Flamelet based chemical reduction techniques are very promising methods for efficient and accurate modeling of premixed flames. Over the years the Flamelet Generated Manifold (FGM) technique has been ...developed by the Combustion Technology Group of Eindhoven University of Technology. Current state-of-the-art of FGM for the modeling of premixed and partially-premixed flames is reviewed. The fundamental basis of FGM consists of a generalized description of the flame front in a (possibly moving) flame-adapted coordinate system. The basic nature of the generalized flamelet model is that effects of strong stretch in turbulent flames are taken into account by resolving the detailed structure of flame stretch and curvature inside the flame front. The generalized flamelet model, which forms the basis on which FGM is built, is derived in Part I. To be able to validate numerical results of flames obtained with full chemistry and obtained from FGM, it is important that the generalized flamelet model is analyzed further. This is done by investigating the impact of strong stretch, curvature and preferential diffusion effects on the flame dynamics as described by the local mass burning rate. This so-called strong stretch theory is derived and analyzed in Part I, as well as multiple simplifications of it, to compare the strong stretch theory with existing stretch theories. The results compare well with numerical results for flames with thin reaction layers, but described by multiple-species transport and chemistry. This opens the way to use the generalized flamelet model as a firm basis for applying FGM in strongly stretched laminar and turbulent flames in Part II. The complete FGM model is derived first and the use of FGM in practice is reviewed. The FGM model is then validated by studying effects of flame stretch, heat loss, and changes in elements, as well as NO formation. The application to direct numerical simulations of turbulent flames is subsequently studied and validated using the strong stretch theory. It is shown that the generalized flamelet model still holds even in case of strong stretch and curvature effects, at least as long as the reaction layer is dominated by reaction and diffusion phenomena and not perturbed too much by stretch related perturbations. The FGM model then still performs very well with a low number of control variables. Turbulent flames with strong preferential diffusion effects can also be modeled efficiently with an FGM model using a single additional control variable for the changes in element mass fractions and enthalpy. Finally FGM is applied to the modeling of turbulent flames using LES and RANS flow solvers. For these cases, the flame front structure is not resolved anymore and unresolved terms need to be modeled. A common approach to include unresolved turbulent fluctuations is the presumed probability density function (PDF) approach. The validity of this FGM-PDF approach is discussed for a few test cases with increasing level of complexity.
The hydrodynamic force exerted by a fluid on small isolated rigid spherical particles are usually well described by the Maxey–Riley (MR) equation. The most time-consuming contribution in the MR ...equation is the Basset history force which is a well-known problem for many-particle simulations in turbulence. In this paper a novel numerical approach is proposed for the computation of the Basset history force based on the use of exponential functions to approximate the tail of the Basset force kernel. Typically, this approach not only decreases the cpu time and memory requirements for the Basset force computation by more than an order of magnitude, but also increases the accuracy by an order of magnitude. The method has a temporal accuracy of O(Δt2) which is a substantial improvement compared to methods available in the literature. Furthermore, the method is partially implicit in order to increase stability of the computation. Traditional methods for the calculation of the Basset history force can influence statistical properties of the particles in isotropic turbulence, which is due to the error made by approximating the Basset force and the limited number of particles that can be tracked with classical methods. The new method turns out to provide more reliable statistical data.
We present a unified mathematical framework for sixteen fundamental optical systems. The systems have a parallel or point source and a parallel, point, near-field or far-field target. These choices ...give eight configurations if we use reflectors only and take the minimum number of freeform surfaces required. Similarly, we get eight lens systems if we only use lens surfaces. The mathematical model for each system is based on Hamilton’s characteristic functions and conservation of luminous flux. Some configurations lead to standard or generalized Monge-Ampère equations. The remaining systems are described by so-called generated Jacobian equations.
In this paper we propose a method to compute a freeform reflector system for collimating and shaping a beam from a point source. We construct these reflectors such that the radiant intensity of the ...source is converted into a desired target. An important generalization in our approach compared to previous research is that the output beam can be in an arbitrary direction. The design problem is approached by using a generalized Monge-Ampère equation. This equation is solved using a least-squares algorithm for non-quadratic cost functions. This algorithm calculates the optical map, from which we can then compute the surfaces. We test our algorithm on two cases. First we consider a uniform source and target distribution. Next, we use the model of a laser diode light source and a ring-shaped target distribution.
In this article, we present a formulation for the design of double freeform lens surfaces to control the intensity distribution of a laser beam with plane wavefronts. Double freefrom surfaces are ...utilized to shape collimated beams. Two different layouts of the freeform lens optical system are introduced, i.e., a single lens with double freeform surfaces, and two separate lenses with two flat and two freeform surfaces. The freeform lens design problem can be formulated as a Monge–Ampère type differential equation with transport boundary condition, expressing conservation of energy combined with the law of refraction and the constraint imposed on the optical path length between source and target planes. Numerical solutions are computed using a generalized least-squares algorithm which is presented by Yadav et al. (2018). The algorithm is capable to compute two solutions of the Monge–Ampère boundary value problem, corresponding to either c-convex or c-concave freeform surfaces for both layouts. The freeform surfaces are validated for several numerical examples using a ray-tracer based on Quasi-Monte Carlo simulation.
•Presented a geometrical formulation for double freeform surfaces optical systems.•Transportation problem with a non-quadratic cost function.•The problem is equivalent to a Monge–Ampère type equation.•Very efficient numerical method, can handle complex target distributions.•Efficiently computed the freeform surfaces for beam shaping problems.
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