel to the quay wall. This study evaluates the wall effects under these motion conditions using numerical analysis. Table 2. Computed cases. Case Direction of hull motion Water depth ([m]) Quay type Quay d epth [m] Quay length [m] Remarks 1 - 1 Parallel motion to the quay wall Deep (7.48) w/o wall - - Baseline w/o quay 1-2 Deep (7.48) Full depth wall 7.48 Overall Reference 1-3 AMB (4.5) Mock 1.0 Overall MQ depth variations 1-4 AMB (4.5) Mock 3.0 Overall MQ depth variations 1-5 AMB (4.5) Mock 2.0 20.0 Close to the experiment 2-1 Berthing and Unberthing with lateral motion AMB (4.5) w/o - - Baseline w/o quay 2-2 AMB (4.5) Full depth wall 4.5 Overall Reference 2-3 AMB (4.5) Mock 1.0 20.0 MQ depth variations 2-4 AMB (4.5) Mock 3.0 20.0 MQ depth variations MQ: Mock quay Figure 1. Hull motions near a quay wall, straight-ahead run parallel to the quay wall (left), lateral motion while maintaining the hull parallel to the quay wall (right). 2.3. Computational setup All numerical analyses are carried out on a model scale, dimensioned by 𝐿 = 3.740 [m]. The kinematic viscosity and the density of water are set to 𝜈 = 1.1386 × 10−6 [m2/s] and 𝜌 = 999.1026 [kg/m3], respectively, from ITTC Recommended Procedures [15]. For a turbulence model, EASM [13] model with wall functions is applied. Therefore, the minimum grid spacings on the surface of a hull, a quay wall and basin walls are set to 0.187 × 10−2 which satisfies 𝑦≅ 100 for the wall function boundary condition. The free surface is treated using a single-phase level set method, and the grid spacings in the vertical direction are clustered near the water surface. Numerical analyses are carried out using an unsteady condition with a time increment of ∆𝑡 = 0.05 [s]. To accurately evaluates the wall effects, the ship speed and the distance to the quay wall are set to stringent conditions such as berthing or unberthing operations. In the numerical analyses of the ship moving parallel to the quay wall, the ship speed is set to 0.333 [m/s], which is equivalent to 25% of the design speed in model scale. For lateral unberthing and berthing with the bow and stern parallel to the quay wall, the maximum unberthing and berthing speed is set to 0.133 [m/s], which is 10% of the design speed in model scale. During unberthing, the ship accelerates from a stationary position to its maximum velocity in 10 seconds. The ship speed increases, and the ship leaves from the quay. Similarly, during berthing, the ship accelerates from rest to its maximum speed in 10 seconds and moves. The ship slows down from its maximum speed in 10 seconds and stops near the quay wall. The dynamic overset grid method [11] is applied to update DCI at each computational step. In numerical analyses, the motion of the hull is prescribed in the surge direction for parallel motion cases, and in the sway direction for lateral motion cases. During the numerical analysis, all degrees of freedom except for the prescribed motion are fixed, as the induced motions are expected to be negligible under the low-speed condition. 4 The minimum distance between the ship and the quay wall Δ𝑏 is 1.2. Δ𝑏 is calculated by Δ𝑏 = 𝑏/(0.5𝐵), based on the ratio of the distance between the ship centerline and the quay wall, 𝑏. Figure 2 shows the position of the hull and quay wall. The distance between the quay wall surface and the port side surface of the hull is 0.054 [m] in model scale, and 5.1 [m] in actual scale. In the case of parallel, Δ𝑏 remains constant. For lateral motion, Δ𝑏 is the minimum value at the point of the closest approach to the quay wall. Figure 3 shows the CFD coordinate and the direction of the forces and moments. The CFD coordinates are righthanded, with coordinate X positive from bow to stern and coordinate Z positive upward. The origin is located at midship on the still