A multidimensional characteristic-based method for making incompressible flow calculations on unstructured grids

The unsteady preconditioned characteristic boundary conditions (UPCBCs) based on the artificial compressibility (AC) method are formulated and applied at artificial boundaries for the direct numerical simulation (DNS) of incompressible flows. The compatibility equations including the unsteady terms are mathematically derived in the generalized curvilinear coordinates and then incorporated as boundary conditions (BCs) in a high-order accurate incompressible flow solver. The spatial derivative terms of the system of equations are discretized using the fourth-order compact finite difference (FD) scheme, consistent with the high-order accuracy required for the DNS. The time integration is carried out using an implicit dual time-stepping method, and the robustness and performance of the numerical algorithm are enhanced by applying the preconditioning technique and calculating the automated AC parameter based on the local flow conditions. At first, the steady viscous incompressible flow for a two-dimensional plane jet is computed to assess the accuracy and performance of the solution method by applying the UPCBCs in comparison with the simplified BCs. Then, the simulation of the unsteady incompressible viscous flow around a two-dimensional NACA0012 airfoil at the Reynolds number 800 and the angle of attack 20° is performed, and it is indicated that the computational cost of the solution is considerably decreased by implementing the UPCBCs at the artificial far-field boundary compared with the simplified BCs, which is remarkably valuable for the DNSs. Finally, the computation of the transition to unsteady for the incompressible flow around a three-dimensional NACA0012 wing at the same conditions of the two-dimensional case is successfully performed, and the results obtained are compared with those of previous studies that exhibit good agreement. Indications are that the present solution method by applying the UPCBCs at the artificial far-field boundary can be effectively used for the DNSs of the incompressible flows.

Numerical simulation of incompressible flows using adaptive unstructured meshes and the pseudo-compressibility hypothesis

The parallelization of the least-squares spectral element formulation of the Stokes problem has recently been discussed for incompressible flow problems on structured grids. In the present work, the extension to unstructured grids is discussed. It will be shown that, to obtain an efficient and scalable method, two different kinds of distribution of data are required involving a rather complicated parallel conversion between the data. Once the data conversion has been performed, a large symmetric positive definite algebraic system has to be solved iteratively. It is well known that the Conjugate Gradient method is a good choice to solve such systems. To improve the convergence rate of the Conjugate Gradient process, both Jacobi and Additive Schwarz preconditioners are applied. The Additive Schwarz preconditioner is based on domain decomposition and can be implemented such that a preconditioning step corresponds to a parallel matrix-by-vector product. The new results reveal that the Additive Schwarz preconditioner is very suitable for the p-refinement version of the least-squares spectral element method. To obtain good portable programs which may run on distributed-memory multiprocessors, networks of workstations as well as shared-memory machines we use MPI (Message Passing Interface). Numerical simulations have been performed to validate the scalability of the different parts of the proposed method. The experiments entailed simulating several large scale incompressible flows on a Cray T3E and on an SGI Origin 3800 with the number of processors varying from one to more than one hundred. The results indicate that the present method has very good parallel scaling properties making it a powerful method for numerical simulations of incompressible flows.

A high-order spectral difference (SD) method for solving the Navier-Stokes equations on moving, deformable unstructured grids has been developed [1]. In this paper, the SD method and the artificial compressibility method (ACM) are integrated with a dual time-stepping scheme to model unsteady incompressible viscous flow past an inline tube bundle of cylinders equally sized (diameter = d) and spaced (spacing = 2.1*d) over an unstructured grid. Flow simulation results are obtained using a fourth-order space accurate SD method. Two forced oscillation cases are considered; (1) 1st cylinder oscillation and (2) 2nd cylinder oscillation. The Reynolds number used for both cases is 100 and the flow is laminar. Forced oscillation is performed in the tranverse direction, and the subsequent altering of the flow physics of the system is studied. The frequency of vortex shedding behind each cylinder is the same. Root mean square results show that the lift coefficient is greatest for the 4th inline cylinder in both cases. Furthermore, a reduction in both lift and drag coefficients is seen from case (1) to case (2).

In this paper, we apply a simple finite element numerical scheme, proposed in an earlier work (Liu in Math Comput 70(234):579–593, 2000), to perform a high resolution numerical simulation of incompressible flow over an irregular domain and analyze its boundary layer separation. Compared with many classical finite element fluid solvers, this numerical method avoids a Stokes solver, and only two Poisson-like equations need to be solved at each time step/stage. In addition, its combination with the fully explicit fourth order Runge–Kutta (RK4) time discretization enables us to compute high Reynolds number flow in a very efficient way. As an application of this robust numerical solver, the dynamical mechanism of the boundary layer separation for a triangular cavity flow with Reynolds numbers $$Re=10^4$$ and $$Re=10^5$$ , including the precise values of bifurcation location and critical time, are reported in this paper. In addition, we provide a super-convergence analysis for the simple finite element numerical scheme, using linear elements over a uniform triangulation with right triangles.

