Boundary-fitted Coordinate System Research Articles

A finite volume formulation for one-dimensional Stefan problems

This paper presents a numerical formulation that combines the finite-volume method with a time-dependent boundary-fitted coordinate system to tackle the moving boundary problems for the one-dimensional (1-D) heat equation. Following the formulation process, an algorithm was created with the help of the linear algebra package Eigen to generate the data for the numerical solution. Four benchmark cases of heat conduction were investigated by this approach and compared to the analytical solution: a fixed boundary case, the liquid phase of a melting process, the solid phase of a melting process, and an ablating process. The results demonstrate the capability of the presented numerical formulation in efficiently solving the general 1-D heat problems, especially the one-phase Stefan problems, as well as the consistency of the numerical and analytical solutions. Future works can strive to extend the formulations to Stefan problems of two phases or higher dimensions. The work can then be extended to real-life applications, such as the spacecraft re-entry ablation process, and the process of iceberg melting, demonstrating its critical value in solving practical problems.

Diffraction and Refraction of Nonlinear Waves by the Green–Naghdi Equations

Abstract Diffraction and refraction of nonlinear shallow water waves due to uneven bathymetry are studied by use of the Green–Naghdi (GN) equations in three dimensions. A numerical wave tank consisting of deep, transitional, and shallow regions is created. Various forms of three-dimensional bathymetry, consisting of ramps with nonuniform profiles and large slopes, are used to connect the deep-water side of the tank to the shallow water shelf. A wavemaker is placed at the upwave side of the domain, capable of generating solitary and cnoidal waves of the GN equations. A numerical wave absorber is located downwave of the domain to minimize the wave reflection back into the domain. The system of equations is solved numerically in time domain by use of a second-order finite-difference approach for spatial discretization, and in a boundary-fitted coordinate system, and by use of the modified Euler method for time marching. Results include solitary and cnoidal wave surface elevation and particle velocities and are compared with the existing solutions where possible. Overall, very good agreement is observed. Discussion is provided on the nonlinearity and dispersion effects on the wave diffraction and refraction by the various forms of the ramps, as well as on the performance of the GN equations in solving these problems.

Bend stiffener linear viscoelastic thermo-mechanical analysis, Part II: Numerical solution and case study

The viscoelastic energy dissipated by the bend stiffener when subjected to cyclic loading increases the polyurethane temperature and may affect the system curvature distribution in a coupled manner. The steady-state thermo-mechanical mathematical formulation and material experimental characterization has been presented in a companion paper (Part I). In this work (Part II), the rate of viscoelastic mechanical energy dissipation is obtained from the steady-state mechanical formulation employing the shooting method combined with the Runge–Kutta method for the two-point boundary value problem numerical solution. The temperature field distribution within the bend stiffener volume is then estimated with the three-dimensional steady-state thermal model employing the finite difference technique with a boundary-fitted coordinate system that transforms the physical domain into a structured computational domain grid. The partial differential equations governing heat transfer are solved in the computational domain and transformed back into the physical domain by transformation relations. A detailed description of the iterative thermo-mechanical numerical solution procedure is presented and a case study carried out demonstrating the loading frequency and oscillation amplitude influence on the bend stiffener temperature field distribution. It is shown that the bend stiffener may be exposed to high internal temperatures for some loading conditions.

Best groyne series as countermeasure against river bend scour

A new approach for finding a cost-effective combination of groynes in a channel bend for the prevention of bend scour is presented. A linked simulation–optimisation model is proposed for this purpose. The morphological flow simulation model was developed by solving the shallow-water equations and the Exner sediment continuity equation in a boundary-fitted coordinate system. The optimisation model was developed utilising a binary-coded genetic algorithm. The proposed model can automatically provide a cost-effective groyne series in terms of their number, length and position in any vulnerable channel bend. A constraint in terms of bend scour is also incorporated in the optimisation model, which allows the optimisation model to obtain the best groyne combination for no bend scour. To appraise the performance of the model, it was applied to a hypothetical vulnerable channel bend. The model produced the minimised cost of installation of the groyne system at that location, at the same time as making the location free from bend scour.

