{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,19]],"date-time":"2026-03-19T02:42:28Z","timestamp":1773888148108,"version":"3.50.1"},"reference-count":35,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2021,6,20]],"date-time":"2021-06-20T00:00:00Z","timestamp":1624147200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Modeling of wall-bounded turbulent flows is still an open problem in classical physics, with relatively slow progress in the last few decades beyond the log law, which only describes the intermediate region in wall-bounded turbulence, i.e., 30\u201350 y+ to 0.1\u20130.2 R+ in a pipe of radius R. Here, we propose a fundamentally new approach based on fractional calculus to model the entire mean velocity profile from the wall to the centerline of the pipe. Specifically, we represent the Reynolds stresses with a non-local fractional derivative of variable-order that decays with the distance from the wall. Surprisingly, we find that this variable fractional order has a universal form for all Reynolds numbers and for three different flow types, i.e., channel flow, Couette flow, and pipe flow. We first use existing databases from direct numerical simulations (DNSs) to lean the variable-order function and subsequently we test it against other DNS data and experimental measurements, including the Princeton superpipe experiments. Taken together, our findings reveal the continuous change in rate of turbulent diffusion from the wall as well as the strong nonlocality of turbulent interactions that intensify away from the wall. Moreover, we propose alternative formulations, including a divergence variable fractional (two-sided) model for turbulent flows. The total shear stress is represented by a two-sided symmetric variable fractional derivative. The numerical results show that this formulation can lead to smooth fractional-order profiles in the whole domain. This new model improves the one-sided model, which is considered in the half domain (wall to centerline) only. We use a finite difference method for solving the inverse problem, but we also introduce the fractional physics-informed neural network (fPINN) for solving the inverse and forward problems much more efficiently. In addition to the aforementioned fully-developed flows, we model turbulent boundary layers and discuss how the streamwise variation affects the universal curve.<\/jats:p>","DOI":"10.3390\/e23060782","type":"journal-article","created":{"date-parts":[[2021,6,20]],"date-time":"2021-06-20T22:00:02Z","timestamp":1624226402000},"page":"782","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":10,"title":["Variable-Order Fractional Models for Wall-Bounded Turbulent Flows"],"prefix":"10.3390","volume":"23","author":[{"given":"Fangying","family":"Song","sequence":"first","affiliation":[{"name":"College of Mathematics and Computer Science, Fuzhou University, Fuzhou 350108, China"}]},{"given":"George Em","family":"Karniadakis","sequence":"additional","affiliation":[{"name":"Division of Applied Mathematics, School of Engineering, Brown University, Providence, RI 02912, USA"}]}],"member":"1968","published-online":{"date-parts":[[2021,6,20]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"123","DOI":"10.1098\/rsta.1895.0004","article-title":"On the dynamical theory of incompressible viscous fluids and the determination of the criterion","volume":"186","author":"Reynolds","year":"1895","journal-title":"Philos. Trans. R. Soc. Lond. A"},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Pope, S.B. (2000). Turbulent Flows, Cambridge University Press.","DOI":"10.1017\/CBO9780511840531"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"136","DOI":"10.1002\/zamm.19250050212","article-title":"Bericht uber Untersuchungen zur ausgebildeten Turbulenz","volume":"5","author":"Prandtl","year":"1925","journal-title":"Z. Angew. Math. Mech."},{"key":"ref_4","first-page":"709","article-title":"Atmospheric diffusion shown on a distance-neighbour graph","volume":"110","author":"Richardson","year":"1926","journal-title":"R. Soc. Lond."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"1048","DOI":"10.1063\/1.1711320","article-title":"Direct-Interaction Approximation for Shear and Thermally Driven Turbulence","volume":"7","author":"Kraichnan","year":"1964","journal-title":"Phys. Fluids"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"241","DOI":"10.1002\/zamm.19420220502","article-title":"Bemerkungen zur Theorie der freien Turbulenz","volume":"22","author":"Prandtl","year":"1942","journal-title":"ZAMM-J. Appl. Math. Mech. Angew. Math. Mech."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"889","DOI":"10.1002\/pamm.201610433","article-title":"A nonlocal zero-Equation turbulence model and a deficit-power law of the wall with a dynamical critical phenomenon","volume":"16","author":"Egolf","year":"2016","journal-title":"Proc. Appl. Math. Mech."},{"key":"ref_8","first-page":"171","article-title":"Fractional calculus and continuous-time finance III: The diffusion limit. Mathematical finnance","volume":"2001","author":"Gorenflo","year":"2001","journal-title":"Trends Math. Birkh\u00e4user"},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Shlesinger, M., West, B., and Klafter, J. (1987). L\u00e9vy dynamics of enhanced diffusion: Application to turbulence. Phys. Rev. Lett., 58.","DOI":"10.1103\/PhysRevLett.58.1100"},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Egolf, P.W., and Hutter, K. (2017). Fractional Turbulence Models. Progress in Turbulence VII, Springer.","DOI":"10.1007\/978-3-319-57934-4_18"},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Chen, W. (2006). A speculative study of 2\/3-order fractional Laplacian modeling of turbulence: Some thoughts and conjectures. Chaos Interdiscip. J. Nonlinear Sci., 16.","DOI":"10.1063\/1.2208452"},{"key":"ref_12","unstructured":"Epps, B.P., and Cushman-Roisin, B. (2018). Turbulence modeling via the fractional Laplacian. arXiv."