Physics of Turbulence

Understanding transition from laminar to turbulent flow is a long standing challenge in fluid mechanics. We are interested in exploring various aspects of turbulence, in all its manifestations. We are trying to understand turbulence from the point of view of Theory, Computations and Experiments. It's in our opinion , that each of these techniques have their own inherent limitations, and a clear understanding of turbulence requires a strong inter play between all these three methods. As of now there is no complete explanation for transition to turbulence in planar Couette flow as this flow is proved to be linearly stable for all Reynolds number in the infinite dimensional case. We propose a completely new HOT route to transition to turbulence where in the flow has no instabilities (not even non linear) at all. However due to huge amplification of external disturbances (deterministic and stochastic) with time scaling with Reynolds Number and energy scaling with square of Reynolds number the huge transients and the wandering of the trajectory in a complicated phase space creates the complicated turbulent flow pattern. This also accounts for the lack of experimental evidence of Ruelle-Takens route to transition to turbulence in open channel flows. We have shown that Stream-wise boundary layer vortices (a primary turbulence production and sustaining mechanisms), which have long been a mystery as to why they should exist (as they are not the eigen functions of the operator), come naturally on our framework of generalized hydro dynamic stability.  We have also shown that there is a steep falling of Hankel singular values for channel flows and hence very low order models exist and these are high gain systems. We have also shown that the operator in Taylor-Couette flow turbulence is non-normal and huge amplifications can exist even in this flow like channel flows. We also argue that there is a fundamental difference between transition scenarios in Rayleigh-Benard convection, Taylor-Couette flow and Boundary layer flow by connecting it to the Highly Optimized Tolerance. We will argue that there are some fundamental limitation (conservation of robustness) in understanding and controlling transition to turbulence. We are also working on large eddy simulations (LES) using Kalman filtering etc.

Generalized Hydrodynamic Stability

Many of traditional hydrodynamic stability notions had been developed on relatively superficial model for uncertainty and no good theory of Hydrodynamic stability has resulted. However understanding uncertainty is the key to many fluids problem and we lay the broad mathematical foundations for stability/instability under uncertainty. In this new framework we take into account the structure of the operators and structure of the uncertainty. We treat the uncertainty in initial condition, model, numerical error, experiments, BC and various inputs like stochastic noise, periodic-a periodic signals etc. We propose various generalized stability and instability measures and not just eigen values taking the input-output view of fluids. We have shown that even though the linearization is stable the existence of large transients (H2 norm), large frequency singular value plots (Hinfinity norm), small stability margins with respect to un modeled dynamics, non normal operators and large amplification of disturbances are very important in predicting the response of Navier-Stokes equations. These effects will have profound implications in transition prediction and control. Even though we are right now focussing on transition to turbulence in channel flows, the potential applications include Aero-Acoustics, Shear layers, Bluff body flows, Compressible flows etc. In future, the plan is to do non-linear stability using Arnold's Energy-Casimir method, energy-momentum method, etc.

The behavior of acoustic fields in ducts is a problem of considerable interest. For instance, there is a need to develop an understanding of the manner in which a mean axial temperature gradient in the presence of mean flow effects the propagation of sound waves and the stability of small amplitude disturbances in a duct. Such understanding will improve existing capabilities for controlling combustion instabilities in propulsion and power generation systems, designing pulse combustors and automotive mufflers, analyzing the behavior of resonating thermal systems, and measuring the impedance's of high temperature systems like flames. Here we derived some exact solutions in terms of special functions for the sound propagation in ducts with temperature gradient, mean flow effects etc. The problem of Aero-Acoustics is also of particular interest to me.