Theoretical and Experimental Study of Transient Behavior of Spiral-Groove Thrust Bearings during Start-Up

In this study, a transient mixed lubrication model that considers mass-conserving cavitation and deterministic asperity contacts is established to study the evolution of the tribological behavior of spiral-groove thrust bearings during start-up. To overcome oil film thickness discontinuity at the edges and the complicated boundary shape of spiral grooves, a boundary-fitted coordinate system transformation is adopted. The evolution of the tribological behavior, including the asperity contact force, load-carrying capacity, minimum oil film thickness, and friction torque during start-up, is presented using the current model. To illustrate more clearly, the evolution of hydrodynamic pressure and cavitation fraction is also displayed. Experiments are performed to verify the proposed theoretical model. The results demonstrate that this model is able to capture the evolution of the tribological behavior of spiral-groove thrust bearings during start-up. To improve start-up performance, a parametric analysis is conducted to identify the effects of the main design parameters (groove depth, groove width ratio, and spiral angle) on the lift-off speed.

A 3D non-hydrostatic model for wave interactions with structures using immersed boundary method

Immersed boundary (IB) method is incorporated in a three-dimensional (3D) non-hydrostatic model for predicting wave-structure interactions. The model solves the Navier–Stokes equations by means of an explicit projection method. The 3D grid system is built based upon a horizontal Cartesian grid framework and a vertical general boundary-fitted coordinate system. The vertical grid system not only has the boundary-fitted properties of better fitting the bed and free surface, but also allows for a great adaptability of the vertical discretization. The numerical model is validated against the experimental data of wave interactions with various types of structures including submerged bottom-mounted structure, suspended structure, and surface-piercing bottom-mounted structure. Good agreement between the model results and experimental data are achieved, demonstrating the capability of the developed model on resolving wave-structure interactions. Thus, the proposed non-hydrostatic model with IB method can provide an alternative to studying wave interactions with complex structures.

Semi-implicit non-hydrostatic model for 2D nonlinear wave interaction with a floating/suspended structure

A non-hydrostatic model is developed to simulate two-dimensional (2D) nonlinear wave interaction with a fixed floating/suspended structure. The immersed boundary method is incorporated in the model to address arbitrary body surfaces. The model employs a semi-implicit, fractional step algorithm to solve the Reynolds-averaged Navier–Stokes (RANS) equations based on a grid system, which is built from a horizontal rectangular grid by adding dozens of horizontal layers. The horizontal layers are distributed following a general boundary-fitted vertical coordinate system. This grid system may overcome the lack of flexibility of other boundary-fitted coordinate systems (e.g., the σ-coordinate system). The model is validated by using three test cases, namely, a solitary wave interacting with a floating rectangular obstacle, a solitary wave incident on a suspended horizontal plate and a solitary wave incident on a coastal-bridge deck with girders. Numerical results of the free surface elevation and wave force exerted on each structure are compared with the results of other models or experimental data. Generally, good agreements are obtained, demonstrating the proposed model’s capability to resolve nonlinear wave interactions with floating/suspended structures. The velocity fields around the structures are also presented and discussed.

Numerical simulations and experimental visualizations of the vortex characteristics for a solitary wave interacting with a bottom-mounted vertical plate

This study is aimed to investigate numerically and experimentally the interaction of a solitary wave with a bottom-mounted vertical thin plate with focuses on the wave induced fluid kinematics and vortex flow patterns. A streamfunction–vorticity free-surface (SVFS) based fully nonlinear viscous wave model solved in a transient boundary-fitted coordinate system with locally overlaid grids is applied to study this wave-plate interaction problem. Qualitative comparisons between numerically generated flow patterns around the plate and the experimental observations, including images from both the particle-tracking and the laser-dye visualization methods, are presented. The shear-layer velocity profiles are compared quantitatively with the experimental measurements. The detailed flow characteristics and formed vortices including the effect of the shape of the plate top on the vortex formation and development are numerically investigated. For a thin vertical plate, the flow characteristics that are affected by the dimensionless parameters of incident-wave height and plate height are also investigated. The numerical visualizations illustrated by the virtual transport of particle-tracing, streamlines, and equi-vorticity lines are useful to understand the kinematic behaviors of the induced vortical motions. Furthermore, the pressure gradient, shear stress, and maximum wave force are examined to detail the hydrodynamic impacts on the submerged plate.

