Calculations

Calculations

• Rail heights from cant

  Each rail's height above the profile for an applied cant and a pivot rule.

  center: ∓E ÷ 2; inner rail fixed: 0 and E; low rail fixed: max(0, ±E)

  Railway cant, cant ramps and track quantities rail.js

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• Cant ramp gradient and rate

  The steepest cant gradient of each ramp in mm per meter and the rate of change of cant in mm per second.

  gradient = |E| ÷ length (m) × peak factor; rate = gradient × v; ramp on a straight: length = |E| ÷ gradient limit

  Railway cant, cant ramps and track quantities rail.js

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• Cant, deficiency and excess checks

  Compares the applied cant, the cant deficiency and the cant excess with the limits.

  deficiency = E_eq − E; excess = E − E_eq(slowest train); flagged when over the limit + 1e-9

  Railway cant, cant ramps and track quantities rail.js

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• Cant segments along the alignment

  The cant layout as constant and ramp segments, with each rail's height at both ends of each.

  Railway cant, cant ramps and track quantities rail.js

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• Cant at a station

  The applied cant and each rail's height at any station.

  u = (sta − ramp start) ÷ ramp length; E = cant × g(u) on the way in, cant × (1 − g(u)) on the way out

  Railway cant, cant ramps and track quantities rail.js

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• Track cross section

  The rails, sleeper, ballast and sub-ballast outline square to the alignment, and the ballast and sub-ballast areas.

  slope = (h_R − h_L) ÷ base; toe distance d = (z_shoulder − z₀ + |o|·fall) ÷ (1 ÷ ballast slope − fall)

  Railway cant, cant ramps and track quantities rail.js

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• Area of a section polygon

  The area of the ballast or sub-ballast outline in the cross section.

  A = Σ (o_i·z_j − o_j·z_i) ÷ 2

  Railway cant, cant ramps and track quantities rail.js

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• Track quantities

  Track and rail lengths, sleeper count and volume, and gross and net ballast and sub-ballast volumes.

  volume = Σ (A_i + A_(i+1)) ÷ 2 × Δsta; sleepers = ⌊length ÷ spacing + 1e-9⌋ + 1; net ballast = gross − sleepers × L × W × (depth − crib)

  Railway cant, cant ramps and track quantities rail.js

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• Section prepared for the solver

  Splits a cross section at the banks, roughness changes and ineffective areas, and adds a point at each split.

  z(o) = z_p + (z_q − z_p) × (o − o_p) ÷ (o_q − o_p)

  River water surface profiles by the standard step method river-solve.js

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• Area, wetted perimeter, conveyance and velocity coefficient

  The hydraulic properties of a section at a water level.

  A = Σ (d_a + d_b) ÷ 2 × Δo; P = Σ hypot(Δo, Δz); K = (k ÷ n) × A × (A ÷ P)^(2/3); n_c = (Σ P_i·n_i^1.5 ÷ P)^(2/3); α = max(1, A²·Σ(K_i³ ÷ A_i²) ÷ K³)

  River water surface profiles by the standard step method river-solve.js

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• Root finding

  Solves the energy balance for the water level between two bracketing levels.

  stop when |bracket ÷ 2| ≤ 2ε|x| + tol ÷ 2, tol = 1e-6

  River water surface profiles by the standard step method river-solve.js

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• Minimum search

  Refines the level of least specific energy.

  River water surface profiles by the standard step method river-solve.js

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• Specific energy

  The energy level at a section for a water level and a flow.

  E = WS + α·Q² ÷ (2g·A²)

  River water surface profiles by the standard step method river-solve.js

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• Critical water surface

  The water level of least specific energy at a section.

  River water surface profiles by the standard step method river-solve.js

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• Normal depth boundary

  The water level at which the section carries the flow at a given energy slope.

  K(WS) = Q ÷ √S

  River water surface profiles by the standard step method river-solve.js

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• Velocity head and flow split

  The velocity head, energy level and the flow in the left overbank, channel and right overbank.

  h_v = α·Q² ÷ (2g·A²); Q_part = Q × K_part ÷ K

  River water surface profiles by the standard step method river-solve.js

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• Energy loss between two sections

  The friction loss and the contraction or expansion loss over a reach.

  L = (L_lob·Q_lob + L_ch·Q_ch + L_rob·Q_rob) ÷ Q; S_f = ((Q₁ + Q₂) ÷ (K₁ + K₂))²; h_e = L·S_f + C·|h_v2 − h_v1|

  River water surface profiles by the standard step method river-solve.js

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• Results at a section

  Depth, velocities, Froude numbers, energy slope and specific force for the results table.

