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Jeffrey Yepez

Computational physics



Overview of computational fluid dynamics projects

This page describes information conserving computational physics methods for fluid turbulence studies. The turbulent flow results in three space dimensions that are presented below were obtained by classical and quantum simulations on supercomputers. My contribution was the development of the entropic lattice Boltzmann equation method, the quantum lattice Boltzmann equation method, and the quantum lattice gas method. My longtime collaborators in this high performance computing work were Profs. George and Linda Vahala, Dr. Min Soe and Dr. Sean Ziegeler. Parallel message passing interface (MPI) coding was done by Dr. Min Soe, the supercomputer runs were done by Profs. George and Linda Vahala, and the supercomputer based graphics routines were implemented by Dr. Sean Ziegeler.


Fluid models

Streamline plot of turbulent flow arising from an initial Kida profile simulated with the entropic lattice Boltzmann method
Streamline plot of turbulent flow arising from an initial Kida profile simulated with the entropic lattice Boltzmann method
Finding an analytical solution of hydrodynamic field equations that represents turbulent flow remains the oldest and most prominent of classical grand challenge problems, open now for over 150 years. It has withstood theoretical attempts employing advanced statistical methods and perturbation methods, borrowed from quantum field theory and statistical mechanics [Kraichnan, PRL 31, 744 (1973)]. Kraichnan began his study of turbulence while at Princeton when he and Einstein noticed great similarity between the classical theories of hydrodynamics and gravity. That classical turbulence emerges from a turbulent condensate at small wave numbers evidenced in the kinetic energy spectrum of the fluid velocity.

I study fluid turbulence from the perspectives of information conservation and entropy conservation. As one of my primary goals was to achieve stable and accurate high Reynolds number computational fluid dynamics simulation of classical turbulence, I pioneered the entropic lattice Boltzmann equation method. Navier-Stokes fluid dynamics is represented at the kinetic level by the entropic lattice Boltzmann equation. A mesoscopic simulation of a many-body classical lattice-gas system, whose collision process is represented by the entropic method, is well suited to run on a conventional supercomputer.

Here are some group papers on the entropic Boltzmann equation:

Magnetized fluids

Quantum simulation in the high dissipation regime. No Gibbs oscillations occur since the shock front is spatially resolved. Image taken from Yepez et al., SPIE Quantum Information and Computation III, <b>5815</b> 227 (2005).
Quantum simulation in the high dissipation regime. No Gibbs oscillations occur since the shock front is spatially resolved. Image taken from Yepez et al., SPIE Quantum Information and Computation III, 5815 227 (2005)
Turbulence is plagued by spatiotemporal intermittency involving coherent structures -- structures that are at odds with the simple scale-similarity arguments. Insight into these coherent structures comes by examining simplified models and the simplest model exhibiting Alfvenization (the exchange of fluid and magnetic energies) are the 1D equations for the velocity field and the magnetic field. The MHD equations contain transport coefficients, one for the fluid viscosity and another for the plasma resistivity. The 1D MHD equations can be derived from the full MHD equations when the fluid density length scales are much longer than those of the magnetic field [Thomas, Physics of Fluids 11, 1245 (1968)]. To leading order, this derivation results in a constant density fluid so that the 1D MHD equations form a closed set of equations for only the velocity and magnetic fields. Adding a y-component of the magnetic field gives a 1D model for solar flares [Galtier, The Astrophysical Journal, 521, 483 (1999)].

In Elsasser variables, the 1D MHD equtions can be written in symmetric form. If the shear viscosity and resistivity transport coefficients are equal, then Elsasser's form of the 1D MHD equations reduces to two uncoupled nonlinear Burgers equations. Furthermore, in the limit of zero magnetic field, Elsasser's form of the 1D MHD equations reduces to the Burgers equation for the velocity field. The Burgers equation is a well-known paradigm for Navier-Stokes turbulence and has also been studied extensively for many decades as a simplified model for boundary layer behavior, shock wave formation, mass transport, self-organized criticality and growing interfaces. It is also a test-bed for numerical methods since a general analytic solution exists. In Burgers turbulence, regions where the velocity gradient is negative steepen into shock singularities, while regions where the velocity gradient is positive become smoother. [Hopt, Communications on Pure & Applied Mathematics 3, 201 (1950) and Cole Quarterly of Applied Mathematics 9, 225 (1951)]. However, with the inclusion of the magnetic field, there is now a magnetic back-pressure in 1D MHD equations that is absent from Burgers equation. In particular, the magnetic field will concentrate in regions of the velocity shock, softening the shock front. In the general case when the transport coefficient are not equal, 1D MHD equations are nonintegrable. It is this general case where highly numerical integration of the MHD flow dynamics is needed.

