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A simple algebraic multigrid (AMG) solver for linear equations is presented, and its performance compared with a conjugate gradient scheme. This multigrid method is extended to solve the discrete Navier—Stokes equations, obtained by applying a finite volume approach to three‐dimensional incompressible flow on a finite element mesh. The resulting multigrid solver is incorporated into a general purpose flow code (ASTEC), where it proves faster than the original solution algorithm, based upon SIMPLE. The linear AMG solver is both efficient and robust, but the extension to include coupling in the Navier—Stokes equations does not converge on all problems.

Applicability of two k‐ω hybrid RANS/LES and a k‐ω RANS models is studied for simulation of round impinging jets at nozzle‐plate distance H/D=2 with Reynolds number 70000, H/D=2 with Reynolds number 5000 and H/D=10 with Reynolds number 5000 (D is the nozzle exit diameter). The aim is to verify two concepts of unified hybrid RANS/LES formulations, one of DES (Detached Eddy Simulation) type and one of LNS (Limited Number Scales) type in analysis of impinging jet flow and heat transfer. The grid resolution requirements are also discussed.

The simulations are performed with two k‐ω based hybrid RANS/LES models of very different nature, one of DES type and one of LNS type, and the RANS k‐ω model. For the lower Reynolds number (5000), also dynamic Smagorinsky LES is done. Both hybrid model formulations converge to the same RANS k‐ω model in the near‐wall region and have the same Smagorinsky limit on fine isotropic grids in the LES mode of the hybrid models.

With the hybrid RANS/LES models, improved fluid flow and heat transfer results are obtained compared to RANS, in the impact region and in the developing wall‐jet region. For accurate predictions at low nozzle‐plate distance, where the impact region is in the core of the jet, it is necessary to sufficiently resolve the formation and breakup of the near‐wall vortices in the jet impingement region and the developing wall‐jet region, as these determine largely the level of fluctuating velocity and the heat transfer. This requires high grid resolution for high Reynolds number, while the grid resolution requirements stay modest for low Reynolds number.

The paper demonstrates that two formulations of hybrid RANS/LES models of different nature, one of DES type and one of LES type, lead to equivalent results. Consistency has been guaranteed in the sense that the RANS limit of both models is the same and that the LES limit on fine, isotropic, grids is the same. In the intermediate range, however, the repartition into resolved and modelled fluctuations may differ considerably.

The purpose of this paper is to develop a new finite difference method for solving the seismic wave propagation in fluid-solid media, which can be described by the acoustic and viscoelastic wave equations for the fluid and solid parts, respectively.

In this paper, the authors introduced a coordinate transformation method for seismic wave simulation method. In the new method, the irregular fluid–solid interface is transformed into a horizontal interface. Then, a multi-block coordinate transformation method is proposed to mesh every layer to curved grids and transforms every interface to horizontal interface. Meanwhile, a variable grid size is used in different regions according to the shape and the velocity within each region. Finally, a Lebedev-standard staggered coupled grid scheme for curved grids is applied in the multi-block coordinate transformation method to reduce the computational cost.

The instability in the auxiliary coordinate system caused by the standard staggered grid scheme is resolved using a curved grid viscoelastic wave field separation strategy. Several numerical examples are solved using this new method. It has been shown that the new method is stable, efficient and highly accurate in solving the seismic wave equation defined on domain with irregular fluid–solid interface.

First, the irregular fluid–solid interface is transformed into a horizontal interface by using the coordinate transformation method. The conversion between pressures and stresses is easy to implement and adaptive to different irregular fluid–solid interface models, because the normal stress and shear stress vanish when the normal angle is 90° in the interface. Moreover, in the new method, the strong false artificial boundary reflection and instability caused by ladder-shaped grid discretion are resolved as well.

A comparative study is performed to reveal the convergence characteristics and the robustness of four variants in the semi‐implicit method for pressure‐linked equations (SIMPLE)‐family: SIMPLE, SIMPLE revised (SIMPLER), SIMPLE consistent (SIMPLEC), and SIMPLE extrapolation (SIMPLEX). The focus is concentrated in the solution at fine grid system. Four typical fluid flow and heat transfer problems are taken as the numerical examples (lid‐driven cavity flow, flow in an axisymmetric sudden expansion, flow in an annulus with inner surface rotating and the natural convection in a square enclosure). It is found that an appropriate convergence condition should include both mass conservation and momentum conservation requirements. For the four problems computed, the SIMPLEX always requires the largest computational time, the SIMPLER comes the next, and the computational time of SIMPLE and SIMPLEC are the least. As far as the robustness is concerned, the SIMPLE algorithm is the worst, the SIMPLER comes the next and the robustness of SIMPLEX and SIMPLEC are superior to the others. The SIMPLEC algorithm is then recommended, especially for the computation at a fine grid system. Brief discussion is provided to further reveal the reasons which may account for the difference of the four algorithms.

