Interfacial dynamics‐based modelling of turbulent cavitating flows,Part‐2: Time‐dependent computations

The interfacial dynamics‐based cavitation model, developed in Part‐1, is further employed for unsteady flow computations. The pressure‐based operator‐splitting algorithm (PISO) is extended to handle the time‐dependent cavitating flows with particular focus on the coupling of the cavitation and turbulence models, and the large density ratio associated with cavitation. Furthermore, the compressibility effect is important for unsteady cavitating flows because in a water–vapour mixture, depending on the composition, the speed of sound inside the cavity can vary by an order of magnitude. The implications of the issue of the speed of the sound are assessed with alternative modelling approaches. Depending on the geometric confinement of the nozzle, compressibility model and cavitation numbers, either auto‐oscillation or quasi‐steady behaviour is observed. The adverse pressure gradient in the closure region is stronger at the maximum cavity size. One can also observe that the mass transfer process contributes to the cavitation dynamics. Compared to the steady flow computations, the velocity and vapour volume fraction distributions within the cavity are noticeably improved with time‐dependent computations. Copyright © 2004 John Wiley & Sons, Ltd.     

An efficient semi‐implicit subgrid method for free‐surface flows on hierarchical grids

We present a new modelling strategy for improving the efficiency of computationally intensive flow problems in environmental free‐surface flows. The approach combines a recently developed semi‐implicit subgrid method with a hierarchical grid solution strategy. The method allows the incorporation of high‐resolution data on subgrid scale to obtain a more accurate and efficient hydrodynamic model. The subgrid method improves the efficiency of the hierarchical grid method by providing better solutions on coarse grids. The method is applicable to both steady and unsteady flows, but we particularly focus on river flows with steady boundary conditions. There, the combined hierarchical grid–subgrid method reduces the computational effort to obtain a steady state with factors up to 43. For unsteady models, the method can be used for efficiently generating accurate initial conditions on high‐resolution grids. Additionally, the method provides automatic insight in grid convergence. We demonstrate the efficiency and applicability of the method using a schematic test for the vortex shedding around a circular cylinder and a real‐world river case study. Copyright © 2015 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.     

A robust multi‐grid pressure‐based algorithm for multi‐fluid flow at all speeds

This paper reports on the implementation and testing, within a full non‐linear multi‐grid environment, of a new pressure‐based algorithm for the prediction of multi‐fluid flow at all speeds. The algorithm is part of the mass conservation‐based algorithms (MCBA) group in which the pressure correction equation is derived from overall mass conservation. The performance of the new method is assessed by solving a series of two‐dimensional two‐fluid flow test problems varying from turbulent low Mach number to supersonic flows, and from very low to high fluid density ratios. Solutions are generated for several grid sizes using the single grid (SG), the prolongation grid (PG), and the full non‐linear multi‐grid (FMG) methods. The main outcomes of this study are: (i) a clear demonstration of the ability of the FMG method to tackle the added non‐linearity of multi‐fluid flows, which is manifested through the performance jump observed when using the non‐linear multi‐grid approach as compared to the SG and PG methods; (ii) the extension of the FMG method to predict turbulent multi‐fluid flows at all speeds. The convergence history plots and CPU‐times presented indicate that the FMG method is far more efficient than the PG method and accelerates the convergence rate over the SG method, for the problems solved and the grids used, by a factor reaching a value as high as 15. Copyright © 2003 John Wiley & Sons, Ltd.     

A multiblock operator‐splitting algorithm for unsteady flows at all speeds in complex geometries

This paper describes a non‐iterative operator‐splitting algorithm for computing all‐speed flows in complex geometries. A pressure‐based algorithm is adopted as the base, in which pressure, instead of density, is a primary variable, thus allowing for a unified formulation for all Mach numbers. The focus is on adapting the method for (a) flows at all speeds, and (b) multiblock, non‐orthogonal, body‐fitted grids for very complex geometries. Key features of the formulation include special treatment of mass fluxes at control volume interfaces to avoid pressure–velocity decoupling for incompressible (low Mach number limit) flows and to provide robust pressure–velocity–density coupling for compressible (high‐speed) flows. The method is shown to be robust for all Mach number regimes for both steady and unsteady flows; it is found to be stable for CFL numbers of order ten, allowing large time steps to be taken for steady flows. Enhancements to the method which allow for stable solutions to be obtained on non‐orthogonal grids are also discussed. The method is found to be very reliable even in complex engineering applications such as unsteady rotor–stator interactions in turbulent, all‐speed turbomachinery flows. Copyright © 2004 John Wiley & Sons, Ltd.     

