Fluid dynamics: stagnation, static and dynamics pressures

Question

I would like to ask how energy changes forms in a flow.

I understand the concept of stagnation pressure, static and dynamic. I was wondering how energy changes form from static to dynamic.

What physical phenomena maintain the total energy of the system but convert static pressure to dynamic pressure, or visa versa?

Now, what phenomena increase the stagnation pressure by virtue of increasing static or dynamic pressure whilst holding the other constant. For example, to what extent does the static and dynamic pressure increase? In the ideal world does the dynamic pressure increase with the static remaining constant?

Answer 1

Stagnation Pressure is analogous the amount of total energy fluid volume have in a system. In absence of any external work feeding mechanism to this fluid volume and if losses can be neglected i.e inviscid flow, total pressure at each and every location remains constant.

Think of flow in a pump, where the impeller is doing external work on the fluid, thus increasing the total energy of the fluid parcel or total pressure.

Static pressure is analogous to potential energy of the fluid volume and dynamic pressure is similar to kinetic energy. Consider flow in a pipe where you are also moving with the fluid velocity, you will be subjected to some force by fluid particles. This force you is because of static pressure but if you were stationary , you will experience a higher force, that is because fluid molecules hitting you will come to stop and their dynamic pressure will be converted into static pressure, thus increasing net force on you.

If you have fluid flowing through a converging or diverging system with zero losses, the static pressure will convert into dynamic pressure keeping total pressure constant.

Now coming back to the pump example, if you were to sit on the impeller blade that is inertial frame of reference, to you impeller is not imparting any momentum to the fluid , thus total pressure in the frame of reference of the impeller remains constant if we are again to neglect losses due to viscosity or any recirculation.

Hope this helps

Answer 2

I believe for incompressible fluids the energy changes are simply between potential (for static conditions) and kinetic (for flow conditions) energy.

Hope this helps

Answer 3

First a small gripe: I know the term "dynamic pressure" is used by engineers, but I don't find it particularly helpful. In the physics literature, we usually state that there is a pressure $P$ which is isotropic and frame independent (the same in the lab frame and the rest frame of the fluid). This pressure is a thermodynamic function, related to the temperature and density of the fluid, $P=P(\rho,T)$.

There is also a stress tensor $$ \Pi_{ij} = P\delta_{ij} + \rho v_iv_j $$ where $\rho$ is the mass density and $v_i$ is the fluid velocity. The force on an area element is $$ F_i =\Pi_{ij}\hat{n}_j $$ where $\hat{n}$ is the surface normal. This force can be used to define a "dynamic pressure" by the usual expression $P_{dyn}=F/A$, which will depend on the orientation of the surface. Clearly, this "dynamic" pressure (really, the force) has an extra contribution from fluid inertia, $\rho v_i v_j$.

The energy density of the fluid is $$ {\cal E} = {\cal E}_0 + \frac{1}{2}\rho v^2 $$ where ${\cal E}_0$ is the fluid rest frame energy density, and the second term is the kinetic energy of the fluid. Note that ${\cal E}_0$ is related to the pressure by the equation of state. For a free gas $$ {\cal E}_0 = \frac{3}{2} P $$ So we see that the inertia of the fluid appears in both the stress and the energy density. For the $xx$ stress in an ideal fluid $$ \Pi_{xx} = P+\rho v_x^2\hspace{1cm} {\cal E} = \frac{3}{2}P + \frac{1}{2}\rho v^2 $$ The two things are related, but not equal. This is because these really are two different concepts, and the equation of energy conservation is yet another relation between these quantities $$ \partial_t {\cal E} + \vec{\nabla}\cdot [\vec{v}({\cal E}+P)]=0 $$