Numerical simulation of oscillatory flows over a rippled bed by immersed boundary method

In this paper, a well-developed numerical model based on the immersed boundary (IB) method is used to study oscillatory flows over a bed with large-amplitude ripples in a systematic manner. The work shows that the complex flow over the rippled bed can be numerically dealt with in Cartesian coordinate by the IB method and that the IB method is able to provide main features of the flows near the ripples. An accurate simulation of vortices generation as a result of flow separation at the rippled bed is obtained. It is found that the oscillatory flows start to separate during the flow deceleration when the Keulegan–Carpenter (KC) number is small. The steady streaming for various ripple steepness is simulated and the criterion for separating the single and double structure streaming is also discussed. Moreover, a new type of steady streaming which consists of a pair of embedded recirculations in the vicinity of the ripple trough is obtained for relatively steep ripples in this work. The numerical results, including the steady streaming in particular, may be helpful to improve the understanding of the sediment transport and the seabed evolution with natural ripples under sea waves.     

Bottom friction caused by boundary layer streaming beneath random waves for laminar and smooth turbulent flow

The effect of boundary layer streaming on the sea bed shear stresses, beneath random waves, is investigated for laminar flow as well as smooth turbulent flow. It is demonstrated how bottom friction formulas for regular waves can be used to obtain the bed shear stresses resulting from steady streaming under random waves. As a result, friction factors for steady streaming under random waves are provided, and the effect of streaming versus the effect of linear waves is discussed. For laminar flow the effect of second order Stokes waves is also included. Examples are included to illustrate the applicability of the present practical method, and results are obtained using data typical for field conditions.     

Turbulence-induced steady streaming in an oscillating boundary layer: On the reliability of turbulence closure models

The accuracy of several closure models of the Reynolds-Averaged Navier–Stokes Equations in predicting the characteristics of an oscillating turbulent wall boundary layer is analyzed. The analysis involves four low Reynolds number k − ε models and a k − ω model and it is carried out by comparing the model results both with experimental data and with data obtained by a Direct Numerical Simulation (DNS) of the Navier–Stokes equations. The boundary layer is generated by a spatially constant time-oscillating pressure gradient given by the sum of two harmonic components characterized by angular frequencies Ω^∗ and 2Ω^∗ respectively, which generates a steady streaming because of the asymmetry of turbulence intensity during the cycle. Thus the results are relevant to the boundary layer at the bottom of nonlinear sea waves. The attention is therefore focused on the accuracy of the models in reproducing the period averaged profiles of the hydrodynamic characteristics of the steady streaming. The instantaneous quantities, such as time development of the wall shear stress, profiles of the streamwise velocity, Reynolds stresses and turbulent kinetic energy are also considered and analyzed. The results shows that a model can be judged better or worse than other models depending on the specific flow characteristic under investigation. However, an approach has been adopted which allowed to rank the models according to their accuracy in predicting the values of the hydrodynamic quantities involved in the present study.     

A numerical model for onset of scour below offshore pipelines

A numerical model is developed to predict the onset of local scour below offshore pipelines in steady currents and waves. The scour is assumed to start when the pressure gradient underneath the pipeline exceeds the floatation gradient of the sediments. In this model, the water flow field above the bed is determined by solving the two-dimensional (2-D) Reynolds-averaged Navier–Stokes equations with a k-ω turbulence closure. The seepage flow below the seabed is calculated by solving the Darcy's law (Laplace's equation) with known pressure distribution along the common boundaries of the flow domains-seabed. The numerical method used for both the turbulent flow around the pipeline and Darcy's flow in the seabed is a fractional finite element method. The average pressure gradient along the buried pipe surface is employed in the evaluation of onset condition with a calibration coefficient. The numerical model is validated against experimental data available in literature. A unified onset condition for steady currents and waves is proposed. Influences of flow parameters, including water depth, embedment depth, boundary layer thickness, Reynolds number (Re) and Keuleagan–Carpenter (KC) number, on the pressure drop coefficient over the pipeline are studied systematically.     

