NACA 2412 Airfoil — CFD Analysis Report

01 Executive Summary

Steady incompressible RANS simulations of a NACA 2412 airfoil were performed using OpenFOAM's simpleFoam solver across four angles of attack (α = 0°, 15°, 30°, 45°). This report documents the pressure and velocity field topology, surface pressure coefficient distributions, and flow physics at each condition — from clean attached flow through leading-edge stall to the bluff-body dominated regime.

02 NACA 2412 Airfoil Geometry

The NACA 4-digit naming convention encodes the airfoil's key geometric parameters directly in its designation.

Geometric Equations

The trailing-edge half-angle of NACA 2412 is approximately 15.6°, which makes it susceptible to trailing-edge boundary-layer thickening at high AoA before full leading-edge stall develops. This is characteristic of moderate-thickness airfoils: they exhibit combined trailing-edge/leading-edge stall rather than the abrupt "thin-airfoil stall" seen in highly cambered, thin sections.

03 CFD Methodology

Governing Equations — Steady RANS

simpleFoam solves the Reynolds-Averaged Navier–Stokes (RANS) equations under the incompressible assumption (Ma < 0.3, density constant). The Reynolds decomposition splits the instantaneous velocity u into a time-mean U and a fluctuation u', yielding closure problem solved by a two-equation turbulence model.

SIMPLE Algorithm

SIMPLE (Semi-Implicit Method for Pressure-Linked Equations) iteratively couples the pressure and velocity fields. At each iteration:

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  Momentum predictor: Solve the momentum equation for an intermediate velocity field U* using the current pressure field p.
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  Pressure correction: Enforce continuity (∇·U=0) by solving a Poisson equation for a pressure correction p'. This updates p → p + α_pp' where α_p is the pressure relaxation factor (typically 0.3).
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  Velocity correction: Correct U* using ∇p' to satisfy continuity. Under-relaxation factor α_U ≈ 0.7 stabilises convergence.
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  Turbulence quantities: Solve transport equations for k (turbulent kinetic energy) and ω (specific dissipation rate) with the k-ω SST model.

k-ω SST Turbulence Model

The k-ω SST model is the industry standard for external aerodynamics because its Shear Stress Transport formulation (Menter, 1994) limits μ_t in adverse-pressure-gradient flows via the Bradshaw constraint, preventing premature separation compared to plain k-ε. However, it still tends to under-predict separation extent at high AoA — a known limitation for post-stall RANS.

Mesh Strategy — blockMesh O-Topology

The mesh was generated by aerofoil2DBlockMeshGen.py followed by blockMesh. The airfoil is physically rotated in the mesh for each AoA (the inflow direction remains along the +x axis). This approach avoids skewed inlet/outlet patches but requires re-meshing for each AoA, as visible in the ParaView visualisations where the airfoil appears tilted in the domain.

Key Simulation Parameters

04 Case: α = 0° — Attached Subcritical Flow

The baseline case establishes the attached-flow behaviour of the cambered airfoil. Despite zero geometric angle of attack, the 2% camber of NACA 2412 (α_L0 ≈ −2.1°) produces measurable lift through an asymmetric circulation field.

