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April 30, 2001, 02:33 
Grashof Number

#1 
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For a 2D flow in a horizontal rectangular enclosure, does the characteristic length /temperature differemce as defined in Grashof Number refer to the length or the height? I am confusing myself, please englighten.


April 30, 2001, 11:11 
Re: Grashof Number

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The characteristic dimension is the dimension PARALLEL to the body force vector  in this case, the "height". (Centrifugal forces get more interesting)


April 30, 2001, 11:53 
Re: Grashof Number

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Oh i see....so that would mean that with the differentially heated vertical walls (insulated horizontal walls), the characteristic length would refer to the height of the rectangle. The temperature difference, i assume, would still be the difference in temperature between the two walls? Thank you.


April 30, 2001, 15:22 
Re: Grashof Number

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(1). It is up to you to pick the characteristic length for your analysis. It can be the height, the length or the sum of the two, etc... (2). Since the geometry is fixed, the conversion between different length scales is also fixed. (3). The Reynolds number can be defined using either the height or the width. It is up to you. Re,h=Re,w * (h/w) ; G,h=G,w * (h/w)^3.


April 30, 2001, 23:08 
Re: Grashof Number

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I think i understand what you mean. However, i have encountered journals, which involves the rectangular geometry that i have mentioned in my earlier post, most of which uses the height as the chracteristic length while the temperature difference is the temp between the two differentially heated vertical walls. Are there any reasons for this?


May 1, 2001, 00:21 
Grashof Number maybe not

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As John points out the length scale is often picked in a way that is not obviously motivated. In fact the Grashof number may not even be the correct dimensionless group unless your Prandtl number is much less than 1, that is in molten metals say. For Prandtl numbers of 1 or greater, the Rayleigh and Prandtl numbers are the appropriate groups, and for a Prandtl number of greater than say 10 to 100, even it doesn't matter much as the thermal boundary layer lies well within the velocity boundary layer.
This is discussed with some "emphasis" in the book "Convection Heat Transfer" by Adrian Bejan. A very nice book that provides a very intuitive presentation to scaling analysis. I strongly recommend it (except the chapter on turbulence has been controversial). 

May 1, 2001, 08:24 
Re: Grashof Number maybe not

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Thanks. I will go look for that book. In the mean time, i assume it would be safe to assume that the height is the characteristic length?


May 1, 2001, 17:02 
Re: Grashof Number maybe not

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I am not sure if the characteristic length is the right one. If the length of your geometry is 1 meter but the width only 0.001, no change you will find buoyancy effects. Therefore, as George pointed out, perhaps you should use the Rayleighnumber Pr*Gr*dh/L with dh the hydrodynamics diameter and L the length. Check an oldie for this: Metais & Eckert, J. Heat Transfer, 86C , 1964, 295.
Cheers, Astrid 

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