Wave boundary layer over a stone-covered bed

This paper summarizes the results of an experimental investigation on wave boundary layers over a bed with large roughness, simulating stone/rock/armour block cover on the sea bottom. The roughness elements used in the experiments were stones the size of 1.4cm and 3.85cm in one group of experiments and regular ping-pong balls the size 3.6cm in the other. The orbital-motion-amplitude-to-roughness ratio at the bed was rather small, in the range a/k_s = 0.6–3. The mean and turbulence properties of the boundary-layer flow were measured. Various configurations of the roughness elements were used in the ping-pong ball experiments to study the influence of packing pattern, packing density, number of layers and surface roughness of the roughness elements. The results show that the friction factor seems to be not extremely sensitive to these factors. The results also show that the friction factor for small values of the parameter a/k_s does not seem to tend to a constant value as a/k_s → 0 (contrary to the suggestion made by some previous investigators). The present friction-factor data indicates that the friction factor constantly increases with decreasing a/k_s. An empirical expression is given for the friction factor for small values of a/k_s. The results further show that the phase lead of the bed friction velocity over the surface elevation does not seem to change radically with a/k_s, and found to be in the range 12°–23°. Furthermore the results show that the boundary-layer turbulence also is not extremely sensitive to the packing pattern, the packing density, the number of layers and the surface roughness of the roughness elements. There exists a steady streaming near the bed in the direction of wave propagation, in agreement with the existing work. The present data indicate that the steady streaming is markedly smaller in the case where the ping-pong balls are aligned at 45° to the wave direction than in the case with 90° alignment. Likewise, it is found that the steady streaming is relatively smaller in the case of the one-layer ping-pong-ball roughness than in the case of the two-layer situation.     

Numerical and experimental study on seakeeping performance of ship in finite water depth

Vessels operating in shallow waters require careful observation of the finite-depth effect. In present study, a Rankine source method that includes the shallow water effect and double body steady flow effect is developed in frequency domain. In order to verify present numerical methods, two experiments were carried out respectively to measure the wave loads and free motions for ship advancing with forward speed in head regular waves. Numerical results are systematically compared with experiments and other solutions using the double body basis flow approach, the Neumann-Kelvin approach with simplified m-terms, and linearized free surface boundary conditions with double-body m-terms. Furthermore, the influence of water depths on added mass and damping coefficients, wave excitation forces, motions and unsteady wave patterns are deeply investigated. It is found that finite-depth effect is important and unsteady wave pattern in shallow water is dependent on both of the Brard number τ and depth Froude number F_h.     

URANS investigation of the interaction between the free surface and a shallowly submerged underwater vehicle at steady drift

An axisymmetric underwater vehicle (UV) at a steady drift angle experiences the complex three-dimensional crossflow separation. This separation arises from the unfavorable circumferential pressure gradient developed from the windward side toward the leeward side. As is well known, the separated flow in the leeward side gives rise to the formation of a pair of vortices, which affects considerably the forces and moments acting on the UV. In this regard, the main purpose of the present study is to evaluate the role of the leeward vortical flow structure in the hydrodynamic behavior of a shallowly submerged UV at a moderate drift angle traveling beneath the free surface. Accordingly, the static drift tests are performed on the SUBOFF UV model using URANS equations coupled with a Reynolds stress turbulence model. The simulations are carried out in the commercial code STARCCM+ at a constant advance velocity based on Froude number equal to F_n = 0.512 over submergence depths and drift angles ranging from h = 1.1D to h = ∞ and from β = 0 to β = 18.11°, respectively. The validation of the numerical model is partially conducted by using the existing experimental data of the forces and moment acting on the totally submerged bare hull model. Significant interaction between the low-pressure region created by the leeward vortical flow structure and the free surface is observed. As a result of this interaction, the leeward vortical flow structure appears to be largely responsible for the behavior of the forces and moments exerted on a shallowly submerged UV at steady drift.     

