## Publications

26 results found

Wood S, Burridge H, Craske J, 2023, Relating quanta conservation and compartmental epidemiological models of airborne disease outbreaks in buildings, *Scientific Reports*, Vol: 13, ISSN: 2045-2322

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.

Andrian V, Craske J, 2023, Stochastic models of ventilation driven by opposing wind and buoyancy, *Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences*, ISSN: 1364-5021

Stochastic versions of a classical model for natural ventilation are proposedand 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 buoyancyand wind admits multiple steady states can be misleading. With fluctuations in the buoyancy exchanged with an external environment modelledas a Wiener process, such systems tend to reside in the vicinity of globalminima of their potential, rather than states associated with meta-stableequilibria. For a heated space with a leeward low-level and windward highlevel opening, sustained buoyancy-driven flow opposing the wind directionis unlikely for wind strengths exceeding a statistically critical value, whichis slightly larger than the critical value of the wind strength at which bifurcation in the deterministic system occurs. When fluctuations in theapplied wind strength are modelled as an Ornstein-Uhlenbeck process,the topology of the system’s potential is effectively modified due to thenonlinear role that wind strength has in the equation for buoyancy conservation. Consequently, large fluctuations in the wind of sufficiently shortduration rule out the possibility of sustained ventilation opposing the winddirection at large base wind strengths.

Arslan A, Fantuzzi G, Craske J,
et al., 2023, Rigorous scaling laws for internally heated convection at infinite Prandtl number, *Journal of Mathematical Physics*, Vol: 64, Pages: 1-24, ISSN: 0022-2488

We prove rigorous scaling laws for measures of the vertical heat transport enhancement in two models of convection driven by uniform internal heating at infinite Prandtl number. In the first model, a layer of incompressible fluid is bounded by horizontal plates held at the same constant temperature and convection reduces the fraction of the total dimensionless heat input per unit volume and time escaping the layer through the bottom boundary. We prove that this fraction decreases no faster than O(R−2), where R is a “flux” Rayleigh number quantifying the strength of the internal heating relative to diffusion. The second model, instead, has a perfectly insulating bottom boundary, so all heat must escape through the top one. In this case, we prove that the Nusselt number, defined as the ratio of the total-to-conductive vertical heat flux, grows no faster than O(R4). These power-law bounds improve on exponential results available for fluids with finite Prandtl number. The proof combines 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 available at infinite Prandtl number.

Hossain MR, Craske J, van Reeuwijk M, 2022, Reconstructing wall shear stress from thermal wall imprints, *International Journal of Heat and Fluid Flow*, Vol: 95, Pages: 108976-108976, ISSN: 0142-727X

We reconstruct the wall shear stress of plane Couette flow from thermal wall imprints generated by direct numerical simulation at using an imposed surface temperature flux and fixed temperature at the bottom and top boundary, respectively. We explore the strong correlation between wall shear stress and wall temperature by analysing their joint probability density function and cross variance spectrum, before developing a spectral model based on linear regression. We then use observed symmetries in the estimator parameters to reduce the degrees of freedom of the model. The reconstructed wall shear stress reproduces streamwise streaky structures well. The relative error in the -norm of is primarily associated with the absence of local maxima in the reconstructed wall shear stress.

Arslan A, Fantuzzi G, Craske J, et al., 2022, Rigorous scaling laws for internally heated convection at infinite Prandtl number, Publisher: arXiv

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.

Kumar A, Arslan A, Fantuzzi G,
et al., 2022, Analytical bounds on the heat transport in internally heated convection, *Journal of Fluid Mechanics*, ISSN: 0022-1120

We obtain an analytical bound on the mean vertical convective heat flux$\langle w T \rangle$ between two parallel boundaries driven by uniforminternal heating. We consider two configurations, one with both boundaries heldat the same constant temperature, and the other one with a top boundary held atconstant temperature and a perfectly insulating bottom boundary. For the firstconfiguration, Arslan et al. (J. Fluid Mech. 919:A15, 2021) recently providednumerical evidence that Rayleigh-number-dependent corrections to the only knownrigorous bound $\langle w T \rangle \leq 1/2$ may be provable if the classicalbackground method is augmented with a minimum principle stating that thefluid's temperature is no smaller than that of the top boundary. Here, weconfirm this fact rigorously for both configurations by proving bounds on$\langle wT \rangle$ that approach $1/2$ exponentially from below as theRayleigh number is increased. The key to obtaining these bounds are innerboundary layers in the background fields with a particular inverse-powerscaling, which can be controlled in the spectral constraint using Hardy andRellich inequalities. These allow for qualitative improvements in the analysisnot available to standard constructions.

