Analysis of a parallelized nonlinear elliptic boundary value problem solver with application to reacting flows

A parallelized finite difference code based on the Newton method for systems of nonlinear elliptic boundary value problems in two dimensions is analyzed in terms of computational complexity and parallel efficiency. An approximate cost function depending on 15 dimensionless parameters is derived for algorithms based on stripwise and boxwise decompositions of the domain and a one-to-one assignment of the strip or box subdomains to processors. The sensitivity of the cost functions to the parameters is explored in regions of parameter space corresponding to model small-order systems with inexpensive function evaluations and also a coupled system of nineteen equations with very expensive function evaluations. The algorithm was implemented on the Intel Hypercube, and some experimental results for the model problems with stripwise decompositions are presented and compared with the theory. In the context of computational combustion problems, multiprocessors of either message-passing or shared-memory type may be employed with stripwise decompositions to realize speedup of O(n), where n is mesh resolution in one direction, for reasonable n

A comparison of computational and experimental lift-off heights of coflow laminar diffusion flames

As a sensitive marker of changes in flame structure, the number densities of excited-state CH (denoted CH*), and excited-state OH (denoted OH*) are imaged in coflow laminar diffusion flames. Measurements are made both in normal gravity and on the NASA KC-135 reduced-gravity aircraft. The spatial distribution of these radicals provides information about flame structure and lift-off heights that can be directly compared with computational predictions. Measurements and computations are compared over a range of buoyancy and fuel dilution levels. Results indicate that the lift-off heights and flame shapes predicted by the computations are in excellent agreement with measurement for both normal gravity (1g) and reduced gravity flames at low dilution levels. As the fuel mixture is increasingly diluted, however, the 1g lift-off heights become underpredicted. This trend continues until the computations predict stable flames at highly dilute fuel mixtures beyond the 1g experimental blow-off limit. To better understand this behavior, an analysis was performed, which indicates that the lift-off height is sensitive to the laminar flame speed of the corresponding premixed mixture at the flame edge. By varying the rates of two key "flame speed" controlling reactions, we were able to modify the predicted lift-off heights so as to be in closer agreement with the experiments. The results indicate that reaction sets that work well in low dilution systems may need to be modified to accommodate high dilution flames

A Computational and Experimental Study of Coflow Laminar Methane/Air Diffusion Flames: Effects of Fuel Dilution, Inlet Velocity, and Gravity

The influences of fuel dilution, inlet velocity, and gravity on the shape and structure of laminar coflow CH4-air diffusion flames were investigated computationally and experimentally. A series of nitrogen-diluted flames measured in the Structure and Liftoff in Combustion Experiment (SLICE) on board the International Space Station was assessed numerically under microgravity (mu g) and normal gravity (1g) conditions with CH4 mole fraction ranging from 0.4 to 1.0 and average inlet velocity ranging from 23 to 90 cm/s. Computationally, the MC-Smooth vorticity-velocity formulation was employed to describe the reactive gaseous mixture, and soot evolution was modeled by sectional aerosol equations. The governing equations and boundary conditions were discretized on a two-dimensional computational domain by finite differences, and the resulting set of fully coupled, strongly nonlinear equations was solved simultaneously at all points using a damped, modified Newton's method. Experimentally, flame shape and soot temperature were determined by flame emission images recorded by a digital color camera. Very good agreement between computation and measurement was obtained, and the conclusions were as follows. (1) Buoyant and nonbuoyant luminous flame lengths are proportional to the mass flow rate of the fuel mixture; computed and measured nonbuoyant flames are noticeably longer than their 1g counterparts; the effect of fuel dilution on flame shape (i.e., flame length and flame radius) is negligible when the flame shape is normalized by the methane flow rate. (2) Buoyancy-induced reduction of the flame radius through radially inward convection near the flame front is demonstrated. (3) Buoyant and nonbuoyant flame structure is mainly controlled by the fuel mass flow rate, and the effects from fuel dilution and inlet velocity are secondary

A Numerical and Experimental Study of Coflow Laminar Diffusion Flames: Effects of Gravity and Inlet Velocity

In this work, the influence of gravity, fuel dilution, and inlet velocity on the structure, stabilization, and sooting behavior of laminar coflow methane-air diffusion flames was investigated both computationally and experimentally. A series of flames measured in the Structure and Liftoff in Combustion Experiment (SLICE) was assessed numerically under microgravity and normal gravity conditions with the fuel stream CH4 mole fraction ranging from 0.4 to 1.0. Computationally, the MC-Smooth vorticity-velocity formulation of the governing equations was employed to describe the reactive gaseous mixture; the soot evolution process was considered as a classical aerosol dynamics problem and was represented by the sectional aerosol equations. Since each flame is axisymmetric, a two-dimensional computational domain was employed, where the grid on the axisymmetric domain was a nonuniform tensor product mesh. The governing equations and boundary conditions were discretized on the mesh by a nine-point finite difference stencil, with the convective terms approximated by a monotonic upwind scheme and all other derivatives approximated by centered differences. The resulting set of fully coupled, strongly nonlinear equations was solved simultaneously using a damped, modified Newton's method and a nested Bi-CGSTAB linear algebra solver. Experimentally, the flame shape, size, lift-off height, and soot temperature were determined by flame emission images recorded by a digital camera, and the soot volume fraction was quantified through an absolute light calibration using a thermocouple. For a broad spectrum of flames in microgravity and normal gravity, the computed and measured flame quantities (e.g., temperature profile, flame shape, lift-off height, and soot volume fraction) were first compared to assess the accuracy of the numerical model. After its validity was established, the influence of gravity, fuel dilution, and inlet velocity on the structure, stabilization, and sooting tendency of laminar coflow methane-air diffusion flames was explored further by examining quantities derived from the computational results