An overlapping control volume method for the Navier–Stokes equations on non‐staggered grids

An algorithm, based on the overlapping control volume (OCV) method, for the solution of the steady and unsteady two‐dimensional incompressible Navier–Stokes equations in complex geometry is presented. The primitive variable formulation is solved on a non‐staggered grid arrangement. The problem of pressure–velocity decoupling is circumvented by using momentum interpolation. The accuracy and effectiveness of the method is established by solving five steady state and one unsteady test problems. The numerical solutions obtained using the technique are in good agreement with the analytical and benchmark solutions available in the literature. On uniform grids, the method gives second‐order accuracy for both diffusion‐ and convection‐dominated flows. There is little loss of accuracy on grids that are moderately non‐orthogonal. Copyright © 1999 John Wiley & Sons, Ltd.     

A fractional two‐step implicit algorithm using curvilinear collected grids

An efficient fractional two‐step implicit algorithm is reported to simulate incompressible fluid flows in a boundary‐fitted curvilinear collocated grid system. Using the finite volume method, the convection terms are discretized by the high‐accuracy Roe's scheme to minimize numerical diffusion. An implicitness coefficient Π is introduced to accelerate the rate of convergence. It is demonstrated that the proposed algorithm links the fractional step method to the pressure correction procedure, and the SIMPLEC method could be considered as a special case of the fractional two‐step implicit algorithm (when Π=1). The proposed algorithm is applicable to unsteady flows and steady flows. Three benchmark two‐dimensional laminar flows are tested to evaluate the performance of the proposed algorithm. Performance is measured by sensitivity analyses of the efficiency, accuracy, grid density, grid skewness and Reynolds number on the solutions. Results show that the model is efficient and robust. Copyright © 2006 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.     

Multigrid convergence of inviscid fixed‐ and rotary‐wing flows

The affect of multigrid acceleration implemented within an upwind‐biased Euler method is presented, and applied to fixed‐wing and rotary‐wing flows. The convergence of fixed‐ and rotary‐wing computations is shown to be vastly different, and multigrid is shown to be less effective for rotary‐wing flows. The flow about a hovering rotor suffers from very slow convergence of the inner blade region, where the flow is effectively incompressible. Furthermore, the vortical wake must develop over several turns before convergence is achieved, whereas for fixed‐wing computations the far‐field grid and solution have little significance. Results are presented for single mesh and two, three, four, and five level multigrid, and using five levels a reduction in required CPU time of over 80 per cent is demonstrated for rotary‐wing computations, but 94 per cent for fixed‐wing computations. It is found that a simple V‐cycle is the most effective, smoothing in the decreasing mesh density direction only, with a relaxed trilinear prolongation operator. Copyright © 2002 John Wiley & Sons, Ltd.     

On the treatment of transient area variation in 1D discontinuous Galerkin simulations of train‐induced pressure waves in tunnels

To simulate the pressure wave generated by a train travelling through a tunnel, we implement a discontinuous Galerkin (DG) method for the solution of the one‐dimensional equations of variable area flow. This formulation uses a spatial discretisation via Legendre polynomials of arbitrary degree, and the resulting semi‐discrete system is integrated using an explicit Runge–Kutta scheme. A simulation of subsonic steady flow in a nozzle shows that the scheme produces stable solutions, without the need for artificial dissipation, and that its performance is optimal for polynomial degrees between 5 and 7. However, when dealing with an unsteady area, we report the presence of numerical oscillations that are not due to the steep pressure fronts in the flow but rather to the projection of a moving area, with piecewise continuous derivatives onto a fixed grid. We propose a reformulation of the DG method to eliminate these oscillations that, put in simple terms, amount to splitting the integrals where the derivatives of the cross‐sectional area are discontinuous into subintegrals where they are continuous. The resulting method does not exhibit oscillations, and it is applied here to two practical cases involving train‐induced pressure waves in a tunnel. The first application is a validation of the DG method through comparison of its computational results with pressure data measured during transit at the Patchway tunnel near Bristol (UK). The second application is a study of the influence of the nose shape and length on the pressure wave gradients responsible for sonic boom at tunnel exit portals to show that the proposed modification is able to deal with realistic train shapes. Copyright © 2012 John Wiley & Sons, Ltd.     

