Lattice Boltzmann method (LBM)-based large eddy simulations (LES) are increasingly used in applied engineering applications characterised by large Reynolds numbers. Classically, the computational grid of a well-conditioned LES resolves 80% of the energy spectrum, with a subgrid-scale (SGS) turbulence model used to parameterise the unresolved scales. Despite being widely adopted, this philosophy is increasingly challenged by an alternate approach, wherein numerical damping mechanisms replace an explicit SGS model, termed implicit LES (ILES). Generally, there remains a lack of consensus on the efficacy of these approaches within both the LBM and broader LES communities. This study investigates the qualitative and quantitative differences between various sub-grid scale treatments for a D3Q27 cumulant-LBM-based LES solver, including commonly used SGS models, cumulant regularisation, and pure ILES, for both a single-mode Kelvin–Helmholtz instability (KHI) and homogeneous decaying turbulence (HDT). For the KHI case, validation against experimental data at Re=1.6×103 demonstrates strong agreement in both momentum thickness and vortex structure. Additionally, we compare the LBM solution with the numerical results of a fifth-order Godunov scheme using a low Mach number correction (LMC), demonstrating that cumulant-LBM achieves comparable resolution requirements for capturing the same wavenumber features. Furthermore, we analyse instability development at increasingly high Reynolds numbers, Re=1.0×104, and 1.0×106, providing insight into the numerical characteristics of each method. The HDT case builds upon this foundation, giving further insight into the scale-dependent dissipation characteristics of each LBM approach. The results highlight that the cumulant-LBM maintains sensible dissipation as grid resolution decreases and remains stable up to at least Re=1.0×106 on very coarse meshes, demonstrating its applicability to high Reynolds number flows.

Direct numerical simulations of stratified open channel flows subject to a varying surface heat flux are performed. The influence of the diurnal heating time on the spatial and temporal variation of mixing in the flow and the characteristics of the mean flow state are examined. The control parameters are the bulk stability parameter λ_B , defined through the ratio of the channel height δ and a bulk Obukhov length scale ℒ_B , and the diurnal time scale t̂ , defined as the ratio of the heating time to an eddy turnover time. The Prandtl number Pr and Reynolds number Re_τ have values of 1 and 400. Simulations are performed over t̂ = 1 to 24 and λ_B = 0.6 to 26. Two key flow features are used to classify the flow regimes observed, namely the laminar layer depth (LLD) and stratified layer depth (SLD) where the LLD is defined as the depth from the free surface when the buoyancy Reynolds number Re_B≈ 7 and the SLD is the depth from the free surface when the turbulent Froude number Fr ≈ 1 . This study attempts to characterise how these length scales vary across the diel cycle. The LLD is a viscous length scale and a regime map of a viscous parameter, the bulk Obhukov Reynolds number Re_ℒ , and t̂ is presented to classify the LLD behaviour. A regime map of λ _B and t̂ is presented to classify the behaviour of the SLD. Three classifications for each layer depth behaviour within a diel cycle form the basis of the regime maps for this paper: a neutral flow where the LLD or SLD does not exist (denoted by NL and NS), a stratified flow where the LLD or SLD are diurnally varying (denoted as DL and DS) and a persistent layer of the LLD or SLD (denoted as PL and PS). The transition between the NL to DL is t̂∝ Re_ℒ^4.5 , DL to PL is t̂∝ Re_ℒ^- 0.5 , NS to DS is t̂∝λ _B^0 and DS to PS is t̂∝λ _B^1 . The regime maps may be used as a predictive tool to determine when suppressed mixing regimes occur in rivers. At each flow depth, the flow sweeps though a range of mixing states across the diel cycle. The local mixing efficiency are briefly assessed and found to scale well with the instantaneous Fr number according to the regimes proposed by Garanaik and Venayagamoorthy (J. Fluid Mech., vol. 867, 2019, pp. 323-333). This paper reports on direct numerical simulations of stratified open channel flows subject to a varying surface heat flux. The results have found that:

