Operation and Response of Flow in Open-channel with Step Works

Abstract

Theoretical approach was tried for evaluating the dead time D (s) and time constant T (s) as parameters of the lead time of the response of flow due to operation in an open-channel with step works in a series.
The main procedures are shown in terms (1)-(3) as follows.
(1) The nom-dimemsional discharge per unit width K_q (x, t) was defined as Eq.(A).
_??_(A)
in which, q: discharge per umit width (m³·s^-1·m^-1), x: distamce from upstream boundary im the openchannel system (m), t: time afer operation (s), subscript b: quasi-uniform flow in initial state and subscript e: quasi-uniform flow in the final state.
(2) Input fumction K_q (0, t) at the upstream boumdary wasgivem as a step fumctiom of time t, and output k_q (x, t) at the downstream boundary was obtained as the calculated results.
(3) The Eqs.(B) snd (C) were apPlied to evaluate the dead time D and time comstamt T in the indicial respomse of flow im an open-chammel under various conditions.
D+T=l/w₀ (B)
_??_(C)
in which, l: partial Iength of chanmel bound with each step work (m), w₀: velocity of hydraulic bore (m/s), μ₀: equivalent to q_b/2i (m²/s), i: bed slope of the opem-channel.
Eq.(B) was derived from the condition of continuity, and Eq.(C) was derived from the diffusion analog model.
The mathematical model for evaluating dead time D and time constant T of the flow in an openchannel with step works in a series was presented and checked using a numericalanalysis.
The validity of this model was confirmed within about 10 step works in an open-channel.