Authors:

Stuti Shah

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'Turbulence' is a characterization of fluid flow by the occurrence of irregular flow patterns and unpredictable fluid behavior - random fluctuations in the velocity of the fluid at a given time and position, for instance. A turbulent flow is said to be 'homogenous' when all its statistical properties (such as the time average of the velocity of the fluid) are invariant with respect to the position in the fluid, 'incompressible' when the density of the fluid remains constant, and 'isotropic' when the properties associated with a turbulent flow are independent of the direction of progression of the fluid. Turbulence is a phenomenon that occurs abundantly around us, for example, the motion of air around aircraft wingtips and in the oceans. However, it lacks a satisfactory mathematically rigorous description. Studying the idealized concept of homogenous, incompressible, isotropic turbulence is a step toward understanding it. While investigating homogenous, incompressible, isotropic turbulence, we determined that the governing equations (established from the incompressibility condition and the Navier-Stokes equation) involving the velocity of a fluid, its density, and the coefficient of viscosity were correlated and obeyed specific recursion relations. Hence, we solved a simpler dynamical system consisting of coupled differential equations known as the Bogdanov system using a numeric computing environment as our first approach to understanding 'the bootstrap method,' i.e., computing recursive correlation functions. The solutions to the said system could be constructed in the form of what is referred to as a positive semi-definite matrix, which with the aid of Convex Optimization Theory and numerical computation, could let us resolve the unique aspects of its behavior. Following this, an attempt was made to model homogenous, incompressible, isotropic turbulence with the governing system of equations aligning with the Bogdanov system. However, we found an exponential growth in the parameters, and therefore devising a method to condense these infinite variables to a workable number poses our next challenge. Subsequently, we looked at the quantum anharmonic oscillator and how the same approach of obtaining recursion relations between the Hamiltonian and an operator symmetric with respect to the Hamiltonian could be taken to understand this system and, further, a quantum spin system.

Source:

Purdue University / 2023

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Co-authors:

Stuti Shah