





























In addition to the 8 postings of E&A below, click here to download the file VVCSE98_errata.doc. This is a MicroSoft Word document that contains corrections to the text in MarkUp format. Turn on MarkUp in your Word to see the changes. 



E & A #8 posted 11/22/99 



The article by R. Cai (1998) cited in E&A #2 below contains some typographical errors. See ASME Journal of Fluids Engineering, Vol. 121, September
1999, page 604 for 6 errata items on this article. 




E & A #7 posted 07/13/99 



Hayase, T. (1999), "Monotonic Convergence Property of Turbulent Flow Solution with Central Difference and QUICK Schemes", ASME Journal of Fluids
Engineering, vol. 121, June 1999, pp. 351358. The principal result of this article by Prof. Toshiyuki Hayase of Tohoku University, Sendai, Japan, is the demonstration that QUICK is superior to central
differencing on yet another account: QUICK converges monotonically for the problem of unsteady turbulent flow in a square duct (calculated without a turbulence model) whereas central differencing does not, making error
estimation/banding problematical. Additionally, the article provides further motivation for the use of Fs = 3 in the definition of the GCI (Chapter 5), and for the use of p = 2 in the QUICK method. More often than
not, papers using the GCI have confused it with an Error Estimator. The GCI is not an error estimator, but an error band. In fact, it is defined as Fs = 3 times the (generalized) Richardson error estimator.
Asymptotically, one would expect the GCI to be conservative by a factor of Fs = 3. The rationale for this conservatism is primarily based on the common practice of performing grid convergence tests by using only
two grids. For studies with many grids which establish conclusively a constant observed rate of convergence p, I recommend the value Fs = 1.25. (See pages 120123.) Several good researchers have criticized me for
using such a conservative value as Fs
= 3, but Hayase’s results tend to confirm my approach. As he pointed out, the fine grid GCI for mean bulk velocity with central differencing (in his Grid C) is half the real error. This would be bad enough, but bearing in mind that
Fs = 3, the (generalized) Richardson Error Estimate is thus 6 times the real error. I think that this again shows that the conservatism of Fs = 3 is more than justified for the difficult problem computed.
Although the author did not mention it, his calculation of the finegrid GCI for QUICK on grid B, like the result for central differencing on Grid C, is less than the real error (15 < 21). However, this is due to the
use of the theoretical rate of convergence (p = 3) for QUICK rather than observed rate. For QUICK, there is ambiguity in the rate of convergence p. Although the advection term is 3rd order for 1D
problems, its order degrades for multidimensions, or at least is sensitive to the implementation. Also, other discretizations in the code may be second order. Examination of Hayase’s Figure 9(a) indicates that in fact
the observed convergence rate for the QUICK calculation is precisely p = 2 for the mean bulk velocity. (Applying Equation 5.10.6.1, page 131 of Verification and Validation in Computational Science and Engineering
to the values read as f3 = 1.64, f2 = 1.2, f1 = 1.09 with r = 2 gives exactly p
= 2.) If the observed rate of convergence p = 2 is used, the new GCI for QUICK is 7/3 times the old value of 15, or 35, which is now conservative compared to the real error (35 > 21). Again, the GCI would not be conservative even with
p = 2 unless Fs = 3 or at least 1.8. 




E & A #6 posted 07/13/99 



Roache, P. J. (1998), "Uncertainties and CFD Code Validation", ASME Journal of Fluids Engineering, Vol. 120, September 1998, p. 635636.
This note, based on Chapter 10, Section 10.23, pp. 331 335, has been published in the "Discussion" section of JFE along with the reply by H. W. Coleman and F. Stern. The authors accept my point about the
validity of "trend capturing" in Figure 10.23.1, p. 332, but only for a "very special case." They also accept the need for including some required error tolerance in the definition of validation, and
agree that the situation that I describe on page 335 should also be considered a successful validation even though it does not fit their original definition; however, they do not agree with my use of Eqn. 10.23.7, page
335. 




E & A #5 posted 03/13/99 



The following E&A are due to the courtesy of Prof. Dominique Pelletier. Page 162. The paper by Pelletier and Ignat (1995) already considers the komega
and ktau models as well as the kepsilon model for a shear layer. In all cases the fields of u, v, p were identical. The fields of k are identical, while those of omega or tau were adjusted so as to yield the same eddy
viscosity distribution. Page 202. The argument that the ratio of specific heat > 1 as Mach number > 0 cannot hold, even though it is a commonly held belief by too many people. See R. L. Panton's book
Incompressible Flows, John Wiley & Sons, New York, 1984, pp. 258260. The error arises because of inappropriate nondimensionalization. Page 222. The consideration of homogeneous boundary conditions was used
in the subject paper, but is not necessary to the methodology described. These are necessary if we use collocated variables. Page 227. My (PJR) interpretation of the example given on the inherent limitation of
the energy norm used in common singlegrid error estimators is debatable. Prof. D. Pelletier maintains (as others would) that even if nodal values are exact, this does not mean that the approximation of the solution is
exact. A nonzero energy norm for the simple problem is the right answer, because in the FEM, the approximation is given by the pairing of nodal values and interpolation functions. Even if the nodal values are exact,
linear interpolation between nodes yields an error, which is measured by the energy norm; it is an interpolation error estimate. Linear interpolation is error free for only linear functions over the domain; it will
display interpolation errors for all other functions. I (PJR) am still of the opinion that the example demonstrates the inadequacy of the error measure. When all node values are exact for all discretizations, yet the
error measure shows nonzero values, something is amiss. The interpretation hinges on a longstanding difference of approach between FDM and FEM. 




