Unsteady RANS method for surface ship boundary layer and wake and wave field

Results are reported of an unsteady Reynolds‐averaged Navier–Stokes (RANS) method for simulation of the boundary layer and wake and wave field for a surface ship advancing in regular head waves, but restrained from body motions. Second‐order finite differences are used for both spatial and temporal discretization and a Poisson equation projection method is used for velocity–pressure coupling. The exact kinematic free‐surface boundary condition is solved for the free‐surface elevation using a body‐fitted/free‐surface conforming grid updated in each time step. The simulations are for the model problem of a Wigley hull advancing in calm water and in regular head waves. Verification and validation procedures are followed, which include careful consideration of both simulation and experimental uncertainties. The steady flow results are comparable to other steady RANS methods in predicting resistance, boundary layer and wake, and free‐surface effects. The unsteady flow results cover a wide range of Froude number, wavelength, and amplitude for which first harmonic amplitude and phase force and moment experimental data are available for validation along with frequency domain, linear potential flow results for comparisons. The present results, which include the effects of turbulent flow and non‐linear interactions, are in good agreement with the data and overall show better capability than the potential flow results. The physics of the unsteady boundary layer and wake and wave field response are explained with regard to frequency of encounter and seakeeping theory. The results of the present study suggest applicability for additional complexities such as practical ship geometry, ship motion, and maneuvering in arbitrary ambient waves. Copyright © 2001 John Wiley & Sons, Ltd.     

Least‐squares meshfree method for incompressible Navier–Stokes problems

A least‐squares meshfree method based on the first‐order velocity–pressure–vorticity formulation for two‐dimensional incompressible Navier–Stokes problem is presented. The convective term is linearized by successive substitution or Newton's method. The discretization of all governing equations is implemented by the least‐squares method. Equal‐order moving least‐squares approximation is employed with Gauss quadrature in the background cells. The boundary conditions are enforced by the penalty method. The matrix‐free element‐by‐element Jacobi preconditioned conjugate method is applied to solve the discretized linear systems. Cavity flow for steady Navier–Stokes problem and the flow over a square obstacle for time‐dependent Navier–Stokes problem are investigated for the presented least‐squares meshfree method. The effects of inaccurate integration on the accuracy of the solution are investigated. Copyright © 2004 John Wiley & Sons, Ltd.     

Coupling methods between finite element–based Boussinesq‐type wave and particle‐based free‐surface flow models

We develop one‐way coupling methods between a Boussinesq‐type wave model based on the discontinuous Galerkin finite element method and a free‐surface flow model based on a mesh‐free particle method to strike a balance between accuracy and computational cost. In our proposed model, computation of the wave model in the global domain is conducted first, and the nonconstant velocity profiles in the vertical direction are reproduced by using its results. Computation of the free‐surface flow is performed in a local domain included within the global domain with interface boundaries that move along the reproduced velocity field in a Lagrangian fashion. To represent the moving interfaces, we used a polygon wall boundary model for mesh‐free particle methods. Verification and validation tests of our proposed model are performed, and results obtained by the model are compared with theoretical values and experimental results to show its accuracy and applicability.     

A two‐dimensional vertical non‐hydrostatic σ model with an implicit method for free‐surface flows

An implicit finite difference model in the σ co‐ordinate system is developed for non‐hydrostatic, two‐dimensional vertical plane free‐surface flows. To accurately simulate interaction of free‐surface flows with uneven bottoms, the unsteady Navier–Stokes equations and the free‐surface boundary condition are solved simultaneously in a regular transformed σ domain using a fully implicit method in two steps. First, the vertical velocity and pressure are expressed as functions of horizontal velocity. Second, substituting these relationship into the horizontal momentum equation provides a block tri‐diagonal matrix system with the unknown of horizontal velocity, which can be solved by a direct matrix solver without iteration. A new treatment of non‐hydrostatic pressure condition at the top‐layer cell is developed and found to be important for resolving the phase of wave propagation. Additional terms introduced by the σ co‐ordinate transformation are discretized appropriately in order to obtain accurate and stable numerical results. The developed model has been validated by several tests involving free‐surface flows with strong vertical accelerations and non‐linear waves interacting with uneven bottoms. Comparisons among numerical results, analytical solutions and experimental data show the capability of the model to simulate free‐surface flow problems. Copyright © 2004 John Wiley & Sons, Ltd.     

