2D Incompressible Navier–Stokes Simulation on the FDA Benchmark Nozzle (Projection · MAC) — SVG
v0.1.4
Simulation Controls
Ready.
Nx/Ny control spatial resolution. ν and Uin control inertia versus diffusion (Re).
\(U_{in}(t)=U_0[1+A\sin(2\pi f t)]\) for pulsatile mode.
2D channel height is scaled by max(Din, Dt, Dout) and centered. Sudden expansion occurs at d = Lc+Lt.
Term maps correspond to the momentum equation: advection, pressure-gradient forcing, and viscous diffusion.
Arrows are sampled sparsely for readability.
Both pressure traces are rendered; selection controls emphasis only (thicker/stronger opacity).
Pressure in an incompressible projection method is defined up to an arbitrary constant; “Non‑negative” applies a constant shift so that min(p)=0, without changing velocity or Δp.
2D CFD Heatmap (FDA nozzle)
Velocity magnitudeKinematic pressureShear & NS term maps
Staggered MAC discretization: \(u\) on vertical faces, \(v\) on horizontal faces, \(p\) at cell centers.
Explicit momentum step (advection–diffusion):
\[
\mathbf{u}^\*=\mathbf{u}^n+\Delta t\Bigl(-(\mathbf{u}\!\cdot\!\nabla)\mathbf{u}+\nu\nabla^2\mathbf{u}\Bigr)
\]
using first‑order upwind advection and central Laplacians.
Pressure Poisson equation:
\[
\nabla^2 p = \frac{1}{\Delta t}\,\nabla\!\cdot\!\mathbf{u}^\*
\]
solved by masked red‑black SOR (Neumann‑like at solid boundaries).
Transient start‑up: early frames show a developing jet and evolving recirculation; a quasi‑steady pattern emerges under steady inflow.
Convergent + throat: acceleration increases |v| and deepens the pressure dip near the throat; advection dominates in the jet core.
Sudden expansion: separation and a recirculation region typically form just downstream; vorticity bands thicken as inertia increases.
Downstream: partial pressure recovery may occur, but net Δp remains positive due to viscous dissipation and separated mixing.
Pulsatile mode: strongest separation and wall shear often occur during deceleration; term maps fluctuate over the cycle.
FDA benchmark nozzle geometry (2D surrogate)
Convergent: diameter transitions from Din to Dt over Lc.
Finite throat: constant Dt over Lt.
Sudden expansion: step from Dt to Dout at d = Lc+Lt.
Downstream pipe: constant Dout over Ld.
Pressure is stored as kinematic pressure (per density; effectively \(p/\rho\) with \(\rho=1\)) and is defined up to an additive constant in incompressible flow. The “Non‑negative (min p = 0)” reference applies a constant shift for display/interpretation and does not change pressure gradients or the velocity field. This 2D laminar solver is intended for mechanistic intuition (jetting, shear, separation), not quantitative FDA validation.
Mechanisms visible in the term maps
Advection |(u·∇)u|: strongest in the throat jet and shear layers.
|∇p|: peaks near acceleration (convergent) and across the expansion region.
Viscous term |ν∇²u|: concentrates at walls and in separated shear layers.
Vorticity |ωz|: highlights recirculation and mixing downstream of expansion.
Practical stability notes
If instability is observed: increase ν, reduce Uin, or increase Ny.
For stronger post‑expansion recirculation: decrease ν and/or increase Uin.