Discussion: “Dynamic Stability of Periodic Pipes Conveying Fluidâ€‌ (Yu, D. L., Paأ¯doussis, M. P., Shen, H. J., and Wan, L., 2013, ASME J. Appl. Mech., 81, p. 011008)

Abstract

In the paper by Yu et al. [1] a study on the dynamic stability of periodic pipes conveying fluids is reported. The authors utilize the transfer matrix method developed by Ma et al. [2] to treat the pipe's periodicity in geometric and/or material properties in view of the pipe's dynamic stability. In particular, the pipe consists of a finite number of binary repetition cells containing alternating segments: segment A of length lA and segment B of length lB, so that the lattice constant is a = lA + lB. The authors of Ref. [1] note the “need to validate the correctness of the transfer matrix method for the periodic pipe.â€‌ They argue: “Because no other work appears in the open literature on the stability of a periodic pipe system conveying fluid, the algorithm will be validated by calculating the stability of a “degradedâ€‌ periodic pipe, in the following sense: if the geometrical and material properties for segments A and B are the same, the periodic pipe will degrade to a uniform pipe, whose stability has been determined exactly [3] and are validated by experimental results in Ref. [4].â€‌ Thus, they report in their Fig. 2 the stability curves for the uniform pipe and note that “they are in good agreement withâ€‌ the curves obtained by Gregory and Paأ¯doussis [3]. Note that the curves in Figs. 2(a) and 2(b) of Ref. [1] depict, respectively, the dimensionless critical flow velocity uC and the dimensionless critical frequency د‰C as functions of the parameter خ²â€‰= mf /(mf + mp), where mf and mp are the fluid and pipe masses per unit length. Figures 2(a) and 2(b) in Ref. [1] exhibit “jump pointsâ€‌ (in the terminology of Yu et al. [1]) and what can be dubbed is an Stype behavior, suggesting that there is no unique relationship between the critical velocity uC and the mass ratio خ². We reproduce the curves by Yu et al. [1] in Fig. 1Fig. 1Apparently two regions of nonmonotonicity in the study by Yu et al.: the arrows are designated by Yu et al. as jump points; arrow 1 appears to be in the incorrect locationGrahic Jump LocationFig. 2Stability curves for both a uniform pipe (dashed line) and a geometrically periodic pipe (continuous line): (a) the dimensionless critical flow velocity uC as a function of خ², and (b) the dimensionless critical frequency د‰C as a function of خ²Grahic Jump Location. This figure has been modified in order to show the regions of nonmonotonicity by boxes. In Ref. [3] there are three regions of nonmonotonicity, whereas in Fig. 2 in the paper by Yu et al. [1] there appear to be only two regions of nonmonotonicity. Thus, the agreement between the curves in Refs. [1,3] cannot be characterized as good. Yu et al. [1] stress that “near “jump pointsâ€‌ (خ²â€‰= 0.3, 0.7, etc.), matrix A always “behaves poorly.â€‌ One should also stress that Yu et al. [1] identify three “jump points.â€‌ For the convenience of the reader, these are marked by 1, 2, and 3 by us. We agree with the location of “jump pointsâ€‌ 2 and 3, but not point 1. Indeed, jumps occur at the location of the curves where the tangent is vertical.