Unsteady hydrodynamic forces of solid objects vertically entering the water surface

SUMMARY The paper presents and evaluates a principle for interpreting forces on bodies in oscillatory flow, based on assuming that the drag coefficient is the same as in stationary flow, provided wake velocities are properly corrected for. The resulting formulae describe general trends correctly, and can in some cases also be used for quantitative calculation of oscillatory flow phenomena. INTRODUCTION For many types of offshore structures drag forces are a major part of the design loads, or they can be of decisive importance to the operational performance of the structures. The drag forces can be due to current, to the orbital motion in the waves, to the oscillatory motions of the structure itself, or to various combinations of these effects. Until now normal design procedures have been based on calculation of drag forces by the wellknown Morison formulation, using drag coefficients determined experimentally for each structure, or taken from empirical data. The present paper introduces and discusses an alternative way of dealing with drag forces in oscillatory flow. It is based on assuming that the drag coefficient itself is the same as in stationary flow, provided the correct relative velocity is applied. This means that the flow velocity has to be corrected for the effect of the wake generated by the previous oscillations. This correction can be done on the basis of momentum considerations and turbulent mixing theory. Based on these principles one can now calculate how drag forces in oscillatory flow are influenced by for instance Keulegan- Carpenter number, geometry of motion pattern, effect of superposed current, effect of interaction between several bodies etc. SCOPE OF INVESTIGATION The scope of the present investigation is:To develop analytical formulae for practical calculations of drag in oscillatory flow based on wake considerations, momentum balance and turbulent mixing theory.To develop computer programs for more extensive calculations.To investigate ranges of applicability of the theory by verification against published experimental data, and by means of new experiments.To discuss possible practical applications of the theory. STATE-OF-THE-ART SURVEY In practical design and analysis of offshore structures the hydrodynamic forces acting on a well submerged, oscillating body are calculated by the Morison formula. (Mathematical equation available in full paper) where the mass coefficient Cm and the drag coefficient Cd depend on:–the geometry of body–the Reynolds number–The Keulegan-Carpenter number. In all practical design and analysis work these dependencies are obtained from experimental data. For the stationary flow case there is a lot of data available from wind tunnel testing of bodies of various geometries, see for instance Refs. 1 and 2. These stationary flow data can also be applied to oscillatory flow in cases of very large Keulegan-Carpenter number. An efficient tool for determining coefficients in oscillatory flow in general is the U-tube. From such experiments there are extensive data available on hydrodynamic coefficients of simple bodies such as two-dimensional cylinders as functions of Rn, Kc, roughness etc., see for instance Ref. 3.

A boundary-integral method is developed for computing first-order and mean second-order wave forces on floating bodies with small forward speed in three dimensions. The method is based on applying Green's theorem and linearizing the Green function and velocity potential in the forward speed. The velocity potential on the wetted body surface is then given as the solution of two sets of integral equations with unknowns only on the body. The equations contain no water-line integral, and the free-surface integral decays rapidly. The Timman-Newman symmetry relations for the added mass and damping coefficients are extended to the case when the double-body flow around the body is included in the free-surface condition. The linear wave exciting forces are found both by pressure integration and by a generalized far-field form of the Haskind relations. The mean drift force is found by far-field analysis. All the derivations are made for an arbitrary wave heading. A boundary-element program utilizing the new method has been developed. Numerical results and convergence tests are presented for several body geometries. It is found that the wave exciting forces and the mean drift forces are most influenced by a small forward speed. Values of the wave drift damping coefficient are computed. It is found that interference phenomena may lead to negative wave drift damping for bodies of complicated shape.

In this paper, we experimentally study the water entry of a set of buoyant bodies with a circular cylindrical shape, which is relevant for several engineering applications, being related to aircraft fuselages in sea landing, solid rocket boosters, and bulbous bows of large ships. Free fall experiments are conducted for three different specimen weights and four different falling heights. The hydrodynamics of the interaction of the buoyant cylinders with the water free surface upon impact is analyzed through an accelerometer, a high-frequency potentiometer and a high-speed camera. Results are presented in terms of impact velocity, kinematics of the cylinder penetration and free surface dynamics, comprising the pile-up and water jet dynamics. Further, the air cavity development is analyzed after the cylinder is completely immersed below the undisturbed free surface.

