{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T21:40:44Z","timestamp":1760218844667,"version":"build-2065373602"},"reference-count":31,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2014,5,16]],"date-time":"2014-05-16T00:00:00Z","timestamp":1400198400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>A computational procedure is developed to determine initial instabilities within a three-dimensional laminar boundary layer and to follow these instabilities in the streamwise direction through to the resulting intermittency exponents within a fully developed turbulent flow. The fluctuating velocity wave vector component equations are arranged into a Lorenz-type system of equations. The nonlinear time series solution of these equations at the fifth station downstream of the initial instabilities indicates a sequential outward burst process, while the results for the eleventh station predict a strong sequential inward sweep process. The results for the thirteenth station indicate a return to the original instability autogeneration process. The nonlinear time series solutions indicate regions of order and disorder within the solutions. Empirical entropies are defined from decomposition modes obtained from singular value decomposition techniques applied to the nonlinear time series solutions. Empirical entropic indices are obtained from the empirical entropies for two streamwise stations. The intermittency exponents are then obtained from the entropic indices for these streamwise stations that indicate the burst and autogeneration processes.<\/jats:p>","DOI":"10.3390\/e16052729","type":"journal-article","created":{"date-parts":[[2014,5,16]],"date-time":"2014-05-16T12:09:04Z","timestamp":1400242144000},"page":"2729-2755","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Transitional Intermittency Exponents Through Deterministic Boundary-Layer Structures and Empirical Entropic Indices"],"prefix":"10.3390","volume":"16","author":[{"given":"LaVar","family":"Isaacson","sequence":"first","affiliation":[{"name":"Mechanical Engineering, University of Utah, 2067 Browning Avenue, Salt Lake City, UT 84108, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2014,5,16]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"051705","DOI":"10.1063\/1.3589842","article-title":"Edge states in a boundary layer","volume":"23","author":"Cherubini","year":"2011","journal-title":"Phys. Fluids"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"034107","DOI":"10.1063\/1.3696303","article-title":"Laminar-turbulent separatrix in a boundary layer flow","volume":"24","author":"Blau","year":"2012","journal-title":"Phys. Fluids"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"221","DOI":"10.1017\/jfm.2011.412","article-title":"The minimal seed of turbulent transition in the boundary layer","volume":"689","author":"Cherubini","year":"2011","journal-title":"J. Fluid Mech"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"041301","DOI":"10.1063\/1.2717527","article-title":"Hairpin vortex organization in wall turbulence","volume":"19","author":"Adrian","year":"2007","journal-title":"Phys. Fluids"},{"key":"ref_5","unstructured":"Cebeci, T., and Bradshaw, P. (1977). Momentum Transfer in Boundary Layers, Hemisphere."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"402","DOI":"10.3390\/e13020402","article-title":"Spectral Entropy in a Boundary Layer Flow","volume":"13","author":"Isaacson","year":"2011","journal-title":"Entropy"},{"key":"ref_7","unstructured":"Hansen, A.G. (1964). Similarity Analyses of Boundary Value Problems in Engineering, Prentice-Hall, Inc."},{"key":"ref_8","unstructured":"Townsend, A.A. (1976). The Structure of Turbulent Shear Flow, Cambridge University Press. [2nd ed.]."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"6","DOI":"10.1063\/1.867010","article-title":"Chaotic behavior of interacting elliptical instability modes","volume":"31","author":"Hellberg","year":"1988","journal-title":"Phys. Fluids"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"53","DOI":"10.3390\/e15010053","article-title":"Ordered Regions within a Nonlinear Time Series Solution of a Lorenz Form of the Townsend Equations for a Boundary-Layer Flow","volume":"15","author":"Isaacson","year":"2013","journal-title":"Entropy"},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Kapitaniak, T. (1996). Controlling Chaos: Theoretical and Practical Methods in Non-linear Dynamics, Academic Press Inc.","DOI":"10.1016\/B978-012396840-1\/50027-8"},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Kapitaniak, T. (1996). Controlling Chaos: Theoretical and Practical Methods in Non-linear Dynamics, Academic Press Inc.","DOI":"10.1016\/B978-012396840-1\/50027-8"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"4134","DOI":"10.3390\/e15104134","article-title":"Spectral Entropy, Empirical Entropy and Empirical Exergy for Deterministic Boundary-Layer Structures","volume":"15","author":"Isaacson","year":"2013","journal-title":"Entropy"},{"key":"ref_14","unstructured":"Tsallis, C. (2009). Introduction to Nonextensive Statistical Mechanics, Springer."