Classical de Laval Nozzle Model — Subsonic CME (SVG Edition) v0.3.3

Inputs & Geometry (Subsonic)

1.30
Ratio of specific heats \( \gamma=c_p/c_v \) (dimensionless).
287
Specific gas constant \(R\); with γ sets \(a=\sqrt{\gamma R T}\).
500.0
Stagnation (total) pressure used by isentropic relations.
3000
Stagnation (total) temperature \(T_0\).
0.20
Inlet Mach at \(x=-L_c\) (used in Subsonic mode only).
900
Number of axial stations for the computation.

Area Function A(x) · A* at x=0

0.010
Geometric throat area at \(x=0\).
3.00
Upstream area ratio at \(x=-L_c\).
6.00
Downstream area ratio at \(x=+L_d\).
0.40
Converging‑section half‑length (upstream).
0.40
Diverging‑section half‑length (downstream).
2.00
Diffuser shape exponent in \( (x/L_d)^q \).
Subsonic quasi‑1D, inviscid, adiabatic. Back‑pressure not enforced. Sonic/Supersonic modes follow the isentropic area–Mach relation about the throat.

Pressure & Velocity vs distance d (with A(d) overlay)

Pressure (kPa) Velocity (m/s) d (m)
Pressure P(d) — solid, blue (current)
Velocity V(d) — solid, yellow (current)
Area A(d) — solid, green (current)
Previous pressure — dotted blue
Previous velocity — dotted yellow
Previous area — dotted green
Subsonic mode · Ready.

Governing relations (steady, 1‑D, inviscid, adiabatic)

  • Continuity: \( \rho V A = \dot{m} = \text{const},\quad d\ln\rho + d\ln V + d\ln A = 0.\)
  • Momentum (Euler): \( dP + \rho V\, dV = 0 \;\Rightarrow\; \frac{dP}{P} = -\gamma M^2 \frac{dV}{V}.\)
  • Energy (stagnation preserved): \( h_0 = c_p T_0 = c_p T + \frac{V^2}{2}\Rightarrow T = T_0 - \frac{V^2}{2c_p}.\)
  • Area–velocity (from C+M): \( (1 - M^2)\,\frac{dV}{V} = -\,\frac{dA}{A}, \quad M^2 = \frac{V^2}{\gamma R T}.\)
  • Isentropic gas: \( P = P_0 \Big(\frac{T}{T_0}\Big)^{\gamma/(\gamma-1)}, \;\; a=\sqrt{\gamma R T}, \;\; V = M a.\)
What to observe in the plot (shock‑free, isentropic):
  • Subnozzle (x<0, converging):
    • Subsonic — \(V\uparrow\) (accel.), \(P\downarrow\).
    • Supersonic — \(V\downarrow\) (decel.), \(P\uparrow\).
  • Supranozzle (x>0, diverging):
    • Subsonic — \(V\downarrow\) (decel.), \(P\uparrow\) (recovery).
    • Supersonic — \(V\uparrow\) (accel.), \(P\downarrow\).
  • Sonic (choked): \(M=1\) at the throat; upstream behaves subsonically, downstream follows the supersonic branch for \(A/A^*>1\).
Why Subsonic and Sonic curves may look identical under default settings:
  • When the chosen parameters \((M_{\mathrm{in}}, P_0, T_0, A_{\mathrm{in}}/A^*, A_{\mathrm{out}}/A^*)\) produce a choked condition, both Subsonic and Sonic modes follow the same unique isentropic solution: subsonic upstream, \(M=1\) at the throat, and supersonic downstream.
  • Therefore, under such default choking conditions, the pressure and velocity curves in both modes appear nearly identical in the sub- and supra-nozzle regions.
  • To visualize a truly subsonic diffuser (no choking), lower \(M_{\mathrm{in}}\) or reduce the area ratios. Example: set \(A_{\mathrm{in}}/A^*\) to about 2.1; Subsonic mode then shows \(V\uparrow,P\downarrow\) upstream and \(V\downarrow,P\uparrow\) downstream, while Sonic/Supersonic modes display a supersonic diffuser downstream \((V\uparrow,P\downarrow)\).
CME = Continuity · Momentum · Energy
The model enforces all three governing laws. The mass-flow rate (ṁ) is shown because it is the conserved quantity in the Continuity equation (\(\rho V A = \dot m\)). Momentum and Energy are verified through their residuals (RMS, Δmax), rather than displayed as standalone values.