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relation between stagnation pressure and naviewstoke equations 

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July 22, 2015, 15:24 
relation between stagnation pressure and naviewstoke equations

#1 
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HI,
I have a question regarding relation between stagnation pressure and navierstoke equation: For incompressible invisid flow, stagnation pressure is constant, p+0.5*rho*v^2=c, if I do partial derivative, I got dp/dx + rho*v*dv/dx = 0, which is same as steady state navierstoke equation without viscous effect. For compressible invisid flow, navierstoke equation is the same (except additional bulk viscosity term), but stagnation pressure is = p+0.5*rho*v^2+0.5*rho*v^2*mach^2/4+.... If I use navierstoke equation to solve a nozzle problem, the stagnation pressure could rise at exit based on this definition, which is also observed on my simulation. So what is wrong here? Should stagnation pressure be conserved? Anything wrong with navierstoke equation? Thanks. Yan Liu 

July 22, 2015, 16:50 

#2 
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Filippo Maria Denaro
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p0 is no longer a constant when dissipative effects take place as for the NS equations...


July 22, 2015, 17:57 

#3 
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Thanks for the reply.
Yes, if there is viscous loss, I guess p0 will reduce from inlet to exit of a nozzle. But for compressible, if the viscous loss is small, I might get p0 increasing from inlet to exit in the converging nozzle from NS equation. This confuses me! Yan Liu 

July 22, 2015, 18:03 

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I guess my question relates the difference between mechanical pressure and thermodynamic pressure. But I am not sure.


July 22, 2015, 18:04 

#5 
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Filippo Maria Denaro
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That depends...for example, even for inviscid flows P0 is no longer constant is a shock appears...
The change in p0 depends on the entropy variation 

July 22, 2015, 18:17 

#6 
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Yes, for an isentropic condition, I would expect stagnation pressure remains constant in the converging duct without shock. but I can not get this result from NS equation for compressible flow


July 22, 2015, 18:24 

#7 
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Filippo Maria Denaro
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What about the difference ? Even for small viscosity the walls of the duct can produce effects ... Have you tried to compute p0 along the centerline axis?


July 22, 2015, 18:40 

#8 
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If I calculate p+0.5*rho*v^2 along the converging duct, the value is decreasing because of the viscous loss.
If I calculate p0=p*(1+0.5*(kappa1)*M^2)^(k/(k1))=p+0.5*rho*v^2+0.5*rho*v^2*M^2/4+H.O.T., the value is increasing This makes me think NS equation only includes the derivatives of the first two terms, not the terms with Mach number 

July 22, 2015, 18:52 

#9  
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Filippo Maria Denaro
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Quote:
you cannot write Bernoulli that has the constant density...for compressible flows you should write the Crocco integral ... 

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compressible, navierstokes, stagnation pressure 
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