Turbulent Froude Number Research Articles

Direct numerical simulation of turbulent intermediate Froude number fountain flow

Fountains are an important flow in many industrial and geophysical problems but the understanding of this flow at low Froude numbers is limited. In this study we use direct numerical simulation to examine two regimes of turbulent fountain flow, weak flow at Fr=2.2 and very weak flow at Fr=0.45. At Fr=0.45 buoyancy stabilises the flow and there is little entrainment of ambient fluid. Kelvin--Helmholtz instabilities are generated at the interface between the upwelling flow and the ambient fluid. At Fr= 2.2 the shear interaction between the rising and falling flow streams becomes important as it drives both large and small scale vortex structures. Large scale structures draw ambient fluid into the fountain while small scale structures exchange mass between the upflow and downflow. Periodic ejection of mass from the top of the fountain is shown to be an important dynamic in the flow. The results provided here can assist in analytical model development. References J. S. Turner. Jets and plumes with negative or reversing buoyancy. J. Fluid Mech., 26:779--792, 1966. doi:10.1017/S0022112066001526. T. Mizushina, F. Ogino, H. Takeuchi, and H. Ikawa. An experimental study of vertical turbulent jet with negative buoyancy. Warme-und Stoffubertragung, 16:15--21, 1982. doi:10.1007/BF01322802. R. W. Cresswell and R. T. Szczepura. Experimental investigation into a turbulent jet with negative buoyancy. Phys. Fluids A, 5:2865--2878, 1993. doi:10.1063/1.858749. N. B. Kaye and G. R. Hunt. Weak fountains. J. Fluid Mech., 558:319--328, 2006. doi:10.1017/S0022112006000383. N. Williamson, N. Srinarayana, S. W. Armfield, G. D. McBain, and W. Lin. Low-reynolds-number fountain behaviour. J. Fluid Mech., 608:297--317, 2008. doi:10.1017/S0022112008002310. W. Lin and S. W. Armfield. Very weak axisymmetric fountains in a homogeneous fluid. Numer. Heat Transfer A, 38:377--396, 2000. doi:10.1080/104077800750022520. P. D. Friedman, V. D. Vadokoot, W. J. Meyer, and S. Carey. Instability threshold of a negatively buoyant fountain. Exp. Fluids, 42:751--759, 2007. doi:10.1007/s00348-007-0283-5. H. Zhang and R. E. Baddour. Maximum penetration of vertical round dense jets at small and large {F}roude numbers. J. Hydraulic Eng., 124:550--553, 1998. doi:10.1061/(ASCE)0733-9429(1998)124:5(550). B. R. Morton, G. I. Taylor, and J. S. Turner. Turbulent gravitational convection from maintained and instantaneous sources. Proc. R. Soc. Lond., 234:1--23, 1956. http://www.jstor.org/stable/99936. 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. Engng, 30:729--766, 1990. doi:10.1002/nme.1620300412. I. H. Campbell and J. S. Turner. Fountains in magma chambers. J. Petrol., 30:885--923, 1989. http://petrology.oxfordjournals.org/cgi/content/abstract/30/4/885 W. D. Baines, J. S. Turner, and I. H. Campbell. Turbulent fountains in an open chamber. J. Fluid Mech., 212:557--592, 1990. doi:10.1017/S0022112090002099. W. Lin and S. W. Armfield. Direct simulation of weak axisymmetric fountains in a homogeneous fluid. J. Fluid Mech., 403:67--88, 2000. doi:10.1017/S0022112099006953. E. Kaminski, S. Tait, and G. Carazzo. Turbulent entrainment in jets with arbitrary buoyancy. J. Fluid Mech., 526:361--376, 2005. doi:10.1017/S0022112004003209.

