In the context of pipeline hydraulics, timereversal has been successfully used historically for “valve stroking” (using the desired response to determine the necessary cause) and has recently been shown by the PI’s research group to make possible efficient, robust, high resolution and noisetolerant schemes for the detection of leaks, blockages, and bursts in real pipe systems where damping (e.g., friction and viscoelasticity) is present.
Herein lies a crucial and unresolved paradox; although timereversal breaks down due to the inevitable damping of real waves, it still produces superior results than other defect detection techniques. This same paradox is described in the classic waterhammer book by Wylie et al (1993), page 223: “the [waterhammer] equations are remarkably robust in calculating backward in time.... [However] the idea of calculating backwards in time defies logic since it cannot be done physically. It cannot because of the irreversibility of losses in real systems.”
This proposed research will address this paradox using theoretical and experimental means. Theoretically, this research will seek a mathematical transformation that transforms the damped wave equation into an undamped one; that is, an approach to transform a nontime reversible problem (damped wave equation) into a timereversible problem (undamped wave equation). The damping effects that will be considered include steady and unsteady friction and viscoelasticity. The resulting transformation will be solved so that the undamped response can be obtained from the measured one and used in conjunction with timereversal schemes for defect detection. Physically, the required transformation must be an amplification operation that adds back the energy that was lost due to damping. Unfortunately, this means that measurement errors will also be amplified under the transformation making the problem illposed. Two regularization techniques will be tested in this research. To supplement this theoretical work, the experimental aim is to provide the firstever proof that timereversal provides efficient, robust, high resolution, and noisetolerant techniques for the detection of defects in pipes and to test the range of validity of the proposed theoretical transformation.
Objective:

Seek a mathematical transformation that transforms a nontime reversible problem (damped wave equation) into a timereversible problem (damped wave equation). The damping effects that will be considered include steady and unsteady friction and viscoelasticity.

Devise stable inversion techniques for the transformation derived in (1) so as to obtain dampingfree pressure signals from the measured pressure signals.

Experimentally test the timereversible techniques by conducting transient tests in the lab with single and multiple. The testing will involve measuring the pressure, applying to it the inversion technique developed in (2) to obtain the undamped pressure, and using the resulting undamped pressure in conjunction with timereversal techniques for defect detection.