Implicit application of non‐reflective boundary conditions for Navier–Stokes equations in generalized coordinates

The non‐reflective boundary conditions (NRBC) for Navier–Stokes equations originally suggested by Poinsot and Lele (J. Comput. Phys. 1992; 101 :104–129) in Cartesian coordinates are extended to generalized coordinates. The characteristic form Navier–Stokes equations in conservative variables are given. In this characteristic‐based method, the NRBC is implicitly coupled with the Navier–Stokes flow solver and are solved simultaneously with the flow solver. The calculations are conducted for a subsonic vortex propagating flow and the steady and unsteady transonic inlet‐diffuser flows. The results indicate that the present method is accurate and robust, and the NRBC are essential for unsteady flow calculations. Copyright © 2005 John Wiley & Sons, Ltd.     

Determination of the anisotropic permeability of a carbon cloth gas diffusion layer through X‐ray computer micro‐tomography and single‐phase lattice Boltzmann simulation

An investigation of the anisotropic permeability of a carbon cloth gas diffusion layer (GDL) based on the integration of X‐ray micro‐tomography and lattice Boltzmann (LB) simulation is presented. The method involves the generation of a 3D digital model of a carbon cloth GDL as manufactured using X‐ray shadow images acquired through X‐ray micro‐tomography at a resolution of 1.74 µm. The resulting 3D model is then split into 21 volumes and integrated with a LB single‐phase numerical solver in order to predict three orthogonal permeability tensors when a pressure difference is prescribed in the through‐plane direction. The 21 regions exhibit porosity values in the range of 0.910–0.955, while the average fibre diameter is 4 µm. The results demonstrate that the simulated through‐plane permeability is about four times higher than the in‐plane permeability for the sample imaged and that the corresponding degrees of anisotropy for the two orthogonal off‐principal directions are 0.22 and 0.27. The results reveal that flow channelling can play an important role in gas transport through the GDL structure due to the non‐homogeneous porosity distribution through the material. The simulated results are also applied to generate a parametric coefficient for the Kozeny–Carman (KC) method of determining permeability. The current research reveals that by applying the X‐ray tomography and LB techniques in a complementary manner, there is a strong potential to gain a deeper understanding of the microscopic fluidic phenomenon in representative models of porous fuel cell structures and how this can influence macroscopic transport characteristics which govern fuel cell performance. Copyright © 2010 John Wiley & Sons, Ltd.     

Numerical simulation of free‐surface flow using the level‐set method with global mass correction

A new numerical method that couples the incompressible Navier–Stokes equations with the global mass correction level‐set method for simulating fluid problems with free surfaces and interfaces is presented in this paper. The finite volume method is used to discretize Navier–Stokes equations with the two‐step projection method on a staggered Cartesian grid. The free‐surface flow problem is solved on a fixed grid in which the free surface is captured by the zero level set. Mass conservation is improved significantly by applying a global mass correction scheme, in a novel combination with third‐order essentially non‐oscillatory schemes and a five stage Runge–Kutta method, to accomplish advection and re‐distancing of the level‐set function. The coupled solver is applied to simulate interface change and flow field in four benchmark test cases: (1) shear flow; (2) dam break; (3) travelling and reflection of solitary wave and (4) solitary wave over a submerged object. The computational results are in excellent agreement with theoretical predictions, experimental data and previous numerical simulations using a RANS‐VOF method. The simulations reveal some interesting free‐surface phenomena such as the free‐surface vortices, air entrapment and wave deformation over a submerged object. Copyright © 2009 John Wiley & Sons, Ltd.     

