9 results
Mixing efficiency in stratified turbulence
- A. Maffioli, G. Brethouwer, E. Lindborg
-
- Journal:
- Journal of Fluid Mechanics / Volume 794 / 10 May 2016
- Published online by Cambridge University Press:
- 05 April 2016, R3
-
- Article
- Export citation
-
We consider mixing of the density field in stratified turbulence and argue that, at sufficiently high Reynolds numbers, stationary turbulence will have a mixing efficiency and closely related mixing coefficient described solely by the turbulent Froude number $Fr={\it\epsilon}_{k}/(Nu^{2})$, where ${\it\epsilon}_{k}$ is the kinetic energy dissipation, $u$ is a turbulent horizontal velocity scale and $N$ is the Brunt–Väisälä frequency. For $Fr\gg 1$, in the limit of weakly stratified turbulence, we show through a simple scaling analysis that the mixing coefficient scales as ${\it\Gamma}\propto Fr^{-2}$, where ${\it\Gamma}={\it\epsilon}_{p}/{\it\epsilon}_{k}$ and ${\it\epsilon}_{p}$ is the potential energy dissipation. In the opposite limit of strongly stratified turbulence with $Fr\ll 1$, we argue that ${\it\Gamma}$ should reach a constant value of order unity. We carry out direct numerical simulations of forced stratified turbulence across a range of $Fr$ and confirm that at high $Fr$, ${\it\Gamma}\propto Fr^{-2}$, while at low $Fr$ it approaches a constant value close to ${\it\Gamma}=0.33$. The parametrization of ${\it\Gamma}$ based on $Re_{b}$ due to Shih et al. (J. Fluid Mech., vol. 525, 2005, pp. 193–214) can be reinterpreted in this light because the observed variation of ${\it\Gamma}$ in their study as well as in datasets from recent oceanic and atmospheric measurements occurs at a Froude number of order unity, close to the transition value $Fr=0.3$ found in our simulations.
A numerical study of the unstratified and stratified Ekman layer
- Enrico Deusebio, G. Brethouwer, P. Schlatter, E. Lindborg
-
- Journal:
- Journal of Fluid Mechanics / Volume 755 / 25 September 2014
- Published online by Cambridge University Press:
- 26 August 2014, pp. 672-704
-
- Article
- Export citation
-
We study the turbulent Ekman layer at moderately high Reynolds number, $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}1600 < \mathit{Re} = \delta _{E}G/\nu < 3000$, using direct numerical simulations (DNS). Here, $\delta _{E} = \sqrt{2\nu /f}$ is the laminar Ekman layer thickness, $G$ the geostrophic wind, $\nu $ the kinematic viscosity and $f$ is the Coriolis parameter. We present results for both neutrally, moderately and strongly stably stratified conditions. For unstratified cases, large-scale roll-like structures extending from the outer region down to the wall are observed. These structures have a clear dominant frequency and could be related to periodic oscillations or instabilities developing near the low-level jet. We discuss the effect of stratification and $\mathit{Re}$ on one-point and two-point statistics. In the strongly stratified Ekman layer we observe stable co-existing large-scale laminar and turbulent patches appearing in the form of inclined bands, similar to other wall-bounded flows. For weaker stratification, continuously sustained turbulence strongly affected by buoyancy is produced. We discuss the scaling of turbulent length scales, height of the Ekman layer, friction velocity, veering angle at the wall and heat flux. The boundary-layer thickness, the friction velocity and the veering angle depend on $Lf/u_\tau $, where $u_\tau $ is the friction velocity and $L$ the Obukhov length scale, whereas the heat fluxes appear to scale with $L^+=L u_\tau /\nu $.
