Karthikeyan Duraisamy

Stanford University Unstructured (SU 2 ): An open-source integrated computational environment for multi-physics simulation and design

This paper describes the history, objectives, structure, and current capabilities of the Stanford University Unstructured (SU 2 ) tool suite. This computational analysis and design software collection is being developed to solve complex, multi-physics analysis and optimization tasks using arbitrary unstructured meshes, and it has been designed so that it is easily extensible for the solution of Partial Differential Equation-based (PDE) problems not directly envisioned by the authors. At its core, SU 2 is an open-source collection of C++ software tools to discretize and solve problems described by PDEs and is able to solve PDE-constrained optimization problems, including optimal shape design. Although the toolset has been designed with Computational Fluid Dynamics (CFD) and aerodynamic shape optimization in mind, it has also been extended to treat other sets of governing equations including potential flow, electrodynamics, chemically reacting flows, and several others. In our experience, capabilities for computational analysis and optimization have improved considerably over the past two decades. However, the ability to integrate the resulting software packages into coupled multi-physics analysis and design optimization solvers has remained a challenge: the variety of approaches chosen for the independent components of the overall problem (flow solvers, adjoint solvers, optimizers, shape parameterization, shape deformation, mesh adaption, mesh deformation, etc) make it difficult to (a) expand the range of applicability to situations not originally envisioned, and (b) to reduce the overall burden of creating integrated applications. By leveraging well-established object-oriented software architectures (using C++) and by enabling a common interface for all the necessary components, SU 2 is able to remove these barriers for both the beginner and the seasoned analyst. In this paper we attempt to describe our efforts to develop SU 2 as an integrated platform. In some senses, the paper can also be used as a software reference manual for those who might be interested in modifying it to suit their own needs. We carefully describe the C++ framework and object hierarchy, the sets of equations that can be currently modeled by SU 2 , the available choices for numerical discretization, and conclude with a set of relevant validation and verification test cases that are included with the SU 2 distribution. We intend for SU 2 to remain open source and to serve as a starting point for new capabilities not included in SU 2 today, that will hopefully be contributed by users in both academic and industrial environments.

A machine learning strategy to assist turbulence model development

Turbulence modeling in a Reynolds Averaged Navier–Stokes (RANS) setting has traditionally evolved through a combination of theory, mathematics, and empiricism. The problem of closure, resulting from the averaging process, requires an infusion of information into the various models that is often managed in an ad-hoc way or that is focused on particular classes of problems, thus diminishing the predictive capabilities of a model in other flow contexts. In this work, a proof-of-concept of a new data-driven approach of turbulence model development is presented. The key idea in the proposed framework is to use supervised learning algorithms to build a representation of turbulence modeling closure terms. The learned terms are then inserted into a Computational Fluid Dynamics (CFD) numerical simulation with the aim of offering a better representation of turbulence physics. But while the basic idea is attractive, modeling unknown terms by increasingly large amounts of data from higher-fidelity simulations (LES, DNS, etc) or even experiment, the details of how to make the approach viable are not at all obvious. In this work, we investigate the feasibility of such an approach by attempting to reproduce, through a machine learning methodology, the results obtained with the well-established SpalartAllmaras model. In other words, the key question that we seek to answer is the following: Given a number of observations of CFD solutions using the Spalart-Allmaras model (our truth model), can we reproduce those solutions using machine-learning techniques without knowledge of the structure, functional form, and coefficients of the actual model? We discuss the challenges of applying machine learning techniques in a fluid dynamic setting and possible successful approaches. We also explore the potential for machine learning as an enhancement to or replacement for traditional turbulence models. Our results highlight the potential and viability of machine learning approaches to aid turbulence model development.

NUMERICAL SIMULATION OF THE EFFECTS OF SPANWISE BLOWING ON WING-TIP VORTEX FORMATION AND EVOLUTION

A high resolution computational methodology is developed for the solution of the compressible Reynolds-averaged Navier-Stokes equations. This methodology is used to study the effect of spanwise blowing as a method of tip vortex control. The numerical error is reduced by using high order accurate schemes on appropriately refined meshes. For vortex evolution problems, the equations are solved on multiple overset grids that ensure adequate resolution in an efficient manner. Reliable validation of the mean flowfield for the baseline and control configurations is obtained by adding a simple correction to the production term in the Spalart-Allmaras turbulence model. A detailed study of the underlying physics of the effects of spanwise blowing is presented.