A domain decomposition approach to compute wave breaking (wave‐breaking flows)

A heterogeneous domain decomposition approach is followed to simulate the unsteady wavy flow generated by a body moving beneath a free surface. Attention being focused on complex free surface configurations, including wave‐breaking phenomena, a two‐fluid viscous flow model is used in the free surface region to capture the air–water interface (via a level‐set technique), while a potential flow approximation is adopted to describe the flow far from the interface. Two coupling strategies are investigated, differing in the transmission conditions. Both the adopted approaches make use of the inviscid velocity field as boundary condition in the Navier–Stokes solution. For validation purposes, two different two‐dimensional non‐breaking flows are simulated. Domain decomposition results are compared with both fully viscous and fully inviscid results, obtained by solving the corresponding equations in the whole fluid domain, and with available experimental data. Finally, the unsteady evolution of a steep breaking wave is followed and some of the physical phenomena, experimentally observed, are reproduced. Copyright © 2003 John Wiley & Sons, Ltd.     

A pseudo‐viscoelastic approach for transient polymeric flow exiting a channel

The influence of elasticity of a fluid exiting a channel is examined on transient coating downstream. A hybrid spectral/boundary element approach is proposed to solve the problem. The flow inside the channel is assumed to be fully developed. A viscoelastic instability of one‐dimensional plane Couette flow is first determined for a large class of Oldroyd fluids with added viscosity, which typically represent polymer solutions composed of a Newtonian solvent and a polymeric solute. The Johnson–Segalman equation is used as the constitutive model. The velocity profile inside the channel is taken as the exit profile for the emerging free‐surface flow. The flow is assumed to be Newtonian as it emerges from the channel. An estimate of the magnitude of the rate‐of‐strain tensor components in the free‐surface region reveals that they are generally smaller than the shear rate inside the channel. The evolution of the flow front is simulated using the boundary element method. For the channel flow, the problem is reduced to a nonlinear dynamical system using the Galerkin projection method. Stability analysis indicates that the channel velocity may be linear or non‐linear depending on the range of the Weissenberg number. The evolution of the coating flow at the exit is examined for steady as well as transient (monotonic and oscillatory) channel flow. It is found that adverse flow can exist as a result of fluid elasticity, which can hinder the process of blade coating. Copyright © 2003 John Wiley & Sons, Ltd.     

Quasi‐steady aerodynamic analysis of propeller–wing interaction

A quasi‐steady scheme for the analysis of aerodynamic interaction between a propeller and a wing has been developed. The quasi‐steady analysis uses a 3D steady vortex lattice method for the propeller and a 3D unsteady panel method for the wing. The aerodynamic coupling is represented by periodic loads, which are decomposed into harmonics and the harmonic amplitudes are found iteratively. Each stage of the iteration involves the solution of an isolated propeller or wing problem, the interaction being done through the Fourier transform of the induced velocity field. The propeller analysis code was validated by comparing the predicted velocity field about an isolated propeller with detailed laser Doppler velocimeter measurements, and the quasi‐steady scheme by comparison with mean loads measured in a wing–propeller experiment. Comparisons have also been made among the fluctuating loads predicted by the present method, an unsteady panel scheme and a quasi‐steady vortex lattice scheme. Copyright © 1999 John Wiley & Sons, Ltd.     