0°

Fully Attached Boundary Layer

Leading-edge stagnation at x/c = 0 · Smooth suction bubble · No separation

Attached Flow

Pressure (p) · −2.7e+03 → 4.9e+03 m²/s² · Streamlines overlaid

|U| Magnitude · 0 → 120 m/s · Streamlines overlaid

▶ P Convergence Animation ▶ U Convergence Animation

Pressure Field Analysis — P_0.png

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  Stagnation Point (red spot, p ≈ +4 908 m²/s², C_p = 1.0): The flow comes to rest at the leading-edge nose exactly at x/c = 0 for this AoA=0° case. By Bernoulli's theorem, the full dynamic head q_∞ converts to static pressure: p_stag = p_∞ + ½ρU_∞² = q_∞. This is the maximum pressure on the entire airfoil surface.
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  Upper Surface Suction Envelope (blue, p ≈ −2 734 m²/s², C_p,min ≈ −0.557): Flow accelerates sharply around the convex upper surface, governed by the Venturi effect. The suction peak occurs at x/c ≈ 0.22, which is forward of the geometric maximum-thickness location (x/c ≈ 0.30) — consistent with inviscid panel-method predictions. A negative C_p means local static pressure has fallen below freestream: this is the primary lift-generating mechanism for the upper surface.
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  Lower Surface Mild Suction (light-green/cyan, C_p ≈ −0.36 max): Even the lower surface of the cambered airfoil experiences suction in the first ~30% chord, because the camber-induced circulation accelerates flow on both surfaces near the leading edge. The upper surface suction is always stronger, producing net upward lift at α = 0°.
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  Trailing-Edge Pressure Recovery: Pressure increases monotonically from C_p,min ≈ −0.557 back toward C_p ≈ 0.136 at the trailing edge on the upper surface. This adverse pressure gradient (∂p/∂x > 0) decelerates the boundary layer. At α = 0° the gradient is moderate and the turbulent BL remains attached throughout.
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  Far-Field (green, p ≈ 0): Freestream pressure equals the reference (zero kinematic gauge). The pressure disturbance decays as ~1/r away from the airfoil — this is the elliptic character of the incompressible pressure field (governed by the Laplacian, not a wave equation).

Velocity Field Analysis — U_0.png

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  Stagnation Point (blue, U = 0): Precisely at the nose where all kinetic energy has converted to pressure. The thin blue sliver at the LE is the actual stagnation streamline. Streamlines bifurcate here: the dividing streamline separates flow going over the upper surface from flow going under.
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  Upper Surface Acceleration (yellow/green, U ≈ 110–120 m/s): The velocity magnitude exceeds freestream by ~20–30% at the suction peak. By inviscid Bernoulli, the local C_p = 1 − (U_local/U_∞)² ≈ 1 − (1.25)² ≈ −0.56, which exactly matches the pressure plot — confirming numerical consistency between the p and U fields.
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  Thin Boundary Layer: The abrupt colour change from yellow (high velocity) to dark (low velocity) immediately adjacent to the wall is the viscous boundary layer. At Re ≈ 6.8×10⁶ the turbulent BL thickness at mid-chord is approximately δ ≈ 0.37 c / Re_x^0.2 ≈ 0.003 m — barely visible at this zoom level but correctly resolved by the near-wall mesh.
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  Narrow Wake (red region trailing the TE): The velocity deficit behind the airfoil marks the momentum-deficit wake. Its width and depth encode the profile drag. For this attached-flow case the wake is thin and symmetric — characteristic of a low-drag configuration.

05 Surface C_p Distribution — α = 0°

Quantitative surface pressure data was extracted from postProcessing/airfoilSample/100/airfoil.xy (converged at iteration 100 for the α = 0° case). Cp = p_surface / q_∞ where q_∞ = 4 908 m²/s².

Physical Interpretation of the C_p Curve

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  Upper Surface Leading-Edge Suction Peak (C_p,min = −0.557 at x/c = 0.22): This peak arises because the camber line forces flow to accelerate rapidly over the convex forward upper surface. The location at x/c = 0.22 (not at the very nose x/c = 0) indicates a front-loaded pressure distribution — the suction starts just aft of the nose and the peak occurs before maximum thickness. This is benign at α = 0°, but as α increases, this same peak intensifies and the leading-edge radius becomes insufficient to maintain attached flow, ultimately causing stall.
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  Lower Surface Anomaly at x/c ≈ 0.05: The lower surface shows negative C_p (suction) beginning at x/c ≈ 0.022, with a local minimum of −0.363 at x/c ≈ 0.046. This is physically correct: camber induces a small leading-edge suction on the lower surface because the streamlines must curve concavely near the lower leading edge. The effect disappears by x/c ≈ 0.70 where the lower surface transitions to mild positive pressure.
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  Pressure Recovery: Both surfaces recover to positive C_p near the trailing edge, with C_p,TE ≈ 0.136 on the upper and ≈ 0.152 on the lower. A perfect airfoil would have C_p,TE = 1.0 (full stagnation recovery), but viscous dissipation in the boundary layer means only partial recovery is achieved. The trailing-edge pressure difference (C_p upper ≠ C_p lower at TE) indicates the Kutta condition is satisfied — circulation is present.

