Mauro Valorani

CSP analysis of a transient flame-vortex interaction: time scales and manifolds

Abstract The interaction of a two-dimensional counter-rotating vortex-pair with a premixed methane-air flame is analyzed with the Computational Singular Perturbation (CSP) method. It is shown that, as the fastest chemical time scales become exhausted, the solution is attracted towards a manifold, whose dimension decreases as the number of exhausted time scales increases. A necessary condition for a chemical time scale to become exhausted is that it must be much faster than the locally prevailing diffusion and convection time scales. Downstream of the flame, the hot products are in a regime of near-equilibrium, characterized by a large number of exhausted fast chemical time scales and the development of a low dimensional manifold, where the dynamics are locally controlled by slow transport processes and slow kinetics. In the flame region, where intense chemical and transport activity takes place, the number of exhausted chemical time scales is relatively small. The manifold has a large dimension and the driving time scale is set by chemical kinetics. In the cold flow region, where mostly reactants are present, the flow regime can be described as frozen, as the active chemical time scales are much slower than the diffusion and convection time scales; the driving scale set by diffusion. The algebraic relations among the elementary rates, which describe the manifold, are discussed along with a classification of the unknowns in three classes: i) CSP radicals; ii) trace; and, iii) major species. It is established that the optimal CSP radicals must be: i) strongly affected by the exhausted fast chemical time scales; and, ii) significant participants in the algebraic relations describing the manifold. The identification of CSP radicals, trace and major species, is a prerequisite for simplification or reduction of chemical kinetic mechanisms.

An efficient iterative algorithm for the approximation of the fast and slow dynamics of stiff systems

The relation between the iterative algorithms based on the computational singular perturbation (CSP) and the invariance equation (IE) methods is examined. The success of the two methods is based on the appearance of fast and slow time scales in the dynamics of stiff systems. Both methods can identify the low-dimensional surface in the phase space (slow invariant manifold, SIM), where the state vector is attracted under the action of fast dynamics. It is shown that this equivalence of the two methods can be expressed by simple algebraic relations. CSP can also construct the simplified non-stiff system that models the slow dynamics of the state vector on the SIM. An extended version of IE is presented which can also perform this task. This new IE version is shown to be exactly similar to a modified version of CSP, which results in a very efficient algorithm, especially in cases where the SIM dimension is small, so that significant model simplifications are possible.

The G-Scheme: A framework for multi-scale adaptive model reduction

The numerical solution of mathematical models for reaction systems in general, and reacting flows in particular, is a challenging task because of the simultaneous contribution of a wide range of time scales to the system dynamics. However, the dynamics can develop very-slow and very-fast time scales separated by a range of active time scales. An opportunity to reduce the complexity of the problem arises when the fast/active and slow/active time scales gaps becomes large. We propose a numerical technique consisting of an algorithmic framework, named the G-Scheme, to achieve multi-scale adaptive model reduction along-with the integration of the differential equations (DEs). The method is directly applicable to initial-value ODEs and (by using the method of lines) PDEs. We assume that the dynamics is decomposed into active, slow, fast, and when applicable, invariant subspaces. The G-Scheme introduces locally a curvilinear frame of reference, defined by a set of orthonormal basis vectors with corresponding coordinates, attached to this decomposition. The evolution of the curvilinear coordinates associated with the active subspace is described by non-stiff DEs, whereas that associated with the slow and fast subspaces is accounted for by applying algebraic corrections derived from asymptotics of the original problem. Adjusting the active DEs dynamically during the time integration is the most significant feature of the G-Scheme, since the numerical integration is accomplished by solving a number of DEs typically much smaller than the dimension of the original problem, with corresponding saving in computational work. To demonstrate the effectiveness of the G-Scheme, we present results from illustrative as well as from relevant problems.

Analysis of Methane-Air Edge Flame Structure.

We study the structure of a methane–air edge flame stabilized against an incoming mixing layer. The flame is computed using detailed chemical kinetics, and the analysis is based on computational singular perturbation theory. We focus on examination of the dynamical fast/slow structure of the flame, exploring the distribution of time-scales, the composition of the related specific modes and the effective low-dimensional structure. We also study the importance of chemical/transport processes for both major species and radicals in the flame, analyzing the information available from slow/fast importance indices as compared to reaction flux analysis. Results provide enhanced understanding of the flame, outlining the role of different chemical and transport processes in its observed structure.