The hydrogen–air burning rate near the lean flammability limit

This paper investigates the inner structure of the thin reactive layer of hydrogen–air fuellean deflagrations close to the flammability limit. The analysis, which employs seven elementary reactions for the chemistry description, uses the ratio of the characteristic radical and fuel concentrations as a small asymptotic parameter, enabling an accurate analytic expression for the resulting burning rate to be derived. The analysis reveals that the steady-state assumption for chemical intermediaries, applicable on the hot side of the reactive layer, fails, however, as the crossover temperature is approached, providing a nonnegligible higher-order correction to the burning rate. The results can be useful, for instance, in future investigations of hydrogen deflagration instabilities near the lean flammability limit

Efficient computation of high index Sturm-Liouville eigenvalues for problems in physics

Finding the eigenvalues of a Sturm-Liouville problem can be a computationally challenging task, especially when a large set of eigenvalues is computed, or just when particularly large eigenvalues are sought. This is a consequence of the highly oscillatory behaviour of the solutions corresponding to high eigenvalues, which forces a naive integrator to take increasingly smaller steps. We will discuss some techniques that yield uniform approximation over the whole eigenvalue spectrum and can take large steps even for high eigenvalues. In particular, we will focus on methods based on coefficient approximation which replace the coefficient functions of the Sturm-Liouville problem by simpler approximations and then solve the approximating problem. The use of (modified) Magnus or Neumann integrators allows to extend the coefficient approximation idea to higher order methods

An Experimental and Computational Study on Soot Formation in a Coflow Jet Flame Under Microgravity and Normal Gravity

Upon the completion of the Structure and Liftoff in Combustion Experiment (SLICE) in March 2012, a comprehensive and unique set of microgravity coflow diffusion flame data was obtained. This data covers a range of conditions from weak flames near extinction to strong, highly sooting flames, and enabled the study of gravitational effects on phenomena such as liftoff, blowout and soot formation. The microgravity experiment was carried out in the Microgravity Science Glovebox (MSG) on board the International Space Station (ISS), while the normal gravity experiment was performed at Yale utilizing a copy of the flight hardware. Computational simulations of microgravity and normal gravity flames were also carried out to facilitate understanding of the experimental observations. This paper focuses on the different sooting behaviors of CH4 coflow jet flames in microgravity and normal gravity. The unique set of data serves as an excellent test case for developing more accurate computational models.Experimentally, the flame shape and size, lift-off height, and soot temperature were determined from line-of-sight flame emission images taken with a color digital camera. Soot volume fraction was determined by performing an absolute light calibration using the incandescence from a flame-heated thermocouple. Computationally, the MC-Smooth vorticity-velocity formulation was employed to describe the chemically reacting flow, and the soot evolution was modeled by the sectional aerosol equations. The governing equations and boundary conditions were discretized on an axisymmetric computational domain by finite differences, and the resulting system of fully coupled, highly nonlinear equations was solved by a damped, modified Newtons method. The microgravity sooting flames were found to have lower soot temperatures and higher volume fraction than their normal gravity counterparts. The soot distribution tends to shift from the centerline of the flame to the wings from normal gravity to microgravity

Determining Why Before How: Crafting Mission, Vision, and Values Statements to Facilitate Undergraduate Music Theory Curriculum Reimagining

The Department of Music Theory at the Peabody Conservatory of the Johns Hopkins University recently joined many other institutions across North America in reimagining our undergraduate curriculum. Following practices more commonly found in the nonprofit sector, we began this undertaking by crafting mission, vision, and values statements. These newly articulated ideals for our department then served as a polestar, effectively guiding the rest of the curriculum review and revision. The process we effected created an environment that facilitated creative thinking while simultaneously engendering clear consensus among our colleagues. In this paper, we—as co-chairs of the committee charged with reviewing the undergraduate music theory curriculum—discuss the steps that we followed in this process, as well as the results. We believe that this way of organizing curriculum review provides a useful framework that other schools can mirror when they reassess their curricula. Thinking critically about why we teach specific topics helped us to clarify exactly what should be required in the new curriculum and what could be optional for the students. As a case study, we present our mission, vision, and values statements, as well as the new curriculum that these led us to create. In so doing, we illustrate how adopting this approach from the nonprofit sector and applying it to curriculum reform guided us in unexpected directions that have been broadly supported within the Department and the Conservatory as a whole