Parallel computation of viscous incompressible flows using Godunov‐projection method on overlapping grids

The Godunov‐projection method is implemented on a system of overlapping structured grids for solving the time‐dependent incompressible Navier–Stokes equations. This projection method uses a second‐order fractional step scheme in which the momentum equation is solved to obtain the intermediate velocity field which is then projected on to the space of divergence‐free vector fields. The Godunov procedure is applied to estimate the non‐linear convective term in order to provide a robust discretization of this terms at high Reynolds number. In order to obtain the pressure field, a separate procedure is applied in this modified Godunov‐projection method, where the pressure Poisson equation is solved. Overlapping grids are used to discretize the flow domain, as they offer the flexibility of simplifying the grid generation around complex geometrical domains. This combination of projection method and overlapping grid is also parallelized and reasonable parallel efficiency is achieved. Numerical results are presented to demonstrate the performance of this combination of the Godunov‐projection method and the overlapping grid. Copyright © 2002 John Wiley & Sons, Ltd.     

A projection method for solving incompressible viscous flows on domains with moving boundaries

In this paper, a projection method is presented for solving the flow problems in domains with moving boundaries. In order to track the movement of the domain boundaries, arbitrary‐Lagrangian–Eulerian (ALE) co‐ordinates are used. The unsteady incompressible Navier–Stokes equations on the ALE co‐ordinates are solved by using a projection method developed in this paper. This projection method is based on the Bell's Godunov‐projection method. However, substantial changes are made so that this algorithm is capable of solving the ALE form of incompressible Navier–Stokes equations. Multi‐block structured grids are used to discretize the flow domains. The grid velocity is not explicitly computed; instead the volume change is used to account for the effect of grid movement. A new method is also proposed to compute the freestream capturing metrics so that the geometric conservation law (GCL) can be satisfied exactly in this algorithm. This projection method is also parallelized so that the state of the art high performance computers can be used to match the computation cost associated with the moving grid calculations. Several test cases are solved to verify the performance of this moving‐grid projection method. Copyright © 2004 John Wiley Sons, Ltd.     

A SOLVER USING NEWTON‘S METHOD FOR UNSTEADY VISCOUS FLOWS

A solver is developed for time-accurate computations of viscous flows based on the conception of Newton‘s method. A set of pseudo-time derivatives are added into governing equations and the discretized system is solved using GMRES algorithm. Due to some special properties of GMRES algorithm, the solution procedure for unsteady flows could be regarded as a kind of Newton iteration. The physical-time derivatives of governing equations are discretized using two different approaches, I.e., 3-point Euler backward, and Crank-Nicolson formulas, both with 2nd-order accuracy in time but with different truncation errors. The turbulent eddy viscosity is calculated by using a version of Spalart~Allmaras one-equation model modified by authors for turbulent flows. Two cases of unsteady viscous flow are investigated to validate and assess the solver, I.e., low Reynolds number flow around a row of cylinders and transonic bi-circular-arc airfoil flow featuring the vortex shedding and shock buffeting problems, respectively. Meanwhile, comparisons between the two schemes of timederivative discretizations are carefully made. It is illustrated that the developed unsteady flow solver shows a considerable efficiency and the Crank-Nicolson scheme gives better results compared with Euler method.     

The solution of viscous incompressible jet flows using non‐staggered boundary fitted co‐ordinate methods

A new approach for the solution of the steady incompressible Navier–Stokes equations in a domain bounded in part by a free surface is presented. The procedure is based on the finite difference technique, with the non‐staggered grid fractional step method used to solve the flow equations written in terms of primitive variables. The physical domain is transformed to a rectangle by means of a numerical mapping technique. In order to design an effective free solution scheme, we distinguish between flows dominated by surface tension and those dominated by inertia and viscosity. When the surface tension effect is insignificant we used the kinematic condition to update the surface; whereas, in the opposite case, we used the normal stress condition to obtain the free surface boundary. Results obtained with the improved boundary conditions for a plane Newtonian jet are found to compare well with the available two‐dimensional numerical solutions for Reynolds numbers, up to Re=100, and Capillary numbers in the range of 0≤Ca<1000. Copyright © 2001 John Wiley & Sons, Ltd.     

A finite volume unstructured multigrid method for efficient computation of unsteady incompressible viscous flows

An unstructured non‐nested multigrid method is presented for efficient simulation of unsteady incompressible Navier–Stokes flows. The Navier–Stokes solver is based on the artificial compressibility approach and a higher‐order characteristics‐based finite‐volume scheme on unstructured grids. Unsteady flow is calculated with an implicit dual time stepping scheme. For efficient computation of unsteady viscous flows over complex geometries, an unstructured multigrid method is developed to speed up the convergence rate of the dual time stepping calculation. The multigrid method is used to simulate the steady and unsteady incompressible viscous flows over a circular cylinder for validation and performance evaluation purposes. It is found that the multigrid method with three levels of grids results in a 75% reduction in CPU time for the steady flow calculation and 55% reduction for the unsteady flow calculation, compared with its single grid counterparts. The results obtained are compared with numerical solutions obtained by other researchers as well as experimental measurements wherever available and good agreements are obtained. Copyright © 2004 John Wiley & Sons, Ltd.     

