# Rocket Nozzle Divergence Loss > Use when you must compute the rocket-nozzle-divergence-loss bookkeeping from the nozzle geometry and the ideal attached-flow state: apply the conical divergence factor lambda = (1 + cos(alpha))/2 for the half-angle to the ideal axial momentum thrust, or the bell contour efficiency (0.98 typical for the 80-percent bell) in its place; grow the turbulent boundary layer over the divergent length to the displacement thickness and convert it with the 1-D mass-flux correction to the momentum loss fraction; assemble the delivered thrust from the loss-corrected momentum term and the unchanged pressure term. Produces the shape factor, the loss fraction, the effective exit area ratio, the delivered thrust and the nozzle-delivered-isp fraction of the ideal in SI units, replacing the band multiplier. Trigger: rocket-nozzle-divergence-loss, conical-nozzle-thrust-correction, bell-nozzle-efficiency, nozzle-delivered-isp, divergence loss, half-angle thrust loss. - Skill: `ashfordeou/rocket-nozzle-divergence-loss` (Agent Skill, multi-file: 3 files) - Install (CLI): `npx skillmds@latest add ashfordeou/rocket-nozzle-divergence-loss` - Raw SKILL.md: https://api.skillmd.com/api/skills/ashfordeou/rocket-nozzle-divergence-loss/raw - Safety review: pending - Works with: Claude Code, Claude.ai, OpenAI Codex - Category: Coding & Dev Tools - License: Apache-2.0 - Author: ashfordeOU (https://skillmd.com/u/ashfordeou) - Updated: 2026-09-17 - Page: https://skillmd.com/skills/ashfordeou/rocket-nozzle-divergence-loss --- # Rocket Nozzle Divergence Loss (propulsion/rocket/rocket-nozzle-divergence-loss) Use when the task is the deterministic delivered-thrust loss bookkeeping of an attached-flow rocket nozzle: how much of the ideal axial momentum thrust a conical or bell divergent contour actually delivers, what the turbulent boundary layer over the divergent wall removes, and the delivered Isp that results. The model computes the conical divergence factor lambda = (1 + cos(alpha))/2 for the half-angle, the equivalent 80-percent-length bell contour efficiency in its place, the boundary-layer displacement thickness grown over the divergent length, the 1-D mass-flux momentum loss fraction, and the combined delivered thrust and delivered Isp. It pairs with propulsion/rocket/nozzle-design, which supplies the ideal attached-flow exit state at the geometry that this leaf consumes as its loss-free baseline. ## Domain quick reference - Conical divergence factor: lambda = (1 + cos(alpha))/2 for the half-angle alpha, the axial projection of the momentum flux leaving a conical wall. lambda(0 deg) tends to 1, lambda(60 deg) = 0.75 exactly, lambda(15 deg) = 0.982963. - Bell equivalence: the parabolic-contour bell at about 80 percent of the equivalent 15-degree conical length carries the contour efficiency eta_bell (0.98 typical, allowed as a parameter); the contour-only comparison with the cone it replaces is eta_bell / lambda(alpha), exactly 1.0 when eta_bell = lambda(15 deg). - Turbulent growth over the divergent length L (flat plate, edge conditions at the exit): Re_L = rho_e * ve * L / mu, delta = 0.37 * L * Re_L**(-0.2), delta* = delta/8 and theta = (7/72) * delta = (7/9) * delta* for the 1/7-power profile. - Displaced-core area: the wall layer shrinks the inviscid core, so eps_eff = eps * (1 - delta*/r_exit)**2 (internal flow); the throat boundary layer is neglected and the external-flow plus-sign displaced-wall form is not used. - 1-D mass-flux correction: the layer removes the momentum flux through the exit annulus of momentum thickness theta, so the fractional loss of the momentum term is 2*theta/r_exit = (14/9)*delta*/r_exit to first order in delta*/r_exit; the pressure term (pe - pa)*Ae acts over the full geometric exit area and is unchanged. - Delivered thrust and Isp: F_del = shape_factor * (1 - xi_bl) * mdot * ve + (pe - pa) * Ae and Isp_del = F_del / (mdot * g0), g0 = 9.80665 m/s^2, with shape_factor the conical lambda or the bell contour efficiency. At shape 1 and xi_bl = 0 the chain reproduces the ideal thrust exactly (the no-loss identity). - Units are SI throughout: N, kg/s, m/s, Pa, m^2, s. - ECSS frames the space-propulsion context; the relations above are standard engineering methodology, summary-only. The claim does not extend past attached flow: off-design overexpansion to the separation limit invalidates the chain, and gamma and R are caller inputs from the hot-product state. ## Workflow 1. Collect the geometry and the ideal attached-flow state: the half-angle alpha or the bell length fraction, the divergent length L (for a conical nozzle, L = (r_exit - r_throat)/tan(alpha)), the exit radius r_exit, and the ideal state mdot, ve, pe and Ae from nozzle-design at the same area ratio. Attached flow at the operating ambient is assumed. 