06 Case: α = 15° — Pre-Stall / Leading-Edge Suction Peak

At 15°, NACA 2412 is operating near its maximum lift coefficient. The flow is still nominally attached but the leading-edge suction peak has intensified dramatically. This is the most structurally efficient condition — high C_L with moderate drag.

15°

Strong Leading-Edge Suction · Incipient Separation

C_p,min ≈ −3.6 · Stagnation displaced to lower LE · Elongated wake

Near-Stall

Pressure (p) · −1.9e+04 → 5.3e+03 m²/s² · Note expanded colorbar scale

|U| Magnitude · 0 → 140 m/s · Streamlines overlaid

▶ P Convergence Animation ▶ U Convergence Animation

Pressure Field Analysis — P_15.png

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  Massive Upper-Surface Suction (deep purple, p ≈ −19 000 m²/s², C_p ≈ −3.58): At 15° AoA, flow must negotiate a much sharper effective turn around the leading edge. The centripetal acceleration requirement (U²/R for the curved streamlines) forces a dramatic pressure drop. C_p ≈ −3.6 implies local velocity U_local = U_∞√(1 − C_p) = 99 × √4.58 ≈ 212 m/s — more than twice the freestream. This intense suction is the engine of high lift but also the precursor to stall: any further AoA increase makes the local adverse pressure gradient too steep for the boundary layer to overcome.
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  Stagnation Point Displaced to Lower Leading Edge (red dot on lower surface): At α = 15°, the geometric leading edge is no longer the aerodynamic stagnation point. The incoming flow, angled at 15° to the chord, impacts the lower-leading-edge surface. The stagnation pressure C_p = 1.0 is now at a point ~15–20% arc-length below the nose tip. This displacement is directly related to the circulation Γ around the airfoil: larger Γ → stagnation moves further toward the pressure side.
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  Pressure Differential Across Airfoil (~24 300 Pa): The difference between max lower-surface pressure (+5 300 m²/s²) and max upper-surface suction (−19 000 m²/s²) is ≈ 24 300 m²/s² kinematic — equivalent to ~29 800 Pa when multiplied by ρ = 1.225 kg/m³. This enormous pressure difference acts over the chord length to produce the section lift force.
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  Tight Streamline Spacing Over Upper Surface: The streamlines are tightly packed over the upper surface, indicating the flow tube has contracted (velocity increased). By the stream-tube area–velocity relationship (Av = const), the local velocity is inversely proportional to the tube cross-section. This confirms the high upper-surface velocities implied by the strong suction.

Velocity Field Analysis — U_15.png

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  Stagnation Region (blue/dark, U ≈ 0) on Lower Leading Edge: Clearly shifted from the nose tip down to the lower surface. The upstream stagnation streamline approaches the lower LE at a 15° angle to the chord.
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  Very High Velocity on Upper Surface (bright yellow/green, U > 140 m/s): The local-to-freestream velocity ratio of ~1.4 corresponds exactly to C_p = 1 − 1.4² ≈ −0.96 by inviscid Bernoulli. The actual C_p,min ≈ −3.6 (from pressure plot) implies peak velocity ~212 m/s, which would appear as bright yellow/white if within the colorbar scale — likely clipped in the display.
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  Elongated Momentum-Deficit Wake: Compared to the thin wake at α = 0°, the 15° wake shows significantly more entrainment and downward deflection (downwash). The wake angle is consistent with the induced downwash angle ε ≈ C_L/(π AR) for finite wings, but in this 2-D case the "wake" represents the profile drag contribution only.
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  Possible Trailing-Edge Separation: The velocity field near the trailing edge shows signs of boundary-layer thickening or incipient separation — the velocity does not recover as smoothly as in the α = 0° case. This trailing-edge separation bubble, if present, would manifest as a recirculation region of very low (or slightly negative) velocity just upstream of the TE. Steady RANS may smear this into a thickened wake rather than capturing discrete separation.

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Proximity to Stall: At α ≈ 15°, NACA 2412 is within 1–3° of the maximum lift angle (α_stall ≈ 16–18° depending on Re and surface roughness). The steep leading-edge suction peak (C_p,min ≈ −3.6) creates an extreme adverse pressure gradient on the upper surface immediately aft of the peak. Any further increase in α intensifies this gradient until the turbulent boundary layer separates from the LE, causing an abrupt C_L drop — the stall event.

