Clark C transvalvular pressure–loss and recovery model — Aortic stenosis fluid mechanics (SVG edition) v0.3.1

Inputs & Geometry

Clark (1976): station model, not a computed field

Clark’s approach treats a stenosed valve as a nozzle-like constriction and evaluates discrete station pressures: upstream (1), at/just distal to the stenosis (2), and after mixing/recovery (3). The educational objective is to understand how \(C_d\), area ratios, and unsteadiness (\(dQ/dt\)) shape station-to-station pressure differences and pressure recovery in pulsatile flow.

Governing relations (education-aligned to Clark’s station framework)

1. Transvalvular drop (1→2)

   \[ \Delta p_{1\to 2} \approx \frac{\rho}{2}\Big(\frac{Q}{C_d A_2}\Big)^2 \Big[1 - \Big(\frac{A_2}{A_1}\Big)^2\Big] \;+\; \rho\,\frac{dQ}{dt}\,\beta_{12}, \qquad \beta_{12}\approx \int_{1}^{2}\frac{dx}{A(x)}. \]

2. Quasi‑steady recovered pressure prediction (2→3, Borda–Carnot)

   \[ \Delta p_{2\to 3}^{(\mathrm{qs})} \approx 2\Big(\frac{A_2}{A_3}\Big)\Big(1-\frac{A_2}{A_3}\Big)\,\frac{1}{2}\rho v_2^2, \qquad v_2=\frac{Q}{A_2}. \]

   This value is displayed as the violet tick in the bar chart (prediction).

3. Non‑recoverable loss (1→3)

   \[ \Delta p_{1\to 3}^{(\mathrm{loss})} = \frac{1}{2}\rho v_2^2 \left[ \Big(\frac{A_3}{A_2}\Big)^2\Big(1-\Big(\frac{A_2}{A_1}\Big)^2\Big)\Big(\frac{1}{C_d^2}-1\Big) +\Big(\frac{A_3}{A_2}-1\Big)^2 \right]. \]

4. Effective recovery used for the green bar

   \[ \Delta p_{2\to 3}^{(\mathrm{eff})} \;=\; \Delta p_{1\to2} \;-\; \Delta p_{1\to3}^{(\mathrm{loss})}. \]

   The effective recovered pressure differs from the quasi‑steady prediction when nozzle/expansion losses are large and when the unsteady term shifts \(\Delta p_{1\to2}\).

5. Mean-data valve area estimator (Paper II)

   \[ A_2 \simeq 0.0242\,\frac{\mathrm{CO}}{\sqrt{k\,\langle\Delta p\rangle}}\quad [\mathrm{cm}^2]. \]

What to learn from the current configuration

Reading the plots

• Blue p(d): falls to station 2, then rises toward station 3 but remains at or below upstream (ends at \(-\Delta p^{(\text{loss})}_{1\to3}\)).
• Green v(d): rises from \(A_1\to A_2\), then decays with mixing toward \(A_3\).
• Bars: red = \(\Delta p_{1\to2}\); green = effective recovery \((\Delta p_{1\to2}-\text{Loss}_{1\to3})\); blue = non‑recoverable loss; violet tick = Eq.(2) prediction.

How to induce pressure recovery

• Increase A₃/A₂: make the downstream area larger than the valve area; recovery requires A₃ > A₂.
• Maintain A₁ ≫ A₂: stronger upstream contraction → stronger jet → more recoverable static pressure.
• Use moderate C_d: higher C_d leaves more Δp₁→₂ available for recovery; very low C_d suppresses recovery.
• Flow effects: larger Q strengthens both Δp₁→₂ and potential recovery; large +dQ/dt boosts Δp₁→₂ but does not enhance recovery.

Meaning of the violet tick in the pressure bars

• Violet tick = Clark Eq.(2) prediction: theoretical recovery based solely on geometry \(2(A₃/A₂-1)\) and throat velocity \(v₂\).
• Not actual recovery: it ignores separation and turbulent-expansion losses, so it may exceed the green bar.

Relationship between red, green, and blue bars

• Red ≈ Green + Blue: Δp₁→₂ ≈ Δp₂→₃(eff) + Δp₁→₃(loss).
• Blue (loss): computed directly from Clark’s loss formula using A₁,A₂,A₃ and C_d.
• Green (effective recovery): whatever remains of Δp₁→₂ after subtracting the loss.
• Why not exact? rounding, clamping, and the fact that Δp₁→₂ includes unsteady terms whereas loss uses only geometric/convective terms.