Omar M. Knio

Numerical Challenges in the Use of Polynomial Chaos Representations for Stochastic Processes

This paper gives an overview of the use of polynomial chaos (PC) expansions to represent stochastic processes in numerical simulations. Several methods are presented for performing arithmetic on, as well as for evaluating polynomial and nonpolynomial functions of variables represented by PC expansions. These methods include {Taylor} series, a newly developed integration method, as well as a sampling-based spectral projection method for nonpolynomial function evaluations. A detailed analysis of the accuracy of the PC representations, and of the different methods for nonpolynomial function evaluations, is performed. It is found that the integration method offers a robust and accurate approach for evaluating nonpolynomial functions, even when very high-order information is present in the PC expansions.

Uncertainty quantification in reacting-flow simulations through non-intrusive spectral projection

Abstract A spectral formalism has been developed for the “non-intrusive” analysis of parametric uncertainty in reacting-flow systems. In comparison to conventional Monte Carlo analysis, this method quantifies the extent, dependence, and propagation of uncertainty through the model system and allows the correlation of uncertainties in specific parameters to the resulting uncertainty in detailed flame structure. For the homogeneous ignition chemistry of a hydrogen oxidation mechanism in supercritical water, spectral projection enhances existing Monte Carlo methods, adding detailed sensitivity information to uncertainty analysis and relating uncertainty propagation to reaction chemistry. For 1-D premixed flame calculations, the method quantifies the effect of each uncertain parameter on total uncertainty and flame structure, and localizes the effects of specific parameters within the flame itself. In both 0-D and 1-D examples, it is clear that known empirical uncertainties in model parameters may result in large uncertainties in the final output. This has important consequences for the development and evaluation of combustion models. This spectral formalism may be extended to multidimensional systems and can be used to develop more efficient “intrusive” reformulations of the governing equations to build uncertainty analysis directly into reacting flow simulations.

Room-temperature soldering with nanostructured foils

Self-propagating formation reactions in nanostructured multilayer foils provide rapid bursts of heat and can act as local heat sources to melt solder layers and join materials. This letter describes the room-temperature soldering of stainless steel specimens using freestanding, nanostructured Al/Ni foils. The products, heats, and velocities of the reactions are described, and the microstructure and the mechanical properties of the resulting joints are characterized. A tensile shear strength of 48 MPa was measured for the reactive foil joints, compared to 38 MPa for conventional joints. Both numerical predictions and infrared measurements show limited heat exposure to the components during reactive joining.

Joining of stainless-steel specimens with nanostructured Al/Ni foils

We describe the joining of stainless-steel specimens at room temperature using free-standing Al/Ni foils as local heat sources for melting AuSn solder layers. The foils contain many nanoscale layers of Al and Ni that react exothermically, generating a self-propagating reaction. The heats, velocities, and products of the reactions are described, and the microstructure and the mechanical properties of the resulting joints are characterized. Increasing the foil thickness, and thereby increasing the total heat released, can improve the strength of the joints until foil thickness reaches 40 μm. For thicker foils, the shear strength is almost constant at 48 MPa, compared to 38 MPa for conventional solder joints. The higher strength is due to finer microstructures in the solder layers of reactive joints. A numerical study of heat transfer during reactive joining and experimental results suggest that the solder layers need to melt completely and remain molten for at least 0.5 ms to form a strong joint.

Reactive nanostructured foil used as a heat source for joining titanium

We have joined titanium alloy (Ti-6Al-4V) specimens at room temperature and in air by using free-standing nanostructured Al∕Ni multilayer foils to melt a silver-based braze. The foils are capable of undergoing self-sustaining exothermic reactions and thus act as controllable local heat sources. By systematically controlling the properties of the foils and by numerically modeling the reactive joining process, we are able to conclude that the temperatures reached by the foils during reaction are critical in determining the success of joining when using higher melting temperature braze layers.