Direct numerical simulations of flow over two-dimensional and three-dimensional ripples and implication to sediment transport: Steady flow

A well resolved and highly accurate direct numerical simulation (DNS) solver has been developed to understand the implication of hydrodynamics to sediment transport. In the first part of the study we focus on steady flow over two-dimensional and three-dimensional ripples at two Reynolds numbers Re_τ = 180 and 400 (defined by channel half-height and wall-friction velocity) in a channel geometry. The DNS scheme is based on a fourth-order vertical velocity and second-order vertical vorticity formulation, which resolves the difficulties in pressure boundary condition encountered when solving the Navier–Stokes equations. The complex boundary introduced due to the ripples has been imposed in the Cartesian domain using an elegant immersed boundary method. Detailed hydrodynamic analysis has revealed turbulence statistics (in particular, the higher order) and henceforth, the flow structures are sensitive — whether the ripples are two-dimensional or three-dimensional. The importance of fluctuating component of the bottom stress in addition to its mean component; and its significance to sediment transport and ripple migration speed have been investigated.     

Numerical study on scale effect of form factor

The scale effect of form factor is investigated via a numerical approach in this paper, where the turbulent ship flow is computed by solving the steady and incompressible Reynolds-averaged Navier-Stokes and continuity equations. A wall function approach is employed to bridge the near-wall and outer turbulent flow region. The numerical scheme based on a finite-volume formulation is applied to discretize the coupled governing equation. For the sake of numerical stability, accuracy and economy, an identical grid is employed to compute ship flow at different Reynolds number, where the grid is optimized for the medium Reynolds number of the investigated range. Four surface ships and two sub-bodies with notably different geometrical characteristics are chosen as the investigated cases, where double-model flow without appendages is computed. The calculated total resistance coefficient shows a decreasing tendency against Reynolds number among all studied hulls. Similar to the calculated total resistance coefficient, the calculated friction resistance coefficient decreases with the Reynolds number and varies relatively little for a given Reynolds number among different hulls. The viscous pressure resistance coefficient is less insensitive to the Reynolds number but apparently depends on hull form. Compared with the form factor calculation based on empirical friction lines, the flat-plate friction prediction based on CFD approach clearly gives smaller Re-dependent form factor, which should more realistically reflect the scale effect of form factor. The form factor exhibits a near linear and increasing dependence on Reynolds number. The numerical results show that the dependence of r_P on Reynolds number mainly governs the scale effect of form factor.     

The influence of corner radius on the forces experienced by cylindrical bluff bodies in oscillatory flow

Measurements are presented of the effects of corner radius on the hydrodynamic forces experienced by cylindrical bluff bodies in oscillatory flow over the Keulegan Carpenter number range from 1 to 100. The range of Reynolds number covered is from 200 to 2 × 10⁴ and two basic cross-sectional shapes, namely a square and a diamond, were used in the investigation. Drag coefficients in oscillatory flow are found to be more sensitive to corner radius than in steady flow. At a fixed Keulegan Carpenter number the drag coefficient of a diamond section decreases with increasing corner radius and the reduction is more marked at low Keulegan Carpenter numbers. The drag of a square section does not follow a consistent trend with corner radius and for high Keulegan Carpenter numbers the rounded square with a corner radius of 0.265D (where D is the width of the model normal to the flow) can have a drag coefficient equal to half the value for a circular cylinder.     

Numerical simulation of a partially buried pipeline in a permeable seabed subject to combined oscillatory flow and steady current

Hydrodynamic forces exerting on a pipeline partially buried in a permeable seabed subjected to combined oscillatory flow and steady current are investigated numerically. Two-dimensional Reynolds-Averaged Navier-Stokes equations with a k−ω turbulent model closure are solved to simulate the flow around the pipeline. The Laplace equation is solved to calculate the pore pressure below the seabed with the simulated seabed hydrodynamic pressure as boundary conditions. The numerical model is validated against the experimental data of a fully exposed pipeline resting on a plane boundary under various flow conditions. Then the flow with different embedment depths, steady current ratios and KC numbers is simulated. The amplitude of seepage velocity is much smaller than the amplitude of free stream velocity as expected. The normalized Morison inertia, drag and lift coefficients based on the corresponding force coefficients of a fully exposed pipeline are investigated. The normalized Morison force coefficients reduce almost linearly with the increase of embedment depth and that the KC only has minor effect on the normalized Morison coefficients. It is also found that the permeable seabed condition causes a slight increase on the inline force and has a little effect on the lift force, compared with corresponding conditions in an impermeable bed.     

