## Publications

59 results found

Grylls T, Suter I, Van Reeuwijk M, 2020, Steady-state large-eddy simulations of convective and stable urban Boundary layers, *Boundary-Layer Meteorology: an international journal of physical and biological processes in the atmospheric boundary layer*, Vol: 175, Pages: 309-341, ISSN: 0006-8314

A comprehensive investigation is carried out to establish best practice guidelines for the modelling of statistically steady-state non-neutral urban boundary layers (UBL) using large-eddy simulation (LES). These steady-state simulations enable targeted studies under realistic non-neutral conditions without the complications associated with the inherently transient nature of the UBL. An extensive set of simulations of convective and stable conditions is carried out to determine which simplifications, volumetric forcings, and boundary conditions can be applied to replicate the mean and turbulent (variance and covariance) statistics of this intrinsically transient problem most faithfully. In addition, a new method is introduced in which a transient simulation can be ‘frozen’ into a steady state. It is found that non-neutral simulations have different requirements to their neutral counterparts. In convective conditions, capping the boundary-layer height h with the top of the modelled domain to h/5 and h/10 (which is common practice in neutral simulations) reduces the turbulent kinetic energy by as much as 61% and 44%, respectively. Consistent with the literature, we find that domain heights lz≥5|L| are necessary to reproduce the convective-boundary-layer dynamics, where L is the Obukhov length. In stably stratified situations, the use of a uniform momentum forcing systematically underestimates the mechanical generation of turbulence over the urban canopy layer, and therefore leads to misrepresentations of both the inner- and outer-layer dynamics. The new ‘frozen-transient’ method that is able to maintain a prescribed flow state (including entrainment at the boundary-layer top) is shown to work well in both stable and convective conditions. Guidelines are provided for future studies of the capped and uncapped convective and stable UBL.

Nair V, Heus T, Van Reeuwijk M, 2020, Dynamics of subsiding shells in actively growing clouds with vertical updrafts, *Journal of the Atmospheric Sciences*, Vol: 77, Pages: 1353-1369, ISSN: 0022-4928

The dynamics of a subsiding shell at the edges of actively growing shallow cumulus clouds with updrafts is analyzed using direct numerical simulation. The actively growing clouds have a fixed in-cloud buoyancy and velocity. Turbulent mixing and evaporative cooling at the cloud edges generate a subsiding shell which grows with time. A self-similar regime is observed for first and second order moments when normalized with respective maximum values. Internal scales derived from integral properties of the flow problem are identified. Self-similarity analysis conducted by normalizing using these scales reveal that contrary to classical self similar flows, the turbulent kinetic energy budget terms and velocity moments scale according to the buoyancy and not with the mean velocity. The shell thickness is observed to increase linearly with time. The buoyancy scale remains time-invariant and is set by the initial cloud-environment thermodynamics. The shell accelerates ballistically with a magnitude set by the saturation value of the buoyancy of the cloud-environment mixture. In this regime, the shell is buoyancy driven and independent of the in-cloud velocity. Relations are obtained for predicting the shell thickness and minimum velocities by linking the internal scales with external flow parameters. The values thus calculated are consistent with the thickness and velocities observed in typical shallow cumulus clouds. The entrainment coefficient is a function of the initial state of the cloud and the environment, and is shown to be of the same order of magnitude as fractional entrainment rates calculated for large scale models.

Verso L, van Reeuwijk M, Liberzon A, 2019, Transient stratification force on particles crossing a density interface, *INTERNATIONAL JOURNAL OF MULTIPHASE FLOW*, Vol: 121, ISSN: 0301-9322

Reeuwijk MV, Holzner M, Caulfield CP, 2019, Mixing and entrainment are suppressed in inclined gravity currents, *Journal of Fluid Mechanics*, Vol: 873, Pages: 786-815, ISSN: 0022-1120