2. Choose the shape factor: conical_divergence_factor for a conical half-angle, or bell_contour_efficiency for the 80-percent bell contour in its place. 3. Judge the bell-to-conical contour comparison with bell_relative_to_conical and note whether the shorter bell contour closes the gap to the cone it replaces. 4. Grow the turbulent boundary layer over the divergent length with turbulent_displacement_thickness(length_m, vel_ms, rho_kgm3, mu_pas) to the exit displacement thickness. 5. Convert the displacement thickness with boundary_layer_loss_fraction to the momentum loss fraction xi_bl and with effective_exit_area_ratio to the displaced-core effective exit area ratio eps_eff. 6. Assemble the delivered thrust with delivered_thrust(mdot, ve, pe, pa, Ae, shape_factor, xi_bl) and the delivered Isp with delivered_isp(...); the pressure term keeps the full geometric exit area. 7. Read off the delivered fraction of the ideal (F_del over the ideal thrust at the same geometry) and the bell-versus-conical verdict: the 80-percent-length bell delivers within about 0.1 percent of the full-length 15-degree conical, so its benefit is the 20 percent length and mass saving, not an Isp gain. 8. Confirm the deterministic checks with the contract test scripts/test_rocket_nozzle_divergence_loss.py. ## Worked example Representative LOX/RP-1 upper-stage nozzle: pc = 7.0 MPa, Tc = 3672 K, Mw = 22.1 (R = 376.220 J/(kg K)), gamma = 1.24, c-star = 1791.2 m/s, area ratio Ae/At = 70, throat radius 0.150 m, hot-product viscosity mu = 8.0e-5 Pa s at the exit. The ideal attached-flow state at the geometry (nozzle-design domain) is Me = 4.931384, pe = 6036.738 Pa, rho_e = 0.017122 kg/m^3, ve = 3260.673 m/s, mdot = 276.24 kg/s, r_exit = 1.2550 m, ideal vacuum thrust F_id = 930598.1 N and ideal vacuum Isp = 343.522 s. Module outputs with these inputs are real: - Conical, half-angle 15 deg: lambda = 0.982963 over the divergent length L = (r_exit - r_throat)/tan(15 deg) = 4.124 m. Turbulent growth at Re_L = 2.879e6 gives delta* = 0.00974 m and the 1-D mass-flux loss xi_bl = (14/9)*delta*/r_exit = 0.01207 (1.21 percent of the momentum term); the displaced core eps_eff = 68.9176 (from 70). Delivered vacuum thrust F_del = 904562.3 N, delivered Isp = 333.911 s, fraction of the geometry ideal = 0.97202, inside the plausibility band [0.93, 0.98]. The prep anchor (full-precision unrounded ideal state) reports F_del = 904558.9 N, 3.4 N lower: the rounding of the printed mdot input, a 4e-6 relative difference. - Equivalent bell (GVC 80-percent convention): Rao-class parabolic contour at L = 0.8 * 4.124 = 3.299 m with contour efficiency eta_bell = 0.98. The shorter wall grows delta* = 0.00815 m, xi_bl = 0.01010, eps_eff = 69.0940. Delivered thrust F_del = 903668.2 N, delivered Isp = 333.581 s, fraction = 0.97106 (anchor 903664.7 N, same 4e-6 input-rounding band). - Bell/conical delivered ratio = 0.99901: the 80-percent-length bell delivers within 0.10 percent of the full-length 15-degree conical (contour-only ratio 0.996986), the design verdict that the bell's benefit is its 20 percent length and mass saving, not an Isp gain. - Half-angle sensitivity at fixed area ratio 70 (conical): the delivered fraction peaks near 10 deg where the shallow cone's longer wall starts to cost more boundary layer than the divergence saves: 8 deg 0.97580, 10 deg 0.97644, 12 deg 0.97551, 15 deg 0.97202, 20 deg 0.96194, 25 deg 0.94751, 28 deg 0.93699, 30 deg 0.92926. The typical design band 12-25 deg holds inside [0.93, 0.98]; only the steep 30 deg cone dips below the band. - Read-off: this 70:1 nozzle loses about 2.8 percent of its ideal vacuum Isp at the geometry to divergence plus boundary layer (15-deg conical), or 2.9 percent with the shorter 80-percent bell; the nozzle-side share of the cea-rocket-combustion "80 to 95 percent of ideal" band is about 0.97, with the rest of the band owned by combustion (c-star) and finite-expansion accounting in the sibling leaves. ## Verification - Confirm conical_divergence_factor(15.0) = 0.982963 within 1e-5 and conical_divergence_factor(60.0) = 0.75 exactly; lambda falls monotonically from 8 to 30 deg. - Confirm bell_contour_efficiency() = 0.98, bell_relative_to_conical() = 