Direct numerical simulation of flow around a heated/cooled isolated sphere up to a Reynolds number of 300 under subsonic to supersonic conditions

In this study, an analysis of the flow properties around an isolated sphere under isothermal conditions for flows with high Mach numbers and low Reynolds numbers is conducted via direct numerical simulation (DNS) of the three-dimensional compressible Navier–Stokes equations. The calculations are performed with a boundary-fitted coordinate system. The Reynolds number based on the diameter of the sphere and the freestream quantities is varied from 100 to 300, the freestream Mach number is varied between 0.3 and 2.0, and the temperature ratio between the sphere surface and the freestream is varied between 0.5 and 2.0. We focus on the effects of the Mach number and the temperature ratio on the flow properties. The results show the following characteristics: (1) unsteady vortex shedding from the sphere is promoted (suppressed) when the temperature ratio is less (greater) than unity; (2) the drag coefficient increases with the temperature ratio, but previous drag relations give poor prediction on effect of the temperature ratio on the drag coefficient in the continuum regime; (3) Nusselt number relations proposed in previous studies can be applied if the temperature ratio is close to unity under subsonic conditions; (4) the changes in several flow properties can be characterized by a separation point in the range investigated.

Nonlinear Wave Kinematics near the Ocean Surface

AbstractEstimation of second-order, near-surface wave kinematics is important for interpretation of ocean surface remote sensing and surface-following instruments, determining loading on offshore structures, and understanding of upper-ocean transport processes. Unfortunately, conventional wave theories based on Stokes-type expansions do not consider fluid motions at levels above the unperturbed fluid level. The usual practice of extrapolating the fluid kinematics from the unperturbed free surface to higher points in the fluid is generally reasonable for narrowband waves, but for broadband ocean waves this results in dramatic (and nonphysical) overestimation of surface velocities. Consequently, practical approximations for random waves are at best empirical and are often only loosely constrained by physical principles. In the present work, the authors formulate the governing equations for water waves in an incompressible and inviscid fluid, using a boundary-fitted coordinate system (i.e., sigma or s coordinates) to derive expressions for near-surface kinematics in nonlinear random waves from first principles. Comparison to a numerical model valid for highly nonlinear waves shows that the new results 1) are consistent with second-order Stokes theory, 2) are similar to extrapolation methods in narrowband waves, and 3) greatly improve estimates of surface kinematics in random seas.

A 3D unstructured non-hydrostatic ocean model for internal waves

A 3D non-hydrostatic model is developed to compute internal waves. A novel grid arrangement is incorporated in the model. This not only ensures the homogenous Dirichlet boundary condition for the non-hydrostatic pressure can be precisely and easily imposed but also renders the model relatively simple in its discretized form. The Perot scheme is employed to discretize horizontal advection terms in the horizontal momentum equations, which is based on staggered grids and has the conservative property. Based on previous water wave models, the main works of the present paper are to (1) utilize a semi-implicit, fractional step algorithm to solve the Navier-Stokes equations (NSE); (2) develop a second-order flux-limiter method satisfying the max–min property; (3) incorporate a density equation, which is solved by a high-resolution finite volume method ensuring mass conservation and max–min property based on a vertical boundary-fitted coordinate system; and (4) validate the developed model by using four tests including two internal seiche waves, lock-exchange flow, and internal solitary wave breaking. Comparisons of numerical results with analytical solutions or experimental data or other model results show reasonably good agreement, demonstrating the model’s capability to resolve internal waves relating to complex non-hydrostatic phenomena.