We explore the dynamics of inclined temporal gravity currents using directnumerical simulation, and find that the current creates an environment in which the flux Richardson number $Ri_f$, gradient Richardson number $Ri_g$, and turbulent flux coefficient $\Gamma$ are constant across a large portion of the depth. Changing the slope angle $\alpha$ modifies these mixing parameters, and the flow approaches a maximum Richardson number $Ri_\textrm{max}\approx 0.15$ as $\alpha \rightarrow 0$ at which the entrainment coefficient $E \rightarrow 0$. The turbulent Prandtl number remains $O(1)$ for all slope angles, demonstrating that $E\rightarrow 0$ is not caused by a switch-off of the turbulent buoyancy flux as conjectured by Ellison (1957). Instead, $E\rightarrow 0$ occurs as the result of the turbulence intensity going to zero as $\alpha\rightarrow 0$, due to the flow requiring larger and larger shear to maintain the same level of turbulence. We develop an approximate model valid for small $\alpha$ which is able to predict accurately $Ri_f$, $Ri_g$ and $\Gamma$ as a function of $\alpha$ and their maximum attainable values. The model predicts an entrainment law of the form $E=0.31(Ri_\textrm{max}-Ri)$, which is in good agreement with the simulation data. The simulations and model presented here contribute to a growing body of evidence that an approach to a marginally or critically stable, relatively weakly stratified equilibrium for stratified shear flows may well be a generic property of turbulent stratified flows.

Grylls T, Le Cornec CMA, Salizzoni P,
et al., Evaluation of an operational air quality model using large-eddy simulation, *Atmospheric Environment: X*, Vol: 3, ISSN: 2590-1621

The large-eddy simulation (LES) model uDALES is used to evaluate the predictive skill of the operational air quality model SIRANE. The use of LES in this study presents a novel approach to air quality model evaluation, avoiding sources of uncertainty and providing numerical control that permits systematic analysis of targeted parametrisations and assumptions.A case study is conducted over South Kensington, London with the morphology, emissions, meteorological conditions and boundary conditions carefully matched in both models. The dispersion of both inert (NOx) and reactive (NO, NO2 and O3) pollutants under neutral, steady-state conditions is simulated for a south-westerly and westerly wind direction. A quantitative comparison between the two models is performed using statistical indices (the fractional bias, FB, the normalised mean squared error, NMSE, and the fraction in a factor of 2, FAC2).SIRANE is shown to successfully capture the dominant trends with respect to canyon-averaged concentrations of inert NOx (FB = -0.08, NMSE = 0.08 and FAC2 = 1.0). The prediction of along-canyon velocities is shown to exhibit sources of systematic error dependant on the angle of incidence of the mean wind (FB = -0.18). The assumption of photostationarity within SIRANE (deviations from equilibrium of up to 170% exist close to busy roads) is also identified as a significant source of systematic bias resulting in over- and underpredictions of NO2 (FB = -0.18) and O3 (FB = 0.14) respectively. The validity of the assumed uniform in-canyon concentration is assessed by analysing the pedestrian, leeward and windward concentrations resolved in uDALES. The use of canyon-averaged concentrations to predict pedestrian level exposure is shown to result in significant underestimations. Linear regression is used to effectively capture the relationship between pedestrian- and canyon-averaged concentrations in uDALES. Correction factors are derived (m ≈ 1.62 and R 2 = 0.92 for inert NOx) th

Fouxon I, Schmidt L, Ditlevsen P,
et al., 2018, Inhomogeneous growth of fluctuations of concentration of inertial particles in channel turbulence, *PHYSICAL REVIEW FLUIDS*, Vol: 3, ISSN: 2469-990X

Alpresa P, Sherwin S, Weinberg P,
et al., 2018, Orbitally shaken shallow fluid layers. II. An improved wall shear stress model, *PHYSICS OF FLUIDS*, Vol: 30, ISSN: 1070-6631

A new model for the analytical prediction of wall shear stress distributions at the base of orbitally shaken shallow fluid layers is developed. This model is a generalisation of the classical extended Stokes solution and will be referred to as the potential theory-Stokes model. The model is validated using a large set of numerical simulations covering a wide range of flow regimes representative of those used in laboratory experiments. It is demonstrated that the model is in much better agreement with the simulation data than the classical Stokes solution, improving the prediction in 63% of the studied cases. The central assumption of the model—which is to link the wall shear stress with the surface velocity—is shown to hold remarkably well over all regimes covered.