To improve the operating stability of a cryogenic liquid turbine expander, the influence of flow behavior on rotor radial force and critical speed is investigated numerically. At both design and off-design conditions, unsteady flow simulation is conducted in the entire expander environment where the physical model is constituted by a volute, nozzle, impeller and diffuser duct. The asymmetric flow characteristics are captured at both design and off-design conditions and they are responsible for the significant radial force on rotor. The radial component of flow-induced resultant force is calculated with direct integration approach and it is significant due to apparent asymmetric impeller flow. The influence of such radial component of flow-induced resultant force on the rotor critical speed is further investigated, where the flow-induced radial force is considered as an equivalent mass and superposed on the impeller mass integrity to obtain a resultant mass of impeller, which is used in the prediction of rotor critical speed. To predict rotor critical speed, a finite element method is developed and incorporated into a FORTRAN code, and it is validated and then used to predict the rotor critical speed with and without consideration of the radial component of flow-induced resultant force respectively. The following is described: at both design condition and off-design conditions, the predicted critical speeds with consideration of flow-induced radial force are significantly below that without flow-induced radial force. The influence of impeller flow behavior on rotor dynamics of the turbine expander is not negligible.

The experimental investigation of the wall effect on the cluster particle settling in the Newtonian and non-Newtonian fluid medium is carried out in three different diameter flow channels. The cluster usually forms during the sedimentation of particles in a fluid medium, so it becomes necessary to study the behavior of the cluster particles. Different cluster particles are considered depending on the number of spheres (N) attached to the cluster and the cluster's shape. The present experiment covers the following range of conditions: 0.05 ≤ deq/D ≤ 0.24, 2 ≤ N ≤ 7, 0.64 ≤ n ≤ 1, and 0.14 ≤ K ≤ 1.81. The results reveal that the terminal velocity varies with the blockage ratio (deq/D), N, and cluster shape. For a particular deq/D ratio and the same N, the terminal velocity of polyhedron particles is high compared to the planar and chain shape particles. The blockage ratio and the Reynolds number affect the wall factor of the cluster particles. However, from the experiment, it is observed that the wall effect also depends on the orientation of the particles. The impact of the wall on the cluster particle is high in Newtonian fluid when compared to cluster particles in non-Newtonian fluid. The present work additionally investigates the influence of drag on cluster particles in the presence and absence of the wall effect. The numerical relationships are developed to predict experimental results in Newtonian and non-Newtonian fluid mediums.

An alternative linear flow model and boundary integral flow representation for ship motions in regular waves

Using energy-based modeling techniques, we propose a nonlinear, time-dependent, parametric motion model for an underwater vehicle maneuvering near an otherwise undisturbed free surface. By augmenting the system Lagrangian used to derive Kirchhoff's equations for a rigid body moving through an unbounded fluid, we directly incorporate the free surface into the derivation of the equations of motion. This is done using a <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">free-surface Lagrangian</i> , which accounts for the instantaneous energy stored within the free surface due to an impulsive vehicle motion as well as fluid memory effects. The system Lagrangian then enables us to derive the six-degree-of-freedom nonlinear equations of motion using the Euler–Lagrange equations. The model structure is similar to standard maneuvering models for surface ships, although additional complexities are present since the hydrodynamic parameters are shown to depend on the vessel position and orientation relative to the free surface. For the proposed model, the vessel motion is unrestricted. This is in contrast to traditional seakeeping models, which use convolution integrals to incorporate memory effects for a vessel, which experiences only small perturbations from steady, forward motion. The proposed motion model is amenable to real-time simulation, design performance analysis, and nonlinear control design. Other important hydrodynamic effects due to viscous flow, for example, may then be incorporated into a robust, nonlinear, closed-loop control system as lumped parameter effects or model uncertainties.

