Calculations

Calculations

• Boundary links as linear terms

  Writes every wall, moving-wall and slip link as two coefficients and a constant, for the time step's inner loop.

  f = c₁ f₁ + c₂ f₂ + a; q < ½: c₁ = 2q, c₂ = 1 − 2q; q ≥ ½: c₁ = 1 ÷ (2q), c₂ = (2q − 1) ÷ (2q)

  Flow study: lattice Boltzmann solver flow.js

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• One time step: stream and collide

  Pulls the populations into each cell, takes density and velocity, and relaxes toward equilibrium.

  ρ = Σ fᵢ; u = (Σ cᵢ fᵢ + F ÷ 2) ÷ ρ

  Flow study: lattice Boltzmann solver flow.js

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• Collision with an eddy viscosity

  The collision used in turbulent runs, with a per-cell viscosity and a stabilizer.

  τ = ½ + 3(ν + ν_t); τ_s = ¼ + √(1/16 + 4.5 √2 C² |Π| ÷ ρ); τ = max(τ, τ_s)

  Flow study: lattice Boltzmann solver flow.js

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• Mass through boundaries and force on walls

  Sums the net mass rate per boundary group and the momentum-exchange force on the part.

  mass = Σ (f_in − f_out); F = −Σ cᵢ (f_in + f_out)

  Flow study: lattice Boltzmann solver flow.js

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• Stepping to a steady state

  Repeats time steps until the velocity field stops changing.

  res = √(Σ |u − u_prev|² ÷ Σ |u|²) ÷ steps between checks

  Flow study: lattice Boltzmann solver flow.js

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• Lattice units from physical ones

  Chooses the lattice speed, viscosity, relaxation time and time step for a cell size, speed and fluid.

  Re_cell = U dx ÷ ν; u = min(0.05, √(0.03 ÷ k), 0.5 Re_cell); τ = 3 u ÷ Re_cell + ½; dt = u dx ÷ U

  Flow study: lattice Boltzmann solver flow.js

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• Face normal and area

  The area-weighted outward normal of a face, its area and how flat it is.

  area = Σ |nₜ| ÷ 2; flatness = |Σ nₜ| ÷ (2 × area)

  Flow study: fluid cells, openings and wall distances flow-domain.js

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• Area of an opening's holes

  The area of the holes in a flat face, which is the opening's true area.

  loop area = ½ Σ (pₐ × p_b) · n; holes = Σ −area over clockwise loops

  Flow study: fluid cells, openings and wall distances flow-domain.js

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• Lid cells over an opening

  Finds the cells just behind a flat face that cover its holes.

  raster area = hole pixels × (h ÷ 2)²

  Flow study: fluid cells, openings and wall distances flow-domain.js

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• Wall distance along a link

  Where the link from a fluid cell to a solid cell crosses the part's surface.

  t = (e₂ · (tv × e₁)) ÷ (e₁ · (d × e₂))

  Flow study: fluid cells, openings and wall distances flow-domain.js

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• Frontal area

  The area of the part seen along the flow, used as the reference area for drag and lift coefficients.

  area = covered pixels × pixel size², pixel size = larger extent ÷ 1024

  Flow study: fluid cells, openings and wall distances flow-domain.js

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• Automatic cell size

  Picks the cell size that gives about a target number of cells in the grid box.

  h = ∛(volume ÷ target), rounded up to 1, 1.25, 1.5, 2, 2.5, 3, 4, 5, 6, 8 or 10 times a power of ten

  Flow study: fluid cells, openings and wall distances flow-study.js

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• Cell count estimate

  The number of grid cells a study will use.

  n = ⌈extent ÷ h − 1e-6⌉ + margin cells, per axis

  Flow study: fluid cells, openings and wall distances flow-study.js

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• Lift direction

  The direction lift is measured along, made square to the flow.

  l = l₀ − (l₀ · d) d, then divided by its length

  Flow study: fluid cells, openings and wall distances flow-study.js

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• Inlet flow and speed

  Each inlet's flow rate, mean speed and the speed set on its lattice links.