Alpresa P, Sherwin S, Weinberg P,
et al., 2018, Orbitally shaken shallow fluid layers. I. Regime classification, *PHYSICS OF FLUIDS*, Vol: 30, ISSN: 1070-6631

Orbital shakers are simple devices that provide mixing, aeration, and shear stress at multiple scales and high throughput. For this reason, they are extensively used in a wide range of applications from protein production to bacterial biofilms and endothelial cell experiments. This study focuses on the behaviour of orbitally shaken shallow fluid layers in cylindrical containers. In order to investigate the behaviour over a wide range of different conditions, a significant number of numerical simulations are carried out under different configuration parameters. We demonstrate that potential theory—despite the relatively low Reynolds number of the system—describes the free-surface amplitude well and the velocity field reasonably well, except when the forcing frequency is close to a natural frequency and resonance occurs. By classifying the simulations into non-breaking, breaking, and breaking with part of the bottom uncovered, it is shown that the onset of wave breaking is well described by Δh/(2R) = 0.7Γ, where Δh is the free-surface amplitude, R is the container radius, and Γ is the container aspect ratio; Δh can be well approximated using the potential theory. This result is in agreement with standard wave breaking theories although the significant inertial forcing causes wave breaking at lower amplitudes.

van Reeuwijk M, Jonker HJJ, 2018, Understanding Entrainment Processes in the Atmosphere: The Role of Numerical Simulation, 10th ERCOFTAC Workshop on Direct and Large Eddy Simulation (DLES), Publisher: SPRINGER INTERNATIONAL PUBLISHING AG, Pages: 53-60, ISSN: 1382-4309

Verso L, van Reeuwijk M, Liberzon A, 2017, Steady state model and experiment for an oscillating grid turbulent two-layer stratified flow, *PHYSICAL REVIEW FLUIDS*, Vol: 2, ISSN: 2469-990X

Turbulence generated by an oscillating grid in a two-layer stably stratified system is a classical flow utilized to study various aspects of turbulence in presence of stratification without mean shear. This flow evolves in a quasisteady state, in which the layer thickness and density difference evolves in a quasisteady manner due to the large separation of timescales between the turbulence and the setup. We present an extension of the classical setup that enables full steady state conditions and in which the entrainment velocity can be prescribed separately from the Richardson number. We develop a theoretical box-model and show that the model is in good agreement with the experiments. The model allows to predict the transient response of the system for a variety of initial conditions and the imposed steady state. The new setup is necessary to obtain the steady position of the density interface, for example, when using advanced optical techniques to measure the small-scale features of turbulence near the interface.

Ghim M, Alpresa P, Yang S,
et al., 2017, Visualization of three pathways for macromolecule transport across cultured endothelium and their modification by flow., *AJP - Heart and Circulatory Physiology*, Vol: 313, Pages: H959-H973, ISSN: 1522-1539

Transport of macromolecules across vascular endothelium and its modification by fluid mechanical forces are important for normal tissue function and in the development of atherosclerosis. However, the routes by which macromolecules cross endothelium, the hemodynamic stresses that maintain endothelial physiology or trigger arterial disease, and the dependence of transendothelial transport on hemodynamic stresses are controversial. Here we visualised pathways for macromolecule transport and determined the effect on these pathways of different types of flow. Endothelial monolayers were cultured under static conditions or on an orbital shaker producing different flow profiles in different parts of the wells. Fluorescent tracers that bound to the substrate after crossing the endothelium were used to identify transport pathways. Maps of tracer distribution were compared with numerical simulations of flow to determine effects of different shear stress metrics on permeability. Albumin-sized tracers dominantly crossed the cultured endothelium via junctions between neighbouring cells, high-density-lipoprotein-sized tracers crossed at tricelluar junctions whilst low-density-lipoprotein-sized tracers crossed through cells. Cells aligned close to the angle that minimised shear stresses across their long axis. The rate of paracellular transport under flow correlated with the magnitude of these minimised transverse stresses, whereas transport across cells was uniformly reduced by all types of flow. These results contradict the long-standing two-pore theory of solute transport across microvessel walls and the consensus view that endothelial cells align with the mean shear vector. They suggest that endothelial cells minimise transverse shear, supporting its postulated pro-atherogenic role. Preliminary data show that similar tracer techniques are practicable in vivo.

