Adjoint sensitivity analysis of chaotic systems using cumulant truncation

We describe a simple and systematic method for obtaining approximate sensitivity information from a chaotic dynamical system using a hierarchy of cumulant equations. The resulting forward and adjoint systems yield information about gradients of functionals of the system and do not suffer from the convergence issues that are associated with the tangent linear representation of the original chaotic system. The functionals on which we focus are ensemble-averaged quantities, whose dynamics are not necessarily chaotic; hence we analyse the system’s statistical state dynamics, rather than individual trajectories. The approach is designed for extracting parameter sensitivity information from the detailed statistics that can be obtained from direct numerical simulation or experiments. We advocate a data-driven approach that incorporates observations of a system’s cumulants to determine an optimal closure for a hierarchy of cumulants that does not require the specification of model parameters. Whilst the sensitivity information from the resulting surrogate model is approximate, the approach is designed to be used in the analysis of turbulence, whose degrees of freedom and complexity currently prohibits the use of more accurate techniques. Here we apply the method to obtain functional gradients from low-dimensional representations of Rayleigh-Bénard convection

Stochastic models of ventilation driven by opposing wind and buoyancy

Stochastic versions of a classical model for natural ventilation are proposed and investigated to demonstrate the effect of random fluctuations on stability and predictability. In a stochastic context, the well-known deterministic result that ventilation driven by the competing effects of buoyancy and wind admits multiple steady states can be misleading. With fluctuations in the buoyancy exchanged with an external environment modelled as a Wiener process, such systems tend to reside in the vicinity of global minima of their potential, rather than states associated with meta-stable equilibria. For a heated space with a leeward low-level and windward highlevel opening, sustained buoyancy-driven flow opposing the wind direction is unlikely for wind strengths exceeding a statistically critical value, which is slightly larger than the critical value of the wind strength at which bifurcation in the deterministic system occurs. When fluctuations in the applied wind strength are modelled as an Ornstein-Uhlenbeck process, the topology of the system’s potential is effectively modified due to the nonlinear role that wind strength has in the equation for buoyancy conservation. Consequently, large fluctuations in the wind of sufficiently short duration rule out the possibility of sustained ventilation opposing the wind direction at large base wind strengths

Energy dispersion in turbulent jets. Part 2. A robust model for unsteady jets

In this paper we develop an integral model for an unsteady turbulent jet that incorporates longitudinal dispersion of two distinct types. The model accounts for the difference in the rate at which momentum and energy are advected (type I dispersion) and for the local deformation of velocity profiles that occurs in the vicinity of a sudden change in the momentum flux (type II dispersion). We adapt the description of dispersion in pipe flow by Taylor (Proc. R. Soc. Lond. A, vol. 219, 1953, pp. 186–203) to develop a dispersion closure for the longitudinal transportation of energy in unsteady jets. We compare our model’s predictions to results from direct numerical simulation and find a good agreement. The model described in this paper is robust and can be solved numerically using a simple central differencing scheme. Using the assumption that the longitudinal velocity profile in a jet has an approximately Gaussian form, we show that unsteady jets remain approximately straight-sided when their source area is fixed. Straight-sidedness provides an algebraic means of reducing the order of the governing equations and leads to a simple advection–dispersion relation. The physical process responsible for straight-sidedness is type I dispersion, which, in addition to determining the local response of the area of the jet, determines the growth rate of source perturbations. In this regard the Gaussian profile has the special feature of ensuring straight-sidedness and being insensitive to source perturbations. Profiles that are more peaked than the Gaussian profile attenuate perturbations and, following an increase (decrease) in the source momentum flux, lead to a local decrease (increase) in the area of the jet. Conversely, profiles that are flatter than the Gaussian amplify perturbations and lead to a local increase (decrease) in the area of the jet

The entrainment and energetics of turbulent plumes in a confined space

We analyse the entrainment and energetics of equal and opposite axisymmetric tur-bulent air plumes in a vertically confined space at a Rayleigh number of1.24×107using theory and direct numerical simulation. On domains of sufficiently large aspectratio, the steady-state consists of turbulent plumes penetrating an interface betweentwo layers of approximately uniform buoyancy. As described by Baines & Turner (J.Fluid Mech.vol. 37, 1969, pp. 51-80), upon penetrating the interface the flow in eachplume becomes forced and behaves like a constant-momentum jet, due to a reduction inits mean buoyancy relative to the local environment. To observe the behaviour of theplumes we partition the domain into sub-domains corresponding to each plume. Domainsof relatively small aspect ratio produce a single primary mean-flow circulation betweenthe sub-domains that is maintained by entrainment into the plumes. At larger aspectratios the mean flow between the sub-domains bifurcates, indicating the existence of asecondary circulation within each layer associated with entrainment into the jets. Thelargest aspect ratios studied here exhibit an additional, tertiary, circulation in the vicinityof the interface. Consistency between independent calculations of an effective entrainmentcoefficient allows us to identify aspect ratios for which the flow can be modelled usingplume theory, under the assumption of a two-layer stratification.To study the flow’s energetics we use a local definition of available potential energy(APE). For plumes with Gaussian velocity and buoyancy profiles, the theory we developsuggests that the kinetic energy dissipation is split equally between the jets and theplumes and, collectively, accounts for almost half of the input of APE at the boundaries.In contrast,1/4of the APE dissipation and background potential energy (BPE) pro-duction occurs in the jets, with the remaining3/4occurring in the plumes. These bulktheoretical predictions agree with observations of BPE production from simulations towithin1%and form the basis of a similarity solution that models the vertical dependenceof APE dissipation and BPE production. Unlike results concerning the dissipation ofbuoyancy variance and the strength of the circulations described above, the model forthe flow’s energetics does not involve an entrainment coefficient

