CFD-based Determination of Load Cell Capacity for Submarine HPMM Model Tests
https://doi.org/10.1007/s11804-025-00637-0
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Abstract
Captive model tests are one of the most common methods to calculate the maneuvering hydrodynamic coefficients and characteristics of surface and underwater vehicles. Considerable attention must be paid to selecting and designing the most suitable laboratory equipment for towing tanks. A computational fluid dynamics (CFD) -based method is implemented to determine the loads acting on the towing facility of the submarine model. A reversed topology is also used to ensure the appropriateness of the load cells in the developed method. In this study, the numerical simulations were evaluated using the experimental results of the SUBOFF benchmark submarine model of the Defence Advanced Research Projects Agency. The maximum and minimum loads acting on the 2.5-meter submarine model were measured by determining the body's lightest and heaviest maneuvering test scenarios. In addition to having sufficient endurance against high loads, the precision in measuring the light load was also investigated. The horizontal planar motion mechanism (HPMM) facilities in the National Iranian Marine Laboratory were developed by locating the load cells inside the submarine model. The results were presented as a case study. A numerical-based method was developed to obtain the appropriate load measurement facilities. Load cells of HPMM test basins can be selected by following the two-way procedure presented in this study.Article Highlights● The maximum and minimum loads acting on the 2.5-meter submarine model were measured by determining the body's lightest and heaviest maneuvering test scenarios.● The horizontal planar motion mechanism (HPMM) facilities in the National Iranian Marine Laboratory were developed by locating the load cells inside the submarine model.● Load cells of HPMM test basins can be selected by following the two-way procedure presented in this study. -
1 Introduction
With the development of the marine industry, high-precision instruments are fundamental in designing and analyzing marine vehicles in marine laboratories. Maneuverability is a major factor affecting the performance of marine underwater vehicles. Maneuver simulations based on mathematical models are widely used to predict motion parameters. A key stage in applying a mathematical model is the determination of the hydrodynamic coefficients. Captive model tests are well-known methods for calculating the hydrodynamic coefficients of ships and submarines. Owing to the port-starboard symmetry of the conventional marine vehicles, dependence on some movements is reduced, and the body's movement can be studied separately in two horizontal and vertical planes. Captive model tests are currently the most reliable method to obtain the maneuvering hydrodynamic coefficients of marine vehicles. In these tests, the body movement is restrained to the carriage mechanism by one or more arms. Owing to the high sensitivity of forces in light tests, the laboratory should be equipped with highprecision equipment that can cover the high forces in heavy-duty tests. The correct selection of load cells and the design of the arm structure require the prediction of the forces with the highest possible accuracy.
Many researchers have conducted numerical and experimental studies on submarine maneuvers. Owing to the high costs of the model tests, numerical methods are considered an appropriate method for estimating the hydrodynamic derivatives and loads acting on the model in the initial state of design. Computational fluid dynamics (CFD) has provided a wide environment for various hydrodynamic analyses and new methods, such as dynamic tests conducted (Ziabari and Mousavizadegan, 2024). Therefore, the current study prefers it over other numerical methods. To date, less attention has been paid to the design process of test mechanisms in towing tanks. The hydrodynamic coefficients of the SUBOFF body was investigated numerically using the finite volume method to solve Reynolds Average Navier– Stokes (RANS) equations (Pan et al., 2012). numerically simulated the captive model tests in horizontal and vertical planes to investigate the maneuvering derivatives of a submarine with cross-shaped fins. Except for surge motion, all the body movements were defined in an internal region near the body surface to precisely simulate the planar motions (Cho et al., 2020). The effects of side walls was investigated in circulating flow channels and strut arms on the hydrodynamic coefficients of an autonomous submarine. Dimensionless width parameters were used to reduce the dependence between channel width and hydrodynamic coefficients of the model. The results demonstrated that the hydrodynamic coefficients increased with the tank width (Huang et al., 2020). The velocity and amplitude of the body motion in planar motion mechanism (PMM) tests for an ellipse numerically investigated (Ardeshiri et al., 2020). Furthermore, several numerical and experimental approaches were adopted to generate a captive model for surface vessels in 3-DOF (Yoon et al., 2015a; Yoon et al., 2015b; Zhu et al., 2022).
PMM tests are classified into two main categories, namely, horizontal planar motion mechanism (HPMM) and vertical plane motion mechanisms (VPMM). Considering the facilities of the National Iranian Marine Laboratory (NIMALA), this study investigated the horizontal planar motion mechanism. The HPMM mechanism must be able to move the model in three degrees of freedom and measure the moment and forces acting on the model's hull. In the HPMM of David Taylor's laboratory, two upper arms were tied to the model to provide translational and rotational motions (Roddy, 1990). Another study use model binding with two arms and two double-component load cells (Kim, 2023). Also a research, used a strut stretched to the model tail to carry the model, in which the moment and forces were measured by a three-component load cell (Park et al., 2017).
