Floodwall Nappe Flow Response

Floodwall Nappe Flow Response

Source: CHESS_Library/Structure_Reponses/Equations/floodwall_nappe_response.m, called from compute_structure_response.m (PROS, DET, LCS) and from chess_pros_frequency_base.m (PROS-FB). Applies to: struc_type = 2 (Floodwall) only. Enable flag: config.compute_nappe (forces config.compute_q = 1, because the jet velocity needs the overtopping rate — see EurOtop Overtopping Discharge q).

What it models

When water flows over a floodwall it leaves the crest as a free-falling jet (the "nappe") and lands on the protected side some distance behind the wall. This response gives the geometry of that landing zone (near and far landing distances, jet angles), the width and velocity of the jet where it lands, and the force it exerts per metre of wall. These are the inputs for sizing scour protection behind the wall and for checking loads on the landward ground.

Conceptual model

• The wall acts as a sharp-crested weir. The nappe trajectory is the surge-overflow nappe of a sharp-crested weir, with the trajectory coefficients of TR-21-15 Report 1, Appendix F (Figure F-8).
• The head over the crest includes the wave crest. The head is h1 = SWL + Hm0 − crest_elevation. Adding the full Hm0 is deliberate and conservative. In combined wave and surge overtopping the flow pulses, and the jet lands farther from the wall when a wave crest passes over, so protection has to extend farther than for surge alone (HSDRRS Design Guidelines, 2010, Ch. 4 "Wave and Surge Overtopping", pp. 4-18/4-19). HSDRRS places the wave crest at 0.7·H above SWL (Eq. 4.25); CHESS uses the full Hm0, which is more conservative.
• No head, no nappe. When h1 ≤ 0 (SWL plus wave crest below the wall crest) there is no jet, and every output is NaN for that time step or storm. Outputs are also NaN wherever q is NaN (for example a dry toe).
• Coordinates and sign convention (Report 1, Sec. 5.6 and Fig. 53): x = 0 at the flood face of the wall, positive landward. Angles are reported positive, as the angle below the leeside horizontal. The report's slope angles (F.32–F.33) are negative, so CHESS reports θ = −θ_report, and F.34 becomes Bjet = Bx·sin(θ_center). Every output is ≥ 0.

Conceptual diagram

The equations

TR-21-15 Report 1, Appendix F, Eqs. F.29–F.36:

h1 = SWL + Hm0 − crest_elevation                       head over the crest   (h1 ≤ 0 → NaN)
Hw = crest_elevation − wall_bottom_elevation           wall height

A = −0.425 ; B = 0.055 ; C = 0.150 ; D = 0.559         trajectory coefficients

X_low  = h1 · (−B − √(B² − 4A(C + Hw/h1))) / (2A)          (F.29)  near landing distance
X_up   = h1 · (−B − √(B² − 4A(C + D + Hw/h1))) / (2A)      (F.30)  far landing distance
X_c_surge = (X_up + X_low) / 2                             (F.31)  centre of the landing zone
θ_low  = −atan( 2A·X_low/h1 + B )                          (F.32)  degrees, below horizontal
θ_up   = −atan( 2A·X_up/h1  + B )                          (F.33)  degrees, below horizontal
θ_center = (θ_low + θ_up) / 2                                      (text above F.32)
Bx     = X_up − X_low                                              landing zone width

Bjet = Bx · sin(θ_center)                                  (F.34)  jet width at impact
Vjet = q / Bjet                                            (F.35)  jet velocity
Fjet = ρw · Bjet · Vjet²                                   (F.36)  force per metre of wall

q is the mean overtopping rate already in m³/s per m (the library's unit for q); there is no unit conversion.

Parameters

Symbol	Meaning	Units	Source
SWL	Still water level	m, datum	Project forcing
Hm0	Significant wave height at the toe	m	Project forcing
crest_elevation	Top of wall elevation	m, datum	structure.crest_elevation
Hw	Wall height, crest_elevation − wall_bottom_elevation	m	structure.crest_elevation, structure.wall_bottom_elevation
q	Mean overtopping rate	m³/s per m	Resp.q (EurOtop Overtopping Discharge q)
ρw	Water density	kg/m³	config.water_density

Outputs

Field	Meaning	Units
X_low, X_up	Near and far landing distances from the flood face of the wall	m
X_c_surge	Distance from the wall to the centre of the landing zone	m
Bx	Width of the landing zone	m
theta_low, theta_up, theta_center	Jet angles below the leeside horizontal	deg
Bjet	Jet width at impact	m
Vjet	Jet velocity at impact	m/s
Fjet	Jet force per metre of wall	N/m

Notes / limitations

• In PROS-FB (workflow = 4) the nappe is computed from the q hazard curve rather than storm by storm; see CHESS PROS.
• The coefficients A–D are fixed. Their validity range is that of the TR-21-15 weir formulation.
• The file keeps a commented-out block of candidate formulas (e.g. an X_c_wave distance) marked "PARAMETERS NOT FOUND IN TR-21-15 Appendix F". They are not used.

References

• ERDC/CHL TR-21-15, Report 1, Appendix F, Eqs. F.29–F.36 and Figure F-8 (sharp-crested weir nappe); Sec. 5.6 and Fig. 53 for the coordinate convention.
• HSDRRS Design Guidelines (2010), Chapter 4 "Wave and Surge Overtopping", Eqs. 4.12–4.15 (the same surge nappe as TR-21-15 F.29–F.30) and Eq. 4.25 (wave crest at 0.7·H above SWL).