Added Mass Force Research Articles

Intermittent swimming of two self-propelled flexible fins with laterally constrained heaving motions in a side-by-side configuration

Inspired by the intermittent locomotion of fish schools, numerical simulations are performed with two self-propelled flexible fins in a side-by-side configuration with anti-phase oscillation actuated by laterally constrained heaving motions. For an intermittent swimming gait, one type of the half-tail-beating mode (HT mode) and two types of multiple-tail-beating modes coasting at the smallest (MTS mode) and largest (MTL mode) lateral gap distances are applied. Similar to the continuous-tail-beating mode (CT mode), equilibrium lateral gap distances between two fins with HT and MTL modes exist, whereas two fins with MTS mode do not maintain a lateral equilibrium state. Although the cycle-averaged lateral force acting on two fins with CT and MTL modes is mostly determined by an outward deflected jet and enhanced positive pressure between two fins, an added-mass lateral force related to an asymmetric flapping kinematics by passive flexibility also plays an important role in MTL mode to achieve a stable state with a lateral gap distance smaller than that in CT mode. When the cruising speed or the cycle-averaged input power is identical in a stable state, the cost of transport (COT) for two fins with MTL mode is smaller than that with CT mode due to not only a benefit from the intermittent swimming gait but also an enhanced schooling benefit with a small equilibrium lateral gap distance. The COT for two fins with CT mode is reduced further when the bending rigidity increases, whereas it is opposite with MTL mode.

Acoustically driven translation of a single bubble in pulsed traveling ultrasonic waves

The acoustic radiation force has been proven as an effective mechanism for displacing particles and bubbles, but it has been mainly applied in a standing wave mode in microfluidics. Alternatively, the use of pulsed traveling acoustic waves could enable new options, but its transient dynamic, which entails the additional complexities of pulse timing, reflections, and the type of waveform, has not yet been fully investigated. To better understand these transient effects, a transient numerical solution and an experimental testbed were developed to gain insights into the displacement of microbubbles when exposed to on- and off-periods of pulsed traveling waves. In this study, a practical sinusoid tone burst excitation at a driving frequency of 0.5 MHz is investigated. Our numerical and experimental results were found to be in good agreement, with only a 13% deviation in the acoustically driven velocity. With greater detail from the numerical solution at a sampling rate of 1 GHz, the fundamental mechanism for the bubble translation was revealed. It was found that the added mass force, gained through the on-period of the pulse, continued to drive the bubble throughout the off-period, enabling a large total displacement, even in the case of low duty-cycle (2%) pulsing. In addition, the results showed greater translational velocity is possible with a lower number of cycles for the same input acoustic energy (constant duty cycle and acoustic pressure amplitude). Overall, this study proposes a new, practical, and scalable approach for the acoustic manipulation of microbubbles for scientific, biomedical, and industrial applications.

Interaction between the oil droplet in water and wetted wall: Force model and motion law

To investigate the force model and motion law of oil droplets in water near the wetted wall, oil droplets with R1 = 0.29–0.62 mm and oil films with R2 = 1–6 mm are solved numerically. In addition to buoyancy, flow resistance, and added mass force, the film-induced force triggered by the wetted wall constraint is also introduced into the force model. The drainage process is described using the Stokes–Reynolds equation, and the Young–Laplace equation is used to calculate the pressure within the water film. The results show that the force model can be coupled with the Stokes–Reynolds–Young–Laplace model equation to better describe the drainage dynamics near the wetted wall. The pressure distribution law is closely related to the shape of the water film, especially when the oil–water interface is in the shape of a dimple, which can lead to the formation of negative pressure zones within the water film. The maximum pressure first grows in an exponential, then logarithmic pattern and eventually approaches the equivalent Laplace pressure. Around the critical size, the direction of the film-induced force changes and the form of action switches between driving and drag forces. The film-induced force's dominant effect is strongest when the curvature radius of the oil film is comparable to the droplet size.

