scholarly journals Artificial Compressibility Methods for the Incompressible Navier–Stokes Equations Using Lowest-Order Face-Based Schemes on Polytopal Meshes

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Riccardo Milani ◽  
Jérôme Bonelle ◽  
Alexandre Ern
Keyword(s):  
Stokes Equations ◽  
Time Discretization ◽  
Second Order ◽  
Navier Stokes ◽  
Time Step ◽  
Navier Stokes Equations ◽  
Artificial Compressibility ◽  
Time Stepping ◽  
Monolithic Approach ◽  
Polytopal Meshes

Abstract We investigate artificial compressibility (AC) techniques for the time discretization of the incompressible Navier–Stokes equations. The space discretization is based on a lowest-order face-based scheme supporting polytopal meshes, namely discrete velocities are attached to the mesh faces and cells, whereas discrete pressures are attached to the mesh cells. This face-based scheme can be embedded into the framework of hybrid mixed mimetic schemes and gradient schemes, and has close links to the lowest-order version of hybrid high-order methods devised for the steady incompressible Navier–Stokes equations. The AC time-stepping uncouples at each time step the velocity update from the pressure update. The performances of this approach are compared against those of the more traditional monolithic approach which maintains the velocity-pressure coupling at each time step. We consider both first-order and second-order time schemes and either an implicit or an explicit treatment of the nonlinear convection term. We investigate numerically the CFL stability restriction resulting from an explicit treatment, both on Cartesian and polytopal meshes. Finally, numerical tests on large 3D polytopal meshes highlight the efficiency of the AC approach and the benefits of using second-order schemes whenever accurate discrete solutions are to be attained.

scholarly journals A Second-Order Time-Stepping Scheme for Simulating Ensembles of Parameterized Flow Problems

2019 ◽  
Vol 19 (3) ◽  
pp. 681-701 ◽  
Author(s):  
Max Gunzburger ◽  
Nan Jiang ◽  
Zhu Wang
Keyword(s):  
Stokes Equations ◽  
Second Order ◽  
Navier Stokes ◽  
Conditional Stability ◽  
Initial Condition ◽  
Time Step ◽  
Navier Stokes Equations ◽  
Flow Simulations ◽  
Flow Problems ◽  
Time Stepping

AbstractWe consider settings for which one needs to perform multiple flow simulations based on the Navier–Stokes equations, each having different initial condition data, boundary condition data, forcing functions, and/or coefficients such as the viscosity. For such settings, we propose a second-order time accurate ensemble-based method that to simulate the whole set of solutions, requires, at each time step, the solution of only a single linear system with multiple right-hand-side vectors. Rigorous analyses are given proving the conditional stability and establishing error estimates for the proposed algorithm. Numerical experiments are provided that illustrate the analyses.

Numerical Solution of Incompressible Unsteady Flows in Turbomachinery

2000 ◽  
Vol 123 (3) ◽  
pp. 680-685 ◽  
Author(s):  
L. He ◽  
K. Sato
Keyword(s):  
Stokes Equations ◽  
Three Dimensional ◽  
Unsteady Flows ◽  
Navier Stokes ◽  
Time Step ◽  
Navier Stokes Equations ◽  
Flow Solver ◽  
Time Stepping ◽  
Incompressible Viscous Flow ◽  
Dual Time Stepping

A three-dimensional incompressible viscous flow solver of the thin-layer Navier-Stokes equations was developed for the unsteady turbomachinery flow computations. The solution algorithm for the unsteady flows combines the dual time stepping technique with the artificial compressibility approach for solving the incompressible unsteady flow governing equations. For time accurate calculations, subiterations are introduced by marching the equations in the pseudo-time to fully recover the incompressible continuity equation at each real time step, accelerated with a multi-grid technique. Computations of test cases show satisfactory agreements with corresponding theoretical and experimental results, demonstrating the validity and applicability of the present method to unsteady incompressible turbomachinery flows.