Investigation on a coupled Navier–Stokes–Direct Simulation Monte Carlo method for the simulation of plume flowfield of a conical nozzle

To assess the plume effects of space thrusters, the accurate plume flowfield is indispensable. The plume flow of thrusters involves both continuum and rarefied flow regimes. Coupled Navier–Stokes–Direct Simulation Monte Carlo (NS–DSMC) method is a major approach to the simulation of continuum‐rarefied flows. An axisymmetric coupled NS–DSMC solver, possessing adaptive‐interface and two‐way coupling features, is investigated in this paper for the simulation of the nozzle and plume flows of thrusters. The state‐based coupling scheme is adopted, and the gradient local Knudsen number is used to indicate the breakdown of continuum solver. The nitrogen flows in a conical nozzle and its plume are chosen as the reference case to test the coupled solver. The threshold value of the continuum breakdown parameter is studied based on both theoretical kinetic velocity sampling and coupled numerical tests. Succeeding comparisons between coupled and full DSMC results demonstrate their conformities, meanwhile, the former saves 58.8% computational time. The pitot pressure evaluated from the coupled simulation result is compared with the experimental data proposed in literatures, revealing that the coupled method makes precise predictions on the experimental pitot pressure. Copyright © 2014 John Wiley & Sons, Ltd.     

3D free‐surface flow computation using a RANSE/Fourier–Kochin coupling

A coupling method for numerical calculations of steady free‐surface flows around a body is presented. The fluid domain in the neighbourhood of the hull is divided into two overlapping zones. Viscous effects are taken in account near the hull using Reynolds‐averaged Navier–Stokes equations (RANSE), whereas potential flow provides the flow away from the hull. In the internal domain, RANSE are solved by a fully coupled velocity, pressure and free‐surface elevation method. In the external domain, potential‐flow theory with linearized free‐surface condition is used to provide boundary conditions to the RANSE solver. The Fourier–Kochin method based on the Fourier–Kochin formulation, which defines the velocity field in a potential‐flow region in terms of the velocity distribution at a boundary surface, is used for that purpose. Moreover, the free‐surface Green function satisfying this linearized free‐surface condition is used. Calculations have been successfully performed for steady ship‐waves past a serie 60 and then have demonstrated abilities of the present coupling algorithm. Copyright © 2003 John Wiley & Sons, Ltd.     

Application of the lattice‐Boltzmann method to study flow and dispersion in channels with and without expansion and contraction geometry

The lattice‐Boltzmann (LB) method, derived from lattice gas automata, is a relatively new technique for studying transport problems. The LB method is investigated for its accuracy to study fluid dynamics and dispersion problems. Two problems of relevance to flow and dispersion in porous media are addressed: (i) Poiseuille flow between parallel plates (which is analogous to flow in pore throats in two‐dimensional porous networks), and (ii) flow through an expansion–contraction geometry (which is analogous to flow in pore bodies in two‐dimensional porous networks). The results obtained from the LB simulations are compared with analytical solutions when available, and with solutions obtained from a finite element code (FIDAP) when analytical results are not available. Excellent agreement is found between the LB results and the analytical/FIDAP solutions in most cases, indicating the utility of the lattice‐Boltzmann method for solving fluid dynamics and dispersion problems. Copyright © 1999 John Wiley & Sons, Ltd.     

Simulation of three‐dimensional flows over moving objects by an improved immersed boundary–lattice Boltzmann method

An improved immersed boundary–lattice Boltzmann method (IB–LBM) developed recently [28] was applied in this work to simulate three‐dimensional (3D) flows over moving objects. By enforcing the non‐slip boundary condition, the method could avoid any flow penetration to the wall. In the developed IB–LBM solver, the flow field is obtained on the non‐uniform mesh by the efficient LBM that is based on the second‐order one‐dimensional interpolation. As a consequence, its coefficients could be computed simply. By simulating flows over a stationary sphere and torus [28] accurately and efficiently, the proposed IB–LBM showed its ability to handle 3D flow problems with curved boundaries. In this paper, we further applied this method to simulate 3D flows around moving boundaries. As a first example, the flow over a rotating sphere was simulated. The obtained results agreed very well with the previous data in the literature. Then, simulation of flow over a rotating torus was conducted. The capability of the improved IB–LBM for solving 3D flows over moving objects with complex geometries was demonstrated via the simulations of fish swimming and dragonfly flight. The numerical results displayed quantitative and qualitative agreement with the date in the literature. Copyright © 2011 John Wiley & Sons, Ltd.     