Verso L, van Reeuwijk M, Gurka R,
et al., 2017, Experimental study of the initial growth of a localized turbulent patch in a stably stratified fluid, *International Journal of Heat and Fluid Flow*, Vol: 66, Pages: 127-136, ISSN: 0142-727X

We present a laboratory experiment of the initial growth of a turbulent patch in a stably stratified fluid. The patch is created due to a localized source of turbulence, generated by a horizontally oriented and vertically oscillating grid much smaller than the tank size and far from solid boundaries. Synchronized and overlapping particle image velocimetry(PIV) and planar laser induced fluorescence (PLIF) measurements capture the evolution of the patch through its initial growth until it reached a maximum size. The simultaneous measurements of density and velocity fields allow for a direct quantification of the distribution of kinetic energy, buoyancy and degree of mixing within the patch. We can also relate the propagation speed of the turbulent/non-turbulent interface and its thickness to the properties of the turbulent fluid inside the evolving patch. The velocity measurements in this setup indicate significant transient effects inside the patch during its growth. A local analysis of the turbulent/non-turbulent interface provides direct measurements of the entrainment velocity we as compared to the local vertical velocity and turbulent intensity at the proximity of the interface. The detailed information about the growth of localized sources of turbulence in stratified environment might be of use in stealth design of autonomous underwater vehicles.

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: 1469-7645

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.

Krug D, Holzner M, Marusic I,
et al., 2017, Fractal scaling of the turbulence interface in gravity currents, *Journal of Fluid Mechanics*, Vol: 820, ISSN: 1469-7645

It was previously observed by Krug et al. (J. Fluid Mech., vol. 765, 2015, pp. 303–324) that the surface area of the turbulent/non-turbulent interface (TNTI) in gravity currents decreases with increasing stratification, significantly reducing the entrainment rate. Here, we consider the multiscale properties of this effect using direct numerical simulations of temporal gravity currents with different gradient Richardson numbers . Our results indicate that the reduction of is caused by a decrease of the fractal scaling exponent , while the scaling range remains largely unaffected. We further find that convolutions of the TNTI are characterized by different length scales in the streamwise and wall-normal directions, namely the integral scale and the shear scale (formed using the mean shear and the turbulent kinetic energy ) respectively. By recognizing that the anisotropy implied by the different scaling relations increases with increasing , we are able to model the dependence of in good agreement with the data.

Bozovic R, Maksimovic C, Mijic A, et al., 2017, Blue Green Solutions. A Systems Approach to Sustainable and Cost-Effective Urban Development

This guide presents an innovative framework to systematically unlock the multiple benefits of city natural infrastructure; thus producing resilient, sustainable and cost-effective solutions. The framework is applicable at a building, neighbourhood and city-scale and is suitable both for new and retrofit developments.

Van Reeuwijk M, Krug D, Holzner M, 2017, Small-scale entrainment in inclined gravity currents, *Environmental Fluid Mechanics*, Vol: 18, Pages: 225-239, ISSN: 1573-1510

We investigate the effect of buoyancy on the small-scale aspects ofturbulent entrainment by performing direct numerical simulation of a gravity cur-rent and a wall jet. In both flows, we detect the turbulent/nonturbulent interfaceseparating turbulent from irrotational ambient flow regions using a range of en-strophy iso-levels spanning many orders of magnitude. Conform to expectation,the relative enstrophy isosurface velocityvnin the viscous superlayer scales withthe Kolmogorov velocity for both flow cases. We connect the integral entrainmentcoefficientEto the small-scale entrainment and observe excellent agreement be-tween the two estimates throughout the viscous superlayer.The contribution ofbaroclinic torque tovnis negligible, and we show that the primary reason forreduced entrainment in the gravity current as compared to the wall-jet is thereduction in the surface area of the isosurfaces.

Suter I, Maksimovic C, Van Reeuwijk M, 2017, A neighbourhood-scale estimate for the cooling potential of green roofs, *Urban Climate*, Vol: 20, Pages: 33-45, ISSN: 2212-0955

Green roofs offer the possibility to mitigate multiple environmental issues in an urban environment. A common benefit attributed to green roofs is the temperature reduction through evaporation. This study focuses on evaluating the effect that evaporative cooling has on outdoor air temperatures in an urban environment. An established urban energy balance model was modified to quantify the cooling potential of green roofs and study the scalability of this mitigation strategy. Simulations were performed for different climates and urban geometries, with varying soil moisture content, green roof fraction and urban surface layer thickness. All simulations show a linear relationship between surface layer temperature reduction ΔTs and domain averaged evaporation rates from vegetation mmW, i.e. ΔTs = eW ⋅ mmW, where eW is the evaporative cooling potential with a value of ∼ −0.35 Kdaymm−1. This relationship is independent of the method by which water is supplied. We also derive a simple algebraic relation for eW using a Taylor series expansion.