High-resolution computational and experimental study of rotary-wing tip vortex formation

The formation and rollup of a tip vortex trailed from a hovering helicopter rotor blade is studied in detail using both computations and measurements. The compressible Reynolds-averaged Navier-Stokes equations are computationally solved on an overset mesh system. The flow measurements are made using stereoscopic particle image velocimetry. The high resolution of both the numerics and the measurements reveal multiple coherent structures in the evolving rotor tip vortex flowfield. Secondary and tertiary vortices that result from crossflow separations near the blade tip are identified. These vortices, along with a part of the trailed wake, are ultimately entrained into the tip vortex that is formed downstream of the blades trailing edge. The simulations clearly demonstrate the resolution required to accurately represent the complex three-dimensional flowfield. The advantage of particle image velocimetry, which has the ability to make planar measurements at a given instant of time, has been fully used to validate the computational fluid dynamics predictions. Even though linear eddy viscosity models are expected to inadequately represent the details of the turbulent quantities, good agreement is seen to be achieved with the particle image velocimetry measurements of the mean flowfield. The various sources of computational and measurement uncertainties are discussed.

Mechanics of viscous vortex reconnection

This work is motivated by our long-standing claim that reconnection of coherent structures is the dominant mechanism of jet noise generation and plays a key role in both energy cascade and fine-scale mixing in fluid turbulence [F. Hussain, Phys. Fluids 26, 2816 (1983); J. Fluid Mech. 173, 303 (1986)]. To shed further light on the mechanism involved and quantify its features, the reconnection of two antiparallel vortex tubes is studied by direct numerical simulation of the incompressible Navier–Stokes equations over a wide range (250–9000) of the vortex Reynolds number, Re (=circulation/viscosity) at much higher resolutions than have been attempted. Unlike magnetic or superfluid reconnections, viscous reconnection is never complete, leaving behind a part of the initial tubes as threads, which then undergo successive reconnections (our cascade and mixing scenarios) as the newly formed bridges recoil from each other by self-advection. We find that the time tR for orthogonal transfer of circulation scales as ...

High resolution wake capturing methodology for hovering rotors

A high-resolution Reynolds-averaged Navier‐Stokes (RANS) solver is applied to study the evolution of tip vortices from rotary blades. The numerical error is reduced by using high-order accurate schemes on appropriately refined meshes. To better resolve the vortex evolution, the equations were solved on multiple overset grids that ensured adequate resolution in an efficient manner. For the RANS closure, a one equation wall-based turbulence model was used with a correction to the production term to account for the stabilizing effects of rotation in the core of the tip vortex. While experimental comparison of the computed vortex structure beyond a few chord lengths downstream of the trailing edge is lacking in the literature, reasonable validation of the vortex velocity profiles is demonstrated up to a distance of 50 chord lengths of evolution for a single-bladed rotor. For the two-bladed rotor case, the tip vortex could be tracked up to two rotor revolutions with minimal diffusion. The accuracy of the computed blade pressures and vortex trajectories confirm that the inflow distribution and blade-vortex interaction are represented correctly. The accuracy achieved in the validation studies establishes the viability of the methodology as a reliable tool that can be used to predict vortex evolution and the aerodynamic performance of hovering rotors.