High‐order numerical simulations of flow‐induced noise

In this paper, the flow/acoustics splitting method for predicting flow‐generated noise is further developed by introducing high‐order finite difference schemes. The splitting method consists of dividing the acoustic problem into a viscous incompressible flow part and an inviscid acoustic part. The incompressible flow equations are solved by a second‐order finite volume code EllipSys2D/3D. The acoustic field is obtained by solving a set of acoustic perturbation equations forced by flow quantities. The incompressible pressure and velocity form the input to the acoustic equations. The present work is an extension of our acoustics solver, with the introduction of high‐order schemes for spatial discretization and a Runge–Kutta scheme for time integration. To achieve low dissipation and dispersion errors, either Dispersion‐Relation‐Preserving (DRP) schemes or optimized compact finite difference schemes are used for the spatial discretizations. Applications and validations of the new acoustics solver are presented for benchmark aeroacoustic problems and for flow over an NACA 0012 airfoil. Copyright © 2010 John Wiley & Sons, Ltd.     

Unstructured pressure‐correction solver based on a consistent discretization of the Poisson equation

A finite volume incompressible flow solver is presented for three‐dimensional unsteady flows based on an unstructured tetrahedral mesh, with collocation of the flow variables at the cell vertices. The solver is based on the pressure‐correction method, with an explicit prediction step of the momentum equations followed by a Poisson equation for the correction step to enforce continuity. A consistent discretization of the Poisson equation was found to be essential in obtaining a solution. The correction step was solved with the biconjugate gradient stabilized (Bi‐CGSTAB) algorithm coupled with incomplete lower–upper (ILU) preconditioning. Artificial dissipation is used to prevent the formation of instabilities. Flow solutions are presented for a stalling airfoil, vortex shedding past a bridge deck and flow in model alveoli. Copyright © 2000 John Wiley & Sons, Ltd.     

A finite volume shock‐capturing solver of the fully coupled shallow water‐sediment equations

This paper describes a numerical solver of well‐balanced, 2D depth‐averaged shallow water‐sediment equations. The equations permit variable horizontal fluid density and are designed to model water‐sediment flow over a mobile bed. A Godunov‐type, Harten–Lax–van Leer contact (HLLC) finite volume scheme is used to solve the fully coupled system of hyperbolic conservation laws that describe flow hydrodynamics, suspended sediment transport, bedload transport and bed morphological change. Dependent variables are specially selected to handle the presence of the variable density property in the mathematical formulation. The model is verified against analytical and semi‐analytical solutions for bedload transport and suspended sediment transport, respectively. The well‐balanced property of the equations is verified for a variable‐density dam break flow over discontinuous bathymetry. Simulations of an idealised dam‐break flow over an erodible bed are in excellent agreement with previously published results, validating the ability of the model to capture the complex interaction between rapidly varying flow and an erodible bed and validating the eigenstructure of the system of variable‐density governing equations. Flow hydrodynamics and final bed topography of a laboratory‐based 2D partial dam breach over a mobile bed are satisfactorily reproduced by the numerical model. Comparison of the final bed topographies, computed for two distinct sediment transport methods, highlights the sensitivity of shallow water‐sediment models to the choice of closure relationships. Copyright © 2016 John Wiley & Sons, Ltd.     

