Hydrodynamic Force Model Research Articles

A general solution for the water entry of an arbitrary curved wedge

ABSTRACT The present study proposes a general solution for the water entry of an arbitrary curved wedge with a constant speed. The solution is based on the assumption that the impact flow is an incompressible potential flow with negligible effects of gravity and surface tension. The slamming and transition stages of the water entry of the curved wedge are addressed theoretically. For the slamming stage, pressure and hydrodynamic force models are derived from a quasi self-similar assumption based on the wetted length and a similarity solution of the water entry of linear wedges. The solution can be applied to the water entry of linear and curved wedges, where the curvature effects are absorbed into the wetted length and the derivative of the wetted length with respect to the penetration depth. For the transition stage, where a fixed separation point of flow detachment is located at the knuckle of the wedge, a hydrodynamic force model is proposed by extending the transition stage model of the water entry of linear wedges to that of curved wedges. The present solution can quickly predict the pressure distributions and hydrodynamic forces for the water entry of curved wedges. The predictions are in good agreement with the experimental and numerical results for the slamming and transition stages.

A comparative analysis of numerically simulated and experimentally measured static responses of a floating dock

ABSTRACT Two numerical methods, dynamic and static analyses, are proposed to calculate the static responses of a floating dock under different ballast water distributions. Model-scale experimental tests were conducted to compare with these numerical methods. The dynamic analysis includes a 6-degree-of-freedom (6-DOF) model, a hydrostatic force model and a hydrodynamic force model to simulate the dock's freely floating processes. The dock's equilibrium position is identified when the difference in the dock’s motions between two successive time steps is below a specified tolerance value. In the static analysis, the static equilibrium equations in draught, heel, and trim are solved using the Newton-Raphson method. Both dynamic and static results of the draughts at the four corners, heel, and trim are in good agreement with the corresponding experimental results, which shows the reliability of the proposed numerical methods. Moreover, the static analysis exhibits quicker convergence, requiring fewer iteration steps than the dynamic analysis.

Flow past a random array of statistically homogeneously distributed stationary Platonic polyhedrons: Data analysis, Probability maps and Deep Learning models

We perform particle-resolved direct numerical simulations (PR-DNS) of the flow past a random array of stationary Platonic polyhedrons seeded in a tri-periodic box at Reynolds number Re=1, 10, and 100 and solid volume fraction 0.05, 0.1 and 0.2. Using Platonic polyhedrons enables us to examine the effects of particle angularity on the hydrodynamic force and torque exerted on the stationary particles in a controlled manner. We report the average drag, lift, and torque coefficients as well as their respective distribution. We explore the relationship between the particle angularity, particle angular position, flow disturbances created by the neighboring particles, and the hydrodynamic force and torque exerted on each individual particle. We attempt to extend our probability map based model (MPP), our Physics Informed neural network (PINN) model previously developed for spheres to Platonic polyhedrons, and propose an additional NN architecture. We show that ignoring the angular position and actual shape of the reference and neighboring particles constitutes a reasonable first order approximation that significantly simplifies the construction of deterministic models of hydrodynamic forces and torques at low Re=1 to moderate Re=10. At high Re=100, both the MPP model and the PINN model without any additional knowledge of the particle shape and angular position fail to properly predict the force and torque fluctuations. We then design a Convolutional Neural Network (CNN) architecture that is able to maintain a satisfactory performance at Re=100 but requires additional input data in the form of a coarse-grained velocity field around each particle. Overall, predicting the torque fluctuation remains a significant challenge.

Microstructure-based prediction of hydrodynamic forces in stationary particle assemblies

In this work, we derive novel hydrodynamic force models to describe the interaction of a flow with particles in an assembly when only an averaged resolution of the flow is available. These force models are able to predict the average drag on the particle assembly, as well as the deviations from the average drag force and the lift force for each individual particle in the assembly. To achieve this, PR-DNS of various particle assemblies and flow regimes are carried out, varying the particle volume fraction up to 0.6, and the mean particle flow Reynolds number up to 300. To characterize the structure of the particles in the assembly, a Voronoi tessellation is carried out, and a number of scalars, vectors and tensors are defined based upon this tessellation. The microstructure informed hydrodynamic force models are based on symbolic regressions of these quantities derived from the Voronoi tessellation, the global particle volume fraction of the particle assembly and the flow regime represented by the Reynolds number, and the forces on the individual particles in the assembly.The resulting hydrodynamic force models are single expressions and can be directly employed in a Lagrangian particle tracking (LPT) or computational fluid dynamics/discrete element model (CFD/DEM) framework. By comparing the results of the newly proposed hydrodynamic force models with an averaged force model, as is usually adopted in Lagrangian particle tracking simulations, we show that, potentially, a significant increase in accuracy can be achieved, with only a relatively small increase in computational cost, compared to the cost of the CFD/DEM simulation.