E & A #4 posted 01/23/99 



Recent references Verification and Validation are the following. AIAA (1998), Guide for the Verification and Validation of Computational Fluid
Dynamics Simulations, AIAA G0771998. Available through
 AIAA
 1801 Alexander Bell Drive
 Suite 500
 Reston, VA 22091
 for US$24.95 + shipping. 19 pages.
Oberkampf, W. L. (1998), “Bibliography for Verification and Validation in Computational Simulation, Sandia Report SAND982041, September 1998. Available through
 Sandia National Laboratories
 P. O. Box 5800
 Albuquerque, NM 871850825
111 pages + distribution list. This bibliography contains abstracts or brief Table of Contents, a listing of authors by year, and brief evaluation notes. It is highly recommended. Blottner, F. G. and Lopez, A. R.
(1998), Determination of Solution Accuracy of Numerical Schemes as Part of Code and Calculation Verification, Sandia Report SAND982222, October 1998. Available through
 Sandia National Laboratories
 P. O. Box 5800
 Albuquerque, NM 871850825
77 pages + distribution list. Rigorous application of Richardson extrapolation not only for accuracy estimation and local truncation error but also to determine if governing equations are well posed. Study of
restrictions on nonuniform mesh variation for various families of discretization schemes, and sequential 1D refinement. Keller, H. B. and Cebeci, T. (1972), “Accurate Numerical Methods for Boundary
Layers, II: TwoDimensional Turbulent Flows”, AIAA Journal, Vol. 1, September 1972. An early paper using rigorous application of Richardson Extrapolation in difficult 2D turbulent boundary layer
problems, this reference was overlooked in Verification and Validation in Computational Science and Engineering. (Cited in Blottner and Lopez, 1998, above.) 




E & A #3 posted 01/21/99 



An additional reference for what is herein called the “Method of Manufactured Solutions” is the “Prescribed Solution Forcing Method” of Dee (1991). See also
Wang (1996). References
 Dee, D. P. (1991), “Prescribed Solution Forcing Method for Model Verification in Hydraulic Engineering”, Proc. 1991 National Conference on Hydraulic Engineering, ASCE, Nashville, July 29August 2, 1991.
 Wang, S. S. Y. (1996), Final Report of the ASCE Task Committee on 3D FreeSurface Flow Model Verifications, to be published as an ASCE monograph.





E & A #2 — Chapter 3. 



Recent references on benchmark solutions are the following.
 Mohamad, A. A. (1998), “Benchmark Solution for Unsteady State CFD Problems”, Numerical Heat Transfer, Part A., Vol. 34, 1998, pp. 653672.
The author suggests (1) the timedependent differentially heated cavity, and (2) the driven cavity with a sinusoidally oscillating lid, for timedependent benchmarks. These appear to be excellent choices, since they
are easily reproducible and contain no vorticity singularities.
 Cai, R. (1998), “Some Explicit Analytical Solutions of Unsteady Compressible Flow”, ASME Journal of Fluids Engineering, Vol. 120, December 1998, p. 760.
Note added 11/22/99. See ASME Journal of Fluids Engineering, Vol. 121, September 1999, page 604 for 6 errata items on this article. The author gives a few 1D solutions and a 2D solution of isentropic unsteady
flow, useful for benchmarking. 




E & A #1 — Chapter 3. 



In retrospect, it seems that early instances of the use of what we now call the Method of Manufactured Solutions were cited in Roache (1972, p. 363365; see
also the recent edition in Roache, 1998B, Fundamentals of CFD). These include Greenspan (1967) and Gourlay and Morris (1968A,B). Although the general method was not presented, it seems clear that these
authors used the approach to generate an ad hoc exact solution for timedependent model equations. Greenspan solved only the linear parabolic 1D heat conduction equation with a source term, but Gourlay and Morris
(1968A) solved a 1D nonlinear advection + source PDE (with both a “manufactured” source term and a “manufactured” timedependent nonlinear advection term) with a solution form of (They did not include diffusion terms.) In Gourlay and Morris (1968B), they solved a 2D nonlinear conservationform advection + source equation
(again, no diffusion terms) with a “manufactured” source term; they compared the discrete solution only in the steady state to the exact steady state solution of Obviously, the simple solution form was chosen first, then passed through the PDE to generate the problem. (Undoubtedly, many of the
noninfiniteseries classical solutions in engineering were obtained this way, i.e. beginning with a solution form.) See also B. K. Crowley (1967), cited on p. 366 of Roache (1972).
What is strange is that the notion persisted, often repeated, that we did not have any nontrivial solutions to the full nonlinear NavierStokes equations,
when all we have to do is “complicate” the problem a little with the addition of a source term, and we can generate all the solutions we want. The key concept is that, for code Verification, these solutions need not be
physically realistic. References
 Roache, P. J. (1972), Computational Fluid Dynamics, Hermosa Publishers, Albuquerque, New Mexico.
 Roache, P. J. (1998B), Fundamentals of Computational Fluid Dynamics, Hermosa Publishers, P. O. Box 9110, Albuquerque, New Mexico, U.S.A.
 Greenspan, D. (1967), “Approximate Solution of InitialBoundary Parabolic Problems by Boundary Value Techniques”, MRCTSR792, Mathematics Research Center, The University of
Wisconsin, Madison, Wisconsin, August 1967.





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