07 Case: α = 30° — Deep Stall / Massive Separation

At 30°, the airfoil is deep into the post-stall regime. Leading-edge separation is fully established; the upper-surface boundary layer has detached from the leading edge and the separated shear layer rolls into large-scale wake vortices. Aerodynamic efficiency L/D collapses.

30°

Leading-Edge Separation · Large-Scale Wake Vortex

Bluff-body-like wake · L/D degraded · Steady RANS approaching solver limits

Deep Stall

Pressure (p) · Strong bi-polar distribution · Wake vortex structure visible

|U| Magnitude · 0 → ~135 m/s · Large recirculation visible

▶ P Convergence Animation ▶ U Convergence Animation

Pressure Field Analysis — P_30.png

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  Circular Streamline Contours in Wake (Standing Vortex): The most striking feature is the large region of circular, concentric pressure isobars immediately downstream of the trailing edge. This is the pressure signature of a standing recirculation vortex — the steady-RANS mean-flow approximation of what is physically a shed vortex that would oscillate unsteadily. The core of this vortex is a low-pressure region (blue/purple), consistent with the centrifugal balance requirement inside a vortex: p_core − p_∞ = −ρΓ²/(4πr)².
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  Uniform Low-Pressure Separated Region on Upper Surface: Unlike the sharp suction peak at α = 15°, the upper surface now shows a broad, nearly uniform low-pressure plateau. This is the hallmark of a separated dead-air region — flow has detached from the leading edge and forms a recirculation bubble over essentially the entire upper surface. The pressure inside this separated zone approximates the base pressure C_p,base ≈ −0.8 to −1.2 (bluff-body coefficient).
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  High Positive Pressure on Lower Surface: The lower surface continues to see strong stagnation-like pressure (red/orange), as the geometry now presents a large projected frontal area to the incoming flow. The force balance shifts: the dominant lift mechanism is now the high pressure on the windward (lower) surface rather than upper-surface suction — analogous to flat-plate aerodynamics.
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  Stagnation Point Far into Lower Surface: At 30° AoA the geometric stagnation point has moved well past the leading edge onto the lower surface. The arc-length from the suction side leading edge to the stagnation point roughly equals sin(30°) × (chord/2) by geometry.

Velocity Field Analysis — U_30.png

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  Large Recirculation Zone Behind Trailing Edge: The velocity field shows a large swirling structure with a clockwise sense of rotation visible through the streamlines. Inside this vortex, local velocities can be both positive (entrained freestream) and reversed (recirculating). The steady-RANS solver captures the mean position of this structure as a quasi-steady feature.
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  Separated Shear Layer from Leading Edge: The leading edge acts as a separation line. The shear layer (high velocity gradient) extends from the LE, arcing over the separated bubble and reattaching to the freestream flow. This shear layer is Kelvin–Helmholtz unstable and physically rolls into shed vortices at a Strouhal number St ≈ 0.15–0.2 — but the steady solver freezes this into the mean recirculation.
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  Effective Blockage and Downward Deflection: The streamlines in the freestream are significantly deflected upward approaching the airfoil and downward in the wake. The net downward momentum imparted to the flow equals the lift force by Newton's 3rd law — and at 30° AoA, this downward deflection is still present despite deep stall.

✕

Flight Safety Note: The 30° AoA condition represents a deep stall — a potentially unrecoverable state in certain aircraft configurations (particularly T-tail aircraft where the horizontal stabiliser enters the wing's wake, eliminating pitch-down authority). The L/D ratio at this condition is typically 0.5–2.0 compared to a design L/D of 15–20 in cruise. The severe drag increase comes from the massive separated wake replacing attached boundary-layer flow.

08 Case: α = 45° — Bluff-Body Dominated Flow

At 45°, the airfoil behaves aerodynamically as a bluff body. Vortex shedding dominates — the steady RANS solver cannot fully converge because the physics are inherently time-periodic (Kármán-like vortex street). The RANS solution represents an unstable mean-flow approximation of the actual time-averaged field.

