TY - CHAP

T1 - Sufficient condition for finite-time singularity and tendency towards self-similarity in a high-symmetry flow

AU - Ng, C. S.

AU - Bhattacharjee, A.

PY - 2004

Y1 - 2004

N2 - A highly symmetric Euler flow, first proposed by Kida (1985), and recently simulated by Boratav and Pelz (1994) is considered. It is found that the fourth order spatial derivative of the pressure (pxxxx) at the origin is most probably positive. It is demonstrated that if pxxxxgrows fast enough, there must be a finite-time singularity (FTS). For a random energy spectrum E(k) ∝ k-v, a FTS can occur if the spectral index v < 3. Furthermore, a positive pxxxx has the dynamical consequence of reducing the third derivative of the velocity uxxx at the origin. Since the expectation value of Uxxx is zero for a random distribution of energy, an ever decreasing uxxx means that the Kida flow has an intrinsic tendency to deviate from a random state. By assuming that Uxxx reaches the minimum value for a given spectral profile, the velocity and pressure are found to have locally self-similar forms similar in shape to what are found in numerical simulations. Such a quasi self-similar solution relaxes the requirement for FTS to v < 6. A special self-similar solution that satisfies Kelvin's circulation theorem and exhibits a FTS is found for v = 2.

AB - A highly symmetric Euler flow, first proposed by Kida (1985), and recently simulated by Boratav and Pelz (1994) is considered. It is found that the fourth order spatial derivative of the pressure (pxxxx) at the origin is most probably positive. It is demonstrated that if pxxxxgrows fast enough, there must be a finite-time singularity (FTS). For a random energy spectrum E(k) ∝ k-v, a FTS can occur if the spectral index v < 3. Furthermore, a positive pxxxx has the dynamical consequence of reducing the third derivative of the velocity uxxx at the origin. Since the expectation value of Uxxx is zero for a random distribution of energy, an ever decreasing uxxx means that the Kida flow has an intrinsic tendency to deviate from a random state. By assuming that Uxxx reaches the minimum value for a given spectral profile, the velocity and pressure are found to have locally self-similar forms similar in shape to what are found in numerical simulations. Such a quasi self-similar solution relaxes the requirement for FTS to v < 6. A special self-similar solution that satisfies Kelvin's circulation theorem and exhibits a FTS is found for v = 2.

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U2 - 10.1007/0-306-48420-x_38

DO - 10.1007/0-306-48420-x_38

M3 - Chapter

AN - SCOPUS:84859856529

SN - 1402009801

SN - 9781402009808

T3 - Fluid Mechanics and its Applications

SP - 317

EP - 328

BT - Tubes, Sheets and Singularities in Fluid Dynamics

PB - Kluwer Academic Publishers

ER -