Arslan A, Fantuzzi G, Craske J,
et al., 2021, Bounds on internally heated convection with fixed boundary heat flux, *Journal of Fluid Mechanics*, Vol: 992, Pages: R1-R1, ISSN: 0022-1120

We prove a new rigorous bound for the mean convective heat transport ⟨wT⟩, where w and T are the non-dimensional vertical velocity and temperature, in internally heated convection between an insulating lower boundary and an upper boundary with a fixed heat flux. The quantity ⟨wT⟩ is equal to half the ratio of convective to conductive vertical heat transport, and also to 12 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 ⟨wT⟩≤12(12+13√)−1.6552R−(1/3) uniformly in the Prandtl number, where R is the non-dimensional 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, Internally Heated Convection and Rayleigh–Bénard Convection, Springer, 2016, 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 12 at high R, which corresponds to the top boundary being hotter than the bottom one on average.

Craske J, 2021, Decomposition of available potential energy for networks of connected volumes, *Journal of Fluid Mechanics*, Vol: 920, Pages: 1-29, ISSN: 0022-1120

A decomposition of available potential energy is derived for Boussinesq fluid flow in networks of connected control volumes. The two constituent parts of the decomposition are positive definite and therefore meaningful representations of available energy. The first (inner) part accounts for available potential energy that is intrinsic to each control volume, while the second (outer) part accounts for the context provided by the larger parent volume to which each smaller control volume belongs. While the intended application casts the control volumes as connected rooms in a building, the formulation can be applied to any domain that is partitioned by either physical boundaries or abstract zones and can be invoked recursively to clarify the hierarchical dependence of available potential energy on scale and context. By deriving budgets for the decomposition, two ways in which available potential energy can be redistributed between its inner and outer parts are identified. The first accounts for an apparent generation of available potential energy due to diapycnal mixing within a control volume that is constrained by removable boundaries. The second involves the reversible conversion between inner and outer parts that occurs when mass or heat is transported between control volumes and accounts for the concomitant change in context. Analytical expressions are derived for the hierarchy of contributions to available potential energy in an example involving three connected spaces, before budgets for the decomposition from a direct numerical simulation are analysed. Finally, the dependence of mixing efficiency on remote regions that was identified by Davies Wykes et al. (J. Fluid Mech., vol. 781, 2015, pp. 261–275) is revisited to demonstrate the precise way in which the proposed decomposition quantifies context.

Arslan A, Fantuzzi G, Craske J,
et al., 2021, Bounds on heat transport for convection driven by internal heating, *Journal of Fluid Mechanics*, Vol: 919, Pages: 1-34, ISSN: 0022-1120

The mean vertical heat transport ⟨wT⟩ 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 ⟨wT⟩≤1/2 can only be improved up to finite R. Indeed, we demonstrate analytically that ⟨wT⟩≤2−21/5R1/5 and therefore prove that ⟨wT⟩<1/2 for R<65536. 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=3.4×105. While the computational results suggest that the best bound on ⟨wT⟩ approaches 1/2 asymptotically from below as R→∞, we prove that typical analytical constructions cannot be used to prove this conjecture.

Mader J, Van Reeuwijk M, Craske J, 2021, Confined turbulent convection driven by a combination of line and distributed sources of buoyancy, *Physical Review Fluids*, Vol: 6, Pages: 1-25, ISSN: 2469-990X

We study the flow and thermal stratification of a closed domain subjected to different combinations of line and distributed surface heating and cooling. Our observations are drawn from a set of direct numerical simulations in which the ratio of the strength of the distributed sources to the localised sources \HfRb is varied and shown to play a decisive role in determining the system’s statistically steady state. Domains of sufficient horizontal extent that are (\HfRb=0) produce a stable two-layer stratification. The planar plumes generated by each line source are connected by a large scale circulation over the full depth of the domain and induce secondary circulations within each layer. As the distributed component of the heating, and therefore \HfRb, increases, the buoyancy difference between the layers decreases, before being destroyed when \HfRb>1. For increasing \HfRb∈[0,1], we observe an increasing tilt of the interface between the layers and the eventual disappearance of the secondary circulation cells. The mean buoyancy transport between the two layers of the stable stratification is dominated by the plumes for all $

van Reeuwijk M, Vassilicos JC, Craske J, 2021, Unified description of turbulent entrainment, *Journal of Fluid Mechanics*, Vol: 908, Pages: 1-22, ISSN: 0022-1120

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 V¯g to be the fluid motion across an isosurface of an averaged scalar, we find that for a slender flow, V¯g=u¯ζ−D¯ht/D¯t, where D¯/D¯t is the material derivative of the average flow field and u¯ζ is the average velocity perpendicular to the flow direction across the interface located at ζ=ht. 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.