45°

Bluff-Body Shedding · RANS Limit · Vortex Street

Alternating vortex structures · Steady solver oscillating · Strouhal shedding

Bluff Body

Pressure (p) · Near-field · Alternating vortex pressure signatures

|U| Magnitude · Near-field · Complex multi-vortex wake

Extended domain views:

Pressure (p) · −9.7e+03 → 5.3e+03 m²/s² · Far-field view

|U| Magnitude · Far-field · Mean separated wake extent

▶ P_45 Convergence Animation ▶ U_45 Convergence Animation ▶ P_45 Far-field Animation ▶ U_45 Far-field Animation

Pressure Field Analysis — P_45.png (Near-field)

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  Alternating Pressure "Blobs" in Wake (Vortex Street Signature): The near-field pressure plot shows alternating high and low pressure regions arranged in a staggered pattern downstream of the airfoil. This is the pressure signature of a von Kármán vortex street — alternately shed vortices of opposite sign that trail downstream. In reality, these vortices are shed periodically at a Strouhal number St ≈ f·t/U_∞ ≈ 0.15 (where t is the projected thickness). The steady RANS solution captures a frozen snapshot of this oscillating pattern, yielding poor convergence because there is no true steady state.
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  Symmetric Pressure Distribution on Leading Faces (P_45_extra.png): The extended-domain view in P_45_extra reveals a cleaner far-field structure: a strong positive pressure region (red) on the face of the airfoil presented to the flow (lower/front at 45°) and a deep suction region (dark blue) immediately behind. This dipole structure is characteristic of bluff bodies and is essentially the same mechanism as a flat plate normal to flow — the airfoil's streamlined shape no longer helps at this extreme angle.
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  Base Pressure Region (deep blue, C_p,base ≈ −1.0 to −2.0): The base pressure on the leeward (upper at 45°) face of the airfoil is approximately C_p,base = (p_base − p_∞)/q_∞ ≈ −1.0 to −2.0. For comparison, a circular cylinder has C_p,base ≈ −1.2 at high Re. The pressure drag is dominated by the high-pressure front minus the low-pressure base: C_D,pressure ≈ |C_p,front| + |C_p,base| acting over the projected area.

Velocity Field Analysis — U_45.png & U_45_extra.png

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  Multi-Vortex Wake Structure (U_45.png): The near-field velocity plot shows two dominant counter-rotating vortical structures behind the airfoil, with a complex shear-layer interaction. In the physical (unsteady) flow, these would alternate in dominance at the shedding frequency, producing the characteristic oscillating drag and lift that drives structural fatigue in engineering applications.
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  Broad Velocity-Deficit Wake (U_45_extra.png): The extended view shows the wake width growing as ~√x downstream due to turbulent diffusion. The wake width at one chord-length downstream is already comparable to the airfoil chord itself — confirming massive blockage effect. The freestream is significantly deflected around this thick wake.
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  High Velocity Strips at Leading and Trailing Edges: The flow accelerates sharply around the two corners of the 45°-inclined airfoil (the leading-edge tip and trailing-edge tip), similar to the corner flow around a tilted flat plate. These high-velocity strips carry little energy compared to the separated-wake loss.

⚙

Solver Limitation at α = 45°: SimpleFoam solves the steady RANS equations. When the flow is inherently time-periodic (vortex shedding), the steady solver oscillates between different snapshot states and cannot converge. The forces.dat convergence history confirms this — large-amplitude oscillations persist beyond iteration 500.

Recommendation: For α ≥ 25–30°, use pimpleFoam (transient RANS with PIMPLE algorithm) with time steps satisfying the Courant condition Co = U·Δt/Δx < 1. Alternatively, Detached Eddy Simulation (DES) or wall-resolved LES would better capture the separated-wake dynamics, though at significantly higher cost.

09 Force Convergence History

The wingForces/0/forces.dat file records the integrated aerodynamic forces at each solver iteration for the final simulation run. The data reveals the convergence (or lack thereof) of the RANS solution.

