Effects of Schmidt number on the short-wavelength instabilities in stratified vortices

2019 ◽  
Vol 867 ◽  
pp. 765-803
Author(s):  
Suraj Singh ◽  
Manikandan Mathur
Keyword(s):  
Growth Rate ◽  
Stagnation Point ◽  
Short Wavelength ◽  
Schmidt Number ◽  
Maximum Growth ◽  
Stable Stratification ◽  
Oscillatory Instability ◽  
Inviscid Limit ◽  
Small Diffusion ◽  
Axisymmetric Vortex

We present a local stability analysis to investigate the effects of differential diffusion between momentum and density (quantified by the Schmidt number $Sc$) on the three-dimensional, short-wavelength instabilities in planar vortices with a uniform stable stratification along the vorticity axis. Assuming small diffusion in both momentum and density, but arbitrary values for $Sc$, we present a detailed analytical/numerical analysis for three different classes of base flows: (i) an axisymmetric vortex, (ii) an elliptical vortex and (iii) the flow in the neighbourhood of a hyperbolic stagnation point. While a centrifugally stable axisymmetric vortex remains stable for any $Sc$, it is shown that $Sc$ can have significant effects in a centrifugally unstable axisymmetric vortex: the range of unstable perturbations increases when $Sc$ is taken away from unity, with the extent of increase being larger for $Sc\ll 1$ than for $Sc\gg 1$. Additionally, for $Sc>1$, we report the possibility of oscillatory instability. In an elliptical vortex with a stable stratification, $Sc\neq 1$ is shown to non-trivially influence the three different inviscid instabilities (subharmonic, fundamental and superharmonic) that have been previously reported, and also introduce a new branch of oscillatory instability that is not present at $Sc=1$. The unstable parameter space for the subharmonic (instability IA) and fundamental (instability IB) inviscid instabilities are shown to be significantly increased for $Sc<1$ and $Sc>1$, respectively. Importantly, for sufficiently small and large $Sc$, respectively, the maximum growth rate for instabilities IA and IB occurs away from the inviscid limit. The new oscillatory instability (instability III) is shown to occur only for sufficiently small $Sc<1$, the signature of which is nevertheless present with zero growth rate in the inviscid limit. The Schmidt number is then shown to play no role in the evolution of transverse perturbations on the flow around a hyperbolic stagnation point with a stable stratification. We conclude by discussing the physical length scales associated with the $Sc\neq 1$ instabilities, and their potential relevance in various realistic settings.

scholarly journals Horizontal shear instabilities in rotating stellar radiation zones

2020 ◽  
Vol 635 ◽  
pp. A133 ◽  
Author(s):  
J. Park ◽  
V. Prat ◽  
S. Mathis
Keyword(s):  
Growth Rate ◽  
Thermal Diffusion ◽  
Turbulent Transport ◽  
Stable Stratification ◽  
Shear Instability ◽  
Inviscid Limit ◽  
Horizontal Shear ◽  
Wide Range ◽  
Inertial Instability ◽  
Shear Instabilities