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Samiee, M., Akhavan-Safaei, A., and Zayernouri, M. (2020). A fractional subgrid-scale model for turbulent flows: Theoretical formulation and a priori study. Phys. Fluids, 32.","DOI":"10.1063\/1.5128379"},{"key":"ref_14","first-page":"789","article-title":"Is there a universal log law for turbulent wall-bounded flows?","volume":"365","author":"George","year":"2007","journal-title":"Philos. Trans. R. Soc. Lond. A Math. Phys. Eng. Sci."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Lischke, A., Pang, G., Gulian, M., Song, F., Glusa, C., Zheng, X., Mao, Z., Cai, W., Meerschaert, M.M., and Ainsworth, M. (2020). What is the fractional Laplacian? A comparative review with new results. J. Comput. Phys., 404.","DOI":"10.1016\/j.jcp.2019.109009"},{"key":"ref_16","unstructured":"Song, F., and Karniadakis, G.E. (2018). A universal fractional model of wall-turbulence. arXiv."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"1675","DOI":"10.1515\/fca-2019-0086","article-title":"Discovering a universal variable-order fractional model for turbulent Couette flow using a physics-informed neural network","volume":"22","author":"Mehta","year":"2019","journal-title":"Fract. Calc. Appl. Anal."},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Marusic, I., McKeon, B., Monkewitz, P., Nagib, H., Smits, A., and Sreenivasan, K. (2010). Wall-bounded turbulent flows at high Reynolds numbers: Recent advances and key issues. Phys. Fluids, 22.","DOI":"10.1063\/1.3453711"},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Johnson, P.L., and Meneveau, C. (2017). Turbulence intermittency in a multiple-time-scale Navier-Stokes-based reduced model. Phys. Rev. Fluids, 2.","DOI":"10.1103\/PhysRevFluids.2.072601"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"14","DOI":"10.1016\/j.jcp.2014.04.024","article-title":"Tempered fractional calculus","volume":"293","author":"Sabzikar","year":"2015","journal-title":"J. Comput. Phys."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"1533","DOI":"10.1016\/j.jcp.2007.02.001","article-title":"Finite difference\/spectral approximations for the time-fractional diffusion equation","volume":"225","author":"Lin","year":"2007","journal-title":"J. Comput. Phys."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"395","DOI":"10.1017\/jfm.2015.268","article-title":"Direct numerical simulation of turbulent channel flow up to Re\u03c4=5200","volume":"774","author":"Lee","year":"2015","journal-title":"J. Fluid Mech."},{"key":"ref_23","unstructured":"White, F.M. (1991). Viscous Fluid Flow, MacGraw-Hill."},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Tang, Y. (2016). A Nested-LES Approach for Computation of High-Reynolds Number, Equilibrium and Non-Equilibrium Turbulent Wall-Bounded Flows. [Ph.D. Thesis, University of Michigan].","DOI":"10.1007\/978-3-319-20388-1_11"},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Schultz, M.P., and Flack, K.A. (2013). Reynolds-number scaling of turbulent channel flow. Phys. Fluids, 25.","DOI":"10.1063\/1.4791606"},{"key":"ref_26","unstructured":"Comte-Bellot, G. (1963). Turbulent Flow between Two Parallel Walls. [Ph.D. Thesis, University of Grenoble]."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"33","DOI":"10.1017\/S0022112098002419","article-title":"Mean-flow scaling of turbulent pipe flow","volume":"373","author":"Zagarola","year":"1998","journal-title":"J. Fluid Mech."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"1608","DOI":"10.1088\/0957-0233\/13\/10\/314","article-title":"Static pressure correction in high Reynolds number fully developed turbulent pipe flow","volume":"13","author":"McKeon","year":"2002","journal-title":"Meas. Sci. Technol."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1093\/biomet\/87.1.1","article-title":"Predicting the output from a complex computer code when fast approximations are available","volume":"87","author":"Kennedy","year":"2000","journal-title":"Biometrika"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"81","DOI":"10.1017\/S0022112008002085","article-title":"A direct numerical simulation study on the mean velocity characteristics in turbulent pipe flow","volume":"608","author":"Wu","year":"2008","journal-title":"J. Fluid Mech."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"988","DOI":"10.1115\/1.1571084","article-title":"Turbulent plane Couette flow and scalar transport at low Reynolds number","volume":"125","author":"Liu","year":"2003","journal-title":"J. Heat Transf."},{"key":"ref_32","doi-asserted-by":"crossref","unstructured":"Avsarkisov, V., Hoyas, S., Oberlack, M., and Garc\u00eda-Galache, J. (2014). Turbulent plane Couette flow at moderately high Reynolds number. J. Fluid Mech., 751.","DOI":"10.1017\/jfm.2014.323"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"1171","DOI":"10.1061\/JMCEA3.0001310","article-title":"Turbulence structure in plane Couette flow","volume":"96","author":"Robertson","year":"1970","journal-title":"J. Eng. Mech. Div."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"116","DOI":"10.1017\/S0022112010003113","article-title":"Assessment of direct numerical simulation data of turbulent boundary layers","volume":"659","author":"Schlatter","year":"2010","journal-title":"J. Fluid Mech."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"251","DOI":"10.1016\/j.ijheatfluidflow.2009.12.011","article-title":"Simulations of spatially evolving turbulent boundary layers up to Re\u03b8= 4300","volume":"31","author":"Schlatter","year":"2010","journal-title":"Int. J. Heat Fluid Flow"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/23\/6\/782\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T06:19:31Z","timestamp":1760163571000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/23\/6\/782"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,6,20]]},"references-count":35,"journal-issue":{"issue":"6","published-online":{"date-parts":[[2021,6]]}},"alternative-id":["e23060782"],"URL":"https:\/\/doi.org\/10.3390\/e23060782","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,6,20]]}}}