This study utilizes a detailed analysis of the first direct numerical simulation (DNS) of thermally stratified open-channel flows through idealized sine-generated meanders with varying sinuosity to enhance our understanding of the impact of thermal stratification on the mean flow, secondary currents, and boundary shear stresses. Four different channel sinuosities Si = (1.07; 1.34; 2.12; 3.70) at a moderate curvature of R-min/B = 2.5 and channel aspect ratio B/H = 10 with friction Reynolds number Re-tau = 200 in neutral and density stratified flows are presented. The results indicate that thermal stratification has a significant effect on the characteristics of the mean flow, resulting in differences in the separated shear layers and separation zones for different sinuosity, as well as in the bed and sidewall shear stress distributions for both neutral and stratified flows. (c) 2023 American Society of Civil Engineers.

A “fountain filling box” flow produced by discharging a weak laminar plane fountain in a confined open channel is studied numerically. Two‐dimensional direct numerical simulations were performed for weak plane fountains. The development of the fountain flow experiences five stages; the initial upflow and the subsequent downflow after the fountain penetrates to the maximum height, followed by the outward movement of the intrusion of the fallen fountain fluid on the channel bottom, and then the wall fountain formed by the impingement of the intrusion on the vertical sidewall, which results in the reversed flow, and finally the gradual stratification of the fluid. The behavior of the intrusion can be approximately described with the plane gravity current theory. The period for the intrusion to reach the bounded side wall increases with increasing Re or decreasing Fr . Three regimes are found for the wall fountain behavior; “no‐falling,” “slumping down,” and “rolling down” behavior. Convection, mixing, conduction, and filling all contribute to the formation and development of stratification, but their effects vary at different stages. For the initial stages, convection and mixing play a key role, resulting in an increasing bulk entrainment rate, while conduction and filling are dominant after quasi‐steady stratification is created, presenting a decreasing bulk entrainment rate.

In order to unravel the effects of meander bends on the mechanism by which the turbulence kinetic energy (TKE) is produced, redistributed and dissipated in a thermally stratified curved open-channel flow, Direct Numerical Simulation (DNS) was carried out through an idealized sharp meander with and without an internal heat source. The heat source models radiative heating from above and varies with height due to progressive absorption. In both cases, we find a pronounced local increase in TKE starting in the region approximately 25% of the distance through the bend. Here a local extremum of TKE is found near the channel bed at the inner bank and close to the free surface at the outer bank. This grows to a more generalised increase in turbulence that starts at a location approximately 35% of the distance through the bend and peaks at the meander apex before gradually weakening in the downstream part of the bend. However, while only showing a marked increase in the mid-depth region near the inner bank in the neutral case, turbulence is amplified more strongly in the stratified case with the high TKE values spread out across the whole channel width. This is due to the two observed separated shear layers converging in the bend apex in this case. There is only one SSL observed in the neutral case. An investigation of the TKE budget terms for the stratified case at the turbulence amplification location was undertaken to elucidate the mechanisms leading to the production, dissipation and transport of TKE in these regions.

Thermal stratification can lead to the damping of turbulence, which reduces the mixing of solutes in a fluid body. A series of direct numerical simulation (DNS) solutions sweeping through a range of four different meandering channel curvatures, from a sharp to mild curvature range, are obtained to investigate the effect of curvature on stratification in meandering thermally stratified turbulent open channel flow with an internal heat source that models radiative heating from above. Based on the DNS results, the present paper addresses two issues. First, the influence of changing curvature on the complex bi-cellular pattern of the secondary flow is investigated, including the distribution of the temperature field. Second, the effects of changing curvature on the degree of stratification are analyzed. Stratification can be characterized by the friction Richardson number Riτ and the bulk Richardson number Rib. Stratification can also be viewed in terms of the transfer of energy from mean flow kinetic energy to potential energy via buoyancy fluxes. We study the effect of curvature on stratification by investigating its effect on the friction and bulk Richardson numbers. We also study the transfers between the global potential and kinetic energy reservoirs, including the global available Ea, background Eb, and total potential energy Ep, and the domain-averaged mean kinetic and turbulent kinetic energy. It is found that, in meandering channels, with the increase in curvature, Ep increases and Riτ and Rib decrease, indicating that increasing curvature leads to a decrease in the level of stratification. On the other hand, we also find that a low curvature meandering channel has a higher level of stratification than a straight channel.