A collection of wet beam models for wave–ice interaction

Abstract. Theoretical models for the prediction of decay rate and dispersion process of gravity waves traveling into an integrated ice cover expanded over a long way are introduced. The term “wet beam” is chosen to refer to these models as they are developed by incorporating water-based damping and added mass forces. Presented wet beam models differ from each other according to the rheological behavior considered for the ice cover. Two-parameter viscoelastic solid models accommodating Kelvin–Voigt (KV) and Maxwell mechanisms along with a one-parameter elastic solid model are used to describe the rheological behavior of the ice layer. Quantitative comparison between the landfast ice field data and model predictions suggests that wet beam models, adopted with both KV and Maxwell mechanisms, predict the decay rate more accurately compared to a dry beam model. Furthermore, the wet beam models, adopted with both KV and Maxwell mechanisms, are found to construct decay rates of disintegrated ice fields, though they are built for a continuous ice field. Finally, it is found that wet beam models can accurately construct decay rate curves of freshwater ice, though they cannot predict the dispersion process of waves accurately. To overcome this limitation, three-parameter solid models, termed standard linear solid (SLS) mechanisms, are suggested to be used to re-formulate the dispersion relationship of wet beam models, which were seen to construct decay rates and dispersion curves of freshwater ice with an acceptable level of accuracy. Overall, the two-parameter wet beam dispersion relationships presented in this research are observed to predict decay rates and dispersion process of waves traveling into actual ice covers, though three-parameter wet beam models were seen to reconstruct the those of freshwater ice formed in a wave flume. The wet beam models presented in this research can be implemented in spectral models on a large geophysical scale.

Second-order inertial forces and torques on a sphere in a viscous steady linear flow

We compute the full set of second-order inertial corrections to the instantaneous force and torque acting on a small spherical rigid particle moving unsteadily in a general steady linear flow. This is achieved by using matched asymptotic expansions and formulating the problem in a coordinate system co-moving with the background flow. Effects of unsteadiness and fluid-velocity gradients are assumed to be small, but to dominate in the far field over those of the velocity difference between the body and fluid, making the results essentially relevant to weakly positively or negatively buoyant particles. The outer solution (which at first order is responsible for the Basset–Boussinesq history force at short time and for shear-induced forces such as the Saffman lift force at long time) is expressed via a flow-dependent tensorial kernel. The second-order inner solution brings a number of different contributions to the force and torque. Some are proportional to the relative translational or angular acceleration between the particle and fluid, while others take the form of products of the rotation/strain rate of the background flow and the relative translational or angular velocity between the particle and fluid. Adding the outer and inner contributions, the known added-mass force or the spin-induced lift force are recovered, and new effects involving the velocity gradients of the background flow are revealed. The resulting force and torque equations provide a rational extension of the classical Basset–Boussinesq–Oseen equation incorporating all first- and second-order fluid inertia effects resulting from both unsteadiness and velocity gradients of the carrying flow.

Coupling immersed boundary and lattice Boltzmann method for modeling multi‐body interactions subjected to pulsatile flow

This paper numerically investigates the effect of pulsating flow on the settling dynamics of rigid circular particles. This is an interdisciplinary subject and spans several areas ranging from mathematical and numerical modeling to fluid mechanics. For this purpose, pulsatile flow characteristics are embedded in the combination of the direct‐forcing immersed boundary method and the split‐forcing lattice Boltzmann method. Inter‐collision forces between the solid boundaries (particles and boundaries) and the added mass force due to acceleration are considered. Adequate verification tests are done to ensure the credibility of the findings. The critical parameters of pulsating flow, such as amplitude and frequency of pulsation, are investigated in detail. The paper especially puts emphasis on the interaction between particles and studies the well‐known drafting, kissing, and tumbling (DKT) phenomena. Two different scenarios are taken into account and also compared with the stationary flow. The first case is when the pulsating flow is in the direction of gravity (co‐flow), while in the latter, there is an opposing flow (counterflow). The sedimentation manners of 12 particles in a vertical channel are also presented. The findings shed light on the importance of pulsating flow and the extension of the proposed computational method for such problems. It is also revealed that pulsation and its variables can alter DKT by either postponing or speeding up the process. Also, in some cases, the cycle of DKT can be maintained incompletely, and particles would just stick together. The results can be useful for various engineering problems like filtration and particle sorting.