Relating quanta conservation and compartmental epidemiological models of airborne disease outbreaks in buildings

We investigate the underlying assumptions and limits of applicability of several documented models for outbreaks of airborne disease inside buildings by showing how they may each be regarded as special cases of a system of equations which combines quanta conservation and compartmental epidemiological modelling. We investigate the behaviour of this system analytically, gaining insight to its behaviour at large time. We then investigate the characteristic timescales of an indoor outbreak, showing how the dilution rate of the space, and the quanta generation rate, incubation rate and removal rate associated with the illness may be used to predict the evolution of an outbreak over time, and may also be used to predict the relative performances of other indoor airborne outbreak models. The model is compared to a more commonly used model, in which it is assumed the environmental concentration of infectious aerosols adheres to a quasi-steady-state, so that the the dimensionless quanta concentration is equal to the the infectious fraction. The model presented here is shown to approach this limit exponentially to within an interval defined by the incubation and removal rates. This may be used to predict the maximum extent to which a case will deviate from the quasi steady state condition

Rigorous scaling laws for internally heated convection at infinite Prandtl number

New bounds are proven on the mean vertical convective heat transport, ⟨wT⟩¯¯¯¯¯¯¯¯¯¯¯, for uniform internally heated (IH) convection in the limit of infinite Prandtl number. For fluid in a horizontally-periodic layer between isothermal boundaries, we show that ⟨wT⟩¯¯¯¯¯¯¯¯¯¯¯≤12−cR−2, where R is a nondimensional `flux' Rayleigh number quantifying the strength of internal heating and c=216. Then, ⟨wT⟩¯¯¯¯¯¯¯¯¯¯¯=0 corresponds to vertical heat transport by conduction alone, while ⟨wT⟩¯¯¯¯¯¯¯¯¯¯¯>0 represents the enhancement of vertical heat transport upwards due to convective motion. If, instead, the lower boundary is a thermal insulator, then we obtain ⟨wT⟩¯¯¯¯¯¯¯¯¯¯¯≤12−cR−4, with c≈0.0107. This result implies that the Nusselt number Nu, defined as the ratio of the total-to-conductive heat transport, satisfies Nu≲R4. Both bounds are obtained by combining the background method with a minimum principle for the fluid's temperature and with Hardy--Rellich inequalities to exploit the link between the vertical velocity and temperature. In both cases, power-law dependence on R improves the previously best-known bounds, which, although valid at both infinite and finite Prandtl numbers, approach the uniform bound exponentially with R

Turbulent transport and entrainment in jets and plumes: A DNS study

We present a new DNS data set for a statistically axisymmetric turbulent jet, plume and forced plume in a domain of size 40r$_{0}$ x 40r$_{0}$ x 60r$_{0}$, where r$_{0}$ is the source diameter. The data set supports the validity of the Priestley and Ball entrainment model in unstratified environments (excluding the region near the source), which is corroborated further by the Wang and Law and Ezzamel et al. experimental data sets, the latter being corrected for a small but influential co-flow that affected the statistics. We show that the second-order turbulence statistics in the core region of the jet and the plume are practically indistinguishable, although there are significant differences near the plume edge. The DNS data indicates that the turbulent Prandtl number is about 0.7 for both jets and plumes. For plumes, this value is a result of the difference in the ratio of the radial turbulent transport of radial momentum and buoyancy. For jets however, the value originates from a different spread of the buoyancy and velocity profiles, in spite of the fact that the ratio of radial turbulent transport terms is approximately unity. The DNS data does not show any evidence of similarity drift associated with gradual variations in the ratio of buoyancy profile to velocity profile widths

A Critical Examination of Sedation Withdrawal Assessment in Children.

Background Sedation withdrawal is one of the terms used to describe the behavioural response to stopping or reducing sedative drugs in physically dependent patients. Withdrawal behaviours differ according to the drug involved and may be unpleasant and interfere with recovery. Recognition of sedation withdrawal is challenging due to differences in patient presentation and may be further complicated by the patient’s condition and concomitant drug therapy. Overall Aim of the full thesis To improve the accuracy of sedation withdrawal assessment in critically ill children. Objectives and Methods A mixed methods interactive approach comprising six studies. Study 1 evaluates the psychometric properties of the Sedation Withdrawal Score, Studies 2 and 3 examine the complexities/challenges of withdrawal assessment by critiquing existing tool validation studies, A further three studies examine the nurse and parent perspectives of sedation withdrawal assessment in critically ill children. Study 4 investigates how nurses use a sedation withdrawal tool, Studies 5 and 6 investigate what behavioural signs parents recognise and ascertain parents’ willingness to participate in withdrawal assessments. Key findings Nurses found withdrawal behaviours difficult to interpret in critically ill children and there were differences in how these behaviours were construed. Parents identified a broader range of behaviours than included in existing tools. Most parents were eager to participate in the assessment. The elusive theoretical basis for the existing approach to withdrawal assessment may account for the lack of a standardisation and poor accuracy of the current tools. A model of the causal relationship between dependence and withdrawal is proposed. Recommendations The model identifies the diagnostic criteria upon which a definition for Pediatric Withdrawal Syndrome may be based. These criteria also provide a novel framework for withdrawal assessment. Focussing on the shared diagnostic criteria and including the parent perspective of the child’s behaviours may aid the assessment and support decision-making