The equipment for HPMM tests should be selected based on laboratory specifications, such as tank dimensions and towing speed. Given that heavy and light loads are measured by load cells during the HPMM tests in towing tanks, improper estimation of the force values may cause untended consequences affecting the accuracy of the test results. Selecting too low-capacity load cells will cause damage to the facilities. However, if the selected load cells are too high-capacity, the accuracy of the load measurement in the light tests may decrease.
This topology offers a proper estimation of the hydrodynamic loads acting on the model's body according to the specified test scenarios. The whole test process is simulated in the CFD numerical environment to consider the specifications of the laboratory facilities and the probable bodies to be tested. Some of the benefits of using CFD in this work include its precision in the direct calculation of the loads and its consideration of the special geometry and various test conditions of the laboratory. Therefore, it is preferred over common empirical estimation methods.
Owing to the lack of a specific guideline for the HPMM load cell selection in marine laboratories, the instructions presented in this paper can be followed for similar works.
In this study, the HPMM mechanism is selected by considering the predicted loads obtained from the CFD simulations. The load cells are controlled in terms of force tolerance and sufficient accuracy using the criteria calculated by numerical analysis and those set by the manufacturer of the load cells. A mechanism in which the load cells are located inside the body is designed to remove the hydrodynamic forces of the carriage arms. Analysis is performed to predict whether the force on the load cells originates only from the hydrodynamic forces acting on the body.
2 Governing equations
2.1 Equations of motion
In general, the motion equations of a marine vehicle can be expressed in six degrees of freedom. Two coordinate systems are defined for the vessel, namely, the earth-fixed coordinate system and the body-fixed coordinate system that is stuck to the body and always moves with it. These two coordinate systems are shown in Figure 1. The equations of motion in the body-fixed coordinate system are as follows (Fossen, 1994):
$$ \boldsymbol{M}_{\mathrm{RB}} \dot{\boldsymbol{v}}+\boldsymbol{C}_{\mathrm{RB}} \boldsymbol{v}=\boldsymbol{\tau}_{\mathrm{RB}} $$ (1) 
Figure 1 Earth-fixed and body-fixed coordinate systemswhere MRB is the inertial matrix, and CRB is the Coriolis matrix of the body. The values of these two matrices depend on the body's weight distribution. v = {u, v, w, p, q, r}T is the velocity vector, in which the first three terms are translational velocities in the direction of the x, y, and z axes, and the second three terms are the respective rotational velocities. $ \dot{\boldsymbol{v}}=\{\dot{u}, \dot{v}, \dot{w}, \dot{p}, \dot{q}, \dot{r}\}^{\mathrm{T}}$ is the vector of translational and rotational accelerations. τRB is the external force and moment matrix including hydrostatic and hydrodynamic forces.
Under the following assumptions, the hydrodynamic forces and moments of the body due to the movement speed and acceleration under the water can be separated from the other external forces.
• The submarine moves deep enough that the free surface does not affect its forces and moments.
• Motion is only provided by the propulsion system.
• Control surfaces are in equilibrium condition and do not change the direction of motion.
Under the above assumptions, the matrix of external forces and moments is equal to that of force and hydrodynamic moments as follows:
$$ \boldsymbol{\tau}_{\mathrm{RB}}=\boldsymbol{\tau}_h=\{X, Y, Z, K, M, N\}^{\mathrm{T}} $$ (2) Therefore, X, Y, and Z are the external forces acting on the body in the x (surge), y (sway), and z (heave) directions, respectively; K, M, and N are the external moments in ϕ (roll), θ (pitch), and ψ (yaw), respectively. expressing these forces and moments using velocities and hydrodynamic coefficients is as the following forms (Getler and Hagen, 1967):