CFD‐based multi‐objective optimization method for ship design

This paper concerns development and demonstration of a computational fluid dynamics (CFD)‐based multi‐objective optimization method for ship design. Three main components of the method, i.e. computer‐aided design (CAD), CFD, and optimizer modules are functionally independent and replaceable. The CAD used in the present study is NAPA system, which is one of the leading CAD systems in ship design. The CFD method is FLOWPACK version 2004d, a Reynolds‐averaged Navier–Stokes (RaNS) solver developed by the present authors. The CFD method is implemented into a self‐propulsion simulator, where the RaNS solver is coupled with a propeller‐performance program. In addition, a maneuvering simulation model is developed and applied to predict ship maneuverability performance. Two nonlinear optimization algorithms are used in the present study, i.e. the successive quadratic programming and the multi‐objective genetic algorithm, while the former is mainly used to verify the results from the latter. For demonstration of the present method, a multi‐objective optimization problem is formulated where ship propulsion and maneuverability performances are considered. That is, the aim is to simultaneously minimize opposite hydrodynamic performances in design tradeoff. In the following, an overview of the present method is given, and results are presented and discussed for tanker stern optimization problem including detailed verification work on the present numerical schemes. Copyright © 2006 John Wiley & Sons, Ltd.     

An implementation of the Spalart–Allmaras DES model in an implicit unstructured hybrid finite volume/element solver for incompressible turbulent flow

In this paper, we describe an implicit hybrid finite volume (FV)/element (FE) incompressible Navier–Stokes solver for turbulent flows based on the Spalart–Allmaras detached eddy simulation (SA‐DES). The hybrid FV/FE solver is based on the segregated pressure correction or projection method. The intermediate velocity field is first obtained by solving the original momentum equations with the matrix‐free implicit cell‐centered FV method. The pressure Poisson equation is solved by the node‐based Galerkin FE method for an auxiliary variable. The auxiliary variable is closely related to the real pressure and is used to update the velocity field and the pressure field. We store the velocity components at cell centers and the auxiliary variable at vertices, making the current solver a staggered‐mesh scheme. The SA‐DES turbulence equation is solved after the velocity and the pressure fields have been updated at the end of each time step. The same matrix‐free FV method as the one used for momentum equations is used to solve the turbulence equation. The turbulence equation provides the eddy viscosity, which is added to the molecular viscosity when solving the momentum equation. In our implementation, we focus on the accuracy, efficiency and robustness of the SA‐DES model in a hybrid flow solver. This paper will address important implementation issues for high‐Reynolds number flows where highly stretched elements are typically used. In addition, some aspects of implementing the SA‐DES model will be described to ensure the robustness of the turbulence model. Several numerical examples including a turbulent flow past a flat plate and a high‐Reynolds number flow around a high angle‐of‐attack NACA0015 airfoil will be presented to demonstrate the accuracy and efficiency of our current implementation. Copyright © 2008 John Wiley & Sons, Ltd.     

A high‐order discontinuous Galerkin method for all‐speed flows

In this article, we present a discontinuous Galerkin (DG) method designed to improve the accuracy and efficiency of steady solutions of the compressible fully coupled Reynolds‐averaged Navier–Stokes and k ? ω turbulence model equations for solving all‐speed flows. The system of equations is iterated to steady state by means of an implicit scheme. The DG solution is extended to the incompressible limit by implementing a low Mach number preconditioning technique. A full preconditioning approach is adopted, which modifies both the unsteady terms of the governing equations and the dissipative term of the numerical flux function by means of a new preconditioner, on the basis of a modified version of Turkel's preconditioning matrix. At sonic speed the preconditioner reduces to the identity matrix thus recovering the non‐preconditioned DG discretization. An artificial viscosity term is added to the DG discretized equations to stabilize the solution in the presence of shocks when piecewise approximations of order of accuracy higher than 1 are used. Moreover, several rescaling techniques are implemented in order to overcome ill‐conditioning problems that, in addition to the low Mach number stiffness, can limit the performance of the flow solver. These approaches, through a proper manipulation of the governing equations, reduce unbalances between residuals as a result of the dependence on the size of elements in the computational mesh and because of the inherent differences between turbulent and mean‐flow variables, influencing both the evolution of the Courant Friedrichs Lewy (CFL) number and the inexact solution of the linear systems. The performance of the method is demonstrated by solving three turbulent aerodynamic test cases: the flat plate, the L1T2 high‐lift configuration and the RAE2822 airfoil (Case 9). The computations are performed at different Mach numbers using various degrees of polynomial approximations to analyze the influence of the proposed numerical strategies on the accuracy, efficiency and robustness of a high‐order DG solver at different flow regimes. Copyright © 2014 John Wiley & Sons, Ltd.     

