Why friction increase subsonic speed through pipe?

Question

I know equations of Fanno flow but can anyone explain physically how friction increases subsonic velocity and decreases supersonic velocity through pipe ? is that for sake of boundary layer displacement thickness ?

Answer 1

This can be answered better without equations:

Boundary conditions: Subsonic flow, so we have little change in density, and no energy is added, thus we can apply Bernoulli's law. For simplicity, we use a horizontal pipe, so no gravity effects need to be considered.

Put simply, the boundary layer becomes thicker with the length of the pipe, and to satisfy the continuity condition (what goes in on one end per unit of time must come out at the other end, too), the core velocity must go up so the mass flow over the length of the pipe stays constant. Remember that we have little change in density, so what changes in the flow parameters are velocity and pressure. Given a high enough flow speed, also the temperature increase from friction cannot be neglected.

The energy for that core flow speed increase is taken from pressure, which drops over the length of the pipe.

Now over to supersonic flow: Here, density changes are higher than velocity changes, so the narrowing core flow needs to become denser in order to satisfy the continuity condition. This requires the flow to become slower, so here both kinetic energy and pressure are converted to an increase in density.

Answer 2

This is quite an interesting question. Indeed, often times a fluid dynamicist finds themself trying to decipher their equations in order to figure out the physics hiding behind the maths, and this problem is one example of that. In this answer I'll only be giving my own guess to what's going on, so take it with a big grain of salt. For a rigorous explanation based on entropy arguments, you can check out the book of Landau & Lifshitz, and you can also check this other question on the site. I'll be presenting a different argument that's hopefully more intuitive.

Well, to begin with, let's define precisely the problem at hands. The Fanno flow is not just a flow experiencing the effects of friction, but there are two more conditions imposed on it, namely the adiabatic condition and the steady flow condition. So, we're running a viscous fluid in a pipe and we're letting it figure out what to do in order to maintain an adiabatic steady flow. As we shall see shortly, this often involves pressure gradients developing inside the flow to counter the effects of friction. Thus, friction is not the only force acting on the fluid, but it is being countered by pressure.

The momentum balance equation for the Fanno flow, as given by Anderson's "Modern Compressible Flow" is: $$ dp + \rho udu = - \rho u^2 \frac{2f}{D}dx, $$ where $f$ is the friction coefficient and $D$ is the tube diameter. One can see that, due to the one-dimensionality of the problem, friction is modeled as a body force, just like gravity. Let's take the extreme case of an incompressible flow. For this flow we have $\rho = \text{constant}$, which along with the steady flow condition $\rho u = \text{constant}$ gives $u = \text{constant}$, i.e $du = 0$. Thus the above equation reduces to $$dp = - \rho u^2 \frac{2f}{D}dx,$$ and one can immediately see how this equation is identical to the equation of hydrostatic equilibrium in a water tank for instance: $$dp = -\rho g dz.$$ In a water tank, pressure increases with depth since each fluid element needs to carry the weight of all fluid elements above it. Similarly for an incompressible Fanno flow, the pressure increases towards the inlet since each fluid element needs to push against the friction acting on all fluid elements downstream of it. Therefore, one obtains a favorable pressure gradient; the highest pressure is at the inlet, and the pressure decreases towards the outlet.

The same view remains valid for a subsonic compressible flow, except that this time, when the pressure decreases downstream, the gas will expand and thus it will accelerate! One can see this easily from the steady flow condition $\rho u = \text{constant}$: when $\rho$ decreases due to the flow expansion, $u$ increases proportionately.

Therefore, from this viewpoint, it is very intuitive why the subsonic flow accelerates due to friction. The less intuitive case is that of a supersonic flow, in which pressure increases and the flow slows down as we move downstream! So why is that the case?

Well, I do not have a definite answer for that, but my feeling is that this has something to do with the speed at which information travels in a supersonic flow. In a subsonic flow, the upstream-traveling characteristic at each point in the flow will be traveling upstream in the lab frame at a speed $a - u$, where $a$ is the speed of sound. Thus, a fluid element at the inlet is able to "feel" the presence of all fluid elements downstream of it and will react instantaneously to the friction acting on them by getting compressed. In a supersonic flow, however, the upstream-traveling characteristic will be traveling downstream in the lab frame at a speed $u-a$. Thus, a fluid element at the inlet will be unable to "see" the flow downstream of it, and so it won't react for it. Only when the fluid element (traveling at speed $u$) starts overtaking the characteristics (traveling at speed $u-a$) at some later downstream locations, will it know that the fluid elements ahead of it are facing a friction force, and so it will react accordingly by getting compressed and slowing down in response (and, of course, it will also slow down and get compressed due to friction acting on it directly). This, I think, leads to the adverse pressure gradient in a supersonic flow.

Answer 3

The answer by @Peter is completely incorrect. The answer by @Tofi in better but I do not think is completely correct either. The reason is not the effect of pressure on density alone (additionally, note that the momentum equation is not a constraint for Fanno flow).

Density is to be taken as a function of two other thermodynamic variables -- the Fanno line is typically shown on a Mollier diagram with entropy as the x axis. While pressure is changing along the duct with friction, enthalpy (linear function of temperature for perfect gas) is also changing. Density will respond to both variables: for a perfect gas, $\frac{dp}{p}=\frac{d\rho}{\rho}+\frac{dT}{T}; \frac{dh}{h}=\frac{dT}{T}$. Again, the Fanno curve (also energy equation) shows that as a subsonic flow accelerates, enthalpy decreases; for a decelerating supersonic flow, enthapy increases: $dT<0$ for accelerating subsonic flow and $dT>0$ for decelerating supersonic flow. In both supersonic and subsonic flow, the velocity will respond directly to changes in density (continuity). Density decreases in the subsonic case and increases in the supersonic case. In either case, entropy increases.

I will edit the answer if I come up with a more physical explanation. But the previous responses are incorrect/incomplete.