Hydrodynamic force and torque models for cylindrical particles in a wide range of aspect ratios

During the pneumatic conveyance of biomass in a coal-fired power station boiler, biomass particles have cylindrical shapes with different aspect ratios. They move through the fluid at any angle and rotate strongly. However, highly accurate and general models of the drag, lift, and torque coefficients (CD, CL, and CT) for biomass particles in a wide range of aspect ratios, especially the CT model and the high aspect ratios, are currently lacking. This paper presents detailed direct numerical simulations of the flow around cylindrical cylinders with varying aspect ratios (6 ≤ AR ≤ 22), Reynolds numbers (100 ≤ Re ≤ 2000), and angles of incidence (0° ≤ θ ≤ 90°). The simulation was conducted using the OpenFOAM solver with the body-fitted mesh method. The flow characteristics and force coefficients of cylindrical particles with different AR were systematically analyzed. New functional correlations between CD, CL, and CT and AR, Re, and θ values were established. The mean squared errors for CD, CL, and CT were 8.8 × 10–2, 2.4 × 10–2, and 4.7 × 10–2, with average relative errors of 5.8%, 3.5%, and 8.17%, respectively. A comparison of the results with other experimental and simulation data in previous literatures showed that the new CD and CL models have considerable higher predictive ability. The generality of the new CD model expanding to low ARs of 1.5 and 3 is verified finally. The new force and torque models are expected to improve the accuracy of Eulerian–Lagrangian simulations of various cylindrical particle-laden flows in the utility of biomass energy.

Numerical Study on the Automatic Ballast Control of a Floating Dock

Abstract The ballast control of a floating dock mainly relies on manual operations, which can be time-consuming and requires skilled workers. This study proposes an automatic ballast control system for floating docks, which improves operational efficiency and safety during the vessel docking process. A numerical model is developed to simulate the dynamic process of the floating dock's operations, which includes a six degrees-of-freedom (6DOF) model, a hydrostatic force model, a hydrodynamic force model, and a hydraulic model. The hydrostatic force model is developed using the Archimedes law and a strip theory along the longitudinal direction. The hydrodynamic model is made based on the effects of added mass and dynamic damping. The hydraulic model is proposed to deal with the hydraulic calculation of the ballast water system. The present automatic ballast control is designed based on a modified proportional controller (P-controller) to control the valve opening angles when the pitch or roll angles are larger than the corresponding threshold values. Without using controllers, the roll angles of the dock can reach 8.9 deg and 13 deg during the ballasting and de-ballasting operations, respectively. The present modified P-controller with optimized control parameters can stabilize the dock during the de-ballasting operation and keep the maximum pitch and roll angles no larger than 0.016 deg and 0.0783 deg, respectively. During the ballasting operation with the same control parameters, the roll and pitch are below 0.0604 deg and 0.0145 deg, respectively. The present automatic control will be further implemented in the vessel docking cases and can significantly improve the stability of the dock.

Dynamic response of a floating dock under corrosion-induced accidents

Dynamic responses of a floating dock under corrosion-induced accidents are studied using a numerical method. The numerical model is proposed to calculate the dynamic responses of the floating dock during operations. It includes a six-degree-of-freedom (6-DOF) model, a hydrostatic force model, a hydrodynamic force model, and a hydraulic model. The effects of the corrosion-induced holes on the stability of the floating dock are investigated and the results show that the maximum pitch and roll angles are 0.18° and 0.69° respectively when there is one hole located at one tank. The maximum pitch and roll angles become 0.42° and 2.04° respectively when there are two holes located at different tanks. The results indicate that situations involving more than one corroded hole result in large roll and pitch angles, which ultimately increase the risk of the vessel capsizing. This analysis not only emphasizes potential hazards but also presents an opportunity for the maritime sector to enhance safety, operational efficiency, and environmental responsibility.