Implicit preconditioned high‐order compact scheme for the simulation of the three‐dimensional incompressible Navier–Stokes equations with pseudo‐compressibility method

This paper combines the pseudo‐compressibility procedure, the preconditioning technique for accelerating the time marching for stiff hyperbolic equations, and high‐order accurate central compact scheme to establish the code for efficiently and accurately solving incompressible flows numerically based on the finite difference discretization. The spatial scheme consists of the sixth‐order compact scheme and 10th‐order numerical filter operator for guaranteeing computational stability. The preconditioned pseudo‐compressible Navier–Stokes equations are marched temporally using the implicit lower–upper symmetric Gauss–Seidel time integration method, and the time accuracy is improved by the dual‐time step method for the unsteady problems. The efficiency and reliability of the present procedure are demonstrated by applications to Taylor decaying vortices phenomena, double periodic shear layer rolling‐up problem, laminar flow over a flat plate, low Reynolds number unsteady flow around a circular cylinder at Re = 200, high Reynolds number turbulence flow past the S809 airfoil, and the three‐dimensional flows through two 90°curved ducts of square and circular cross sections, respectively. It is found that the numerical results of the present algorithm are in good agreement with theoretical solutions or experimental data. Copyright © 2011 John Wiley & Sons, Ltd.     

High‐order strand grid methods for low‐speed and incompressible flows

A novel high‐order finite volume scheme using flux correction methods in conjunction with structured finite differences is extended to low Mach and incompressible flows on strand grids. Flux correction achieves a high order by explicitly canceling low‐order truncation error terms across finite volume faces and is applied in unstructured layers of the strand grid. The layers are then coupled together using a source term containing summation‐by‐parts finite differences in the strand direction. A preconditioner is employed to extend the method to low speed and incompressible flows. We further extend the method to turbulent flows with the Spalart–Allmaras model. Laminar flow test cases indicate improvements in accuracy and convergence using the high‐order preconditioned method, while turbulent body‐of‐revolution flow results show improvements in only some cases, perhaps because of dominant errors arising from the turbulence model itself. Copyright © 2016 John Wiley & Sons, Ltd.     

Unsteady shock‐fitting for unstructured grids

An unstructured, shock‐fitting algorithm, originally developed to simulate steady flows, has being further developed to make it capable of dealing with unsteady flows. The present paper discusses and analyses the additional features required to extend to unsteady flows, the steady algorithm. The properties of the unsteady version of this novel, unstructured shock‐fitting technique, are tested by reference to the inviscid interaction between a vortex and a planar shock: a comparative assessment of shock‐capturing and shock‐fitting is made for the same test problem. Copyright © 2015 John Wiley & Sons, Ltd.     

On high‐resolution schemes for solving unsteady compressible two‐phase dilute viscous flows

A high‐resolution numerical scheme based on the MUSCL–Hancock approach is developed to solve unsteady compressible two‐phase dilute viscous flow. Numerical considerations for the development of the scheme are provided. Several solvers for the Godunov fluxes are tested and the results lead to the choice of an exact Riemann solver adapted for both gaseous and dispersed phases. The accuracy of the scheme is proven step by step through specific test cases. These simulations are for one‐phase viscous flows over a flat plate in subsonic and supersonic regimes, unsteady flows in a low‐pressure shock tube, two‐phase dilute viscous flows over a flat plate and, finally, two‐phase unsteady viscous flows in a shock tube. The results are compared with well‐established analytical and numerical solutions and very good agreement is achieved. Copyright © 1999 John Wiley & Sons, Ltd.     