Convergence Interpretation

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  Iterations 0–30 (Initial Transient — excluded from chart): The solver starts from initial conditions (U = U_∞, p = 0 everywhere). The first few iterations produce huge force spikes (Fy reaching ~140 000 N at iteration 5) as the pressure field adjusts from the uniform initial condition to the actual aerodynamic state. These are numerical transients, not physical values.
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  Iterations 30–200 (Settling Phase): Forces decrease rapidly from the initial spike and settle into an oscillatory pattern. This represents the solver finding the approximate mean separated-flow state.
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  Iterations 200–500 (Persistent Oscillation): Large-amplitude oscillations (Fy varying between ~4 800 N and ~13 100 N) confirm that no steady state exists. For a separated flow, each SIMPLE iteration effectively "time-marches" the flow through different vortex configurations, with the pressure field unable to find a fixed point.
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  Convergence Criterion: A well-converged simpleFoam run shows force residuals less than 1% of the mean value. Here the peak-to-peak amplitude is ~50–100% of the mean — orders of magnitude above acceptance. This should trigger a switch to pimpleFoam for proper time-accurate simulation.

10 Comparative Aerodynamic Analysis

Summary of flow topology evolution across the four angle-of-attack cases.

Stall Mechanism Progression

Physical Mechanisms: Bernoulli, Circulation and the RANS Closure Problem

The four cases illustrate the interplay between three key mechanisms:

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  Circulatory Lift (Kutta–Joukowski): L = ρ U_∞ Γ per unit span, where Γ is the bound circulation. At α = 0° and 15°, Γ is well-defined and the Kutta condition (smooth trailing-edge departure) is satisfied. At α = 30° and 45°, the concept of a single bound circulation breaks down — the flow separates and Γ is not conserved as a single vortex.
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  Adverse Pressure Gradient (APG) and Boundary Layer Separation: The BL integral momentum equation (von Kármán) shows separation occurs when the shape factor H = δ*/θ exceeds ~2.4–2.6 for turbulent flow. As AoA increases, the suction peak intensifies and the APG steepens, progressively increasing H until the BL separates. The RANS k-ω SST model moderately delays this (over-predicts attached-flow extent) — compare to experiment or DES for accurate stall prediction.
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  Turbulence Modelling Limitation: All RANS models solve for the mean flow and model turbulent fluctuations via μ_t. In massively separated regions, the eddy-viscosity concept breaks down — the turbulent stress tensor is no longer aligned with the mean strain rate (Boussinesq hypothesis failure). This is why RANS systematically predicts different (often smaller) separation zones than DES or LES, which resolve the large eddies directly.

Effect of AoA on Colorbar Scaling

Notice that the ParaView colorbars use different scales for each AoA case:

The expanding colorbar range at higher AoA reflects the intensifying pressure gradients as more flow energy is redirected into circulation and ultimately lost to the separated wake. The asymmetry between min and max (min always larger in magnitude) shows that suction forces are always stronger than stagnation overpressure — this is a fundamental characteristic of low-speed aerodynamics where the suction side does more aerodynamic work than the pressure side.

11 Conclusions & Recommendations

Summary of Findings

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  α = 0°: Fully attached flow with textbook RANS behaviour. C_p,min = −0.557 on the upper surface at x/c = 0.22. The CFD Cp curve from airfoilSample data is physically consistent with a moderately cambered airfoil at zero incidence. Steady simpleFoam is the correct tool here.
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  α = 15°: Strong leading-edge suction (C_p,min ≈ −3.6) characteristic of pre-stall. Stagnation point displaced to lower leading edge. This is near C_L,max for NACA 2412. Steady RANS gives a reasonable mean-flow picture but may under-predict trailing-edge separation extent.
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  α = 30°: Deep stall with leading-edge separation, large standing wake vortex, and essentially bluff-body pressure distribution on the upper surface. The standing-vortex pressure structure visible in P_30.png is the RANS-averaged representation of physically shed vortices. Steady RANS gives marginal convergence; results should be treated as qualitative only.
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  α = 45°: Fully bluff-body regime with Kármán-type vortex shedding. The steady solver oscillates without converging, as demonstrated by the forces.dat history. The P_45_extra and U_45_extra views provide the clearest picture of the mean flow structure. Only time-accurate methods (pimpleFoam, DES, LES) can properly resolve this condition.

Recommendations for Future Work