Sheet V2 · Verification

Gradual closure: three durations and the Michaud formula

Source : Independently written reference MOC (verification/cvb/run_cvb.js, cases CV-003, CV-004a, CV-004b) and the Michaud formula (verification/tests)

1. The question

When the closure lasts less than, as long as, or longer than a wave round trip, the pressure rise changes in nature. Does CBELIER follow a reference computation in all three regimes, and the Michaud formula for slow closures?

2. The case

upstream reservoirL = 600 m, D = 0.5 m, a = 1,200 m/svalvedownstream, H = 100 mJ

Same pipe as sheet V1. The valve closes linearly in 0.5, 1 and then 4 times 2L/a. For slow closures the verification suite also compares the pressure rise with the Michaud formula ΔH = 2·L·V₀/(g·tf), on another reference pipe.

3. Data and status

DataValueStatus
Pipe: length, diameter600 m; 0.50 mDEFINED
Wave speed a1,200 m/sDEFINED
Upstream reservoir; downstream of the valve150 m; 100 mDEFINED
Initial velocity V₀ (small: friction negligible)0.10 m/sDEFINED
Mesh: 20 reaches, Courant number = 120DEFINED
Reference: separately written frictionless MOC (same scheme)—DEFINED
Linear closure durations (2L/a = 1 s)0.5; 1; 4 sDEFINED

4. Results

1401451501551600246time (s)head (m)CBELIER, tf = 0.5 sreference, tf = 0.5 sCBELIER, tf = 1 sreference, tf = 1 sCBELIER, tf = 4 sreference, tf = 4 s
Head at the valve for three closure durations: CBELIER (lines) and reference MOC (circles).

5. Differences

IndicatorCBELIERReferenceDifference
Maximum pressure rise, tf = 0.5 s12.24 m12.26 m−0.13 %
Maximum pressure rise, tf = 1 s12.24 m12.26 m−0.15 %
Maximum pressure rise, tf = 4 s2.81 m2.83 m−0.76 %
Michaud, slow closure tf = 4 s——1.6 %
Michaud, slow closure tf = 6 s——2.3 %
Michaud, slow closure tf = 10 s——3.0 %

All three regimes follow the reference computation within 1 % on the maximum. Against Michaud the difference stays below 3 %: the formula assumes a linearly decreasing flow, which is only approximate.

6. Limitations and assumptions