Abstract
The method of tracking waves in the fluid flows goes back to the suggestion of Allievi (General theory of the variable flow of water in pressure conducts. Riccardo Garoni, Rome, (1902)) in the earlier time. The method, in effect, relies on the d’Alembert solution of the one-dimensional hyperbolic wave equation and represents a Lagrangian approach in fluid mechanics. The method is simply based on tracking the shock wave propagation in the time series. This indicates that both the shock pressure and the flow velocity at any interested position and the components in a hydraulic system can be directly computed. The latter may for instance be hydraulic machines, regulation valves or surge tanks . The approach clearly represents a significant advantage against the method of characteristics, which always requires the predefined characteristic grids in the time-space domain.
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Notes
- 1.
In combining two equations for eliminating \(\sqrt h\), one obtains alternatively
$$\frac{{1 - \left( {{{\varphi A_{\text{D0}} } / A}} \right)^{2} }}{{2g\left( {\varphi A_{\text{D0}} } \right)^{2} }}Q^{2} + \frac{a}{gA}Q - \left( {h_{0}^{*} + 2f} \right) = 0.$$(7.26)This is the determination equation in form of a quadratic polynomial for discharge. The fact to be mentioned is that this equation is undefined, as soon as \(\varphi = 0\), i.e., Q = 0 is reached by closing the injector. The pressure head is then directly computed from Eq. (7.24) to \(h = h_{0}^{*} + 2f\).
In addition, the approximation \(1 - \left( {{{\phi A_{\text{D0}} } / A}} \right)^{2} \approx 1\) can be made in both Eqs. (7.24) and (7.26).
- 2.
- 3.
The impeller diameter is denoted by \(d_{{ 2 {\text{D}}}}\) with subscript 2 for impeller exit and D for impeller as a rotating disc. This designation is meaningful because another designation \(d_{2}\) is often used for the pipe diameter of the pump exit or at the exit of a unified pump and spherical valve.
References
Allievi, L. (1902). General theory of the variable flow of water in pressure conducts, see “Theory of Water Hammer”, (E. E. Halmos, Trans.). Typography Riccardo Garoni, Rome, 1925.
Wood, D. J., Dorsch, R. & Lightner, C. (1966). Wave analysis of unsteady flow in conduits. Journal of the Hydraulics Division ASCE, 92(HY2), 83–110.
Zhang, Z. (2016). Transient flows in pipe system with pump shut-down and the simultaneous closing of the spherical valve. In IAHR, 28th Symposium on Hydraulic Machinery and Systems. Grenoble, France, see also IOP Conference Series: Earth and Environmental Science (Vol. 49, p. 052001). https://doi.org/10.1088/1755-1315/49/5/052001.
Zhang, Z. (2018, August). Wave tracking method of hydraulic transients in pipe systems with pump shut-off under simultaneous closing of spherical valves, Journal of Renewable Energy, 132, 157–166.
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Zhang, Z. (2020). Wave Tracking Method. In: Hydraulic Transients and Computations. Springer, Cham. https://doi.org/10.1007/978-3-030-40233-4_7
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DOI: https://doi.org/10.1007/978-3-030-40233-4_7
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