Measurement of Scalar Variance Dissipation from Lagrangian Floats

Simultaneous measurements of temperature, salinity, their vertical gradients, and the vertical gradient of velocity across a 1.4-m-long Lagrangian float were used to investigate the accuracy with which the dissipation of scalar variance χ can be computed using inertial subrange methods from such a neutrally buoyant float. The float was deployed in a variety of environments in Puget Sound; χ varied by about 3.5 orders of magnitude. A previous study used an inertial subrange method to yield accurate measurements of ε, the rate of dissipation of kinetic energy, from this data. Kolmogorov scaling predicts a Lagrangian frequency spectrum for the rate of change of a scalar as ΦDσ/Dt(ω) = βsχ, where βs is a universal Kolmogorov constant. Measured spectra of the rate of change of potential density σ were nearly white at frequencies above N, the buoyancy frequency. Deviations at higher frequency could be modeled quantitatively using the measured deviations of the float from perfect Lagrangian behavior, yielding an empirical nondimensional form ΦDσ/Dt = βsχH(ω/ωL) for the measured spectra, where L is half the float length, ω3L = ε/L2, and H is a function describing the deviations of the spectrum from Kolmogorov scaling. Using this empirical form, estimates of χ were computed and compared with estimates derived from ε. The required mixing efficiency was computed from the turbulent Froude number ω0/N, where ω0 is the large-eddy frequency. The results are consistent over a range of ε from 10−8 to 3 × 10−5 W kg−1 implying that χ can be estimated from float data to an accuracy of least a factor of 2. These methods for estimating ε, χ, and the Froude number from Lagrangian floats appear to be unbiased and self-consistent for ε > 10−8 W kg−1. They are expected to fail in less energetic turbulence both for instrumental reasons and because the Reynolds number typically becomes too small to support an inertial subrange. The value of βs is estimated at 0.6 to within an uncertainty of less than a factor of 2.

Modelling the effects of horizontal and vertical shear in stratified turbulent flows

Abstract Direct numerical simulations (DNS) and model results from a number of one-point turbulence models are compared for homogeneous, stably stratified flows. Because of their wide spread use in numerical ocean modelling, only explicit algebraic second-moment models are investigated. Considered are two types of shear flows with either purely vertical or purely horizontal shear. The dissipation rate is evaluated from the observation that the shear-number becomes independent of stratification for low to moderate Richardson numbers as soon as the flow approaches self-similarity. For the cases with vertical shear, it is found that all statistical models essentially reproduced the DNS results, though with different accuracy. In contrast, only the most recent model was able to predict the salient features of horizontally sheared flows, i.e. a steady-state Richardson number that is about an order of magnitude larger and a vertical mixing efficiency that is about twice as large compared to the case with vertical shear. This model also reproduced other key parameters like the turbulent Froude number and the turbulent Prandtl number with good accuracy, but it failed to predict quantitatively the reduction of the shear anisotropy with increasing stratification. For strong stratification, none of the models was able to describe the rapid decrease of the mixing efficiency associated with the collapse and fossilisation of turbulence.

The effect of stable stratification on turbulence anisotropy in uniformly sheared flow

Direct numerical simulation of uniform shear flow is used to study the anisotropy of fluctuating motion in a stably stratified medium with uniform mean shear. Turbulence is found to be three dimensional over a wide range of gradient Richardson numbers in the two flows investigated here: vertical mean shear ( d U dz and horizontal mean shear ( d U dy . The role of the turbulent Froude number in establishing the regime of stratified turbulence observed here is described. The fluctuating velocity gradients are examined. The vertical of streamwise velocity is found to dominate the other components of turbulent dissipation in both horizontal and vertical shear flows.

Interfacial mixing in a highly stratified estuary 1. Characteristics of mixing

This article is Part I of a set of papers addressing mixing in a highly stratified estuary. Measurements of interfacial turbulence were made in Columbia River estuary under conditions of moderate river flow and neap tides. A flux correlation method was used to determine buoyancy fluxes, and fits to theoretical velocity and temperature variance spectra were used to measure turbulent kinetic and potential energy dissipation rates. At the measurement site in the interior of the estuary, mixing along the top of the salt wedge is found to occur only on ebb, during periods when the internal Froude number is supercritical. The wedge cannot, therefore, be in a quasiequilibrium state, and vertical mixing plays a dominant role in its ebb retreat. The turbulent Froude number FrT, which represents the ratio of effects of shear and stratification on the mixing, was found to be close to FrT ≈ 1 during mixing episodes. This suggests an interplay between shear and buoyancy effects that corresponds to a condition of maximum mixing efficiency Γ. Γ is measured using both direct calculations of the buoyancy and dissipation terms in the turbulent kinetic energy (TKE) balance (this estimate of Γ is referred to as Γ0), and by estimating the buoyancy term B by its relationship to χT, the dissipation of temperature variance (this estimate of Γ is referred to as Γd). Averages over several mixing events give Γ0 ≈ 0.22 and Γd ≈ 0.36. Sources of bias in these measurements result from (1) approximations to the full TKE and available potential energy (APE) balances, (2) empirical constants used in fits of measured spectra in inertial subranges, (3) frame motion, and (4) high‐pass filtering. The sign of the biases suggest that 0.22 < Γ < 0.36. Flood‐ebb asymmetry in mixing also results in a time dependent force balance (discussed further by Kay and Jay [2003]). This asymmetry is analyzed using a new method to determine profiles of the momentum balance during ebb mixing events.

