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 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.

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. 351-358.

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 120-123.) 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 fine-grid 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 1-D 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.

Roache, P. J. (1998), "Uncertainties and CFD Code Validation", ASME Journal of Fluids Engineering, Vol. 120, September 1998, p. 635-636.

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.

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 k-omega and k-tau models as well as the k-epsilon 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. 258-260. 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 single-grid 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 non-zero 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 non-zero values, something is amiss. The interpretation hinges on a long-standing difference of approach between FDM and FEM.

Recent references Verification and Validation are the following.

AIAA (1998), Guide for the Verification and Validation of Computational Fluid Dynamics Simulations, AIAA G-077-1998. Available through

• AIAA
• 1801 Alexander Bell Drive
• Suite 500
• Reston, VA 22091
• for US$24.95 + shipping. 19 pages.

An early paper using rigorous application of Richardson Extrapolation in difficult 2-D turbulent boundary layer problems, this reference was overlooked in Verification and Validation in Computational Science and Engineering. (Cited in Blottner and Lopez, 1998, above.)

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 29-August 2, 1991.
• Wang, S. S. Y. (1996), Final Report of the ASCE Task Committee on 3D Free-Surface Flow Model Verifications, to be published as an ASCE monograph.

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. 653-672.

The author gives a few 1-D solutions and a 2-D solution of isentropic unsteady flow, useful for benchmarking.

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. 363-365; 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 time-dependent model equations. Greenspan solved only the linear parabolic 1-D heat conduction equation with a source term, but Gourlay and Morris (1968A) solved a 1-D nonlinear advection + source PDE (with both a “manufactured” source term and a “manufactured” time-dependent nonlinear advection term) with a solution form of

• 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 Initial-Boundary Parabolic Problems by Boundary Value Techniques”, MRC-TSR-792, Mathematics Research Center, The University of Wisconsin, Madison, Wisconsin, August 1967.