water line. The longitudinal force is denoted as FX, the lateral force is denoted as FY, and the yaw moment is denoted as MZ. The yaw moment is calculated around the midship. The hull grid is generated for both sides. This hull grid is used in all simulations. Figure 4 shows the computational grid and boundary conditions for cases 1-1, 1-2, 1-3, and 1-5 in Table 2. The computational domain of case 1-2 is the y > 0 part of case 1-1. In case 1-2, the position of the minimum y in the basin grid corresponds to the quay wall surface (full depth wall). In cases 1-1 through 1-4, the hull grid is fixed, and a uniform flow velocity is given. Accordingly, a moving wall condition is applied as a boundary condition on the surface of the quay wall in cases 1-2, 1-3 and 1-4. In case 1-3 and 1-4, a part of walls and bottom of the experimental tank, AMB is reproduced in the computational grid. In case1-5, a section of the AMB is reproduced on the computational grid and the hull is navigated along the mock quay wall in an AMB. Figure 5 shows the computational grids and boundary conditions for cases 2-1, 2-2, and 2-4. The AMB grids for case 2-1, 2-3 and 2-4 are the same as in case 1-5. In case 2-2, the basin grid of case 1-2 is extended into the AMB shape to create DCI in the hull and basin grids. Figure 6 shows the enlarged view of the grids around the hull. In case 2-2, the number of grid points is increased around the hull in the basin grid from case 1-2. Table 3 shows the grid size and the cell number for each grid, and Table 4 shows the grid combinations for each calculation case. The purpose of these computations is to preliminarily investigate the overview of the forces and the flow field around the mock quay prior to its construction. Since it is not necessary to verify the accuracy for estimating the performance of the ship, the verification of grid dependency is not carried out. The grid dimension and number of grid points are determined by referring to the preceding overset computations [16]. Figure 2. Position of the hull and quay wall. Figure 3. The CFD coordinates and the direction of the force and moment. The longitudinal force is denoted as FX, the lateral force as FY, and the yaw moment as MZ. 5 Figure 4. Schematic view of the computational grids and boundary conditions for cases 1-1(top left), 1-2(top right), 1-3(bottom left) and 1-5(bottom right). Figure 5. Schematic view of the computational grids and boundary conditions for cases 2-1(top left), 2-2(top right) and 24(bottom) in the unberthing condition. X[m] -20 -15 -10 -5 0 5 10 15 20 Y[m] 0 5 10 15 Z [m] -4 -2 0 XY Z X[m] -20 -15 -10 -5 0 5 10 15 20 Y[m] 0 5 10 15 Z [m] -4 -2 0 XY Z 2-1 2-2 X[m] -20 -15 -10 -5 0 5 10 15 20 Y[m] 0 5 10 15 Z [m] -4 -2 0 XY Z 2-4 Wall of mock quay Wall of AMB Wall of hull Z symmetry Farfield Wall of AMBWall of AMB Wall of hull Wall of hull Z symmetry Z symmetry Farfield Farfield 6 Figure 6. Enlarged view of the computational grids and boundary conditions around the hull for cases 2-1(top), 2-2(bottom) in the unberthing condition. Table 3. Grid size and number of cells. Grid Grid size [m] Number of cells Hull -3.74 ≤ x ≤ 3.74, -1.65 ≤ y ≤ 1.65, - 1.65 ≤ z ≤ 0.374 1,032,192 Basin -7.48 ≤ x ≤ 11.22, -11.22 ≤ y ≤ 11.22, - 7.48 ≤ z ≤ 0.374 1,277,952 Basin with quay -7.48 ≤ x ≤ 11.22, 0.00 ≤ y ≤ 11.22, - 7.48 ≤ z ≤ 0.374 638,976 A part of AMB -10.0 ≤ x ≤ 10.0, 0.0 ≤ y ≤ 18.7, - 4.50 ≤ z ≤ 0.374 182,784 A section of AMB -20.0 ≤ x ≤ 20.0, 0.0 ≤ y ≤ 18.7, -4.50 ≤ z ≤ 0.374 (*case2 - 2, 4.0 ≤ y ≤ 18.7 ) 1,204,224 (*case2-2, 442,368) MQ, L=overall, d= 1.0m -10.0 ≤ x ≤ 10.0, 0.11 ≤ y ≤ 7.74, - 4.74 ≤ z ≤ 0.374 1,240,320 MQ, L=overall, d= 3.0m -10.0 ≤ x ≤ 10.0, 0.11 ≤ y ≤ 7.74, - 6.74 ≤ z ≤ 0.374 1,969,920 MQ, L=20.0m, d= 1.0m -13.8 ≤ x ≤ 13.8, 0.11 ≤ y ≤ 7.74, - 4.74 ≤ z ≤ 0.374 1,452,480 