Bernoulli–Darcy–Borda model on FDA nozzle v0.2.8

Inputs & Geometry

Fluid density ρ (kg·m⁻³)
Dynamic viscosity μ (mPa·s)	1.00 mPa·s ≡ 0.001 Pa·s
Volumetric flow Q (mL/s)

Nozzle Geometry

Inlet diameter D_in (mm)
Throat diameter D_t (mm)
Outlet diameter D_out (mm)	Gradual convergent, finite‑length throat, sudden expansion to D_out.
Convergent length L_c (mm)
Throat length L_t (mm)
Downstream length L_d (mm)
Axial samples N
Friction model

No Canvas • SVG Charts

Component toggles (live)

Include Convergent friction Distributed Darcy–Weisbach loss \(\Delta p_c=\int(½\rho V^2 f\,ds/D).\)

Include Throat friction Short straight section \(\Delta p_{\text{throat}}\approx ½\rho V_t^2 f_t(L_t/D_t).\)

Include Sudden expansion (Borda–Carnot) \(\Delta p_{\text{exp}} = ½\rho V_t^2(1-A_t/A_{\text{out}})^2.\) Removing this removes the pressure jump at \(d=L_t\).

Include Downstream friction Constant-diameter pipe \(\Delta p_d\approx ½\rho V_{\text{out}}^2 f_{\text{out}}(L_d/D_{\text{out}}).\)

Bars

Velocity Bars (m/s) — enlarged

Δp Components (Pa) — enlarged

Δp Waterfall (cumulative drop: inlet → outlet)

Axial Profiles

Green curve is A(d) rescaled to the pressure axis for overlay; flat segments correspond to constant‑diameter sections (throat and downstream). EGL is plotted as (p + ½ρV²)/1000 so it shares the pressure (kPa) axis.

Hover the chart to inspect local D, V, Re, f, and incremental loss.

Quantity	Inlet	Throat	Outlet
Area A (mm²)	—	—	—
Mean velocity V (m/s)	—	—	—
Reynolds number Re	—	—	—
Dynamic pressure ½ρV² (Pa)	—	—	—

Total Δp (Pa) —

Breakdown: —

Loss coefficient K_total = Δp / (½ρV_in²) —

Pressure‑recovery coefficient PR —

Expansion factor K_exp —

Jet breakdown estimate x_b/D_out —

Flow regime: —

Ready.

Governing relations by section

The 1-D model combines a Bernoulli-type axial energy balance, Darcy–Weisbach friction, and Borda–Carnot sudden-expansion loss on the FDA benchmark nozzle geometry.

Common relations (apply in all regions)

Continuity and kinematic definitions
\[ A(d)=\frac{\pi D(d)^2}{4},\qquad V(d)=\frac{Q}{A(d)},\qquad Re(d)=\frac{\rho\,V(d)\,D(d)}{\mu},\qquad q(d)=\tfrac12\rho V(d)^2. \]
Darcy–Weisbach friction factor and incremental loss
\[ f(d)= \begin{cases} 64/Re(d), & \text{laminar}\\[3pt] 0.3164\,Re(d)^{-1/4}, & \text{Blasius (smooth, moderate turbulent)} \end{cases} \] \[ d(\Delta p_f)=\tfrac12\rho V(d)^2\,f(d)\,\frac{ds}{D(d)}. \]

Sub-nozzle (convergent, \(-L_c\le d<0\))

1-D Bernoulli balance with Darcy–Weisbach loss
\[ \frac{dp}{ds}\approx -\rho\,V\frac{dV}{ds} \;-\; \tfrac12\rho V^2\,\frac{f}{D}, \]
Kinematic area–velocity relation
\[ \frac{dV}{ds}=-\frac{Q}{A^2}\frac{dA}{ds}. \] Static pressure decreases due to acceleration (Bernoulli term); viscous loss adds a continuous Darcy–Weisbach contribution.