Investigation on subsonic to supersonic flow around a sphere at low Reynolds number of between 50 and 300 by direct numerical simulation

In this study, analysis of flow properties around a sphere and its aerodynamic coefficients in the high-Mach-and-low-Reynolds-numbers conditions is carried out by direct numerical simulations solving the three-dimensional compressible Navier–Stokes equations. The calculation is performed on a boundary-fitted coordinate system with a high-order scheme of sufficient accuracy. The analysis is conducted by assuming a rigid sphere with a Reynolds number of between 50 and 300, based on the diameter of the sphere and the freestream velocity and a freestream Mach number of between 0.3 and 2.0, together with the adiabatic wall boundary condition. The calculation shows the following yields: (1) unsteady fluctuation of hydrodynamic forces become smaller as the Mach number increases under the same Reynolds number condition, (2) the drag coefficient increases with the Mach number due to an increase in the pressure drag by the shock wave, and (3) an accurate prediction of the drag coefficient in the supersonic regime using traditional models might be difficult.

Numerical study on three-dimensional waves produced by a bottom jet

The eruption of an underwater volcano can initiate the disturbances of the sea surface and subsequently generate a group of outward-propagating tsunamis. The theme of this study is to introduce a three-dimensional (3D) fully nonlinear wave model for the simulation of wave motions induced by a bottom jet. A boundary-fitted coordinate system is utilized to conveniently adjust grids according to the transient moving free surface. The governing Laplace equation of the velocity potential is solved by an implicit finite-difference scheme while a mixed explicit/implicit iteration procedure is applied to solve the free-surface boundary conditions. In addition, a set of generalized Boussinesq equations are solved for comparison with the fully nonlinear model. Good agreements in comparisons with the existing numerical and analytical solutions are achieved for cases investigated. Waves induced by three types of bottom jets: namely (1) sudden eruption, (2) initial transient, and (3) periodic transient are discussed in this paper. For the case of sudden erupted jet, a system of 3D outgoing waves as the cylindrical wave pattern are presented and discussed. For the initial transient types, it shows the transition in the incipient stage has a great influence on the initial rising of the water surface and the induced leading waves. Furthermore, an interesting up-down phenomenon in the center of disturbed free surface due to the type of periodic jet is revealed.

Accurate Predictions of Velocity Profiles and Frictional Pressure Losses in Annular YPL-Fluid Flow

Summary Eccentricity of the annulus can greatly affect the velocity profile, especially in extended-reach wells and slimholes. While frictional pressure losses in concentric and fully eccentric annuli have been studied before, this paper focuses on the effect of arbitrary eccentricities on velocity profile and corresponding influence on frictional pressure losses. In this study, axial flow of yield-power-law (YPL) fluids in eccentric annuli for a 2D steady-state flow has been investigated numerically and verified against experiments. A boundary-fitted coordinate system is used to discretize the flow equations and generate the mesh network. Fluid-flow equations were solved adopting an iterative method. The results of simulation include the effects of fluid rheology, flow rate, annulus dimensions, and eccentricity on velocity profile and frictional pressure losses in the annulus. Numerical results were compared with the available extensive experimental investigations on the flow of a variety of drilling fluids that have a strong shear-thinning property and high yield stress (e.g., polymer-based and bentonite fluids). The tests were implemented over a wide range of flow rates using a flow loop that is equipped with a pipe viscometer and several annular test sections with various sizes and eccentricities. Field observations, experimental data, and the analytical approach in this study indicate that increasing eccentricity lowers frictional pressure drop in the annulus. A good agreement was observed between the numerical-simulation results and experimental measurements. Comparison of the present and past studies with analytical solutions of Newtonian fluids in an eccentric annulus shows that the current study provides more-accurate results. Detailed numerical simulation and the equations that are presented in this study can be used by investigators. The application of the Cartesian and boundary-fitted coordinate systems and algebraic correlations for geometry transformations are presented. Moreover, the exact method of flow-rate calculation based on the velocities at each gridpoint is shown in the provided details. The application of findings in this study includes more-accurate predictions of velocity profile and frictional pressure loss in the annulus, which lead to better predictions of barite sag, cuttings transport, equivalent circulating density, and mud cement displacement in special cases of cementing operations.

