Archive Tutorial This page provides a classification of uncertainties and errorsthat cause CFD simulation results to differ from their true or exactvalues. This discussion not only applies to the CFD code, but othercomputer programs used in the analysis process such as CAD packages,grid generators, and flow visualizers.
Defining Uncertainty and Error
Uncertainty and Error are commonly used interchangeably in everyday language. Here we follow the definitions of the AIAA Guidlines:
Uncertainty is defined as:
"A potential deficiency in any phase or activity of themodeling process that is due to the lack of knowledge." (AIAAG-077-1998)
Error is defined as:
A recoqnizable deficiency in any phase or activity of modeling and simulation that is not due to lack of knowledge. (AIAA G-077-1998)
The key phrase differentiating the definitions of uncertainty and error is lack of knowledge.
The key word in the definition of uncertainty is potential, which indicates that deficiencies may or may not exist. Lack ofknowledge has primarily to do with lack of knowledge about thephysical processes that go into building the model. Sensitivity anduncertainty analyses can be used to better determine uncertainty. Uncertainty applies to describing deficiencies in turbulence modeling. There is a lot about turbulence modeling that is not understood. One approach for determining the level of uncertainty and it effect on one's analysis is to run a number of simulations with a variety of turbulence models and see how the modeling affects the results.
The definition for error implies that the deficiency is identifiable upon examination. Errors can also be classified as acknowledged or unacknowledged:
Acknowledged errors (examples include round-offerror and discretization error) have procedures for identifying themand possibly removing them. Otherwise they can remain in the code withtheir error estimated and listed.
Unacknowledged errors (examples include computerprogramming errors or usage errors) have no set procedures for findingthem and may continue within the code or simulation.
One can differentiate between local and globalerrors. Local errors refer to errors at a grid point or cell, whereasglobal errors refer to errors over the entire flow domain. We areinterested here in the global error of the solution thataccounts for the local error at each grid point but is morethan just the sum of the local errors. Local errors are transported,advected, and diffused throughout the grid.
The definition of error presented here is different than that anexperimentalist may use, which is "the difference between the measuredvalue and the exact value". Experimentalist usually define uncertaintyas "the estimate of error". These definitions are inadequate forcomputational simulations because the exact value is typically notknown. Further these definitions link error with uncertainty. Thedefintions provided in the above paragraphs are more definite becausethey differentiate error and uncertainty according to what is known.
Classification of Errors
Here we provide a classification or taxonomy of error.
Acknowledged Error
- Physical approximation error
- Physical modeling error
- Geometry modeling error
Computer round-off error Iterative convergence error Discretization error Unacknowledged Error
- Spatial discretization error
- Temporal discretization error
- Computer programming error
- Usage error
Each of these types of errors are discussed below.
Physical Approximation Error
Physical modeling errors are those due to uncertainty in theformulation of the model and deliberate simplifications of the model.These errors deal with the continuum model only. Converting the modelto discrete form for the code is discussed as part of discretizationerrors. Errors in the modeling of the fluids or solids problem areconcerned with the choice of the governing equations which are solvedand models for the fluid or solid properties. Further, the issue ofproviding a well-posed problem can contribute to modeling errors. Oftenmodeling is required for turbulence quantities, transistion, andboundary conditions (bleed, time-varying flow, surface roughness). Mehta lists sources of uncertainty inphysical models as 1) the phenomenon is not thoroughly understood; 2)parameters used in the model are known but with some degree ofuncertainty; 3) appropriate models are simplified, thus introducinguncertainty; and 4) an experimental confirmation of the models is notpossible or is incomplete. Even when a physical process is known to ahigh level of accuracy, a simplified model may be used within the CFDcode for the convenience of a more efficient computation. Physicalmodeling errors are examined by performing validation studies thatfocus on certain models (i.e. inviscid flow, turbulent boundary layers,real-gas flows, etc...).
Computer Round-Off Error
Computer round-off errors develop with the representation offloating point numbers on the computer and the accuracy at whichnumbers are stored. With advanced computer resources, numbers aretypically stored with 16, 32, or 64 bits. Round-off errors are notconsidered significant when compared with other errors. If computerround-off errors are suspected of being significant, one test is to runthe code at a higher precision or on a computer known to store floatingpoint numbers at a higher precision. One can attempt to iterate acoarse grid solution to a residual of machine zero; however, this maynot be possible for more complex algorithms.
