General Formulations for Rhie-Chow Interpolation

2004 ◽  
Author(s):  
Sijun Zhang ◽  
Xiang Zhao
Keyword(s):  
Small Time ◽  
Unsteady State ◽  
Flow Conditions ◽  
Step Size ◽  
Time Step ◽  
Time Step Size ◽  
Relaxation Factor ◽  
Numerical Discretization ◽  
Small Time Step

In this paper, general formulations for Rhie-Chow interpolation on co-located grid are derived. Unlike the standard Rhie-Chow interpolation, the general formulations are applicable for any flow conditions. It has observed that the original momentum-based interpolation due to Rhie and Chow has serious deficiency, such as under-relaxation factor dependence, failure to suppress saw-tooth pressure solutions with small time step size for unsteady state problems and wrong solutions or divergence with discontinuities. Thus the derivation of Rhie-Chow interpolation is first recalled, then the errors of numerical discretization are analyzed, finally, new formulations and some improvements are given.

Load Discontinuity and Its Remedy in Step-by-Step Integration

2006 ◽  
Author(s):  
Shuenn-Yih Chang ◽  
Chiu-Li Huang
Keyword(s):  
Numerical Solution ◽  
Small Time ◽  
Step Size ◽  
Time Step ◽  
Integration Procedure ◽  
Amplitude Distortion ◽  
Small Time Step

The discontinuity at the end of an impulse will lead to an extra impulse and thus an extra displacement. Consequently, an amplitude distortion is introduced in the numerical solution. The difficulty arising from the discontinuity at the end of an impulse can be overcome by using a very small time step to perform the step-by-step integration since it reduces the extra impulse and thus extra displacement. However, computational efforts might be significantly increased since the small time step is performed for a complete step-by-step integration procedure. A remedy is devised to computationally efficiently overcome this difficulty by using a very small time step immediately upon termination of the applied impulse. This is because that the extra impulse caused by the discontinuity is almost proportional to the discontinuity value at the end of the impulse and the step size. The feasibility of this proposed remedy is analytically and numerically confirmed herein.

Simulation of Red Blood Cell Passing Through Constrictions Using Modified Moving Particle Semi-Implicit Method

2011 ◽  
Author(s):  
K. Firoozbakhsh ◽  
M. T. Ahmadian ◽  
M. Hasanian
Keyword(s):  
Mechanical Behavior ◽  
Experimental Studies ◽  
Small Time ◽  
Implicit Method ◽  
Parallel Plates ◽  
Time Step ◽  
Mps Method ◽  
Time Step Size ◽  
Small Time Step ◽  
Moving Particle

During the circulation of RBC it undergoes elastic deformation as it passes through micro-capillaries where the inner diameter of the constriction can be about 3 micro meters. It means RBC shape must be changed in order to pass through these narrow channels. The role of mechanical behavior of RBC and the deformability traits of RBC are observed with the several experimental studies [1]. Several methods were implemented to simulate the mechanical behavior of RBCs in micro-capillaries [1, 2]. One of the most recent methods is Moving Particle Semi-implicit method (MPS) which is a Lagrangian method with semi-implicit algorithm that guaranties the incompressibility of the fluid. MPS method was implemented for simulation of RBC motion through parallel plates by Tsubota et al. 2006 [3]. Due to small Reynolds number and the Diffusion number restrictions, implementation of small time step size would be necessary which leads to long time simulation. By the way in case of complex geometries or FSI problems, standard MPS method has a delicate pressure solver which leads to diverge the solution. So in these cases using a small time step can help to overcome the problem. Some studies have applied a new approach for time integration and the fractional time step method is employed to overcome the noticed problem. Yohsuke Imai and coworkers (2010) have developed the former studies with two main new approaches [4]. Firstly, evaluation of viscosity is upgraded and secondly boundary condition is assumed to be periodic. Although the developments are really impressive and MPS method has turned into a practical method for simulation of RBC motion in micro-capillaries, but still there are some considerations about using large time steps and error of the velocity profile consequently.

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.

Pseudodynamic Response to Impulsive Loading

2009 ◽  
Author(s):  
Shuenn-Yih Chang ◽  
Chiu-Li Huang ◽  
Ching-Hao Yang
Keyword(s):  
Error Propagation ◽  
Small Time ◽  
Impulsive Loading ◽  
Small Displacement ◽  
Step Size ◽  
Time Step ◽  
Pseudodynamic Test ◽  
Experimental Errors ◽  
Small Time Step ◽  
Test Result

The application of a pseudodynamic technique to yield a shock response from an impulse might encounter a difficulty caused by a significant load discontinuity at the end of the impulse since this load discontinuity at the end of an impulse will result in an extra impulse and then an extra amplitude distortion. This extra impulse is linearly proportional to the step size and thus it is natural to consider the use of a very small time step for a whole pseudodynamic test to overcome the difficulty. However, a series of computer simulations reveal that this approach might not be feasible. This is because that the use of a small time step will lead to a very small displacement increment and it may be contaminated by experimental errors as its magnitude is less than or close to the magnitude of the experimental errors. Thus, an inaccurate test result is obtained. A technique is proposed to overcome this difficulty. This novel technique is to perform a single small time step immediately upon the termination of the applied impulse while the other time steps are conducted by using the time step determined from general considerations. This single small time step will not lead to a significant error propagation problem since only this time step is performed by using a very small step size for a complete pseudodynamic test. The feasibility of this technique was confirmed by a series of pseudodynamic tests.

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.

