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Linear variation of gradient of a scalar in a cell. 

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March 15, 2020, 08:45 
Linear variation of gradient of a scalar in a cell.

#1 
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Mandeep Shetty
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If the temperature (or any scalar) is varying linearly in a cell, will the variation of the gradient of the temperature also be linear?
If we say that temperature, T, is varying linearly in a square plate given by T = Tp +(xxp)grad(Tp) where Tp is the Temperature at the centroid of the square plate, xp is the centroid, x is any point on the plate so (xxp) is the position (so a vector) of any point on the plate from the centroid, grad(Tp) is the temperature gradient at the centroid. Is it correct to write, a) grad(T) = grad(Tp)+(xxp)grad(grad(Tp)) as (xxp), even though is a vector, doesn't actually change with x,y,z OR b) grad(T) = grad(Tp)+grad[(xxp)grad(Tp)] as we cannot consider (xxp) to be a constant with respect to x, y, z. Ref: Prof.Hrvoje Jasak Thesis, Error Analysis and Estimation for the Finite Volume Method with Applications to Fluid Flows 

March 15, 2020, 10:53 

#2  
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Filippo Maria Denaro
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Quote:
Once the temperature is assumed linear, its derivative is a constant in the cell: T(x)=T0+T1*x > dT/dx=T1 

March 15, 2020, 13:11 

#3 
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Mandeep Shetty
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March 15, 2020, 13:16 

#4  
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Filippo Maria Denaro
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Well, be careful that grad is also a vector operator. Work on the components dT/dx and dT/dy. But this way you are assuming a quadratic law for the temperature 

March 15, 2020, 13:26 

#5 
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Mandeep Shetty
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I am trying to use a secondorder FVM method. I was told that using the secondorder FVM method means all the variables (which I am starting to understand just means the primitive variables and not the gradient of the said variables) would be assumed to vary linearly within the cell. So as I am already assuming the temperature to vary linearly, and so I cannot assume the grad(T) to vary linearly again.


March 15, 2020, 13:41 

#6  
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Filippo Maria Denaro
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Quote:
Indeed in a FVM you have a second order accuracy using a linear assumption for the function. The derivative is constant on each of the face so that you can evaluate their difference 

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finite volume calculus, gradient 
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