Anatomy of turbulence in a narrow and strongly stratified estuary

[1] More than 400 profiles of (salinity) density and velocity microstructure with 923 turbulent patches were collected over 4 days from an anchored boat with the Portable Flux Profiler (PFP) in the Swan River estuary during the wet season. By using the PFP the buoyancy flux was measured directly and estimated indirectly for this flow. Overturns were suppressed by the strong stratification, and large-scale anisotropy was correlated with turbulent Froude number. The direct measurements of the buoyancy flux exhibited both positive (downgradient) and negative (upgradient) values within the turbulent patches. The net average mixing efficiency was 0.04, less than the commonly used value of 0.17. Therefore the rate of vertical transport was small even though the dissipation levels within the turbulent patches were relatively high. The magnitude of the above mentioned competing fluxes increased with decreasing turbulent Froude numbers even though the net flux when bin-averaged with respect to turbulent Froude number remained close to zero.

Anatomy of turbulence in a narrow and weakly stratified estuary

A weakly stratified straight channel in the Swan River estuary was examined using a CTD probe and a microstructure profiler capable of measuring two components of velocity, temperature and conductivity. In addition, a pair of acoustic Doppler current profilers were deployed at the bottom to record profiles of the horizontal velocity. The results indicated that the mean flow was barotropic with little shear. The turbulent overturns were mostly active and viscous forces were negligible in their energetics. Although the stratification was too weak to influence the mean motion, it was found that the turbulence was influenced by buoyancy, and a collapse of the vertical anisotropy was documented with decreasing turbulent Froude number, Frt. Using the results from laboratory and numerical studies, we infer that the mixing efficiencies associated with these turbulent events ranged between 0 and 0.3, with a mean value of 0.07.

Turbulent flow over a wavy surface: Stratified case

Stably stratified turbulent flow over wavy terrain is investigated using large‐eddy simulation for laboratory‐scale domains. Basic near‐wall turbulence structure and production mechanisms appear similar to the neutral case, though dampened. The state of turbulence in the stratified fluid is analyzed by graphing the relationships between key nondimensional parameters. A shifting and smearing of the curve of flux Richardson number Rif versus turbulent Froude number Frt caused by the wavy boundary (compared with a flat boundary) demonstrates the higher level of turbulence near the wavy wall and shows strong mixing in the detached shear layer over the trough. The relationship describing the state of turbulence in a stratifed fluid given by Ivey and Imberger [1991] is approximately satisfied. Conditional sampling is used to analyze the nature of Reynolds stress events.

