Laminar flow in a circular pipe driven by periodic pressure variations

G.4 Laminar flow in a circular pipe driven by periodic pressure variations

The problem considered Many engineering problems involve unsteady behaviour. In some cases,

for instance paint mixing, the flow may be steady but the distribution of a transported scalar variable changes with time. In the present example we consider one of the simplest cases of the class of problems with genuinely unsteady flow fields: the periodic oscillations of an incompressible laminar flow in a circular pipe driven by harmonic pressure variations between inlet and outlet. Blood flows in veins and arteries, pressure waves in oil pipelines and air flows in intake manifolds of internal combustion engines can be modelled as periodic duct flows.

The applied pressure difference between the pipe ends is varied according to

∆P = K cos nt

(G.1)

APPENDIX G

The amplitude K is taken to be 50 000 (Pa) and the circular frequency n is equal to 2 π Hz, giving an oscillation period of 1 s. Schlichting (1979) gives

the analytical solution for the axial velocity component u(r, t) as a function of radius r and time t for periodic laminar flow in a very long pipe as the real part of the following expression:

In this formula ρ, ν and L are the fluid density, kinematic viscosity and the length of the pipe respectively, J 0 denotes the Bessel function of the first kind of order 0, and i is −1. The general features of the velocity distribu- tions are dependent on the value of the non-dimensional parameter (n/ ν)R.

The solution behaviour for small and large values of this parameter will be discussed below. Here we calculate the flow for two intermediate values of n by taking the pipe radius R equal to 0.01 m and frequency n as 2 π Hz in

conjunction with a constant fluid density of 1000 kg/m 3 and dynamic vis- cosity of 0.4 and 0.1 kg/m.s. This yields values of (n/ ν)R of 1.253 and

2.507 respectively.

CFD simulation In order to set up a valid comparison between the analytical and finite

volume solution of this problem we need to consider a pipe of sufficient length. The boundary layer flow near the inlet of a pipe changes in the down- stream direction, and in a steady flow the velocity distribution becomes fully

developed after a distance l E given by (Schlichting, 1979)

l E X = 0.25 R

(G.3)

An estimate of the maximum possible mean velocities X (approximately 4 m/s here) can be obtained from the Hagen–Poiseuille formula (Schlichting, 1979). This leads to a maximum Reynolds number of 800 and hence a value

of l E of 1 m. Since the flow switches direction in the course of a cycle it is necessary to employ a computational domain of length greater than two times l E to ensure that there is always a section of fully developed flow half way along the duct. In this simulation we use a domain with a length L equal to 2.5 m and consider the solution in a cross-sectional plane at a distance of

1.25 m from its ends. The flow is axisymmetric, and we use a grid of 250 axial and 20 radial nodes distributed uniformly in the z- and r-directions. Figure G.8 shows a sketch of the solution domain and part of the mesh used. At r = 0 a symme- try boundary condition ensures that there is no flow across the axis and that the gradients of all variables in the radial direction are locally zero. At r =R

= 0.01 m the usual wall boundary condition is maintained. The cosinusoidal driving pressure difference given by Figure G.9 and equation 10.36 is applied by means of prescribed pressure boundary conditions at z = 0 and

APPENDIX G

Figure G.8 Solution domain and a part of the mesh for simulation of periodic laminar pipe flow

z = L = 2.5 m. The solution procedure is SIMPLER with fully implicit time marching; the time step is 1 ms. The parabolic velocity profile of a steady laminar pipe flow is used as the initial velocity field.

Specimen results Figures G.9 and G.10 compare the numerical and analytical solutions half

way along the pipe at time intervals of 0.125 s. The finite volume solution is studied after three pressure cycles, allowing time for the initial transients to die out. It is clear from the solution that the agreement between numerical and analytical solutions is generally excellent. There are minor discrepancies in the simulation with (n/ ν)R = 2.507 during those parts of the solution cycle where the flow near the boundary moves in the opposite direction to that in the core of the pipe. These can be explained by the fact that the local pressure gradient ∂p/∂x is somewhat different from the overall pres- sure gradient ∆p/L due to energy losses between the inlet and the solution cross-section.

Figure G.9 Imposed transient pressure cycle

APPENDIX G

Figure G.10 Velocity distribution for periodic laminar pipe flow with

(n/ ν)R = 1.253

Overall flow behaviour can be explained by considering appropriate expressions (Abramowitz and Stegun, 1964) of the Bessel function J 0 in the

analytical solution.

For very slow oscillations (n/ ν)R → 0 we obtain

u(r, t) =

(R 2 −r 2 ) cos nt

(G.4)

This exhibits the parabolic velocity distribution of a steady, fully developed, laminar pipe flow with a periodic time variation. The amplitude depends on the fluid viscosity, and the oscillations are in phase with the driving pressure difference. For fast oscillations (n/ ν)R → ∞ we have

u(r, t) = H sin nt

− n exp −

B (R

− r) sin nt − n E B (R − r) E K (G.5)

F L Expression (G.5) contains two sinusoidal terms, the first of which is

independent of viscosity. It describes the flow in the central core of the pipe, which has a uniform velocity distribution with an amplitude inversely pro- portional to the oscillation frequency and a phase lag of π/2 radians behind

the excitation force. The amplitude and the phase of the second term are viscosity dependent. The term decays quickly with distance (R − r) from the

pipe wall due to the exponential factor. It can be shown that this boundary layer flow lags behind the driving pressure difference by π/4 radians. The

phase difference between the core and the boundary layer gives rise to an annular flow pattern during fast oscillations. It is clear that the results of Figures G.10 and G.11 exhibit the main characteristics of the slow and fast solution respectively.

APPENDIX G

Figure G.11 Velocity distribution for periodic laminar pipe flow with

(n/ ν)R = 2.507

The above flow can be comfortably calculated on a workstation, but this success should not mislead the prospective user of commercial CFD codes. Other types of unsteady flow problems with complex geometries and/or fluid physics such as turbulent intake manifold flows (Chen, 1994), pulsed combustion (Benelli et al., 1992), transient free convective cooling of warm crude oil in storage tanks (Cotter and Charles, 1993) or hydrodynamic insta- bilities such as vortex shedding require very large computing resources. Often such flow calculations are only practical within reasonable time limits on dedicated large computers with advanced architecture and specially adapted algorithm structures.