In an earlier study by Milton-McGruk et al. 2020, Int. J. Heat Fluid Flow , we implemented an experimental procedure to correct for the effects of nonuniform and unsteady laser power on laser induced fluorescence (LIF) measurements in an aqueous medium. The method involves passing the laser sheet through a reference box filled with fluorescent dye at a known concentration before it enters the test rig. The instantaneous LIF images of the reference box and the test rig are then used to correct for the effects of temporal and spatially varying laser power. This paper outlines a simple analytical model that can estimate the experimental error in concentration measurements for statistically stationary flows once the probability density function of time-varying laser power is known. The analytical model has been tested using concentration data in a turbulent jet, and the results reveal that non-uniform and time-varying laser power affects the higher-order statistics such as concentration variance but not the mean. Further, the error in concentration variance as predicted by the model is found to agree very well with the error estimated from the experimental data.

Direct numerical simulation (DNS) results for turbulent open-channel flow through an idealized sine-generated meander with and without an internal heat source that models radiative heating from above are used to analyze the effect of a very sharp meander configuration and thermal stratification on the turbulence structure in the channel with friction Reynolds number Reτ=200. Spatial distributions of temperature, mean velocities, vorticity, mean-flow kinetic energy, and turbulent kinetic energy (TKE) are presented. In both cases, the cross-sectional motion is characterized by three circulation cells: a center-region cell and two weaker outer bank and inner bank cells. However, there is also a small cell observed near the corner of the channel bed inner bank at the channel outlet and the channel bed outer bank at the channel inlet. The tri-cellular cross-stream motions control the distributions of temperature and kinetic energy. In the stratified case, two separated shear layers (SSLs) are found: the first one is formed before the bend apex, and the second one is observed in the wake after the bend apex. In the neutral case, only the first SSL is observed. Turbulent amplification can be seen in both cases; however, in the stratified case, the second SSL stretches out to the channel outlet and is introduced back to the channel inlet by an anti-symmetric periodic boundary condition and then follows the outer bank line. The two SSLs converge in the region before the bend apex and amplify the turbulence more strongly there than in the neutral case. The turbulence kinetic energy budget terms for the stratified case are analyzed to determine the characteristics of production, dissipation and transport of TKE in thermally stratified meandering flow.

Abstract High-fidelity measurements of velocity and concentration are carried out in a neutral jet (NJ) and a negatively buoyant jet (NBJ) by injecting a jet of fresh water vertically downwards into ambient fresh and saline water, respectively. The Reynolds number ($Re$) based on the pipe inlet diameter ($d$) and the source velocity ($W_o$) is approximately 5900 in all the experiments, while the source Froude number based on density difference is approximately 30 in the NBJ experiments. Velocity and concentration measurements are obtained in the region $17 \leq z/d \leq 40$ ($z$ being the axial coordinate) using particle image velocimetry and planar laser induced fluorescence techniques, respectively. Consistent with the literature on jets, the centreline velocity ($W_c$) decays as $z^{-1}$ in the NJ, but in the NBJ, $W_c$ decays faster along $z$ due to the action of negative buoyancy. Nonetheless, the mean velocity ($W$) and concentration ($C$) profiles in both the flows exhibit self-similar Gaussian form, when scaled by the local centreline parameters ($W_c,C_c$) and the jet half-widths ($r^\ast _{W},r^\ast _{C}$). On the other hand, the turbulence statistics and Reynolds stress in the NBJ do not scale with $W_c$. The results of autocorrelation functions, integral length scales and two-dimensional correlation maps show the similarity of turbulence structure in the NJ and the NBJ when the axial and radial distances are normalised by the local jet half-width. Further, the spectra and probability density functions are similar on the axis and only minor differences are seen near the jet interface. The above findings are fundamentally consistent with our recent analysis (Milton-McGurk et al., J. Fluid Mech., 2020b), where we observed that the mean and turbulence statistics in the NBJ have different development characteristics. Overall, we find that the turbulence structure of the NBJ (when scaled by local velocity and length scales) is very similar to the momentum-driven NJ, and the differences (e.g. spreading rate, scaling of turbulence intensities, etc.) between the NJ and the NBJ seem to be of secondary importance.

