2D Incompressible Navier–Stokes (Projection · MAC) — SVG Heatmap v0.3.2

Simulation Controls

2D CFD Heatmap (velocity / pressure / shear / NS terms)

Estimator (throat)

Model, intuition, and clinical interpretation

Governing equations (incompressible, Newtonian)

\[ \nabla\!\cdot\!\mathbf{u}=0,\qquad \frac{\partial\mathbf{u}}{\partial t}+(\mathbf{u}\!\cdot\!\nabla)\mathbf{u} = -\nabla p + \nu \nabla^2 \mathbf{u}. \]

Navier–Stokes (two useful forms)

1. Convective (non‑conservative) form \[ \nabla\!\cdot\!\mathbf{u}=0,\quad \frac{\partial\mathbf{u}}{\partial t}+(\mathbf{u}\!\cdot\!\nabla)\mathbf{u} = -\,\nabla p + \nu \nabla^2 \mathbf{u}. \]
   Observed via heatmap toggles: |(u·∇)u| (advection), |∇p| (pressure), |ν∇²u| (viscous). Advection dominates the throat jet; viscous terms and vorticity concentrate at walls and separation zones.
2. Conservative (flux/stress) form \[ \frac{\partial(\rho\mathbf{u})}{\partial t} + \nabla\!\cdot\!\big(\rho\,\mathbf{u}\otimes\mathbf{u} + p\,\mathbf{I} - \mu(\nabla\mathbf{u}+\nabla\mathbf{u}^{\!\top})\big) = \mathbf{0},\quad(\rho{=}1,\ \mu{=}\nu). \]
   Momentum fluxes (convective + pressure + viscous stresses) balance over any control volume. The measured Δp_net reflects irreversible loss from viscous dissipation and separation; large recirculation and high τ_w predict reduced pressure recovery.

Reading pressure recovery

• Green curve: centerline p_c(x). Dashed green shows cross‑section mean p̄(x). Recovery is most evident on p̄(x).
• Markers: up‑, throat‑, and down‑stream stations are highlighted on p̄(x) when the downstream reference is active.
• Reference choice: downstream mean set to zero aids interpretation of recovery along the diffuser; zero‑mean remains available.

Enhancing pressure recovery

• Lower ν: mildly reduce viscosity to delay separation and strengthen recovery.
• Longer diffuser: increase L_d relative to L_c.
• Smoother walls: use the cosine‑smooth profile.
• Moderate throat: avoid an excessively narrow minimum height.
• Moderate inflow: use a stable U_in (≈0.20–0.30).

Interpreting the four throat estimates

• Peak Doppler (e.g., 2.5): the largest value because it uses the peak jet velocity at the throat with the classic 4V² formulation and applies no corrections.
• Modified (e.g., 2.3): lower because it uses the same throat velocity but subtracts the inlet kinetic term (V_in²), reducing the computed gradient; this is not a peak-based V but an adjusted difference.
• Doppler + pressure-recovery (e.g., 2.4): a partially corrected value that applies a pressure-recovery factor based on area-ratio, giving a value below the peak Doppler but above the modified estimate.
• CFD net Δp (e.g., 2.8): typically the highest because it reflects the full Navier–Stokes net pressure drop, including viscous losses and recirculation, serving as the model’s internal reference.

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