arXiv subjects
Publications and source records attributed to John Craske.
A probability density function (PDF) of a spatially dependent field provides a means of calculating moments of the field or, equivalently, the proportion of a spatial domain that is mapped to a given set of values. This paper describes a finite element approach to estimating the PDF of a spatially dependent field and its numerical implementation in the Python package NumDF.
A partial differential equation governing the global evolution of the joint probability distribution of an arbitrary number of local flow observations, drawn randomly from a control volume, is derived and applied to examples involving irreversible mixing. Unlike local probability density methods, this work adopts a global integral perspective by regarding a control volume as the sample space. Doing so enables the divergence theorem to be used to expose contributions made by uncertain or stochastic boundary fluxes and internal cross-gradient mixing in the equation governing the joint probability distribution's evolution. Advection and diffusion across the control volume's boundary result in source and drift terms, respectively, whereas internal mixing, in general, corresponds to the sign-indefinite diffusion of probability density. Several typical circumstances for which the corresponding diffusion coefficient is negative semidefinite are identified and discussed in detail. The global joint probability perspective is the natural setting for available potential energy and the incorporation of uncertainty into bulk, volume integrated, models of transport and mixing. Finer-grained information in space can be readily obtained by treating coordinate functions as observables. By extension, the framework can be applied to networks of interacting control volumes of arbitrary size.
We consider an internally heated fluid between parallel plates with fixed thermal fluxes. For a large class of heat sources that vary in the direction of gravity, we prove that $\langle\delta T \rangle_h \geq \sigma R^{-1/3} - \mu$, where $\langle\delta T \rangle_h$ is the average temperature difference between the bottom and top plates, $R$ is a `flux' Rayleigh number and the constants $\sigma,\mu >0$ depend on the geometric properties of the internal heating. This result implies that mean downward conduction (for which $\langle\delta T \rangle_h< 0$) is impossible for a range of Rayleigh numbers smaller than a critical value $R_0$. The bound demonstrates that $R_0$ depends on the heating distribution and can be made arbitrarily large by concentrating the heating near the bottom plate. However, for any given fixed heating profile of the class we consider, the corresponding value of $R_0$ is always finite. This points to a fundamental difference between internally heated convection and its limiting case of Rayleigh-B\'enard convection with fixed flux boundary conditions, for which $\langle\delta T \rangle_h$ is known to be positive for all $R$.
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 metastable equilibria. For a heated space with a leeward low-level and windward high-level 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 sufficient duration rule out the possibility of sustained ventilation opposing the wind direction at large base wind strengths.
We present and analyse observational data from a highly instrumented classroom computer laboratory and develop a multizone model to describe its mechanical ventilation and mixing regime. The laboratory houses 70 workstations that are used heterogeneously in time and space, in a manner similar to a generic office environment. Our model predicts CO2 concentration in the laboratory, accounting for air exchange between the occupied classroom and its ceiling plenum and by parametrising irreversible mixing in each zone. Applying the model to our measurements helps identify critical components in the ventilation network, as highlighted by a strong separation of the time scales characterising the flow response. On the one hand, this time scale separation leads to a simplified model describing the CO2 transport. On the other hand, it suggests that the forced exchange of volume between the room and the plenum is 'overdriven' in that reduced energy operation could be achieved without compromising air quality. More generally, our modelling approach offers a systematic method to enhance energy efficient ventilation of multi-zone systems.
New bounds are proven on the mean vertical convective heat transport, $\overline{\langle wT \rangle}$, 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 $\overline{\langle wT \rangle} \leq \frac12 - c R^{-2}$, where $R$ is a nondimensional `flux' Rayleigh number quantifying the strength of internal heating and $c = 216$. Then, $\overline{\langle wT \rangle} = 0$ corresponds to vertical heat transport by conduction alone, while $\overline{\langle wT \rangle} > 0$ represents the enhancement of vertical heat transport upwards due to convective motion. If, instead, the lower boundary is a thermal insulator, then we obtain $\overline{\langle wT \rangle} \leq \frac12 - c R^{-4}$, with $c\approx 0.0107$. This result implies that the Nusselt number $Nu$, defined as the ratio of the total-to-conductive heat transport, satisfies $Nu \lesssim R^{4}$. 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 $R$.
We obtain an analytical bound on the mean vertical convective heat flux $\langle w T \rangle$ between two parallel boundaries driven by uniform internal heating. We consider two configurations, one with both boundaries held at the same constant temperature, and the other one with a top boundary held at constant temperature and a perfectly insulating bottom boundary. For the first configuration, Arslan et al. (J. Fluid Mech. 919:A15, 2021) recently provided numerical evidence that Rayleigh-number-dependent corrections to the only known rigorous bound $\langle w T \rangle \leq 1/2$ may be provable if the classical background method is augmented with a minimum principle stating that the fluid's temperature is no smaller than that of the top boundary. Here, we confirm this fact rigorously for both configurations by proving bounds on $\langle wT \rangle$ that approach $1/2$ exponentially from below as the Rayleigh number is increased. The key to obtaining these bounds are inner boundary layers in the background fields with a particular inverse-power scaling, which can be controlled in the spectral constraint using Hardy and Rellich inequalities. These allow for qualitative improvements in the analysis not available to standard constructions.