A new method of treating incompressible flows with nonslip boundaries is proposed as an extension of the Fourier spectral method. This is characteristic in using the function subspace that is a hyperplane in the Fourier-transformed velocity space, prescribed by the boundary condition, as well as in taking the solenoidal field representation in the Fourier space so that the pressure term need not be involved in the main dynamics and then time-integration can simply be made by the high-order Runge-Kutta scheme. The method can be applied in a more complicated case with an active scalar. As examples, the flow transitions to turbulence in a channel and in a rectangular duct heated from below are treated.

Numerical simulation of incompressible flows through granular porous media

Parallel direct Poisson solver for discretisations with one Fourier diagonalisable direction

Purpose The purpose of this paper is to develop a combined method for three-dimensional incompressible flow and heat transfer by the spectral collocation method (SCM) and the artificial compressibility method (ACM), and further to study the performance of the combined method SCM-ACM for three-dimensional incompressible flow and heat transfer. Design/methodology/approach The partial differentials in space are discretized by the SCM with Chebyshev polynomial and Chebyshev–Gauss–Lobbatto collocation points. The unsteady artificial compressibility equations are solved to obtain the steady results by the ACM. Three-dimensional exact solutions with trigonometric function form and exponential function form are constructed to test the accuracy of the combined method. Findings The SCM-ACM is developed successfully for three-dimensional incompressible flow and heat transfer with high accuracy that the minimum value of variance can reach. The accuracy increases exponentially along with time marching steps. The accuracy is also improved exponentially with the increasing of nodes before stable accuracy is achieved, while it keeps stably with the increasing of the time step. The central processing unit time increases exponentially with the increasing of nodes and decreasing of the time step. Research limitations/implications It is difficult for the implementation of the implicit scheme by the developed SCM-ACM. The SCM-ACM can be used for solving unsteady impressible fluid flow and heat transfer. Practical implications The SCM-ACM is applied for two classic cases of lid-driven cavity flow and natural convection in cubic cavities. The present results show good agreement with the published results with much fewer nodes. Originality/value The combined method SCM-ACM is developed, firstly, for solving three-dimensional incompressible fluid flow and heat transfer by the SCM and ACM. The performance of SCM-ACM is investigated. This combined method provides a new choice for solving three-dimensional fluid flow and heat transfer with high accuracy.

A massively parallel hybrid scheme for direct numerical simulation of turbulent viscoelastic channel flow

A novel meshless method for incompressible flow calculations

Simulation of environmental flow problems in geometrically complex domains. Part I: A general coordinate transformation

Mixing is the process that homogenizes initially segregated miscible constituents, increases the volume occupied by a solute, and decreases concentration peaks. It is important for the assessment of contamination levels and biogeochemical reactions in groundwater and soils. Mixing processes are governed by the interplay of fluid advection, molecular diffusion and local-scale dispersion at Darcy scale. Here we study the mechanisms of mixing in three-dimensional Darcy scale porous media with different heterogeneity structure. We analyze the role of medium and flow topology on the mixing and dispersion behavior. To this end, we perform Darcy-scale numerical simulations of incompressible flow and transport in heterogeneous three-dimensional porous media. Hydraulic conductivity is represented as a multi-Gaussian random field with lognormal marginal distribution. We consider isotropic and anisotropic correlation structures and scalar and tensorial conductivity. Flow is solved using a finite volume two-point method and transport using a Lagrangian approach. The flow topology is quantified by the helicity of the velocity field. We consider a planar injection of particles. Dispersion is quantified by the longitudinal and transverse dispersion coefficient, which are determined by the evolution along time of the position’s variance in the respective direction divide by two. It is also quantified by the breakthrough curves, which measure the distribution of arrival times at a given position from the initial one. Mixing is quantified by the ability of the flow to stretch and elongate a fluid strip which enhances diffusion through the creation and sustaining of concentration gradients. Results show that for a helical flow, a finite transverse dispersion coefficient is observed at long times and that the elongation of elemental strips follow an exponential stretching  (for large logK variances). On the contrary, on non-helical flows, transverse dispersion tends asymptotically to zero and the stretching rate is algebraic. The longitudinal dispersion coefficient seems unaffected by the helicity of the flow. These results shed light on the relation between medium structure and flow topology on mixing, making an important step towards the control, upscaling and large scale representation of mixing in porous media .  

A distributed parallel direct simulator for pore-scale two-phase flow on digital rock images using a finite difference implementation of the phase-field method

This paper concerns the mixed Laguerre–Legendre spectral approximation and its application to numerical simulation of incompressible flow in an infinite strip. Some approximation results in weighted Sobolev spaces are given. A Laguerre–Legendre spectral scheme for the stream function form of Navier–Stokes equations is constructed. The stability and the convergence of the proposed scheme are proved. The numerical experiments show the high accuracy of this method. The main techniques used in this paper are also applicable to other nonlinear partial differential equations in an infinite strip.