  Fr = V ÷ √(g·A ÷ T); S_f = (Q ÷ K)²; specific force = Q² ÷ (g·A) + A·ȳ

  River water surface profiles by the standard step method river-solve.js

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• Subcritical run

  Steps upstream from the downstream boundary, solving each section's water level.

  River water surface profiles by the standard step method river-solve.js

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• Supercritical run

  Steps downstream from the upstream boundary, solving each section's water level below critical depth.

  River water surface profiles by the standard step method river-solve.js

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• Mixed regime

  Keeps, at each section, the subcritical or supercritical answer with the larger specific force, and marks hydraulic jumps.

  River water surface profiles by the standard step method river-solve.js

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• Decimals in the results table

  The number of decimals each result is shown with in the table, the text report and the CSV.

  River water surface profiles by the standard step method river-report.js

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• Table rows

  Formats each section's results for the table and the CSV.

  text = v.toFixed(d)

  River water surface profiles by the standard step method river-report.js

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• Water surface at a plan point

  The water level at any point from the results at the sections either side.

  rs = centerline length − distance along; WS = WS_a + (WS_b − WS_a) × (rs − rs_a) ÷ (rs_b − rs_a)

  Flood extent on the ground surface river-extent.js

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• Behind a levee

  Whether a point is kept dry by a levee the water does not top.

  offset and crest interpolated between the two sections; dry when beyond the levee and WS < crest

  Flood extent on the ground surface river-extent.js

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• Depth at a point

  The depth of water at a plan point.

  depth = WS − ground

  Flood extent on the ground surface river-extent.js

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• Waterline point on a triangle edge

  Where the depth is zero along an edge between a wet corner and a dry corner.

  u = d_a ÷ (d_a − d_b); point = a + u × (b − a)

  Flood extent on the ground surface river-extent.js

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• Flooded area and greatest depth

  Adds up the wet part of each triangle and tracks the greatest depth at a corner.

  piece area = |Σ (x_i·y_j − x_j·y_i)| ÷ 2; area = Σ piece areas

  Flood extent on the ground surface river-extent.js

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• Barrel flow area and top width

  The flow area, water surface width and full-barrel hydraulic radius of one culvert barrel at a given depth.

  circle: θ = 2·acos((r − y) ÷ r); A = r²(θ − sin θ) ÷ 2; T = 2√(y(D − y)); R = D ÷ 4. box: A = B·y; R = B·D ÷ (2(B + D))

  Culverts, bridges and junctions in a river model river-structures.js

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• Critical depth in a barrel

  The depth at which the flow in one barrel is critical, capped at the rise.

  Q²·T(y) − g·A(y)³ = 0

  Culverts, bridges and junctions in a river model river-structures.js

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• Headwater for one barrel's flow

  The inlet-control and outlet-control headwater depths for a flow in one barrel, and which one governs.

  HWw = (Q ÷ (Cw·B))^(2/3); HWo = Q² ÷ (Cd²·A²·2g) + D ÷ 2; φ = clamp((HWo ÷ D − 1) ÷ 0.5, 0, 1); HWin = (1 − φ)·HWw + φ·HWo; Kf = 2g·n²·L ÷ (k²·R^(4/3)); HWout = (1 + Ke + Kf)·V² ÷ 2g + ho − L·S₀

  Culverts, bridges and junctions in a river model river-structures.js

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• Culvert headwater with road overtopping

  The headwater at which the flow through the barrels plus the flow over the road equals the total flow.

  n·Qbarrel(HW) + Cr·Lr·(HW − crest)^1.5 = Q

  Culverts, bridges and junctions in a river model river-structures.js

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• Bridge opening section

  The downstream cross-section clipped to the abutments, with each pier as a strip that carries no flow.

  z(o) = z₀ + (o − o₀) ÷ (o₁ − o₀) × (z₁ − z₀)

  Culverts, bridges and junctions in a river model river-structures.js

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• Water surface above a bridge

  The water surface in the opening and at the section above the bridge, in low flow, pressure flow or flow over the deck.

  WS + α·Q² ÷ (2g·A²) = E₀ + L·(Q ÷ ((K₁ + K₂) ÷ 2))² + c·|Δ velocity head|; pressure: Q = Cd·An·√(2g·head); deck: Q = Cw·L·(EGL − deck)^1.5

  Culverts, bridges and junctions in a river model river-structures.js

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• Stepping a profile across a structure

  Takes the water surface below a structure, works out the level above it and uses that as the start of the next reach upstream.

  TW = max(0, WSd − invert down); WSu = invert up + HW

  Culverts, bridges and junctions in a river model river-structures.js

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• Flows and water levels at a junction

  Adds the flows of the reaches that join and carries the water surface at the head of the downstream reach up to them.