Third-order structure functions in rotating and stratified turbulence: a comparison between numerical, analytical and observational results
- Enrico Deusebio, P. Augier, E. Lindborg
-
- Journal:
- Journal of Fluid Mechanics / Volume 755 / 25 September 2014
- Published online by Cambridge University Press:
- 19 August 2014, pp. 294-313
-
- Article
- Export citation
-
First, we review analytical and observational studies on third-order structure functions including velocity and buoyancy increments in rotating and stratified turbulence and discuss how these functions can be used in order to estimate the flux of energy through different scales in a turbulent cascade. In particular, we suggest that the negative third-order velocity–temperature–temperature structure function that was measured by Lindborg & Cho (Phys. Rev. Lett., vol. 85, 2000, p. 5663) using stratospheric aircraft data may be used in order to estimate the downscale flux of available potential energy (APE) through the mesoscales. Then, we calculate third-order structure functions from idealized simulations of forced stratified and rotating turbulence and compare with mesoscale results from the lower stratosphere. In the range of scales with a downscale energy cascade of kinetic energy (KE) and APE we find that the third-order structure functions display a negative linear dependence on separation distance $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}} r $, in agreement with observation and supporting the interpretation of the stratospheric data as evidence of a downscale energy cascade. The spectral flux of APE can be estimated from the relevant third-order structure function. However, while the sign of the spectral flux of KE is correctly predicted by using the longitudinal third-order structure functions, its magnitude is overestimated by a factor of two. We also evaluate the third-order velocity structure functions that are not parity invariant and therefore display a cyclonic–anticyclonic asymmetry. In agreement with the results from the stratosphere, we find that these functions have an approximate $ r^{2} $-dependence, with strong dominance of cyclonic motions.
The route to dissipation in strongly stratified and rotating flows
- Enrico Deusebio, A. Vallgren, E. Lindborg
-
- Journal:
- Journal of Fluid Mechanics / Volume 720 / 10 April 2013
- Published online by Cambridge University Press:
- 27 February 2013, pp. 66-103
-
- Article
-
- You have access Access
- Open access
- Export citation
-
We investigate the route to dissipation in strongly stratified and rotating systems through high-resolution numerical simulations of the Boussinesq equations (BQs) and the primitive equations (PEs) in a triply periodic domain forced at large scales. By applying geostrophic scaling to the BQs and using the same horizontal length scale in defining the Rossby and the Froude numbers, $\mathit{Ro}$ and $\mathit{Fr}$, we show that the PEs can be obtained from the BQs by taking the limit ${\mathit{Fr}}^{2} / {\mathit{Ro}}^{2} \rightarrow 0$. When ${\mathit{Fr}}^{2} / {\mathit{Ro}}^{2} $ is small the difference between the results from the BQ and the PE simulations is shown to be small. For large rotation rates, quasi-geostrophic dynamics are recovered with a forward enstrophy cascade and an inverse energy cascade. As the rotation rate is reduced, a fraction of the energy starts to cascade towards smaller scales, leading to a shallowing of the horizontal spectra from ${ k}_{h}^{- 3} $ to ${ k}_{h}^{- 5/ 3} $ at the small-scale end. The vertical spectra show a similar transition as the horizontal spectra and we find that Charney isotropy is approximately valid also at larger wavenumbers than the transition wavenumber. The high resolutions employed allow us to capture both ranges within the same simulation. At the transition scale, kinetic energy in the rotational and in the horizontally divergent modes attain comparable values. The divergent energy is several orders of magnitude larger than the quasi-geostrophic divergent energy given by the $\Omega $-equation. The amount of energy cascading downscale is mainly controlled by the rotation rate, with a weaker dependence on the stratification. A larger degree of stratification favours a downscale energy cascade. For intermediate degrees of rotation and stratification, a constant energy flux and a constant enstrophy flux coexist within the same range of scales. In this range, the enstrophy flux is a result of triad interactions involving three geostrophic modes, while the energy flux is a result of triad interactions involving at least one ageostrophic mode, with a dominant contribution from interactions involving two ageostrophic and one geostrophic mode. Dividing the ageostrophic motions into two classes depending on the sign of the linear wave frequency, we show that the energy transfer is for the largest part supported by interactions within the same class, ruling out the wave–wave–vortex resonant triad interaction as a mean of the downscale energy transfer. The role of inertia-gravity waves is studied through analyses of time-frequency spectra of single Fourier modes. At large scales, distinct peaks at frequencies predicted for linear waves are observed, whereas at small scales no clear wave activity is observed. Triad interactions show a behaviour which is consistent with turbulent dynamics, with a large exchange of energy in triads with one small and two large comparable wavenumbers. The exchange of energy is mainly between the modes with two comparable wavenumbers.
