The Caltech helicopter control experiment

This report describes the Caltech helicopter control experiment. The experiment consists of an electric model helicopter interfaced to and controlled by a PC. We describe the hardware and software. A state-space model for the angular position is identified from experimental data near hover, using the prediction error method. An LQR controller with integrators for set point tracking is designed for the system. We also undertake a separate identification and loop shaping control for the yaw dynamics

Micro structure and Lagrangian statistcs of the scalar field with a mean gradient in isotropic turbulence

This paper presents an analysis and numerical study of the relations between the small-scale velocity and scalar fields in fully developed isotropic turbulence with random forcing of the large scales and with an imposed constant mean scalar gradient. Simulations have been performed for a range of Reynolds numbers from Reλ = 22 to 130 and Schmidt numbers from Sc = 1/25 to 144. The simulations show that for all values of Sc [gt-or-equal, slanted] 0.1 steep scalar gradients are concentrated in intermittently distributed sheet-like structures with a thickness approximately equal to the Batchelor length scale η/Sc[fraction one-half] with η the Kolmogorov length scale. We observe that these sheets or cliffs are preferentially aligned perpendicular to the direction of the mean scalar gradient. Due to this preferential orientation of the cliffs the small-scale scalar field is anisotropic and this is an example of direct coupling between the large- and small-scale fluctuations in a turbulent field. The numerical simulations also show that the steep cliffs are formed by straining motions that compress the scalar field along the imposed mean scalar gradient in a very short time period, proportional to the Kolmogorov time scale. This is valid for the whole range of Sc. The generation of these concentration gradients is amplified by rotation of the scalar gradient in the direction of compressive strain. The combination of high strain rate and the alignment results in a large increase of the scalar gradient and therefore in a large scalar dissipation rate. These results of our numerical study are discussed in the context of experimental results (Warhaft 2000) and kinematic simulations (Holzer & Siggia 1994). The theoretical arguments developed here follow from earlier work of Batchelor & Townsend (1956), Betchov (1956) and Dresselhaus & Tabor (1991)

The influence of the thermal diffusivity of the lower boundary on eddy motion in convection

The paper presents new concepts and results for the eddy structure of turbulent convection in a horizontal fluid layer of depth h which lies above a solid base with thickness hb. The fluid parameters are the kinematic viscosity ν, the thermal diffusivity κ, which is taken to be comparable with ν, the density ρ, the specific heat cp and the expansion parameter β. The thermal diffusivity of the solid is κb. The results are an extension of the more commonly studied cases, where a constant heat flux or constant temperature is applied at the interface between the fluid and the base. The buoyancy forces induce eddy motions with a typical velocity w∗∼(gβFθh)1/3 where ρcpFθ is the average heat flux and Fθ the covariance of the fluctuations of the temperature and of the vertical velocity. At moderate Reynolds numbers (Re=w∗h/ν), say less than about 103, an order-of-magnitude analysis shows that for the case of high diffusivity of the base (i.e. κb≫κ) elongated ‘plumes’ form at the surface and extend to the top of the fluid layer. When the base diffusivity is low (i.e. κb≤κ) the surface cools below the developing ‘plume’ and either the plume breaks up into elongated puffs or, if κb≪κ, horizontal pressure gradients form so that only small-scale puffs can form near the surface. At very high Reynolds numbers, approximately greater than 104, the surface boundary layer below each puff/plume is highly turbulent with a local logarithmic velocity and temperature profile. An approximate analysis indicates for this case that there is insufficient buoyancy flux from the base, irrespective of its diffusivity, to maintain plumes, because of the high turbulent heat transfer. So puffs dominate high-Reynolds-number thermal convection as numerical simulations and field experiments demonstrate. However, when the surface heat flux is uniform, for example as a result of radiant heat transfer or by forcing with a constant heat flux below a very thin conducting base, plumes are the dominant form of eddy motion, as is commonly observed. In the numerical solutions presented here, where Re∼3×102 and the slab thickness hb=h, it is shown that the spatial scales of eddy structures in the fluid layer close to the surface become significantly smaller as κb/κ is reduced from 100 to 0.1. At the same time in the core of the convective layer the change in the autocorrelation and spatial correlation function indicates that there is a transition from long-duration plumes into shorter-duration and smaller-length-scale elongated puffs. The simulations show that the largest temperature fluctuations near the surface occur when a constant heat flux is applied at the bottom of the fluid layer. The smallest temperature fluctuations are associated with the constant-temperature boundary condition. The finite base diffusivity cases lie in between these limits, with the largest fluctuations occurring when the thermal diffusivity of the base is small. The hypothesis introduced above has been tested qualitatively in a laboratory set-up when the effective diffusivity of the base was varied. The flow structure was observed as it changed from being characterized by nearly steady plumes, into unsteady plumes and finally into puffs when the thickness of the conducting base was first increased and then the diffusivity was decreased

Turbulent channel flow near maximum drag reduction: simulations, experiments and mechanisms

