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Computational basis

Status: Phase 0 research basis closed 2026-09-09. HDS-5 equation and table transcriptions used by the current code have been checked. Research completion does not make the combined solver engineering-validated; the remaining validation and external comparison tasks are tracked separately.

Source hierarchy and scope

Use FHWA HDS-5 (third edition, April 2012) as the starting culvert reference. Evaluate refinements against the publications in references. External software is an implementation comparison, not the definition of truth.

The completed source review adopts the following method dispositions:

  • Bodhaine's six types define physical classification evidence, not production equations or a one-to-one mapping to every extended HY-8 profile label.
  • Corrected FHWA-HRT-06-138 coefficients apply only to their recorded box inlet geometry, barrel count, span-to-rise range, fillets, bevels, and skew. CS-013 implements that context and fails closed on unmatched Figure 93 configurations. Its Table 12 polynomial is limited to approximately 0.4 < HW/D < 2.3; it is not HY-8's recomputed South Dakota-box curve.
  • NCHRP 734 supports the current representative-barrel total-flow architecture within its tested limitations. Its Borda-Carnot outlet refinement is deferred because the initial reservoir/pool boundary has no downstream channel area. Slipline, buried-invert, depth-varying, and composite roughness are separated into CS-030 through CS-033. Original NCHRP embedded- culvert coefficients are rejected for implementation because the HY-8-bundled correction documents a dimensionless-discharge error and false 50% embedded beveled data. The correction's synthetic high-flow extension is also not adopted as primary evidence.
  • Austroads AGRD05B-23 supplies Australian design context, including allowable headwater, blockage, outlet velocity, scour, multiple-event checking, and reporting. It does not displace the HDS-5 computational baseline. Its convention of assigning a near-crown Froude number to full flow is rejected for the computational core: Froude number remains undefined for a pressurised closed section.
  • HY-8 v8.0 method documentation and executable 8.0.1.2 results are pinned comparison evidence. They do not define a local coefficient, transition, or acceptance tolerance.

Fixture-ready source records and the limits of their use are in research/fixture_candidates.md.

The first hydraulic scope was steady, forward flow through straight, prismatic circular and rectangular barrels with specified absolute headwater/tailwater elevations in a common datum. Negligible reservoir approach and receiving-water velocity remain explicit simplifications; channel approach sections and their energy correction require a later contract. Roadway overtopping was outside that first solver release and has since been added under CS-028 with explicit applicability limits. Adverse-slope solutions, reverse flow and barrel junctions remain outside the supported scope. Models may represent a horizontal barrel; normal depth at zero slope must not be invented for positive discharge.

These are project decisions, not claims that the excluded cases lack solutions. An unsupported case must raise a structured error rather than silently selecting a different hydraulic model. NCHRP 734 supports representative-barrel superposition for most total-flow calculations, but reports nonuniform-approach differences up to 10%, depressed-barrel average-flow differences up to about 4%, and individual-barrel differences up to about 7%. These limitations matter where local barrel performance is the design criterion.

SI hydraulic primitives to implement after geometry

With flow area A, wetted perimeter P, free-surface top width T, discharge Q, gravity g and Manning roughness n:

R = A / P
V = Q / A
hydraulic_depth = A / T
velocity_head = V^2 / (2*g)
Fr^2 = Q^2*T / (g*A^3)                 [uniform velocity distribution]
E = y + Q^2 / (2*g*A^2)               [alpha = 1]
Q_normal = A*R^(2/3)*sqrt(S0) / n     [SI Manning, uniform flow]
Sf = (n*Q / (A*R^(2/3)))^2
hf = L*Sf                             [constant area/roughness]
h_local = K*V_reference^2 / (2*g)
M(y) = Q^2/(g*A(y)) + integral(A(z), z=0..y) [momentum function]

HDS-5 section 3.1.4, printed pages 3.9–3.12 (local PDF pages 91–94), provides velocity, entrance/friction/exit losses and energy balance, equations 3.1–3.7. Critical depth is the stationary-specific-energy solution, equivalently Fr = 1 under the assumptions above; it must be solved from geometry and discharge. Normal-depth solvers must identify admissible branches rather than assuming the circular Manning conveyance curve is monotone up to the crown.

For a circle of radius r at depth y, the segment angle is theta = 2*acos((r-y)/r), area is r^2*(theta-sin(theta))/2, wetted perimeter is r*theta, and surface chord is 2*sqrt(y*(2*r-y)) for 0 < y < 2*r. These analytical geometry identities need cancellation/limiting-case tests.

Geometry conventions are a Phase 2 gate: distinguish a surface approaching the crown from a pressurised closed section. Do not reuse the old circular top width of one diameter at full depth. Hydraulic depth and Froude number are undefined for a pressurised section and should not be manufactured from full-barrel radius.

Outlet losses and profile requirements

HDS-5 equation 3.4b lists friction prefactors 29 in English units and 19.63 in SI. The new solver will use the direct SI Manning friction slope above instead of transplanting a prefactor from an English-unit implementation.

Exit loss requires an explicitly identified downstream velocity/reference area. HDS-5 equations 3.4c–3.4e describe different assumptions. Do not derive barrel outlet velocity merely by dividing flow by the tailwater-wetted barrel area.

