Calculations

Calculations

• Rain depth of a storm

  The total depth of rain over the storm's duration.

  depth (mm) = Σ i × Δt ÷ 60

  Rain on the ground: overland flow, ponding and outflow rain.js

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• Design storm by alternating blocks

  Builds a storm from an intensity-duration curve with the heaviest block in the middle.

  depth_k = idf(k·Δt) × k·Δt ÷ 60 − depth_(k−1); i_k = depth_k × 60 ÷ Δt

  Rain on the ground: overland flow, ponding and outflow rain.js

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• Time step

  The length of each simulation step.

  dt = min(dt_max, α × Δx ÷ √(g × max(h_max, 0.01)))

  Rain on the ground: overland flow, ponding and outflow rain.js

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• Flow across a cell face

  The unit flow between two neighboring cells for one time step.

  q = (q − g·h_f·dt·Δη ÷ Δx) ÷ (1 + g·dt·n²·|q| ÷ h_f^(7/3)); |q| ≤ h_f·√(g·h_f)

  Rain on the ground: overland flow, ponding and outflow rain.js

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• Flow leaving at an open edge

  The unit flow out of the grid at an open edge or beside a hole in the ground.

  q = h^(5/3) × √S ÷ n, S = max(edge slope, bed slope to the cell behind); q ≤ h·√(g·h)

  Rain on the ground: overland flow, ponding and outflow rain.js

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• Keeping depths from going negative

  Scales the outflows of a cell so it cannot give away more water than it has.

  when Σ out × dt ÷ Δx > available: out ← out × available ÷ (Σ out × dt ÷ Δx)

  Rain on the ground: overland flow, ponding and outflow rain.js

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• New depth, deepest water, fastest flow and first-wet time

  Updates each cell's depth and records its greatest depth, greatest speed and the time it first got wet.

  h ← h + rain − losses + (dt ÷ Δx) × (q_W − q_E + q_S − q_N); speed = |q| ÷ h

  Rain on the ground: overland flow, ponding and outflow rain.js

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  32-bit in part

• Where the records are kept

  Sets up the arrays that hold the per-cell maximum depth, maximum speed and first-wet time.

  Rain on the ground: overland flow, ponding and outflow rain.js

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  32-bit in part

• Drains

  Takes water out of a drain's cell up to its capacity each step.

  take = min(capacity × dt, depth × cell area)

  Rain on the ground: overland flow, ponding and outflow rain.js

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• Volumes of rain, loss and outflow

  Adds each step's rain, losses and outflow to the running totals.

  out += Σ q × Δx × dt; rain += Σ depth of rain × cell area

  Rain on the ground: overland flow, ponding and outflow rain.js

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• Mass balance and the results

  The stored volume, the mass balance error, the pond volumes at the end and the peak outflow.

  error = (rain − loss − out − drained − stored) ÷ rain

  Rain on the ground: overland flow, ponding and outflow rain.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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  64-bit

• 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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  64-bit

• 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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  64-bit

• 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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• Gravity and Manning constant per unit system

  The gravity and the Manning equation constant every river calculation uses.

  m: k = 1, g = 9.80665; ft: k = 1.486, g = 32.174

  River model: stations, cut sections and reach lengths river.js

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• River station of a point

  The station of a point, measured along the centerline from its downstream end, and its offset to the side.

  rs = L − d

  River model: stations, cut sections and reach lengths river.js

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• Cross-section from a cut line

  Turns a line drawn across the river into a cross-section: station, ground points by offset, and first bank positions.

  rs = round₆(L − d_crossing); o = round₆(offset); banks = lowest point ± bank distance, clipped to the section

  River model: stations, cut sections and reach lengths river.js

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• Sections at a spacing along the centerline

  Cuts square cross-sections every set distance along the centerline and at both ends.

  dₖ = k × spacing while k × spacing < L − 1e-9; rs = round₆(L − dₖ)

  River model: stations, cut sections and reach lengths river.js

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• Reach lengths between sections

  The channel and overbank distances from each section to the next one downstream.

  channel = rsᵢ − rsᵢ₋₁; overbank = √(Δx² + Δy²) between the middles of the overbank parts of the two cut lines

  River model: stations, cut sections and reach lengths river.js

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