Numerical simulations have been carried out to investigate the unsteady natural convection flow in a cavity subjected to a sidewall heat flux varying sinusoidally with time. With all walls non-slip and the upper and lower boundaries and the other sidewall adiabatic, the heating and cooling produces an alternating direction natural convection boundary layer that discharges hot fluid to the top and cold fluid to the bottom of the cavity, generating a time-varying thermal stratification in the cavity interior. Scaling analysis has been conducted for different flow regimes based on the forcing frequency, with the characteristic time scales being the forcing period and the boundary layer development time. The scaling relations are then verified using the simulations, with the results showing overall good agreement with the derived scaling relations.

In this study, direct numerical simulation (DNS) results for turbulent stratified open channel flow through an idealized meander with an internal heat source that models radiative heating from above are used to investigate the effect of bends on the degree of stratification and the mechanism by which the turbulence kinetic energy is produced and redistributed along the channel centreline. The pronounced local increase of TKE gains strength in the area entering halfway in the bend where a local extremum of TKE is found near the outer bank. The curve is sharp with the ratio of minimum radius of curvature to depth Rmin/B= 1.67, the maximum deviation angle of θ0 = 110◦ with the ratio of width over depth B/H = 10, the meander wavelength to depth λ/H = 11, and the friction Reynolds number Reτ = 180. We find that there is an explosion of turbulence at the location of 35% of the distance along the channel centreline which gradually weakens in the downstream part of the bend. Possible causes of this feature are discussed.

Transient natural convection flow is investigated in a two-dimensional square enclosure, subjected to periodic heating and cooling on one sidewall. The thermal forcing applied on the heated sidewall varies with time as a sine wave around a zero mean at a frequency f, with all other walls adiabatic. The time-varying heating/cooling produces an alternating direction vertical natural convection boundary layer that entrains fluid from the cavity interior and discharges it alternatively at the top and bottom of the cavity. The behaviour of the flow is governed by three characteristic time scales, the forcing period, 1/f, the development time for the boundary layer and the filling time of the cavity. For low forcing frequency the filling time is less than the forcing period and the time average stratification g is well approximated by (S) over bar f(4/5). For high forcing frequency the forcing period is smaller than the boundary layer development time and (S) over bar similar to f(-2). The maximum (S) over bar occurs in a transition region, between the low and high frequency ranges. The maximum value of (S) over bar increases with increasing Rayleigh number, approaching (S) over bar congruent to 1.0 for the highest values of Rayleigh number considered. (C) 2019 Elsevier Ltd. All rights reserved.

Turbulent negatively buoyant jets occur when the buoyancy of a jet opposes its source momentum. In these flows, the fluid will rise until it reaches a stagnation point and a return flow is established, forming a fountain (Hunt and Burriclge, 2015). This study looks at both the initial negatively buoyant jet stage of this flow, before the return flow has established, and the fully developed fountain stage. Two-dimensional particle image velocimetry (PIV) and planar laser induced fluorescence (PLIF) are used to simultaneously measure the velocity and scalar concentration fields. An experimental and image processing procedure for the PLIF is introduced that accounts for pulse-to-pulse variations in laser power and beam profile for an Nd:YAG laser, which has been demonstrated to reduce the error in scalar concentration measurements. The flow is investigated experimentally using a 1m(3) tank of salt-water ambient with freshwater + ethanol negatively buoyant jets, allowing for measurements to be taken at Fr-o = 30 and Re-o = 5900. The entrainment coefficient for a negatively buoyant jet has been estimated as alpha congruent to 0.054, lower than a neutral jet at alpha congruent to 0.058. A finding consistent with existing literature (Bloomfield and Kerr, 2000; McDougall, 1981).