Unmanned Underwater Vehicle Autonomy and Control near Submarines Using Actively Sampled Surrogates

Many tools have been developed to simulate unmanned underwater vehicle (UUV) motion and autonomous behaviors to evaluate UUV capabilities. However, there is no simulator that performs real-time modeling of the complex hydrodynamic interaction forces that a UUV experiences when operating near a moving submarine. These hydrodynamic interactions must be determined in real time to simulate the launch and recovery of UUVs from submarines. Potential flow models may be fast enough to solve the hydrodynamic interactions in real time, but by oversimplifying the physics and neglecting viscosity, they introduce inaccuracies into the simulations. Computational fluid dynamics (CFD) is capable of accurately modeling these hydrodynamic interactions, but simulations take hours or days to solve. To overcome this obstacle, a machine learning method known as Gaussian process (GP) regression is used to create a surrogate reduced-order-model that predicts the hydrodynamic interactions in real time. The GP regression model is trained by actively sampling CFD simulations in order to accurately model complex hydrodynamic interactions. This new approach allows the GP regression model to be incorporated into a UUV motion simulator and evaluate how the UUV is affected by the hydrodynamic interactions. Operating envelopes are developed that outline regions where the UUV safely overcomes the hydrodynamic interactions and where the UUV is overpowered and collides with the submarine. By incorporating this surrogate model into the autonomy architecture, new autonomous behaviors are created that compensate for the hydrodynamic interactions by adjusting the desired UUV heading and speed which allows it to better stay on course.

Hydrodynamic force and torque fluctuations in a random array of polydisperse stationary spheres

We report data on the fluctuations of hydrodynamic force and torque exerted on polydisperse stationary spheres randomly distributed in a cubic tri-periodic box. We generate our data with an accurate Particle-Resolved/Direct Numerical Simulation (PR-DNS) method that relies on an Immersed Boundary/lattice Boltzmann method implemented on a Cartesian octree grid (Cheng and Wachs, 2022). The local grid refinement capability offered by the octree grid enables us to properly resolve the flow dynamics even around the smallest stationary spheres in the system in the case of a wide sphere diameter distribution. We verify that our data are indeed accurate and reliable through a thorough comparison with results produced by other numerical methods and with data published in the literature. We evaluate the accuracy and limitations of various average drag models on different levels of polydispersity. We consider two Reynolds numbers Re=1 and Re=100 representative of low and moderate inertia. We analyze results in terms of hydrodynamic force and torque fluctuations around the average value and relate these fluctuations to the flow disturbances created by the neighboring particles, i.e., the local microstructure. Finally, we discuss the implications of the presented results on the extension of hydrodynamic force and torque models recently proposed in the literature that take advantage of the local microstructure to predict fluctuations in the case of arrays of monodisperse stationary spheres to the case of arrays of polydisperse stationary spheres.

An integrated modeling approach to predict critical flow rate for fines migration initiation in sandstone reservoirs and water-bearing formations

Fines migration causes formation damage in sandstone reservoirs and water-bearing subsurface formations. Several factors affect fines migration such as composition and salinity of permeating brine, flow rate, and system pH. Electrostatic forces of van der Waals attraction and electric double layer repulsion, gravitational force and hydrodynamic force all contribute to fines detachment and migration under certain conditions. The current research presents a new integrated model of electrostatic, gravitational, and hydrodynamic forces to estimate the critical injection rate of NaCl brine for fines migration initiation in subsurface formations.The DLVO modeling approach was modified by considering electrostatic, hydrodynamic, and gravitational forces. Parameters such as injection brine salinity, average fine particle size, and zeta potential of sand-brine systems were used, and effective forces were quantified. A microscopic force and torque balance approach was applied to model hydrodynamic forces to modify the DLVO model. The developed models were validated with experimental results.The profiles of injection velocity versus fine particle radius were generated and the models predicted critical flow rates of 0.9, 2.6, and 3.8 cc/min for 0.15 M, 0.2 M, and 0.25 M NaCl, respectively. For model validation, the coreflood experiments were performed on Berea sandstone core samples under given salinities and critical flow rates were determined. Comparable experimental results were obtained with critical flow rates of 1, 3, and 4 cc/min for 0.15 M, 0.2 M, and 0.25 M NaCl, respectively. In addition, sensitivity analysis assesses the impact of different forces on the fines migration phenomenon.Formation damage caused by fines migration is a well-documented problem in sandstone reservoirs and water-bearing formations. Regulating the injection/ permeating fluid rate and salinity can avoid fines migration and associated permeability impairment. Our study presents a novel approach incorporating electrostatic, gravitational, and hydrodynamic forces to accurately predict fines mobilization in fine-containing subsurface formations.

