Steady state incompressible flows using explicit schemes with an optimal local preconditioning

Solving large systems of equations from CFD problems by the explicit pseudo-temporal scheme requires a very low amount of memory and is highly parallelizable, but the CPU time largely depends on the conditioning of the system. For advective systems it is shown that the rate of convergence depends on a condition number defined as the ratio of the maximum and the minimum group velocities of the continuum system. If the objective is to reach the steady state, the temporal term can be modified in order to reduce this condition number. Another possibility consists in the addition of a local preconditioning mass matrix. In this paper an optimal preconditioning for incompressible flow is presented, also applicable to compressible ones with locally incompressible zones, like stagnation points, in contrast with the artificial compressibility method. The preconditioned system has a rate of convergence independent from Mach number. Moreover, the discrete solution is highly improved, eliminating spurious oscillations frequently encountered in incompressible flows

Numerical simulation of liquid sloshing in a partially filled container with inclusion of compressibility effects

A numerical scheme of study is developed to model compressible two-fluid flows simulating liquid sloshing in a partially filled tank. For a two-fluid system separated by an interface as in the case of sloshing, not only a Mach-uniform scheme is required, but also an effective way to eliminate unphysical numerical oscillations near the interface. By introducing a preconditioner, the governing equations expressed in terms of primitive variables are solved for both fluids (i.e. water, air, gas etc.) in a unified manner. In order to keep the interface sharp and to eliminate unphysical numerical oscillations in unsteady fluid flows, the non-conservative implicit Split Coefficient Matrix Method (SCMM) is modified to construct a flux difference splitting scheme in the dual time formulation. The proposed numerical model is evaluated by comparisons between numerical results and measured data for sloshing in an 80% filled rectangular tank excited at resonance frequency. Through similar comparisons, the investigation is further extended by examining sloshing flows excited by forced sway motions in two different rectangular tanks with 20% and 83% filling ratios. These examples demonstrate that the proposed method is suitable to capture induced free surface waves and to evaluate sloshing pressure loads acting on the tank walls and ceiling

New numerical solver for flows at various Mach numbers

Many problems in stellar astrophysics feature flows at low Mach numbers. Conventional compressible hydrodynamics schemes frequently used in the field have been developed for the transonic regime and exhibit excessive numerical dissipation for these flows. While schemes were proposed that solve hydrodynamics strictly in the low Mach regime and thus restrict their applicability, we aim at developing a scheme that correctly operates in a wide range of Mach numbers. Based on an analysis of the asymptotic behavior of the Euler equations in the low Mach limit we propose a novel scheme that is able to maintain a low Mach number flow setup while retaining all effects of compressibility. This is achieved by a suitable modification of the well-known Roe solver. Numerical tests demonstrate the capability of this new scheme to reproduce slow flow structures even in moderate numerical resolution. Our scheme provides a promising approach to a consistent multidimensional hydrodynamical treatment of astrophysical low Mach number problems such as convection, instabilities, and mixing in stellar evolution.Comment: 16 pages, 8 figures, accepted for publication by A&

Preconditioned methods for solving the incompressible and low speed compressible equations

Acceleration methods are presented for solving the steady state incompressible equations. These systems are preconditioned by introducing artificial time derivatives which allow for a faster convergence to the steady state. The compressible equations in conservation form with slow flow are also considered. Two arbitrary functions, alpha and beta, are introduced in the general preconditioning. An analysis of this system is presented and an optimal value for beta is determined given a constant, alpha. It is further shown that the resultant incompressible equations form a symmetric hyperbolic system and so are well posed. Several generalizations to the compressible equations are presented which generalize previous results

A Preconditioned Lattice Boltzmann Flux Solver for Steady Flows on Unstructured Hexahedral Grids