Craske J, Davies Wykes M, 2020, The entrainment and energetics of turbulent plumes in a confined space, *Journal of Fluid Mechanics*, Vol: 883, Pages: 1-37, ISSN: 0022-1120

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 observatio

Craske J, Hughes G, 2019, On the robustness of emptying filling boxes to sudden changes in the wind, *Journal of Fluid Mechanics*, Vol: 868, ISSN: 0022-1120

We determine the smallest instantaneous increase in the strength of an opposing windthat is necessary to permanently reverse the forward displacement flow that is drivenby a two-layer thermal stratification. With an interpretation in terms of the flow’s ener-getics, the results clarify why the ventilation of a confined space with a stably-stratifiedbuoyancy field is less susceptible to being permanently reversed by the wind than theventilation of a space with a uniform buoyancy field. For large opposing wind strengthswe derive analytical upper and lower bounds for the system’s marginal stability, which ex-hibit 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,Build-ing and Env.44, pp. 666-673) by accounting for the transient dynamics and energeticsassociated with the homogenisation of the interior, which prove to play a significant rolein buffering temporal variations in the wind.

Craske J, 2019, Adjoint sensitivity analysis of chaotic systems using cumulant truncation, *Chaos, Solitons and Fractals*, Vol: 119, Pages: 243-254, ISSN: 0960-0779

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.

Craske J, Salizzoni P, van Reeuwijk M, 2017, The turbulent Prandtl number in a pure plume is 3/5, *Journal of Fluid Mechanics*, Vol: 822, Pages: 774-790, ISSN: 0022-1120

We derive a new expression for the entrainment coefficient in a turbulent plume usingan equation for the squared mean buoyancy. Consistency of the resulting expressionwith previous relations for the entrainment coefficient implies that the turbulent Prandtlnumber in a pure plume is equal to 3/5 when the mean profiles of velocity and buoyancyhave a Gaussian form of equal width. Entrainment can be understood in terms of thevolume flux, the production of turbulence kinetic energy or the production of scalarvariance for either active or passive variables. The equivalence of these points of viewindicates how the entrainment coefficient and the turbulent Prandtl and Schmidt numbersdepend on the Richardson number of the flow, the ambient stratification and the relativewidths of the velocity and scalar profiles. The general framework is valid for self-similarplumes, which are characterised by a power-law scaling. For jets and pure plumes it isshown that the derived relations are in reasonably good agreement with results fromdirect numerical simulations and experiments.

Van Reeuwijk M, sallizoni P, Hunt GR,
et al., 2016, Turbulent transport and entrainment in jets and plumes: A DNS study, *Physical Review Fluids*, Vol: 1, ISSN: 2469-990X

We present a direct numerical simulation (DNS) data set for a statistically axisymmetric turbulent jet, plume, and forced plume in a domain of size 40r0×40r0×60r0, where r0 is the source diameter. The data set supports the validity of the Priestley-Ball entrainment model in unstratified environments (excluding the region near the source) [Priestley and Ball, Q. J. R. Meteor. Soc. 81, 144 (1955)], which is corroborated further by the Wang-Law and Ezzamel et al. experimental data sets [Wang and Law, J. Fluid Mech. 459, 397 (2002); Ezzamel et al., J. Fluid Mech. 765, 576 (2015)], the latter being corrected for a small but influential coflow 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 from each other, although there are significant differences near the plume edge. The DNS data indicate 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 do not show any evidence of similarity drift associated with gradual variations in the ratio of buoyancy profile to velocity profile widths.

