Numerical Experiments on Application of Richardson Extrapolation With Nonuniform Grids

1997 ◽  
Vol 119 (3) ◽  
pp. 584-590 ◽  
Author(s):  
Ismail Celik ◽  
Ozgur Karatekin
Keyword(s):  
Viscous Sublayer ◽  
Standard Uncertainty ◽  
Richardson Extrapolation ◽  
Grid Refinement ◽  
Uncertainty Measure ◽  
Nonuniform Grids ◽  
Wall Functions ◽  
Convergence Error ◽  
Grid Convergence ◽  
Grid Convergence Index

Some unresolved problems related to Richardson extrapolation (RE) are elucidated via examples, and possible remedies are suggested. The method is applied to the case of turbulent flow past a backward facing step using nonuniform grid distributions. It is demonstrated that RE can be used to obtain grid independent solutions using the same grid refinement factors in both coordinate directions. The use of generalized wall functions together with the standard k-ε model seems to work well even if the grid refinement extends into the viscous sublayer. In addition, the grid convergence index and other standard uncertainty measures are compared, and a new uncertainty measure is suggested which seems to be a better indicator for the grid convergence error.

Perspective: A Method for Uniform Reporting of Grid Refinement Studies

1994 ◽  
Vol 116 (3) ◽  
pp. 405-413 ◽  
Author(s):  
P. J. Roache
Keyword(s):  
Richardson Extrapolation ◽  
Coarse Grid ◽  
Error Estimator ◽  
Grid Refinement ◽  
Order Method ◽  
A Posteriori ◽  
Asymptotic Approach ◽  
Grid Convergence ◽  
Grid Convergence Index ◽  
Do So

This paper proposes the use of a Grid Convergence Index (GCI) for the uniform reporting of grid refinement studies in Computational Fluid Dynamics. The method provides an objective asymptotic approach to quantification of uncertainty of grid convergence. The basic idea is to approximately relate the results from any grid refinement test to the expected results from a grid doubling using a second-order method. The GCI is based upon a grid refinement error estimator derived from the theory of generalized Richardson Extrapolation. It is recommended for use whether or not Richardson Extrapolation is actually used to improve the accuracy, and in some cases even if the conditions for the theory do not strictly hold. A different form of the GCI applies to reporting coarse grid solutions when the GCI is evaluated from a “nearby” problem. The simple formulas may be applied a posteriori by editors and reviewers, even if authors are reluctant to do so.

Richardson Extrapolation-Based Discretization Uncertainty Estimation for Computational Fluid Dynamics

2014 ◽  
Vol 136 (12) ◽  
Author(s):  
Tyrone S. Phillips ◽  
Christopher J. Roy
Keyword(s):  
Fluid Dynamics ◽  
Computational Fluid Dynamics ◽  
Convergence Rates ◽  
Richardson Extrapolation ◽  
Local Solution ◽  
Navier Stokes ◽  
Grid Convergence ◽  
Computational Fluid Dynamics Cfd ◽  
Grid Convergence Index ◽  
2D And 3D

This study investigates the accuracy of various Richardson extrapolation-based discretization error and uncertainty estimators for problems in computational fluid dynamics (CFD). Richardson extrapolation uses two solutions on systematically refined grids to estimate the exact solution to the partial differential equations (PDEs) and is accurate only in the asymptotic range (i.e., when the grids are sufficiently fine). The uncertainty estimators investigated are variations of the grid convergence index and include a globally averaged observed order of accuracy, the factor of safety method, the correction factor method, and least-squares methods. Several 2D and 3D applications to the Euler, Navier–Stokes, and Reynolds-Averaged Navier–Stokes (RANS) with exact solutions and a 2D turbulent flat plate with a numerical benchmark are used to evaluate the uncertainty estimators. Local solution quantities (e.g., density, velocity, and pressure) have much slower grid convergence on coarser meshes than global quantities, resulting in nonasymptotic solutions and inaccurate Richardson extrapolation error estimates; however, an uncertainty estimate may still be required. The uncertainty estimators are applied to local solution quantities to evaluate accuracy for all possible types of convergence rates. Extensions were added where necessary for treatment of cases where the local convergence rate is oscillatory or divergent. The conservativeness and effectivity of the discretization uncertainty estimators are used to assess the relative merits of the different approaches.