Hydrodynamic coefficients of a forced oscillating flexible pipe with energy competition model

The objective of this study is to reveal the characteristics of hydrodynamic force and coefficients undergoing vortex-induced vibration (VIV) of a flexible pipe in an oscillatory flow. The flexible pipe is forced to oscillate with different amplitudes and periods in the still water to simulate the equivalent oscillatory flow. Different from the traditional model, the hydrodynamic forces for the flexible pipe under the new energy competition force model in both in-line (IL) and cross-flow (CF) directions are divided into drag, excitation and added mass force. The identification methods of the hydrodynamic coefficients for the energy competition force model are then established. The detailed features of the hydrodynamic coefficients under the new force model in an oscillatory are illustrated. The distribution comparison of the two forces, i.e., drag force plays a role in energy dissipation and excitation force acts as energy input, explains the differences of the VIV response in magnitudes under different oscillatory flows. By summarizing the hydrodynamic coefficients, the drag and excitation coefficients are found to be inverse proportion to the Keulegan-Carpenter (KC) number. The independence of the added mass coefficients and KC number is revealed. On the basis of the observation, the empirical models for hydrodynamic coefficients are preliminarily proposed. The effects of the oscillatory flow on the hydrodynamics are established to be determined by the KC numbers under the same reduced velocity. The present investigations will provide a better guideline for developing flow-induced vibration prediction methods in future work.

The composite mechanism of damping and elasticity for a fish-like flexible propeller

Fish in nature adjust their swimming modes by regulating muscle stiffness under different ocean conditions. Peduncle stiffness plays a significant role in propulsive performance. However, flexible structures designed with single-mechanism elastic components produce optimal propulsive performance only at specific frequencies. Existing structures were difficult to achieve effective propulsion over a wide range of frequencies. To this end, this article firstly proposed a flexible fish-like propeller based on the composite mechanism of damping and elasticity. Combining with the compliance characteristics of the elastic structure at a specific frequency and damping characteristics that the damping force raised rapidly with frequency, the biological propeller maintained a reasonable phase and flapping amplitude in a wide frequency range, ensuring that the passive flexible tail promoted propulsive performance over a wide frequency range. Secondly, based on the blade element theory, fluid forces exerted on the biological propeller were calculated by drag forces and added mass forces. A hydrodynamic-twisting-damping torque (HTDT) model was established to analyze the motion of the passive flexible joint. Finally, the propulsive performance was tested over a grid of torsional stiffness and dynamic damping viscosity. The results showed that torsional stiffness and dynamic damping viscosity significantly influenced the phase and flapping amplitude of the caudal fin. The experimental results further found the optimized combination with the optimal torsional stiffness and dynamic damping viscosity. This work further offers valuable insights into the development and optimization of the fish-like propeller.

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.

Tailoring the bending pattern of non-uniformly flexible pitching hydrofoils enhances propulsive efficiency

We present new measurements of non-uniformly flexible pitching foils fabricated with a rigid leading section joined to a flexible trailing section. This construction enables us to vary the bending pattern and resonance condition of the foils independently. A novel effective flexibility, defined as the ratio of added mass forces to elastic forces, is proposed and shown to provide a scaling for the natural frequencies of the fluid-structural system. Foils with very flexible trailing sections of EI < 1.81 × 10−5 N m2 do not show a detectable resonance and are classified as ‘non-resonating’ as opposed to ‘resonating’ foils. Moreover, the non-resonating foils exhibit a novel bending pattern where the foil has a discontinuous hinge-like deflection instead of the smooth beam-like deflection of the resonating foils. Performance measurements reveal that both resonating and non-resonating foils can achieve high propulsive efficiencies of around 50% or more. It is discovered that non-uniformly flexible foils outperform their rigid and uniformly flexible counterparts, and that there is an optimal flexion ratio from 0.4 ⩽ λ ⩽ 0.7 that maximizes the efficiency. Furthermore, this optimal range coincides with the flexion ratios observed in nature. Performance is also compared under the same dimensionless flexural rigidity, R*, which highlights that at the same flexion ratio more flexible foils achieve higher peak efficiencies. Overall, to achieve high propulsive efficiency non-uniformly flexible hydrofoils should (1) oscillate above their first natural frequency, (2) have a flexion ratio in the range of 0.4 ⩽ λ ⩽ 0.7 and (3) have a small dimensionless rigidity at their optimal flexion ratio. Scaling laws for rigid pitching foils are found to be valid for non-uniformly flexible foils as long as the measured amplitude response is used and the deflection angle of the trailing section β is < 45°. This work provides guidance for the development of high-performance underwater vehicles using simple purely pitching bio-inspired propulsive drives.