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"3237","DOI":"10.1103\/PhysRevE.61.3237","article-title":"Analysis of fully developed turbulence in terms of Tsallis statistics","volume":"61","author":"Arimitsu","year":"2000","journal-title":"Phys. Rev. E"},{"key":"ref_16","unstructured":"Dorrance, W.H. (1962). Viscous Hypersonic Flow, McGraw-Hill Book Company, Inc."},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Mathieu, J., and Scott, J. (2000). An Introduction to Turbulent Flow, Cambridge University Press.","DOI":"10.1017\/CBO9781316529850"},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Sagaut, P., and Cambon, C. (2008). Homogeneous Turbulence Dynamics, Cambridge University Press.","DOI":"10.1017\/CBO9780511546099"},{"key":"ref_19","unstructured":"Manneville, P. (1990). Dissipative Structures and Weak Turbulence, Academic Press Inc."},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Kapitaniak, T. (1996). Controlling Chaos: Theoretical and Practical Methods in Non-linear Dynamics, Academic Press Inc.","DOI":"10.1016\/B978-012396840-1\/50027-8"},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Feng, J.C., and Tse, C.K. (2008). Reconstruction of Chaotic Signals with Applications to Chaos-Based Communications, World Scientific Publishing Co Pte. Ltd.","DOI":"10.1142\/9789812771148"},{"key":"ref_22","unstructured":"Chen, C.H. (1982). Digital Waveform Processing and Recognition, CRC Press Inc."},{"key":"ref_23","unstructured":"Press, W.H., Teukolsky, S.A., Vetterling, W.T., and Flannery, B.P. (1992). Numerical Recipes in C: The Art of Scientific Computing, Cambridge University Press. [2nd ed.]."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"2053","DOI":"10.1088\/0305-4470\/12\/11\/017","article-title":"A spectral entropy method for distinguishing regular and irregular motion for Hamiltonian systems","volume":"12","author":"Powell","year":"1979","journal-title":"J. Phys. Math. Gen"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"2591","DOI":"10.1103\/PhysRevA.28.2591","article-title":"Estimation of the Kolmogorov entropy from a chaotic signal","volume":"28","author":"Grassberger","year":"1983","journal-title":"Phys. Rev. A"},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Holmes, P., Lumley, J.L., Berkooz, G., and Rowley, C.W. (2012). Turbulence, Coherent Structures, Dynamical Stations and Symmetry, Cambridge University Press. [2nd ed.].","DOI":"10.1017\/CBO9780511919701"},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Greven, A., Keller, G., and Warnecke, G. (2003). Entropy, Princeton University Press.","DOI":"10.1515\/9781400865222"},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Rissanen, J. (2007). Information and Complexity in Statistical Modeling, Springer.","DOI":"10.1007\/978-0-387-68812-1"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"409","DOI":"10.1016\/0375-9601(92)90339-N","article-title":"On the irreversible nature of the Tsallis and Renyi entropies","volume":"165","author":"Mariz","year":"1992","journal-title":"Phys. Lett. A"},{"key":"ref_30","unstructured":"Glansdorff, P., and Prigogine, I. (1971). Thermodynamic Theory of Structure, Stability and Fluctuations, John Wiley & Sons Ltd."},{"key":"ref_31","doi-asserted-by":"crossref","unstructured":"Schmid, P.J., and Henningson, D.S. (2001). Stability and Transition in Shear Flows, Springer-Verlag New York, Inc.","DOI":"10.1007\/978-1-4613-0185-1"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/16\/5\/2729\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T21:11:29Z","timestamp":1760217089000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/16\/5\/2729"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2014,5,16]]},"references-count":31,"journal-issue":{"issue":"5","published-online":{"date-parts":[[2014,5]]}},"alternative-id":["e16052729"],"URL":"https:\/\/doi.org\/10.3390\/e16052729","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2014,5,16]]}}}