Iterative Convergence Error
The iterative convergence error exists because the iterative methods used in the simulation must have a stopping point eventually. The error scales to the variation in the solution at the completion of the simulations.
Discretization Errors
Discretization errors are those errors that occur from therepresentation of the governing flow equations and other physicalmodels as algebraic expressions in a discrete domain of space(finite-difference, finite-volume, finite-element) and time. Thediscrete spatial domain is known as the grid or mesh. Thetemporal discreteness is manifested through the time step taken.Discretization error is also known as numerical error. Aconsistent numerical method will approach the continuum representationof the equations and zero discretization error as the number of gridpoints increases and the size of the grid spacing tends to zero. Asthe mesh is refined, the solution should become less sensitive to thegrid spacing and approach the continuum solution. This is gridconvergence. Such thinking also applies to the time step. The grid convergence study is a useful procedure for determining thelevel of discretization error existing in a CFD solution. "Ordered"discretization errors are those dependent on the grid size and vanishas the grid size approaches zero. These are the errors that areaddressed by a grid convergence study. Further details can be found on the pages entitled Examining Spatial (Grid) Convergence and Examining Temporal Convergence.
The discretization error is of most concern to a CFD code userduring an application. Discretization errors are of major concernbecause they are dependent on the quality of the grid; however, it isoften difficult to precisely indicate the relationship between aquality grid and an accurate solution prior to beginning thesimulation. The level of discretization error is dependent on gridquality. The grid should be generated with consideration of suchthings as resolution, density, aspect ratio, stretching, orthogonality,grid singularities, and zonal boundary interfaces.
The level of discretization error is dependent on the features of theflow as resolved by the grid. Errors may develop due to representationof discontinuities (shocks, slip surfaces, interfaces, ...) on a grid.Interpolation errors come about at zonal interfaces where the solutionof one zone is approximated on the boundary of the other zone.
The truncation error is the difference between the partialdifferential equation (PDE) and the finite equation. The truncationerror is a function of the grid quality and flow gradients.Dispersive error terms causes oscillations in the solution. One fixto this is adding artificial dissipation to decrease the size of thedispersive errors. Dissipation error terms cause a smoothing ofgradients. However, a level of dissipation comparable to the actualphysical viscosity may contaminate the solution. Boundary layers maythicken. The truncation error terms are those of the expansion whichare not used in the discretized equation. If the order of the leadingterm of the truncation error is of second-order, it is known as anumerical viscosity (dimensions of length2 / time), which is dimensionsof kinematic viscosity. A positive viscous term will indicate thaterrors will be damped whereas a negative viscous term will indicatethat errors will grow (unstable).
Included in the discretization error are errors due to not properlyconverging the solution with respect to the iterations to thesteady-state solution or within a time step. This is reffered to asiterative convergence.
Computer Programming Errors
Programming errors are "bugs" and mistakes made inprogramming or writing the code. They are the responsibility of theprogrammers. These type of errors are discovered by systematicallyperforming verification studies of subprograms of the code and theentire code, reviewing the lines of code, and performing validationstudies of the code. The programming errors shouldbe removed from the code prior to release.
Usage Errors
Usage errors are due to the application of the code in aless-than-accurate or improper manner. Usage errors may actually showup as modeling and discretization errors. The user sets the models,grid, algorithm, and inputs used in a simulation, which thenestablishes the accuracy of the simulation. There may be blatanterrors, such as attempting to compute a known turbulent flow with anassumption of inviscid flow. A converged solution may be obtained;however, the conclusions drawn from the simulation may be incorrect.The errors may not be as evident, such as proper choice of turbulencemodel parameters for separated flows with shocks. The potential forusage errors increases with an increased level of options available ina CFD code. Usage errors are minimized through proper training and theaccumulation of experience.
The user may intentionally introduce modeling and discretizationerror as an attempt to expedite the simulation at the expense ofaccuracy. This may be proper in the conceptual stage of a design studywhere more general information is needed at less accuracy. Even in thelater stages, there may not be proper computational resources tosimulate at the proper grid density. One has to understand the levelof accuracy accompanying the results.
Usage errors should be controlable throughproper training and analysis.
Usage errors can exist in the CAD, grid generation, and post-processing software, in addition to the CFD code.
Last Updated: Wednesday, 10-Feb-2021 09:38:58 EST