$$ \begin{aligned} X= & \frac{1}{2} \rho l^4\left(X_{q q}^{\prime} q^2+X_{r r}^{\prime} r^2+X_{r p}^{\prime} r p\right)+ \\ & \frac{1}{2} \rho l^3\left(X_{\dot{u}}^{\prime} \dot{u}+X_{v r}^{\prime} v r+X_{w q}^{\prime} w q\right)+ \\ & \frac{1}{2} \rho l^2\left(X_{u u}^{\prime} u^2+X_{v v}^{\prime} r^2+X_{w w}^{\prime} w^2\right) \end{aligned} $$ (3) $$ \begin{aligned} Y= & \frac{1}{2} \rho l^4\left(Y_{\dot{r}}^{\prime} \dot{r}+Y_{\dot{p}}^{\prime} \dot{p}+Y_{p|p|}^{\prime} p|p|+Y_{q r}^{\prime} q r\right)+ \\ & \frac{1}{2} \rho l^3\left(Y_{\dot{v}}^{\prime} \dot{v}+Y_{v q}^{\prime} v q+Y_{w p}^{\prime} w p+Y_{w r}^{\prime} w r+Y_r^{\prime} u r+\right. \\ & \left.Y_p^{\prime} u p+Y_{v|r|}^{\prime} \frac{v}{v|r|}\left|\left(v^2+w^2\right)^{\frac{1}{2}}\right||r|\right)+ \\ & \frac{1}{2} \rho l^2\left(Y_*^{\prime} u^2+Y_v^{\prime} u v+Y_{v|v|}^{\prime} v\left|\left(v^2+w^2\right)^{\frac{1}{2}}\right|+Y_{v w}^{\prime} v w\right) \end{aligned} $$ (4) $$ \begin{aligned} Z= & \frac{1}{2} \rho l^4\left(Z_{\dot{q}}^{\prime} \dot{q}+Z_{p p}^{\prime} p^2+Z_{r r}^{\prime} r^2+Z_{r p}^{\prime} r p\right)+\frac{1}{2} \rho l^3\left(Z_{\dot{w}}^{\prime} \dot{w}+\right. \\ & \left.Z_{v r}^{\prime} v r+Z_{v p}^{\prime} v p+Z_q^{\prime} u q+Z_{w|q|}^{\prime} \frac{w}{|w|}\left|\left(v^2+w^2\right)^{\frac{1}{2}}\right||q|\right)+ \\ & \frac{1}{2} \rho l^2\left(Z_*^{\prime} u^2+Z_v^{\prime} u v+Z_w^{\prime} u w+Z_{w|w|}^{\prime}\left|w\left(v^2+w^2\right)^{\frac{1}{2}}\right|+Z_{v v}^{\prime} v^2\right) \end{aligned} $$ (5) $$ \begin{aligned} K= & \frac{1}{2} \rho l^5\left(K_{\dot{p}}^{\prime} \dot{p}+K_{\dot{r}}^{\prime} \dot{r}+K_{q r}^{\prime} q r+K_{p q}^{\prime} p q+K_{p|p|}^{\prime} p|p|\right)+ \\ & \frac{1}{2} \rho l^4\left(K_{\dot{v}}^{\prime} \dot{v}+K_{v q}^{\prime} v q+K_{w p}^{\prime} w p+K_{w r}^{\prime} w r\right)+ \\ & \frac{1}{2} \rho l^3\left(K_*^{\prime} u^2+K_v^{\prime} u v+K_{v|v|}^{\prime}\left|v\left(v^2+w^2\right)^{\frac{1}{2}}\right|+K_{v w}^{\prime} v w\right) \end{aligned} $$ (6) $$ \begin{aligned} M= & \frac{1}{2} \rho l^5\left(M_{\dot{q}}^{\prime} \dot{q}+M_{p p}^{\prime} p^2+M_{r r}^{\prime} r^2+M_{r p}^{\prime} r p+\right. \\ & \left.M_{p \mid p}^{\prime} p|p|\right)+\frac{1}{2} \rho l^4\left(M_{\dot{w}}^{\prime} \dot{w}+M_{v r}^{\prime} v r+M_{v p}^{\prime} v p+\right. \\ & \left.M_q^{\prime} u q+M_{q|w|}^{\prime}\left|\left(v^2+w^2\right)^{\frac{1}{2}}\right| q\right)+ \\ & \frac{1}{2} \rho l^3\left(M_*^{\prime} u^2+M_w^{\prime} u w+M_{w|w|}^{\prime} w\left|\left(v^2+w^2\right)^{\frac{1}{2}}\right|+\right. \\ & \left.M_{|w|}^{\prime} u|w|+M_{w w}^{\prime}\left|w\left(v^2+w^2\right)^{\frac{1}{2}}\right|+M_{v v}^{\prime} v^2\right) \end{aligned} $$ (7) $$ \begin{aligned} N= & \frac{1}{2} \rho l^5\left(N_{\dot{r}}^{\prime} \dot{r}+N_{\dot{p}}^{\prime} \dot{p}+N_{p q}^{\prime} p q+N_{q r}^{\prime} q r+N_{r|r|}^{\prime} r|r|\right)+ \\ & \frac{1}{2} \rho l^4\left(N_{\dot{v}}^{\prime} \dot{v}+N_{w r}^{\prime} w r+N_{w p}^{\prime} w p+N_{v q}^{\prime} v q+N_p^{\prime} u p+\right. \\ & \left.N_r^{\prime} u r+N_{v|r|}^{\prime}\left|\left(v^2+w^2\right)^{\frac{1}{2}}\right| r\right)+\frac{1}{2} \rho l^3\left(N_*^{\prime} u^2+\right. \\ & \left.N_v^{\prime} u v+N_{v|v|}^{\prime} v\left|\left(v^2+w^2\right)^{\frac{1}{2}}\right|+N_{v w}^{\prime} v w\right) \end{aligned} $$ (8) where m expresses the mass, and Ix, Iy, and Iz are the inertia moment of each axis. The translational velocities in the directions of surge, sway, and heave are expressed by u, v, and w, respectively, and the angular velocities around these axes are expressed by p, q, and r, respectively. Translational and rotational accelerations are displayed in the formulas with the derivative sign on the velocity components. $ Y_{\dot{r}}^{\prime}$, $ X_{\dot{u}}^{\prime}, Z_{\dot{q}}^{\prime}$, … are the added mass hydrodynamic coefficients, and X'rr, M'p|p|, N'p, … are the damping hydrodynamic coefficients presented in a non-dimensional form.