  Q = v × A (or the given Q); U = Q ÷ A; link speed = Q × (A_face ÷ A) ÷ (lattice link area × dx²); D = √(4A ÷ π)

  Flow study: results in physical units flow-worker.js

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• Speed and pressure scales

  The factors between lattice and physical speed and pressure, and the density set at each outlet.

  c_u = dx ÷ dt; c_p = ρ c_u²; ρ_outlet = 1 + 3 (p − p_ref) ÷ c_p

  Flow study: results in physical units flow-worker.js

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• Second steady-state test

  Watches the pressure drop or the drag force between checks before the run is accepted as steady.

  |q − q_prev| ÷ max(|q|, 1e-30) < 1e-4 on two checks in a row

  Flow study: results in physical units flow-worker.js

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• Mass flow, mean pressure and pressure drop

  The mass and volume flow through each opening, its mean pressure, and the pressure drop between inlets and outlets.

  ṁ = lattice mass × ρ dx³ ÷ dt; p = p_ref + (mean ρ − 1) ÷ 3 × c_p; Δp = flow-weighted mean inlet p − outlet p

  Flow study: results in physical units flow-worker.js

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• Force, drag and lift coefficients

  The force on the part in newtons and, for external flow, drag, lift and their coefficients.

  F = lattice force × ρ dx⁴ ÷ dt²; C_d = F · d ÷ (½ ρ U² A)

  Flow study: results in physical units flow-worker.js

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• Reynolds number

  The Reynolds number of the study.

  Re = U √(4A ÷ π) ÷ ν (internal); Re = U L ÷ ν (external)

  Flow study: results in physical units flow-worker.js

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• Inflow eddy viscosity

  The turbulence level given to inflows in a turbulent run.

  ν_t = max(3ν, √1.5 × 0.09^0.25 × I × u × ℓ), I = 5 % (internal) or 1 % (external)

  Flow study: results in physical units flow-worker.js

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• Mass balance

  The share of the inflow that does not leave through the outlets.

  balance = (ṁ_in − ṁ_out) ÷ ṁ_in

  Flow study: results in physical units flow-ui.js

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• The mesh handed to the flow solver

  Builds the job for a flow study from the part and the study's settings.

  Flow study: 32-bit steps in the mesh and the result fields flow-study.js

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• Cell fields, maximum speed and pressure range

  Velocity and pressure of every cell in physical units, and the largest speed and lowest and highest pressure the results state.

  v = u × c_u; p = p_ref + (ρ − 1) ÷ 3 × c_p; v_max = max |v|

  Flow study: 32-bit steps in the mesh and the result fields flow-worker.js

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• Pressure on the part's surface

  The static pressure at each vertex of the plot mesh, sampled a little way into the fluid.

  p = Σ (wᵢ + 1e-6) pᵢ ÷ Σ (wᵢ + 1e-6) over the fluid cells around the sample point

  Flow study: 32-bit steps in the mesh and the result fields flow-worker.js

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• Result probe

  Shows the speed, velocity and pressure of the cell under the pointer, or the surface pressure.

  speed = √(v_x² + v_y² + v_z²)

  Flow study: 32-bit steps in the mesh and the result fields flow-ui.js

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• Lattice Mach number warning

  The fastest cell's speed as a share of the lattice speed of sound.

  Ma = v_max × dt ÷ dx × √3

  Flow study: 32-bit steps in the mesh and the result fields flow-ui.js

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• Model constants

  The constants of the turbulence model.

  c_w1 = c_b1 ÷ κ² + (1 + c_b2) ÷ σ

  Flow study: turbulence model and wall functions flow-turb.js

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• Spalding's law of the wall

  y⁺ as a function of u⁺.

  y⁺ = u⁺ + e^(−κB) (e^(κu⁺) − 1 − κu⁺ − (κu⁺)² ÷ 2 − (κu⁺)³ ÷ 6)

  Flow study: turbulence model and wall functions flow-turb.js

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• Friction velocity

  Solves the wall law for the friction velocity from the speed at a distance from the wall.

  find u_τ with spalding(U ÷ u_τ) = y u_τ ÷ ν

  Flow study: turbulence model and wall functions flow-turb.js

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• Inverse wall law

  u⁺ as a function of y⁺.

  Flow study: turbulence model and wall functions flow-turb.js

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• Wall cells: normal and wall distance

  Each wall cell's wall normal and its true distance to the wall, and the model's starting fields.

  n = Σ wᵢ nᵢ ÷ |Σ wᵢ nᵢ|; y = Σ wᵢ c_n (q c_n) ÷ Σ wᵢ c_n over links with c_n > 0.3, y ≥ 0.05

  Flow study: turbulence model and wall functions flow-turb.js

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• Wall distance of every cell

  The distance from each fluid cell to the nearest wall point.

  d(m) = |v(n) − c| carried from a settled neighbor n along the link c

  Flow study: turbulence model and wall functions flow-turb.js

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• Model update in a cell

  The wall function in wall cells and one pseudo-time step of the model equation elsewhere.

