Compressible Pipe Flow with Friction and Gravity

A viscous one-dimensional compressible pipe flow under gravity effect is studied analytically. The compressible one-dimensional pipe flow with friction is called Fanno flow and the solution is given by analytical formula. In gas dynamics, the gravity effect is minimal and it is not included in the equations. However, it was shown by the present author that the elevation of a pipe could change the flow conditions in a one-dimensional compressible potential flow under gravity. The sonic condition is reached at the maximum height for an inviscid pipe flow. In this paper, the gravity effect is extended to the viscous onedimensional pipe flow. Subsonic–supersonic transition is also possible by up and down of the pipe as in the inviscid flow, and it is found that the sonic condition deviates from the peak position of the pipe.


Introduction
In gas dynamics, the gravity effect is negligible and not included in the governing equations. However, the effect is evident in astrophysics, i.e. Bondi flow [1], [2]. It was also shown that an inviscid compressible onedimensional pipe flow could be accelerated from subsonic to supersonic by the elevation of the pipe in theory [3]. Although the gravity effect is not evident in an ordinary air flow, it can be noticeable near the sonic condition and in the low acoustic velocity like cryogenics. This sonic condition occurs at the peak location of a pipe and the gravity has a similar effect to that of the throat of a Laval nozzle [3].
In order to apply this gravity effect in the real world problem, the viscous effect should be taken into account. A one-dimensional pipe flow with friction is called Fanno flow and the analytical solution is available [4], [5]. In the present analysis, the gravity term is added to the Fanno flow equations.

Governing equations
The continuity equation for a steady one-dimensional compressible pipe flow in Fig.1 where R is the gas constant and γ is the specific heat ratio.

Modified equations
Equations (1)-(5) are modified as follow: and the sonic condition is reaced after the pipe peak elevation. From Eqs.(6)-(10), the following relations are derived.
where 0 p is the local total pressure and s is the specific entropy. The local total pressure 0 p can be obtained from the local static pressure p by assuming isentropic compression. The first terms of the right hand side of Eqs.

Solutions in Mach numbers
From Eqs. (8) and (10), From Eqs. (11) and (18), the gravitational terms can be eliminated, and 2 2 2 Similarly,  (28) The elevation of pipe position has the same effect as that of the cross section of a Lavel nozzle. The sonic condition occurs at the peak height of the pipe.

Viscous flow acceleration
The viscous flow solutions Eqs. (22)-(26) are effective when the Mach number distribution is assumed. For example, a simple distribution from subsonic to supersonic transition can be given as follows: The integral in Eqs.  Figure 3 shows the pipe geometry for the given Mach number distribution with skin friction coefficients as parameters.
As mentioned in Eq. (13) the subsonic-supersonic transition occurs at * / / 0 l D l D = = after the peak elevation of the pipe for 0 f C > , while the inviscid flow reaches the sonic condition at the peak elevation of the pipe. It means that the flow has to be accelerated by gravity to overcome the pipe friction to be supersonic.
From Eq. (11), it is clear that the following equation is satisfied at the pipe peak position 0 dz = .

Viscous flow deceleration
Flow deceleration is also studied analytically. The Mach number might be assumed as follows: The integral in Eqs.    inviscid supersonic flow goes upwards to decelerate to the sonic condition and downwards for subsonic deceleration.
The subsonic-supersonic transition might be possible by up and down of a pipe for both inviscid and viscous flows in theory. Although it is not an easy task to achieve this condition in a laboratory scale, it could be possible in cryogenics, planetary and astrophysics conditions.