2.2 Fluid dynamic equations
The governing equations are the conservation of mass and Navier–Stokes equations in three dimensions, which can be expressed as RANS equations by assuming viscosity and incompressibility. RANS equations are used to simulate the fluid flow around a submarine body in CFD for calculating force and torque. The variables of velocity and pressure are divided into average and fluctuating terms. RANS equations are expressed as follows (Ferizer and Peric, 2002):
$$ \begin{aligned} \frac{\partial\left(\rho \overline{u_i}\right)}{\partial t}+ & \frac{\partial}{\partial x_j}\left(\rho \overline{u_i} \overline{u_j}+\rho \overline{u_i^{\prime} u_j^{\prime}}\right)= \\ & \frac{\partial \bar{p}}{\partial x_i}+\rho g_i+\mu \frac{\partial}{\partial x_j}\left(\frac{\partial \overline{u_j}}{\partial x_i}+\frac{\partial \overline{u_i}}{\partial x_j}\right) \end{aligned} $$ (9) $$ \frac{\partial\left(\rho \bar{u}_i\right)}{\partial x_i}=0 $$ (10) where μ, ρ, and p represent viscosity, density, and timedaverage pressure, respectively. In Equation (10), $ \tau_{i j}=\overline{u_i^{\prime} u_j^{\prime}}$ is the Reynolds stress term that includes the effects of turbulence on the average stress. In this study, the two equations model k-ε is chosen to solve the Reynolds stress equation (Shih et al., 1994).
3 Verification
The captive model test is applied to calculate the hydrodynamic coefficients of damping and added mass on the SUBOFF bare hull body with 4.35 m length and 0.508 m diameter (Groves et al., 1989). The simulations are performed in static and dynamic conditions. For checking the precision of the numerical solution, the static drift test is selected among the static tests and the pure yaw test is selected among the dynamic tests.
3.1 Computational domain and mesh study
For the numerical solution, creating a computational domain around the body and defining its boundary conditions are necessary. Given that the selected scenarios for verification are based on the tests conducted in David Taylor's laboratory, the width of the computing domain is also adopted from the same tests. The computational domain and boundary conditions of the numerical solution are specified in Figure 2. The same computational domain is specified for the static and dynamic test simulations.
Figure 2 Static drift test boundary conditionsSimulations in CFD are highly dependent on the generated mesh. This study uses the trimmer mesh produced by Star CCM+, as shown in Figure 3. In the static drift test simulation, which is in steady-state, the domain mesh is fixed, and the forward movement of the model is simulated by the inlet flow. However, in the pure yaw test, a dynamic mesh in which the whole computational domain moves within the body movements is used due to the submarine motion. In the dynamic mesh simulation, the generated mesh is modified during the simulation to accommodate the geometrical changes, such as translating and rotating boundaries. Compared with selecting a fixed domain and moving the body surface, this procedure dramatically reduces the calculation time.
Figure 3 DARPA SUBOFF maneuver simulation gridTable 1 Summary of numerical setupSetting Description Time Implicit unsteady Time step (for pure yaw) (s) 0.005 Iteration per time step 5 Flow Segregated flow Equation of state Constant density Turbulence model Realizable K-Epsilon Pressure link SIMPLE Verification must be performed to ensure that appropriate results are calculated from the numerical simulations. In this work, a mesh convergence study is applied to verify the numerical simulations (ITTC, 2017). Three different meshes with a ratio of $ \sqrt{2}$ are considered. The force in the longitudinal and transverse directions and the yaw moment at the drift angle of 12° are calculated in the three mentioned states. In Figure 4, the results of the numerical simulations are shown in different meshing modes for the calculated forces and moments. Starting from the second case, increasing the number of grids does not significantly affect the forces and moments, and the solution is independent of the number of grids.
Figure 4 Mesh independency for surge and sway forceIn this study, the solver results and the quality of the grid are evaluated by the grid convergence index (GCI), which estimates the discretization error. GCI is defined as follows (Celik et al., 2008):
$$ \mathrm{GCI}_{i, i+1}=\frac{F_s\left|e_{i, i+1}\right|}{r_{i, i+1}^p-1} $$ (11) where Fs is the safety factor equal to 1.25. For checking the convergence of more than two states, ei, i+1 shows the relative error between two states, and r is the correction factor. The convergence ratio and discretization order are indicated by R and P, respectively.