The lattice Boltzmann flux solver (LBFS), first introduced by Shu et al. (2014) on structured meshes, allows fluid flow problems to be solved on unstructured meshes discretised by the finite volume method. The solver calculates the macroscopic fluxes at the cell interfaces from a local reconstruction of the lattice Boltzmann solution. In this paper the LBFS is extended to three-dimensional unstructured hexahedral meshes and a preconditioned lattice Boltzmann flux solver (PLBFS) is presented. The PLBFS involves applying the preconditioning technique proposed by Guo (2004) to the LBFS and is achieved by modifying the equilibrium distribution function used to calculate the macroscopic fluxes at the cell interface. When the PLBFS is applied to steady flow problems, it is shown that convergence is significantly accelerated and the accuracy of predictions with unstructured grids is greatly improved when compared to the LBFS. This paper also introduces a strategy for choosing the optimal value of preconditioning factor with unstructured hexahedral meshes

Time-derivative preconditioning for viscous flows

A time-derivative preconditioning algorithm that is effective over a wide range of flow conditions from inviscid to very diffusive flows and from low speed to supersonic flows was developed. This algorithm uses a viscous set of primary dependent variables to introduce well-conditioned eigenvalues and to avoid having a nonphysical time reversal for viscous flow. The resulting algorithm also provides a mechanism for controlling the inviscid and viscous time step parameters to be of order one for very diffusive flows, thereby ensuring rapid convergence at very viscous flows as well as for inviscid flows. Convergence capabilities are demonstrated through computation of a wide variety of problems

Time-domain and harmonic balance turbulent Navier-Stokes analysis of wind turbine aerodynamics using a fully coupled low-speed preconditioned multigrid solver

The research work reported in this thesis stems from the development of an accurate and computationally efficient Reynolds-Averaged Navier-Stokes (RANS) research code, with a particular emphasis on the steady and unsteady aerodynamics analysis of complex low speed turbulent flows. Such turbulent flow problems include horizontal axis wind turbine (HAWT) and vertical axis wind turbine (VAWT) operating at design and off-design conditions. On the algorithmic side, the main contribution of this research is the successful development of a rigorous novel approach to low-speed preconditioning (LSP) for the multigrid fully coupled integration of the steady, time-domain and harmonic balance RANS equations coupled to the two-equation shear stress transport (SST) turbulence model. The design of the LSP implementation is such that each part of the code affected by LSP can be validated individually against the baseline solver by suitably specifying one numerical input parameter of the LSP-enhanced code. The thesis has investigated several important issues on modelling and numerical aspects which are seldom thoroughly analysed in the computational fluid dynamics problems of the type presented herein. The first and most important modelling issue is the necessity of applying the low speed preconditioning to both RANS and SST equations and maintaining the turbulent kinetic energy in the definition of the total energy, which, to the best knowledge of author, has never been seen in any published literature so far. Based on the results obtained in the analysis of the vertical axis wind turbine application, we have demonstrated that by preconditioning the SST turbulence equations, one can significantly improve the convergence rate; and keeping the turbulence kinetic energy in the total energy has a great positive effect on the solution accuracy. The other modelling issue to be analysed is the sensitivity of the flow solution to the farfield boundary conditions, particularly for low speed problems. The analyses reported in the thesis highlight that with a small size of the computational domain, the preconditioned farfield boundary conditions are crucial to improve the solution accuracy. As for the numerical aspects, we analyse the impact of using the relative velocity to build the preconditioning parameter on the flow solutions of an unsteady moving-grid problem. The presented results demonstrate that taking into account the grid motion in building the preconditioning parameter can achieve a noticeable enhancement of the solution accuracy. On the other hand, the nonlinear frequency-domain harmonic balance approach is a fairly new technology to solve the unsteady RANS equations, which yields significant reduction of the run-time required to achieve periodic flows with respect to the conventional time-domain approach. And the implementation of the LSP approach into the turbulent harmonic balance RANS and SST formulations is another main novelty presented herein, which is also the first published research work on this aspect. The newly developed low speed turbulent flow predictive capabilities are comprehensively validated in a wide range of tests varying from subsonic flow with slight compressibility to user-defined extremely low speed incompressible flows. The solutions of our research code with LSP technology are compared with experiment data, theoretical solutions and numerical solutions of the state-of-the-art CFD research code and commercial package. The main computational results of this research consist of the analyses of HAWT and VAWT applications. The first one is a comparative analysis of 30% and 93.5% blade sections of a VESTAS multi-megawatt HAWT working in various regimes. The steady, time-domain and frequency-domain results obtained with the LSP solver are used to analyse in great detail the steady and unsteady aerodynamic characteristics in those regimes. The main motivation is to highlight the predictive capabilities and the numerical robustness of the LSP-enhanced turbulent steady, time-domain and frequency domain flow solvers for realistic complex and even more challenging problems, to quantify the effects of flow compressibility on the steady and yawed wind-induced unsteady aerodynamics in the tip region of a 82-m HAWT blade in rated operating condition, and to assess the computational benefits achieved by using the harmonic balance method rather than the conventional time-domain method. The second application is the comparative aerodynamic analyses of the NREL 5MW HAWT working in the inviscid steady flow condition. The main motivation of this analysis is to further demonstrate the predictive capabilities of the LSP solver to simulate the threedimensional wind turbine flows. The last application is the time-domain turbulent flow analysis of the VAWT to the aim of demonstrating the accuracy enhancement of the LSP solver for this particular problem, the necessity of applying the full preconditioning strategy, the important effect of the turbulent kinetic energy on the solution accuracy and the proper implementation of the preconditioning parameter required for an accurate numerical solution to an unsteady moving grid low-speed problem