Context. Rotational mixing transports angular momentum and chemical elements in stellar radiative zones. It is one of the key processes for modern stellar evolution. In the past two decades, an emphasis has been placed on the turbulent transport induced by the vertical shear instability. However, instabilities arising from horizontal shear and the strength of the anisotropic turbulent transport that they may trigger remain relatively unexplored. The weakest point of this hydrodynamical theory of rotational mixing is the assumption that anisotropic turbulent transport is stronger in horizontal directions than in the vertical one. Aims. This paper investigates the combined effects of stable stratification, rotation, and thermal diffusion on the horizontal shear instabilities that are obtained and discussed in the context of stellar radiative zones. Methods. The eigenvalue problem describing linear instabilities of a flow with a hyperbolic-tangent horizontal shear profile was solved numerically for a wide range of parameters. When possible, the Wentzel–Kramers–Brillouin–Jeffreys (WKBJ) approximation was applied to provide analytical asymptotic dispersion relations in both the nondiffusive and highly diffusive limits. As a first step, we consider a polar f-plane where the gravity and rotation vector are aligned. Results. Two types of instabilities are identified: the inflectional and inertial instabilities. The inflectional instability that arises from the inflection point (i.e., the zero second derivative of the shear flow) is the most unstable when at a zero vertical wavenumber and a finite wavenumber in the streamwise direction along the imposed-flow direction. While the maximum two-dimensional growth rate is independent of the stratification, rotation rate, and thermal diffusivity, the three-dimensional inflectional instability is destabilized by stable stratification, while it is stabilized by thermal diffusion. The inertial instability is rotationally driven, and a WKBJ analysis reveals that its growth rate reaches the maximum value of √f(1 − f) in the inviscid limit as the vertical wavenumber goes to infinity, where f is the dimensionless Coriolis parameter. The inertial instability for a finite vertical wavenumber is stabilized as the stratification increases, whereas it is destabilized by the thermal diffusion. Furthermore, we found a selfsimilarity in both the inflectional and inertial instabilities based on the rescaled parameter PeN2 with the Péclet number Pe and the Brunt–Väisälä frequency N.

Saturation mechanism and generated viscosity in gravito-turbulent accretion disks

2020 ◽  
Vol 640 ◽  
pp. A53
Author(s):  
L. Löhnert ◽  
S. Krätschmer ◽  
A. G. Peeters
Keyword(s):  
Growth Rate ◽  
Numerical Simulations ◽  
Accretion Disks ◽  
Gravitational Instability ◽  
Turbulent Viscosity ◽  
Radial Distance ◽  
Maximum Growth ◽  
Wave Number ◽  
Radiative Cooling ◽  
Mixing Length

Here, we address the turbulent dynamics of the gravitational instability in accretion disks, retaining both radiative cooling and irradiation. Due to radiative cooling, the disk is unstable for all values of the Toomre parameter, and an accurate estimate of the maximum growth rate is derived analytically. A detailed study of the turbulent spectra shows a rapid decay with an azimuthal wave number stronger than ky−3, whereas the spectrum is more broad in the radial direction and shows a scaling in the range kx−3 to kx−2. The radial component of the radial velocity profile consists of a superposition of shocks of different heights, and is similar to that found in Burgers’ turbulence. Assuming saturation occurs through nonlinear wave steepening leading to shock formation, we developed a mixing-length model in which the typical length scale is related to the average radial distance between shocks. Furthermore, since the numerical simulations show that linear drive is necessary in order to sustain turbulence, we used the growth rate of the most unstable mode to estimate the typical timescale. The mixing-length model that was obtained agrees well with numerical simulations. The model gives an analytic expression for the turbulent viscosity as a function of the Toomre parameter and cooling time. It predicts that relevant values of α = 10−3 can be obtained in disks that have a Toomre parameter as high as Q ≈ 10.

A note on growth curves of rabbit lines selected on growth rate or litter size

1993 ◽  
Vol 57 (2) ◽  
pp. 332-334 ◽  
Author(s):  
A. Blasco ◽  
E. Gómez
Keyword(s):  
Growth Rate ◽  
Litter Size ◽  
Growth Curves ◽  
Live Weight ◽  
Maximum Growth ◽  
Maximum Growth Rate ◽  
Standard Errors ◽  
Adult Weight ◽  
Weight Growth ◽  
Using Data

Two synthetic lines of rabbits were used in the experiment. Line V, selected on litter size, and line R, selected on growth rate. Ninety-six animals were randomly collected from 48 litters, taking a male and a female each time. Richards and Gompertz growth curves were fitted. Sexual dimorphism appeared in the line V but not in the R. Values for b and k were similar in all curves. Maximum growth rate took place in weeks 7 to 8. A break due to weaning could be observed in weeks 4 to 5. Although there is a remarkable similarity of the values of all the parameters using data from the first 20 weeks only, the higher standard errors on adult weight would make 30 weeks the preferable time to take data for live-weight growth curves.