HDS-5 describes the effective outlet depth (dc + D)/2 as an approximation; section 3.1.4 states that backwater calculations are required for low headwater. Therefore it does not replace a free-surface profile in the production solver.

Direct step is the adopted production method for monotonic S1, S2, M1, M2, and H2 profiles in the currently supported prismatic circular and rectangular barrels. Depth is a stable integration coordinate for those branches and directly locates normal-depth and crown limits. Standard step is reserved as an independent spatial-coordinate validation method and for future non-prismatic geometry; it is not currently a second production solver. Hydraulic jumps use momentum matching between independently routed S1 and S2 branches rather than attempting to integrate through the discontinuity.

HEC-RAS's outlet-control energy formulation is an independent methodology reference for boundary energy accounting and candidate conditions.

Empirical relationships and transition policy

The implemented inlet constants were checked against HDS-5 Appendix A, Table A.1 (local PDF page 197), and the implemented entrance-loss constants against Appendix C, Table C.2 (local PDF page 216). Equation form, unit conversion, chart/scale, entrance geometry, locator and shape applicability are represented in typed records. The box-report (FHWA-HRT-06-138) and NCHRP 734 coefficients have been extracted into docs/research/, providing expanded parameters for slip-lined culverts and modern box geometries. Transcription is not the same as engineering acceptance; assess applicability before adding these to the catalogue.

HDS-5 Appendix A, printed page A.1 (local PDF page 190), describes the transition as a smooth curve drawn between and tangent to the unsubmerged and submerged curves. Printed page A.6 states that the nomograph transition was drawn by hand, so no reproducible digital algorithm is prescribed. The adopted project method is a cubic Hermite bridge over 3.5 <= q* <= 4.0, using the value and first derivative of each bounding curve. For Form 1, the lower tangent includes the exact critical-specific-energy derivative. It is C1 continuous and monotonic for every catalogued circular and rectangular coefficient set. This is a deterministic implementation of the HDS-5 tangency requirement, not a claim of exact HY-8 polynomial or hand-drawn-nomograph equivalence.

HDS-5 Section 3.5.2, printed page 3.39 (local PDF page 121), states that its laboratory inlet-control curves cover 0.5 <= HW/D <= 3.0. Below 0.5, a general weir equation is fitted at the boundary; above 3.0, a general orifice equation is fitted at the boundary. The project evaluates the direct Appendix A Equation A.3 relationship above that range but marks every such result with inlet_control_high_head_extension. Results above HW/D = 10 receive the additional inlet_control_extreme_headwater review warning. These numeric extensions are available for sensitivity and comparison work; the warnings prevent them from appearing ordinarily validated. HY-8's version-specific ** report marker motivated the conservative extreme-review threshold but is not the source of the HW/D = 3 applicability boundary.

The HEC-RAS inlet page renders the submerged slope term with an apparent equality-sign error. Verify the typeset primary equation before use. SWMM's inlet transition is an implementation choice to investigate, not an automatically adopted continuous-regime model.

Current implementation boundary

Default selection follows the typed-context policy in CS-002. Current MRWA Design Procedure version 2E, Table 2.1 supplies distinct smooth reinforced-concrete pipe and box ranges; Table 2.2 supplies diameter/corrugation-specific CSP values. The MRWA Part 5B supplement requires plastic-pipe Manning roughness from the applicable manufacturer. Accordingly, generic concrete roughness and missing material context fail explicitly, while a plastic HDS-5 laboratory value is available only through a deliberate fallback flag with structured applicability notices. Hydraulic defaults and construction compliance remain separate decisions. No HY-8 fallback value has been adopted.

The repository implements circular and rectangular geometry, hydraulic primitives, critical and normal depth, selected HDS-5 inlet and loss coefficients, provisional inlet/outlet control, direct-step profiles, barrel/group/crossing solvers, rating curves, and roadway overtopping for both constant and piecewise-linear irregular crests. Roadway flow follows HDS-5 Section 3.1.5 Equation 3.9 in SI units:

Q_roadway = C_d L H_wr^1.5

The caller must supply the SI coefficient selected for the actual roadway geometry and overtopping depth. Constant crests use the direct equation. Irregular profiles are split at hydraulic-level crossings and integrated horizontally using the four-point Gaussian procedure documented in FHWA HY8; each weighted contribution retains its source interval, integration station, effective length, crest elevation, local heads, discharge and source provenance. The crossing solver adds roadway flow to the independently calculated culvert-group flows at a common headwater.

When downstream water is above a local roadway crest, paved or gravel submergence uses the source-traceable digital ordinates attributed to FHWA/RD-86/108 and recorded by EPA SWMM. The implemented interpolation is deliberately bounded to downstream_head / upstream_head <= 0.99. The source digitisation also contains an ordinate at ratio 1.00, but applying its nonzero factor in this static capacity solver would imply positive discharge when the upstream and downstream water levels are equal. That ordinate is therefore not used. Ratios between 0.99 and equal stage fail closed rather than extrapolating or inventing a decay relation; exactly equal upstream and downstream water levels return zero roadway flow. A submerged roadway requires an explicit RoadwaySurface.PAVED or RoadwaySurface.GRAVEL. Reverse roadway flow is not supported.