Solute transport from geosphere to biosphere: Modeling results from the Forsmark site, Sweden
- J. Barescut, D. Lariviere, T. Stocki, S. Berglund, E. Bosson, T. Lindborg, M. Sassner
-
- Journal:
- Radioprotection / Volume 46 / Issue 6 / 2011
- Published online by Cambridge University Press:
- 09 January 2012, pp. S539-S545
- Print publication:
- 2011
-
- Article
- Export citation
-
This paper describes transport modeling performed by the Swedish Nuclear Fuel and Waste Management Company (SKB) within the recently completed SR-Site safety assessment. SR-Site is a part of the license application for building a deep geological repository for spent nuclear fuel in Sweden; this application was submitted to the authorities in March 2011. In the safety assessment, a suite of transport models was used to quantify radionuclide transport through the engineered and geological barriers and in the biosphere. The present paper is focused on the transport in the upper part of the bedrock and in the regolith (the Quaternary deposits overlying the rock). The results presented provide input to the assessment of uncertainties in calculated discharge locations for radionuclide-bearing groundwater from the considered repository volume and to analyses of solute spreading in near-surface and surface systems.
Numerical study of vertical dispersion by stratified turbulence
- G. BRETHOUWER, E. LINDBORG
-
- Journal:
- Journal of Fluid Mechanics / Volume 631 / 25 July 2009
- Published online by Cambridge University Press:
- 17 July 2009, pp. 149-163
-
- Article
- Export citation
-
Numerical simulations are carried out to investigate vertical fluid particle dispersion in uniformly stratified stationary turbulent flows. The results are compared with the analysis of Lindborg & Brethouwer (J. Fluid Mech., vol. 614, 2008, pp. 303–314), who derived long- and short-time relations for the mean square vertical displacement 〈δz〉 of fluid particles. Several direct numerical simulations (DNSs) with different degrees of stratification and different buoyancy Reynolds numbers are carried out to test the long-time relation 〈δz2〉 = 2ϵPt/N2. Here, ϵP is the mean dissipation of turbulent potential energy; N is the Brunt–Väisälä frequency; and t is time. The DNSs show good agreement with this relation, with a weak dependence on the buoyancy Reynolds number. Simulations with hyperviscosity are carried out to test the relation 〈δz2〉 = (1+πCPL)2ϵPt/N2, which should be valid for shorter time scales in the range N−1 ≪ t ≪ T, where T is the turbulent eddy turnover time. The results of the hyperviscosity simulations come closer to this prediction with CPL about 3 with increasing stratification. However, even in the simulation with the strongest stratification the growth of 〈δz2〉 is somewhat slower than linear in this regime. Based on the simulation results it is argued that the time scale determining the evolution of 〈δz2〉 is the eddy turnover time, T, rather than the buoyancy time scale N−1, as suggested in previous studies. The simulation results are also consistent with the prediction of Lindborg & Brethouwer (2008) that the nearly flat plateau of 〈δz2〉 observed at t ~ T should scale as 4EP/N2, where EP is the mean turbulent potential energy.
Vertical dispersion by stratified turbulence
- E. LINDBORG, G. BRETHOUWER
-
- Journal:
- Journal of Fluid Mechanics / Volume 614 / 10 November 2008
- Published online by Cambridge University Press:
- 16 October 2008, pp. 303-314
-
- Article
- Export citation
-
We derive a relation for the growth of the mean square of vertical displacements, δz, of fluid particles of stratified turbulence. In the case of freely decaying turbulence, we find that for large times 〈δz2〉 goes to a constant value 2(EP(0) + aE(0))/N2, where EP(0) and E(0) are the initial mean potential and total turbulent energy per unit mass, respectively, a < 1 and N is the Brunt–Väisälä frequency. In the case of stationary turbulence, we find that 〈δz2〉 = 〈δb2〉/N2 + 2εPt/N2, where εP is the mean dissipation of turbulent potential energy per unit mass and 〈δb2〉 is the Lagrangian structure function of normalized buoyancy fluctuations. The first term is the same as that obtained in the case of adiabatic fluid particle dispersion. This term goes to the finite limit 4EP/N2 as t → ∞. Assuming that the second term represents irreversible mixing, we show that the Osborn & Cox model for vertical diffusion is retained. In the case where the motion is dominated by a turbulent cascade with an eddy turnover time T ≫ N−1, rather than linear gravity waves, we suggest that there is a range of time scales, t, between N−1 and T, where 〈δb2〉 = 2πCPLεPt, where CPL is a constant of the order of unity. This means that for such motion the ratio between the adiabatic and the diabatic mean-square displacement is universal and equal to πCPL in this range. Comparing this result with observations, we make the estimate CPL ≈ 3.