On coupling the Reynolds‐averaged Navier–Stokes equations with two‐equation turbulence model equations

Two methods for coupling the Reynolds‐averaged Navier–Stokes equations with the q–ω turbulence model equations on structured grid systems have been studied; namely a loosely coupled method and a strongly coupled method. The loosely coupled method first solves the Navier–Stokes equations with the turbulent viscosity fixed. In a subsequent step, the turbulence model equations are solved with all flow quantities fixed. On the other hand, the strongly coupled method solves the Reynolds‐averaged Navier–Stokes equations and the turbulence model equations simultaneously. In this paper, numerical stabilities of both methods in conjunction with the approximated factorization‐alternative direction implicit method are analysed. The effect of the turbulent kinetic energy terms in the governing equations on the convergence characteristics is also studied. The performance of the two methods is compared for several two‐ and three‐dimensional problems. Copyright © 2005 John Wiley & Sons, Ltd.     

SIMPLE‐type preconditioners for cell‐centered,colocated finite volume discretization of incompressible Reynolds‐averaged Navier–Stokes equations

This paper contains a comparison of four SIMPLE‐type methods used as solver and as preconditioner for the iterative solution of the (Reynolds‐averaged) Navier–Stokes equations, discretized with a finite volume method for cell‐centered, colocated variables on unstructured grids. A matrix‐free implementation is presented, and special attention is given to the treatment of the stabilization matrix to maintain a compact stencil suitable for unstructured grids. We find SIMPLER preconditioning to be robust and efficient for academic test cases and industrial test cases. Compared with the classical SIMPLE solver, SIMPLER preconditioning reduces the number of nonlinear iterations by a factor 5–20 and the CPU time by a factor 2–5 depending on the case. The flow around a ship hull at Reynolds number 2E9, for example, on a grid with cell aspect ratio up to 1:1E6, can be computed in 3 instead of 15 h.Copyright © 2012 John Wiley & Sons, Ltd.     

A high‐order compact finite‐difference lattice Boltzmann method for simulation of steady and unsteady incompressible flows

A high‐order compact finite‐difference lattice Boltzmann method (CFDLBM) is proposed and applied to accurately compute steady and unsteady incompressible flows. Herein, the spatial derivatives in the lattice Boltzmann equation are discretized by using the fourth‐order compact FD scheme, and the temporal term is discretized with the fourth‐order Runge–Kutta scheme to provide an accurate and efficient incompressible flow solver. A high‐order spectral‐type low‐pass compact filter is used to stabilize the numerical solution. An iterative initialization procedure is presented and applied to generate consistent initial conditions for the simulation of unsteady flows. A sensitivity study is also conducted to evaluate the effects of grid size, filtering, and procedure of boundary conditions implementation on accuracy and convergence rate of the solution. The accuracy and efficiency of the proposed solution procedure based on the CFDLBM method are also examined by comparison with the classical LBM for different flow conditions. Two test cases considered herein for validating the results of the incompressible steady flows are a two‐dimensional (2‐D) backward‐facing step and a 2‐D cavity at different Reynolds numbers. Results of these steady solutions computed by the CFDLBM are thoroughly compared with those of a compact FD Navier–Stokes flow solver. Three other test cases, namely, a 2‐D Couette flow, the Taylor's vortex problem, and the doubly periodic shear layers, are simulated to investigate the accuracy of the proposed scheme in solving unsteady incompressible flows. Results obtained for these test cases are in good agreement with the analytical solutions and also with the available numerical and experimental results. The study shows that the present solution methodology is robust, efficient, and accurate for solving steady and unsteady incompressible flow problems even at high Reynolds numbers. Copyright © 2014 John Wiley & Sons, Ltd.     