  Q = Σ Q of the joining reaches; WS of each joining reach's downstream end = WS at the head of the reach below

  Culverts, bridges and junctions in a river model river-structures.js

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• Area and conveyance table of a section

  Flow area and conveyance of one cross-section as functions of water level, read from a table.

  levels hᵢ = h₀ + (hi − h₀)·(i ÷ 200)²; between levels: linear interpolation; below h₀: A·f, K·f^1.6667 with f = depth ÷ (h₀ − bed)

  Unsteady flow routing down a river river-unsteady.js

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• Banded linear solve

  Solves the Newton step for all sections at once.

  L·U·x = b with row pivoting inside the band

  Unsteady flow routing down a river river-unsteady.js

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• Continuity and momentum residuals of one interval

  The two equations that must be zero between a pair of neighboring sections for one time step.

  continuity: Δ(Aᵢ + Aⱼ) ÷ (2Δt) + (θ·ΔQ_new + (1 − θ)·ΔQ_old) ÷ Δx; momentum: Δ(Qᵢ + Qⱼ) ÷ (2Δt) + Δ(Q|Q| ÷ A) ÷ Δx + g·Ā·Δh ÷ Δx + g·Ā·S̄f, Sf = Q|Q| ÷ K²

  Unsteady flow routing down a river river-unsteady.js

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• Newton iteration for one time step

  Finds the water levels and flows at the end of a time step.

  J·Δu = −F; J by forward differences with steps 1e-6·Href and 1e-6·Qref; stop when max(|Δh| ÷ Href, |ΔQ| ÷ Qref) < 1e-8

  Unsteady flow routing down a river river-unsteady.js

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• Time stepping, peaks and Froude check

  Advances the run step by step, records the peak level and flow at each section and flags supercritical flow.

  V_in += Δt·(θ·Q_new + (1 − θ)·Q_old); T ≈ (A(h + 0.001·Href) − A(h)) ÷ (0.001·Href); flag when |Q| ÷ A > √(g·A ÷ T)

  Unsteady flow routing down a river river-unsteady.js

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• Volume balance error

  The share of water not accounted for between inflow, outflow and the change of storage in the reach.

  storage = Σ ½(Aᵢ + Aᵢ₊₁)·Δxᵢ; error = (V_in − V_out − (S₁ − S₀)) ÷ max(V_in, V_out, |S₁ − S₀|, 1e-12)

  Unsteady flow routing down a river river-unsteady.js

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• US gallon in cubic feet

  The factor that turns gallons into cubic feet in feet projects.

  1 gal = 0.133680556 ft³

  Sanitary sewer design flows and pipe sizes sewer.js

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• Peaking factor

  The ratio of peak to average sanitary flow for a population.

  Harmon: 1 + 14 ÷ (4 + √P); Babbitt: 5 ÷ P^0.2; P in thousands; held between pfMin and pfMax

  Sanitary sewer design flows and pipe sizes sewer.js

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• Average flow from people

  People times per-person daily flow, in project flow units.

  m: P × L/day ÷ 1000 ÷ 86400; ft: P × gal/day × 0.133680556 ÷ 86400

  Sanitary sewer design flows and pipe sizes sewer.js

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• Infiltration flow

  Tributary area times the infiltration allowance, in project flow units.

  m: ha × L/s per ha ÷ 1000; ft: ac × gal/day per ac × 0.133680556 ÷ 86400

  Sanitary sewer design flows and pipe sizes sewer.js

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• Design flow, slope and velocity per pipe

  For each pipe in flow order: the people and area above it, the peak flow, the slope, the full capacity, the normal depth and the velocity.

  S = (inv up − inv down) ÷ length; Q = q_dom × pf + q_inf; ratio = Q ÷ Q_full; V = Q ÷ A(yn)

  Sanitary sewer design flows and pipe sizes sewer.js

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• Pipe size for the design flow

  The smallest listed pipe size whose allowed share of full capacity carries the design flow.

  first size with Q_full(size) × maxFull ≥ Q

  Sanitary sewer design flows and pipe sizes sewer.js

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• Catchment runoff hydrograph

  The runoff from one catchment at each time step for a constant storm.

  q(t) = k·C·A·(depth(t) − depth(t − w)) ÷ w; depth(τ) = i·clamp(τ, 0, duration); k = 1 ÷ 360 (m), 1 (ft)

  Storm hydrographs routed through the pipe network storm-route.js

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• Muskingum coefficients

  The three weights of the routing equation for a reach.

  d = K(1 − X) + Δt ÷ 2; c₀ = (−KX + Δt ÷ 2) ÷ d; c₁ = (KX + Δt ÷ 2) ÷ d; c₂ = (K(1 − X) − Δt ÷ 2) ÷ d

  Storm hydrographs routed through the pipe network storm-route.js

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