On the Influence of Inflow Model Selection for Time-Domain Tiltrotor Aeroelastic Analysis

2021 ◽  
Author(s):  
Ethan Corle ◽  
Matthew Floros ◽  
Sven Schmitz
Keyword(s):  
Experimental Data ◽  
Particle Method ◽  
Comprehensive Analysis ◽  
Solution Stability ◽  
Step Size ◽  
Time Step ◽  
Time Step Size ◽  
Vortex Particle Method ◽  
Whirl Flutter ◽  
Vortex Particle

The methods of using the viscous vortex particle method, dynamic inflow, and uniform inflow to conduct whirl-flutter stability analysis are evaluated on a four-bladed, soft-inplane tiltrotor model using the Rotorcraft Comprehensive Analysis System. For the first time, coupled transient simulations between comprehensive analysis and a vortex particle method inflow model are used to predict whirl-flutter stability. Resolution studies are performed for both spatial and temporal resolution in the transient solution. Stability in transient analysis is noted to be influenced by both. As the particle resolution is refined, a reduction in simulation time-step size must also be performed. An azimuthal time step size of 0.3 deg is used to consider a range of particle resolutions to understand the influence on whirl-flutter stability predictions. Comparisons are made between uniform inflow, dynamic inflow, and the vortex particle method with respect to prediction capabilities when compared to wing beam-bending frequency and damping experimental data. Challenges in assessing the most accurate inflow model are noted due to uncertainty in experimental data; however, a consistent trend of increasing damping with additional levels of fidelity in the inflow model is observed. Excellent correlation is observed between the dynamic inflow predictions and the vortex particle method predictions in which the wing is not part of the inflow model, indicating that the dynamic inflow model is adequate for capturing damping due to the induced velocity on the rotor disk. Additional damping is noted in the full vortex particle method model, with the wing included, which is attributed to either an interactional aerodynamic effect between the rotor and the wing or a more accurate representation of the unsteady loading on the wing due to induced velocities.

A Strategy to Accelerate the Numerical Integration in the Dynamic Simulation of Multibody Systems

1997 ◽  
Author(s):  
Jesús Cardenal ◽  
Javier Cuadrado ◽  
Eduardo Bayo
Keyword(s):  
Multibody Systems ◽  
Equations Of Motion ◽  
Stiff Systems ◽  
Step Size ◽  
Step Method ◽  
Integral Of Motion ◽  
Time Step ◽  
Time Step Size ◽  
Variable Time ◽  
Numerical Integrator

Abstract This paper presents a multi-index variable time step method for the integration of the equations of motion of constrained multibody systems in descriptor form. The basis of the method is the augmented Lagrangian formulation with projections in index-3 and index-1. The method takes advantage of the better performance of the index-3 formulation for large time steps and of the stability of the index-1 for low time steps, and automatically switches from one method to the other depending on the required accuracy and values of the time step. The variable time stepping is accomplished through the use of an integral of motion, which in the case of conservative systems becomes the total energy. The error introduced by the numerical integrator in the integral of motion during consecutive time steps provides a good measure of the local integration error, and permits a simple and reliable strategy for varying the time step. Overall, the method is efficient and powerful; it is suitable for stiff and non-stiff systems, robust for all time step sizes, and it works for singular configurations, redundant constraints and topology changes. Also, the constraints in positions, velocities and accelerations are satisfied during the simulation process. The method is robust in the sense that becomes more accurate as the time step size decreases.

Numerical Simulation of Proppant Transport in Hydraulically Fractured Reservoirs

2021 ◽  
Author(s):  
Seyhan Emre Gorucu ◽  
Vijay Shrivastava ◽  
Long X. Nghiem
Keyword(s):  
Hydraulic Fracturing ◽  
Hydraulic Fracture ◽  
Fractured Reservoirs ◽  
Injection Timing ◽  
Fracture Permeability ◽  
Step Size ◽  
Time Step ◽  
Time Step Size ◽  
Proppant Transport ◽  
Fracture Closure

Abstract An existing equation-of-state compositional simulator is extended to include proppant transport. The simulator determines the final location of the proppant after fracture closure, which allows the computation of the permeability along the hydraulic fracture. The simulation then continues until the end of the production. During hydraulic fracturing, proppant is injected in the reservoir along with water and additives like polymers. Hydraulic fracture gets created due to change in stress caused by the high injection pressure. Once the fracture opens, the bulk slurry moves along the hydraulic fracture. Proppant moves at a different speed than the bulk slurry and sinks down by gravity. While the proppant flows along the fracture, some of the slurry leaks off into the matrix. As the fracture closes after injection stops, the proppant becomes immobile. The immobilized proppant prevents the fracture from closing and thus keeps the permeability of the fracture high. All the above phenomena are modelled effectively in this new implementation. Coupled geomechanics simulation is used to model opening and closure of the fracture following geomechanics criteria. Proppant retardation, gravitational settling and fluid leak-off are modeled with the appropriate equations. The propped fracture permeability is a function of the concentration of immobilized proppant. The developed proppant simulation feature is computationally stable and efficient. The time step size during the settling adapts to the settling velocity of the proppants. It is found that the final location of the proppants is highly dependent on its volumetric concentration and slurry viscosity due to retardation and settling effects. As the location and the concentration of the proppants determine the final fracture permeability, the additional feature is expected to correctly identify the stimulated region. In this paper, the theory and the model formulation are presented along with a few key examples. The simulation can be used to design and optimize the amount of proppant and additives, injection timing, pressure, and well parameters required for successful hydraulic fracturing.

Sign in / Sign up

Export Citation Format

Share Document