scholarly journals Stability of Finite Difference Discretizations of Multi-Physics Interface Conditions

2013 ◽  
Vol 13 (2) ◽  
pp. 386-410 ◽  
Author(s):  
Björn Sjögreen ◽  
Jeffrey W. Banks
Keyword(s):  
Stokes Equations ◽  
Normal Mode Analysis ◽  
Linear Equations ◽  
Mode Analysis ◽  
Navier Stokes ◽  
Computational Domain ◽  
Interface Conditions ◽  
Step Size ◽  
Time Step ◽  
Time Step Size

AbstractWe consider multi-physics computations where the Navier-Stokes equations of compressible fluid flow on some parts of the computational domain are coupled to the equations of elasticity on other parts of the computational domain. The different subdomains are separated by well-defined interfaces. We consider time accurate computations resolving all time scales. For such computations, explicit time stepping is very efficient. We address the issue of discrete interface conditions between the two domains of different physics that do not lead to instability, or to a significant reduction of the stable time step size. Finding such interface conditions is non-trivial.We discretize the problem with high order centered difference approximations with summation by parts boundary closure. We derive L2 stable interface conditions for the linearized one dimensional discretized problem. Furthermore, we generalize the interface conditions to the full non-linear equations and numerically demonstrate their stable and accurate performance on a simple model problem. The energy stable interface conditions derived here through symmetrization of the equations contain the interface conditions derived through normal mode analysis by Banks and Sjögreen in [8] as a special case.

scholarly journals Analysis of a Decoupled Time-Stepping Scheme for Evolutionary Micropolar Fluid Flows

2016 ◽  
Vol 2016 ◽  
pp. 1-13 ◽  
Author(s):  
S. S. Ravindran
Keyword(s):  
Micropolar Fluid ◽  
Stokes Equations ◽  
Fluid Model ◽  
Navier Stokes ◽  
Optimal Order ◽  
Step Size ◽  
Time Step ◽  
Order Error ◽  
Time Step Size ◽  
Time Stepping

Micropolar fluid model consists of Navier-Stokes equations and microrotational velocity equations describing the dynamics of flows in which microstructure of fluid is important. In this paper, we propose and analyze a decoupled time-stepping algorithm for the evolutionary micropolar flow. The proposed method requires solving only one uncoupled Navier-Stokes and one microrotation subphysics problem per time step. We derive optimal order error estimates in suitable norms without assuming any stability condition or time step size restriction.

scholarly journals Studying Inertia Effects in Open Channel Flow Using Saint-Venant Equations

Water ◽  
2018 ◽  
Vol 10 (11) ◽  
pp. 1652
Author(s):  
Dong-Sin Shih ◽  
Gour-Tsyh Yeh
Keyword(s):  
Stokes Equations ◽  
Field Application ◽  
Benchmark Problems ◽  
Algebraic Equations ◽  
Step Size ◽  
Time Step ◽  
Cross Sectional ◽  
Inertia Effects ◽  
Time Step Size ◽  
Saint Venant Equations

One-dimensional (1D) Saint-Venant equations, which originated from the Navier–Stokes equations, are usually applied to express the transient stream flow. The governing equation is based on the mass continuity and momentum equivalence. Its momentum equation, partially comprising the inertia, pressure, gravity, and friction-induced momentum loss terms, can be expressed as kinematic wave (KIW), diffusion wave (DIW), and fully dynamic wave (DYW) flow. In this study, the method of characteristics (MOCs) is used for solving the diagonalized Saint-Venant equations. A computer model, CAMP1DF, including KIW, DIW, and DYW approximations, is developed. Benchmark problems from MacDonald et al. (1997) are examined to study the accuracy of the CAMP1DF model. The simulations revealed that CAMP1DF can simulate almost identical results that are valid for various fluvial conditions. The proposed scheme that not only allows a large time step size but also solves half of the simultaneous algebraic equations. Simulations of accuracy and efficiency are both improved. Based on the physical relevance, the simulations clearly showed that the DYW approximation has the best performance, whereas the KIW approximation results in the largest errors. Moreover, the field non-prismatic case of the Zhuoshui River in central Taiwan is studied. The simulations indicate that the DYW approach does not ensure achievement of a better simulation result than the other two approximations. The investigated cross-sectional geometries play an important role in stream routing. Because of the consideration of the acceleration terms, the simulated hydrograph of a DYW reveals more physical characteristics, particularly regarding the raising and recession of limbs. Note that the KIW does not require assignment of a downstream boundary condition, making it more convenient for field application.

