scholarly journals Richardson Number Criterion for the Nonlinear Stability of Three-Dimensional Stratified Flow

1984 ◽  
Vol 52 (26) ◽  
pp. 2352-2355 ◽  
Author(s):  
Henry D. I. Abarbanel ◽  
Darryl D. Holm ◽  
Jerrold E. Marsden ◽  
Tudor Ratiu
Keyword(s):  
Nonlinear Stability ◽  
Richardson Number ◽  
Three Dimensional ◽  
Stratified Flow

scholarly journals Nonlinear stability analysis of stratified fluid equilibria

1986 ◽  
Vol 318 (1543) ◽  
pp. 349-409 ◽  
Keyword(s):  
Nonlinear Stability ◽  
Richardson Number ◽  
Three Dimensional ◽  
Stratified Fluid ◽  
Stationary Solutions ◽  
Three Dimensions ◽  
Stability Criteria ◽  
Nonlinear Stability Analysis ◽  
Hamiltonian Structures ◽  
Linearized Stability

Nonlinear stability is analysed for stationary solutions of incompressible inviscid stratified fluid flow in two and three dimensions. Both the Euler equations and their Boussinesq approximations are treated. The techniques used were initiated by Arnold around 1965. These techniques combine energy methods, conserved quantities and convexity estimates. The resulting nonlinear stability criteria involve standard quantities, such as the Richardson number, but they differ from the linearized stability criteria. For example, the full three-dimensional problem has nonlinearly stable stationary solutions with Richardson number greater than unity, provided the gradients of the variations in density satisfy explicitly given bounds. Specific examples and the associated Hamiltonian structures for the problems are given.

Modification of the k-epsilon scheme and its application for describing turbulence in inland water bodies

2021 ◽  
Author(s):  
Irina Soustova ◽  
Yuliya Troitskaya ◽  
Daria Gladskikh
Keyword(s):  
Prandtl Number ◽  
Turbulent Mixing ◽  
Richardson Number ◽  
Three Dimensional ◽  
Water Bodies ◽  
Inland Water ◽  
Dependent Relation ◽  
Vertical Distributions ◽  
Computing Center ◽  
Inland Water Bodies

<p>A parameterization of the Prandtl number as a function of the gradient Richardson number is proposed in order to correctly take into account stratification when calculating the thermohydrodynamic regime of inland water bodies. This parameterization allows the existence of turbulence at any values ​​of the Richardson number.</p><p>The proposed function is used to calculate the turbulent thermal conductivity coefficient in a k-epsilon mixing scheme. Modification is implemented in the three-dimensional hydrostatic model developed at the Research Computing Center of Moscow State University.</p><p>It is demonstrated that the proposed modification (in contrast to the standard scheme with a constant Prandtl number) leads to smoothing all sharp changes in vertical distributions of turbulent mixing parameters (turbulent kinetic energy, temperature and thickness of the shock layer) and imposes a Richardson number-dependent relation on the empirical constants of k-epsilon turbulent mixing scheme.</p><p>The work was supported by grants of the RF President’s Grant for Young Scientists (MK-1867.2020.5) and by the RFBR (19-05-00249, 20-05-00776). </p>

Turbulent Mixing With Density Differences

2011 ◽  
Author(s):  
Jos Derksen
Keyword(s):  
Turbulent Mixing ◽  
Richardson Number ◽  
Kinematic Viscosity ◽  
Three Dimensional ◽  
Flow Regimes ◽  
Reynolds Numbers ◽  
Flow Structures ◽  
Three Dimensional Flow ◽  
Flow Equations ◽  
Miscible Liquids

Homogenization of initially segregated and stably stratified systems consisting of two miscible liquids with different density and the same kinematic viscosity in an agitated tank was studied computationally. Reynolds numbers were in the range of 3,000 to 12,000 so that it was possible to solve the flow equations without explicitly modeling turbulence. The Richardson number that characterizes buoyancy was varied between 0 and 1. The stratification clearly lengthens the homogenization process. Two flow regimes could be identified. At low Richardson numbers large, three-dimensional flow structures dominate mixing, as is the case in non-buoyant systems. At high Richardson numbers the interface between the two liquids largely stays intact. It rises due to turbulent erosion, gradually drawing down and mixing up the lighter liquid.

Transient growth in strongly stratified shear layers

2014 ◽  
Vol 758 ◽  
Author(s):  
A. K. Kaminski ◽  
C. P. Caulfield ◽  
J. R. Taylor
Keyword(s):  
Shear Layer ◽  
Normal Mode ◽  
Richardson Number ◽  
Mixing Layer ◽  
Three Dimensional ◽  
Buoyancy Frequency ◽  
Transient Growth ◽  
Time Intervals ◽  
Optimal Perturbation ◽  
Target Time

AbstractWe investigate numerically transient linear growth of three-dimensional perturbations in a stratified shear layer to determine which perturbations optimize the growth of the total kinetic and potential energy over a range of finite target time intervals. The stratified shear layer has an initial parallel hyperbolic tangent velocity distribution with Reynolds number $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\mathit{Re}=U_0 h/\nu =1000$ and Prandtl number $\nu /\kappa =1$, where $\nu $ is the kinematic viscosity of the fluid and $\kappa $ is the diffusivity of the density. The initial stable buoyancy distribution has constant buoyancy frequency $N_0$, and we consider a range of flows with different bulk Richardson number ${\mathit{Ri}}_b=N_0^2h^2/U_0^2$, which also corresponds to the minimum gradient Richardson number ${\mathit{Ri}}_g(z)=N_0^2/(\mathrm{d}U/\mathrm{d} z)^2$ at the midpoint of the shear layer. For short target times, the optimal perturbations are inherently three-dimensional, while for sufficiently long target times and small ${\mathit{Ri}}_b$ the optimal perturbations are closely related to the normal-mode ‘Kelvin–Helmholtz’ (KH) instability, consistent with analogous calculations in an unstratified mixing layer recently reported by Arratia et al. (J. Fluid Mech., vol. 717, 2013, pp. 90–133). However, we demonstrate that non-trivial transient growth occurs even when the Richardson number is sufficiently high to stabilize all normal-mode instabilities, with the optimal perturbation exciting internal waves at some distance from the midpoint of the shear layer.

Computational and experimental investigations in a cyclone dust separator

1997 ◽  
Vol 211 (4) ◽  
pp. 247-257 ◽  
Author(s):  
S M Fraser ◽  
A M Abdel-Razek ◽  
M Z Abdullah
Keyword(s):  
Flow Field ◽  
Richardson Number ◽  
Swirling Flow ◽  
Laser Doppler Anemometry ◽  
Experimental Studies ◽  
Three Dimensional ◽  
Experimental Results ◽  
Experimental Investigations ◽  
Cyclone Chamber ◽  
Dust Separator

Three-dimensional turbulent flow in a model cyclone has been simulated using PHOENICS code and experimental studies carried out using a laser Doppler anemometry (LDA) system. The experimental results were used to validate the computed velocity distributions based on the standard and a modified k-∊ model. The standard k-∊ model was found to be unsatisfactory for the prediction of the flow field inside the cyclone chamber. By considering the strong swirling flow and the streamlined curvature, a k-∊ model, modified to take account of the Richardson number, provided better velocity distributions and better agreement with the experimental results.

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