MQ, L=20.0m, d= 2.0m -13.8 ≤ x ≤ 13.8, 0.11 ≤ y ≤ 7.74, - 5.74 ≤ z ≤ 0.374 1,875,456 MQ, L=20.0m, d= 3.0m -13.8 ≤ x ≤ 13.8, 0.11 ≤ y ≤ 7.74, - 6.74 ≤ z ≤ 0.374 2,301,696 MQ: Mock quay XY Z 2-1 Wall of hull Wall of AMB Z symmetry XY Z 2-2 Wall of hull Wall of AMB Z symmetry 7 Table 4. Grid combinations for each calculation case. Grid Case 1 - 1 1 - 2 1 - 3 1 - 4 1 - 5 2 - 1 2 - 2 2 - 3 2 - 4 Hull ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ Basin ✓ Basin with quay ✓ A part of AMB ✓ ✓ A section of AMB ✓ ✓ ✓* ✓ ✓ MQ, L=overall, d= 1.0m ✓ MQ, L=overall, d= 3.0m ✓ MQ, L=20.0, d= 1.0m ✓ MQ, L=20.0, d= 2.0m ✓ MQ, L=20.0, d= 3.0m ✓ MQ: Mock quay *In case 2-2, AMB grid is modified for generating DCI. 3. Results and Discussions 3.1. Parallel motion to the quay wall Numerical analyses are performed to investigate the influence of the wall effect on the hydrodynamic forces acting on a hull advancing along a straight path parallel to the quay wall. The analyses focused on three conditions: without a quay wall (case 1-1), with a quay wall (case 1-2), and with mock quay walls of varying submerged depths (cases 1-3 to 1-5). A comparison between case 1-1 and case 1-2 (Table 5) revealed that the presence of the quay wall increases the longitudinal force (FX), lateral force (FY), and yaw moment (MZ). This phenomenon can be attributed to the wall effect, wherein the proximity of the hull to the wall generates asymmetric pressure fields around the hull. The direction of the forces agrees with previous experimental and theoretical findings: a lateral force acting toward the quay and a yaw moment that causes the bow to turn outward (bow-out moment), both of which are reproduced by numerical analysis. Figure 7 shows the hull surface pressure distributions divided by density. for case1-1 and case1-2. The results of case 1-2 show that a large negative pressure area is generated on the port side. Table 5. Comparison of hydrodynamic forces acting on the hull. Case F X [N] F Y [N] M Z [Nm] 1 - 1 0.6 38 0.0 0 0 0.00 0 1 - 2 0.6 66 - 0.53 3 - 0.27 2 Figure 7. Comparison of hull surface pressure distribution divided by density for cases 1-1 and 1-2, port side(top) and starboard side(bottom). 8 To further examine the role of wall depth, additional simulations are conducted with varying depths of the quay wall. Table 6 shows the relative magnitudes of the hydrodynamic forces with respect to case 1-2. Case 1-5 is the average of the computed values in a stable range, about 30 seconds in real time. Comprehensive results for additional quay depth configurations can be found in [9]. In case 1-3, where the wall depth is shallow, FY and MZ decrease. In contrast, cases 1-4 and 1-5 exhibit hydrodynamic forces nearly equivalent to those in case 1-2, indicating that the effect of the wall persists when the submerged depth is sufficient. Figure 8 shows the pressure distribution divided by density on the XY cross-section at the midship position of case 1-2, 1-3 and 1-4. In case 1-3, the pressure distribution extends down to the bottom edge of the mock quay wall, and the pattern around the hull differs significantly from those observed in other cases. This deviation is considered to be the primary reason why FY in case 1-3 is smaller than in the other cases. In contrast, the pressure distribution in case 1-4 is similar to case 1-2, including near the hull and around the mock quay. These results suggest that numerical analyses are capable of qualitatively capturing the wall effect and the associated hydrodynamic forces acting on a hull navigating near a quay wall. The magnitude of these forces appears to be sensitive to the quay wall depth, which affects the degree of flow blockage and the resulting asymmetry in the pressure distribution around the hull. Table 6. Comparison of the ratio of hydrodynamic forces acting on the hull for case 1-2. Ratio to c ase 1 - 2 Case F X F Y M Z 1 - 3 0.997 0.978 0.979 1 - 4 1.001 0.993 0.997 1 - 5 * 0.999 0.994 1.003 *Average of about 30 seconds of stable range Figure 8. Comparison of the pressure distribution divided by density around the hull and the mock quay at midship for cases 1-2, 1-3 and 1-4 (Δ𝑝/𝜌 = -0.0005). 