Numerical Study on the Effect of a Volute on Surge Phenomena in a Centrifugal Compressor

2016 ◽  
Author(s):  
S. H. Jeon ◽  
D. H. Hwang ◽  
J. H. Park ◽  
C. H. Kim ◽  
J. H. Baek ◽  
...  
Keyword(s):  
Centrifugal Compressor ◽  
Stokes Equations ◽  
Flow Instability ◽  
Numerical Study ◽  
Pressure Ratio ◽  
Unstable Flow ◽  
Step Size ◽  
Time Step ◽  
Time Step Size ◽  
Blade Passing Frequency

Numerical investigation of the effect of the volute on stall flow phenomenon is presented by solving three-dimensional Reynolds-averaged compressible Navier-Stokes equations. Two different configurations of a centrifugal compressor were used to compare their performance: One is an original centrifugal compressor which is composed of impeller, splitter, vaned diffuser and a volute and the other is the one without a volute. Steady calculations were performed to predict aerodynamic performance in terms of the pressure ratio, efficiency and mass flow rate. The results show that the operating range of the compressor with a volute is narrower than that of the compressor without a volute. This can be interpreted that flow instability is strongly influenced by the tongue of a volute which is highly asymmetric. Unsteady calculations were also performed with a time-step size of 38μs corresponding to a pitch angle of 5 degrees at the given rotational speed. The flow characteristics for two configurations are analyzed and compared at various instantaneous times showing unsteady dynamic features. Based on the unsteady flow simulation, fast Fourier transform at several discrete points in semi-vaneless space was performed at peak efficiency and near surge point in order to illustrate the unstable flow physics in both configurations. It is found that the blade passing frequency is dominant, indicating that diffuser passages have a periodicity of 40 degrees due to the rotational blades. Besides blade passing frequency, there were several noticeable frequencies which affect the instability of the whole system. Those frequencies in both configurations are compared and analyzed in various aspects.

scholarly journals An Unconditionally Energy Stable Immersed Boundary Method with Application to Vesicle Dynamics

2013 ◽  
Vol 3 (3) ◽  
pp. 247-262 ◽  
Author(s):  
Wei-Fan Hu ◽  
Ming-Chih Lai
Keyword(s):  
Immersed Boundary Method ◽  
Navier Stokes ◽  
Immersed Boundary ◽  
Step Size ◽  
Time Step ◽  
Time Step Size ◽  
Boundary Method ◽  
Stretching Factor ◽  
Vesicle Dynamics ◽  
Energy Stable

AbstractWe develop an unconditionally energy stable immersed boundary method, and apply it to simulate 2D vesicle dynamics. We adopt a semi-implicit boundary forcing approach, where the stretching factor used in the forcing term can be computed from the derived evolutional equation. By using the projection method to solve the fluid equations, the pressure is decoupled and we have a symmetric positive definite system that can be solved efficiently. The method can be shown to be unconditionally stable, in the sense that the total energy is decreasing. A resulting modification benefits from this improved numerical stability, as the time step size can be significantly increased (the severe time step restriction in an explicit boundary forcing scheme is avoided). As an application, we use our scheme to simulate vesicle dynamics in Navier-Stokes flow.

