2010/09/21 by V. A. Vladimirov, Vladimirov, V. A. · 2 citations
Computer Science · Engineering · #Computational Fluid Dynamics and Aerodynamics #FOS: Physical sciences #Fluid Dynamics (physics.flu-dyn) #Fluid Dynamics and Turbulent Flows #Nonlinear Dynamics and Pattern Formation
paper · pdf · doi:10.48550/arxiv.1009.4058
openalex publication_date 2010/09/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The motions of a passive scalar a in a general high-frequency oscillating flow are studied. Our aim is threefold: (i) to obtain different classes of general solutions; (ii) to identify, classify, and develop related asymptotic procedures; and (iii) to study the notion of drift motion and the limits of its applicability. The used mathematical approach combines a version of the two-timing method, the Eulerian averaging procedure, and several novel elements. Our main results are: (i) the scaling procedure produces two independent dimensionless scaling parameters: inverse frequency 1/ω and displacement amplitude δ; (ii) we propose the inspection procedure that allows to find the natural functional forms of asymptotic solutions for 1/ω→ 0, δ→ 0 and leads to the key notions of critical, subcritical, and supercritical asymptotic families of solutions; (iii) we solve the asymptotic problems for an arbitrary given oscillating flow and any initial data for a; (iv) these solutions show that there are at least three different drift velocities which correspond to different asymptotic paths on the plane (1/ω,δ); each velocity has dimensionless magnitude O(1); (v) the obtained solutions also show that the averaged motion of a scalar represents a pure drift for the zeroth and first approximations and a drift combined with pseudo-diffusion for the second approximation; (vi) we have shown how the changing of a time-scale produces new classes of solutions; (vii) we develop the two-timing theories of a drift based on both the GLM-theory and the dynamical systems approach; (viii) examples illustrating different options of drifts and pseudo-diffusion are presented.