Study of a staggered fourth‐order compact scheme for unsteady incompressible viscous flows

The objective of this paper is the development and assessment of a fourth‐order compact scheme for unsteady incompressible viscous flows. A brief review of the main developments of compact and high‐order schemes for incompressible flows is given. A numerical method is then presented for the simulation of unsteady incompressible flows based on fourth‐order compact discretization with physical boundary conditions implemented directly into the scheme. The equations are discretized on a staggered Cartesian non‐uniform grid and preserve a form of kinetic energy in the inviscid limit when a skew‐symmetric form of the convective terms is used. The accuracy and efficiency of the method are demonstrated in several inviscid and viscous flow problems. Results obtained with different combinations of second‐ and fourth‐order spatial discretizations and together with either the skew‐symmetric or divergence form of the convective term are compared. The performance of these schemes is further demonstrated by two challenging flow problems, linear instability in plane channel flow and a two‐dimensional dipole–wall interaction. Results show that the compact scheme is efficient and that the divergence and skew‐symmetric forms of the convective terms produce very similar results. In some but not all cases, a gain in accuracy and computational time is obtained with a high‐order discretization of only the convective and diffusive terms. Finally, the benefits of compact schemes with respect to second‐order schemes is discussed in the case of the fully developed turbulent channel flow. Copyright © 2008 John Wiley & Sons, Ltd.     

Calculation of curved open channel flow using physical curvilinear non‐orthogonal co‐ordinates

There are two main difficulties in numerical simulation calculations using FD/FV method for the flows in real rivers. Firstly, the boundaries are very complex and secondly, the generated grid is usually very non‐uniform locally. Some numerical models in this field solve the first difficulty by the use of physical curvilinear orthogonal co‐ordinates. However, it is very difficult to generate an orthogonal grid for real rivers and the orthogonal restriction often forces the grid to be over concentrated where high resolution is not required. Recently, more and more models solve the first difficulty by the use of generalized curvilinear co‐ordinates (ξ,η). The governing equations are expressed in a covariant or contra‐variant form in terms of generalized curvilinearco‐ordinates (ξ,η). However, some studies in real rivers indicate that this kind of method has some undesirable mesh sensitivities. Sharp differences in adjacent mesh size may easily lead to a calculation stability problem oreven a false simulation result. Both approaches used presently have their own disadvantages in solving the two difficulties that exist in real rivers. In this paper, the authors present a method for two‐dimensional shallow water flow calculations to solve both of the main difficulties, by formulating the governing equations in a physical form in terms of physical curvilinear non‐orthogonal co‐ordinates (s,n). Derivation of the governing equations is explained, and two numerical examples are employed to demonstrate that the presented method is applicable to non‐orthogonal and significantly non‐uniform grids. Copyright © 2004 John Wiley & Sons, Ltd.     

Simulation of unsteady turbulent flows around moving aerofoils using the pseudo‐time method

The pseudo‐time formulation of Jameson has facilitated the use of numerical methods for unsteady flows, these methods have proved successful for steady flows. The formulation uses iterations through pseudo‐time to arrive at the next real time approximation. This iteration can be used in a straightforward manner to remove sequencing errors introduced when solving mean flow equations together with another set of differential equations (e.g. two‐equation turbulence models or structural equations). The current paper discusses the accuracy and efficiency advantages of removing the sequencing error and the effect that building extra equations into the pseudo‐time iteration has on its convergence characteristics. Test cases used are for the turbulent flow around pitching and ramping aerofoils. The performance of an implicit method for solving the pseudo‐steady state problem is also assessed. Copyright © 2000 John Wiley & Sons, Ltd.     

Optimization of unsteady incompressible Navier–Stokes flows using variational level set method

This paper presents the optimization of unsteady Navier–Stokes flows using the variational level set method. The solid–liquid interface is expressed by the level set function implicitly, and the fluid velocity is constrained to be zero in the solid domain. An optimization problem, which is constrained by the Navier–Stokes equations and a fluid volume constraint, is analyzed by the Lagrangian multiplier based adjoint approach. The corresponding continuous adjoint equations and the shape sensitivity are derived. The level set function is evolved by solving the Hamilton–Jacobian equation with the upwind finite difference method. The optimization method can be used to design channels for flows with or without body forces. The numerical examples demonstrate the feasibility and robustness of this optimization method for unsteady Navier–Stokes flows.Copyright © 2012 John Wiley & Sons, Ltd.     

