[yanliu]

HI,
I have a question regarding relation between stagnation pressure and navier-stoke 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 navier-stoke equation without viscous effect.
For compressible invisid flow, navier-stoke 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 navier-stoke 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 navier-stoke equation?
Thanks.
Yan Liu

[FMDenaro]

p0 is no longer a constant when dissipative effects take place as for the NS equations...

[yanliu]

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

[yanliu]

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

[FMDenaro]

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

[yanliu]

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

[FMDenaro]

Originally Posted by yanliu

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

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?

[yanliu]

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*(kappa-1)*M^2)^(k/(k-1))=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

[FMDenaro]

Originally Posted by yanliu

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*(kappa-1)*M^2)^(k/(k-1))=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

you cannot write Bernoulli that has the constant density...for compressible flows you should write the Crocco integral ...