Dimensionless Numbers: Difference between revisions
Jump to navigation
Jump to search
No edit summary |
No edit summary |
||
| Line 1: | Line 1: | ||
<div id="Reynolds Number"></div> | <div id="Reynolds Number"></div> | ||
;'''Reynolds Number''' | ;'''Reynolds Number''' | ||
: <math> Re = \frac{UL}{\nu} </math>, where <math> U </math> and <math> L </math> are the characteristic velocity and length scales, and <math> \nu </math> is the [[#Viscosity|dynamic viscosity]]. | * Definition: <math> Re = \frac{UL}{\nu} </math>, where <math> U </math> and <math> L </math> are the characteristic velocity and length scales, and <math> \nu </math> is the [[#Viscosity|dynamic viscosity]]. | ||
* Interpretation: The ratio of inertial force to viscous force. | |||
* Analysis: Small Reynolds numbers are often associated with viscous flows, whereas large Reynolds numbers are typically found in turbulent flows. | |||
<div id=" | <div id="Richardson Number"></div> | ||
;'''Richardson Number''' | ;'''Gradient Richardson Number''' | ||
: <math> Ri = N^2/U_z^2 </math>, where <math> N </math> is the [[#Buoyancy Frequency|buoyancy frequency]] and <math> U </math> is the background horizontal velocity. | * Definition: <math> Ri = N^2/U_z^2 </math>, where <math> N </math> is the [[#Buoyancy Frequency|buoyancy frequency]] and <math> U </math> is the background horizontal velocity. | ||
* Interpretation: The ratio between the strength of stratification and velocity shear. | |||
* Analysis: A sufficient condition for the flow to be ''linearly stable'' is that the local Richardson number exceed 0.25 throughout the flow. However, <math> Ri < 0.25 </math> does not mean the flow is necessarily unstable (the criterion is not sufficient). When the flow is not a parallel shear flow, the meaning of <math> Ri </math> is not clear. | |||
<div id="Schmidt Number"></div> | <div id="Schmidt Number"></div> | ||
;'''Schmidt Number''' | ;'''Schmidt Number''' | ||
: <math> Sc = \nu/\kappa </math>, where <math> \nu </math> is the [[#Viscosity|viscosity]], and <math> \kappa </math> is the [[#Diffusivity|diffusivity]]. | * Definition: <math> Sc = \nu/\kappa </math>, where <math> \nu </math> is the [[#Viscosity|viscosity]], and <math> \kappa </math> is the [[#Diffusivity|diffusivity]]. | ||
* Interpretation: The ratio between the momentum diffusivity (i.e. viscosity) and mass diffusivity. | |||
* Analysis: The typical Schmidt number for water is around 500, depending on the temperature and salinity. For [[#DNS|direct numerical simulations]], <math> Sc = 1 </math> is commonly used in the literature. See also [[#Prandtl Number|Prandtl number]]. | |||
Revision as of 09:38, 28 May 2015
- Reynolds Number
- Definition: , where and are the characteristic velocity and length scales, and is the dynamic viscosity.
- Interpretation: The ratio of inertial force to viscous force.
- Analysis: Small Reynolds numbers are often associated with viscous flows, whereas large Reynolds numbers are typically found in turbulent flows.
- Gradient Richardson Number
- Definition: , where is the buoyancy frequency and is the background horizontal velocity.
- Interpretation: The ratio between the strength of stratification and velocity shear.
- Analysis: A sufficient condition for the flow to be linearly stable is that the local Richardson number exceed 0.25 throughout the flow. However, does not mean the flow is necessarily unstable (the criterion is not sufficient). When the flow is not a parallel shear flow, the meaning of is not clear.
- Schmidt Number
- Definition: , where is the viscosity, and is the diffusivity.
- Interpretation: The ratio between the momentum diffusivity (i.e. viscosity) and mass diffusivity.
- Analysis: The typical Schmidt number for water is around 500, depending on the temperature and salinity. For direct numerical simulations, is commonly used in the literature. See also Prandtl number.