Holzner M, van Reeuwijk M, 2017, The turbulent/nonturbulent interface in penetrative convection, *JOURNAL OF TURBULENCE*, Vol: 18, Pages: 260-270, ISSN: 1468-5248

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.

Schmidt L, Fouxon I, Krug D,
et al., 2016, Clustering of particles in turbulence due to phoresis, *Physical Review E*, Vol: 93, ISSN: 1539-3755

We demonstrate that diffusiophoretic, thermophoretic, and chemotactic phenomena in turbulence lead to clustering of particles on multifractal sets that can be described using one single framework, valid when the particle size is much smaller than the smallest length scale of turbulence l0. To quantify the clustering, we derive positive pair correlations and fractal dimensions that hold for scales smaller than l0. For scales larger than l0 the pair-correlation function is predicted to show a stretched exponential decay towards 1. In the case of inhomogeneous turbulence we find that the fractal dimension depends on the direction of inhomogeneity. By performing experiments with particles in a turbulent gravity current we demonstrate clustering induced by salinity gradients in conformity to the theory. The particle size in the experiment is comparable to l0, outside the strict validity region of the theory, suggesting that the theoretical predictions transfer to this practically relevant regime. This clustering mechanism may provide the key to the understanding of a multitude of processes such as formation of marine snow in the ocean and population dynamics of chemotactic bacteria.

Craske J, van Reeuwijk M, 2016, Generalised unsteady plume theory, *Journal of Fluid Mechanics*, Vol: 792, Pages: 1013-1052, ISSN: 1469-7645

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.

Morel CRG, van Reeuwijk M, Graf T, 2015, Systematic investigation of non-Boussinesq effects in variable-density groundwater flow simulations, *JOURNAL OF CONTAMINANT HYDROLOGY*, Vol: 183, Pages: 82-98, ISSN: 0169-7722

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Van Reeuwijk M, Craske J, 2015, Energy-consistent entrainment relations for jets and plumes, *Journal of Fluid Mechanics*, Vol: 732, Pages: 333-355, ISSN: 1469-7645

We discuss energetic restrictions on the entrainment coefficient for axisymmetric jets and plumes. The resulting entrainment relation includes contributions from the mean flow, turbulence and pressure, fundamentally linking 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, Taylor and Turner (Proc. Roy. Soc. London A vol. 234, 1955, pp.1{23) and Priestley and Ball (Q. J. R. Meteorol. Soc. vol. 81, 1954, pp. 144{157). Datafor pure jets and plumes in unstratified environments indicate that to first order the physics are 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 for a pure plume is roughly a factor 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)

van Reeuwijk M, Hadziabdic M, 2015, Modelling high Schmidt number turbulent mass transfer, *INTERNATIONAL JOURNAL OF HEAT AND FLUID FLOW*, Vol: 51, Pages: 42-49, ISSN: 0142-727X

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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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- Citations: 25

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

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van Reeuwijk M, Holzner M, 2014, The turbulence boundary of a temporal jet, *J. Fluid Mech.*, Vol: accepted

Lari KS, van Reeuwijk M, Maksimovic C, 2013, The role of geometry in rough wall turbulent mass transfer (vol 49, pg 1191, 2013), *HEAT AND MASS TRANSFER*, Vol: 49, Pages: 1523-1523, ISSN: 0947-7411

Jonker HJJ, van Reeuwijk M, Sullivan PP,
et al., 2013, On the scaling of shear-driven entrainment: a DNS study, *Journal of Fluid Mechanics*, Vol: 732, Pages: 150-165, ISSN: 0022-1120

The deepening of a shear-driven turbulent layer penetrating into stably stratified quiescent layer is studied using Direct Numerical Simulation (DNS). The simulation design mimics the classical laboratory experiments by Kato and Phillips (J. Fluid Mech. 37, 643–655, 1969) in that it starts with linear stratification and applies a constant shear stress at the lower boundary, but avoids inherent side wall and rotation effects of that experiment. It is found that the layers universally deepen as a function of the square root of time, independent of the initial stratification and the Reynolds number of the simulations, provided that the Reynolds number is large enough. Consistent with this finding, the dimensionless entrainment velocity varies with the bulk Richardson number as Ri−1/2 . In addition it is observed that all cases evolve in a self-similar fashion. A selfsimilarity analysis of the conservation equations shows that only a square root growth law is consistent with self-similar behaviour.

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