Computation of head–disk interface gap micro flowfields using DSMC and continuum–atomistic hybrid methods

This paper discusses computational modeling of micro flow in the head–disk interface (HDI) gap using the direct simulation Monte Carlo (DSMC) method. Modeling considerations are discussed in detail both for a stand‐alone DSMC computation and for the case of a hybrid continuum–atomistic simulation that couples the Navier–Stokes (NS) equation to a DSMC solver. The impact of the number of particles and number of cells on the accuracy of a DSMC simulation of the HDI gap is investigated both for two‐ and three‐dimensional configurations. An appropriate implicit boundary treatment method for modeling inflow and outflow boundaries is used in this work for a three‐dimensional DSMC micro flow simulation. As the flow outside the slider is in the continuum regime, a hybrid continuum–atomistic method based on the Schwarz alternating method is used to couple the DSMC model in the slider bearing region to the flow outside the slider modeled by NS equation. Schwarz coupling is done in two dimensions by taking overlap regions along two directions and the Chapman–Enskog distribution is employed for imposing the boundary condition from the continuum region to the DSMC region. Converged hybrid flow solutions are obtained in about five iterations and the hybrid DSMC–NS solutions show good agreement with the exact solutions in the entire domain considered. An investigation on the impact of the size of the overlap region on the convergence behavior of the Schwarz method indicates that the hybrid coupling by the Schwarz method is weakly dependent on the size of the overlap region. However, the use of a finite overlap region will facilitate the exchange of boundary conditions as the hybrid solution has been found to diverge in the absence of an overlap region for coupling the two models. Copyright © 2009 John Wiley & Sons, Ltd.     

Flow simulation on moving boundary‐fitted grids and application to fluid–structure interaction problems

We present a method for the parallel numerical simulation of transient three‐dimensional fluid–structure interaction problems. Here, we consider the interaction of incompressible flow in the fluid domain and linear elastic deformation in the solid domain. The coupled problem is tackled by an approach based on the classical alternating Schwarz method with non‐overlapping subdomains, the subproblems are solved alternatingly and the coupling conditions are realized via the exchange of boundary conditions. The elasticity problem is solved by a standard linear finite element method. A main issue is that the flow solver has to be able to handle time‐dependent domains. To this end, we present a technique to solve the incompressible Navier–Stokes equation in three‐dimensional domains with moving boundaries. This numerical method is a generalization of a finite volume discretization using curvilinear coordinates to time‐dependent coordinate transformations. It corresponds to a discretization of the arbitrary Lagrangian–Eulerian formulation of the Navier–Stokes equations. Here the grid velocity is treated in such a way that the so‐called Geometric Conservation Law is implicitly satisfied. Altogether, our approach results in a scheme which is an extension of the well‐known MAC‐method to a staggered mesh in moving boundary‐fitted coordinates which uses grid‐dependent velocity components as the primary variables. To validate our method, we present some numerical results which show that second‐order convergence in space is obtained on moving grids. Finally, we give the results of a fully coupled fluid–structure interaction problem. It turns out that already a simple explicit coupling with one iteration of the Schwarz method, i.e. one solution of the fluid problem and one solution of the elasticity problem per time step, yields a convergent, simple, yet efficient overall method for fluid–structure interaction problems. Copyright © 2005 John Wiley & Sons, Ltd.     

Semi‐coupled air/water immersed boundary approach for curvilinear dynamic overset grids with application to ship hydrodynamics

For many problems in ship hydrodynamics, the effects of air flow on the water flow are negligible (the frequently called free surface conditions), but the air flow around the ship is still of interest. A method is presented where the water flow is decoupled from the air solution, but the air flow uses the unsteady water flow as a boundary condition. The authors call this a semi‐coupled air/water flow approach. The method can be divided into two steps. At each time step the free surface water flow is computed first with a single‐phase method assuming constant pressure and zero stress on the interface. The second step is to compute the air flow assuming the free surface as a moving immersed boundary (IB). The IB method developed for Cartesian grids (Annu. Rev. Fluid Mech. 2005; 37 :239–261) is extended to curvilinear grids, where no‐slip and continuity conditions are used to enforce velocity and pressure boundary conditions for the air flow. The forcing points close to the IB can be computed and corrected under a sharp interface condition, which makes the computation very stable. The overset implementation is similar to that of the single‐phase solver (Comput. Fluids 2007; 36 :1415–1433), with the difference that points in water are set as IB points even if they are fringe points. Pressure–velocity coupling through pressure implicit with splitting of operators or projection methods is used for water computations, and a projection method is used for the air. The method on each fluid is a single‐phase method, thus avoiding ill‐conditioned numerical systems caused by large differences of fluid properties between air and water. The computation is only slightly slower than the single‐phase version, with complete absence of spurious velocity oscillations near the free surface, frequently present in fully coupled approaches. Validations are performed for laminar Couette flow over a wavy boundary by comparing with the analytical solution, and for the surface combatant model David Taylor Model Basin (DTMB) 5512 by comparing with Experimental Fluid Dynamics (EFD) and the results of two‐phase level set computations. Complex flow computations are demonstrated for the ONR Tumblehome DTMB 5613 with superstructure subject to waves and wind, including 6DOF motions and broaching in SS7 irregular waves and wind. Copyright © 2008 John Wiley & Sons, Ltd.     