Here we have developed three new methods for simulation of compressible flows. They are Kinetic Smooth Particle Hydrodynamics (KSPH), Viscous Kinetic Flux Vector Splitting (KFVS) and Viscous Peculiar Velocity Upwinding (PVU) methods. Kinetic Smooth Particle Hydrodynamics method (KSPH) is based on a mixture of Lagrangian description of flow and kinetic treatmentof pressure gradient terms. The advective terms in Euler equations of fluid dynamics are simulated by movement of a finite number of particles while the pressure gradient terms are distretized by using concepts from Peculiar velocity based upwinding and Least square kinetic upwind methods. This new method is robust, versatile, grid free, a particle based method, needs no addition of artificial viscosity and works on arbitrary distribution of particles. This method has been tested on 1-D shock tube problem. Kinetic Flux Vector Splitting (KFVS) and Peculiar Velocity Upwinding (PVU) methods are Boltzmann schemes for the numerical simulation of viscous compressible flows. They are based on Chapman-Enskog distribution. Both the methods have been tested in 2-D for shock wave-boundary layer interaction. The simulations agree well with experiments. We are also interested in applying our Generalized Hydrodynamic Stability theory to other compressible flow problems like compressible shear layers, compressible boundary layers, Rayleigh-Benard flow; compressible vortices like Batchelors Dipole, Kerchoff rotating vortex, Patches etc.

A close observation of turbulence reveals (universality phenomena) some features common to second order phase transition, directed percolation processes and lattices of coupled mappings etc.  Here we explore the relationship between highly streamlined high shear  flows and Highly Optimized Tolerance (HOT), which was recently introduced as a conceptual framework to study the fundamental  nature of complexity. HOT emphasizes certain features of complex systems: 1) highly structured, non generic, self dissimilar internal  configurations 2) robust, yet fragile external behavior 3) power laws.  Essentially this theory focusses on measure zero perturbations  unlike the traditional theories like Self Organized Criticality (SOC) of Bak or Edge of Chaos of Kauffman where in they study generic  configurations. Here we propose a HOT route to turbulence, where streamlining eliminates the bifurcation transitions to turbulence  that occur in bluff body flows, resulting in a flow which is robustly stable to arbitrary changes in Reynolds number but highly fragile in amplifying any small perturbation.

We are interested in understanding, modeling and controlling uncertainty in various fluid problems with emphasis on transition to turbulence. For example we are interested in accurately modeling the uncertainty in real experiment and take this into account in simulations and theory. We are also interested in understanding how new uncertainties are created and evolved after discretization of equations in numerical simulations. This has lots of applications to LES where in as of now there is no accurate way understanding the role of uncertainty in various models.  Currently we are working on LES with Kalman filtering and Weiner filtering etc.

Here we are interested in applications of Robust, Nonlinear and Modern control theories to fluids. The traditional transition control strategies on aero plane (civil and military) wings are blowing and suction based on some empirical criteria. These passive techniques and empirical criterion are not very effective in preventing transition and hence were unable to reduce drag etc. The goal is to predict and control transition to turbulence using robust and nonlinear control theories. This research has enormous potential application in real life transition prediction and control on aero plane wings in particular. The techniques being developed here can be applied to other general fluid problems like shear layers, bluff body flows, jets etc. which have various industrial applications.

We are interested in Geometric Mechanics, Functional analysis, Numerical analysis and Dynamical Systems theory. We are also interested in traditional applied mathematics techniques like PDE, Perturbation techniques, Asymptotics, Special functions.

Central to understanding turbulence is understanding the dynamics of vortices as this might tell us the underlying mechanisms in real world turbulence. We are also interested in understanding the sources of vorticity in  a given flow filed as this will help us to control the flow by placing controllers at strategic places and also understand the physics of the problem. We are studying the stochastic evolution  of elliptical vortex in a turbulent flow field using Fokker-Plank equation unlike the traditional deterministic evolutions of Saffman, Moore, Kida etc. Understanding this problem may shed light on vortex based sub-grid stress models in LES(based on discussion with Dale Pullin). The ultimate aim is to be able to write an functional Fokekr-Plank equation for vorticity equation and study turbulence analogous to Hopf's functional formulation or Lundgren's pdf formulation of Navier-Stokes equations. Also we are looking at homogenization of vorticity equation etc. so as to capture accurate large scale dynamics. We are also studying some exact solutions (generalizations of incompressible vortices) of compressible vortices like Batchelors Dipole, Kerchoff rotating vortex, Patches etc. and their stability characteristics (finite time).