Application of grid convergence index in FE computation

2013 ◽  
Vol 61 (1) ◽  
pp. 123-128 ◽  
Author(s):  
L. Kwaśniewski
Keyword(s):  
Fluid Dynamics ◽  
Finite Element ◽  
Order Of Convergence ◽  
Richardson Extrapolation ◽  
Simple Problem ◽  
Plastic Response ◽  
Grid Convergence ◽  
The Difference ◽  
Grid Convergence Index ◽  
Selection Of

Abstract This paper presents an application of the grid convergence index (GCI) concept based on the Richardson extrapolation to a selected simple problem of a cantilever beam loaded with vertical forces at the tip end. The GCI method, popular in computational fluid dynamics, has been recently recommended for finite element (FE) applications in solid and structural mechanics. Based on the results obtained usually for three meshes, the GCI method enables one to determine, in an objective manner, the order of convergence to estimate the asymptotic solution and the bounds for discretization error. The example shows that the characteristics of the convergence depend on the selection of the quantity of interest, which can be local or a global functional such as the deflection considered here. The results differ for different FE formulations, and the difference is bigger when the nonlinearities (e.g., due to plastic response) are taken into account

Numerical Study of Wall Shear and Velocities Using a Commercial CFD Code: Some Crucial Aspects to Consider

2017 ◽  
Vol 21 ◽  
pp. 45
Author(s):  
Binaya Baidar ◽  
Michel Cervantes
Keyword(s):  
Numerical Study ◽  
Turbulence Models ◽  
Wall Functions ◽  
Energy And Environment ◽  
Commercial Code ◽  
Index Approach ◽  
Grid Convergence ◽  
Sst Model ◽  
Wall Shear ◽  
Grid Convergence Index

<p>This paper presents the numerical prediction of wall shear and velocities in steady and superposed pulsatile turbulent flow in a pipe, the phenomena that can be observed in hydropower. The previously conducted experiment is a base for this study and some crucial aspects of CFD while using a commercial code have been emphasized. The widely-accepted grid convergence index approach is adopted to quantify the discretization uncertainty and the results are validated against the experiment. The influence of the wall functions applied in the code is also studied with two turbulence models: standard <em>k-ε </em>and based SST model. The time-averaged results of superposed flow with small amplitude unsteadiness are equivalent to results from the steady flow. The results and the method used in this paper may be useful for the CFD simulations in hydropower applications like penstock and bifurcations designs. </p><p> </p><p><strong>HYDRO Nepal Journal</strong></p><p><em>Journal of Water Energy and Environment</em></p><p>Issue: 21, July, 2017</p><p>Page: 45-49</p><p>Upload Date: July 18, 2017</p>

Factors of Safety for Richardson Extrapolation

2010 ◽  
Vol 132 (6) ◽  
Author(s):  
Tao Xing ◽  
Frederick Stern
Keyword(s):  
Correction Factor ◽  
Confidence Level ◽  
Factor Of Safety ◽  
Richardson Extrapolation ◽  
Time Step ◽  
Order Of Accuracy ◽  
P Values ◽  
Grid Convergence ◽  
Small Uncertainty ◽  
Grid Convergence Index

A factor of safety method for quantitative estimates of grid-spacing and time-step uncertainties for solution verification is developed. It removes the two deficiencies of the grid convergence index and correction factor methods, namely, unreasonably small uncertainty when the estimated order of accuracy using the Richardson extrapolation method is greater than the theoretical order of accuracy and lack of statistical evidence that the interval of uncertainty at the 95% confidence level bounds the comparison error. Different error estimates are evaluated using the effectivity index. The uncertainty estimate builds on the correction factor method, but with significant improvements. The ratio of the estimated order of accuracy and theoretical order of accuracy P instead of the correction factor is used as the distance metric to the asymptotic range. The best error estimate is used to construct the uncertainty estimate. The assumption that the factor of safety is symmetric with respect to the asymptotic range was removed through the use of three instead of two factor of safety coefficients. The factor of safety method is validated using statistical analysis of 25 samples with different sizes based on 17 studies covering fluids, thermal, and structure disciplines. Only the factor of safety method, compared with the grid convergence index and correction factor methods, provides a reliability larger than 95% and a lower confidence limit greater than or equal to 1.2 at the 95% confidence level for the true mean of the parent population of the actual factor of safety. This conclusion is true for different studies, variables, ranges of P values, and single P values where multiple actual factors of safety are available. The number of samples is large and the range of P values is wide such that the factor of safety method is also valid for other applications including results not in the asymptotic range, which is typical in industrial and fluid engineering applications. An example for ship hydrodynamics is provided.

scholarly journals Numerical Study of Wall Shear and Velocities Using a Commercial CFD Code: Some Crucial Aspects to Consider

2017 ◽  
Vol 21 ◽  
pp. 45-49
Author(s):  
Binaya Baidar
Keyword(s):  
Numerical Study ◽  
Turbulence Models ◽  
Cfd Simulations ◽  
Wall Functions ◽  
Commercial Code ◽  
Index Approach ◽  
Grid Convergence ◽  
Sst Model ◽  
Wall Shear ◽  
Grid Convergence Index

This paper presents the numerical prediction of wall shear and velocities in steady and superposed pulsatile turbulent flow in a pipe, the phenomena that can be observed in hydropower. The previously conducted experiment is a base for this study and some crucial aspects of CFD while using a commercial code have been emphasized. The widely-accepted grid convergence index approach is adopted to quantify the discretization uncertainty and the results are validated against the experiment. The influence of the wall functions applied in the code is also studied with two turbulence models: standard k-ε and kω based SST model. The time-averaged results of superposed flow with small amplitude unsteadiness are equivalent to results from the steady flow. The results and the method used in this paper may be useful for the CFD simulations in hydropower applications like penstock and bifurcations designs. HYDRO Nepal JournalJournal of Water Energy and EnvironmentIssue: 21, July, 2017Page: 45-49Upload Date: July 20, 2017

scholarly journals Performance of Turbulence Models and Near-Wall Treatments in Discrete Jet Film Cooling Simulations