A CSP and tabulation-based adaptive chemistry model

We demonstrate the feasibility of a new strategy for the construction of an adaptive chemistry model that is based on an explicit integrator stabilized by an approximation of the Computational Singular Perturbation (CSP)-slow-manifold projector. We examine the effectiveness and accuracy of this technique first using a model problem with variable stiffness. We assess the effect of using an approximation of the CSP-slow-manifold by either reusing the CSP vectors calculated in previous steps or from a pre-built tabulation. We find that while accuracy is preserved, the associated CPU cost was reduced substantially by this method. We used two ignition simulations – hydrogen–air and heptane–air mixtures – to demonstrate the feasibility of using the new method to handle realistic kinetic mechanisms. We test the effect of utilizing an approximation of the CSP-slow-manifold and find that its use preserves the order of the explicit integrator, produces no degradation in accuracy, and results in a scheme that is competitive with traditional implicit integration. Further analysis on the performance data demonstrates that the tabulation of the CSP-slow-manifold provides an increasing level of efficiency as the size of the mechanism increases. From the software engineering perspective, all the machinery developed is Common Component Architecture compliant, giving the software a distinct advantage in the ease of maintainability and flexibility in its utilization. Extension of this algorithm is underway to implement an automated tabulation of the CSP-slow-manifold for a detailed chemical kinetic system either off-line, or on-line with a reactive flow simulation code.

Reactive and reactive-diffusive time scales in stiff reaction-diffusion systems

Two different sets of time scales arising in stiff systems of reaction-diffusion PDEs are examined; the first due to the reaction term alone and the second due to the interaction of the reaction and diffusion terms. The fastest time scales of each set are responsible for the development of a low dimensional manifold, the characteristics of which depend on the set of time scales considered. The advantages and disadvantages of employing these two manifolds for the simplification of large and stiff systems of reaction-diffusion PDEs are discussed. It is shown that the two approaches provide a non-stiff simplified system of similar accuracy. The approach based on the reaction time scales allows for a simpler construction of the simplified system, while that based on the reaction/diffusion time scales allows for a simpler time marching scheme.

Optimization methods for non-smooth or noisy objective functions in fluid design problems

Convergence difficulties may arise during derivative-based optimization involving nonsmooth or noisy objective functions. Such may be the case whcSh shape design problems are attempted using analysis codes for fluid flows. For example, the interaction between the discretization of the design problem and a shock wave in the flow solution may cause the objective function to be non-smooth. In this work, we present a method for robust optimization of non-smoot h objective functions. The optimization begins with the construction of a response surface that smoothly approximates the objective function. Here the response surface is a least-squares polynomial fit to carefully selected design points. By minimiging the response surface we can obtain a first guess for a reasonable design. Optimization may continue in one of two ways. In the first method, we probe a small region of the design space around the minimum and perform another response surface minimization. In the second method, we use a derivative-based optimization where estimates of the sensitivity derivatives are obtained by means of the discrete direct or adjoint formulations. To overcome difficulties associated with the non-smooth objective function, the sensitivity equations are regularized by adding artificial dissipative terms, whereas the flow solution and the objective function are unmodified. Two design problems involving inviscid flow with shock waves are formulated to demonstrate the efficacy and robustness of the two methods.

Smoothed Sensitivity Equation Method for Fluid Dynamic Design Problems

We consider shape optimization problems involving compressible fluid flows, which are characterized by non-smooth and/or noisy objective functions. Such functions are difficult to optimize using derivative-based techniques. To overcome such a difficulty, we suggest an approach for estimating the sensitivity derivatives, based on a suitable smoothing of the sensitivity equations. The smoothing affects only the sensitivity derivatives and not the accuracy of the analysis. The basic mechanism by which the smoothing process achieves this result is illustrated with the help of an inverse design problem involving an inviscid quasi-one-dimensional flow having a closed-form solution. The convergence properties and the computational efficiency of the approach are demonstrated on two inverse design problems involving two-dimensional inviscid, compressible flows

Enforcing positivity in intrusive PC-UQ methods for reactive ODE systems

We explore the relation between the development of a non-negligible probability of negative states and the instability of numerical integration of the intrusive Galerkin ordinary differential equation system describing uncertain chemical ignition. To prevent this instability without resorting to either multi-element local polynomial chaos (PC) methods or increasing the order of the PC representation in time, we propose a procedure aimed at modifying the amplitude of the PC modes to bring the probability of negative state values below a user-defined threshold. This modification can be effectively described as a filtering procedure of the spectral PC coefficients, which is applied on-the-fly during the numerical integration when the current value of the probability of negative states exceeds the prescribed threshold. We demonstrate the filtering procedure using a simple model of an ignition process in a batch reactor. This is carried out by comparing different observables and error measures as obtained by non-intrusive Monte Carlo and Gauss-quadrature integration and the filtered intrusive procedure. The filtering procedure has been shown to effectively stabilize divergent intrusive solutions, and also to improve the accuracy of stable intrusive solutions which are close to the stability limits.

On the numerical integration of multi-dimensional, initial boundary value problems for the Euler equations in quasi-linear form

A matricial formalism to solve multi-dimensional initial boundary values problems for hyperbolic equations written in quasi-linear based on the λ scheme approach is presented. The derivation is carried out for nonorthogonal, moving systems of curvilinear coordinates. A uniform treatment of the integration at the boundaries, when the boundary conditions can be expressed in terms of combinations of time or space derivatives of the primitive variables, is also presented. The methodology is validated against a two-dimensional test case, the supercritical flow through the Hobson cascade n.2, and in three-dimensional test cases such as the supersonic flow about a sphere and the flow through a plug nozzle.