TY - JOUR

T1 - Singularity formation for complex solutions of the 3D incompressible Euler equations

AU - Caflisch, Russel E.

N1 - Funding Information:
1Research supported in part by grant #DAAL03-91-G-0162 from the Army Research Office and grant #AFOSR 89-0003 from the Air Force Office of Scientific Research.

PY - 1993/8/15

Y1 - 1993/8/15

N2 - Moore's approximation method, first formulated for vortex sheets, is generalized and applied to axi-symmetric flow with swirl and with smooth initial data. The approximation preserves the forward cascade of energy but neglects any backflow of energy. It splits the Euler equations into two sets of equations: one for u+ = u+ (r,z,t) containing all non-negative wavenumbers (in z) and the second for u- = u+. The equations for u+ are exactly the Euler equations but with complex initial data. Traveling waves solutions u+ = u+(r,z-iσt) with imaginary wave speed are found numerically for this problem. The asymptotic properties of the resulting Fourier coefficients show a singularity forming in finite time at which the velocity blows up.

AB - Moore's approximation method, first formulated for vortex sheets, is generalized and applied to axi-symmetric flow with swirl and with smooth initial data. The approximation preserves the forward cascade of energy but neglects any backflow of energy. It splits the Euler equations into two sets of equations: one for u+ = u+ (r,z,t) containing all non-negative wavenumbers (in z) and the second for u- = u+. The equations for u+ are exactly the Euler equations but with complex initial data. Traveling waves solutions u+ = u+(r,z-iσt) with imaginary wave speed are found numerically for this problem. The asymptotic properties of the resulting Fourier coefficients show a singularity forming in finite time at which the velocity blows up.

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U2 - 10.1016/0167-2789(93)90195-7

DO - 10.1016/0167-2789(93)90195-7

M3 - Article

AN - SCOPUS:43949163593

VL - 67

SP - 1

EP - 18

JO - Physica D: Nonlinear Phenomena

JF - Physica D: Nonlinear Phenomena

SN - 0167-2789

IS - 1-3

ER -