The results for the grid formed for the numerical solution of the static drift test at the drift angle of 12° are shown in Table 2. According to the values of GCI and R, the convergence criteria for medium and fine meshing are acceptable, and the numerical solution is independent of the number of meshes. A grid system with a number of 2.6 million cells is used for the continuation of the simulations of the SUBOFF body maneuver. The steady run used for the static drift test has taken 4.15, 8.89, and 29.74 h at the 1 500th iteration for the course, medium, and fine grids, respectively.
Table 2 Values of mesh independency analysis parameters based on the roache methodParameter Sway force Yaw moment Surge force N1 (Coarse) 1 060 150 - - N2 (Medium) 2 633 531 - - N2 (Fine) 6 491 859 - - ϕ1 289.773 -1 053.30 -95.056 ϕ2 288.27 -1 054.784 -93.84 ϕ3 287.396 -1 055.51 -93.27 P 0.588 76 0.770 28 0.826 44 R 0.581 5 0.493 2 0.468 8 GCI (coarse-medium) 0.919 65% 0.172 71% 1.444 7% GCI (medium-fine) 0.542 32% 0.086 13% 0.689 59% Owing to the boundary conditions of the no-slip wall for the floating body, the fluid flow near the body is strongly affected and causes large changes in the fluid velocity. Therefore, the wall function and prismatic layered grid are used to accurately simulate the flow near the wall. The value of the prismatic layered network components is determined based on y+ according to Equation (12).
$$ y^{+}=\frac{y \cdot u^*}{v} $$ (12) where y is the distance of the first node to the wall, ν is the dynamic viscosity, and u* is the characteristic velocity of the turbulent flow. According to ITTC recommendation, the value of y+ in turbulent flows should be between 30 and 300 (ITTC, 2014). The value of y+ at the drift angle of 12° for the SUBOFF bare hull is shown in Figure 5. Most parts of the body surfaces are in the log layer zone with y+ value more than 30. The all-y+ treatment is applied to the numerical solver in STAR CCM+ to eliminate the concern of the parts with y+ values in the buffer region (Siemens, 2021). This treatment is suitable for a wide range of boundary layer mesh densities, including the instance when the wall-cell centroid falls in the buffer region of the boundary layer.
Figure 5 y+ distribution on SUBOFF body in static drift test3.2 Calculation of hydrodynamic coefficients
3.2.1 Static drift
The static drift tests and pure yaw of SUBOFF in the simulations are according to the tests conducted in David Taylor's laboratory. In its static drift test, the model is pulled forward by the carriage at a speed of 6.5 kn, and the submarine body is connected to the carriage's arms from -18 to +18. In this study, static drift simulations are performed for the drift angles of - 16° to +16° for the bare hull of the SUBOFF model. The nondimension form of force and torque are as follows (Feldman, 1979):
$$ N^{\prime}=\frac{N}{0.5 \rho L^3 U^2} $$ (13) $$ Y^{\prime}=\frac{Y}{0.5 \rho L^2 U^2} $$ (14) By knowing the drift angle in each simulation and the relation of the sway velocity according to the drift angle, the sway velocity can be calculated using Equation (15).
$$ v=U \sin \beta $$ (15) The numerical simulation results are shown as black dots in Figure 6. The most optimal curve from the numerical results must be crossed to obtain the hydrodynamic coefficients according to the following equation:
$$ y=a x+b x^3 $$ (16) 
Figure 6 Numerical simulation results and fitted charts in static drift testwhere y is the dimensionless sway force or dimensionless yaw moment, a is the linear hydrodynamic coefficient (Y'v or N'v), and b is the nonlinear hydrodynamic coefficient (Y'vvv or N'vvv). The graphs are fitted using the curve fitting tool in MATLAB. The hydrodynamic coefficients obtained from the analysis of computational fluid dynamics results are compared with the experimental results in Table 3.