Throat (finite length \(L_t\); constant \(D_t\))

Darcy–Weisbach loss in a constant-diameter throat
\[ \Delta p_{\text{throat}} \approx \tfrac12\rho V_t^2\, f_t\,\frac{L_t}{D_t}, \qquad Re_t=\frac{\rho V_t D_t}{\mu}. \] Pressure decreases smoothly; no discontinuity occurs in this region.

Sudden expansion at the throat exit (\(d=L_t\)) — Borda–Carnot

Borda–Carnot sudden-expansion loss coefficient
\[ K_{\text{exp}} = \Bigl(1 - \frac{A_t}{A_{\text{out}}}\Bigr)^2. \]
Static-pressure discontinuity (Borda–Carnot pressure jump)
\[ \Delta p_{\text{exp}} = \tfrac12 \rho V_t^2 K_{\text{exp}}, \qquad p(d_{\text{exp}}^+) = p(d_{\text{exp}}^-) - \Delta p_{\text{exp}}. \] This relation models the abrupt jet expansion from area \(A_t\) to \(A_{\text{out}}\), generating the only finite discontinuity in the 1-D formulation. The velocity \(V(d)\), cross-sectional area \(A(d)\), and energy grade line (EGL) remain continuous across the plane.

Supra-nozzle (downstream pipe, \(L_t< d\le L_t+L_d\))

Darcy–Weisbach loss in the downstream pipe
\[ \Delta p_{\text{down}} \approx \tfrac12\rho V_{\text{out}}^2\, f_{\text{out}}\,\frac{L_d}{D_{\text{out}}}. \] Static pressure recovers partially but remains below its pre-throat value due to accumulated losses.

Energy grade line (EGL)

\[ EGL(d)=p(d)+\tfrac12\rho V(d)^2. \]

Constant for ideal, inviscid flow.

Monotonically decreasing with friction and sudden-expansion loss.

Continuous across the expansion plane; the drop equals the imposed loss.

What Region-2 illustrates

Continuity: \(V_t\) largest; velocity bars reflect \(V=Q/A\).
Expansion dominance: sudden-expansion bar grows steeply as \(A_t/A_{\text{out}}\downarrow\).
Re-sensitive friction: friction rises with \(Re\); growth is sub-quadratic compared with expansion loss (∼\(V_t^2\)).
Waterfall: cumulative Δp steps show how each region contributes to the total loss.

Expected trends when modifying Region-1 inputs

Increase Q: \(V\uparrow\), \(Re\uparrow\); \(\alpha_Q\to 2\) when expansion dominates, \(\alpha_Q\to 1\) for laminar long sections.
Increase μ: \(Re\downarrow\); friction factors rise → Δp rises; \(\alpha_\mu>0\).
Decrease D_t: \(V_t\uparrow\), \(Re_t\uparrow\); expansion term strengthens; Δp grows sharply (\(\alpha_{D_t}<0\)).
Increase L_t\) or \(L_d\): total friction scales with \(L/D\).
Increase D_out: larger \(A_{\text{out}}/A_t\) → stronger Borda–Carnot loss \(\propto (1-A_t/A_{\text{out}})^2\).
Toggles: switching components reshapes Δp, relative shares, waterfall steps, and the p(d)/EGL(d) profiles.

Continuity vs. Borda–Carnot step (FDA vs. CFD / de Laval)

Continuous in smooth regions: \(A(d),V(d),p(d),EGL(d)\) vary smoothly in the convergent, throat, and downstream pipe.
Only static pressure is discontinuous: \[ p(d_{\text{exp}}^+)=p(d_{\text{exp}}^-)-\Delta p_{\text{exp}}. \] Velocity, area, and EGL remain continuous.
CFD (Navier–Stokes): resolves the expansion over a finite region → steep but continuous drop.
de Laval nozzle: smooth area, no expansion step → all fields smooth and continuous.