A second‐order time accurate finite element method for quasi‐incompressible viscous flows

A finite element method for quasi‐incompressible viscous flows is presented. An equation for pressure is derived from a second‐order time accurate Taylor–Galerkin procedure that combines the mass and the momentum conservation laws. At each time step, once the pressure has been determined, the velocity field is computed solving discretized equations obtained from another second‐order time accurate scheme and a least‐squares minimization of spatial momentum residuals. The terms that stabilize the finite element method (controlling wiggles and circumventing the Babuska–Brezzi condition) arise naturally from the process, rather than being introduced a priori in the variational formulation. A comparison between the present second‐order accurate method and our previous first‐order accurate formulation is shown. The method is also demonstrated in the computation of the leaky‐lid driven cavity flow and in the simulation of a crossflow past a circular cylinder. In both cases, good agreement with previously published experimental and computational results has been obtained. Copyright © 2010 John Wiley & Sons, Ltd.     

A deconvolution‐based fourth‐order finite volume method for incompressible flows on non‐uniform grids

This paper is concerned with the development of a new high‐order finite volume method for the numerical simulation of highly convective unsteady incompressible flows on non‐uniform grids. Specifically, both a high‐order fluxes integration and the implicit deconvolution of the volume‐averaged field are considered. This way, the numerical solution effectively stands for a fourth‐order approximation of the point‐wise one. Moreover, the procedure is developed in the framework of a projection method for the pressure–velocity decoupling, while originally deriving proper high‐order intermediate boundary conditions. The entire numerical procedure is discussed in detail, giving particular attention to the consistent discretization of the deconvolution operation. The present method is also cast in the framework of approximate deconvolution modelling for large‐eddy simulation. The overall high accuracy of the method, both in time and space, is demonstrated. Finally, as a model of real flow computation, a two‐dimensional time‐evolving mixing layer is simulated, with and without sub‐grid scales modelling. Copyright © 2003 John Wiley & Sons, Ltd.     

A numerical study of projection algorithms in the finite element simulation of three‐dimensional viscous incompressible flow

In this work a comparative study of two versions of the projection algorithm used either for time integration or as an iterative method to solve the three‐dimensional incompressible Navier–Stokes equations is presented. It is also shown that these projection algorithms combined with the finite element method are particularly suited for the treatment of outflow boundary conditions in the context of external flows. This assertion is illustrated by means of some numerical examples in which five types of boundary conditions are compared. The scheme is applied to simulate the flow past a cylinder clamped on two fixed parallel solid walls. Comparison with experimental data available for this problem shows good agreement of the velocity and pressure fields, both computed with continuous piecewise linear elements. Copyright © 1999 John Wiley & Sons, Ltd.     

Simulation of incompressible viscous flows around moving objects by a variant of immersed boundary‐lattice Boltzmann method

A variant of immersed boundary‐lattice Boltzmann method (IB‐LBM) is presented in this paper to simulate incompressible viscous flows around moving objects. As compared with the conventional IB‐LBM where the force density is computed explicitly by Hook's law or the direct forcing method and the non‐slip condition is only approximately satisfied, in the present work, the force density term is considered as the velocity correction which is determined by enforcing the non‐slip condition at the boundary. The lift and drag forces on the moving object can be easily calculated via the velocity correction on the boundary points. The capability of the present method for moving objects is well demonstrated through its application to simulate flows around a moving circular cylinder, a rotationally oscillating cylinder, and an elliptic flapping wing. Furthermore, the simulation of flows around a flapping flexible airfoil is carried out to exhibit the ability of the present method for implementing the elastic boundary condition. It was found that under certain conditions, the flapping flexible airfoil can generate larger propulsive force than the flapping rigid airfoil. Copyright © 2009 John Wiley & Sons, Ltd.     

Three‐dimensional modeling of non‐hydrostatic free‐surface flows on unstructured grids

A three‐dimensional numerical model is presented for the simulation of unsteady non‐hydrostatic shallow water flows on unstructured grids using the finite volume method. The free surface variations are modeled by a characteristics‐based scheme, which simulates sub‐critical and super‐critical flows. Three‐dimensional velocity components are considered in a collocated arrangement with a σ‐coordinate system. A special treatment of the pressure term is developed to avoid the water surface oscillations. Convective and diffusive terms are approximated explicitly, and an implicit discretization is used for the pressure term to ensure exact mass conservation. The unstructured grid in the horizontal direction and the σ coordinate in the vertical direction facilitate the use of the model in complicated geometries. Solution of the non‐hydrostatic equations enables the model to simulate short‐period waves and vertically circulating flows. Copyright © 2015 John Wiley & Sons, Ltd.     