Rotor-Stator Interaction Analysis Using the Navier-Stokes Equations and a Multigrid Method

1995 ◽  
Author(s):  
Andrea Arnone ◽  
Roberto Pacciani
Keyword(s):  
Stokes Equations ◽  
Interaction Analysis ◽  
Time Discretization ◽  
Navier Stokes ◽  
Implicit Time ◽  
Navier Stokes Equations ◽  
Time Stepping ◽  
Fully Implicit ◽  
Rotor Stator Interaction ◽  
Stage Analysis

A recently developed, time-accurate multigrid viscous solver has been extended to the analysis of unsteady rotor-stator interaction. In the proposed method, a fully-implicit time discretization is used to remove stability limitations. By means of a dual time-stepping approach, a four-stage Runge-Kutta scheme is used in conjunction with several accelerating techniques typical of steady-state solvers, instead of traditional time-expensive factorizations. The accelerating strategies include local time stepping, residual smoothing, and multigrid. Two-dimensional viscous calculations of unsteady rotor-stator interaction in the first stage of a modem gas turbine are presented. The stage analysis is based on the introduction of several blade passages to approximate the stator:rotor count ratio. Particular attention is dedicated to grid dependency in space and time as well as to the influence of the number of blades included in the calculations.

Multigrid Computations of Unsteady Rotor-Stator Interaction Using the Navier-Stokes Equations

1995 ◽  
Vol 117 (4) ◽  
pp. 647-652 ◽  
Author(s):  
A. Arnone ◽  
R. Pacciani ◽  
A. Sestini
Keyword(s):  
Stokes Equations ◽  
Time Discretization ◽  
Navier Stokes ◽  
Implicit Time ◽  
Navier Stokes Equations ◽  
Time Stepping ◽  
Fully Implicit ◽  
Local Time Stepping ◽  
Rotor Stator Interaction ◽  
Residual Smoothing

A Navier-Stokes time-accurate solver has been extended to the analysis of unsteady rotor-stator interaction. In the proposed method, a fully-implicit time discretization is used to remove stability limitations. A four-stage Runge-Kutta scheme is used in conjunction with several accelerating techniques typical of steady-state solvers, instead of traditional time-expensive factorizations. Those accelerating strategies include local time stepping, residual smoothing, and multigrid. Direct interpolation of the conservative variables is used to handle the interfaces between blade rows. Two-dimensional viscous calculations of unsteady rotor-stator interaction in a modern gas turbine stage are presented to check for the capability of the procedure.

scholarly journals A TIME ACCURATE PSEUDO-WAVELET SCHEME FOR TWO-DIMENSIONAL TURBULENCE

2005 ◽  
Vol 03 (04) ◽  
pp. 587-599 ◽  
Author(s):  
B. V. RATHISH KUMAR ◽  
MANI MEHRA
Keyword(s):  
Galerkin Method ◽  
Taylor Series ◽  
Stokes Equations ◽  
Taylor Series Expansion ◽  
Navier Stokes ◽  
Two Dimensional ◽  
Time Step ◽  
Navier Stokes Equations ◽  
Time Stepping ◽  
Taylor Galerkin Method

In this paper, we propose a wavelet-Taylor–Galerkin method for solving the two-dimensional Navier–Stokes equations. The discretization in time is performed before the spatial discretization by introducing second-order generalization of the standard time stepping schemes with the help of Taylor series expansion in time step. Wavelet-Taylor–Galerkin schemes taking advantage of the wavelet bases capabilities to compress both functions and operators are presented. Results for two-dimensional turbulence are shown.

Efficient Time-Stepping/Spectral Methods for the Navier-Stokes-Nernst-Planck-Poisson Equations

2017 ◽  
Vol 21 (5) ◽  
pp. 1408-1428 ◽  
Author(s):  
Xiaoling Liu ◽  
Chuanju Xu
Keyword(s):  
Stability Condition ◽  
Time Discretization ◽  
Equation System ◽  
Second Order ◽  
Navier Stokes ◽  
Step Size ◽  
Time Step ◽  
Time Step Size ◽  
First Order ◽  
Time Stepping