Nonlinear free-surface flow around a yawed ship in steady motion is analyzed by the boundary-element method. The solution is obtained by a distribution of singularities on the hull surface and on the undisturbed free surface. The total velocity potential is divided into two parts, i.e., the double-model lifting-flow potential and the free-surface-flow potential. The former is expressed by source and doublet distributions and determined with a Dirichlet-type boundary condition, while the latter is expressed by a simple Rankine-type singularity and determined with nonlinear boundary conditions applied on the free surface. An overview is given for the present approach and numerical results are presented for the Wigley hull, including comparisons between the linear and nonlinear solutions with available experimental data.

: The focus of this paper is to describe and document a recent effort to assess these numerical codes for the prediction of deep-V planing craft hydrodynamic forces and moments and evaluate how well NFA models the complex multiphase flows associated with high Froude number flows, specifically the formation of the spray sheet. This detailed validation effort was composed of three parts. The first part focused on assessing the codes' ability to predict pressures on the surface of a 10 degree deadrise wedge during impact with an undisturbed free surface. Detailed comparisons of NFA to pressure gauges show that NFA accurately predicted pressures during the slamming event. Comparisons of experimental measurements with LAMP predictions showed fairly good agreement, though LAMP tends to overpredict pressures. The second part examines the codes' ability to match sinkage, trim and resistance from Fridsma's experiments performed on constant deadrise planing hulls. Results show good agreement between NFA and experimentally measured values, as well as values calculated using Savitsky's parametric equations. Fairly good agreement was shown between LAMP and experimental values as well, with trim predictions matching well and heave tending to be over-predicted. The final part of the validation study focused on assessing how the codes' ability to accurately model the complex multiphase flow associated with high Froude number flows, specifically the formation of the spray sheet. Comparisons to underwater photographs illustrate NFA's ability to model the formation of the spray sheet and the free surface turbulence associated with planing boat hydrodynamics. Though LAMP was not used to model the complex flow, it accurately predicted the model heave and trim. Overall these three validation studies provide a detailed assessment on the current capabilities of NFA and LAMP to predict the hydrodynamics of a deep-V planing hull.

The linearized water-wave radiation problem for a 2D oscillating source in an inviscid shear flow with a free surface is investigated analytically. The singular source is located at the bottom, with constant fluid depth. The velocity of the basic flow varies linearly with depth, assuming uniform vorticity. The far-field surface waves radiated out from the 2D bottom source are calculated, based on Euler’s equation of motion with the application of radiation conditions. The present analysis extends the work by Tyvand and Sveen to include a nonzero surface flow. Doppler effects will arise in a system at rest with the undisturbed free surface. Resonance with zero group velocity will occur at a critical Froude number for the surface flow. In the presence of a basic surface flow, there are in total four radiated waves when the surface flow is subcritical. With supercritical flow, there are only two emitted waves, with downstream propagation. The critical Froude number depends on the surface velocity, the shear rate and the oscillation frequency of the bottom source.

A physical model of the swimming appendage (second antenna) of a larval Artemia was oscillated and translated through a tank of glycerine to determine how such a shape may be used to generate thrust at the intermediate Reynolds numbers at which it operates. Force on the model was measured by strain gauges and used to calculate coefficients of drag at a series of speeds and frequencies that represented flow regimes of different larval stages. Measured coefficients of drag (Cd) over this Reynolds number range ({approx}1-10) suggest that an expression for a cylinder perpendicular to flow at intermediate Reynolds number (Cd = 1 + 10 Re-2/3) best represents the changes in drag coefficients for this geometry. Unsteady forces were found to be a negligible portion of the force on the model in spite of a high ratio of frequency of oscillation to forward translational velocity (i.e., Strouhal number). Comparison of the thrust generated by the model with its fan of setae rigidly fixed versus passively flexing suggests that passive extension of setae can be influenced by relative limb and body speed.