The initial discharge-dependent tailwater option is a prismatic-channel normal-depth boundary. For bottom width b, left and right horizontal-to-vertical side slopes zL and zR, and depth y, it uses:

A = b y + 0.5 (zL + zR) y^2
P = b + y sqrt(1 + zL^2) + y sqrt(1 + zR^2)
T = b + (zL + zR) y
Q = (1/n) A (A/P)^(2/3) sqrt(Sf)

The solver brackets and solves Q(y) - Qtarget = 0 with Brent's method, then returns tailwater elevation = channel invert elevation + y. Sf remains explicitly the friction/energy slope; using bed slope estimates it only under uniform flow. Consistent with HDS-5 Section 1.4.4, this is an approximation for a downstream channel without a controlling backwater influence. It is not valid as a general reach model for downstream impoundment, constriction, junction, tidal, or other backwater controls.

The boundary's method_source records HDS-5 as the basis for using a normal-depth approximation. It does not attribute the caller's Manning roughness, friction slope, surveyed geometry, or channel invert to HDS-5; those inputs have separate optional source fields. Likewise, TailwaterResolution.depth is normal depth above the downstream channel invert. RatingCurvePoint.tailwater_depth retains the older culvert-relative reporting meaning: depth above the barrel outlet for a barrel curve or above the lowest outlet invert for a crossing curve. These values differ whenever the channel and culvert inverts differ.

A user-supplied tailwater rating curve is a separate empirical boundary, not a Manning calculation. It contains at least two finite (Q, WSE) points, with nonnegative strictly increasing discharge and nondecreasing absolute water-surface elevation. At a supplied discharge the exact elevation is returned. Between adjacent points the implementation uses only linear interpolation:

WSE(Q) = WSE1 + (Q - Q1) (WSE2 - WSE1) / (Q2 - Q1)

Discharge outside the supplied closed range fails explicitly; there is no implicit clamp or extrapolation. The caller's rating_curve_source, complete curve, requested discharge, resolved elevation, and exact-versus-linear interpolation decision are retained in the tailwater resolution. The curve is resolved from barrel flow for a standalone barrel, total group flow for a standalone group, and total crossing flow before allocation for a crossing. Every generated rating point performs an independent boundary resolution.

Inverse capacity with a discharge-dependent tailwater solves the coupled residual

f(Q) = HWforward(Q, TW(Q)) - HWtarget

using the same forward barrel, group, or crossing solver that defines the corresponding rating calculation. TW(Q) is therefore recalculated for every bracket expansion and root evaluation using barrel discharge, total group discharge, or total crossing discharge as applicable. The crossing path retains mixed-group allocation and supported roadway flow, including irregular-profile integration and bounded paved/gravel submergence, inside each evaluation. Fixed-stage inputs retain the earlier direct inverse path. For a bounded user rating curve, the supplied discharge endpoints are hard root bounds; a target outside them fails rather than freezing, clamping, or extrapolating stage.

For HDS-5 Section 3.5 steep-slope inlet-control cases, the solver routes an S2 profile downstream from immediately below critical depth toward normal depth. Tailwater no higher than normal depth directly implies a swept-out jump. For higher sub-crown tailwater, the solver compares the boundary depth with the conjugate depth of the S2 outlet section, found by equating momentum functions. If tailwater cannot sustain that conjugate depth, the S2 outlet depth and velocity remain valid. HDS-5 printed pages 3.36-3.37 (local PDF pages 118-119) describe this momentum comparison and swept-out-jump result.

When the S1 and S2 conjugate-depth profiles intersect, the solver records a JS1 profile, the jump station, and both depths at that duplicated station. If S1 instead reaches the inlet, its free-surface outlet-control headwater remains a physical candidate. The solver still does not resolve all mixed free-surface/pressurised Type 6/7 cases, so the combined solver remains provisional rather than engineering accepted. See progress and validation for the remaining gates.

For outlet-controlled M2 flow that reaches the crown before the inlet, the upstream remainder is continued as a full-section reach. Its headwater is the transition-section energy plus full-flow Manning friction over the reported full_flow_length and the inlet loss. This represents a mixed free-surface/pressurised Type 7 path without treating the whole barrel as full. An explicit Type 6 comparison now exercises nearly full-length pressurisation, but its high-head inlet result does not yet agree sufficiently with HY-8 and remains unsupported.

For a submerged outlet whose full-flow HGL intersects the crown inside the barrel, the intersection defines a downstream pressurised length and an upstream free-surface reach. The upstream reach is routed from immediately below the crown using the same backward S1 and S1/S2 momentum-intersection methods as sub-crown tailwater cases. This distinguishes S1f from JS1f while retaining the full length and jump station in the result.

Supported calculations that retain a known approximation expose stable HydraulicWarningCode values rather than relying on prose or inferred regimes. Current codes identify the inlet-control outlet-depth approximation and unresolved mixed flow. Candidate headwaters and these warnings are calculation evidence; they do not make an unsupported flow state valid.