Optimal Feedback Control of Thermal Networks

An improved approach to the mathematical modeling of feedback control of thermal networks has been devised. Heretofore software for feedback control of thermal networks has been developed by time-consuming trial-and-error methods that depend on engineers expertise. In contrast, the present approach is a systematic means of developing algorithms for feedback control that is optimal in the sense that it combines performance with low cost of implementation. An additional advantage of the present approach is that a thermal engineer need not be expert in control theory. Thermal networks are lumped-parameter approximations used to represent complex thermal systems. Thermal networks are closely related to electrical networks commonly represented by lumped-parameter circuit diagrams. Like such electrical circuits, thermal networks are mathematically modeled by systems of differential-algebraic equations (DAEs) that is, ordinary differential equations subject to a set of algebraic constraints. In the present approach, emphasis is placed on applications in which thermal networks are subject to constant disturbances and, therefore, integral control action is necessary to obtain steady-state responses. The mathematical development of the present approach begins with the derivation of optimal integral-control laws via minimization of an appropriate cost functional that involves augmented state vectors. Subsequently, classical variational arguments provide optimality conditions in the form of the Hamiltonian equations for the standard linear-quadratic-regulator (LQR) problem. These equations are reduced to an algebraic Riccati equation (ARE) with respect to the augmented state vector. The solution of the ARE leads to the direct computation of the optimal proportional- and integral-feedback control gains. In cases of very complex networks, large numbers of state variables make it difficult to implement optimal controllers in the manner described in the preceding paragraph

Tilt-Sensitivity Analysis for Space Telescopes

A report discusses a computational-simulation study of phase-front propagation in the Laser Interferometer Space Antenna (LISA), in which space telescopes would transmit and receive metrological laser beams along 5-Gm interferometer arms. The main objective of the study was to determine the sensitivity of the average phase of a beam with respect to fluctuations in pointing of the beam. The simulations account for the effects of obscurations by a secondary mirror and its supporting struts in a telescope, and for the effects of optical imperfections (especially tilt) of a telescope. A significant innovation introduced in this study is a methodology, applicable to space telescopes in general, for predicting the effects of optical imperfections. This methodology involves a Monte Carlo simulation in which one generates many random wavefront distortions and studies their effects through computational simulations of propagation. Then one performs a statistical analysis of the results of the simulations and computes the functional relations among such important design parameters as the sizes of distortions and the mean value and the variance of the loss of performance. These functional relations provide information regarding position and orientation tolerances relevant to design and operation

Unsplit Schemes for Hyperbolic Conservation Laws with Source Terms in One Space Dimension

The present work is concerned with the extension of the theory of characteristics to conservation laws with source terms in one space dimension, such as the Euler equations for reacting flows. New spacetime curves are introduced on which the equations decouple to the characteristic set of O.D.E's for the corresponding homogeneous laws, thus allowing the introduction of functions analogous to the Riemann Invariants. The geometry of these curves depends on the spatial gradients for the solution. This particular decomposition can be used in the design of efficient unsplit algorithms for the numerical integration of the equations. As a first step, these ideas are implemented for the case of a scalar conservation law with a nonlinear source term. The resulting algorithm belongs to the class of MUSCL-type, shock-capturing schemes. Its accuracy and robustness are checked through a series of tests. The aspect of the stiffness of the source term is also studied. Then, the algorithm is generalized for a system of hyperbolic equations, namely the Euler equations for reacting flows. An extensive numerical study of unstable detonations is performed

