A Contribution for the Assessment of Discretization Error Estimators Based on Grid Refinement Studies

2018 ◽  
Vol 3 (2) ◽  
Author(s):  
L. Eça ◽  
G. Vaz ◽  
M. Hoekstra
Keyword(s):  
Eddy Viscosity ◽  
Discretization Error ◽  
Functional Integral ◽  
Two Dimensions ◽  
Convergence Criterion ◽  
Double Precision ◽  
Grid Refinement ◽  
Mean Velocity ◽  
Data Set ◽  
Local Flow

This paper presents grid refinement studies for statistically steady, two-dimensional (2D) flows of an incompressible fluid: a flat plate at Reynolds numbers equal to 107, 108, and 109 and the NACA 0012 airfoil at angles of attack of 0, 4, and 10 deg with Re = 6 × 106. Results are based on the numerical solution of the Reynolds-averaged Navier–Stokes (RANS) equations supplemented by one of three eddy-viscosity turbulence models of choice: the one-equation model of Spalart and Allmaras and the two-equation models k – ω SST and k−kL. Grid refinement studies are performed in sets of geometrically similar structured grids, permitting an unambiguous definition of the typical cell size, using double precision and an iterative convergence criterion that guarantees a numerical error dominated by the discretization error. For each case, different grid sets with the same number of cells but different near-wall spacings are used to generate a data set that allows more than one estimation of the numerical uncertainty for similar grid densities. The selected flow quantities include functional (integral), surface, and local flow quantities, namely, drag/resistance and lift coefficients; skin friction and pressure coefficients at the wall; and mean velocity components and eddy viscosity at specified locations in the boundary-layer region. An extra set of grids significantly more refined than those proposed for the estimation of the numerical uncertainty is generated for each test case. Using power law extrapolations, these extra solutions are used to obtain an approximation of the exact solution that allows the assessment of the performance of the numerical uncertainty estimations performed for the basis data set. However, it must be stated that with grids up to 2.5 (plate) and 8.46 (airfoil) million cells in two dimensions, the asymptotic range is not attained for many of the selected flow quantities. All this data is available online to the community.

scholarly journals Assessment of Discretization Uncertainty Estimators Based On Grid Refinement Studies

2021 ◽  
Author(s):  
Luis Eca ◽  
Guilherme Vaz ◽  
Martin Hoekstra ◽  
Scott Doebling ◽  
Robert Singleton ◽  
...  
Keyword(s):  
Incompressible Flows ◽  
Turbulence Models ◽  
Double Precision ◽  
Grid Refinement ◽  
Convergence Criteria ◽  
Data Set ◽  
Local Flow ◽  
Uncertainty Estimates ◽  
Wide Range ◽  
Naca 0012

Abstract This paper presents the assessment of the performance of 9 discretization uncertainty estimates based on grid refinement studies including methods that use grid triplets and others that use a largest number of data points, which in the present study was set to five. The uncertainty estimates are performed for the data set proposed for the 2017 ASME Workshop on Estimation of Discretization Errors including functional and local flow quantities from the two-dimensional incompressible flows over a flat plate and the NACA 0012 airfoil. The data were generated with a RANS solver using three eddy-viscosity turbulence models with double precision and sufficiently tight iterative convergence criteria to ensure that the numerical error is dominated by the discretization error. The use of several geometrically similar grid sets with different near-wall cell sizes lead to a wide range of convergence properties for the selected flow quantities. The evaluation of uncertainty estimates is based on the ratio of the estimated uncertainty over the "exact error" that is obtained from an "exact solution" obtained from extra grid sets significantly more refined than those used to generate the Workshop data. Although none of the methods tested fulfilled the goal of bounding the "exact error" 95 times out of 100 that was tested, the results suggest that the methods tested are useful tools for the assessment of the numerical uncertainty of practical numerical simulations even for cases where it is not possible to generate data in the "asymptotic range".