0.996986 within 1e-5, and the ratio is 1.0 within 1e-9 when eta_bell equals lambda(15 deg). - Confirm the no-loss identity: delivered_thrust with shape 1 and xi 0 reproduces mdot*ve + (pe - pa)*Ae with residual below 1e-6 N, and delivered_isp at pe = pa, shape 1, xi 0 equals ve/g0 = 332.496090 s. - Confirm the worked-example chain returns F_del = 904562.3 N (conical 15 deg) and 903668.2 N (80-percent bell, 0.98 contour) within 1 N, delivered Isp 333.911 s and 333.581 s within 0.05 s, fractions 0.97202 and 0.97106 within 1e-4 and inside [0.93, 0.98], and the bell/conical ratio 0.99901 within 1e-4. - Confirm the deterministic checks of step 8 and the ValueError rejections of non-physical inputs: half-angle at or beyond the (0, 90) deg limits, contour efficiency outside (0, 1], displacement thickness at zero or at the exit radius, area ratio at or below 1, non-positive mass flow, velocity, area or g0, shape factor outside (0, 1] and loss fraction outside [0, 1). - Run the contract test offline: python3 scripts/test_rocket_nozzle_divergence_loss.py (30 tests, deterministic). ## Pitfalls - Reading the anchor thrust at full precision: the prep anchor computes the ideal state at full internal precision, so its 904558.9 N sits 3.4 N below the module output fed the spec's rounded inputs (904562.3 N). The 4e-6 relative difference is input rounding, not model error; the fractions, Isp, thicknesses and area ratios agree to the printed digits. - Reporting the finite-expansion shortfall as a nozzle loss: the gap between the geometry ideal (343.522 s) and the fully expanded ceiling (385.276 s) is the area-ratio verdict of nozzle-design, not divergence or boundary-layer loss. - Applying the chain past the separation limit: when the exit area ratio exceeds the separation-station area ratio at the operating ambient, the wall flow detaches and rocket-nozzle-flow-separation's separated-thrust loss replaces these attached-flow corrections. - Using the plus-sign displaced-wall form: the effective area ratio shrinks the core (eps_eff = eps * (1 - delta*/r_exit)**2) for this internal flow; external-flow displacement conventions do not apply. - Double-counting the cea band: cea-rocket-combustion's isp_with_efficiency remains the c-star and user-supplied quick-look band tool; this leaf supplies only the computed divergence and boundary-layer share (about 0.97 of the ideal at the worked example), so applying both to the same ideal Isp double counts the nozzle side. ## Related leaves - propulsion/rocket/nozzle-design: the ideal attached-flow sizing (exit Mach, mass flow, exit velocity, static pressure, ideal thrust at the area ratio) that feeds this leaf its loss-free baseline. - propulsion/rocket/rocket-nozzle-flow-separation: the off-design overexpanded boundary that takes over once the wall flow separates. - propulsion/combustion/cea-rocket-combustion: the ideal Isp ceiling and the c-star side of the efficiency band this leaf computes the nozzle share of. - propulsion/rocket/rocket-gravity-loss: a different loss class (trajectory), kept separate from the nozzle-side bookkeeping. - propulsion/electric/hall-thruster: ion-optics divergence in its electric context only, no gasdynamic overlap. ## Behavior contract (gate 3) Run the deterministic contract test (stdlib unittest, offline): python3 scripts/test_rocket_nozzle_divergence_loss.py The test covers the module constants and 1/7-power profile shape, the conical divergence factor at the reference half-angles and its monotonic fall, the bell contour efficiency and its validation, the bell-to-conical ratio and its unit-ratio identity, the turbulent displacement thickness worked value and growth scaling, the momentum loss fraction and the theta profile identity, the displaced-core effective exit area ratio and its limits, the no-loss thrust and Isp identities, the full worked-example delivered chains for the 15-degree conical and the 80-percent bell, the bell/conical delivered-ratio verdict, the half-angle sensitivity peak and design band, and the ValueError rejection of every non-physical input class across the module, plus import purity and determinism. ## Compliance - Standards referenced, not reproduced: ECSS (ecss) frames the space propulsion context per standards-map.yaml; the divergence, boundary-layer and delivered-thrust relations above are standard engineering methodology, summary-only. - compliance: STANDARDS-REF, gated: false. The model never re-derives the ideal exit Mach or mass flow: those are nozzle-design inputs.