Observed relationships between microstructure patches and the gradient Richardson number in a thermally stratified lake

2013 ◽  
Vol 13 (3) ◽  
pp. 205-226 ◽  
Author(s):  
Peter S. Yeates ◽  
Andrés Gómez-Giraldo ◽  
Jörg Imberger
Keyword(s):  
Richardson Number ◽  
Stratified Lake ◽  
Gradient Richardson Number

Novel Perturbation Analysis for a Widely Applicable Curvature Law of the Wall

1999 ◽  
Author(s):  
Namhyo Kim ◽  
David L. Rhode
Keyword(s):  
Perturbation Analysis ◽  
Richardson Number ◽  
Perturbation Technique ◽  
Solution Approach ◽  
Law Of The Wall ◽  
Streamline Curvature ◽  
Curvature Effects ◽  
Gradient Richardson Number ◽  
Closed Form Analytical Solution ◽  
First Time

Abstract A streamline curvature law of the wall is analytically derived to include very strong curved-channel wall curvature effects through a novel perturbation analysis. The new law allows improved analysis of such flows, and it provides the basis for improved wall function boundary conditions for their computation (CFD) over a wider range of y+, even for very strong curvature cases. The unique derivation is based on the Boussinesq eddy viscosity and curvature-corrected mixing length concepts, which is a linear function of the gradient Richardson number. For the first time, to include more complete curved flow physics, local streamline curvature effects in the gradient Richardson number are kept. To overcome the mathematical difficulty of keeping all of these local streamline curvature terms, a novel perturbation solution approach is successfully developed. This novel perturbation technique allows a closed-form analytical solution to many similar non-linear problems which previously required more complicated techniques. Qualitative and quantitative comparisons with measurements and previous curvature laws of the wall obtained by different approaches reveal that the new law shows improved representation of the wall curvature effects for all of the four test cases.

Les travaux sur la turbulence : les origines, Toucans, Cost-ES0905 et influence de l'entropie

2021 ◽  
pp. 079
Author(s):  
Ivan Bašták Ďurán ◽  
Pascal Marquet
Keyword(s):  
Kinetic Energy ◽  
Turbulent Kinetic Energy ◽  
Richardson Number ◽  
Stable Stratification ◽  
Turbulence Energy ◽  
Third Order ◽  
Length Scales ◽  
Gradient Richardson Number ◽  
Turbulence Length ◽  
Consistent Treatment

Le schéma de turbulence Toucans est utilisé dans la configuration opérationnelle Alaro du modèle Aladin depuis début 2015. Son développement a été initié, guidé et en grande partie conçu par Jean-François Geleyn. Ce développement a commencé avec le prédécesseur du schéma Toucans, le schéma « pseudo-pronostique » en énergie cinétique turbulente, lui-même basé sur l'ancien schéma de turbulence de Louis, mais étendu dans Toucans à un schéma pronostique. Le schéma Toucans a pour objectif de traiter de manière cohérente les fonctions qui dépendent de la stabilité verticale de l'atmosphère, de l'influence de l'humidité et des échelles de longueur de la turbulence (de mélange et de dissipation). De plus, de nouvelles caractéristiques ont été ajoutées : une représentation améliorée pour les stratifications très stables (absence de nombre de Richardson critique), une meilleure représentation de l'anisotropie, un paramétrage unifié de la turbulence et des nuages par l'ajout d'une deuxième énergie turbulente pronostique et la paramétrisation des moments du troisième ordre. The Toucans turbulence scheme is a turbulence scheme that is used in the operational Alaro configuration of the Aladin model since early 2015. Its development was initiated, guided and to a large extend authored by Jean-François Geleyn. The development started with the predecessor of the Toucans scheme, the "pseudo-prognostic" turbulent kinetic energy scheme which itself was built on the "Louis" turbulence scheme, but extended to a prognostic scheme. The Toucans scheme aims for a consistent treatment of stability dependency functions, influence of moisture, and turbulence length scales. Additionally, new features were added to the turbulence scheme: improved representation of turbulence in very stable stratification (absence of critical gradient Richardson number), better representation of anisotropy, unified parameterization of turbulence and clouds via addition of second prognostic turbulence energy, and parameterization of third order moments.

scholarly journals Modulation of Small-Scale Turbulence by Ducted Gravity Waves in the Nocturnal Boundary Layer

2008 ◽  
Vol 65 (4) ◽  
pp. 1414-1427 ◽  
Author(s):  
Y. P. Meillier ◽  
R. G. Frehlich ◽  
R. M. Jones ◽  
B. B. Balsley
Keyword(s):  
Boundary Layer ◽  
Gravity Waves ◽  
Richardson Number ◽  
Potential Temperature ◽  
Nocturnal Boundary Layer ◽  
Small Scale ◽  
Temperature Structure ◽  
Turbulence Measurements ◽  
Gradient Richardson Number ◽  
Scale Turbulence

