The motions of a passive scalar ba 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/Ï‰ and displacement amplitude Î´; (ii) we propose the inspection procedure that allows to find the natural functional forms of asymptotic solutions for 1/Ï‰textrightarrow0, Î´textrightarrow0 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 ba; (iv) these solutions show that there are at least three different drift velocities which correspond to different asymptotic paths on the plane (1/Ï‰, Î´); 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.
|Published - 1 Nov 2010
- Fluid Dynamics, oscillating flows, two-timing method