Particle dispersion and mixing induced by breaking internal gravity waves

The purpose of this paper is to analyze diapycnal mixing induced by the breaking of an internal gravity wave — the primary wave — either standing or propagating. To achieve this aim we apply two different methods. The first method consists of a direct estimate of vertical eddy diffusion from particle dispersion while the second method relies upon potential energy budgets [Winters, K.B., Lombard, P.N., Riley, J.J., D’Asaro, E.A., 1995. J. Fluid Mech. 289, 115–128; Winters, K.B., D’Asaro, E.A., 1996. J. Fluid Mech. 317, 179–193]. The primary wave we consider is of small amplitude and is statically stable, a case for which the breaking process involves two-dimensional instabilities. The dynamics of the waves have been previously analyzed by means of two-dimensional direct numerical simulations [Bouruet-Aubertot, P., Sommeria, J., Staquet, C., 1995. J. Fluid Mech. 285, 265–301; Bouruet-Aubertot, P., Sommeria, J., Staquet, C., 1996. Dyn. Atmos. Oceans 29, 41–63; Koudella, C., Staquet, C., 1998. In: Davis, P. (Ed.), Proceedings of the IMA Conference on Mixing and Dispersion on Stably-stratified Flows, Dundee, September 1996. IMA Publication]. High resolution three-dimensional calculations of the same wave are also reported here [Koudella, C., 1999]. A local estimate of mixing is first inferred from the time evolution of sets of particles released in the flow during the breaking regime. We show that, after an early evolution dominated by shear effects, a diffusion law is reached and the dispersion coefficient is fairly independent of the initial seeding location of the particles in the flow. The eddy diffusion coefficient , K , is then estimated from the diapycnal diffusive flux. A good agreement with the value inferred from particle dispersion is obtained. This finding is of particular interest regarding the interpretation of in situ estimates of K inferred either from tracer dispersion or from microstructure measurements. Computation of the Cox number, equal to the ratio of eddy diffusivity to molecular diffusivity, shows that the Cox number varies within the interval [9, 262], which corresponds to the range of vertical eddy diffusivity measured in the interior of the ocean. The Cox number is found to depend on the turbulent Froude number squared. We show eventually that mixing results in a weak distortion of the initial density profile and we relate this result to observations made at small scale in the ocean. Comparisons between the analysis of the two-dimensional and high resolution (256 3 ) three-dimensional direct numerical simulations of the primary wave were also conducted. We show that the energetics and the amount of mixing are very close when the primary wave is of small amplitude. This results from the fact that, for a statically stable wave, the dynamics of the initially two-dimensional primary wave remains mostly two-dimensional even after the onset of wavebreaking.

Scaling and parameterization of stratified homogeneous turbulent shear flow

Homogeneous sheared stratified turbulence was simulated using a DNS code. The initial turbulent Reynolds numbers (Re) were 22, 44, and 89, and the initial dimensionless shear rate (S*) varied from 2 to 16. We found (similarly to Rogers (1986) for unstratified flows) the final value of S* at high Re to be ∼ 11, independent of initial S*. The final S* varies at low Re, in agreement with Jacobitz et al. (1997). At low Re, the stationary Richardson number (Ris) depends on both Re and S*, but at higher Re, it varies only with Re. A scaling based on the turbulent kinetic energy equation which suggests this result employs instantaneous rather than initial values of flow parameters.At high Re the dissipation increases with applied shear, allowing a constant final S*. The increased dissipation occurs primarily at high wavenumbers due to the stretching of eddies by stronger shear. For the high-Re stationary flows, the turbulent Froude number (Frt) is a constant independent of S*. An Frt-based scaling predicts the final value of S* well over a range of Re. Therefore Frt is a more appropriate parameter for describing the state of developed stratified turbulence than the gradient Richardson number.

Observations of Turbulence in a Partially Stratified Estuary

The authors present a field study of estuarine turbulence in which profiles of Reynolds stresses were directly measured using an ADCP throughout a 25-h tidal day. The dataset that is discussed quantifies turbulent mixing for a water column in northern San Francisco Bay that experiences a sequence of states that includes a weak ebb and flood that are stratified, followed by a strong, and eventually unstratified, ebb and flood. These measurements show that energetic turbulence is confined to a bottom mixed layer by the overlying stratification. Examination of individual Reynolds stress profiles along with profiles of Richardson number and turbulent Froude number shows that the water column can be divided into regions based on the relative importance of buoyancy effects. Using the measured turbulence production rate P, the dissipation rate ϵ is estimated. The observed turbulence had values of ϵ/νN2 > 20 all of the time and ϵ/νN2 > 200 most of the time, suggesting that the observed motions were buoyancy affected turbulence rather than internal waves. However, at times, turbulent Froude numbers in much of the upper-water column were less than one, indicating important stratification effects. Taken as a whole, the data show that stratification affects the turbulent velocity variance q2 most severely; that is, observed reductions in u′w′ are largely associated with small values of q2 rather than with a dramatic reduction in the efficiency with which turbulent motions produce momentum fluxes. Finally, the dataset is compared to predictions made using the popular Mellor–Yamada level 2.5 closure. These comparisons show that the model tends to underestimate the turbulent kinetic energy in regions of strong stratification where the turbulence is strongly inhomogeneous and to overestimate the turbulent kinetic energy in weakly stratified regions. The length scale does not appear to compensate for these errors, and, as a result, similar errors are seen in the eddy viscosity predictions. It is hypothesized that the underestimation of q2 is due to an inaccurate parameterization of turbulence self-transport from the near-bed region to the overlying stratification.

