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Internal Energy equation in viscous compressible floe equations 

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August 17, 2015, 15:34 
Internal Energy equation in viscous compressible floe equations

#1 
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Mihir Makwana
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How can I use FVM to discretize the term
sigma : D in the internal energy equation http://i.imgur.com/3RaYm5p.jpg where sigma and D are given by http://i.imgur.com/6K65Lih.jpg I am not able to expand the term sigma : D Please help. Thanks in Advance  Mihir 

August 17, 2015, 17:03 

#2 
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Filippo Maria Denaro
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If you want using a FVM, you have to recast the equation in the divergence form and integrate over a finite volume each term.
Using the expression of sigma, the term D:D you get is a scalar function that appears as volume source 

August 18, 2015, 01:21 

#3  
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Mihir Makwana
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Quote:
http://i.imgur.com/UmzDCrI.jpg How do i convert this equation to divergence form ? 

August 18, 2015, 02:45 

#4 
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Mihir Makwana
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sir say i split D
then one of the term is http://i.imgur.com/zkbee0D.jpg here Tau is sigma so the 1st term on r.h.s is in divergence form but the second is not 

August 18, 2015, 03:13 

#5 
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Filippo Maria Denaro
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no, you do not have all terms in te internal energy equation in divergence form...that respects the physical fact that such energy form does not have a conservative formulation. The term D: D is a pointwise source term, you have to integrate over the local volume in a FVM and discretize the volume integral.
If you want to work with a fully divergence form then you have to adopt the total energy equation 

August 18, 2015, 04:58 

#6  
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Mihir Makwana
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Quote:
1) As i am using FVM, i cannot integrate the second term in RHS of http://i.imgur.com/zkbee0D.jpg Right ?? or is there a way I can integrate it over the C.V 2) what is D: D ? 

August 18, 2015, 05:02 

#7 
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Filippo Maria Denaro
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you can always integrate each term of the equation over a finite volume...when the term is in divergence form you can apply Gauss and get the surface integral of the fluxes, conversely the integral to discretize remains the volume integral (see for example the book of Peric & Ferziger).
D: D is nothing else that the double dot product between the symmetric velocity gradient 

August 18, 2015, 05:04 

#8  
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Mihir Makwana
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Quote:
2) but i need to find sigma : D and not D : D 

August 18, 2015, 05:11 

#9 
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Filippo Maria Denaro
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sigma: D is nothing else that a product by an isotropic tensor (I: D) added with D: D
for brevity I disregarded the coefficients 

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