Numerical modeling of flow and scour below a pipeline in currents: Part II. Scour simulation

A vertical two-dimensional (2D) numerical model for time dependent local scour below offshore pipelines subject to unidirectional steady flow is developed. The governing equations for the flow and sediment transport are solved by using finite difference method in a general curvilinear coordinate system. The performance of two turbulence models, the standard k–ɛ model and Smagorinsky subgrid scale (SGS) model, on modeling time dependent scour processes is examined. Both suspended load and bed load are considered in the scour model. The suspended-load model is verified against two channel sediment transport cases. The change of bed level is calculated from the continuity equation of total sediment transport. A new time marching scheme and a sand slide scheme are proposed for the scour calculation. It is found that the proposed time marching scheme and sand slide model work well for both clear-water and live-bed scour situations and the standard k–ɛ turbulence closure is more preferable than the SGS model in the 2D scour model developed in this study.     

Numerical simulation of flow around a circular cylinder close to a flat seabed at high Reynolds numbers using a k–ε model

High Reynolds number flows around a circular cylinder close to a flat seabed have been computed using a two-dimensional standard high Reynolds number k–ε turbulence model. The effects of gap to diameter ratio, Reynolds number and flat seabed roughness for a given boundary layer thickness of the inlet flow upstream of the cylinder have been investigated. Hydrodynamic quantities and the resulting bedload transport have been predicted, and the vortex shedding mechanisms have been investigated. Predictions of hydrodynamic quantities around a cylinder located far away from the bed (so that the effect of the bed is negligible) are in satisfactory agreement with published experimental data and numerical results obtained for the flow around an isolated cylinder. Results for lower Reynolds number flows have also been computed for comparison with the high Reynolds number flow results. Overall it appears that the present approach is suitable for design purposes at high Reynolds numbers which are present near the seabed in the real ocean.     

Drag and inertia coefficients for a circular cylinder in steady plus low-amplitude oscillatory flows

Computer simulations of steady plus low-amplitude oscillatory flow about a circular cylinder are reported at a fixed Reynolds number of 150 based on the steady component. The conventional Keleugan–Carpenter number based on the oscillatory component is fixed at π/5. The oscillation frequency is varied so as to study a wide spectrum of flows where inertial forces dominate at one end and viscous drag forces at the other as a function of the modified Keleugan–Carpenter number. The hydrodynamic force on the cylinder in-line with the flow direction is represented by Morison's equation and an extended version with three terms. The drag and inertia coefficients in Morison's equation are determined by least-squares fits to data directly computed from integration of skin friction and pressure distributions around the periphery of the cylinder. The root-mean-square value of the residue of reconstructed minus directly-computed forces varies between 2 and 41% depending on the flow parameters. Comparable results can be obtained with a semi-theoretical approach using inviscid inertia and quasi-steady viscous drag terms. Physical explanations for the variation of the force coefficients are provided and implications for pertinent flow–structure interactions are discussed.     

Potential vorticity estimates of absolute velocities on the Ross Sea shelf

We analyze absolute velocities on the continental shelf off Cape Adare, in the western sector of the Ross Sea (Antarctica). Such a velocity field is here inferred by using a novel inverse method of absolute velocity determination, namely the tracer PV method, related to potential vorticities of temperature and salinity. This theoretical choice allows us to directly use in situ temperature and salinity data. Moreover, it avoids high-order derivatives, which can give large uncertainties that affect estimates made using previous approaches. The tracer PV method also allows us to separately estimate the steady and non-diffusive component and the unsteady and diffusive components of the flow. The western sector of the Ross Sea is characterized by a surface layer of Antarctic Surface Water over layers of Low Salinity Shelf Water and High Salinity Shelf Water, flowing northward with average velocities ～6–7 cm/s. At ～200 m depth an intrusion of warmer and saltier Circumpolar Deep Water is also evident in our data. The steady absolute velocities are in good agreement with those obtained from the classical Margules equation, in particular regarding the northward flux of the High Salinity Shelf Water. Furthermore, velocities due to diffusive processes and mesoscale activity are discussed. Finally, a steady “thermal” approximation is discussed; it allows for a qualitative check of the results by means of temperature horizontal sections only.     