Forces on a circular cylinder in elliptical orbital flows at low Keulegan-Carpenter numbers

The mathematical formulation of the flowing liquid stream around and through confined multiply connected domains brings a complex differential system. Due to this, one cannot provide a complete description of flow field properties. The current pagination provides a numerical remedy by the use of artificial intelligence in this direction. To be more precise, fluid is considered in the rectangular channel. The circular, square, and triangular-shaped cylinders are positioned as an obstacle to the continuous fluid. The channel’s left wall is viewed as an inlet and two different velocity profiles are introduced at an inlet that are constant and parabolic profile. To discretize the computational domain, hybrid meshing is used. The variance in basic variables, namely, the velocity of the liquid and the distribution of the liquid pressure, is recorded via graphs. The nine different meshed grades are tested for the drag and lift coefficients around the right-angle triangle, square, and circular barrier. The neural networking model is constructed by using 99 datasets of sample values for drag coefficient when characteristic length, the density of fluid, the dynamic viscosity of the fluid, and mean flow velocity are taken as inputs. The training of the neural network takes up 69 slots (about 70%), while the testing and validation of the neural network each take up 15 slots (15%). The Levenberg–Marquardt backpropagation algorithm is used for training. We have observed that for the parabolic profile, the drag coefficient is higher in intensity for each obstacle compared to the constant profile, while the lift coefficient shows opposite patterns.

When bottom mounted structures of large displacement are immersed in the sea, earthquake induced hydrodynamic loads may become important design factors. Specifically, as the earth oscillates, a bottom mounted structure is forced to describe time dependent motion in an otherwise still fluid. As a result, hydrodynamic loads in addition to the inertial loads of the structure itself are induced. In this paper, a theoretical approach to the calculation of these hydrodynamic loads is outlined and numerical results are presented for several submerged configurations. Practical geometries considered include a submerged oil storage tank configuration and a conical configuration as has been proposed for offshore drilling rig designs for deployment in the Arctic. Also, computations were carried out for a sphere and vertical circular cylinder and various comparisons with classical results are made. Numerical results for these submerged structures are presented in the form of a dimensionless hydrodynamic load parameter or added mass coefficient. Results corresponding to a number of different water depths are presented to show the rather sizable effect of the relative water depth on the hydrodynamic force. It is shown that for typical earthquake frequencies, the effect of the free water surface is to reduce the hydrodynamic loads in comparison to the corresponding infinite depth values. Experimental results obtained by vibration testing are presented for a submerged sphere and a vertical circular cylinder. These results show excellent agreement with the theoretical results. Introduction Principal attention in earthquake engineering has been given to the generation of tsunami waves, the shoreline run-up and damage caused by these waves, as well as the damage to dry land structures caused by strong ground motion. With the increased deployment of large submerged structures, an additional facet of this important problem has come to light. Namely, if a large bottom mounted structure submerged in the ocean is excited by oscillatory ground motion, hydrodynamic loads in addition to its own inertia forces come into play. As the structure is caused to move through the water, hydrodynamic forces which are dependent upon the size and shape of the structure as well as the water depth and frequency of oscillation arise. On account of the large density of water, these forces are often quite large and have considerable influence on the structural design. When an object is accelerated through a fluid there are, in general, two types of forces that are recognized, one being a drag component and a second, the inertial component. The nature of the flow produced by the time dependent motion of the immersed rigid object and the relative contribution of these two components of force is generally considered to be strongly dependent on the amplitude of the relative fluid motion in comparison to the characteristic lineal dimension of the object. For example, Keulegan and Carpenter [1] found that, for harmonic motion of the fluid past a fixed circular cylinder, major flow separation did not occur and the forces were well represented by potential flow values provided the amplitude of the motion was less than about a half diameter.

In oil and gas well drilling operations, it is of great significance to accurately predict the drag coefficient and settling velocity of drill cuttings in non-Newtonian drilling fluids. In this paper, the free-falling of 172 groups of spheres and 522 groups of irregular-shaped sand particles in Newtonian/non-Newtonian fluids were investigated experimentally. It was found that the drag coefficient calculated based on Newtonian correlations can result in a significant error when the particle settles in the non-Newtonian fluid. Therefore, predictive models of drag coefficient were established respectively for different types of fluids. The validity of the proposed drag coefficient model of spheres was verified by comparing it with the previous works. On this basis, the drag coefficient model of irregular-shaped sand particles was established by introducing a shape factor. The models do not use the shape factor that requires detailed three-dimensional shape and size information. Instead, two-dimensional geometric information (circularity) is obtained via image analysis techniques. The present new models predict the settling velocity of sand particles in the power-law fluid and Herschel–Bulkley fluid accurately with a mean relative error of 5.03% and 6.74%, respectively, which verifies the accuracy of the model.