Stratified turbulence forced in rotational and divergent modes
- E. LINDBORG, G. BRETHOUWER
-
- Journal:
- Journal of Fluid Mechanics / Volume 586 / 10 September 2007
- Published online by Cambridge University Press:
- 14 August 2007, pp. 83-108
-
- Article
- Export citation
-
We perform numerical box simulations of strongly stratified turbulence. The equations solved are the Boussinesq equations with constant Brunt–Väisälä frequency and forcing either in rotational or divergent modes, or, with another terminology, in vortical or wave modes. In both cases, we observe a forward energy cascade and inertial-range scaling of the horizontal kinetic and potential energy spectra. With forcing in rotational modes, there is approximate equipartition of kinetic energy between rotational and divergent modes in the inertial range. With forcing in divergent modes the results are sensitive to the vertical forcing wavenumber kfv. If kfv is sufficiently large the dynamics is very similar to the dynamics of the simulations which are forced in rotational modes, with approximate equipartition of kinetic energy in rotational and divergent modes in the inertial range. Frequency spectra of rotational, divergent and potential energy are calculated for individual Fourier modes. Waves are present at low horizontal wavenumbers corresponding to the largest scales in the boxes. In the inertial range, the frequency spectra exhibit no distinctive peaks in the internal wave frequency. In modes for which the vertical wavenumber is considerably larger than the horizontal wavenumber, the frequency spectra of rotational and divergent modes fall on top of each other. The simulation results indicate that the dynamics of rotational and divergent modes develop on the same time scale in stratified turbulence. We discuss the relevance of our results to atmospheric and oceanic dynamics. In particular, we review a number of observational reports indicating that stratified turbulence may be a prevalent dynamic process in the ocean at horizontal scales of the order of 10 or 100 m up to several kilometres.
Scaling analysis and simulation of strongly stratified turbulent flows
- G. BRETHOUWER, P. BILLANT, E. LINDBORG, J.-M. CHOMAZ
-
- Journal:
- Journal of Fluid Mechanics / Volume 585 / 25 August 2007
- Published online by Cambridge University Press:
- 07 August 2007, pp. 343-368
-
- Article
- Export citation
-
Direct numerical simulations of stably and strongly stratified turbulent flows with Reynolds number Re ≫ 1 and horizontal Froude number Fh ≪ 1 are presented. The results are interpreted on the basis of a scaling analysis of the governing equations. The analysis suggests that there are two different strongly stratified regimes according to the parameter . When , viscous forces are unimportant and lv scales as lv ∼ U/N (U is a characteristic horizontal velocity and N is the Brunt–Väisälä frequency) so that the dynamics of the flow is inherently three-dimensional but strongly anisotropic. When , vertical viscous shearing is important so that (lh is a characteristic horizontal length scale). The parameter is further shown to be related to the buoyancy Reynolds number and proportional to (lO/η)4/3, where lO is the Ozmidov length scale and η the Kolmogorov length scale. This implies that there are simultaneously two distinct ranges in strongly stratified turbulence when : the scales larger than lO are strongly influenced by the stratification while those between lO and η are weakly affected by stratification. The direct numerical simulations with forced large-scale horizontal two-dimensional motions and uniform stratification cover a wide Re and Fh range and support the main parameter controlling strongly stratified turbulence being . The numerical results are in good agreement with the scaling laws for the vertical length scale. Thin horizontal layers are observed independently of the value of but they tend to be smooth for < 1, while for > 1 small-scale three-dimensional turbulent disturbances are increasingly superimposed. The dissipation of kinetic energy is mostly due to vertical shearing for < 1 but tends to isotropy as increases above unity. When < 1, the horizontal and vertical energy spectra are very steep while, when > 1, the horizontal spectra of kinetic and potential energy exhibit an approximate k−5/3h-power-law range and a clear forward energy cascade is observed.