Smoothed Profile Method for Particulate Two-Phase Flow

2008 ◽  
Author(s):  
Xian Luo ◽  
Martin R. Maxey ◽  
George E. Karniadakis
Keyword(s):  
Numerical Simulations ◽  
Stokes Equations ◽  
Integration Scheme ◽  
Force Density ◽  
Step Size ◽  
Time Step ◽  
Element Discretization ◽  
Profile Method ◽  
Time Step Size ◽  
Smoothed Profile Method

We re-formulate and demonstrate a new method for particulate flows, the so-called “Smoothed Profile” method (SPM) first proposed in [1]. The method uses a fixed computational mesh, which does not conform to the geometry of the particles. The particles are represented by certain smoothed indicator profiles to construct a smooth body force density term added into the Navier-Stokes equations. The SPM imposes accurately and efficiently the rigid-body constraint inside the particles. In particular, while the original method employs a fully-explicit time-integration scheme, we develop a high-order semi-implicit splitting scheme, which we implement in the context of spectral/hp element discretization. We show that the modeling error of SPM has a non-monotonic dependence on the time step size Δt. The optimum time step size balances the thickness of the Stokes layer and that of the profile interface. Subsequently, we present several numerical simulations, including flow past three-dimensional complex-shaped particles and two interacting microspheres, which are compared against full direct numerical simulations and the force coupling method (FCM).

scholarly journals Multi-Degree of Freedom Propeller Force Models Based on a Neural Network and Regression

2020 ◽  
Vol 8 (2) ◽  
pp. 89 ◽  
Author(s):  
Bradford Knight ◽  
Kevin Maki
Keyword(s):  
Neural Network ◽  
Stokes Equations ◽  
Equations Of Motion ◽  
Open Water ◽  
Small Time ◽  
Training Data ◽  
Step Size ◽  
Time Step ◽  
Neural Network Approach ◽  
Time Step Size

Accurate and efficient prediction of the forces on a propeller is critical for analyzing a maneuvering vessel with numerical methods. CFD methods like RANS, LES, or DES can accurately predict the propeller forces, but are computationally expensive due to the need for added mesh discretization around the propeller as well as the requisite small time-step size. One way of mitigating the expense of modeling a maneuvering vessel with CFD is to apply the propeller force as a body force term in the Navier–Stokes equations and to apply the force to the equations of motion. The applied propeller force should be determined with minimal expense and good accuracy. This paper examines and compares nonlinear regression and neural network predictions of the thrust, torque, and side force of a propeller both in open water and in the behind condition. The methods are trained and tested with RANS CFD simulations. The neural network approach is shown to be more accurate and requires less training data than the regression technique.

INFLUENCE OF PENALIZATION AND BOUNDARY TREATMENT ON THE STABILITY AND ACCURACY OF HIGH-ORDER DISCONTINUOUS GALERKIN SCHEMES FOR THE COMPRESSIBLE NAVIER-STOKES EQUATIONS

2013 ◽  
Vol 21 (01) ◽  
pp. 1250019
Author(s):  
ANDREAS RICHTER ◽  
EVA BRUSSIES ◽  
JÖRG STILLER
Keyword(s):  
Discontinuous Galerkin ◽  
Stokes Equations ◽  
Critical Time ◽  
High Order ◽  
Condition Numbers ◽  
Navier Stokes ◽  
Time Step ◽  
Navier Stokes Equations ◽  
Time Step Size ◽  
Boundary Treatment

A high-order interior penalty discontinuous Galerkin method for the compressible Navier–Stokes equations is introduced, which is a modification of the scheme given by Hartmann and Houston. In this paper we investigate the influence of penalization and boundary treatment on accuracy. By observing eigenvalues and condition numbers, a lower bound for the penalization term μ was found, whereas convergence studies depict reasonable upper bounds and a linear dependence on the critical time step size. By investigating conservation properties we demonstrate that different boundary treatments influence the accuracy by several orders of magnitude, and propose reasonable strategies to improve conservation properties.