Numerical study of a three-dimensional vortex method

Abstract A three-dimensional vortex method based on the discretization of the vorticity field into vortex vector elements of finite spherical cores is constructed for the simulation of inviscid incompressible flow. The velocity is obtained by summing the contribution of individual elements using the Biot-Savart law desingularized according to the vorticity cores. Vortex elements are transported in Lagrangian coordinates, and vorticity is redistributed, when necessary, among larger number of elements arranged along its direction. The accuracy and convergence of the method are investigated by comparing numerical solutions to analytical results on the propagation and stability of vortex rings. Accurate discretization of the initial vorticity field is shown to be essential for the prediction of the linear growth of azimuthal instability waves on vortex rings. The unstable mode frequency, growth rate and shape are in agreement with analytical results. The late stages of evolution of the instability show the generation of small scales in the form of bair-pin vortex structures. The behavior of the turbulent vortex ring is in good qualitative agreement with experimental data.

Effect of varying bilayer spacing distribution on reaction heat and velocity in reactive Al/Ni multilayers

Self-propagating reactions in Al/Ni nanostructured multilayer foils are examined both experimentally and computationally to determine the impact of variations in reactant spacing on reaction properties. Heats of reaction and reaction velocities have been characterized as a function of average bilayer spacing for sputter-deposited, single-bilayer foils (having a uniform bilayer spacing) and for dual-bilayer foils (having two different bilayer spacings that are labeled thick and thin). In the latter case, the spatial distribution of the thick and thin bilayers is found to have a significant effect on reaction velocity, with coarse distributions leading to much higher reaction velocities than fine distributions. Numerical simulations of reaction velocity match experimental data well for most spatial distributions, with the exception of very coarse distributions or distributions containing very small bilayer spacings. A simple model based on thermal diffusivities and reaction velocities is proposed to predict when the spatial distribution of thick and thin bilayers becomes coarse enough to affect reaction velocity. This combination of experiment and simulation will allow for more effective design and prediction of reaction velocities in both sputter-deposited and mechanically processed reactive materials with variable reactant spacings.

Effect of reactant and product melting on self-propagating reactions in multilayer foils

The evolution of self-propagating reactions along nanostructured multilayer foils is analyzed computationally. A simplified physical model is used that combines a two-dimensional diffusion equation for the atomic concentration with a quasi-one-dimensional form of the energy equation which accounts for the melting of the reactants and products. The model thus generalizes previous formulations which have ignored melting effects. The computations are used to predict the evolution of self-propagating fronts in Ni/Al foils, and analyze the dependence of these fronts on the foil parameters. In particular, the results indicate that melting substantially affects the properties of the unsteady reactions, and generally results in an appreciable reduction of the average front speed.

A Study of Flame Observables in Premixed Methane - Air Flames

The use of particular experimental flame observables as flame markers, and as measures of flame burning and heal release rates requires the establishment of robust correlations between the particular observable and the rate in question. In this work, we use a compilation of results from numerical computations of the interaction of a premixed methane flame with a two-dimensional counter-rotating vortex pair using detailed kinetics. The data set involves the use of two different chemical mechanisms, a two-fold variation in flow time scales, and the examination of both stoichiometric and rich methane flames. Correlations between a number of flame observables and heat release and burning rates are examined. We study HCO, ▽·v, OH, CH, CO, CH3, CH2O, CH2*, and C2H2, as well as various concentration products (surrogates for production rates) including [OH][CH2O], [OH][CH4], and [OH][CO]. Other concentration products expected to relate to chemiluminescent observables such as CH*, OH* and CO2*, are also studied. H...

Spectral stochastic uncertainty quantification in chemical systems

Uncertainty quantification (UQ) in the computational modelling of physical systems is important for scientific investigation, engineering design, and model validation. We have implemented an ‘intrusive’ UQ technique in which (1) model parameters and field variables are modelled as stochastic quantities, and are represented using polynomial chaos (PC) expansions in terms of Hermite polynomial functions of Gaussian random variables, and (2) the deterministic model equations are reformulated using Galerkin projection into a set of equations for the time evolution of the field variable PC mode strengths. The mode strengths relate specific parametric uncertainties to their effects on model outputs. In this work, the intrusive reformulation is applied to homogeneous ignition using a detailed chemistry model through the development of a reformulated pseudospectral chemical source term. We present results analysing the growth of uncertainty during the ignition process. We also discuss numerical issues pertaining to the accurate representation of uncertainty with truncated PC expansions, and ensuing stability of the time integration of the chemical system.