Hydrodynamic force model for flexible pipe based on energy competition and applications into flow induced vibration prediction in uniform flow

A modified hydrodynamic force model based on the energy competition is developed. The model divides the hydrodynamic loads on flexible cylinder pipes undergoing bidirectional VIV into three parts: vortex-induced force, drag force and added mass force. The vortex-induced force is directly defined to be in phase with the VIV velocity response, which makes the key hydrodynamic coefficients of the synchronization model can be identified. The energy roles of the hydrodynamic force components are analyzed. The vortex-induced force acts as an energy input, while the energy is dissipated by the drag force. The energy competition results in flow-induced vibration. To determine the hydrodynamic coefficients, experiments on the flexible pipe are conducted. The identification methods of hydrodynamic force and coefficients are then introduced. The identified results show that the drag and vortex-induced force and coefficients are positively correlated with the VIV response amplitudes. It confirms that the hydrodynamic coefficients are not independent of the VIV response amplitudes. The empirical models of the drag and vortex-induced force coefficients are established by fitting the measured values. By instantaneous phase adjustment between the vortex-induced force and VIV velocity response, the empirical hydrodynamic coefficient models are effectively applied to a time-domain prediction method. Compared with experimental results, the good consistencies of the dominant mode number, dominant frequency, mean deformation and VIV response amplitudes can be found in the numerically simulated results. These consistencies prove the feasibility and advantages of the modified force model and the developed empirical coefficient model. Some discrepancies caused by the preliminary coefficient model should be improved by further optimization through intensive systematical experiments.

Consistent Strain-Based Multifidelity Modeling for Geometrically Nonlinear Beam Structures

Abstract Conventional multifidelity modeling for slender structures such as folding-wing aircraft and offshore wind turbines does not allow the generation of multiple fidelity models that consistently use the same external force model, which complicates simulation program and design process. Although consistent absolute nodal coordinate formulation (ANCF)-based multifidelity modeling was recently proposed to address this inconsistency, it still has the following four problems: (1) a large number of generalized coordinates, (2) a large number of Lagrange multipliers, (3) difficulty in constraining high-frequency axial deformation, and (4) a lack of lower-fidelity models. The lower-fidelity models that have not yet been developed are torsion-only beam, extension-only truss, and bending-only beam models. The objective of this study was to develop a novel consistent strain-based multifidelity modeling framework that addresses these problems by leveraging new vector–strain transformations from ANCF to the strain-based beam formulation. We employed a hydrodynamic force model based on Morison's equation as an example to demonstrate all fidelity models obtained from the proposed strain-based framework consistently use the same external force model. We conducted five simulations to verify the proposed models. The consistent external force model for the hydrodynamic force was then validated by comparison with experimental data. The simulation results concurred with those of conventional models and experiments. Low-fidelity models exhibited over 98% reduction in calculation time compared to high-fidelity models, which helps in conceptual and initial designs that require a large number of parametric simulations.

Comparative studies of hydrodynamic force models for flexible pipe in an oscillatory flow

The objective of this comparison study is to verify and clarify whether the energy competitive force model has significant benefits in eliminating the influence of certain factors on the hydrodynamic coefficient. The typical cases with the smaller Keulegan–Carpenter (KC) number (KC = 31) and larger KC number (KC = 178) under the maximum reduced velocity of 6.5 with the prominent VIV responses are selected for analysis. The characteristics of the ortex-induced vibration (VIV) response, the hydrodynamic force, and the corresponding spatiotemporal coefficients in typical cases are revealed, compared, and discussed. The results suggest that the energy competition force model cannot avoid spatial-temporal distribution characteristics of the hydrodynamic coefficients. Drag and excitation coefficients with energy roles under Sarpkaya's and energy competitive force models are affected by the VIV response and wake effects in an oscillatory flow. The larger coefficients are witnessed under the smaller KC number. When the KC number increases from 31 to 178, the mean drag coefficients decrease from 2.25 to 1.73, and the mean excitation coefficients in in-line (IL) and cross-flow (CF) directions decrease from 1.42 and 1.44 to 0.60 and 0.81, respectively, under the energy competition force model. Changes in wake effects under a larger KC number lead to distinctive differences in drag and excitation coefficients during the acceleration and deceleration phases. Through a quantitative comparative analysis, the current identified drag and excitation coefficients under the energy competition force model can obtain a more accurate prediction result. The present work shows that there exists the blindness in the selection of coefficients within the framework of the energy competition force model.