Craske J, 2016, The properties of integral models for planar and axisymmetric unsteady jets, *IMA Journal of Applied Mathematics*, ISSN: 0272-4960

This article reviews and builds upon recent progress that has been made in understanding the mathematical properties of integral models for unsteady turbulent jets. The focus is on models that describe the evolution of the volume flux and the momentum flux in a jet, whose source conditions are time dependent. A generalized approach that postpones making assumptions about the ‘internal’ properties of the flow, such as the radial dependence of the longitudinal velocity profile, turbulent transport and pressure, allows one to understand how the resulting integral equations are affected by model-specific assumptions. Whereas the assumptions invoked in previous unsteady jet models have resulted in a parabolic system of equations, generalized equations that are derived from first principles have a hyperbolic character and statistical stability that depends sensitively on assumptions that are normally invoked a priori. Unsteady axisymmetric jets with Gaussian velocity profiles have special properties, including a tendency to remain straight-sided (conical) and marginal stability in response to source perturbations. A distinct difference between planar jets and axisymmetric jets is that the mean energy flux, which plays a leading-order role in determining the unsteady dynamics of jets, is significantly lower in planar jets. We hypothesize that in order to maintain marginal stability the turbulence and pressure fields in planar jets adjust themselves, relative to axisymmetric jets, to compensate for the lower mean energy flux.

Craske J, van Reeuwijk M, 2016, Generalised unsteady plume theory, *Journal of Fluid Mechanics*, Vol: 792, Pages: 1013-1052, ISSN: 0022-1120

We develop a generalised unsteady plume theory and compare it with a new direct numerical simulation (DNS) dataset for an ensemble of statistically unsteady turbulent plumes. The theoretical framework described in this paper generalises previous models and exposes several fundamental aspects of the physics of unsteady plumes. The framework allows one to understand how the structure of the governing integral equations depends on the assumptions one makes about the radial dependence of the longitudinal velocity, turbulence and pressure. Consequently, the ill-posed models identified by Scase & Hewitt (J. Fluid Mech., vol. 697, 2012, p. 455) are shown to be the result of anon-physical assumption regarding the velocity profile. The framework reveals that these ill-posed unsteady plume models are degenerate cases amongst a comparatively large set of well-posed models that can be derived from the generalised unsteady plume equations that we obtain. Drawing on the results of DNS of a plume subjected to an instantaneous step change in its source buoyancy flux, we use the framework in a diagnostic capacityto investigate the properties of the resulting travelling wave. In general, the governing integral equations are hyperbolic, becoming parabolic in the limiting case of a `top-hat' model, and the travelling wave can be classified as lazy, pure or forced according to the particular assumptions that are invoked to close the integral equations. Guided by observations from the DNS data, we use the framework in a prognostic capacity to develop a relatively simple, accurate and well-posed model of unsteady plumes that is based on the assumption of a Gaussian velocity profile. An analytical solution is presented for a pure straight-sided plume that is consistent with the key features observed from the DNS.

Van Reeuwijk M, Craske J, 2015, Energy-consistent entrainment relations for jets and plumes, *Journal of Fluid Mechanics*, Vol: 782, Pages: 333-355, ISSN: 0022-1120

We discuss energetic restrictions on the entrainment coefficient α for axisymmetric jets and plumes. The resulting entrainment relation includes contributions from the mean flow, turbulence and pressure, fundamentally linking α to the production of turbulence kinetic energy, the plume Richardson number Ri and the profile coefficients associated with the shape of the buoyancy and velocity profiles. This entrainment relation generalises the work by Kaminski et al. (J. Fluid Mech., vol. 526, 2005, pp. 361–376) and Fox (J. Geophys. Res., vol. 75, 1970, pp. 6818–6835). The energetic viewpoint provides a unified framework with which to analyse the classical entrainment models implied by the plume theories of Morton et al. (Proc. R. Soc. Lond. A, vol. 234, 1955, pp. 1–23) and Priestley & Ball (Q. J. R. Meteorol. Soc., vol. 81, 1954, pp. 144–157). Data for pure jets and plumes in unstratified environments indicate that to first order the physics is captured by the Priestley and Ball entrainment model, implying that (1) the profile coefficient associated with the production of turbulence kinetic energy has approximately the same value for pure plumes and jets, (2) the value of α for a pure plume is roughly a factor of 5/3 larger than for a jet and (3) the enhanced entrainment coefficient in plumes is primarily associated with the behaviour of the mean flow and not with buoyancy-enhanced turbulence. Theoretical suggestions are made on how entrainment can be systematically studied by creating constant- Ri flows in a numerical simulation or laboratory experiment.