On the immersed boundary‐lattice Boltzmann simulations of incompressible flows with freely moving objects

For simulating freely moving problems, conventional immersed boundary‐lattice Boltzmann methods encounter two major difficulties of an extremely large flow domain and the incompressible limit. To remove these two difficulties, this work proposes an immersed boundary‐lattice Boltzmann flux solver (IB‐LBFS) in the arbitrary Lagragian–Eulerian (ALE) coordinates and establishes a dynamic similarity theory. In the ALE‐based IB‐LBFS, the flow filed is obtained by using the LBFS on a moving Cartesian mesh, and the no‐slip boundary condition is implemented by using the boundary condition‐enforced immersed boundary method. The velocity of the Cartesian mesh is set the same as the translational velocity of the freely moving object so that there is no relative motion between the plate center and the mesh. This enables the ALE‐based IB‐LBFS to study flows with a freely moving object in a large open flow domain. By normalizing the governing equations for the flow domain and the motion of rigid body, six non‐dimensional parameters are derived and maintained to be the same in both physical systems and the lattice Boltzmann framework. This similarity algorithm enables the lattice Boltzmann equation‐based solver to study a general freely moving problem within the incompressible limit. The proposed solver and dynamic similarity theory have been successfully validated by simulating the flow around an in‐line oscillating cylinder, single particle sedimentation, and flows with a freely falling plate. The obtained results agree well with both numerical and experimental data. Copyright © 2016 John Wiley & Sons, Ltd.     

Finite volume solution of the Navier–Stokes equations in velocity–vorticity formulation

For the incompressible Navier–Stokes equations, vorticity‐based formulations have many attractive features over primitive‐variable velocity–pressure formulations. However, some features interfere with the use of the numerical methods based on the vorticity formulations, one of them being the lack of a boundary conditions on vorticity. In this paper, a novel approach is presented to solve the velocity–vorticity integro‐differential formulations. The general numerical method is based on standard finite volume scheme. The velocities needed at the vertexes of each control volume are calculated by a so‐called generalized Biot–Savart formula combined with a fast summation algorithm, which makes the velocity boundary conditions implicitly satisfied by maintaining the kinematic compatibility of the velocity and vorticity fields. The well‐known fractional step approaches are used to solve the vorticity transport equation. The paper describes in detail how we accurately impose no normal‐flow and no tangential‐flow boundary conditions. We impose a no‐flux boundary condition on solid objects by the introduction of a proper amount of vorticity at wall. The diffusion term in the transport equation is treated implicitly using a conservative finite update. The diffusive fluxes of vorticity into flow domain from solid boundaries are determined by an iterative process in order to satisfy the no tangential‐flow boundary condition. As application examples, the impulsively started flows through a flat plate and a circular cylinder are computed using the method. The present results are compared with the analytical solution and other numerical results and show good agreement. Copyright © 2005 John Wiley & Sons, Ltd.     

Fully non‐linear free‐surface simulations by a 3D viscous numerical wave tank

A finite difference scheme using a modified marker‐and‐cell (MAC) method is applied to investigate the characteristics of non‐linear wave motions and their interactions with a stationary three‐dimensional body inside a numerical wave tank (NWT). The Navier–Stokes (NS) equation is solved for two fluid layers, and the boundary values are updated at each time step by a finite difference time marching scheme in the frame of a rectangular co‐ordinate system. The viscous stresses and surface tension are neglected in the dynamic free‐surface condition, and the fully non‐linear kinematic free‐surface condition is satisfied by the density function method developed for two fluid layers. The incident waves are generated from the inflow boundary by prescribing a velocity profile resembling flexible flap wavemaker motions, and the outgoing waves are numerically dissipated inside an artificial damping zone located at the end of the tank. The present NS–MAC NWT simulations for a vertical truncated circular cylinder inside a rectangular wave tank are compared with the experimental results of Mercier and Niedzwecki, an independently developed potential‐based fully non‐linear NWT, and the second‐order diffraction computation. Copyright © 1999 John Wiley & Sons, Ltd.     