Changes in the Hydrodynamic Characteristics of Ships During Port Maneuvers

To reach a port, a ship must pass through a shallow water zone where seabed effects alter the hydrodynamics acting on the ship. This study examined the maneuvering characteristics of an autonomous surface ship at 3-DOF (Degree of freedom) motion in deep water and shallow water based on the in-port speed of 1.54 m/s. The CFD (Computational fluid dynamics) method was used as a specialized tool in naval hydrodynamics based on the RANS (Reynolds-averaged Navier-Stoke) solver for maneuvering prediction. A virtual captive model test in CFD with various constrained motions, such as static drift, circular motion, and combined circular motion with drift, was performed to determine the hydrodynamic forces and moments of the ship. In addition, a model test was performed in a square tank for a static drift test in deep water to verify the accuracy of the CFD method by comparing the hydrodynamic forces and moments. The results showed changes in hydrodynamic forces and moments in deep and shallow water, with the latter increasing dramatically in very shallow water. The velocity fields demonstrated an increasing change in velocity as water became shallower. The least-squares method was applied to obtain the hydrodynamic coefficients by distinguishing a linear and non-linear model of the hydrodynamic force models. The course stability, maneuverability, and collision avoidance ability were evaluated from the estimated hydrodynamic coefficients. The hydrodynamic characteristics showed that the course stability improved in extremely shallow water. The maneuverability was satisfied with IMO (2002) except for extremely shallow water, and collision avoidance ability was a good performance in deep and shallow water.

Research on accurate modeling of hydrodynamic interaction forces on AUVs operating in tandem

The hydrodynamic interaction force between autonomous underwater vehicles (AUVs) operating in tandem can adversely affect the hydrodynamic prediction of individual AUVs and the formation control of the tandem fleet. Accurate modeling of hydrodynamic interaction forces is crucial to the model-based controller design of AUVs. In this paper, we innovatively propose a modeling method that can describe the hydrodynamic interaction relationship accurately and continuously (when the spacing between adjacent AUVs changes). First, an accurate hydrodynamic model for individual AUVs in tandem fleets is established by modifying the typical model through the consideration of memory effects. Second, a model of hydrodynamic interaction forces is derived by subtracting hydrodynamic forces acting on a single AUV operating independently from those acting on an AUV operating in tandem. To enable this model to continuously describe the hydrodynamic interaction relationship when the spacing changes, this model is further derived by taking the spacing as an independent variable based on the parameter conversion method. Finally, planar motion mechanism (PMM) tests are carried out using the computational fluid dynamics (CFD) method to verify the accuracy and continuity of the proposed modeling method. The results show that compared with the typical model, the average error of hydrodynamic forces predicted by the proposed model is reduced by 5.36%, which verifies the feasibility and effectiveness of the proposed modeling method.

Cross-flow vortex-induced vibration of a flexible fluid-conveying riser undergoing external oscillatory flow

Cross-flow (CF) vortex-induced vibration (VIV) of a flexible riser considering both internal flow and external oscillatory flow is numerically investigated with consideration of combining the structural model with semi-empirical hydrodynamic force model by using Finite Element Method. The accuracy of the applied model is firstly examined by comparing the numerical results with the experimental data, which proves that the model can reproduce typical characteristics of CF VIV of a flexible riser undergoing external oscillatory flow. Then CF VIV of a flexible fluid-conveying riser subjected to external oscillatory flow is studied while the non-dimensional internal flow velocity and density ratio between internal and external flows are changed. The results show that regardless of the non-dimensional internal flow velocity and density ratio, typical VIV features of a flexible riser, such as intermittent VIV, amplitude modulation, hysteresis, mode and frequency transitions as well as standing and travelling wave responses, can be captured with variation of external oscillatory flow velocity. Moreover, VIV developing process, including building-up, lock-in and dying-out, can be detected for CF VIV. With the increase of the non-dimensional internal flow velocity and density ratio, high mode response can be effortlessly triggered for CF VIV, which is accompanied with occurrence of new vibrating frequencies. In addition, the vibrating frequency of CF VIV decreases while the non-dimensional internal flow velocity and density ratio are increased.