Table 3 Hydrodynamic coefficients obtained from static drift tests using numerical and experimental approachesCoefficient CFD EFD Error (%) Y'v -0.006 128 -0.005 948 3.03 N'v -0.013 61 -0.012 795 6.37 Y'vvv -0.165 6 - - N'vvv 0.037 73 - - 3.2.2 Pure yaw
In obtaining the hydrodynamic coefficients related to yaw movement, the submarine transverse motions must be limited and its head angle must always be tangent to the path. The pure yaw motion is applied to the model and computational domain using the following relations:
$$ \psi=-\psi_0 \cos (2 \pi \omega t) $$ (17) $$ r=\psi_0(2 \pi \omega) \sin (2 \pi \omega t) $$ (18) $$ \dot{r}=\psi_0(2 \pi \omega)^2 \sin (2 \pi \omega t) $$ (19) where ψ0 is the maximum heading angle, ψ is the yaw angle, and ω is the frequency of yaw motion. The pure yaw reference test is performed at a fixed motion frequency of 0.353 Hz and forward speed of 4.5, 5, 6, and 6.5 kn. In this study, only the forward speed of 6.5 kn is investigated. One of the methods for calculating hydrodynamic coefficients from the force and torque output diagrams of dynamic maneuver tests is to apply the linear regression method. The calculation results at each time step are recorded in CFD. By knowing the test scenario, the force, speed, and acceleration matrices can be written as follows (Foroushani and Sabzpooshani, 2021):
$$ \underbrace{\left[\begin{array}{c} \theta_1 \\ \theta_2 \\ \vdots \\ \theta_n \end{array}\right]}_{\boldsymbol{\theta}}=\underbrace{\left[\begin{array}{cccccc} u_1 & \cdots & r_1 & \dot{u}_1 & \cdots & \dot{r}_1 \\ u_2 & \cdots & r_2 & \dot{u}_2 & \cdots & \dot{r}_2 \\ \vdots & & \vdots & \vdots & & \vdots \\ u_n & \cdots & r_n & \dot{u}_n & \cdots & \dot{r}_n \end{array}\right]}_{\boldsymbol{\varphi}} \underbrace{\left[\begin{array}{c} \psi_n \\ \vdots \\ \psi_r \\ \psi_{\dot{u}} \\ \vdots \\ \psi_{\dot{r}} \end{array}\right]}_{\boldsymbol{\psi}} $$ (20) where each row of the θ matrix shows the amount of force or torque in one time step of the solution. n is the number of data points extracted from numerical simulation. The more data points, the more accurate the curve fitting and the extraction of hydrodynamic coefficients. Increasing the number of data points may increase the calculation time without significantly affecting the results. The matrix φ expresses the speed and accelerations at each time step, and ψ is the matrix of hydrodynamic coefficients. The matrix of hydrodynamic coefficients can be calculated as follows using the inverse and determinant of the velocity and acceleration matrix:
$$ \boldsymbol{\psi}=\left(\boldsymbol{\varphi}^{\mathrm{T}} \boldsymbol{\varphi}\right)^{-1} \boldsymbol{\varphi}^{\mathrm{T}} \boldsymbol{\theta} $$ (21) The optimal diagram fitted to the force and moment points obtained from the numerical simulation in the pure yaw test is shown in Figure 7. Hydrodynamic coefficients related to yaw motion are also described in Table 4.
Figure 7 Diagram of the fitted curves and the points obtained from the pure yaw numerical simulationTable 4 Hydrodynamic coefficients obtained from pure yaw test in numerical simulation and laboratory experimentCoefficient CFD EFD Error (%) Y'r 0.001 03 0.001 811 43.13 N'r -0.001 13 -0.001 597 29.24 $ Y_{\dot{r}}^{\prime}$ 0.000 258 0.000 060 330 $ N_{\dot{r}}^{\prime}$ -0.000 76 -0.000 676 12.43 4 Determining the capacity and type of load cells
Test scenario for calculating maximum forces. The submarine model for testing in the NIMALA must have dimensions suitable for the tank dimensions. The goal of this study is to select the most appropriate load cells suitable for this facility. The first step of load cell selection is modeling a submarine model with the largest allowable dimensions from a hydrodynamic point of view. The main specifications of the NIMALA towing tank are given in Table 5. ITTC recommendations (ITTC, 2021) stipulate the framework of the HPMM tests, including the model dimensions and test scenario details.
Table 5 Characteristics of the National Iranian Persian Gulf Towing TankParameter Value Length (m) 400 Width (m) 6 Depth (m) 4 Density of water (kg/m3) 1 002 Kinematic viscosity of water 9.75x10-7 Maximum velocity of the carrier (m/s) 18 The free surface effect is one of the effects that should be avoided in the test. With the distance set as five times the diameter from the top of the submarine model from the water surface, the distance of the top of the submarine model from the surface should be 1.9 m. This value is the lowest allowed depth and makes the floor effects negligible. The geometric characteristics of the submarine model determined for the analysis are described in Table 6.
Table 6 Geometric characteristics of the tested submarine bodyQuantity Value Length (m) 2.5 Diameter (m) 0.254 Number of control surfaces 4 LCG (m) 1.428 68 Mass (kg) 100.62 Izz (kg·m2) 34.702 4 The last limitation of the laboratory environment is the speed limit. According to ITTC instructions, the maximum speed should be less than the $\sqrt{g h} / 2 $ to remove the effects of the wall and the floor. Therefore, with the depth of immersion being known, the selected speed should be less than 2.27 m/s. In this work, the forward speed selected for the tests is 2.25 m/s. According to the ITTC instructions, the ratio of the length of the model to the width of the tank should be less than 0.45 to eliminate the effect of the wall in harmonic tests. Considering the 6-meter width of the tank, the 2.5-meter model is selected.
The maximum forces that may be applied to the selected load cells during various tests should be predicted using numerical simulation. The heaviest scenario for the submarine maneuver test in the NIMALA is the combined yaw and drift test at the frequency of 0.35 Hz and the maximum drift angle of 30°. This test scenario is arranged according to some the criteria to determine the exact capacity of load cells using numerical analysis in CFD (Vantorre, 2000). The debatable point at this stage is determining the selection process of load cells. Two two-component load cells are selected according to the flowchart in Figure 8.