My interest in this field was sparkled by the excellent talk of George Homsy in Galcit Fluids Seminar, and discussions with him later on. The subject of motion of long bubble is thin tubes is central to many MEMS devices. Most of the past theoretical work in this area is studied assuming the wall is smooth. How ever when the dimensions of the walls are so small we argue that the roughness cannot be neglected and can change dramatically the physics of the problem. We went on to derive the modified stochatic Landau-Levich equation governing the bubble movement in rough walls. Exact soln and other aspects are currently under investigation.

The primary interest in this area is developing new schemes or improve existing schemes or use existing schemes to aid understanding the physics of the problem. Here we have developed new scheme Kinetic Smooth Particle Hydrodynamics method (KSPH) based on a mixture of Lagrangian description of flow and kinetic treatmentof pressure gradient terms. The advective terms in Euler equations of fluid dynamics are simulated by movement of a finite number of particles while the pressure gradient terms are distretized by using concepts from Peculiar velocity based upwinding and Least square kinetic upwind methods. This new method is robust, versatile, grid free, a particle based method, needs no addition of artificial viscosity and works on arbitrary distribution of particles. This method has been tested on 1-D shock tube problem. We also developed two new Boltzmann schemes for the numerical simulation of viscous compressible flows. They are Kinetic Flux Vector Splitting (KFVS) and Peculiar Velocity Upwinding (PVU) methods  for viscous flows using Chapman-Enskog distribution. Both the methods have been tested in 2-D for shock wave-boundary layer interaction. The simulations agree well with experiments. We are using right now spectral methods for the turbulence computations. In future, I would like to work on multi symplectic integrators for fluid mechanics problems. These integrators have many nice properties like, preservation of continum symmetries at discrete level too, accurate long term behavior of solutions, etc. The main idea behind these methods is to use the discrete analogue of the continum Lagrangian-Hamiltonian mechanics principles. So, here one writes the discrete Lagrangian and Hamiltonian, and apply the discrete varational principle to get discrete Euler-lagrange equations. This might be the key to answering some of the important questions in turbulence.

Central to any numerical simulation is the problem of representing a given partial differential equation by finite set of ordinary differential equations. This process is achieved through some projection technique. However these finite number of retained modes is very large and it is of considerable interest to project the dynamics of these large number of ordinary differential equations onto a proper low dimensional subspace on which most of the important dynamics evolve. Here we introduced new techniques for getting simple models for fluids which takes into account the underlying input-output properties of fluids and has considerable advantages like rigorous error bounds, transparent physics etc. The main idea behind these methods is deleting the weakly controllable and weakly observable states of the system after the controllability and the observability gramians of the system are aligned through a similarity transformation. The relative importance of a state in the input-output behavior of the system is given by the corresponding Hankel singular value. Application of these ideas include Turbulence, Aero elasticity, Flow-Structure interaction, Combustion etc.

Though lot of turbulence data has been collected in the last 100 or so years the very difficulty experimental conditions for direct study of developed and developing turbulence restricted the quality of the data. Its our belief that getting highly resolved, accurate, 3-dimensional, global, time evolving data is the key to turbulence. Hence we investigate using state of the art Digital Particle Image Velocimetry and digital shear stress sensors. We are also working on improving the capabilities and error management in DPIV. In the experiment we are concerned with questions regarding maximum possible amplification rates in the range of sub critical Reynolds numbers and application of dynamical systems theory. We would like to quantify the contribution of large scale and small-scale structures to the flow dynamics and their properties. We will also study birth, growth and destruction of structures (finite horizon notions of coherent structures) and the events that they lead to. We would also clearly take into account the strength and character of uncertain environment in analyzing the onset of transition.

My interest in this field started in my undergrad sophomore year. Leonardo Pisano or Fibonacci first introduced the famous Fibonacci numbers in 1202. These numbers pop up in very peculiar places where we hardly expect them. Then came the Lucas numbers in 1878. Here we developed the modified Fibonacci numbers (m-numbers) which have some very interesting properties. We showed the relation between m-numbers and Fibonacci  numbers, Lucas numbers, Golden ratio, Euler and Bernouli polynomials etc.  The generating function for m-numbers is derived and used to define the most general recurrence relation. Application of m-numbers to Q-matrix and connection between m-triangle, Pascals triangle, Hoggat steps and Array of Hope are also studied.