  ν̃_new = (ν̃ + Δt × rhs) ÷ (1 + Δt c_w1 f_w ν̃ ÷ d²); wall cell: ν̃ = κ u_τ y

  Flow study: turbulence model and wall functions flow-turb.js

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• Pseudo-time step

  The model's time step from the advection and diffusion limits.

  Δt = max(0.05, min(every, 0.5 ÷ u_max, 0.8 σ ÷ (6 (ν + ν̃_max))))

  Flow study: turbulence model and wall functions flow-turb.js

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• First-cell y⁺ statistics

  The minimum, maximum, mean and percentiles of y⁺ over the wall cells.

  percentile p = sorted[⌊p × n⌋]

  Flow study: turbulence model and wall functions flow-turb.js

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• Wall force correction for slip

  The term added to the momentum-exchange force on walls that carry a slip velocity.

  F += Σ u_w (f_in − f_out)

  Flow study: turbulence model and wall functions flow-turb.js

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• Streamlines

  Traces lines along the flow from seed points for drawing.

  x ← x + ds × v(x + ½ ds v̂) ÷ |v|

  Simulation and flow result plots flow.js

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• Slice of a field

  Copies one layer of cells of a field for the cut-plane picture.

  Simulation and flow result plots flow.js

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• Field for coloring

  Picks the values and the color range for the cut plane.

  Simulation and flow result plots flow-ui.js

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• Plot mesh

  Splits the part's triangles until no edge is longer than the cell size, for drawing a field.

  midpoint = (a + b) ÷ 2

  Simulation and flow result plots fea.js

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• One subdivision step

  New face, edge and vertex points of the cage refined once.

  smooth vertex = Q ÷ n + 2R ÷ n + (n − 3) v ÷ n; crease vertex = (a + 6v + b) ÷ 8; edge = mean of its ends and the two face points

  Freeform bodies: subdivision surface and cage edits form.js

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• Limit position of a vertex

  Where a vertex ends up on the smooth surface.

  smooth: (n² v + 4 Σ e + Σ f) ÷ (n (n + 5)); crease: (a + 4v + b) ÷ 6; corner: v

  Freeform bodies: subdivision surface and cage edits form.js

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• Limit surface normal at a vertex

  The unit normal of the smooth surface at an interior vertex.

  A = 1 + cos(2π ÷ n) + cos(π ÷ n) √(2 (9 + cos(2π ÷ n))); normal = t₁ × t₂ ÷ |t₁ × t₂|

  Freeform bodies: subdivision surface and cage edits form.js

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• Bézier patch of one quad

  The 16 control points of the bicubic patch for one face of the refined cage.

  b = M G Mᵀ with rows of M: (1, 4, 1, 0) ÷ 6, (0, 4, 2, 0) ÷ 6, (0, 2, 4, 0) ÷ 6, (0, 1, 4, 1) ÷ 6; across a sharp edge: 2 × edge − inner

  Freeform bodies: subdivision surface and cage edits form.js

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• Point on a patch

  Evaluates a bicubic Bézier patch at (u, v).

  S(u, v) = Σᵢ Σⱼ Bᵢ(u) Bⱼ(v) b_ij, B = ((1 − u)³, 3u(1 − u)², 3u²(1 − u), u³)

  Freeform bodies: subdivision surface and cage edits form.js

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• Normal of a patch

  The unit normal of a patch at (u, v).

  n = ∂S/∂u × ∂S/∂v, normalized

  Freeform bodies: subdivision surface and cage edits form.js

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• Refinement level for conversion

  How many times the cage is refined before it is turned into patches.

  level = max(1 if any face is not a quad, ⌈largest semi-sharp weight − 1e-12⌉), at most 4

  Freeform bodies: subdivision surface and cage edits form.js

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• How well the patches meet

  The largest gap and the largest angle between normals across shared patch edges.

  gap = max |p₀ − p₁|; angle = acos(clamp(n₀ · n₁, −1, 1)) × 180 ÷ π

  Freeform bodies: subdivision surface and cage edits form.js

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• Fitting a cage to a target surface

  Moves cage vertices until their limit points lie on a sphere or torus.

  p ← p + (proj(L) − L), L the vertex's limit point

  Freeform bodies: subdivision surface and cage edits form.js

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