A high‐order accurate method for two‐dimensional incompressible viscous flows

A high‐order accurate solution method for complex geometries is developed for two‐dimensional flows using the stream function–vorticity formulation. High‐order accurate spectrally optimized compact schemes along with appropriate boundary schemes are used for spatial discretization while a two‐level backward Euler implicit scheme is used for the time integration. The linear system of equations for stream function and vorticity are solved by an inner iteration while contravariant velocities constitute outer iterations. The effect of curvilinear grids on the solution accuracy is studied. The method is used to compute Cartesian and inclined driven cavity, flow in a triangular cavity and viscous flow in constricted channel. Benchmark‐like accuracy is obtained in all the problems with fewer grid points compared to reported studies. Copyright © 2006 John Wiley & Sons, Ltd.     

Low‐cost implicit schemes for all‐speed flows on unstructured meshes

Matrix‐free implicit treatments are now commonly used for computing compressible flow problems: a reduced cost per iteration and low‐memory requirements are their most attractive features. This paper explains how it is possible to preserve these features for all‐speed flows, in spite of the use of a low‐Mach preconditioning matrix. The proposed approach exploits a particular property of a widely used low‐Mach preconditioner proposed by Turkel. Its efficiency is demonstrated on some steady and unsteady applications. Copyright © 2008 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.     

Adaptive strategy of transonic flows over vibrating blades with interblade phase angles

An error indicator and a locally implicit scheme with anisotropic dissipation model on dynamic quadri‐ lateral–triangular mesh are developed to study transonic flows over vibrating blades with interblade phase angles. In the Cartesian co‐ordinate system, the unsteady Euler equations with moving domain effects are solved. The error indicator, in which unified magnitudes of dynamic grid speed, substantial derivative of pressure, and substantial derivative of vorticity magnitude are incorporated to capture the unsteady wave behaviours and vortex‐shedding phenomena due to unsteadiness. To assess the accuracy of the locally implicit scheme with anisotropic dissipation model on quadrilateral–triangular mesh, two flow calculations are performed. Based on the comparison with the related numerical and experimental data, the accuracy of the present approach is confirmed. According to the high‐resolutional result on the adaptive mesh, the unsteady pressure wave, shock and vortex‐shedding behaviours are clearly demonstrated. Copyright © 2003 John Wiley & Sons, Ltd.     

Computation of unsteady viscous incompressible flows in generalized non‐inertial co‐ordinate system using Godunov‐projection method and overlapping meshes

Time‐dependent incompressible Navier–Stokes equations are formulated in generalized non‐inertial co‐ordinate system and numerically solved by using a modified second‐order Godunov‐projection method on a system of overlapped body‐fitted structured grids. The 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 second‐order Godunov method is applied for numerically approximating the non‐linear convection terms in order to provide a robust discretization for simulating flows at high Reynolds number. In order to obtain the pressure field, the pressure Poisson equation is solved. Overlapping grids are used to discretize the flow domain so that the moving‐boundary problem can be solved economically. Numerical results are then presented to demonstrate the performance of this projection method for a variety of unsteady two‐ and three‐dimensional flow problems formulated in the non‐inertial co‐ordinate systems. Copyright © 2002 John Wiley & Sons, Ltd.     

High‐resolution p‐adaptive DG simulations of flows with moving shocks

High‐order accurate DG discretization is employed for Reynolds‐averaged Navier–Stokes equations modeling of complex shock‐dominated, unsteady flow generated by gas issuing from a shock tube nozzle. The DG finite element discretization framework is used for both the flow field and turbulence transport. Turbulent flow in the near wall regions and the flow field is modeled by the Spalart–Allmaras one‐equation model. The effect of rotation on turbulence modeling for shock‐dominated supersonic flows is considered for accurate resolution of the large coherent and vortical structures that are of interest in high‐speed combustion and supersonic flows. Implicit time marching methodologies are used to enable large time steps by avoiding the severe time step limitations imposed by the higher order DG discretizations and the source terms. Sufficiently high mesh density is used to enable crisp capturing of discontinuities. A p ? type refinement procedure is employed to accurately represent the vortical structures generated during the development of the flow. The computed solutions showed qualitative agreement with experiments. Copyright © 2014 John Wiley & Sons, Ltd.