Toward low‐noise synthetic turbulent inflow conditions for aeroacoustic calculations

The accuracy of boundary conditions for computational aeroacoustics is a well‐known challenge, due in part to the necessity of truncating the flow domain and replacing the analytical boundary conditions at infinity with numerical boundary conditions. In particular, the inflow boundary condition involving turbulent velocity or scalar fields is likely to introduce spurious waves into the domain, therefore degrading the flow behavior and deteriorating the physical acoustic waves. In this work, a method to generate low‐noise, divergence‐free, synthetic turbulence for inflow boundary conditions is proposed. It relies on the classical view of turbulence as a superposition of random eddies convected with the mean flow. Within the proposed model, the vector potential and the requirement that the individual eddies must satisfy the linearized momentum equations about the mean flow are used. The model is tested using isolated eddies convected through the inflow boundary and an experimental benchmark data for spatially decaying isotropic turbulence. Copyright © 2013 John Wiley & Sons, Ltd.     

An explicit formulation for the evolution of nonlinear surface waves interacting with a submerged body

An explicit formulation to study nonlinear waves interacting with a submerged body in an ideal fluid of infinite depth is presented. The formulation allows one to decompose the nonlinear wave–body interaction problem into body and free‐surface problems. After the decomposition, the body problem satisfies a modified body boundary condition in an unbounded fluid domain, while the free‐surface problem satisfies modified nonlinear free‐surface boundary conditions. It is then shown that the nonlinear free‐surface problem can be further reduced to a closed system of two nonlinear evolution equations expanded in infinite series for the free‐surface elevation and the velocity potential at the free surface. For numerical experiments, the body problem is solved using a distribution of singularities along the body surface and the system of evolution equations, truncated at third order in wave steepness, is then solved using a pseudo‐spectral method based on the fast Fourier transform. A circular cylinder translating steadily near the free surface is considered and it is found that our numerical solutions show excellent agreement with the fully nonlinear solution using a boundary integral method. We further validate our solutions for a submerged circular cylinder oscillating vertically or fixed under incoming nonlinear waves with other analytical and numerical results. Copyright © 2007 John Wiley & Sons, Ltd.     

A Fourier–Chebyshev spectral collocation method for simulating flow past spheres and spheroids

An accurate Fourier–Chebyshev spectral collocation method has been developed for simulating flow past prolate spheroids. The incompressible Navier–Stokes equations are transformed to the prolate spheroidal co‐ordinate system and discretized on an orthogonal body fitted mesh. The infinite flow domain is truncated to a finite extent and a Chebyshev discretization is used in the wall‐normal direction. The azimuthal direction is periodic and a conventional Fourier expansion is used in this direction. The other wall‐tangential direction requires special treatment and a restricted Fourier expansion that satisfies the parity conditions across the poles is used. Issues including spatial and temporal discretization, efficient inversion of the pressure Poisson equation, outflow boundary condition and stability restriction at the pole are discussed. The solver has been validated primarily by simulating steady and unsteady flow past a sphere at various Reynolds numbers and comparing key quantities with corresponding data from experiments and other numerical simulations. Copyright © 1999 John Wiley & Sons, Ltd.     

Application of a vortex method to free surface flows

Vortex methods have found wide applications in various practical problems. The use of vortex methods in free surface flow problems, however, is still very limited. This paper demonstrates a vortex method for practical computation of non-linear free surface flows produced by moving bodies. The method is a potential flow formulation which uses the exact non-linear free surface boundary condition at the exact location of the instantaneous free surface. The position of the free surface, on which vortices are distributed, is updated using a Lagrangian scheme following the fluid particles on the free surface. The vortex densities are updated by the non-linear dynamic boundary condition, derived from the Euler equations, with an iterative Lagrangian numerical scheme. The formulation is tested numerically for a submerged circular cylinder in unsteady translation. The iteration is shown to converge for all cases. The results of the unsteady simulations agree well with classical linearized solutions. The stability of the method is also discussed.     