Abstract Constant altitude measurements of temperature and velocity in the residual layer of the nocturnal boundary layer, collected by the Cooperative Institute for Research in Environmental Sciences (CIRES) Tethered Lifting System (TLS), exhibit fluctuations identified by previous work (Fritts et al.) as the signature of ducted gravity waves. The concurrent high-resolution TLS turbulence measurements (temperature structure constant C2T and turbulent kinetic energy dissipation rate ɛ) reveal the presence of patches of enhanced turbulence activity that are roughly synchronized with the troughs of the temperature and velocity fluctuations. To investigate the potentially dominant role ducted gravity waves might play on the modulation of atmospheric stability and therefore, on turbulence, time series of the wave-modulated gradient Richardson number (Ri) and of the vertical gradient of potential temperature ∂θ/∂z(t) are computed numerically and compared to the TLS small-scale turbulence measurements. The results of this study agree with the predictions of previous theoretical studies (i.e., wave-generated fluctuations of temperature and velocity modulate the gradient Richardson number), resulting in periodic enhancements of turbulence at Ri minima. The patches of turbulence observed in the TLS dataset are subsequently identified as convective instabilities generated locally within the unstable phase of the wave.

Gradient Richardson number measurements in a stratified shear layer

1999 ◽  
Vol 30 (1) ◽  
pp. 47-63 ◽  
Author(s):  
I.P.D De Silva ◽  
A Brandt ◽  
L.J Montenegro ◽  
H.J.S Fernando
Keyword(s):  
Shear Layer ◽  
Richardson Number ◽  
Gradient Richardson Number

scholarly journals The Influence of Stratification and Nonlocal Turbulent Production on Estuarine Turbulence: An Assessment of Turbulence Closure with Field Observations

2011 ◽  
Vol 41 (1) ◽  
pp. 166-185 ◽  
Author(s):  
Malcolm E. Scully ◽  
W. Rocky Geyer ◽  
John H. Trowbridge
Keyword(s):  
Richardson Number ◽  
Turbulence Closure ◽  
Length Scale ◽  
Field Observations ◽  
Turbulent Length Scale ◽  
Stability Functions ◽  
Gradient Richardson Number ◽  
Simple Scaling ◽  
The Impact ◽  
Distance To The Boundary

Abstract Field observations of turbulent kinetic energy (TKE), dissipation rate ɛ, and turbulent length scale demonstrate the impact of both density stratification and nonlocal turbulent production on turbulent momentum flux. The data were collected in a highly stratified salt wedge estuary using the Mobile Array for Sensing Turbulence (MAST). Estimates of the dominant length scale of turbulent motions obtained from the vertical velocity spectra provide field confirmation of the theoretical limitation imposed by either the distance to the boundary or the Ozmidov scale, whichever is smaller. Under boundary-limited conditions, anisotropy generally increases with increasing shear and decreased distance to the boundary. Under Ozmidov-limited conditions, anisotropy increases rapidly when the gradient Richardson number exceeds 0.25. Both boundary-limited and Ozmidov-limited conditions demonstrate significant deviations from a local production–dissipation balance that are largely consistent with simple scaling relationships for the vertical divergence in TKE flux. Both the impact of stratification and deviation from equilibrium turbulence observed in the data are largely consistent with commonly used turbulence closure models that employ “nonequilibrium” stability functions. The data compare most favorably with the nonequilibrium version of the L. H. Kantha and C. A. Clayson stability functions. Not only is this approach more consistent with the observed critical gradient Richardson number of 0.25, but it also accounts for the large deviations from equilibrium turbulence in a manner consistent with the observations.

Mixing events in a stratified jet subject to surface wind and buoyancy forcing

2011 ◽  
Vol 685 ◽  
pp. 54-82 ◽  
Author(s):  
Hieu T. Pham ◽  
Sutanu Sarkar
Keyword(s):  
Surface Layer ◽  
Wind Stress ◽  
Richardson Number ◽  
Reynolds Shear Stress ◽  
Surface Wind ◽  
Initial Density ◽  
Shear Instability ◽  
Energy Budgets ◽  
Surface Cooling ◽  
Gradient Richardson Number

AbstractThe fine-scale response of a subsurface stable stratified jet subject to the forcing of surface wind stress and surface cooling is investigated using direct numerical simulation. The initial velocity profile consists of a symmetric jet located below a surface layer driven by a constant wind stress. The initial density profile is well-mixed in the surface layer and linearly stratified in both upper and lower flanks of the jet. The minimum value of the gradient Richardson number in the upper flank of the jet exceeds the critical value of 0.25 for linear shear instability. Broadband finite-amplitude fluctuations are introduced to the surface layer to initiate the simulation. Turbulence is generated in the surface layer and deepens into the jet upper flank. Internal waves generated by the turbulent surface layer are observed to propagate downward across the jet. The momentum flux carried by the waves is significantly smaller than the Reynolds shear stress extracted from the background velocity. The wave energy flux is also smaller than the turbulence production by mean shear. Ejections of fluid parcels by horseshoe-like vortices cause intermittent patches of intense dissipation inside the jet upper flank where the background gradient Richardson number is larger than 0.25. Drag due to the wind stress is smaller than the drag caused by turbulent stress in the flow. Analysis of the mean and turbulent kinetic energy budgets suggests that the energy input by surface forcing is considerably smaller than the energy extracted from the initially imposed background shear in the surface layer.