Numerical modeling of nonstationary hydrodynamic forces and induced motions of a coupled offshore floating installation system

Offshore installations always involve multi-body systems in which the hydrodynamic and mechanical interactions are complicated for the properties keep changing under certain operation procedures. Float-over deck (FOD) installation is a typical example, where the draft of barge continuously varies during the load transfer operation. Numerical analysis of such continuous operation is crucial to identify the extreme responses, but it's difficult to model the nonstationary properties involved in the hydrodynamic loads and motions and contact loads of the system. In the study, a numerical modeling methodology is developed to calculate these nonstationary hydrodynamic forces and induced responses, considering an active update of the hydrodynamic properties with the time varying body boundary conditions. The effects of time varying body boundary conditions on hydrodynamic forces and induced nonstationary motions and contact loads of a FOD installation system is studied. The results indicate that the dynamic analysis of the continuous load transfer operation could be effectively carried out with one update of hydrodynamic forces at the end of the operation. The barge roll motions and its natural period are the most affected. The proposed method could be used in intensive simulation of offshore multi-body systems with obvious draft variation during operations.

Model development of tangential hydrodynamic force on particles with pendular liquid bridge of power-law fluid

In this study, new models are proposed for hydrodynamic interaction forces acting on particles through a pendular liquid bridge. The liquid considered is a power-law fluid, and the tangential forces due to both translational and rotational motions of particles are independently modelled, which can be superposed as long as the Reynolds number is sufficiently small. The force models are derived by obtaining improved expressions for the pressure profile inside a liquid bridge which can be reduced to the solutions of the pressure equations based on the lubrication theory in the limiting cases. Direct Numerical Simulation (DNS) is also performed to provide reference data to compare. The forces obtained from the proposed model and DNS are in reasonable agreement compared to the model in the literature where significant overestimation can be seen especially when the power-law index is small. Furthermore, the models proposed are written in a closed-form and can be easily implemented in Discrete Element Method (DEM) for wet particle simulation in the future. • New models of tangential hydrodynamic forces on particles are developed. • The models consider a pendular liquid bridge of power-law fluid between particles. • Good agreement between the model developed and DNS are observed. • The models are written in a closed-form and can be easily implemented in DEM.

Nonlinear dynamic analysis of multiple grooved water-lubricated journal bearings using attenuation rate interpolation method

Purpose The stringent requirements for environmental protection have induced the extensive applications of water-lubricated journal bearings in marine propulsion. The nonlinear dynamic analysis of multiple grooved water-lubricated bearings (MGWJBs) has not been fully covered so far in the literature. This study aims to conduct the nonlinear dynamic analysis of the instability for MGWJBs. Design/methodology/approach An attenuation rate interpolation method is proposed for the determination of the critical instability speed. Based on a structured mesh movement algorithm, the transient hydrodynamic force model of MGWJBs is set up. Furthermore, the parameters’ analysis of nonlinear instability for MGWJBs is conducted. The minimum water film thickness, side leakage, friction torque and power loss of friction are fully analyzed. Findings With the increase of speed, the journal orbits come across the steady state equilibrium motion, sub-harmonic motion and limit circle motion successively. At the limit circle motion stage, the orbits are much larger than that of steady state equilibrium and sub-harmonic motion. The critical instability speed increases when the spiral angle decreases or the groove angle increases. The minimum water film thickness peak is at the rotor speed of 4,000 r/min for the MGWJB with Sa = 0°. As rotor speed increases, the side leakage decreases slightly while the friction torque and the power loss of friction increase gradually. Originality/value Present research provides a beneficial reference for the dynamic mechanism analysis and design of MGWJBs.

Anguilliform Swimming Performance of an Eel-Inspired Soft Robot.

In this article, we propose a soft eel robot design using soft pneumatic actuators that mimic eel muscles. Four pairs of soft actuators are used to construct the eel robot body. Pulse signals with suitable shifting phases are utilized to control delivery of compressed air to the actuators in sequence to create a sinusoidal wave from head to tail of the robot body. A model of hydrodynamic forces acting on an anguilliform swimmer when moving in fluid was built to estimate the thrust force generated by the robot at different tail beat frequencies. Experimental data revealed that the generated thrust force was positively correlated with the beat frequency. Measured data showed that swimming efficiency depended on both generated thrust force and body posture in situ. At the beat frequency of 1.25 Hz, and air pressure at three segments from head to tail of 65, 50, and 30 kPa, respectively, the eel robot body showed the best cost of transport (COT) of 19.21 with velocity of 10.5 cm/s (or 0.198 body length per second [BL/s]), compared to the other's values of operation frequency and air pressure. We also found that control shifting phase strongly affects the swimming speed and COT. The robot body reached the highest velocity at around 19 cm/s (0.36 BL/s) with the COT of 10.72. Obtained result in this research would contribute to development of soft elongated swimming robot and enhance the knowledge on swimming performance of both robot and natural eels.