We prove a new rigorous bound for the mean convective heat transport $\langle w T \rangle$, where $w$ and $T$ are the nondimensional vertical velocity and temperature, in internally heated convection between an insulating lower boundary and an upper boundary with a fixed heat flux. The quantity $\langle wT \rangle$ is equal to half the ratio of convective to conductive vertical heat transport, and also to $\frac12$ plus the mean temperature difference between the top and bottom boundaries. An analytical application of the background method based on the construction of a quadratic auxiliary function yields $\langle w T \rangle \leq \tfrac{1}{2}\big(\tfrac{1}{2}+ \tfrac{1}{\sqrt{3}} \big) - 1.6552\, R^{-\frac13}$ uniformly in the Prandtl number, where $R$ is the nondimensional control parameter measuring the strength of the internal heating. Numerical optimisation of the auxiliary function suggests that the asymptotic value of this bound and the $-1/3$ exponent are optimal within our bounding framework. This new result halves the best existing (uniform in $R$) bound (Goluskin 2016, Springer, Table 1.2) and its dependence on $R$ is consistent with previous conjectures and heuristic scaling arguments. Contrary to physical intuition, however, it does not rule out a mean heat transport larger than $\frac12$ at high $R$, which corresponds to the top boundary being hotter than the bottom one on average.
The mean vertical heat transport $\langle wT \rangle$ in convection between isothermal plates driven by uniform internal heating is investigated by means of rigorous bounds. These are obtained as a function of the Rayleigh number $R$ by constructing feasible solutions to a convex variational problem, derived using a formulation of the classical background method in terms of quadratic auxiliary functions. When the fluid's temperature relative to the boundaries is allowed to be positive or negative, numerical solution of the variational problem shows that best previous bound $\langle wT \rangle \leq 1/2$ can only be improved up to finite $R$. Indeed, we demonstrate analytically that $ \langle wT \rangle \leq 2^{-21/5} R^{1/5}$ and therefore prove that $\langle wT\rangle< 1/2$ for $R < 65\,536$. However, if the minimum principle for temperature is invoked, which asserts that internal temperature is at least as large as the temperature of the isothermal boundaries, then numerically optimised bounds are strictly smaller than $1/2$ until at least $R =3.4\times 10^{5}$. While the computational results suggest that the best bound on $\langle wT\rangle$ approaches $1/2$ asymptotically from below as $R\rightarrow \infty$, we prove that typical analytical constructions cannot be used to prove this conjecture.
We present a mathematical description of turbulent entrainment that is applicable to free shear problems that evolve in space, time or both. Defining the global entrainment velocity $\overline V_g$ to be the fluid motion across an isosurface of an averaged scalar, we find that for a slender flow, $\overline V_g=\overline u_ζ- \overline{D}h_t/\overline{D}t$, where $\overline D/\overline D t$ is the material derivative of the average flowfield and $\overline u_ζ$ is the average velocity perpendicular to the flow direction across the interface located at $ζ=h_t$. The description is shown to reproduce well-known results for the axisymmetric jet, the planar wake and the temporal jet, and provides a clear link between the local (small-scale) and global (integral) descriptions of turbulent entrainment. Application to unsteady jets/plumes demonstrates that, under unsteady conditions, the entrainment coefficient $α$ no longer only captures entrainment of ambient fluid, but also time-dependency effects due to the loss of self-similarity.
We analyse the entrainment and energetics of equal and opposite axisymmetric turbulent air plumes in a vertically confined space at a Rayleigh number of $1.24\times 10^{7}$ using theory and direct numerical simulation. On domains of sufficiently large aspect ratio, the steady state consists of turbulent plumes penetrating an interface between two layers of approximately uniform buoyancy. As described by Baines and Turner (J. Fluid Mech. vol. 37, 1969, pp. 51-80), upon penetrating the interface the flow in each plume becomes forced and behaves like a constant-momentum jet, due to a reduction in its mean buoyancy relative to the local environment. To observe the behaviour of the plumes we partition the domain into sub-domains corresponding to each plume. Domains of relatively small aspect ratio produce a single primary mean-flow circulation between the sub-domains that is maintained by entrainment into the plumes. At larger aspect ratios the mean flow between the sub-domains bifurcates, indicating the existence of a secondary circulation within each layer associated with entrainment into the jets. 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 develop suggests that the kinetic energy dissipation is split equally between the jets and the plumes and, collectively, accounts for almost half of the input of APE at the boundaries. In contrast, $1/4$ of the APE dissipation and background potential energy (BPE) production occurs in the jets, with the remaining $3/4$ occurring in the plumes. These bulk theoretical predictions agree with observations of BPE production from simulations to within $1\%$ and form the basis of a similarity solution that models the vertical dependence of APE dissipation and BPE production.
We determine the smallest instantaneous increase in the strength of an opposing wind that is necessary to permanently reverse the forward displacement flow that is driven by a two-layer thermal stratification. With an interpretation in terms of the flow's energetics, the results clarify why the ventilation of a confined space with a stably-stratified buoyancy field is less susceptible to being permanently reversed by the wind than the ventilation of a space with a uniform buoyancy field. For large opposing wind strengths we derive analytical upper and lower bounds for the system's marginal stability, which exhibit a good agreement with the exact solution, even for modest opposing wind strengths. The work extends a previous formulation of the problem (Lishman & Woods, 2009, Building and Env. 44, pp. 666-673) by accounting for the transient dynamics and energetics associated with the homogenisation of the interior, which prove to play a significant role in buffering temporal variations in the wind.
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 chaotic systems. 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 number of 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.
We present a new DNS data set for a statistically axisymmetric turbulent jet, plume and forced plume in a domain of size $40 r_0 \times 40 r_0 \times 60 r_0$, where $r_0$ is the source diameter. The data set provides evidence of 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 \emph{et al.} experimental data sets, the latter being corrected for a small but influential co-flow that affected the statistics. We show that the turbulence in the core region of the jet and the plume are practically indistinguishable, although the invariants of the anisotropy tensor reveal a significant change in the turbulence 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.