A discontinuous Galerkin‐based sharp‐interface method to simulate three‐dimensional compressible two‐phase flow

A numerical method for the simulation of compressible two‐phase flows is presented in this paper. The sharp‐interface approach consists of several components: a discontinuous Galerkin solver for compressible fluid flow, a level‐set tracking algorithm to follow the movement of the interface and a coupling of both by a ghost‐fluid approach with use of a local Riemann solver at the interface. There are several novel techniques used: the discontinuous Galerkin scheme allows locally a subcell resolution to enhance the interface resolution and an interior finite volume Total Variation Diminishing (TVD) approximation at the interface. The level‐set equation is solved by the same discontinuous Galerkin scheme. To obtain a very good approximation of the interface curvature, the accuracy of the level‐set field is improved and smoothed by an additional P_NP_M‐reconstruction. The capabilities of the method for the simulation of compressible two‐phase flow are demonstrated for a droplet at equilibrium, an oscillating ellipsoidal droplet, and a shock‐droplet interaction problem at Mach 3. Copyright © 2015 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.     

Computations of two passing‐by high‐speed trains by a relaxation overset‐grid algorithm

This paper presents a relaxation algorithm, which is based on the overset grid technology, an unsteady three‐dimensional Navier–Stokes flow solver, and an inner‐ and outer‐relaxation method, for simulation of the unsteady flows of moving high‐speed trains. The flow solutions on the overlapped grids can be accurately updated by introducing a grid tracking technique and the inner‐ and outer‐relaxation method. To evaluate the capability and solution accuracy of the present algorithm, the computational static pressure distribution of a single stationary TGV high‐speed train inside a long tunnel is investigated numerically, and is compared with the experimental data from low‐speed wind tunnel test. Further, the unsteady flows of two TGV high‐speed trains passing by each other inside a long tunnel and at the tunnel entrance are simulated. A series of time histories of pressure distributions and aerodynamic loads acting on the train and tunnel surfaces are depicted for detailed discussions. Copyright © 2004 John Wiley & Sons, Ltd.     

A hybrid pressure‐based solver for nonideal single‐phase fluid flows at all speeds

This paper describes the implementation of a numerical solver that is capable of simulating compressible flows of nonideal single‐phase fluids. The proposed method can be applied to arbitrary equations of state and is suitable for all Mach numbers. The pressure‐based solver uses the operator‐splitting technique and is based on the PISO/SIMPLE algorithm: the density, velocity, and temperature fields are predicted by solving the linearized versions of the balance equations using the convective fluxes from the previous iteration or time step. The overall mass continuity is ensured by solving the pressure equation derived from the continuity equation, the momentum equation, and the equation of state. Nonphysical oscillations of the numerical solution near discontinuities are damped using the Kurganov‐Tadmor/Kurganov‐Noelle‐Petrova (KT/KNP) scheme for convective fluxes. The solver was validated using different test cases, where analytical and/or numerical solutions are present or can be derived: (1) A convergent‐divergent nozzle with three different operating conditions; (2) the Riemann problem for the Peng‐Robinson equation of state; (3) the Riemann problem for the covolume equation of state; (4) the development of a laminar velocity profile in a circular pipe (also known as Poiseuille flow); (5) a laminar flow over a circular cylinder; (6) a subsonic flow over a backward‐facing step at low Reynolds numbers; (7) a transonic flow over the RAE 2822 airfoil; and (8) a supersonic flow around a blunt cylinder‐flare model. The spatial approximation order of the scheme is second order. The mesh convergence of the numerical solution was achieved for all cases. The accuracy order for highly compressible flows with discontinuities is close to first order and, for incompressible viscous flows, it is close to second order. The proposed solver is named rhoPimpleCentralFoam and is implemented in the open‐source CFD library OpenFOAM^®. For high speed flows, it shows a similar behavior as the KT/KNP schemes (implemented as rhoCentralFoam‐solver, Int. J. Numer. Meth. Fluids 2010), and for flows with small Mach numbers, it behaves like solvers that are based on the PISO/SIMPLE algorithm.