Vortex Core Singularity: 1/r
Point vortices have a 1/r velocity singularity at their cores. Classical solvers struggle as they approach the core, requiring infinitesimal timesteps. NNC with k=-1 regularizes this singularity, maintaining stable computation.
The Physics
Point Vortex Dynamics
In 2D inviscid flow, a point vortex with circulation Gamma induces a velocity field:
v(r) = Gamma / (2 * pi * r)This is the famous Biot-Savart law. As r approaches 0 (the vortex core), velocity diverges to infinity.
Turbulence Energy Cascade
Kolmogorov's -5/3 law describes how energy cascades from large to small scales:
E(k) ~ k^(-5/3)Vortex cores are where dissipation occurs. Resolving them accurately is critical for turbulence simulations.
How NNC Regularizes 1/r
Classical (k=0)
v = Gamma / (2 * pi * r)- - Diverges as r approaches 0
- - Requires tiny timesteps near cores
- - Eventually stalls or blows up
NNC (k=-1)
v_nnc = v * r = Gamma / (2 * pi) = constant!- - Regularization factor r^(-k) = r
- - Cancels the 1/r singularity exactly
- - Stable timesteps even at core
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Verified Results
70.9%
Closer to Core
from CASCADE 21-sim suite
k=-1
Optimal Transform
for 1/r singularity
SpectralDNS
Physics Domain
turbulence simulation
Mathematical Foundation
From Grossman & Katz (1972), the NNC regularization factor for a singularity of form r^n is:
regularization_factor = r^(-k)For a 1/r singularity (n=-1), choosing k=-1 gives:
r^(-(-1)) = r^1 = rMultiplying: (1/r) * r = 1, canceling the singularity exactly.