Grashof number

Summary

In fluid mechanics (especially fluid thermodynamics), the Grashof number (Gr, after Franz Grashof) is a dimensionless number which approximates the ratio of the buoyancy to viscous forces acting on a fluid. It frequently arises in the study of situations involving natural convection and is analogous to the Reynolds number (Re). Free convection is caused by a change in density of a fluid due to a temperature change or gradient. Usually the density decreases due to an increase in temperature and causes the fluid to rise. This motion is caused by the buoyancy force. The major force that resists the motion is the viscous force. The Grashof number is a way to quantify the opposing forces. The Grashof number is: for vertical flat plates for pipes for bluff bodies where: g is gravitational acceleration due to Earth β is the coefficient of volume expansion (equal to approximately 1/T for ideal gases) T_s is the surface temperature T_∞ is the bulk temperature L is the vertical length D is the diameter ν is the kinematic viscosity. The L and D subscripts indicate the length scale basis for the Grashof number. The transition to turbulent flow occurs in the range 10^8 < Gr_L < 10^9 for natural convection from vertical flat plates. At higher Grashof numbers, the boundary layer is turbulent; at lower Grashof numbers, the boundary layer is laminar, that is, in the range 10^3 < Gr_L < 10^6. There is an analogous form of the Grashof number used in cases of natural convection mass transfer problems. In the case of mass transfer, natural convection is caused by concentration gradients rather than temperature gradients. where and: g is gravitational acceleration due to Earth C_a,s is the concentration of species a at surface C_a,a is the concentration of species a in ambient medium L is the characteristic length ν is the kinematic viscosity ρ is the fluid density C_a is the concentration of species a T is the temperature (constant) p is the pressure (constant). The Rayleigh number, shown below, is a dimensionless number that characterizes convection problems in heat transfer.

Official source

https://en.wikipedia.org/wiki/Grashof_number

About this result

This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.

Related publications (1)

Related concepts (4)

Related courses (2)

Related lectures (47)

For the modelling of the transport and diffusion of atmospheric pollutants during accidental releases, sophisticated emergency response systems are used. These modelling systems usually consist of three main parts. The atmospheric flow conditions can be simulated with a numerical weather prediction (NWP) model. The evolution of the pollutant cloud is described with a dispersion model of variable complexity. The NWP and the dispersion models have to be coupled with a so-called meteorological pre-processor. This means that all the necessary – in most cases turbulence related – variables which are not available from the standard output of the NWP model have to be diagnosed. The main difficulty of the turbulence coupling is that these subgrid scale variables of NWP models are not routinely verified and thus little is known concerning their quality and impact on dispersion processes. The general aim of the present work is to better understand and improve this coupling mechanism. For this purpose all the three main components of the emergency response system of MeteoSwiss are carefully evaluated and possible improvement strategies are suggested. In the first part, the NWP component of the system, namely the COSMO model, is investigated focusing on the model performance in the Planetary Boundary Layer (PBL). Three case studies, representing both unstable and stable situations, are analyzed and the COSMO simulations are validated with turbulence measurements and Large Eddy Simulation (LES) data. It is shown that the COSMO model is able to reproduce the main evolution of the boundary layer in dry convective situations with the operational parameter setting. However, it is found that the COSMO model tends to simulate a too moist and too cold PBL with shallower PBL heights than observed. During stable conditions the operational parameter setting has to be significantly modified (e.g., the minimum diffusion coefficient) to obtain a good model performance. The turbulence scheme of COSMO, which carries a prognostic equation for Turbulent Kinetic Energy (TKE), is studied in detail to understand the shortcomings of the simulations. The turbulent transport term (third order moment) in the TKE equation is found to be significantly underestimated by the COSMO model during unstable situations. This results in inaccurate TKE profiles and hence missing entrainment fluxes at the top of the PBL. A solution to increase the TKE transport in the PBL is proposed in the present work and evaluated during a three-month continuous period. While improving the TKE profile substantially, the modification is demonstrated to not impair other model output characteristics. The second component of the emergency response system, namely the meteorological pre-processor, is also validated on case studies and a continuous period. The main objective of this analysis is to compare the currently operational coupling approach, which is based on the direct usage of the prognostic TKE from the COSMO model, to a classical approach based on similarity theory considerations, thereby using turbulence measurements on the one hand and LES data on the other hand. To be able to use similarity theory approaches for the determination of turbulence characteristics, the PBL height has first to be diagnosed from the NWP model. In the present study, several approaches for the determination of PBL height have been implemented and validated with radio sounding measurements. Based on the verification results and the operational convenience, the method based on the bulk Richardson number method has been chosen for the diagnosis of the PBL height. Validation results of post-diagnosed turbulence characteristics show that during convective situations, the similarity approach tends to overestimate the turbulence intensity, while the approach based on the direct usage of TKE yields more accurate results. For stable conditions the different approaches are closer to each other and both give reasonable predictions. It is found that the main drawback of the direct approach is the isotropic assumption in the horizontal direction. A new hybrid method is proposed which uses similarity considerations for the partitioning of horizontal TKE between along-wind and cross-wind directions. In the last part, pollutant dispersion in complex terrain is studied using a new scaling approach for TKE that is suited for steep and narrow Alpine valleys. This scaling approach is introduced in the interface between COSMO and a Lagrangian particle dispersion model (LPDM), and its results are compared to those of a classical similarity theory approach and to the operational coupling type, which uses the TKE from the COSMO model directly. For the validation of the modelling system, the TRANSALP-89 tracer experiment is used, which was conducted in highly complex terrain in southern Switzerland. The ability of the COSMO model to simulate the valley-wind system is assessed with several meteorological surface stations. The dispersion simulation is evaluated with the measurements from 25 surface samplers. The sensitivity of the modelling system towards the soil moisture, horizontal grid resolution, and boundary layer height determination is investigated. It is shown that if the flow field is correctly reproduced, the new scaling approach improves the tracer concentration simulation compared to the classical coupling methods.

EPFL2010