Specific dissipation
Hi all!
A question for you expert users: given the total dissipated power that one can calculate through variation of total pressure at inlet/outlet ports, is there any variable that, once integrated over the control volume, gives the same loss? Attempt 1. I built the variable mu*tau:grad(U) (the classical "dissipation function") but the loss calculated by integrating such variable over the whole domain is not even close to the actual loss, for the obvious reason that the solution field contains only the averaged quantities (i.e. Reynolds stresses and velocity fluctuations are missing). Attempt 2. I integrated the variable "Turbulence Eddy Dissipation" and I got somehow closer to the variation of total pressure but it's still around half the expected value. Attempt 3: I reused the tau:grad(U) variable but in combination with Eddy Viscosity: I got a similar result to that in Attempt 2. Attempt 4: I rearranged the equation of mechanical energy to isolate the effect of viscous and Reynolds stresses: by doing so I'm able to match the variation of total pressure calculated using surface integrals with a single volume integral (nice!), but the variable I created for this purpose is not the "specific loss", rather the "power done by viscosity" which, as such, can be either positive or negative. What I'm looking for is a monotonic variable that reproduces the total loss, i.e. something like the tau:grad(U) approach. Am I missing something? Thanks to you all! 
What are you asking for? It sounds like you're asking what volume integral should be exactly equal to some difference in total pressure which is presumably a surface integral, but you didn't define what this difference is exactly.
(total) pressure must be scaled by a mass flux to make it look like a work/power, but this simple analogy only works in 1D uniform flows. For a generalized or realistic 3D flow you don't have a "total pressure at the inlet" vs a "total pressure at the outlet" unless it is the degenerate case of uniform flow AND uniform total pressure. 
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Have you considered starting from the equation of the conservation of the total energy in the integral form? That express a balance between powers and you can see the analogy with the fundamental thermodinamic equation in a system dE/dt= WQ. 
The equation of energy in integral form is what I apply when I mention the variation of total pressure on all inlet/outlet ports. What I actually would like to achieve is to have a specific variable that returns the total loss when integrated over the control volume (i.e. I’m interested in the differential form, aka nonconservative form)

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The equation of the total energy is always conservative by definition, even if you write it in differential form. The pressure acts in it as div(vp), that is a mechanical reversible work, that is not associated to dissipation. On the other hand, you can write the equations for the kinetic energy or the internal energy, they are not conservative and a dissipative term appears. Be careful that in turbulence is used to define different equations for the kinetic energy (mean/residual). Have a look for example to the book of Pope 
"Conservative" and "nonconservative" definitions were referred to the form of corresponding equations rather than to physical quantities being conserved or not. However, I agree that the trick is in the averaging of NS equation, as I also explained in my first post.

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Sagaut is a further textbook to see the several form of the resolved and unresolved kinetic energy equation 
Can you just explain precisely what you mean by loss obtained through total pressure variation and maybe provide a definition in terms of a mathematical equation?

What I’m try to get is the following:
massFlowInt(Total Pressure/Density)@Inlet1...Inlet_n+massFlowInt(Total Pressure/Density)@Outlet1...Outlet_n=Total Dissipated Power=volumeInt(var_x)@Domain where var_x is the variable I’m trying to find or build. 
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I am not sure about the meaning of your formula but if we assume a simple example, that is a pipe having 1 inflow and 1 outflow, I think you will just achive the standard linear pressure decaying law between inlet and outlet that depends on the tangential stress and, for turbulent flows, the fluctuation contribute. 
The pipe is a good example and I believe you got what I want to achieve. My goal is use the “var_x” I defined in my previous post to create, as an example, a contour plot that illustrates where and to which extent the enenrgy loss is taking place.

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There are a lot of papers wherein the analysis of the energy budget is performed for several flow problems. Generally, in case of statistical energy equilibrium you get a direct balance between the production of mechanical energy and its dissipation. That is also the basic assumption of eddyviscosity models. Otherwise their net difference does not vanish and the flow is not in energy equilibrium. However, you can have a look to slide 38 here https://www.researchgate.net/publica...eLES_II_theory but I suggest then to read the more specific literature. 
As I stated before, the massflow*totalpressure gives the power dissipated only in 1D uniform flows.
For general 3D flows, this property doesn't work the same way. For starters, the massflow is just an integrated mass flow. You also have a total pressure at every cell volume and cell face on the boundary. There isn't a single total pressure. You have to decide first the meaning of massflow*totalpressure. You can check out Averaging Nonuniform Flow for a Purpose, to read more about this matter. Sry the article is behind the ASME paywall. 
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