Turbulent mixing in a shear-free stably stratified two-layer fluid

Direct numerical simulation is used to examine turbulent mixing in a shear-free stably stratified fluid. Energy is continuously supplied to a small region to maintain a well-developed kinetic energy profile, as in an oscillating grid flow (Briggs et al. 1996; Hopfinger & Toly 1976; Nokes 1988). A microscale Reynolds number of 60 is maintained in the source region. The turbulence forms a well-mixed layer which diffuses from the source into the quiescent fluid below. Turbulence transport at the interface causes the mixed layer to grow under weakly stratified conditions. When the stratification is strong, large-scale turbulent transport is inactive and pressure transport becomes the principal mechanism for the growth of the turbulence layer. Down-gradient buoyancy flux is present in the large scales; however, far from the source, weak counter-gradient fluxes appear in the medium to small scales. The production of internal waves and counter-gradient fluxes rapidly reduces the mixing when the turbulent Froude number is lower than unity. When the stratification is weak, the turbulence is strong enough to break up the density interface and transport fluid parcels of different density over large vertical distances. As the stratification intensifies, turbulent eddies flatten against the interface creating anisotropy and internal waves. The dominant entrainment mechanism is then scouring. Mixing efficiency, defined as the ratio of buoyancy flux to available kinetic energy, exhibits a similar dependence on Froude number to other stratified flows (Holt et al. 1992; Lienhard & Van Atta 1990). However, using the anisotropy of the turbulence to define an alternative mixing efficiency and Froude number improves the correlation and allows local scaling.

Turbulence within inertia‐buoyancy balanced axisymmetric intrusions

Data from two detailed field experiments were used to investigate the turbulence within inertia‐buoyancy balanced, axisymmetric intrusive gravity currents entering nearly quiescent stratified water bodies. The intrusions were generated by energetic point source bubble plume systems installed within different freshwater reservoirs, and the mixing activity was recorded with microstructure profilers. Close to the bubble plume, the intrusions were found to be very turbulent. The turbulent activity within the intrusions was observed to decay most rapidly in the intrusion core, but the shear at the upper and lower boundaries of the intrusions maintained the turbulence in these regions for a long distance. The results showed that the turbulence began with a length scale equal to the maximum sustainable overturn length for internally generated shear and hence a turbulent Froude number near unity. As the intrusions moved away from the generation region, the Froude number remained near unity, but the Reynolds number decreased with distance; the shear generated by the intrusion thus sustained an active turbulence field, albeit with ever decreasing length scales.

Three-dimensional structure in a tidal convergence front

Lagrangian measurements of the structure of a tidal convergence zone were made in Haro Strait, on the western boundary of the U.S. and Canada. Acoustic imaging of the ocean surface from below was used in combination with a sub-surface, acoustically tracked, neutrally buoyant float and several CTDs. Generally, regions of high acoustic back-scatter were correlated with surface convergence and downwelling, while regions of low back-scatter corresponded to divergence and upwelling. The large scale velocity gradients which appear to form this front were concentrated in a shear zone a few 10 s of m wide with vertical vorticity up to 0.15 s −1 . Associated with this is a 100 m wide region of roughly 0.1 m s −1 average downwelling and maximum downwellings of 0.2 m s −1. A rich structure of three-dimensional eddies and front-like features occurs throughout the tidal mixing region. One common feature, appearing on the surface as a circular boil, extended at least 50 m deep and consisted of a 100–150 m region of upwelling surrounded by a much narrower ring of more intense downwelling. These eddies have a turbulent Froude number near one, so their size is constrained by the stratification. As a whole, the convergence zone acted to mix the waters brought together by the converging tidal streams. Although our data are very limited, the turbulence intensities which we observed are potentially large enough to support all of the mixing required by the overall circulation in Haro Strait.