Numerical studies on non‐linear free surface flow using generalized Schwarz–Christoffel transformation

This paper deals with a technique to transform a free surface flow problem in the physical domain with an unknown boundary to a standard domain that has a fixed boundary. All the difficulties in the physical domain are reduced to finding an unknown mapping function that can be solved iteratively in a standard domain. A derivation is first presented to express an analytic function in terms of the boundary value of its imaginary part. Using a relationship between boundaries of the standard and the physical domains, a formula for the generalized Schwarz–Christoffel transformation is then developed. Based on the generalized Schwarz–Christoffel integral and the Hilbert transform, a pair of non‐linear boundary integro‐differential equations in an infinite strip is formulated for solving fully non‐linear free surface flow problems. The boundary integral equations are then discretized with quadratic elements in an untruncated standard domain and solved by the Levenberg–Marquardt algorithm. Several examples of supercritical flow past obstructions are provided to demonstrate the flexibility and the accuracy of the proposed numerical scheme. Copyright © 2000 John Wiley & Sons, Ltd.     

A pressure–velocity coupling approach for high void fraction free surface bubbly flows in overset curvilinear grids

A methodology for improved robustness in the simulation of high void fraction free surface polydisperse bubbly flows in curvilinear overset grids is presented. The method is fully two‐way coupled in the sense that the bubbly field affects the continuous fluid and vice versa. A hybrid projection approach is used in which staggered contravariant velocities at cell faces are computed for transport and pressure–velocity coupling while the momentum equation is solved on a collocated grid arrangement. Conservation of mass is formulated such that a strong coupling between void fraction, pressure, and velocity is achieved within a partitioned approach, solving each field separately. A pressure–velocity projection solver is iterated together with a predictor stage for the void fraction to achieve a robust coupling. The implementation is described for general curvilinear grids detailing particulars in the neighborhood to overset interfaces or a free surface. A balanced forced method to avoid the generation of spurious currents is extended for curvilinear grids. The overall methodology allows simulation of high void fraction flows and is stable even when strong packing forces accounting for bubble collisions are included. Convergence and stability in one‐dimensional (1D) and two‐dimensional (2D) configurations is evaluated. Finally, a full‐scale simulation of the bubbly flow around a flat‐bottom boat is performed demonstrating the applicability of the methodology to complex problems of engineering interest. Copyright © 2015 John Wiley & Sons, Ltd.     

Finite element and implicit Runge–Kutta implementation of an acoustics–convection upstream resolution algorithm for the time‐dependent two‐dimensional Euler equations

The second of a two‐paper series, this paper details a solver for the characteristics‐bias system from the acoustics–convection upstream resolution algorithm for the Euler and Navier–Stokes equations. An integral formulation leads to several surface integrals that allow effective enforcement of boundary conditions. Also presented is a new multi‐dimensional procedure to enforce a pressure boundary condition at a subsonic outlet, a procedure that remains accurate and stable. A classical finite element Galerkin discretization of the integral formulation on any prescribed grid directly yields an optimal discretely conservative upstream approximation for the Euler and Navier–Stokes equations, an approximation that remains multi‐dimensional independently of the orientation of the reference axes and computational cells. The time‐dependent discrete equations are then integrated in time via an implicit Runge–Kutta procedure that in this paper is proven to remain absolutely non‐linearly stable for the spatially‐discrete Euler and Navier–Stokes equations and shown to converge rapidly to steady states, with maximum Courant number exceeding 100 for the linearized version. Even on relatively coarse grids, the acoustics–convection upstream resolution algorithm generates essentially non‐oscillatory solutions for subsonic, transonic and supersonic flows, encompassing oblique‐ and interacting‐shock fields that converge within 40 time steps and reflect reference exact solutions. Copyright © 2005 John Wiley & Sons, Ltd.