Craske J, Debugne ALR, van Reeuwijk M, 2015, Shear-flow dispersion in turbulent jets, *Journal of Fluid Mechanics*, Vol: 781, Pages: 28-51, ISSN: 0022-1120

We investigate the transport of a passive scalar in a fully developed turbulent axisymmetric jet at a Reynolds number of Re = 4815 using data from direct numerical simulation. In particular, we simulate the response of the concentration field to an instantaneous variation of the scalar flux at the source. To analyse the time evolution of this statisticallyunsteady process we take an ensemble average over 16 independent simulations. We find that the evolution of Cm(z, t), the radial integral of the ensemble-averaged concentration, is a self-similar process, with front position and spread both scaling as √t. The longitudinal mixing of Cm is shown to be primarily caused by shear-flow dispersion.Using the approach developed by Craske & van Reeuwijk (J. Fluid Mech., vol. 763, 2014,pp. 538–566), the classical theory for shear-flow dispersion is applied to turbulent jets to obtain a closure that couples the integral scalar flux to the integral concentration Cm. Model predictions using the dispersion closure are in good agreement with the simulation data. Application of the dispersion closure to a two-dimensional jet results in an integraltransport equation that is fully consistent with that of Landel et al. (J. Fluid Mech., vol.711, 2012, pp. 212–258)

Craske J, van Reeuwijk M, 2015, Energy dispersion in turbulent jets. Part 2. A robust model for unsteady jets, *Journal of Fluid Mechanics*, Vol: 763, Pages: 538-566, ISSN: 0022-1120

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.

Craske J, van Reeuwijk M, 2015, Dispersion in unsteady jets and plumes

We investigate the transport of both passive and active scalars in fully developed turbulent axisymmetric jets and plumes using data from direct numerical simulation. In both cases we simulate the response of the flow to an instantaneous increase in the scalar flux at the source and our focus is on the determination of the rate at which the resulting disturbance propagates and spreads in the longitudinal direction. We apply Taylor’s theory of shear-flow dispersion [9] to free-shear flows and therefore model the way in which departures from self-similarity result in the longitudinal mixing of integral quantities. The resulting integral models exhibit a good agreement with the simulation data and, in the case of passive scalar transport, admit an analytical similarity solution. For the case of active scalar transport we examine the buoyancy flux in an unsteady plume and show that the momentum–energy framework [7], rather than the classical volume–momentum framework [6], provides the natural setting from which to view the effects of dispersion. Consequently, we demonstrate the effect that dispersion has on turbulent entrainment and the way in which a plume responds to source perturbations in its buoyancy flux.

Van Reeuwijk M, Salizzoni P, Craske J, 2015, Turbulent entrainment in jets and plumes, Pages: 175-178

We perform direct simulation of a statistically steady jets, forced plume and pure plume in a neutral environment and present the value of the entrainment coefficient decomposed into 1) turbulence production; 2) buoyancy effects; and 3) deviations from self-similarity. The value of the decomposed entrainment coefficient is in excellent agreement with a direct estimate from the volume conservation equation. It is shown that the Priestley and Ball entrainment model describes the entrainment physics reasonably well.

Craske J, van Reeuwijk M, 2015, Energy dispersion in turbulent jets. Part 1. Direct simulation of steady and unsteady jets, *JOURNAL OF FLUID MECHANICS*, Vol: 763, Pages: 500-537, ISSN: 0022-1120

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van Reeuwijk M, Salizzoni P, Craske J, 2015, Turbulent entrainment in jets and plumes, 8th International Symposium On Turbulence Heat and Mass Transfer (THMT), Publisher: BEGELL HOUSE, INC, Pages: 175-178, ISSN: 2377-2816

Craske JC, van Reeuwijk M, 2013, Robust and accurate open boundary conditions for incompressible turbulent jets and plumes, *Computers and Fluids*

We show that a popular convective open boundary condition (OBC) is unsuitable in the direct simulation of incompressible turbulent jets and plumes, because (1) the boundary condition modifies their spreading rate; (2) the results are domain dependent; and (3) the boundary condition is liable to cause instability and therefore requires domains that are much larger than the area of interest. We demonstrate the accuracy of new axisymmetric OBCs compared to the standard OBC by conducting direct numerical simulation (DNS) of a turbulent plume and a turbulent jet. The new OBCs conform to the fundamental features of statistically axisymmetric turbulent flows, regardless of the computational domains on which they are imposed. They do not contain tunable parameters and are dynamical, accounting for the strength and extent of a flow at a given time, which eliminates the need for calibration to particular cases. The implementation presented herein is computationally efficient and robust in the vicinity of turbulent flows

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