The effect of electrostatic charges on particle-laden duct flows

We report on direct numerical simulations of the effect of electrostatic charges on particle-laden duct flows. The corresponding electrostatic forces are known to affect particle dynamics at small scales and the associated turbophoretic drift. Our simulations, however, predicted that electrostatic forces also dominate the vortical motion of the particles, induced by the secondary flows of Prandtl's second kind of the carrier fluid. Herein we treated flows at two frictional Reynolds numbers ($Re_\mathrm{\tau}=$ 300 and~600), two particle-to-gas density ratios ($\rho_\mathrm{p}/\rho=$ 1000 and 7500), and three Coulombic-to-gravitational force ratios ($F_\mathrm{el}/F_\mathrm{g}=$ 0, 0.004, and 0.026). In flows with a high density ratio at $Re_\mathrm{\tau}=$ 600 and $F_\mathrm{el}/F_\mathrm{g}=$ 0.004, the particles tend to accumulate at the walls. On the other hand, at a lower density ratio, respectively a higher $F_\mathrm{el}/F_\mathrm{g}$ of 0.026, the charged particles still follow the secondary flow structures that are developed in the duct. However, even in this case, the electrostatic forces counteract the particles' inward flux from the wall and, as a result, their vortical motion in these secondary structures is significantly attenuated. This change in the flow pattern results in an increase of the particle number density at the bisectors of the walls by a factor of five compared to the corresponding flow with uncharged particles. Finally, at $Re_\mathrm{\tau}=$ 300, $\rho_\mathrm{p}/\rho=$ 1000, and $F_\mathrm{el}/F_\mathrm{g}=$ 0.026 the electrostatic forces dominate over the aerodynamic forces and gravity and, consequently the particles no longer follow the streamlines of the carrier gas

A two-phase model for compressible granular flows based on the theory of irreversible processes

In this article we introduce a new two-phase model for compressible viscous flows of saturated mixtures consisting of a carrier fluid and a granular material. The mixture is treated as a multicomponent fluid, with a set of thermodynamic variables assigned to each of its constituents. The volume fraction occupied by the granular phase and its spatial gradient are introduced as additional degrees of freedom. Then, by applying the classical theory of irreversible processes we derive algebraic expressions for the viscous stresses and heat flux vectors, the momentum and energy exchanges between the two phases, as well as a parabolic partial differential equation for the volume fraction. In our model, thermal non-equilibrium between the two phases emerges as a source term of the evolution equation for the volume fraction, in contrast with earlier models

Influence of inert particles on the propagation of multidimensional detonation waves

In this article we examine numerically the multidimensional Structure of detonation waves propagating in a mixture consisting of a combustible gas and inert solid particles. First, we provide a brief description of the two-phase flow model that is employed in our study. Subsequently, we present and analyze the results from numerical simulations of three representative cases. These results show in detail the flow structures that are developed behind the leading shock due to the momentum and energy exchanges between the two phases. The article also includes a parametric study on the effect of the particle volume fraction of the Mixture and the particle diameter. It is predicted that the detonation wave speed decreases monotonically as the particle volume fraction increases. In particular, sufficiently high volume fractions can cause detonation quenching. Finally it is predicted that, under constant volume fraction, the decrease in detonation speed is slightly more important for large particles than for small ones. © 2005 The Combustion Institute. Published by Elsevier Inc. All rights reserved

On the applicability of Stokes’ hypothesis to low-Mach-number flows

Stokes’ hypothesis states that the bulk viscosity of a Newtonian fluid can be set to zero. Althoughnot valid for many fluids, it is common practice to invoke this hypothesis in the study of low-Mach-number,variable-density flows. Based on scaling arguments, we provide a necessary condition for neglecting the bulkviscous pressure from the governing equations. More specifically, we show that the Reynolds number definedwith respect to the bulk viscosity must be very large. We further show that even when this condition is notsatisfied, the bulk viscous pressure does not need to be taken explicitly into account in the computation of thevelocity field because it can be combined with the hydrodynamic pressure

Boundary-layer flow in a porous domain above a flat plate

This paper is concerned with the structure and properties of boundary-layer flow in a porous domain above a flat plate. The flow is generated by an incoming uniform stream at the vertical boundary of the porous domain and is maintained by an external pressure forcing. Herein we provide the parametrization of the interphasial drag in terms of a Darcy–Forchheimer law, derive the momentum boundary-layer equation and elaborate on the profile of the free-stream velocity. The boundary-layer equation is then solved numerically via the local similarity method and via two local nonsimilarity methods at different levels of truncation. The accuracy of these methods is compared via a series of numerical tests. For the problem in hand, the free-stream velocity decreases monotonically to a terminal far-field value. Once this value is reached, the velocity profile no longer evolves in the streamwise direction. The computations further reveal that, for sufficiently small external forcing, the boundary-layer thickness initially increases, reaches a peak and then decreases towards its terminal value. This unusual overshoot is attributed to the large variation of the rate of decrease of the free-stream velocity. On the other hand, our computations predict that the wall stress always decreases monotonically in the streamwise direction