Abstract Destratification of thermally stratified open-channel flow by surface cooling is investigated using direct numerical simulation. The initial states are the equilibrium states resulting from radiative heating. Using these states as initial conditions, a series of direct numerical simulations was run with radiative heating removed and a constant, uniform cooling flux applied at the upper surface. The flow evolves until the initial stable stratification is broken down and replaced by unstable stratification driven by surface cooling. The destratification process is described with reference to the evolution of the internal structure of the turbulent flow field. Based on these observations, we conclude that the dominant time scales in the flow from the perspective of destratification are the time scales associated with shear ${t}_{\tau }$, convection ${t}_*$ and stable density stratification ${t}_N$. Scaling arguments are then used to derive a scaling relationship for destratification rate as a function of a friction Richardson number $Ri_{\tau } = ( {t}_{\tau }/ {t}_N)^2$ and a convection Richardson number $Ri_* = ( {t}_*/ {t}_N)^2$. The relationship takes the form ${\mathcal {D}}_N = C_1Ri_{\tau }^{-1} + C_2Ri_*^{-1}$, where ${\mathcal {D}}_N$ is the destratification rate non-dimensionalised with respect to $ {t}_N$ and $C_1$ and $C_2$ are model coefficients. The relationship is compared with simulation results and is shown to accurately predict the destratification rate in the simulations across a range of parameters. This relationship is then integrated to give a formula for the time taken for the flow to destratify.

We present a lattice Boltzmann (LB) method using a rectangular, non-isotropic lattice based on D2Q9 and D3Q27 velocity sets in two and three dimensions. A second order multi-scale expansion ensures that the scheme correctly reproduces hydrodynamic behaviour. A novel set of basis vectors is introduced in order to allow independent adjustment of eigenvalues corresponding to second order moments as required in order to ensure correct hydrodynamic behaviour using the non-isotropic lattice. Errors are reduced compared to other rectangular grid implementations. Linear perturbation analysis indicates that our scheme has similar stability properties to the isotropic LB method. We investigate the error behaviour of our scheme by performing Taylor-Green vortex flow simulations and comparing our results to simulations using a square grid and also to analytical results. We demonstrate that our scheme is well suited to direct numerical simulation of wall bounded turbulent flows and compare to well known benchmark results. (C) 2020 Elsevier Ltd. All rights reserved.

Evolution of thermally stratified open channel flow after removal of a volumetric heat source is investigated using direct numerical simulation. The heat source models radiative heating from above and varies with height due to progressive absorption. After removal of the heat source the initial stable stratification breaks down and the channel approaches a fully mixed isothermal state. The initial state consists of three distinct regions: a near-wall region where stratification plays only a minor role, a central region where stratification has a significant effect on flow dynamics and a near-surface region where buoyancy effects dominate. We find that a state of local energetic equilibrium observed in the central region of the channel in the initial state persists until the late stages of the destratification process. In this region local turbulence parameters such as eddy diffusivity $k_{h}$ and flux Richardson number $R_{f}$ are found to be functions only of the Prandtl number $Pr$ and a mixed parameter ${\mathcal{Q}}$, which is equal to the ratio of the local buoyancy Reynolds number $Re_{b}$ and the friction Reynolds number $Re_{\unicode[STIX]{x1D70F}}$. Close to the top and bottom boundaries turbulence is also affected by $Re_{\unicode[STIX]{x1D70F}}$ and vertical position $z$. In the initial heated equilibrium state the laminar surface layer is stabilised by the heat source, which acts as a potential energy sink. Removal of the heat source allows Kelvin–Helmholtz-like shear instabilities to form that lead to a rapid transition to turbulence and significantly enhance the mixing process. The destratifying flow is found to be governed by bulk parameters $Re_{\unicode[STIX]{x1D70F}}$, $Pr$ and the friction Richardson number $Ri_{\unicode[STIX]{x1D70F}}$. The overall destratification rate ${\mathcal{D}}$ is found to be a function of $Ri_{\unicode[STIX]{x1D70F}}$ and $Pr$.