Computation of unsteady transonic flow over a fighter wing using a zonal Navier–Stokes/full‐potential method

An improved hybrid method for computing unsteady compressible viscous flows is presented. This method divides the computational domain into two zones. In the inner zone, the Navier–Stokes equations are solved using a diagonal form of an alternating‐direction implicit (ADI) approximate factorisation procedure. In the outer zone, the unsteady full‐potential equation (FPE) is solved. The two zones are tightly coupled so that steady and unsteady flows may be efficiently solved. Characteristic‐based viscous/inviscid interface boundary conditions are employed to avoid spurious reflections at that interface. The resulting CPU times are about 60% of the full Navier–Stokes CPU times for unsteady flows in non‐vector processing machines. Applications of the method are presented for a F‐5 wing in steady and unsteady transonic flows. Steady surface pressures are in very good agreement with experimental data and are essentially identical to the full Navier–Stokes predictions. Density contours show that shocks cross the viscous/inviscid interface smoothly, so that the accuracy of full Navier–Stokes equations can be retained with significant savings in computational time. Copyright © 1999 John Wiley & Sons, Ltd.     

A projection method for the spectral solution of non‐homogeneous and incompressible Navier–Stokes equations

This paper is devoted to the development of a parallel, spectral and second‐order time‐accurate method for solving the incompressible and variable density Navier–Stokes equations. The method is well suited for finite thickness density layers and is very efficient, especially for three‐dimensional computations. It is based on an exact projection technique. To enforce incompressibility, for a non‐homogeneous fluid, the pressure is computed using an iterative algorithm. A complete study of the convergence properties of this algorithm is done for different density variations. Numerical simulations showing, qualitatively, the capabilities of the developed Navier–Stokes solver for many realistic problems are presented. The numerical procedure is also validated quantitatively by reproducing growth rates from the linear instability theory in a three‐dimensional direct numerical simulation of an unstable, non‐homogeneous, flow configuration. It is also shown that, even in a turbulent flow, the spectral accuracy is recovered. Copyright © 2012 John Wiley & Sons, Ltd.     

Numerical method for calculation of the incompressible flow in general curvilinear co‐ordinates with double staggered grid

A solution methodology has been developed for incompressible flow in general curvilinear co‐ordinates. Two staggered grids are used to discretize the physical domain. The first grid is a MAC quadrilateral mesh with pressure arranged at the centre and the Cartesian velocity components located at the middle of the sides of the mesh. The second grid is so displaced that its corners correspond to the centre of the first grid. In the second grid the pressure is placed at the corner of the first grid. The discretized mass and momentum conservation equations are derived on a control volume. The two pressure grid functions are coupled explicitly through the boundary conditions and implicitly through the velocity of the field. The introduction of these two grid functions avoids an averaging of pressure and velocity components when calculating terms that are generated in general curvilinear co‐ordinates. The SIMPLE calculation procedure is extended to the present curvilinear co‐ordinates with double grids. Application of the methodology is illustrated by calculation of well‐known external and internal problems: viscous flow over a circular cylinder, with Reynolds numbers ranging from 10 to 40, and lid‐driven flow in a cavity with inclined walls are examined. The numerical results are in close agreement with experimental results and other numerical data. Copyright © 2003 John Wiley & Sons, Ltd.     

Implementing a high‐order accurate implicit operator scheme for solving steady incompressible viscous flows using artificial compressibility method

This paper uses a fourth‐order compact finite‐difference scheme for solving steady incompressible flows. The high‐order compact method applied is an alternating direction implicit operator scheme, which has been used by Ekaterinaris for computing two‐dimensional compressible flows. Herein, this numerical scheme is efficiently implemented to solve the incompressible Navier–Stokes equations in the primitive variables formulation using the artificial compressibility method. For space discretizing the convective fluxes, fourth‐order centered spatial accuracy of the implicit operators is efficiently obtained by performing compact space differentiation in which the method uses block‐tridiagonal matrix inversions. To stabilize the numerical solution, numerical dissipation terms and/or filters are used. In this study, the high‐order compact implicit operator scheme is also extended for computing three‐dimensional incompressible flows. The accuracy and efficiency of this high‐order compact method are demonstrated for different incompressible flow problems. A sensitivity study is also conducted to evaluate the effects of grid resolution and pseudocompressibility parameter on accuracy and convergence rate of the solution. The effects of filtering and numerical dissipation on the solution are also investigated. Test cases considered herein for validating the results are incompressible flows in a 2‐D backward facing step, a 2‐D cavity and a 3‐D cavity at different flow conditions. Results obtained for these cases are in good agreement with the available numerical and experimental results. The study shows that the scheme is robust, efficient and accurate for solving incompressible flow problems. Copyright © 2010 John Wiley & Sons, Ltd.