By Lu Ting, Rupert Klein, Omar M Knio
Addressed to either graduate scholars and researchers this monograph offers in-depth analyses of vortex ruled flows through matched and multiscale asymptotics, and it demonstrates how perception received via those analyses will be exploited within the development of strong, effective, and exact numerical options. The dynamics of narrow vortex filaments is mentioned intimately, together with primary derivations, compressible center constitution, weakly non-linear restrict regimes, and linked numerical tools. equally, the amount covers asymptotic research and computational strategies for weakly compressible flows related to vortex generated sound and thermoacoustics.
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Additional info for Vortex Dominated Flows: Analysis and Computation for Multiple Scale Phenomena
A slender vortex ﬁlament, or in short a ﬁlament, models a special type of vorticity ﬁeld, in which the bulk of vorticity is concentrated in a slender tubular region around a curve C, called its centerline. We assume that the radius of curvature of C and the characteristic length scale of variations of the vorticity ﬁeld along C are O( ). But in a normal plane of C, a crosssectional plane of the ﬁlament, the vorticity decays rapidly as the distance r . We call δ ∗ the typical from C increases, in a typical decaying length δ ∗ core size of the ﬁlament.
6) It was pointed out in Subsect. 3) on the nth moments of vorticity for incompressible ﬂows remain valid for compressible ﬂows. 16b), are identical to those for Ω for incompressible ﬂows for which ρ = 1 and Ξ = Ω. 23), become Moreau’s invariants for the moments of Ω (Moreau, 1948a, 1948b). They are x × Ω = x × Υ = E and r2 Ω = r2 Υ = D . 7) This equation says that the polar moment of vorticity with respect to the origin is time invariant. No additional invariants for n ≥ 3 have been found from integrations of the vorticity evolution equation (Howard, 1957).
The asymptotic analyses of slender ﬁlaments with small parameter were carried out ﬁrst for two-dimensional problems by Ting and Tung (1965), and then for axisymmetric problems by Tung and Ting (1967) to show the eﬀects of curvature and stretching of the centerline. Both eﬀects are absent in two-dimensional problems. For a slender ﬁlament in a threedimensional ﬂow, orthogonal coordinates s, r, and θ, intrinsic to the centerline C at time t, were introduced in Ting (1971), where s denotes the axial variable along C, and r and θ denote the polar coordinates in the normal plane at point s on C.