An Assessment of Three-Dimensional Turbulent Boundary Layer Prediction Methods

1973 ◽  
Vol 95 (3) ◽  
pp. 415-421 ◽  
Author(s):  
A. J. Wheeler ◽  
J. P. Johnston
Keyword(s):  
Boundary Layer ◽  
Eddy Viscosity ◽  
Three Dimensional ◽  
High Sensitivity ◽  
Boundary Layer Flows ◽  
Mean Velocity ◽  
Eddy Viscosity Model ◽  
Separating Flow ◽  
Dimensional Boundary ◽  
Stress Models

Predictions have been made for a variety of experimental three-dimensional boundary layer flows with a single finite difference method which was used with three different turbulent stress models: (i) an eddy viscosity model, (ii) the “Nash” model, and (iii) the “Bradshaw” model. For many purposes, even the simplest stress model (eddy viscosity) was adequate to predict the mean velocity field. On the other hand, the profile of shear stress direction was not correctly predicted in one case by any model tested. The high sensitivity of the predicted results to free stream pressure gradient in separating flow cases is demonstrated.

A revision of Hofstede’s model of national culture: old evidence and new data from 56 countries

2018 ◽  
Vol 25 (2) ◽  
pp. 231-256 ◽  
Author(s):  
Michael Minkov
Keyword(s):  
Cultural Differences ◽  
National Culture ◽  
Well Being ◽  
Two Dimensions ◽  
Internal Reliability ◽  
Data Set ◽  
Content Type ◽  
Gender Egalitarianism ◽  
Fifth Dimension ◽  
National Differences

PurposeHofstede’s model of national culture has enjoyed enormous popularity but rests partly on faith. It has never been fully replicated and its predictive properties have been challenged. The purpose of this paper is to provide a test of the model’s coherence and utility.Design/methodology/approachAnalyses of secondary data, including the World Values Survey, and a new survey across 56 countries represented by nearly 53,000 probabilistically selected respondents.FindingsImproved operationalizations of individualism-collectivism (IDV-COLL) suggest it is a robust dimension of national culture. A modern IDV-COLL index supersedes Hofstede’s 50 year-old original one. Power distance (PD) seems to be a logical facet of IDV-COLL, rather than an independent dimension. Uncertainty avoidance (UA) lacks internal reliability. Approval of restrictive societal rules and laws is a facet of COLL and is not associated with national anxiety or neuroticism. UA is not a predictor of any of its presumed main correlates: importance of job security, preference for a safe job, trust, racism and xenophobia, subjective well-being, innovation, and economic freedom. The dimension of masculinity-femininity (MAS-FEM) lacks coherence. MAS and FEM job goals and broader values are correlated positively, not negatively, and are not related to the MAS-FEM index. MAS-FEM is not a predictor of any of its presumed main correlates: achievement and competition orientation, help and compassion, preference for a workplace with likeable people, work orientation, religiousness, gender egalitarianism, foreign aid. After a radical reconceptualization and a new operationalization, the so-called “fifth dimension” (CWD or long-term orientation) becomes more coherent and useful. The new version, called flexibility-monumentalism (FLX-MON), explains the cultural differences between East Asian Confucian societies at one extreme and Latin America plus Africa at the other, and is the best predictor of national differences in educational achievement.Research limitations/implicationsDifferences between subsidiaries of a multinational company, such as IBM around 1970, are not necessarily a good source of knowledge about broad cultural differences. A model of national culture must be validated across a large number of countries from all continents and its predictions should withstand various plausible controls. Much of Hofstede’s model (UA, MAS-FEM) fails this test while the remaining part (IDV-COLL, PD, LTO) needs a serious revision.Practical implicationsConsultancies and business schools still teach Hofstede’s model uncritically. They need to be aware of its deficiencies.Originality/valueAs UA and MAS-FEM are apparently misleading artifacts of Hofstede’s IBM data set, a thorough revision of Hofstede’s model is proposed, reducing it to two dimensions: IDV-COLL and FLX-MON.