It is well known that the drag in a turbulent flow of a polymer solution is significantly reduced compared to Newtonian flow. Here we consider this phenomenon by means of a direct numerical simulation of a turbulent channel flow. The polymers are modelled as elastic dumbbells using the FENE-P model. In the computations the polymer model is solved simultaneously with the flow equations, i.e. the polymers are deformed by the flow and in their turn influence the flow structures by exerting a polymer stress. We have studied the results of varying the polymer parameters, such as the maximum extension, the elasticity and the concentration. For the case of highly extensible polymers the results of our simulations are very close to the maximum drag reduction or Virk (1975) asymptote. Our simulation results show that at approximately maximum drag reduction the slope of the mean velocity profile is increased compared to the standard logarithmic profile in turbulent wall flows. For the r.m.s. of the streamwise velocity fluctuations we find initially an increase in magnitude which near maximum drag reduction changes to a decrease. For the velocity fluctuations in the spanwise and wall-normal directions we find a continuous decrease as a function of drag reduction. The Reynolds shear stress is strongly reduced, especially near the wall, and this is compensated by a polymer stress, which at maximum drag reduction amounts to about 40% of the total stress. These results have been compared with LDV experiments of Ptasinski et al. (2001) and the agreement, both qualitatively and quantitatively, is in most cases very good. In addition we have performed an analysis of the turbulent kinetic energy budgets. The main result is a reduction of energy transfer from the streamwise direction, where the production of turbulent kinetic energy takes place, to the other directions. A substantial part of the energy production by the mean flow is transferred directly into elastic energy of the polymers. The turbulent velocity fluctuations also contribute energy to the polymers. The elastic energy of the polymers is subsequently dissipated by polymer relaxation. We have also computed the various contributions to the pressure fluctuations and identified how these change as a function of drag reduction. Finally, we discuss some cross-correlations and various length scales. These simulation results are explained here by two mechanisms. First, as suggested by Lumley (1969) the polymers damp the cross-stream or wall-normal velocity fluctuations and suppress the bursting in the buffer layer. Secondly, the ‘shear sheltering’ mechanism acts to amplify the streamwise fluctuations in the thickened buffer layer, while reducing and decoupling the motions within and above this layer. The expression for the substantial reduction in the wall drag derived by considering the long time scales of the nonlinear fluctuations of this damped shear layer, is shown to be consistent with the experimental data of Virk et al. (1967) and Virk (1975)

Strong Modification of the Nonlinear Optical Response of Metallic Subwavelength Hole Arrays

The influence of hole shape on the nonlinear optical properties of metallic subwavelength hole arrays is investigated. It is found that the amount of second harmonics generated can be enhanced by changing the hole shape. In part this increase is a direct result of the effect of hole shape on the linear transmission properties. Remarkably, in addition to enhancements that follow directly from the linear properties of the array, we find a hot hole shape. For rectangular holes the effective nonlinear response is enhanced by more than 1 order of magnitude for one particular aspect ratio. This enhancement can be attributed to slow propagation of the fundamental wavelength through the holes which occurs close to the hole cutoff

Differential Flatness and Absolute Equivalence

In this paper we give a formulation of differential flatness---a concept originally introduced by Fleiss, Levine, Martin, and Rouchon---in terms of absolute equivalence between exterior differential systems. Systems which are differentially flat have several useful properties which can be exploited to generate effective control strategies for nonlinear systems. The original definition of flatness was given in the context of differentiable algebra, and required that all mappings be meromorphic functions. Our formulation of flatness does not require any algebraic structure and allows one to use tools from exterior differential systems to help characterize differentially flat systems. In particular, we shown that in the case of single input control systems (i.e., codimension 2 Pfaffian systems), a system is differentially flat if and only if it is feedback linearizable via static state feedback. However, in higher codimensions feedback linearizability and flatness are *not* equivalent: one must be careful with the role of time as well the use of prolongations which may not be realizable as dynamic feedbacks in a control setting. Applications of differential flatness to nonlinear control systems and open questions will be discussed. Revised 14 Aug 9

Flatness and Monge parameterization of two-input systems, control-affine with 4 states or general with 3 states

This paper studies Monge parameterization, or differential flatness, of control-affine systems with four states and twocontrols. Some of them are known to be flat, and this implies admitting a Monge parameterization. Focusing on systems outside this class, we describe the only possible structure of such a parameterization for these systems, and give a lower bound on the order of this parameterization, if it exists. This lower-bound is good enough to recover the known results about "(x,u)-flatness" of these systems, with much more elementary techniques

Differential flatness and absolute equivalence

In this paper we give a formulation of differential flatness-a concept originally introduced by Fliess, Levine, Martin, and Rouchon (1992)-in terms of absolute equivalence between exterior differential systems. Systems which are differentially flat have several useful properties which can be exploited to generate effective control strategies for nonlinear systems. The original definition of flatness was given in the context of differential algebra, and required that all mappings be meromorphic functions. Our formulation of flatness does not require any algebraic structure and allows one to use tools from exterior differential systems to help characterize differentially flat systems. In particular, we show that in the case of single input control systems (i.e., codimension 2 Pfaffian systems), a system is differentially flat if and only if it is feedback linearizable via static state feedback. However, in higher codimensions feedback linearizability and flatness are not equivalent: one must be careful with the role of time as well the use of prolongations which may not be realizable as dynamic feedbacks in a control setting. Applications of differential flatness to nonlinear control systems and open questions are also discussed