On the Scale-dependence of the Gradient Richardson Number in the Residual Layer

2007 ◽  
Vol 127 (1) ◽  
pp. 57-72 ◽  
Author(s):  
Ben B. Balsley ◽  
Gunilla Svensson ◽  
Michael Tjernström
Keyword(s):  
Richardson Number ◽  
Scale Dependence ◽  
Residual Layer ◽  
Gradient Richardson Number

A note on analytic solutions for entraining stratified gravity currents

2017 ◽  
Vol 836 ◽  
pp. 260-276
Author(s):  
Marcus C. Horsley ◽  
Andrew W. Woods
Keyword(s):  
Froude Number ◽  
Interfacial Layer ◽  
Richardson Number ◽  
Mixing Layer ◽  
Analytic Solutions ◽  
Mixing Zone ◽  
Stable Case ◽  
Gradient Richardson Number ◽  
Convergence Point ◽  
Downstream Flow

High-Reynolds-number steady currents of relatively dense fluid propagating along a horizontal boundary become unstable and mix with the overlying fluid if the gradient Richardson number across the interface is less than 1/4. The process of entrainment produces a deepening mixing layer at the interface, which increases the gradient Richardson number of this layer and eventually may suppress further entrainment. The conservation of the vertically averaged buoyancy and momentum flux, as the current advances along the boundary, leads to two integral constraints relating the downstream flow with that upstream of the mixing zone. These constraints are equivalent to imposing a Froude number in the upstream flow. Using the ansatz that the dowstream velocity and buoyancy profiles in the current have a lower well-mixed region overlain by an interfacial layer of constant gradient, we can use these two constraints to quantify the total entrainment of ambient fluid into the flow as a function of the gradient Richardson number of the downstream flow. This leads to recognition that both subcritical and supercritical currents may develop downstream of the mixing zone. However, as the mixing increases and the interfacial layer gradually deepens, there is a critical point at which these two solution branches coincide. For each upstream Froude number, we can also determine the downstream flow with maximal entrainment. This maximal entrainment solution coincides with the convergence point of the supercritical and subcritical branches. We compare this with the entrainment predicted for those solutions with a gradient Richardson number of 1/4, which corresponds to the marginally stable case. As the upstream Froude number $Fr_{u}$ increases, the maximum depth of the interfacial mixing layer gradually increases until eventually, for $Fr_{u}>2.921$, the whole current may become modified through entrainment. We discuss the relevance of these results for mixing in gravity-driven flows.

scholarly journals The influence of far field stratification on shear-induced turbulent mixing

2021 ◽  
Vol 928 ◽  
Author(s):  
S.F. Lewin ◽  
C.P. Caulfield
Keyword(s):  
Reynolds Number ◽  
Turbulent Flows ◽  
Richardson Number ◽  
Scaling Laws ◽  
Mixing Efficiency ◽  
Far Field ◽  
Mixing Rate ◽  
Gradient Richardson Number ◽  
Background Density ◽  
Secondary Instabilities

We compare the properties of the turbulence induced by the breakdown of Kelvin–Helmholtz instability (KHI) at high Reynolds number in two classes of stratified shear flows where the background density profile is given by either a linear function or a hyperbolic tangent function, at different values of the minimum initial gradient Richardson number ${{Ri}}_0$ . Considering global and local measures of mixing defined in terms of either the irreversible mixing rate $\mathscr {M}$ associated with the time evolution of the background potential energy, or an appropriately defined density variance dissipation rate $\chi$ , we find that the proliferation of secondary instabilities strongly affects the efficiency of mixing early in the flow evolution, and also that these secondary instabilities are highly sensitive to flow perturbations that are added at the point of maximal (two-dimensional) billow amplitude. Nevertheless, mixing efficiency does not appear to depend strongly on the far field density structure, a feature supported by the evolution of local horizontally averaged values of the buoyancy Reynolds number ${Re}_b$ and gradient Richardson number ${Ri}_g$ . We investigate the applicability of various proposed scaling laws for flux coefficients $\varGamma$ in terms of characteristic length scales, in particular discussing the relevance of the overturning ‘Thorpe scale’ in stratified turbulent flows. Finally, we compare a variety of empirical model parameterizations used to compute diapycnal diffusivity in an oceanographic context, arguing that for transient flows such as KHI-induced turbulence, simple models that relate the ‘age’ of a turbulent event to its mixing efficiency can produce reasonably robust mixing estimates.

Gradient Richardson number profiles using SAS

1990 ◽  
Vol 5 (4) ◽  
pp. 207-211
Author(s):  
A.H. Weber ◽  
R.J. Kurzeja
Keyword(s):  
Richardson Number ◽  
Gradient Richardson Number
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