Relevance of transition turbulent model for hydrodynamic characteristics of low Reynolds number propeller

The propeller of an Autonomous Underwater Vehicle (AUV) operates at low Reynolds number in laminar to turbulent transition region. The performance of these propellers can be calculated accurately using RANSE solver with γ − Re_θ transition model. In this study, the global and local hydrodynamic characteristics of open and ducted propeller are investigated using the γ − Re_θ transition model. The capability of the γ − Re_θ transition model to capture laminar to turbulent transition on the surface of the open propeller is demonstrated by comparison with published experimental results. The application of transition model for the propeller K_a-4-70 inside the duct 19A shows that the centrifugal forces are dominant at low Reynolds number and the flow is mainly directed in the radial direction. The transition model is able to predict complex flow physics such as leading-edge separation, tip leakage vortex, and the separation bubble on outer surface of the duct. The accurate prediction of these flow phenomenon can lead to correct calculation of global hydrodynamic forces and moments acting on the propeller at low Reynolds number.     

Buckling of cross-ply cylinders under hydrostatic pressure considering pressure stiffness

Buckling behavior of cross-ply cylinders under hydrostatic pressure is investigated using a semi-analytical finite element based on a consistent first order shear deformable shell theory. Potential loss due to external pressure, also called pressure stiffness (PS) is taken into account by making use of Koiter's related energy expression. A number of verification problems are solved and the numerical results are compared with the analytical results available in the literature and excellent agreement is observed. New numerical results are presented to assess the effect of PS on buckling due to hydrostatic pressure. It is shown that PS causes a decrease in the buckling load and this decrease depends on the size of the cylinder and the material. Also, issues related to thickness optimization are examined and optimal lamina thicknesses are determined for a number of cases with and without PS taken into account.     

On helical vortex motions of moist air

Two results that are fundamentally different from what takes place in a dry atmosphere have been obtained for adiabatic motions of unsaturated moist air: (1) the steady helical motion of moist air with collinear velocity and vorticity vectors everywhere is dynamically impossible; (2) the spontaneous amplification (generation) of helicity in a moist air due to baroclinicity is dynamically and thermodynamically feasible. In the absence of helicity flux through the boundary of the domain occupied by air flows, the difference between the values of integral helicity H at time instant t delaying at a small time interval from the initial instant t ₀ (at which the instantaneous state of air motion is isomorphic either to a steady Beltrami flow or to an irrotational flow) and the initial value of H increases proportionally to (t ? t ₀)⁴. The nonzero value of the proportionality factor is ensured by the difference in values of the Poisson ratio for dry air and water vapor, respectively.     

Hydrodynamic forces on a partially submerged cylinder at high Reynolds number in a steady flow

The hydrodynamic forces on the stationary partially submerged cylinder are investigated through towing test with Reynolds number ranging from 5 × 10⁴ to 9 × 10⁵. Three test groups of partially submerged cylinders with submerged depths of 0.25 D, 0.50 D, and 0.75 D and one validation group of fully submerged cylinders are conducted. During the experiments, the hydrodynamic forces on the cylinders are measured using force sensors. The test results show a considerable difference in the hydrodynamic coefficients for the partially submerged cylinders versus the fully submerged cylinders. A significant mean downward lift force is first observed for the partially submerged cylinders in a steady flow. The maximum of the mean lift coefficients can reach 1.5. Two distinct features are observed due to the effects of overtopping: random distributions in the mean drag coefficients and a clear quadratic relationship between the mean lift coefficients and the Froude number appear in the non-overtopping region. However, the novel phenomenon of a good linear relationship with the Froude number for the mean hydrodynamic coefficients is clearly shown in the overtopping region. In addition, fluctuating hydrodynamic coefficients are further proposed and investigated. These results are helpful to have a better understanding of the problem and to improve related structural designs.