Numerical Modeling of Aerated Cavitation Using a Penalization Approach for Air Bubble Modeling Coupled to Homogeneous Equilibrium Model

2014 ◽  
Author(s):  
Petar Tomov ◽  
Sofiane Khelladi ◽  
Christophe Sarraf ◽  
Farid Bakir
Keyword(s):  
Fluid Flow ◽  
Mixture Model ◽  
Stokes Equations ◽  
Saturation Pressure ◽  
Homogeneous Mixture ◽  
Navier Stokes ◽  
Strain Rate Tensor ◽  
Computational Domain ◽  
Time Step ◽  
Homogeneous Mixture Model

Cavitation is a well-known physical phenomena occurring in various technical applications. It appears when the pressure of the liquid drops below the saturation pressure. Coupling aeration in a cavitating flow is a recent technique to control the overall effect of the cavitation. It is achieved by introducing air bubbles into the flow. In order to reveal and explore the behaviour of air gas in the vicinity of the cavitation region, the paper is oriented towards the physics of the colliding vapor phase bubbles and cavitating regions. The re-entrant jet may influence the dynamics of the bubbles as well as the frequency of cavitation separation. Therefore, a two-way coupling between the fluid flow and the introduced vapor is of capital importance. By penalizing the strain rate tensor in the Homogeneous Mixture Model, the two-way coupling has been achieved. The contact-handling algorithm is based on the projections of the velocity fields of the injected particles over the velocity field of the fluid flow. At each time step the gradient of the distance between the bubbles, is kept non-negative as a guarantee of the physical non overlapping. The bubbles’ collisions are considered as inelastic. The differential equations system is composed of the Navier-Stokes equations, implemented with the Homogeneous Mixture Model. A high-order Finite Volume (FV) solver based on Moving Least Squares (MLS) approximations is used. The code uses a SLAU-type Riemann solver for the accurate calculation of the low Mach numbers. The computational domain is a symmetrical 2D venturi duct with an 18°–8° convergent/divergent angles respectively.

High Order Compact Schemes in Projection Methods for Incompressible Viscous Flows

2011 ◽  
Vol 9 (4) ◽  
pp. 994-1019 ◽  
Author(s):  
Michel Fournié ◽  
Alain Rigal
Keyword(s):  
Numerical Solutions ◽  
Stokes Equations ◽  
Normal Mode Analysis ◽  
Correction Method ◽  
Projection Methods ◽  
High Order ◽  
Mode Analysis ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Von Neumann

AbstractWithin the projection schemes for the incompressible Navier-Stokes equations (namely “pressure-correction” method), we consider the simplest method (of order one in time) which takes into account the pressure in both steps of the splitting scheme. For this scheme, we construct, analyze and implement a new high order compact spatial approximation on nonstaggered grids. This approach yields a fourth order accuracy in space with an optimal treatment of the boundary conditions (without error on the velocity) which could be extended to more general splitting. We prove the unconditional stability of the associated Cauchy problem via von Neumann analysis. Then we carry out a normal mode analysis so as to obtain more precise results about the behavior of the numerical solutions. Finally we present detailed numerical tests for the Stokes and the Navier-Stokes equations (including the driven cavity benchmark) to illustrate the theoretical results.

scholarly journals Energy-corrected FEM and explicit time-stepping for parabolic problems

2019 ◽  
Vol 53 (6) ◽  
pp. 1893-1914
Author(s):  
Piotr Swierczynski ◽  
Barbara Wohlmuth
Keyword(s):  
Finite Element ◽  
Finite Element Approximation ◽  
Regularity Of Solutions ◽  
Parabolic Problems ◽  
Element Approximation ◽  
Computational Domain ◽  
Step Size ◽  
Time Step ◽  
Time Step Size ◽  
Time Stepping

The presence of corners in the computational domain, in general, reduces the regularity of solutions of parabolic problems and diminishes the convergence properties of the finite element approximation introducing a so-called “pollution effect”. Standard remedies based on mesh refinement around the singular corner result in very restrictive stability requirements on the time-step size when explicit time integration is applied. In this article, we introduce and analyse the energy-corrected finite element method for parabolic problems, which works on quasi-uniform meshes, and, based on it, create fast explicit time discretisation. We illustrate these results with extensive numerical investigations not only confirming the theoretical results but also showing the flexibility of the method, which can be applied in the presence of multiple singular corners and a three-dimensional setting. We also propose a fast explicit time-stepping scheme based on a piecewise cubic energy-corrected discretisation in space completed with mass-lumping techniques and numerically verify its efficiency.

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