Numerical simulations are used to investigate the entrainment for forced turbulent fountains over a range of Reynolds numbers and Froude numbers, with ranges based on the fountain source properties. Other fountain properties such as height and width are also examined to provide information on the general structure of the fountains. The results show that the fountains have minimal Reynolds number dependency, while they have a strong linear relation with the Froude number for the cases considered in this study. The entrainment coefficient is obtained as well as scaling constants for height and width in terms of the Froude number. References G. Abraham. Jets with negative buoyancy in homogeneous fluid. J. Hydraul. Res., 5(4):235248, 1967. doi:10.1080/00221686709500209. W. D. Baines, J. S. Turner, and I. H. Campbell. Turbulent fountains in an open chamber. J. Fluid Mech., 212:557592, 1990. doi:10.1017/S0022112090002099. L. J. Bloomfield and R. C. Kerr. A theoretical model of a turbulent fountain. J. Fluid Mech., 424:197216, 2000. doi:10.1017/S0022112000001907. H. C. Burridge and G. R. Hunt. Entrainment by turbulent fountains. J. Fluid Mech., 790:407418, 2016. doi:10.1017/jfm.2016.16. I. H. Campbell and J. S. Turner. Fountains in magma chambers. J. Petrol., 30(4):885923, 1989. doi:10.1093/petrology/30.4.885. P. D. Friedman, V. D. Vadakoot, W. J. Meyer, and S. Carey. Instability threshold of a negatively buoyant fountain. Exp. Fluids, 42(5):751759, 2007. doi:10.1007/s00348-007-0283-5. D. D. Gray and A. Giorgini. The validity of the Boussinesq approximation for liquids and gases. Int. J. Heat Mass Tran., 19(5):545551, 1976. doi:10.1016/0017-9310(76)90168-X. N. B. Kaye and G. R. Hunt. Weak fountains. J. Fluid Mech., 558:319328, 2006. doi:10.1017/S0022112006000383. B. P. Leonard and S. Mokhtari. Beyond first-order upwinding: The ultra-sharp alternative for non-oscillatory steady-state simulation of convection. Int. J. Numer. Meth. Eng., 30(4):729766, 1990. doi:10.1002/nme.1620300412. T. J. McDougall. Negatively buoyant vertical jets. Tellus, 33(3):313320, 1981. doi:10.3402/tellusa.v33i3.10718. T. Mizushina, F. Ogino, H. Takeuchi, and H. Ikawa. An experimental study of vertical turbulent jet with negative buoyancy. Warme Stoffubertrag., 16(1): 1521, 1982. doi:10.1007/BF01322802. B. R. Morton. Forced plumes. J. Fluid Mech., 5(1):151163, 1959. doi:10.1017/S002211205900012X. S. E. Norris. A parallel NavierStokes solver for natural convection and free surface flow. PhD thesis, University of Sydney, 2000. J. S. Turner. Jets and plumes with negative or reversing buoyancy. J. Fluid Mech., 26(4):779792, 1966. doi:10.1017/S0022112066001526. N. Williamson, N. Srinarayana, S. W. Armfield, G. D. McBain, and W. Lin. Low-Reynolds-number fountain behaviour. J. Fluid Mech., 608:297317, 2008. doi:10.1017/S0022112008002310. N. Williamson, S. W. Armfield, and W. Lin. Forced turbulent fountain flow behaviour. J. Fluid Mech., 671:535558, 2011. doi:10.1017/S0022112010005872. H. Zhang and R. E. Baddour. Maximum penetration of vertical round dense jets at small and large Froude numbers. J. Hydraul. Eng., 124(5):550553, 1998. doi:10.1061/(ASCE)0733-9429(1998)124:5(550).

In this study, direct numerical simulation (DNS) and large eddy simulation (LES) of turbulent channel flow beneath a flat surface with imposed wind shear stress are presented. An open channel with friction Reynolds number Reτ= 360 has been considered, while wind shear stresses were imposed as being aligned with the flow direction or not aligned with components in streamwise and spanwise directions. The results indicate that there are differences between wind-driven flow and the case with no wind shear stress. Streaks are present close to the surface in wind-driven flow and profiles of the mean velocity, TKE and mean shear production rate in wind-driven flow are different from those in the unsheared surface flow. Large eddy simulation of wind-driven flow was also carried out to evaluate the performance of sub-grid scale models. It is shown that the features of the flow observed in DNS are well simulated by LES with a relatively small difference near the surface due to shear stress boundary condition.