Numerical simulation of the fluvial processes in the channels by sediment mining

This study aims to experimentally and numerically investigate the fluvial processes of channels disturbed through sand or gravel mining. A two-dimensional numerical model is applied to simulate flume experiments. A moving boundary-fitted coordinate system is used to describe the channel changes. As a numerical scheme, the cubic interpolated pseudoparticle method is used to calculate the flow field and bed changes, which introduces little numerical diffusion. The numerical model is verified with experimental data. The inflow sediment from the upstream is trapped into a pit and the sediment deposit wedge migrated to the downstream with a steep submerged angle of repose. The numerical and experimental results show that the sediment deposit wedge in the pit migrates to the downstream under the condition of steady state. The adjustment time within the disturbed channels increases as the dimension of the pit is increased. The sediment mining pit migrates with speed in the steep channel. Moreover, the numerical experiments show that the migration speed is proportional to the sediment inflow. And the disturbed channels by the sediment mining are affected by the sediment inflow conditions of the upstream and adjusted to the new state of equilibrium.

A general boundary-fitted 3D non-hydrostatic model for nonlinear focusing wave groups

This paper employs a three-dimensional (3D) non-hydrostatic model to simulate nonlinear focusing wave groups. The non-hydrostatic model utilizes an explicit projection method to solve the Navier–Stokes equations. To accurately simulate the steep free surface involved in focusing waves, the model is built upon a general boundary-fitted coordinate system. This grid system allows for a great adaptability of the vertical discretization and meanwhile maintains the boundary-fitted properties of better fitting the bed and free surface. The advantage of the general boundary-fitted model is first validated by two test cases of nonlinear waves, including nonlinear standing waves and two-dimensional (2D) focusing freak wave. Then, the model is applied to simulate 2D focusing waves in deep and intermediate-water depths and 3D focusing waves in deep-water depth. By comparing with experimental data, the model results well reproduce the main characteristics of 2D deep-water focusing waves and 2D intermediate-water focusing waves as well as 3D deep-water focusing waves, demonstrating the model׳s capability to resolve 2D or 3D focusing wave groups. Furthermore, in the test of 2D intermediate-water focusing waves, the downstream shifting of the focusing position and time is also studied numerically, which is not presented in the experiments.

Numerical study of turbulent wall jet over multiple-inclined flat surface

In the present work flow field of a turbulent oblique wall jet with multiple flat inclined wall has been numerically investigated. The Reynolds averaged Navier–Stokes (RANS) equations for incompressible flow are solved in transformed geometry using boundary-fitted coordinate system. The aim of the work is to investigate the influence of three inlet conditions: uniform, parabolic and trapezoidal inlet velocity profiles, on the mean flow and turbulent characteristics of the jet flowing tangentially along a multiple inclined solid wall. The results show that the inlet conditions affect the flow in the vicinity of the nozzle exit area and also the far field region. Mean flow has been presented in terms of evolution of mean velocity profiles, jet spread, decay of streamwise maximum velocity, distribution of wall pressure and skin friction coefficients, and streamwise variations of integrated momentum and energy fluxes. Discontinuities in the boundary slope of the segmented wall produces oscillations in distribution of skin friction and pressure coefficients. Reduced pressure in the region of wall discontinuity appeared to be causing the deflected jet to remain attached to the wall. The growth of the inner layer of the jet is relatively high with uniform inlet velocity profiles in the near-field region, but in the far-field region the inner layer grows with higher rate for the trapezoidal and the parabolic profiles. The rate of decay of maximum streamwise velocity for oblique wall jet on segmented wall is higher than that of plane wall jet. The jet with parabolic and trapezoidal inlet profiles show higher rate of decay than that with a uniform inlet profile. Similarity of streamwise velocity and velocity component parallel to the oblique wall has been observed in the developed region of flow. The volume flow rate of the jet varies linearly along the flow direction due to entrainment of the fluid from the surrounding. Sudden change in slope of the wall causes a sharp rise in entrained volume of fluid. The distribution of near wall velocity in the inner coordinates of the boundary layer is grossly altered and the profiles do not conform to the universal velocity profiles in the form of the law of the wall. Relatively higher position of defect law data signifies comparatively lower turbulence in the outer mixing layer. Cross-stream distribution of Reynolds stresses shows that the effect of inlet velocity profiles is more pronounced in the near field region of flow. Self-similarity is observed in the normalized turbulent shear stress profiles whereas considerable scatter can be noticed in the profiles of turbulent normal stresses for all the inlet velocity conditions. Two distinct peaks in the normalized turbulent kinetic energy profiles characterize the two shear layers of the jet and shifting of the outer peak toward the wall suggests a strong interaction of the outer layer with the inner layer as the jet develops.