Aerodynamic Drag Reduction Investigation for a Simplified Road Vehicle Using Plasma Flow Control

2016 ◽  
Author(s):  
Bahram Khalighi ◽  
Joanna Ho ◽  
John Cooney ◽  
Brian Neiswander ◽  
Thomas C. Corke ◽  
...  
Keyword(s):  
Drag Reduction ◽  
Flow Control ◽  
Plasma Flow ◽  
Aerodynamic Drag ◽  
Force Balance ◽  
Plasma Actuators ◽  
Ground Vehicles ◽  
Mean Velocity ◽  
Local Flow ◽  
Plasma Flow Control

The effect of plasma flow control on reducing aerodynamic drag for ground vehicles is investigated. The experiments were carried out for a simplified ground vehicle using single dielectric barrier discharge (SDBD) plasma actuators. The plasma actuators were designed to alter the flow structure in the wake region behind the vehicle. The Ahmed body was modified to allow eight different vehicle geometries (with backlight or slant angles of 0° and 35°). Each of these were further modified by rounding the edges with different radii. Flow visualizations such as particle streams and surface oil were used to quantify features of the local flow field. The drag on the models was measured using a force balance as well as by integrating the mean velocity profiles in the model wakes. The results indicated that flow modifications needed to be applied symmetrically (upper to lower and/or side to side). This was demonstrated with the 0° backlight angle (square-back) that had all four side-corners rounded. Plasma actuators were applied to all four of the rounded edges to enhance the ability to direct the flow into the wake. Wake measurements showed that steady actuation at a fixed actuator voltage reduced the drag by an average of 20% at the lower velocities (below 15 m/s) and by 3% at the highest velocity tested (20 m/s). Model constraints prevented increasing the plasma actuator voltage that was needed to maintain the higher drag reduction observed at the lower speeds.

scholarly journals Characterization of the Nonaerated Flow Region in a Stepped Spillway by PIV

2006 ◽  
Vol 128 (6) ◽  
pp. 1266-1273 ◽  
Author(s):  
António Amador ◽  
Martí Sánchez-Juny ◽  
Josep Dolz
Keyword(s):  
Flow Region ◽  
Design Guidelines ◽  
Quadrant Analysis ◽  
Stepped Spillway ◽  
Main Stream ◽  
Reynolds Stresses ◽  
Particle Image Velocimetry Technique ◽  
Mean Velocity ◽  
Local Flow ◽  
Stepped Spillways

The development of the roller-compacted concrete (RCC) as a technique of constructing dams and the stepped surface that results from the construction procedure opened a renewed interest in stepped spillways. Previous research has focused on studying the air-water flow down the stepped chute with the objective of obtaining better design guidelines. The nonaerated flow region enlarges as the flow rate increases, and there is a lack of knowledge on the hydraulic performance of stepped spillways at high velocities that undermines its use in fear of cavitation damage. In the present, study the developing flow region in a stepped channel with a slope 1v:0.8h is characterized using a particle image velocimetry technique. An expression for the growth of the boundary layer thickness is proposed based on the streamwise distance from the channel crest and the roughness height. The local flow resistance coefficient is calculated by application of the von Kármán integral momentum equation. The shear strain, vorticity, and swirling strength maps obtained from the mean velocity gradient tensor are presented. Also, the fluctuating velocity field is assessed. The turbulent kinetic energy map indicates the region near the pseudobottom (imaginary line joining two adjacent step edges) as the most active in terms of Reynolds stresses. The turbulence was found to be very intense with maximum levels of turbulence intensity from 0.40 to 0.65 measured near the pseudobottom. Finally, the quadrant analysis of the velocity fluctuations suggests the presence of strong outflows of fluid from the cavities as well as inflows into the cavities. It is conjectured that the mass transfer/exchange between cavities and main stream, play an important role in the high levels of turbulent energy observed.