Figure 8 Flowchart of load cell selection4.1 Numerical results
4.1.1 Hydrodynamic forces acting on the body
Given that all the linear and nonlinear forces and moments are present in the acting loads to the load cells, the estimation method must consist of nonlinear, cross-coupling, and added mass terms. Several empirical and analytical approaches can be employed for calculating the hydrodynamic coefficients of the submarine model, and each of them can estimate a limited set of hydrodynamic coefficients. However, some of the cross-coupling terms will be unestimated, even by combining different methods (Holmes and Papoulias, 1995; Kepler, 2018; Humphreys and Watkinson, 1978). Furthermore, the full appendage model submarine model of this study is different from other benchmark models. Therefore, these methods are not sufficiently precise to calculate the forces and moments acting on the body in the drift–yaw test. In the present study, the loads acting on each load cell are directly calculated by simulating the test scenarios in CFD.
The results of the numerical analysis related to the static drift test, pure sway, pure yaw, and combined drift–yaw tests are used to determine the capacity of the load cells. The graphs in Figure 9 show the results of transverse forces calculated in computational fluid dynamics in the heaviest test scenario. Simulations are stopped when the third cycle of the oscillatory motion is finished at the physical time of 8.57 s. The first two cycles are not taken into account due to a lack of convergence in numerical solving. The total CPU time spent for the drift–yaw simulation equals 78 h, which was done by a 9th generation core I7 CPU with 32 GB RAM.
Figure 9 Sway force graphs in terms of time in the combined drift– yaw tests with frequency of 0.35 rad/sIt should be noted that the minimum load that can be measured by load cells is usually 0.2% of its maximum capacity. Therefore, according to the minimum force acting on the body and considering this percentage, the maximum capacity of the load cell can be calculated. The forces acting on the model in different test scenarios are described in Table 7. As expected, the maximum forces appeared in the combined drift–yaw test, and the minimum forces were recorded in the static drift test.
Table 7 Maximum and minimum loads along longitudinal and transversed axes in different test simulationsTest Axis Max./Min. Value (N) Static drift X Min.
Max.2.85
16.80Y Min.
Max.15.52
551.42Pure sway X Min.
Max.16.59
17.60Y Min.
Max.15.50
305.50Pure yaw X Min.
Max.15.23
17.15Y Min.
Max.0.46
19.90Combined drift-yaw X Min.
Max.-
57.28Y Min.
Max.-
709.984.1.2 Determining the force on each load cell
The load measured by each of the load cells during the test is affected by the two components of force and moment, which are extracted from the numerical simulations. In the predicted mechanism for testing the submarine model, each of the load cells has a distance equal to l/4 from the model's center of gravity, as shown in Figure 10.
Figure 10 Location of the load cells inside the submarine modelGiven the 2.5-meter length of the submarine model and the values of forces and moments applied to the body during the tests, the transverse forces applied to the front and rear load cells can be expressed as follows:
$$ F_1=0.5\left(Y+\frac{N}{\frac{l}{4}}\right) $$ (22) $$ F_2=0.5\left(Y-\frac{N}{\frac{l}{4}}\right) $$ (23) where F1 and F2 are the transverse forces on the front and rear struts, respectively. The force values acting on each of the load cells related to the heaviest test scenario (combined drift–yaw with β0 = 30° and a frequency of 0.35 rad/s) are depicted in Figure 11. In this test, the maximum force applied to the front load cell is 500 N, and the maximum force in the x direction always has a value smaller than 60.
Figure 11 Calculated loads acting on each load cell in the heaviest test scenario versus4.2 Load cell selection
In this study, the capacity of the load cells is first determined based on numerical results and then according to the proposed method for testing the submarine model in the NIMALA. The specifications of the load cell are presented in the next step. The values of the forces acting on the load cells are listed in Table 8.
Table 8 Minimum and maximum forces acting on the load cellsAxis direction Minimum required accuracy(N) Maximum capacity of load cells based on 0.2% ratio (N) Maximum capacity of load cells based on CFD (N) X 2.85 1 425 57.25 Y 15.50 7 750 709.98 The use of two struts with an appropriate profile and two load cells is recommended to perform the captive model test with the HPMM system. The arrangement of the struts is shown in Figure 12.
Figure 12 Overview of the conceptual design for submarine testing using the HPMM systemThe capacity shown in Table 7 is related to the total force on the model. Given that two 2-component load cells are used in the proposed mechanism, the capacity of each load cell must be measured by considering the effect of the yaw moment. The final capacity of each load cell is specified, as shown in Table 9.
Table 9 Final calculation results for the capacity of the load cellsNumber of load cells Load cell type Fx (N) Fy (N) 2 2-components 40 500 5 Conclusions
In this study, the forces acting on the submarine model in HPMM tests were investigated using CFD. The investigations were presented as an instruction flow chart to be used in other similar works. The general results were as follows:
1) Specifying the numerical analysis scenario is essential to determine the appropriate capacity of the load cells. In this scenario, dynamic and static tests must be included, and damping and added mass forces must be determined from the analysis of their results.