Discontinuous Galerkin Methods for inviscid low Mach number flows

In this work we present two preconditioning techniques for inviscid low Mach number flows. The space discretization used is a high-order Discontinuous Galerkin finite element method. The time discretizations analyzed are explicit and implicit schemes. The convective physical flux is replaced by a flux difference splitting scheme. Computations were performed on triangular and quadrangular grids to analyze the influence of the spatial discretization. For the preconditioning of the explicit Euler equations we propose to apply the fully preconditioning approach: a formulation that modifies both the instationary term of the governing equations and the dissipative term of the numerical flux function. For the preconditioning of the implicit Euler equations we propose to apply the flux preconditioning approach: a formulation that modifies only the dissipative term of the numerical flux function. Both these formulations permit to overcome the stiffness of the governing equations and the loss of accuracy of the solution that arise when the Mach number tends to zero. Finally, we present a splitting technique, a proper manipulation of the flow variables that permits to minimize the cancellation error that occurs as an accumulation effect of round-off errors as the Mach number tends to zero

Algorithms for the Euler and Navier-Stokes equations for supercomputers

The steady state Euler and Navier-Stokes equations are considered for both compressible and incompressible flow. Methods are found for accelerating the convergence to a steady state. This acceleration is based on preconditioning the system so that it is no longer time consistent. In order that the acceleration technique be scheme-independent, this preconditioning is done at the differential equation level. Applications are presented for very slow flows and also for the incompressible equations

Artificial compressibility method for high-pressure transcritical fluids at low Mach numbers

Supercritical fluids possess unique properties that makes them relevant in various scientific and engineering applications. However, the experimental investigation of these fluids is challenging due to the high pressures involved and their complex thermophysical behavior. To overcome these limitations, computational researchers employ scale-resolving methods, such as direct numerical simulation and large-eddy simulation to study them. Nonetheless, these methods require substantial computational resources, especially in the case of low-Mach-number regimes due to the disparity between acoustic and hydrodynamic/thermal time scales. This work, therefore, addresses this problem by extending the artificial compressibility method to high-pressure transcritical fluids. This method is based on decoupling the thermodynamic and hydrodynamic parts of the pressure field, such that the acoustic time scales can be externally modified without severely impacting the flow physics of the problem. In addition, the method proposed has two key characteristics: (i) the splitting method presents low computational complexity, and (ii) an automatic strategy for selecting the speedup factor of the approach is introduced. The effectiveness of the resulting methodology is demonstrated through comprehensive numerical tests of increasing complexity, showcasing its ability to accurately simulate a wide range of high-pressure transcritical flows including turbulence. The results obtained indicate that the approach proposed can readily lead to computational speedups larger than without significantly compromising the accuracy of the numerical solutions.This work is funded by the European Union (ERC, SCRAMBLE, 101040379). Views and opinions expressed are however those of the authors only and do not necessarily reflect those of the European Union or the European Research Council. Neither the European Union nor the granting authority can be held responsible for them.Peer ReviewedPostprint (published version