Interaction between a spatially growing turbulent boundary layer and embedded streamwise vortices

1996 ◽  
Vol 326 ◽  
pp. 151-179 ◽  
Author(s):  
Junhui Liu ◽  
Ugo Piomelli ◽  
Philippe R. Spalart
Keyword(s):  
Boundary Layer ◽  
Kinetic Energy ◽  
Turbulent Boundary Layer ◽  
Turbulent Kinetic Energy ◽  
Vortex Core ◽  
Eddy Viscosity ◽  
Shear Stresses ◽  
Streamwise Vortices ◽  
Mean Velocity ◽  
Production Term

The interaction between a zero-pressure-gradient turbulent boundary layer and a pair of strong, common-flow-down, streamwise vortices with a sizeable velocity deficit is studied by large-eddy simulation. The subgrid-scale stresses are modelled by a localized dynamic eddy-viscosity model. The results agree well with experimental data. The vortices drastically distort the boundary layer, and produce large spanwise variations of the skin friction. The Reynolds stresses are highly three-dimensional. High levels of kinetic energy are found both in the upwash region and in the vortex core. The two secondary shear stresses are significant in the vortex region, with magnitudes comparable to the primary one. Turbulent transport from the immediate upwash region is partly responsible for the high levels of turbulent kinetic energy in the vortex core; its effect on the primary stress 〈u′v′〉 is less significant. The mean velocity gradients play an important role in the generation of 〈u′v′〉 in all regions, while they are negligible in the generation of turbulent kinetic energy in the vortex core. The pressure-strain correlations are generally of opposite sign to the production terms except in the vortex core, where they have the same sign as the production term in the budget of 〈u′v′〉. The results highlight the limitations of the eddy-viscosity assumption (in a Reynolds-averaged context) for flows of this type, as well as the excessive diffusion predicted by typical turbulence models.

On Application of Nonlinear k-ε Models for Internal Combustion Engine Flows

2002 ◽  
Vol 124 (3) ◽  
pp. 668-677 ◽  
Author(s):  
G. M. Bianchi ◽  
G. Cantore ◽  
P. Parmeggiani ◽  
V. Michelassi
Keyword(s):  
Internal Combustion Engine ◽  
Eddy Viscosity ◽  
Combustion Engine ◽  
Turbulence Models ◽  
Internal Combustion ◽  
Moment Closure ◽  
Reynolds Stresses ◽  
Mean Velocity ◽  
Numerical Predictions ◽  
Viscosity Models

The linear k-ε model, in its different formulations, still remains the most widely used turbulence model for the solutions of internal combustion engine (ICE) flows thanks to the use of only two scale-determining transport variables and the simple constitutive relation. This paper discusses the application of nonlinear k-ε turbulence models for internal combustion engine flows. Motivations to nonlinear eddy viscosity models use arise from the consideration that such models combine the simplicity of linear eddy-viscosity models with the predictive properties of second moment closure. In this research the nonlinear k-ε models developed by Speziale in quadratic expansion, and Craft et al. in cubic expansion, have been applied to a practical tumble flow. Comparisons between calculated and measured mean velocity components and turbulence intensity were performed for simple flow structure case. The effects of quadratic and cubic formulations on numerical predictions were investigated too, with particular emphasis on anisotropy and influence of streamline curvature on Reynolds stresses.

CORRELATION FUNCTIONS OF σ FIELDS WITH VALUES IN A HYPERBOLIC SPACE

1989 ◽  
Vol 04 (01) ◽  
pp. 267-286 ◽  
Author(s):  
Z. HABA
Keyword(s):  
Field Theory ◽  
Hyperbolic Space ◽  
Half Plane ◽  
Correlation Functions ◽  
Functional Integral ◽  
Two Dimensions ◽  
Hyperbolic Spaces ◽  
Conformal Invariant ◽  
Upper Half Plane

It is shown that the functional integral for a σ field with values in the Poincare upper half-plane (and some other hyperbolic spaces) can be performed explicitly resulting in a conformal invariant noncanonical field theory in two dimensions.

