This article discusses numerical errors in unsteady flow simulations, which may include round-off, statistical, iterative, and time and space discretization errors. The estimation of iterative and discretization errors and the influence of the initial condition on unsteady flows that become periodic are discussed. In this latter case, the goal is to determine the simulation time required to reduce the influence of the initial condition to negligible levels. Two one-dimensional, unsteady manufactured solutions are used to illustrate the interference between the different types of numerical errors. One solution is periodic and the other includes a transient region before it reaches a steady-state. The results show that for a selected grid and time-step, statistical convergence of the periodic solution may be achieved at significant lower error levels than those of iterative and discretization errors. However, statistical convergence deteriorates when iterative convergence criteria become less demanding, grids are refined, and Courant number increased.For statistically converged solutions of the periodic flow and for the transient solution, iterative convergence criteria required to obtain a negligible influence of the iterative error when compared to the discretization error are more strict than typical values found in the open literature. More demanding criteria are required when the grid is refined and/or the Courant number is increased. When the numerical error is dominated by the iterative error, it is pointless to refine the grid and/or reduce the time-step. For solutions with a numerical error dominated by the discretization error, three different techniques are applied to illustrate how the discretization uncertainty can be estimated, using grid/time refinement studies: three data points at a fixed Courant number; five data points involving three time steps for the same grid and three grids for the same time-step; five data points including at least two grids and two time steps. The latter two techniques distinguish between space and time convergence, whereas the first one combines the effect of the two discretization errors.
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June 2019
Research-Article
Numerical Errors in Unsteady Flow Simulations
L. Eça,
L. Eça
Mechanical Engineering Department,
Instituto Superior Técnico,
ULisboa,
Avenida Rovisco Pais 1,
Lisboa 1049 001, Portugal
e-mail: luis.eca@ist.utl.pt
Instituto Superior Técnico,
ULisboa,
Avenida Rovisco Pais 1,
Lisboa 1049 001, Portugal
e-mail: luis.eca@ist.utl.pt
1Corresponding author.
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G. Vaz,
G. Vaz
Maritime Research Institute Netherlands,
P.O. Box 28,
Wageningen 6700AA, The Netherlands
e-mail: G.Vaz@marin.nl
P.O. Box 28,
Wageningen 6700AA, The Netherlands
e-mail: G.Vaz@marin.nl
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S. L. Toxopeus,
S. L. Toxopeus
Maritime Research Institute Netherlands,
Wageningen 6700AA, The Netherlands
e-mail: S.Toxopeus@marin.nl
P.O. Box 28
,Wageningen 6700AA, The Netherlands
e-mail: S.Toxopeus@marin.nl
Search for other works by this author on:
M. Hoekstra
M. Hoekstra
Voorthuizen 3781XK, The Netherlands
Search for other works by this author on:
L. Eça
Mechanical Engineering Department,
Instituto Superior Técnico,
ULisboa,
Avenida Rovisco Pais 1,
Lisboa 1049 001, Portugal
e-mail: luis.eca@ist.utl.pt
Instituto Superior Técnico,
ULisboa,
Avenida Rovisco Pais 1,
Lisboa 1049 001, Portugal
e-mail: luis.eca@ist.utl.pt
G. Vaz
Maritime Research Institute Netherlands,
P.O. Box 28,
Wageningen 6700AA, The Netherlands
e-mail: G.Vaz@marin.nl
P.O. Box 28,
Wageningen 6700AA, The Netherlands
e-mail: G.Vaz@marin.nl
S. L. Toxopeus
Maritime Research Institute Netherlands,
Wageningen 6700AA, The Netherlands
e-mail: S.Toxopeus@marin.nl
P.O. Box 28
,Wageningen 6700AA, The Netherlands
e-mail: S.Toxopeus@marin.nl
M. Hoekstra
Voorthuizen 3781XK, The Netherlands
1Corresponding author.
Manuscript received January 9, 2019; final manuscript received June 7, 2019; published online July 11, 2019. Assoc. Editor: William Rider.
J. Verif. Valid. Uncert. Jun 2019, 4(2): 021001 (10 pages)
Published Online: July 11, 2019
Article history
Received:
January 9, 2019
Revised:
June 7, 2019
Citation
Eça, L., Vaz, G., Toxopeus, S. L., and Hoekstra, M. (July 11, 2019). "Numerical Errors in Unsteady Flow Simulations." ASME. J. Verif. Valid. Uncert. June 2019; 4(2): 021001. https://doi.org/10.1115/1.4043975
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