Experimental investigation on vortex-induced force of a flexible pipe under oscillatory flow

In this paper, vortex-induced force (VIF) characteristics of a flexible pipe under oscillatory flow are experimentally investigated with different KC numbers ranging from 10 to 178, and reduced velocities from 4 to 7.9. The strain information caused by vortex-induced vibration (VIV) at different cross-sections on the flexible cylinder is measured in the model test. Combined with the modal superposition method and inverse analysis method for hydrodynamic force, the hydrodynamic forces on the flexible cylinder are inverted, including the excitation force in phase with velocity and added mass force in phase with acceleration. Then, through a least square method with forgetting factors, the excitation coefficients and added mass coefficients at each time step are identified. The results show that the VIV excitation coefficient of a flexible pipe under oscillatory flow has a significant “acceleration and deceleration” effect, that is, the excitation coefficients are mostly positive at the acceleration section of the oscillatory flow and negative at the deceleration section. This “acceleration and deceleration” effect also explains the response amplitude modulation feature of VIV under oscillatory flow. And this effect can be represented by the dimensionless acceleration parameter γ of the flow field, and the range of γ corresponding to the positive excitation coefficient is affected by the KC number and the maximum reduced velocity. In addition, the added mass coefficients present different time-varying features under oscillatory flow with different maximum reduced velocity. When the maximum reduced velocity is larger than 6.5, the time-varying trend of the added mass coefficient is opposite to that of the flow velocity. While when the maximum reduced velocity is smaller than 6.5, the added mass coefficient is maintained around 2.0 in VIV excitation region; and it is around 1.0 in the non-excited region which is close to the added mass coefficient of a cylinder in still water.

Physics of gust response mitigation in open-loop pitching manoeuvres

This paper experimentally investigates the flow field development and unsteady loading of three force-mitigating pitch manoeuvres during a transverse gust encounter. The manoeuvres are constructed using varying levels of theoretical and simulation fidelity and implemented as open-loop kinematics in a water towing tank. It is found that pitch actuation during a gust encounter results in two important changes in flow topology: (i) early detachment of the leading-edge vortex (LEV) and (ii) formation of an LEV on the pressure side of the wing upon gust exit. Each of the pitch manoeuvres is found to mitigate a significant portion of the circulatory contribution of the lift force while only manoeuvres with accurate modelling of the added-mass force are found to adequately mitigate the total lift force. The penalty of aerodynamic lift mitigation using pitch manoeuvres was a twofold increase in the pitching moment transients experienced by the wing for all cases. By quantifying changes in the vertical gust momentum before and after the encounter, lift-mitigating manoeuvres were found to reduce the disturbance to the gust's flow field, thereby reducing the momentum exchange between the gust and the wing.

Dynamics of suspended sediment transport: A Direct Numerical Simulation study

The dynamics of suspended sediment transport in horizontal open channel flow is analysed using point-particle one-way coupling Direct Numerical Simulations (DNS), with a virtual wall as a simple particle resuspension model. In sediment transport, the bed-load is dominated by the inter-particle interactions, but the suspended sediments are transported essentially in a one-way coupling situation. The validity of one-way coupling DNS with point-particle approach for the transport of suspended sediment is analysed, by comparing the simulations with existing well-designed experiments performed under very similar conditions. The range of the relevant non-dimensional parameters of the simulations is roughly the same as in actual sediment transport, except for the flow Reynolds number. Good agreement is observed between the simulations and the experiments; furthermore, the simulation results of the motion of the suspended sediment are insensitive to the position of the virtual wall, provided this wall is placed in a region where the fluid velocity fluctuation in the wall-normal direction is comparable to the particle settling velocity. Using the simulation results, the interplay between the different fluid–particle interaction forces is analysed, with and without gravity. In the absence of gravity, the dynamics is dominated by the balance between the stress-gradient force and the turbophoretic effects; as the particle-to-fluid density ratio for the sediment particles is on the order of one, the situation is quite different when compared to the dynamics with a density ratio on the order of 1000. When gravity is included, the dynamics is dominated by the interplay between the drag and gravitational forces, and they balance each other. Both with and without gravity, the lift and added-mass forces have only secondary effects and do not play an important role.