Reassessment of Maximum Growth Rates for C3 and C4 Crops

1978 ◽  
Vol 14 (1) ◽  
pp. 1-5 ◽  
Author(s):  
J. L. Monteith
Keyword(s):  
Growth Rate ◽  
Growth Rates ◽  
Growing Season ◽  
Maximum Growth ◽  
Maximum Growth Rate ◽  
Crop Growth ◽  
Close Inspection ◽  
Similar Difference ◽  
Photosynthetic Efficiencies

SUMMARYFigures for maximum crop growth rates, reviewed by Gifford (1974), suggest that the productivity of C3 and C4 species is almost indistinguishable. However, close inspection of these figures at source and correspondence with several authors revealed a number of errors. When all unreliable figures were discarded, the maximum growth rate for C3 stands fell in the range 34–39 g m−2 d−1 compared with 50–54 g m−2 d−1 for C4 stands. Maximum growth rates averaged over the whole growing season showed a similar difference: 13 g m−2 d−1 for C3 and 22 g m−2 d−1 for C4. These figures correspond to photosynthetic efficiencies of approximately 1·4 and 2·0%.

scholarly journals Growth of GaN from elemental Gallium and Ammonia via Modified Sandwich Growth Technique

2004 ◽  
Vol 831 ◽  
Author(s):  
E. Berkman ◽  
R. Collazo ◽  
R. Schlesser ◽  
Z. Sitar
Keyword(s):  
Growth Rate ◽  
Gas Flow ◽  
Transport Model ◽  
Theoretical Calculations ◽  
Substrate Surface ◽  
Maximum Growth ◽  
Direct Reaction ◽  
Pure Nitrogen ◽  
Flow Rates ◽  
Reactor Pressure

ABSTRACTGallium nitride (GaN) films were grown on (0001) sapphire substrates at 1050°C by controlled evaporation of gallium (Ga) metal and reaction with ammonia (NH3) at a total reactor pressure of 800 Torr. Pure nitrogen (N2) was flowed directly above the molten Ga source to prevented direct reaction between the molten Ga and ammonia, which causes Ga spattering and GaN crust formation. At the same time, this substantially enhanced the Ga transport to the substrate. A simple mass-transport model based on total reactor pressure, gas flow rates and source temperature was developed and verified. The theoretical calculations and growth rate measurements at different ammonia flow rates and reactor pressures showed that the maximum growth rate was controlled by transport of both Ga species and reactive ammonia to the substrate surface.

Interpretation of Experimental Data with Regard to the Activated Sludge Model No.1 and Calibration of the Model for Municipal Wastewater Treatment Plants

1992 ◽  
Vol 25 (6) ◽  
pp. 167-183 ◽  
Author(s):  
H. Siegrist ◽  
M. Tschui
Keyword(s):  
Experimental Data ◽  
Growth Rate ◽  
Activated Sludge ◽  
Municipal Wastewater ◽  
Wastewater Treatment Plants ◽  
Treatment Plant ◽  
Maximum Growth ◽  
Maximum Growth Rate ◽  
Municipal Wastewater Treatment ◽  
Activated Sludge Model

The wastewater of the municipal treatment plants Zürich-Werdhölzli (350000 population equivalents), Zürich-Glatt (110000), and Wattwil (20000) have been characterized with regard to the activated sludge model Nr.1 of the IAWPRC task group. Zürich-Glatt and Wattwil are partly nitrifying treatment plants and Zürich-Werdhölzli is fully nitrifying. The mixing characteristics of the aeration tanks at Werdhölzli and Glatt were determined with sodium bromide as a tracer. The experimental data were used to calibrate hydrolysis, heterotrophic growth and nitrification. Problems arising by calibrating hydrolysis of the paniculate material and by measuring oxygen consumption of heterotrophic and nitrifying microorganisms are discussed. For hydrolysis the experimental data indicate first-order kinetics. For nitrification a maximum growth rate of 0.40±0.07 d−1, corresponding to an observed growth rate of 0.26±0.04 d−1 was calculated at 10°C. The half velocity constant found for 12 and 20°C was 2 mg NH4-N/l. The calibrated model was verified with experimental dam of me Zürich-Werdhölzli treatment plant during ammonia shock load.