Interactional aerodynamics and acoustics of a hingeless coaxial helicopter with an auxiliary propeller in forward flight

The aerodynamics and acoustics of a generic coaxial helicopter with a stiff main rotor system and a tail-mounted propulsor are investigated using Browns Vorticity Transport Model. In particular, the model is used to capture the aerodynamic interactions that arise between the various components of the configuration. By comparing the aerodynamics of the full configuration of the helicopter to the aerodynamics of various combinations of its sub-components, the influence of these aerodynamic interactions on the behaviour of the system can be isolated. Many of the interactions follow a simple relationship between cause and effect. For instance, ingestion of the main rotor wake produces a direct effect on the unsteadiness in the thrust produced by the propulsor. The causal relationship for other interdependencies within the system is found to be more obscure. For instance, a dependence of the acoustic signature of the aircraft on the tailplane design originates in the changes in loading on the main rotor that arise from the requirement to trim the load on the tailplane that is induced by its interaction with the main rotor wake. The traditional approach to the analysis of interactional effects on the performance of the helicopter relies on characterising the system in terms of a network of possible interactions between the separate components of its configuration. This approach, although conceptually appealing, may obscure the closed-loop nature of some of the aerodynamic interactions within the helicopter system. It is suggested that modem numerical simulation techniques may be ready to supplant any overt reliance on this reductionist type approach and hence may help to forestall future repetition of the long history of unforeseen, interaction-induced dynamic problems that have arisen in various new helicopter designs.

Analysis of Wind Tunnel Wall Interference Effects on Subsonic Unsteady Airfoil Flows

In this work, the effect of wall interference on steady and oscillating airfoils in a subsonic wind tunnel is studied. A variety of approaches including linear theory, compressible inviscid and viscous computations, and experimental data are considered. Integral transform solutions of the linearized potential equations show an augmentation of the lift magnitude for steady flows when the wall is close to the airfoil surface. For oscillating airfoils, lift augmentation is accompanied by a significant change in the phase of the lift response. Idealized compressible Euler calculations are seen to corroborate the linear theory under conditions that are sufficiently away from acoustic resonance. Further, the theory compares well with compressible Reynolds-averaged Navier-Stokes calculations and experimental measurements over a wide range of attached flows at subsonic Mach numbers. The present methodology can thus be used to predict wall interference effects and also to help extrapolate linear and nonlinear (dynamic stall) wind tunnel data to free-air conditions.

Concepts and Application of Time-Limiters to High Resolution Schemes

A new class of implicit high-order non-oscillatory time integration schemes is introduced in a method-of-lines framework. These schemes can be used in conjunction with an appropriate spatial discretization scheme for the numerical solution of time dependent conservation equations. The main concept behind these schemes is that the order of accuracy in time is dropped locally in regions where the time evolution of the solution is not smooth. By doing this, an attempt is made at locally satisfying monotonicity conditions, while maintaining a high order of accuracy in most of the solution domain. When a linear high order time integration scheme is used along with a high order spatial discretization, enforcement of monotonicity imposes severe time-step restrictions. We propose to apply limiters to these time-integration schemes, thus making them non-linear. When these new schemes are used with high order spatial discretizations, solutions remain non-oscillatory for much larger time-steps as compared to linear time integration schemes. Numerical results obtained on scalar conservation equations and systems of conservation equations are highly promising.

High-resolution simulations of parallel blade-vortex interactions

The physics of a parallel blade―vortex interaction is studied numerically and the predicted pressure and acoustic results are compared with experimental measurements. A high-resolution solution of the compressible Euler equations is performed on structured overset meshes. Initially, a two-dimensional airfoil-vortex interaction is studied for both a case where the vortex misses the blade and a case of direct impact. The vortex is initiated in the flow as a perturbation to the freestream conditions and is free to evolve, thus allowing for the deformation of the vortex as it interacts with the blade to be studied. The simulation is seen to accurately reproduce the experimental results and the emission of the acoustic waves from the airfoil surface is observed in detail. Acoustic energy generated by the interaction is seen to primarily radiate from the leading-edge section of the airfoil with a weaker contribution coming from the trailing edge. The simulations are then extended to three-dimensional moving overset meshes where the vortex generation and convection is also resolved. The numerical methodology is seen to accurately preserve the vortex strength and accurately reproduce the experimentally measured blade surface pressures and acoustics. The computations presented here face similar challenges to that encountered in the simulation of realistic helicopter blade—vortex interaction, but the computational costs are such that the solutions can be well resolved, and comprehensively validated using moderate resources.