Experimental Research on Vortex-Induced Force Characteristics of Flexible Riser with Buoyancy Module and Strakes

In this paper, the vortex-induced force characteristics of a flexible riser with buoyancy modules and strakes were experimentally investigated. The strain information caused by vortex-induced vibration (VIV) at different cross-sections on the flexible cylinder was measured in the model test. Combined with the modal superposition method and inverse analysis method for vortex-induced force, the vortex-induced forces on the flexible cylinder were inverted, including the excitation force in phase with velocity and added mass force in phase with acceleration. Then, the excitation coefficients and added mass coefficients at each time step were identified via a least-squares method. The results show that the added mass coefficient and excitation coefficient of the flexible riser covering buoyancy modules present a “sawtooth” distribution along the length of the pipe, while the total mass (sum of structural mass and added mass) is consistent along the pipe. The strake has little effect on the added mass coefficient of the buoyancy module and the bare pipe section, but when it covers the bare pipe section, the excitation coefficient of the bare pipe section decreases to a negative value, showing a damping effect. In addition, when the bare pipe section of the flexible riser is completely covered with helical strakes, only the vortex-shedding frequency component of the buoyancy modules remains in the VIV response of the pipe. The reason for this phenomenon is that helical strakes completely suppress the vortex shedding of the bare pipe section, resulting in the reduction in the VIV excitation energy of the original bare pipe section to 0.

Added-mass force in dry granular matter.

From two-dimensional (2D) numerical simulations of the motion of a circular intruder into a dry granular packing, we provide evidence for a specific force term in the case of unsteady motion in addition to the force corresponding to a steady motion. We show that this additional term is proportional to the acceleration of the intruder relative to the grains as the added-mass force known for simple fluids. This force term corresponds to a variation in the kinetic energy of the surrounding flow and is characterized by a coefficient C_{AM} which is intrinsically linked to the nature of the granular media. An analytical calculation of the added-mass coefficient C_{AM} is developed based on the specific velocity field known for 2D granular flow around a cylinder. The theoretical value is shown to depend on the grain-cylinder size ratio, in good agreement with the measurements from our unsteady simulations.

Dynamics of bubble formation and ascent motion on submerged orifices under different Mo number of petrol-based random copolymer solutions

The petrol-based random copolymer, e.g., aromatic-hydrocarbon resin, is widely used in industrial production, but the research about its gas–liquid two flow is far from being solved. Herein, the formation, motion, and deformation of a single bubble in six different concentrations of random copolymer solutions were systemically investigated using high-speed photography (1000 fps), combined with digital image processing technology. The departure time of the bubble first decreases and then increases, and maximum of aspect ratio and specific surface area reduce as the proportion of solute increases. The bubble departure time decreases and bubble size increases with viscous force increases in the case of same surface tension. Based on force balance analysis, added mass force is one of the prime reasons to affect bubble oscillation during the acceleration phase after bubble detachment, whereas the effect of added mass force decreases as solute increases or as the bubble reaches stability zone. Also, velocity, drag coefficient and aspect ratio of bubble were discussed in detail by dimensionless analysis. Furthermore, three new shape numerical models (based on We, Ta, and WeEo, respectively) were proposed, which allowed for satisfactory predictions, and new correlations represent well the experimental data in a wider range of Mo numbers from the mentioned literature (−11.6 ≤ logMo ≤ 0.63).