The potential for overall efficiency improvements of modern hydro power turbines is a few percent. A significant part of the losses occurs in the draft tube. To improve the efficiency by analysing the flow in the draft tube, it is therefore necessary to do this accurately, i.e. one must know how large the iterative and the grid errors are. This was done by comparing three different methods to estimate errors. Four grids (122,976 to 4,592 cells) and two numerical schemes (hybrid differencing and CCCT) were used in the comparison. To assess the iterative error, the convergence history and the final value of the residuals were used. The grid error estimates were based on Richardson extrapolation and least square curve fitting. Using these methods we could, apart from estimate the error, also calculate the apparent order of the numerical schemes. The effects of using double or single precision and changing the under relaxation factors were also investigated. To check the grid error the pressure recovery factor was used. The iterative error based on the pressure recovery factor was very small for all grids (of the order 10^–4 percent for the CCCT scheme and 10^–10percent for the hybrid scheme). The grid error was about 10 percent for the finest grid and the apparent order of the numerical schemes were 1.6 for CCCT (formally second order) and 1.4 for hybrid differencing (formally first order). The conclusion is that there are several methods available that can be used in practical simulations to estimate numerical errors and that in this particular case, the errors were too large. The methods for estimating the errors also allowed us to compute the necessary grid size for a target value of the grid error. For a target value of 1 percent, the necessary grid size for this case was computed to 2 million cells.

The averaged Navier‐Stokes and the k‐e turbulence model equations are used to simulate turbulent flows in some internal flow cases. The discrete equations are solved by different variations of Multigrid methods. These include both steady state as well as time dependent solvers. Locally refined grids can be added dynamically in all cases. The Multigrid schemes result in fast convergence rates, whereas local grid refinements allow improved accuracy with rational increase in problem size. The applications of the solver to a 3‐D (cold) furnace model and to the simulation of the flow in a wind tunnel past an object prove the efficiency of the Multigrid scheme with local grid refinement.

An adaptive grid solution method is described for computing the time accurate solution of an unsteady flow problem. The solution method consists of three parts: a grid point redistribution method; an unsteady Euler equation solver; and a temporal coupling routine that links the dynamic grid to the flow solver. The grid movement technique is a direct curve by curve method containing grid controls that generate a smooth grid that resolves the severe solution gradients and the sharp transitions in the solution gradients. By design, the temporal coupling procedure provides a grid that does not lag the solution in time. The adaptive solution method is tested by computing the unsteady inviscid solutions for a one‐dimensional shock tube and a two‐dimensional shock vortex interaction. Quantitative comparisons are made between the adaptive solutions, theoretical solutions and numerical solutions computed on stationary grids. Test results demonstrate the good temporal tracking of the solution by the adaptive grid, and the ability of the adaptive method to capture an unsteady solution of comparable accuracy to that computed on a stationary grid containing significantly more grid points than used in the adaptive grid.

Multiphase flow computations involve coupled momentum, mass and energy transfer between moving and irregularly shaped boundaries, large property jumps between materials, and computational stiffness. In this study, we focus on the immersed boundary technique, which is a combined Eulerian‐Lagrangian method, to investigate the performance improvement using the multigrid technique in the context of the projection method. The main emphasis is on the interplay between the multigrid computation and the effect of the density and viscosity ratios between phases. Two problems, namely, a rising bubble in a liquid medium and impact dynamics between a liquid drop and a solid surface are adopted. As the density ratio increases, the single grid computation becomes substantially more time‐consuming; with the present problems, an increase of factor 10 in density ratio results in approximately a three‐fold increase in CPU time. Overall, the multigrid technique speeds up the computation and furthermore, the impact of the density ratio on the CPU time required is substantially reduced. On the other hand, the impact of the viscosity ratio does not play a major role on the convergence rates.

We develop new high‐order positive, monotone and convex interpolations, which are to be used in the multigrid context. This means that the value of the interpolant is calculated only at the midpoints lying between the locations of the given values. As a consequence, these interpolants can be calculated very efficiently. They are then tested in a time‐dependent very large scale integration process simulation application.

The overall pressure drop in an axisymmetric contraction is minimised using two different grid sizes. The transition region was parameterised with only two design variables to make it possible to create surface plots of the objective function in the design space, which were based on 121 CFD calculations for each grid. The coarse grid showed to have significant numerical noise in the objective function while the finer grid had less numerical noise. The optimisation was performed with two methods, a Response Surface Model (RSM) and a gradient‐based method (the Method of Feasible Directions) to study the influence from numerical noise. Both optimisation methods were able to find the global optimum with the two different grid sizes (the search path for the gradient‐based method on the coarse grid was able to avoid the region in the design space containing local minima). However, the RSM needed fewer iterations in reaching the optimum. From a grid convergence study at two points in the design space the level of noise appeared to be sufficiently low, when the relative step size is 10^–4 for the finite difference calculations, to not influence the convergence if the errors are below 5 per cent for this contraction geometry.