2) The maximum load is applied to the body in the combined drift–yaw test, and the minimum load appears in the low-frequency pure yaw test.
3) In determining the input parameters of the considered analyses, the heaviest scenarios should be considered to obtain the maximum forces for calculating the maximum capacity of the load cells. Given that minimum forces are important in calculating the maneuvering hydrodynamic coefficients, the minimum values of the input parameters must also be included.
This research found that numerical approaches can provide an acceptable estimation of the loads acting on HPMM equipment in marine laboratories. In addition to predicting the loads acting on the load cells, an accurate structural and hydrodynamic analysis must be performed on the struts towing the submarine model. This aspect can be a practical subject for further investigations and design of HPMM facilities.
Competing interest The authors have no competing interests to declare that are relevant to the content of this article. -
Figure 1 Earth-fixed and body-fixed coordinate systems
Figure 2 Static drift test boundary conditions
Figure 3 DARPA SUBOFF maneuver simulation grid
Figure 4 Mesh independency for surge and sway force
Figure 5 y+ distribution on SUBOFF body in static drift test
Figure 6 Numerical simulation results and fitted charts in static drift test
Figure 7 Diagram of the fitted curves and the points obtained from the pure yaw numerical simulation
Figure 8 Flowchart of load cell selection
Figure 9 Sway force graphs in terms of time in the combined drift– yaw tests with frequency of 0.35 rad/s
Figure 10 Location of the load cells inside the submarine model
Figure 11 Calculated loads acting on each load cell in the heaviest test scenario versus
Figure 12 Overview of the conceptual design for submarine testing using the HPMM system
Table 1 Summary of numerical setup
Setting Description Time Implicit unsteady Time step (for pure yaw) (s) 0.005 Iteration per time step 5 Flow Segregated flow Equation of state Constant density Turbulence model Realizable K-Epsilon Pressure link SIMPLE Table 2 Values of mesh independency analysis parameters based on the roache method
Parameter Sway force Yaw moment Surge force N1 (Coarse) 1 060 150 - - N2 (Medium) 2 633 531 - - N2 (Fine) 6 491 859 - - ϕ1 289.773 -1 053.30 -95.056 ϕ2 288.27 -1 054.784 -93.84 ϕ3 287.396 -1 055.51 -93.27 P 0.588 76 0.770 28 0.826 44 R 0.581 5 0.493 2 0.468 8 GCI (coarse-medium) 0.919 65% 0.172 71% 1.444 7% GCI (medium-fine) 0.542 32% 0.086 13% 0.689 59% Table 3 Hydrodynamic coefficients obtained from static drift tests using numerical and experimental approaches
Coefficient CFD EFD Error (%) Y'v -0.006 128 -0.005 948 3.03 N'v -0.013 61 -0.012 795 6.37 Y'vvv -0.165 6 - - N'vvv 0.037 73 - - Table 4 Hydrodynamic coefficients obtained from pure yaw test in numerical simulation and laboratory experiment
Coefficient CFD EFD Error (%) Y'r 0.001 03 0.001 811 43.13 N'r -0.001 13 -0.001 597 29.24 $ Y_{\dot{r}}^{\prime}$ 0.000 258 0.000 060 330 $ N_{\dot{r}}^{\prime}$ -0.000 76 -0.000 676 12.43 Table 5 Characteristics of the National Iranian Persian Gulf Towing Tank
Parameter Value Length (m) 400 Width (m) 6 Depth (m) 4 Density of water (kg/m3) 1 002 Kinematic viscosity of water 9.75x10-7 Maximum velocity of the carrier (m/s) 18 Table 6 Geometric characteristics of the tested submarine body
Quantity Value Length (m) 2.5 Diameter (m) 0.254 Number of control surfaces 4 LCG (m) 1.428 68 Mass (kg) 100.62 Izz (kg·m2) 34.702 4 Table 7 Maximum and minimum loads along longitudinal and transversed axes in different test simulations
Test Axis Max./Min. Value (N) Static drift X Min.
Max.2.85
16.80Y Min.
Max.15.52
551.42Pure sway X Min.
Max.16.59
17.60Y Min.
Max.15.50
305.50Pure yaw X Min.
Max.15.23
17.15Y Min.
Max.0.46
19.90Combined drift-yaw X Min.
Max.-
57.28Y Min.
Max.-
709.98Table 8 Minimum and maximum forces acting on the load cells
Axis direction Minimum required accuracy(N) Maximum capacity of load cells based on 0.2% ratio (N) Maximum capacity of load cells based on CFD (N) X 2.85 1 425 57.25 Y 15.50 7 750 709.98 Table 9 Final calculation results for the capacity of the load cells
Number of load cells Load cell type Fx (N) Fy (N) 2 2-components 40 500 -
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