One-dimensional continuum finite-difference model of the dynamics of a dusty medium in aerodynamic, electric and gravitational fields

This paper presents a mathematical model of the dynamics of a gas suspension under the action of an aerodynamic field, an electric field, and a gravitational field. The continuum model is used to describe the dynamics of the disperse component. The intercomponent momentum exchange included the aerodynamic drag force, the dynamic Archimedes force, and the added mass force. The Coulomb force acting on dispersed particles was taken into account. The model assumes the solution of the equations of conservation of “average density”, momentum and energy for the dispersed phase. The electric field was described by the Poisson equation. The equations of the mathematical model were supplemented with boundary conditions. The equations of mechanics were integrated by explicit finite-difference method of McCormack.

A CFD approach for numerical assessment of hydrodynamic coefficients of an inclined prism near the sea bottom

The hydrodynamic forces and dynamic responses of subsea structures, like manifolds, are studied in the final phase of their installation when they are near the sea bottom. The hydrodynamic coefficients of a submerged 3D prism are predicted through the constant acceleration method, which is implemented by the CFD tool OpenFOAM. In the constant acceleration method, the added mass force can be obtained at the moment that acceleration flow just begins or disappears by subtracting the drag force from the total hydrodynamic force. By comparing the CFD predictions with the corresponding available experimental and numerical results for the prism, a very good agreement indicates the validity of the proposed CFD method. The simulations of the flow past the prism with different inclined angles disclose the effect of the inclined angles on the hydrodynamic forces, pressure distributions, streamlines around the prism and finally the hydrodynamic coefficients. The bottom effect on the hydrodynamic coefficients of the prism is systematically studied through varying spacing ratios. At last, the inclined angle and spacing ratio are found to greatly affect the added mass coefficient and drag coefficient of the prism by changing the pressure distributions around it. The sensitivity of the hydrodynamic coefficients to the variation of the spacing ratio and inclined angles are studied and discussed.

Numerical simulation of vortex-induced vibration of a smooth circular cylinder at the subcritical regime

The present paper focuses on the simulation of vortex-induced vibration (VIV) of a rigid, smooth circular cylinder with elastic supports subject to a cross-flow at the subcritical regime of Reynolds number, 30,000&lt; Re&lt;80,000. The circular cylinder is allowed to move in one degree-of-freedom (DOF), heave. Unsteady Reynolds-averaged Navier-Stokes (URANS) equations are solved with Menter’s k — [Formula: see text] based Shear Stress Transport based Scale-Adaptive Simulation, SAS-SST, turbulence model, and two-equation transition transport γ — θ model. The transport equations are discretized using the Finite Volume Method (FVM). The numerical amplitude and frequency ratio is compared against the experiments conducted in the Marine Renewable Energy Laboratory (MRELab) at the University of Michigan. The angle in which the computed lift leads the displacement in VIV is compared against experimental results reported by Cornell-ONR Water Channel as well. The existence of the initial, upper, and lower VIV response branches is demonstrated. Wake vortex pattern mode has been studied in the different branches of VIV. The time records of the added mass force coefficient and the vortex force coefficient are obtained. Then, the time-averaged phase angle of the vortex and added mass force coefficients are compared against the experimental results. Lastly, the time records of the phase angle in different branches of VIV are shown and analyzed.

Modeling the combined parameter for incipient motion from the force balance on a single proud grain

We assess the validity and limitations of the “combined parameter” as a parametric representative of the torque balance for a single “proud” grain for the threshold of motion for non-cohesive sediments under monochromatic wave forcing conditions. Threshold was determined by balancing the sum of the particle drag force, pressure gradient force, and added mass force against the stabilizing force of gravity adjusted by the lift force. As expected, the magnitude of the particle drag force was relatively large in the balance equation at threshold for all simulated conditions. The pressure gradient force strongly influenced the timing (phase) of threshold for all simulated conditions. The inclusion of the added mass force was found to have a negligible effect for all simulated conditions. Results were sensitive to the magnitude of the lift force through the specification of the lift coefficient. We conclude that the combined parameter, made up of the Shields parameter and Sleath number, should be modified to include a representation of the lift force. Finally, the results suggest a new interpretation of the bed depth variable in the combined parameter that bridges the regimes of motion from the incipient (i.e. single grain) up to sheet flow and plug flow.