Dietary Protein Requirements of Fishes — A Reassessment

1987 ◽  
Vol 44 (11) ◽  
pp. 1995-2001 ◽  
Author(s):  
Stephen H. Bowen
Keyword(s):  
Growth Rate ◽  
Dietary Protein ◽  
Protein Concentration ◽  
Trophic Ecology ◽  
Maximum Growth ◽  
Maximum Growth Rate ◽  
Intake Rate ◽  
Protein Requirement ◽  
Fish Physiology ◽  
Protein Retention

It is widely believed that fishes require more dietary protein than other vertebrates. Many aspects of fish physiology, nutrition, and trophic ecology have been interpreted within the context of this high protein requirement. Here, fishes are compared with terrestrial homeotherms in terms of (1) protein requirement for maintenance, (2) relative protein concentration in the diet required for maximum growth rate, (3) protein intake rate required for maximum growth rate, (4) efficiency of protein retention in growth, and (5) weight of growth achieved per weight of protein ingested. The two animal groups compared differ only in relative protein concentration in the diet required for maximum growth rate. This difference is explained in terms of homeotherms' greater requirement for energy and does not reflect absolute differences in protein requirement. The remaining measures of protein requirement suggest that fishes and terrestrial homeotherms are remarkably similar in their use of protein as a nutritional resource. Reinterpretation of the role of protein in fish physiology, nutrition, and trophic ecology is perhaps in order.

Allometric scaling and taxonomic variation in nutrient utilization traits and maximum growth rate of phytoplankton

2012 ◽  
Vol 57 (2) ◽  
pp. 554-566 ◽  
Author(s):  
Kyle F. Edwards ◽  
Mridul K. Thomas ◽  
Christopher A. Klausmeier ◽  
Elena Litchman
Keyword(s):  
Growth Rate ◽  
Allometric Scaling ◽  
Maximum Growth ◽  
Nutrient Utilization ◽  
Maximum Growth Rate

Single Crystal Silicon Carbide On Silicon Using A Supersonic Gas Jet Of Methylsilane

1997 ◽  
Vol 483 ◽  
Author(s):  
S. A. Ustin ◽  
C. Long ◽  
L. Lauhon ◽  
W. Ho
Keyword(s):  
Growth Rate ◽  
Growth Rates ◽  
Single Crystal Silicon ◽  
Maximum Growth ◽  
X Ray Diffraction ◽  
Crystal Silicon ◽  
Cross Sectional ◽  
Supersonic Molecular Beam ◽  
Transmission Electron

AbstractCubic SiC films have been grown on Si(001) and Si(111) substrates at temperatures between 600 °C and 900 °C with a single supersonic molecular beam source. Methylsilane (H3SiCH3) was used as the sole precursor with hydrogen and nitrogen as seeding gases. Optical reflectance was used to monitor in situ growth rate and macroscopic roughness. The growth rate of SiC was found to depend strongly on substrate orientation, methylsilane kinetic energy, and growth temperature. Growth rates were 1.5 to 2 times greater on Si(111) than on Si(001). The maximum growth rates achieved were 0.63 μm/hr on Si(111) and 0.375μm/hr on Si(001). Transmission electron diffraction (TED) and x-ray diffraction (XRD) were used for structural characterization. In-plane azimuthal (ø-) scans show that films on Si(001) have the correct 4-fold symmetry and that films on Si(111) have a 6-fold symmetry. The 6-fold symmetry indicates that stacking has occurred in two different sequences and double positioning boundaries have been formed. The minimum rocking curve width for SiC on Si(001) and Si(111) is 1.2°. Fourier Transform Infrared (FTIR) absorption was performed to discern the chemical bonding. Cross Sectional Transmission Electron Microscopy (XTEM) was used to image the SiC/Si interface.

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