Dundee Tunnel Research

Academic Research - Wave Propagation

Alan Vardy’s research in fluid mechanics has a strong focus on waves.  There are many types of wave, however, so it is important to focus this statement somewhat.  Vardy’s work has little, if any, meaningful relevance to the sorts of waves that are observed in mid-ocean; these are inherently surface effects that are almost undetectable far beneath the surface.  Vardy’s waves are more like those enjoyed by paddlers on coastal beaches where the water is so shallow that the influence of the wave is felt almost uniformly at all depths.  Even these waves are only loosely relevant, however, because the real focus of the research is on waves propagating inside rigid (or nearly-rigid) ducts where the fluid depth cannot change.  On a beach, the fluid (water) density is the same at all locations and variations in flow rate are associated with changes in depth. In a pipe or tunnel, the cross-sectional size is the same at all locations and variations in flow rate are associated with changes in density - and hence with changes in pressure. 

 Within the confines of a pipe, waves travel much faster than those on a beach.  In water supply pipes and hydro-electric penstocks, for instance, wavespeeds nearly always exceed 1000 m/s.  Even in railway tunnels, air pressure waves travel at about one third of this speed.  In fact, sound waves are simply small-amplitude pressure waves so it is easy to realise that pressure waves will travel at approximately the speed of sound.

Sometimes, engineers are interested in waves where pressures vary along a pipe in a periodic manner.  Most of Vardy’s work is more relevant to non-periodic behaviour such as a sudden increase in pressure followed by a sustained period of relative calm.  Such behaviour can result, for instance from the rapid closure of a valve at the end of a long pipe or from the entry of a train into a railway tunnel.  In these contexts, “rapid” does not necessarily imply an exceptionally short time.  Instead, it typically relates to the time taken for pressure waves to propagate along the duct and back again.  For a water pipe in a modern domestic house this is only a fraction of a second.  For a 2 km long railway tunnel, however, it is about 12 seconds and, for a 300 km long oil pipeline, it is about 10 minutes.

For waves characterised by a sudden increase followed by a sustained period of more gentle change, greatest academic challenge arises close to the leading edge.  Usually, it is easy to relate the amplitude of the pressure change to the amplitude of the velocity change, but it can be much more difficult to predict how the overall picture will evolve.  In railway tunnels, for instance, the front of the wave tends to steepen almost indefinitely when the tunnel surfaces are relatively smooth concrete, whereas excessive steepening is rarely possible in tunnels with conventional ballast track.

As a first approximation, waves may be regarded as inertial phenomena. Most of their behaviour can be predicted using inviscid analyses and, in many practical cases, analyses can be founded on linear equations, at least for initial design purposes.  The academic fun starts when allowance has to be made for large amplitude effects and/or for complicating factors such as viscosity and temperature.  These tend to be relatively small effects, but their cumulative influence over long periods can be decisive.

Hydro-power penstocks Closed surge tank

Selected References

Brown JMB & Vardy AE (1994) Reflections of pressure waves at tunnel portals. J Sound & Vibration, 173(1), 95-111

Chen J-Y & Vardy AE (1995) Surge waves in a free-surface tailrace system. Int J on Hydropower & Dams, 2(2), 63-67

Hu X, Chen J & Vardy AE (1998) Optimal preliminary design of hydro-power waterway systems. Dam Engineering, 9(4), 371-394

Vardy AE (1976) On the use of the method of characteristics for the solution of unsteady flows in networks. Proc 2nd int conf on Pressure Surges, London, UK, BHR Group, H2:15-30.

Vardy AE (1978) Reflection of step-wavefronts from perforated and flared tube extensions. J Sound and Vibration, 59(4), 577-589

Vardy AE (2012) On asymptotic wavefronts in compressible flows, Proc 11th int conf on Pressure Surges, Lisbon, Portugal, 24-26 Oct 2012, BHR Group, 247-260

Vardy AE & Brown JMB (2000) Influence of ballast on wave steepening in tunnels, J Sound & Vibration, 238(4), 595-615

Vardy AE (2008) Method of characteristics in quasi-steady compressible flows. Proc 10th int conf on Pressure Surges, Edinburgh UK, 14-16 May 2008, BHR Group, 505-518

Vardy AE & Mackenzie H (2012) Use of pressure surge for pipeline condition monitoring, Proc 11th int conf on Pressure Surges, Lisbon, Portugal, 24-26 Oct 2012, BHR Group, 57-68

Tijsseling AS & Vardy AE (2015) What is wavespeed?, Proc 12th int conf on Pressures Surges, Dublin, 18-20 Nov 2015, BHR Group, 343-360