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On Transition to Turbulence and Synthesis of Alternative Approaches
Author(s) -
Lin C. C.
Publication year - 2002
Publication title -
studies in applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.164
H-Index - 46
eISSN - 1467-9590
pISSN - 0022-2526
DOI - 10.1111/1467-9590.01420
Subject(s) - context (archaeology) , turbulence , transition (genetics) , epistemology , theoretical physics , statistical physics , mathematics , physics , mechanics , philosophy , geology , paleontology , biochemistry , chemistry , gene
In the study of complex systems, controversial debates often arise among advocates of different schools of thought. In this article, we examine how such controversies should be addressed, with the problem of transition to turbulence as a primary example. It is shown that, in many cases, these controversies may be resolved by first noting that the alternative theories proposed may not be mutually exclusive. Indeed, they may even be mutually complementary, if they were originally developed to address similar issues in different physical contexts. In any case, for the validity of the alternative theories proposed, each should be separately and fully supported from both the theoretical and empirical points of view. Each applies to a specific physical context, and each stands on its own merits and limitations. Synthesis into a broader theory may then be achieved, if commonality is identified among the different alternative theories proposed. To demonstrate this conciliatory approach, we begin with an examination of the move toward resolution of the well‐known controversy over the problem of transition to turbulence from the steady laminar flow in the boundary layer over a flat plate. Several other long‐standing controversies have been successfully addressed on the basis of this approach. In addition to the problem of transition to turbulence, we considered, in some detail, two additional examples: (1) the global structures of spiral galaxies; and (2) the theory of jet noise. In all three cases, it is shown that the model approach is meritorious despite the limitations. Synthesis, with a conciliatory approach to apparent conflicts, will be recommended in general as a new part of an extended paradigm in applied mathematics. It is an approach appropriate to situations where an ideal theory, with universal applicability, is elusive. Parallel development of several alternative theories is natural, and a final synthesis is needed. In contrast, it should be noted that the same perspective is generally not expected useful if the controversies concern the unique solution of well‐defined mathematical issues. The potential success of the application of this conciliatory perception and approach to other areas of science are discussed (see Section 5).