AbstractThis paper is concerned with numerical methods for the Navier-Stokes-Nernst-Planck-Poisson equation system. The main goal is to construct and analyze some stable time stepping schemes for the time discretization and use a spectral method for the spatial discretization. The main contribution of the paper includes: 1) an useful stability inequality for the weak solution is derived; 2) a first order time stepping scheme is constructed, and the non-negativity of the concentration components of the discrete solution is proved. This is an important property since the exact solution shares the same property. Moreover, the stability of the scheme is established, together with a stability condition on the time step size; 3) a modified first order scheme is proposed in order to decouple the calculation of the velocity and pressure in the fluid field. This new scheme equally preserves the non-negativity of the discrete concentration solution, and is stable under a similar stability condition; 4) a stabilization technique is introduced to make the above mentioned schemes stable without restriction condition on the time step size; 5) finally we construct a second order finite difference scheme in time and spectral discretization in space. The numerical tests carried out in the paper show that all the proposed schemes possess some desirable properties, such as conditionally/unconditionally stability, first/second order convergence, non-negativity of the discrete concentrations, and so on.

scholarly journals On Parallel Numerical Algorithms for Simulating Industrial Filtration Problems

2007 ◽  
Vol 7 (2) ◽  
pp. 118-134 ◽  
Author(s):  
R. Čiegis ◽  
O. Iliev ◽  
Z. Lakdawala
Keyword(s):  
Parallel Algorithm ◽  
Stokes Equations ◽  
Data Communication ◽  
Numerical Algorithms ◽  
Time Discretization ◽  
Solid Particles ◽  
Navier Stokes ◽  
Time Step ◽  
Navier Stokes Equations ◽  
Parallel Numerical Algorithms

AbstractThe performance of oil filters used in the automotive industry can be significantly improved, especially when computer simulation is an essential component of the design process. In this paper, we consider parallel numerical algorithms for solving mathematical models describing the process of filtration, filtering solid particles out of liquid oil. The Navier — Stokes — Brinkmann system of equations is used to describe the laminar flow of incompressible isothermal oil. The space discretization in the complicated filter geometry is based on the finite-volume method. Special care is taken for an accurate approximation of the velocity and pressure on the interface between the fluid and the porous media. The time discretization used here is a proper modification of the fractional time step discretization (cf. Chorin scheme) of the Navier- Stokes equations, where the Brinkmann term is considered in both the prediction and the correction substeps. A data decomposition method is used to develop a parallel algorithm, where the domain is distributed among the processors by using a structured reference grid. The MPI library is used to implement the data communication part of the algorithm. A theoretical model is proposed for the estimation of the complexity of the given parallel algorithm and a scalability analysis is done on the basis of this model. The results of the computational experiments are presented, and the accuracy and efficiency of the parallel algorithm is tested on real industrial geometries.

Rotor-Stator Interaction Analysis Using the Navier–Stokes Equations and a Multigrid Method

1996 ◽  
Vol 118 (4) ◽  
pp. 679-689 ◽  
Author(s):  
A. Arnone ◽  
R. Pacciani
Keyword(s):  
Stokes Equations ◽  
Interaction Analysis ◽  
Time Discretization ◽  
Navier Stokes ◽  
Implicit Time ◽  
Navier Stokes Equations ◽  
Time Stepping ◽  
Fully Implicit ◽  
Rotor Stator Interaction ◽  
Stage Analysis

A recently developed, time-accurate multigrid viscous solver has been extended to the analysis of unsteady rotor–stator interaction. In the proposed method, a fully implicit time discretization is used to remove stability limitations. By means of a dual time-stepping approach, a four-stage Runge–Kutta scheme is used in conjunction with several accelerating techniques typical of steady-state solvers, instead of traditional time-expensive factorizations. The accelerating strategies include local time stepping, residual smoothing, and multigrid. Two-dimensional viscous calculations of unsteady rotor–stator interaction in the first stage of a modern gas turbine are presented. The stage analysis is based on the introduction of several blade passages to approximate the stator:rotor count ratio. Particular attention is dedicated to grid dependency in space and time as well as to the influence of the number of blades included in the calculations.

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