why velocity and pressure are decoupled in incompressible flow?
 

Why is it said that velocity and pressure are decoupled in incompressible flow? To my understanding, coupling means mutual influence. From the governing equations, pressure and velocity certainly affect each other, so how should we understand the statement "velocity and pressure are decoupled in incompressible flow"?

In the incompressible limit, there is no thermodynamic pressure. Once you apply the continuity equation and (obtain the density), the pressure is no longer a free variable. The pressure is whatever it needs to be to satisfy continuity. You can interpret this as decoupling or severely constrained coupling.

i understand all your sentences except for the last, i mean: the meaning of "decoupling" and "severely constrained coupling" are antonyms for me.

i understand all your sentences except for the last, i mean: the meaning of "decoupling" and "severely constrained coupling" are antonyms for me.

I think you should focus first on the fact that the pressure is nothing but a potential function whose gradient generates a velocity that accomodates for the divergence-free constraint. There is no EOS for the pressure.
The system of equation is coupled but the link provided by the mass constraint can be shown to generate a lagrangian multiplier.

Then, “decoupling” is a term often used in a numerical formulation, for example in the fractional time step.

If you have a specific reference where you read that, post it.

i understand the fact you said, we can get pressure poisson equation by mass constraint, so for me, velocity field decides the pressure filed, and pressure field is one of the advance-source of the velocity, ie velocity and pressure is coupling not decoupling.

i read that "velocity and pressure is decoupling in the incompressible flows" in a Chinese-language textbook, so even i show you, you can't read.

so i have two more little question to my confusion
1. do you heard the "velocity and pressure is decoupling in the incompressible flows" before?may be it is not a widely accepted knowledge/saying.
2. may be "velocity and pressure is decoupling in the incompressible flows" is only used in CFD not in theory of fluid mechanics? but it sounds impossible too: How can there be a numerical algorithm where the pressure and velocity evolve separately

In CFD there is the historical idea of the "splitting" method that can be applied both for the spatial dimensions and for the physical terms in the equations. For example in reacting flows where the reactions are very fast compared to the other terms.


In case of incompressible flows, the so-called "pressure-free projection method" produces a sequence of steps for an intermediate velocity, the pressure field and the correction step.

It is decoupled because the thermodynamic pressure can be anything.

That remaining p that you solve in the navier-stokes isn't the thermodynamic pressure anymore it is just a field that enforces the continuity equation. Whatever is the thing that remains that we label p is severely constrained.

If it helps, just change what we label p to another name in the incompressible navier stokes, call it pepperonipizza instead. Presure is decoupled, pepperoni is constrained to be on the pizza.

I think I understand your point, and your analogy is great. However, when I think about this issue a little more deeply, there are still some things that bother me. Why can pressure be calculated using an EOS in compressible flow, but not in incompressible flow? For the former, as we know, EOS is used for homogeneous matter, or more precisely, for thermodynamic equilibrium states. But obviously, compressible flow is non-equilibrium, so why it still works? As for the latter, it can be explained from the perspective of equations that adding an EOS is redundant. But is there a more fundamental explanation for why "pressure" changes to "pepperonipizza"?(and why "pressure" in compressible flow is still "pressure")

If you use the EOS for incompressible and omotherma flows, the pressure gradient in the momentum would be zero.

Compressible flow may be in dynamic non-equilibrium, but in thermodynamic equilibrium. Thermodynamic equilibrium doesn't mean that the gas properties are not changing in space and time. It only means that for a infinitesimal fluid parcel (chich can contain billions of molecules) I can define meaningful averages of the properties based on the statistics of the molecular behavior.