It is possible to study MHD turbulence from the perspectives of information conservation where the magnetized fluid dynamics is governed by a unitary equation of motion. The quantum Boltzmann equation method provides highly-accurate and unconditionally stable numerical predictions of the analytically-nonintegrable MHD flow.

A couple papers on the quantum lattice gas and quantum Boltzmann equation methods are given here:

The vorticity isosurfaces at time step t=50K for the low Prandtl number (Pr=0.3) simulation. Some large scale magnetic structures persist, along with corresponding large scale vortex structures. Image taken from Vahala et al., International Sherwood Fusion Theory Conf., 1D-19 (2008).
The vorticity isosurfaces at time step t=50K for the low Prandtl number (Pr=0.3) simulation. Some large scale magnetic structures persist, along with corresponding large scale vortex structures. Image taken from Vahala et al., International Sherwood Fusion Theory Conf., 1D-19 (2008).
The effect of magnetic disturbance from solar activities can have devastating effects on earth-based electronics and satellites. In its own right, magnetic turbulence is critical to understand. We have investigated a mesoscopic representation of the resistive magnetohydrodynamics (MHD) equations. The MHD equations are closed by the continuity equation, the divergence-free magnetic field condition, and an isothermal equation of state. Unlike standard nonspectral computational fluid dynamics algorithms, the mesoscopic lattice Boltzmann algorithm is not only amenable to massive parallelization but automatically enforces the divergence-free magnetic field condition to machine accuracy.

Here are a few group papers on the entropic Boltzmann equation method for modeling MHD turbulence:

Bose-Einstein condensate superfluids

Tangled quantum vortices arising from 12 initially straight vortex lines within a superfluid in the zero-temperature limit simulated with the quantum lattice gas method. Image taken from my PhD dissertation.
Tangled quantum vortices arising from 12 initially straight vortex lines within a superfluid in the zero-temperature limit simulated with the quantum lattice gas method. Image taken from Yepez, Vahala, and Vahala, SPIE, Quantum Information Theory, 7342 73420M (2009).
In 1955, Feynman proposed that the superfluid turbulent state consists of a tangle of quantized vortices. He also correctly predicted the existence of quantized vortex lines as the source of vorticity flows in Helium II and predicted quantized vortex rings. His conjectures, that have been proven to be correct, launched an extremely active theoretical investigation of low-temperature physics that goes on until this day. This investigation has witnessed a more recent revolution driven by the experimental realization of Bose-Einstein condensates (BECs) in cold atomic vapors. Condensed matter theory of BEC superfluidity and Bardeen-Cooper-Schrieffer (BSC) superconductivity, and the interplay of their mutual crossover, is presently being sorted out, elucidating the role of quantum entanglement in the many-body dynamics underlying quantum fluids. Finally, the great mysteries underlying quantum turbulence are being unraveled, presently one of the most prominent and actively pursued areas in low-temperature physics.

Recently, it has been observed that at very low temperatures <100 mK thermal excitations are unimportant in Helium II and effectively the normal fluid component therefore vanishes in the bulk. Thus, Landau's mutual frictional process no longer operates as a source of dissipation in the bulk region of the quantum fluid in the low-temperature limit so only the superfluid component remains. Yet, at these ultracold temperatures, dissipation has been observed by Walmsley in 2007. That is, even a pure superfluid component of Helium II behaves dissipatively.

The recent discovery of BECs of atomic alkali gases provides a new way to explore pristine superfluid behavior. The dissipation mentioned above occurs in ultracold atomic vapor BECs too, at high wave number (at scales below the healing length ~1nm in atomic BECs). I have explored this phenomenon and understand it as the consequence of nonlinear Kelvin wave cascade dynamics and Kelvin mode coupling to phonon modes that escape into the bulk of the condensate.

Here are a few group papers on the quantum lattice gas method for modeling quantum turbulence:

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Contact Information

Jeffrey Yepez, Ph.D.

Department of Physics and Astronomy

University of Hawaiʻi at Mānoa

Watanabe Hall, Rm 336, 2505 Correa Road

Honolulu, HI 96822

(808) 956-2963 (office)

yepez@hawaii.edu

 

 

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