Numerical Solution of Pressure Underground Seepage Flow Using a Technique of Curvilinear Boundary-Fitted Coordinates

The technique of curvilinear boundary-fitted coordinate system is used for the mathematical model of pressure underground seepage flow and the boundary conditions and the method of solving the transformed equations have been presented. The computing example shows that the computed hydraulic parameters conform to the physical laws. Compared with the finite element method, the presented method need not solve the large matrix equations, and the computer memories and the time for computing are less. The mathematical model is effective for numerical solution of pressure underground seepage flows with complicated boundaries, and can be used in practice hydraulic engineering. Keywords-curvilinear boundary-fitted coordinate, pressure underground seepage flow, mathematical model

A three-dimensional hydrodynamic and salinity transport model of estuarine circulation with an application to a macrotidal estuary

A three-dimensional semi-implicit finite volume numerical model has been developed and applied to study tidal circulation and salinity stratification in the region of Oujiang River Estuary, China. The model employs horizontally unstructured grids and boundary-fitted coordinate system in the vertical direction. Governing equations consisting of continuity, momentum, and transport equations are all solved in the integral form of the equations, which provides a better representation of the conservative laws for mass, momentum, and transport in the coastal region with complex geometry and bottom bathymetry. The model performance was firstly quantified with skill assessment statistics on the choice of different parameters and validated with observed tidal elevation, current velocity, direction and salinity data over a spring–neap tidal cycle collected in 2006. Numerical results show that the model with wetting–drying capability successfully simulated the tidal currents and salinity fields with a reasonable accuracy and indicate that the Oujiang River Estuary is a macrotidal estuary with strong tidal mixing. In addition, the model results also show that the Oujiang River Estuary is a well-mixed estuary during spring tide. Then, the numerical simulations were performed to compare the hydrodynamic process and salinity distribution before and after a river training, which was conducted by blocking the south branch of the Oujiang River mouth. The results reveal that with the only north access to the sea, the influence of the blocking project on the flood discharge capacity is limited and the incremental velocity is beneficial to the navigation channel maintenance, although it will cause some scour to the embankment. Furthermore, the redistribution of tidal prism passing in or out the north branch makes a little severe salinity intrusion during high tide or low tide. However, the salinity intrusion is still within acceptable range, although it can cause some adverse effect on water intaking of production and life. The variations of salinity levels in Yueqing Bay situated at the north of the river mouth are not obvious, so the blocking project will not bring damage to local aquiculture. However, significant changes of salinity happen inside or outside of the south branch, so enough attention need to be paid to the changes of environment caused by the salinity variation after the blocking project. Overall, by weighing advantages and disadvantages of the blocking project, it is feasible and the model can be considered as a tool for managing and studying estuarine circulation.

Simulation of Seepage Flow in Earth Dam with a Free Surface

The technique of curvilinear boundary-fitted coordinate system is used for the mathematical model of pressure underground seepage flow and the boundary conditions and the method of solving the transformed equations have been presented. The computing example shows that the computed hydraulic parameters conform to the physical laws. Compared with the finite element method, the presented method need not solve the large matrix equations, and the computer memories and the time for computing are less. The mathematical model is effective for numerical solution of seepage flows with complicated boundaries, and can be used in practice hydraulic engineering.