Abstract
Three particles floating on a fluid surface define a triangle. The aim of this paper is to characterise the shape of the triangle, defined by two of its angles, as the three vertices are subject to a complex or turbulent motion. We consider a simple class of models for this process, involving a combination of a random strain of the fluid and Brownian motion of the particles. Following D.G. Kendall, we map the space of triangles to a sphere, whose equator corresponds to degenerate triangles with colinear vertices, with equilaterals at the poles. We map our model to a diffusion process on the surface of the sphere and find an exact solution for the shape distribution. Whereas the action of the random strain tends to make the shape of the triangles infinitely elongated, in the presence of a Brownian diffusion of the vertices, the model has an equilibrium distribution of shapes. We determine here exactly this shape distribution in the simple case where the increments of the strain are diffusive.
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References
Castiglione, P., Pumir, A.: Evolution of triangles in a two-dimensional turbulent flow. Phys. Rev. E 64, 056303 (2001)
Cressman, J.R., Davoudi, J., Goldburg, W.I., Schumacher, J.: Eulerian and Lagrangian studies in surface flow turbulence. New J. Phys. 6, 53 (2004)
de Chaumont Quitry, A., Kelley, D.H., Ouellette, N.T.: Mechanisms driving shape distortion in two-dimensional flow. Europhys. Lett. 94, 64006 (2011)
Kendall, D.G.: The diffusion of shape. Adv. Appl. Probab. 9, 428–430 (1977)
Kendall, D.G.: Shape manifolds, procrustean metrics, and complex projective spaces. Bull. Lond. Math. Soc. 16, 81–121 (1984)
Kendall, W.S.: A diffusion model for Bookstein triangle shape. Adv. Appl. Probab. 30, 317–334 (1998)
Pumir, A., Shraiman, B., Chertkov, M.: Geometry of Lagrangian dispersion in turbulence. Phys. Rev. Lett. 85, 5324 (2000)
Biferale, L., Boffetta, G., Celani, A., Devenish, B.J., Lanotte, A., Toschi, F.: Multiparticle dispersion in fully developed turbulence. Phys. Fluids 17, 111701 (2005)
Xu, H., Ouellette, N.T., Bodenschatz, E.: Evolution of geometric structures in intense turbulence. New J. Phys. 10, 013012 (2008)
Hackl, J.F., Yeung, P.K., Sawford, B.L.: Multi-particle and tetrad statistics in numerical simulations of turbulent relative dispersion. Phys. Fluids 23, 065103 (2011)
Mydlarski, L., Pumir, A., Shraiman, B.I., Siggia, E.D., Warhaft, Z.: Structure and multi-point correlators for turbulent advection. Phys. Rev. Lett. 81, 4373–4376 (1998)
Shraiman, B.I., Siggia, E.D.: Anomalous scaling for a passive scalar near the Batchelor limit. Phys. Rev. E 57, 2965–2977 (1998)
Shraiman, B.I., Siggia, E.D.: Scalar turbulence. Nature 405, 639–646 (2000)
Falkovich, G., Gawedzki, K., Vergassola, M.: Particles and fields in turbulence. Rev. Mod. Phys. 73, 913–975 (2000)
Gat, O., Zeitak, R.: Multiscaling in passive scalar advection as stochastic shape dynamics. Phys. Rev. E 57, 5511–5519 (1998)
Balkovisky, E., Chertkov, M., Kolokolov, I., Lebedev, V.: Fourth-order correlation function of randomly advected passive scalar. JETP Lett. 61, 1049 (1995)
Pumir, A., Shraiman, B., Siggia, E.D.: Perturbation theory for δ-correlated model of passive scalar advection near the Batchelor limit. Phys. Rev. E 55, R1263–1266 (1997)
Chertkov, M., Kolokolov, I., Lebedev, V.: Strong effect of weak diffusion on scalar turbulence at large scales. Phys. Fluids 19, 101703 (2007)
van Kampen, N.G.: Stochastic Processes in Physics and Chemistry, 2nd edn. North-Holland, Amsterdam (1981)
Acknowledgements
We acknowledge R. Guichardaz for his comments on our manuscript. A.P. has been supported by the grant from A.N.R. “TEC 2”. We acknowledge the support of the European COST Action MP0806.