arXiv subjects

Sai Swetha Venkata Kolluru

Publications and source records attributed to Sai Swetha Venkata Kolluru.

4 recordsLinked to original sources

arXiv2026

Complex-time singular structure of the 1D Hou-Luo model

Starting from smooth initial data, we investigate the complex-time analytic structure of the one-dimensional Hou--Luo (HL) model, a wall approximation of the three-dimensional axisymmetric Euler equations. While the finite-time blow-up in this setting has been already established, here we chart the entire singular landscape. This analysis is enabled by a novel formulation of the HL model in Lagrangian coordinates, in which the time-Taylor coefficients of the flow fields are evaluated symbolically to high truncation order. Our results are threefold. First, we show that the Lagrangian series for the vorticity converges within the complex-time disc of radius~$t_\star >0$ and is free from (early-time) resonances that impede the Eulerian formulation. Second, applying asymptotic analysis on the series, we recover both the blow-up time and the singularity exponent with high accuracy. This also enables a quantitative assessment of the Beale--Kato--Majda criterion, which we find correctly identifies the blow-up time, but washes out the local singularity exponent, as it relies on a spatial supremum. Third, and most importantly, we develop a Lagrangian singularity theory that predicts the eye-shaped singularity profile observed in Eulerian coordinates by exploiting the driving mechanism of the blow-up: The accumulation of multiple fluid particles at the same Eulerian position. The employed techniques extend recently introduced methods for the inviscid Burgers equation [C. Rampf et al., Phys. Rev. Fluids 7 (2022) 10, 104610], and can be further adapted to higher spatial dimensions or other hydrodynamical equations.

physics.flu-dyn↗

arXiv2024

Early-time resonances in the three-dimensional wall-bounded axisymmetric Euler and related equations

We investigate the complex-time analytic structure of solutions of the 3D-axisymmetric, wall-bounded, incompressible Euler equations, by starting with the initial data proposed in Luo and Hou (2014), to study a possible finite-time singularity. We use our pseudospectral Fourier-Chebyshev method, with quadruple-precision arithmetic, to compute the time-Taylor series coefficients of the flow fields, up to a high order. We show that the resulting approximations display early-time resonances; the initial spatial location of these structures is different from that for the tygers, which we have obtained in Kolluru et al. (2022). We then perform asymptotic analysis of the Taylor-series coefficients, by using generalised ratio methods, to extract the location and nature of the convergence-limiting singularities and demonstrate that these singularities are distributed around the origin, in the complex-t2 plane, along two curves that resemble the shape of an eye. We obtain similar results for the 1D wall-approximation (of the full 3D-axisymmetric Euler equation) called the 1D HL model, for which we use Fourier-pseudospectral methods to compute the time-Taylor series coefficients of the flow fields. Our work examines the link between tygers, in Galerkin-truncated pseudospectral studies, and early-time resonances, in truncated time-Taylor expansions of solutions of PDEs, such as those we consider.

physics.flu-dyn↗

arXiv2024

Novel spectral methods for shock capturing and the removal of tygers in computational fluid dynamics

Spectral methods yield numerical solutions of the Galerkin-truncated versions of nonlinear partial differential equations involved especially in fluid dynamics. In the presence of discontinuities, such as shocks, spectral approximations develop Gibbs oscillations near the discontinuity. This causes the numerical solution to deviate quickly from the true solution. For spectral approximations of the 1D inviscid Burgers equation, nonlinear wave resonances lead to the formation of tygers in well-resolved areas of the flow, far from the shock. Recently, Besse(to be published) has proposed novel spectral relaxation (SR) and spectral purging (SP) schemes for the removal of tygers and Gibbs oscillations in spectral approximations of nonlinear conservation laws. For the 1D inviscid Burgers equation, it is shown that the novel SR and SP approximations of the solution converge strongly in L2 norm to the entropic weak solution, under an appropriate choice of kernels and related parameters. In this work, we carry out a detailed numerical investigation of SR and SP schemes when applied to the 1D inviscid Burgers equation and report the efficiency of shock capture and the removal of tygers. We then extend our study to systems of nonlinear hyperbolic conservation laws - such as the 2x2 system of the shallow water equations and the standard 3x3 system of 1D compressible Euler equations. For the latter, we generalise the implementation of SR methods to non-periodic problems using Chebyshev polynomials. We then turn to singular flow in the 1D wall approximation of the 3D-axisymmetric wall-bounded incompressible Euler equation. Here, in order to determine the blowup time of the solution, we compare the decay of the width of the analyticity strip, obtained from the pure pseudospectral method, with the improved estimate obtained using the novel spectral relaxation scheme.

math.NA↗

arXiv2022

Insights from a pseudospectral study of a potentially singular solution of the three-dimensional axisymmetric incompressible Euler equation

We develop a Fourier-Chebyshev pseudospectral direct numerical simulation (DNS) to examine a potentially singular solution of the radially bounded, three-dimensional (3D), axisymmetric Euler equations [G. Luo and T.Y. Hou, Proc. Natl. Acad. Sci. USA, 111.36 (2014)]. We demonstrate that: (a) the time of singularity is preceded, in any spectrally truncated DNS, by the formation of oscillatory structures called tygers, first investigated in the one-dimensional (1D) Burgers and two-dimensional (2D) Euler equations; (b) the analyticity-strip method can be generalized to obtain an estimate for the (potential) singularity time.

physics.flu-dyn↗