1998 ◽  
Author(s):  
Jeffrey D. Ferguson ◽  
Dibbon K. Walters ◽  
James H. Leylek
Keyword(s):  
Experimental Data ◽  
Film Cooling ◽  
Turbulence Models ◽  
Viscous Sublayer ◽  
Reynolds Stress Model ◽  
Quality Data ◽  
Wall Functions ◽  
First Time ◽  
Near Wall ◽  
Non Equilibrium

For the first time in the open literature, code validation quality data and a well-tested, highly reliable computational methodology are employed to isolate the true performance of seven turbulence treatments in discrete jet film cooling. The present research examines both computational and high quality experimental data for two length-to-diameter ratios of a row of streamwise injected, cylindrical film holes. These two cases are used to document the performance of the following turbulence treatments: 1) standard k-ε model with generalized wall functions; 2) standard k-ε model with non-equilibrium wall functions: 3) Renormalization Group k-ε (RNG) model with generalized wall functions; 4) RNG model with non-equilibrium wall functions: 51 standard k-ε model with two-layer turbulence wall treatment; 6) Reynolds Stress Model (RSM) with generalized wall functions; and 7) RSM with non-equilibrium wall functions. Overall, the standard k-ε turbulence model with the two-layer near-wall treatment, which resolves the viscous sublayer, produces results that are more consistent with experimental data.

scholarly journals Grid convergence study of a cyclone separator using different mesh structures

2017 ◽  
Vol 23 (3) ◽  
pp. 311-320 ◽  
Author(s):  
R.A.F. Oliveira ◽  
G.H. Justi ◽  
G.C. Lopes
Keyword(s):  
Pressure Drop ◽  
Collection Efficiency ◽  
Fluid Dynamic ◽  
Two Phase ◽  
Mesh Structure ◽  
Pressure Drops ◽  
Regular Elements ◽  
Grid Convergence ◽  
Grid Convergence Index ◽  
Mesh Structures

In a cyclone design, pressure drop and collection efficiency are two important performance parameters to estimate its implementation viability. The optimum design provides higher efficiencies and lower pressure drops. In this paper, a grid independence study was performed to determine the most appropriate mesh to simulate the two-phase flow in a Stairmand cyclone. Computational fluid dynamic (CFD) tools were used to simulate the flow in an Eulerian-Lagrangian approach. Two different mesh structure, one with wall-refinement and the other with regular elements, and several mesh sizes were tested. The grid convergence index (GCI) method was applied to evaluate the result independence. The CFD model results were compared with empirical correlations from bibliography, showing good agreement. The wall-refined mesh with 287 thousand elements obtained errors of 9.8% for collection efficiency and 14.2% for pressure drop, while the same mesh, with regular elements, obtained errors of 8.7% for collection efficiency and 0.01% for pressure drop.

Numerical Uncertainty in Heat Conduction Calculations for a Tube Immersed in a Fluidized Bed

1995 ◽  
Vol 209 (5) ◽  
pp. 323-336 ◽  
Author(s):  
A I Karamavruc ◽  
N N Clark ◽  
I Celik
Keyword(s):  
Heat Flux ◽  
Heat Conduction ◽  
Horizontal Tube ◽  
Numerical Code ◽  
Uniform Grid ◽  
Grid Refinement ◽  
Transfer Coefficients ◽  
Parabolic Profile ◽  
Grid Convergence ◽  
Spatial Domains

A numerical code has been evaluated with regard to numerical uncertainties involved in calculating heat flux through the wall of a horizontal tube in a bubbling bed of sand. The two-dimensional unsteady heat conduction equation is solved numerically with a non-linearly varying temperature boundary condition prescribed according to measurements. The finite difference method used is an implicit method with a second-order accurate discretization scheme both in temporal and spatial domains. Previous literature dealing with numerical calculations in heat conduction usually reports any detailed study about numerical errors. In the present analysis, a rigorous grid dependence test is applied, and it is shown that the results, in particular heat flux, are very sensitive to the grid size and distribution. Therefore, to achieve better grid convergence when heat flux is sought, the discretization error in the heat flux rather than in the temperature calculations should be considered. This should be done even in cases where temperature is the primary unknown, because it is usually the derivative of temperature which is of any physical importance. The errors are also strongly dependent on the number of iterations which need to be increased as the grid is refined. The present application showed that a non-uniform grid refinement throughout the calculation domain gives a more efficient (less expensive) solution than uniform grid refinement. Furthermore, for calculation of the temperature gradient at the wall, a parabolic profile assumption gives a faster grid convergence compared to a linear profile assumption. The present study shows that the previously published results concerning calculated heat transfer coefficients should be interpreted with caution, unless the authors have provided some measure of grid dependency of their results.

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