3.2. Berthing and unberthing with lateral motion while maintaining the hull parallel to the quay wall In order to investigate the wall effects on the navigation of the hull different from the subsection 3.1, numerical analyses with lateral motion of the hull are performed with no quay (case 2-1), with a deep full depth wall quay (case 2-2) and with different mock quay depths (cases 2-3 and 2-4). Figure 9 and Figure 10 show the position of the hull and the histories of FY and MZ acting on the hull. Figure 9 shows the result in unberthing conditions, and Figure 10 shows the result in berthing conditions. Figure 9 shows that in the presence of the quay wall (case 2-2 to 2-4), the lateral force acting on the hull is larger, and the yaw moment has a smaller initial peak than without the quay wall (case 2-1). This is due to the wall effect. The FY is larger due to the mutual interference between the hull and the wall. In contrast, the turning moment 9 is smaller due to the lateral force. On the other hand, in the case without the quay wall, the yaw moment acts in the direction of turning the bow to port. When the quay wall is present, however, the yaw moment becomes smaller, primarily due to the influence of the increased lateral force and the induced bow-out moment, resulting from the interaction between the hull and the quay wall. A similar phenomenon is observed during berthing in Figure 10. Also, the hydrodynamic forces acting on the hull are larger when the hull is unberthing than when it is berthing to the quay. This can be attributed to two factors. First, the higher relative velocity between the hull and the initially stationary fluid. Secondly, the generation of stronger added mass effects during acceleration. Conversely, during the berthing process, the fluid around the ship is already in motion, and deceleration gradually reduces these effects, resulting in lower hydrodynamic forces. For the mock quays (case 2-3, 2-4), the lateral force and yaw moment acting on the hull are nearly identical under both berthing and unberthing conditions. Additional variations of quay depth, along with the corresponding results, are provided in detail in [10]. Compared to the case2-2, with a sufficiently deep full depth wall quay, case 2-3 and 2-4 show the larger FY, and the oscillatory time histories. The present study further investigates the amplification of these forces and elucidates the mechanisms responsible for the observed oscillations. Figure 9. Position of the hull(top) and the time histories of Y-direction(center), lateral force FY(bottom left) and yaw moment MZ(bottom right) during unberthing for cases 2-1 to 2-4. X Y Z Quay wall t = 8.0 t = 6.0 t = 4.0 t = 2.0 t = 10.0 t = 0.0 Time: t [s] / /// /// ///////// time[s] Port-side clearance [m] 0 1 2 3 4 5 6 7 8 9 10 11 12 0.0 0.0 0.1 0.1 0.2 0.2 0.3 0.3 0.4 0.4 0.5 0.5 0.6 0.6 0.7 0.7 0.8 0.8 0.9 0.9 1.0 1.0 time[s] F Y [N] 0 1 2 3 4 5 6 7 8 9 10 11 12 -7.0 -7.0 -6.0 -6.0 -5.0 -5.0 -4.0 -4.0 -3.0 -3.0 -2.0 -2.0 -1.0 -1.0 0.0 0.0 2-1 2-2 2-3 2-4 time[s] M Z [Nm] 0 1 2 3 4 5 6 7 8 9 10 11 12 0.0 0.0 1.0 1.0 2.0 2.0 2-1 2-2 2-3 2-4 16 [10] M. Hirota, Y. Kitagawa and H. Kobayashi. CFD analysis of bank wall effects on a ship berthing and unberthing near a quay wall (in Japanese), Conference proceedings, the Japan Society of Naval Architects and Ocean Engineers, Japan, Vol. 40, 2025. [11] H. Kobayashi and Y. Kodama. 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