Frequency‐domain acoustic‐wave modeling and inversion of crosshole data: Part II—Inversion method, synthetic experiments and real‐data results

Geophysics ◽  
1995 ◽  
Vol 60 (3) ◽  
pp. 796-809 ◽  
Author(s):  
Zhong‐Min Song ◽  
Paul R. Williamson ◽  
R. Gerhard Pratt
Keyword(s):  
Frequency Domain ◽  
Real Data ◽  
Forward Modeling ◽  
Two Dimensions ◽  
Inversion Method ◽  
Inversion Algorithm ◽  
Data Set ◽  
Wave Inversion ◽  
Full Wave ◽  
And Inversion

In full‐wave inversion of seismic data in complex media it is desirable to use finite differences or finite elements for the forward modeling, but such methods are still prohibitively expensive when implemented in 3-D. Full‐wave 2-D inversion schemes are of limited utility even in 2-D media because they do not model 3-D dynamics correctly. Many seismic experiments effectively assume that the geology varies in two dimensions only but generate 3-D (point source) wavefields; that is, they are “two‐and‐one‐half‐dimensional” (2.5-D), and this configuration can be exploited to model 3-D propagation efficiently in such media. We propose a frequency domain full‐wave inversion algorithm which uses a 2.5-D finite difference forward modeling method. The calculated seismogram can be compared directly with real data, which allows the inversion to be iterated. We use a descents‐related method to minimize a least‐squares measure of the wavefield mismatch at the receivers. The acute nonlinearity caused by phase‐wrapping, which corresponds to time‐domain cycle‐skipping, is avoided by the strategy of either starting the inversion using a low frequency component of the data or constructing a starting model using traveltime tomography. The inversion proceeds by stages at successively higher frequencies across the observed bandwidth. The frequency domain is particularly efficient for crosshole configurations and also allows easy incorporation of attenuation, via complex velocities, in both forward modeling and inversion. This also requires the introduction of complex source amplitudes into the inversion as additional unknowns. Synthetic studies show that the iterative scheme enables us to achieve the theoretical maximum resolution for the velocity reconstruction and that strongly attenuative zones can be recovered with reasonable accuracy. Preliminary results from the application of the method to a real data set are also encouraging.

scholarly journals Assessing Rotation/Curvature Corrections to Eddy-Viscosity Models in the Calculations of Centrifugal-Compressor Flows

2008 ◽  
Vol 130 (9) ◽  
Author(s):  
G. Dufour ◽  
J.-B. Cazalbou ◽  
X. Carbonneau ◽  
P. Chassaing
Keyword(s):  
Centrifugal Compressor ◽  
Eddy Viscosity ◽  
Turbulence Modeling ◽  
Turbulence Models ◽  
Equation Modeling ◽  
Test Case ◽  
Consistency Analysis ◽  
Perfect Agreement ◽  
Local Flow ◽  
The Impact

Rotation and curvature (RC) effects on turbulence are expected to impact losses and flow structure in turbomachines. This paper examines two recent eddy-viscosity-model corrections devised to account for these effects: the Spalart and Shur (1997, “On the Sensitization of Turbulence Models to Rotation and Curvature,” Aerosp. Sci. Technol., 1(5), pp. 297–302) correction to the model of Spalart and Allmaras (1994, “A One-Equation Turbulence Model for Aerodynamic Flows,” Rech. Aerosp., 1, pp. 5–21) and the correction of Cazalbou et al. (2005, “Two-Equation Modeling of Turbulent Rotating Flows,” Phys. Fluids., 17, p. 055110) to the (k,ϵ) model. The method of verification and validation is applied to assess the impact of these corrections on the computation of a centrifugal-compressor test case. First, a review of RC effects on turbulence as they apply to centrifugal compressors is made. The two corrected models are then presented. Second, the Radiver open test case (Ziegler K. U., Gallus, H. E., and Niehuis R., 2003, “A Study on Impeller Diffuser Interaction Part 1: Influence on the Performance,” ASME J. Turbomach, 125, pp. 173–182) is used as a basis for the assessment of the two corrections. After a physical-consistency analysis, the Richardson extrapolation is applied to quantify the numerical errors involved in all the calculations. Finally, experimental data are used to perform validation for both global and local predictions. The consistency analysis shows that both corrections lead to significant changes in the turbulent field, in perfect agreement with the underlying theoretical considerations. The uncertainty analysis shows that the predictions of the global performances are more sensitive to grid refinement than they are to RC turbulence modeling. However, the opposite conclusion is drawn with regard to the prediction of some local flow properties: Improvements are obtained with the RC corrections, the best results being observed for the RC-corrected (k,ϵ) model.

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