Fact-checked by Grok 2 weeks ago

Lift coefficient

The lift coefficient, denoted as C_L, is a dimensionless quantity in aerodynamics that quantifies the lift force generated by an airfoil, wing, or aircraft relative to the dynamic pressure of the surrounding airflow and a reference area, such as the wing planform area.^[1] It serves as a key parameter in the lift equation, L = C_L \cdot \frac{1}{2} \rho V^2 S, where L is the lift force, \rho is the fluid density, V is the freestream velocity, and S is the reference area.^[1] This coefficient encapsulates the complex effects of geometry and flow conditions, allowing engineers to predict aerodynamic performance without resolving every detail of the flow field.^[2] The value of C_L primarily depends on the airfoil shape, angle of attack (the angle between the oncoming flow and the chord line), Reynolds number (characterizing viscous effects), and Mach number (indicating compressibility).^[3] For thin airfoils at low subsonic speeds, C_L approximates a linear relationship with angle of attack: C_L \approx 2\pi \alpha, where \alpha is in radians, yielding values up to about 1.0 before nonlinear effects dominate.^[4] Stall occurs when C_L reaches a maximum, typically 1.5 to 1.7 for clean subsonic wings at angles of 12° to 16°, beyond which flow separation reduces lift sharply.^[3] In cruise conditions for conventional aircraft, C_L is lower, often 0.3 to 0.5, balancing lift against drag for efficiency.^[5] Introduced in early 20th-century aerodynamics through experimental work by pioneers like Otto Lilienthal and the Wright brothers, who employed empirical forms of the lift equation, C_L evolved into its modern dimensionless standard via dimensional analysis and wind tunnel testing to enable scalable predictions across flight regimes.^[6] Today, it is essential for aircraft design, performance analysis, and optimization, influencing everything from takeoff speeds to fuel efficiency, with values determined experimentally or via computational fluid dynamics for specific configurations.^[2]

Fundamentals

Definition

The lift coefficient, denoted as C_L, is a dimensionless quantity in aerodynamics that quantifies the lift generated by a body moving through a fluid, such as air or water.^[1] Lift itself is the aerodynamic force acting perpendicular to the direction of the oncoming fluid flow, counteracting the body's weight in flight applications.^[7] The lift coefficient is defined as the ratio of the lift force to the product of the dynamic pressure (which depends on fluid density and flow speed) and a reference area (typically the wing planform area for aircraft).^[1] This formulation enables the scaling of lift predictions across different body sizes, speeds, and fluid densities without dimensional inconsistencies, making it essential for comparing aerodynamic performance universally.^[1] The dimensionless lift coefficient built on earlier empirical and theoretical foundations in airfoil theory, with formalization for finite wings occurring through developments like Ludwig Prandtl's lifting-line theory, published in 1918–1919.^[8] These efforts addressed the limitations of early empirical lift models, providing a theoretical framework that integrated viscous and inertial effects for practical engineering use.^[6] As a dimensionless parameter, C_L has no units and typically ranges from negative values (indicating downforce, as in some ground vehicles) to positive maxima of about 1.5–2.0 for conventional airfoils before the onset of stall, where flow separation sharply reduces lift.^[9] This range reflects the coefficient's sensitivity to body geometry and flow conditions, allowing engineers to normalize lift data for design optimization.^[9] It is analogous to the drag coefficient C_D, together forming key non-dimensional groups for overall aerodynamic force analysis.^[1]

Mathematical Basis

The lift coefficient C_L is a dimensionless quantity that quantifies the lift force generated by a body in a fluid flow, derived through dimensional analysis to ensure scalability across different conditions. The foundational equation is

C_L = \frac{L}{\frac{1}{2} \rho V_\infty^2 S},

where L is the lift force, \rho is the fluid density, V_\infty is the freestream velocity, and S is the reference area.^[10] This form arises from the Buckingham \pi theorem, which identifies dimensionless groups from the physical parameters governing the lift. Consider the lift per unit span L' for a two-dimensional body as a function of the angle of attack \alpha, freestream density \rho_\infty, freestream velocity V_\infty, chord length c, dynamic viscosity \mu_\infty, and speed of sound a_\infty: L' = f(\alpha, \rho_\infty, V_\infty, c, \mu_\infty, a_\infty). These seven parameters involve three fundamental dimensions (mass, length, time), yielding four dimensionless \pi groups: the lift coefficient \Pi_1 = \frac{L'}{\frac{1}{2} \rho_\infty V_\infty^2 c} = c_\ell, the angle of attack \Pi_2 = \alpha, the Reynolds number \Pi_3 = \frac{\rho_\infty V_\infty c}{\mu_\infty}, and the Mach number \Pi_4 = \frac{V_\infty}{a_\infty}. The theorem thus relates c_\ell = \bar{f}(\alpha, Re, M_\infty), extending to the three-dimensional case by replacing L' and c with L and S.^[10] The reference area S normalizes the lift for geometric scale; for two-dimensional airfoils, it is the chord length c times the span (often taken as unity for per-unit-span analysis), while for finite wings, it is the planform area.^[11] This choice ensures consistency in comparing aerodynamic performance. Nondimensionalization via C_L enables direct comparison of lift characteristics across varying scales, speeds, and fluids, such as air versus water, by collapsing data onto universal curves dependent only on parameters like Reynolds and Mach numbers.^[10] In vector terms, the lift force L is defined as the component of the net aerodynamic force perpendicular to the freestream velocity vector. For small angles of attack, C_L incorporates a cosine factor in its relation to the normal force coefficient, approximating C_L \approx C_n \cos \alpha, where C_n is based on the angle relative to the body.^[12]^[13] This aligns with approximations for infinite-span sections in lifting-line theory.

Aerodynamic Applications

Section Lift Coefficient

The section lift coefficient, denoted as c_l, quantifies the lift generated by a two-dimensional airfoil section and is defined as c_l = \frac{l}{\frac{1}{2} \rho V^2 c}, where l is the lift per unit span, \rho is the fluid density, V is the freestream velocity, and c is the airfoil chord length.^[14] This dimensionless parameter isolates the aerodynamic efficiency of the airfoil shape in an infinite-span (two-dimensional) flow, distinct from the three-dimensional wing lift coefficient C_L, which incorporates the full wing area including span effects.^[14] In thin airfoil theory, the relationship between c_l and the angle of attack \alpha (in radians) is linear for small angles: c_l = 2\pi (\alpha - \alpha_{L=0}), where the lift curve slope \frac{d c_l}{d \alpha} = 2\pi per radian applies to thin, inviscid airfoils regardless of camber.^[15] This theoretical slope, derived from potential flow analysis using vortex sheets along the camber line, closely approximates experimental values for many practical airfoils, typically within 10% accuracy up to moderate angles of attack.^[15] The zero-lift angle \alpha_{L=0} is the angle of attack at which c_l = 0; for symmetric airfoils, it is approximately 0°, while for positively cambered airfoils, it shifts to negative values, such as -2° to -4°, depending on the camber magnitude.^[16] This shift arises because camber generates lift even at zero geometric angle of attack, effectively advancing the lift curve along the angle-of-attack axis.^[14] Data from the NACA airfoil series illustrate how c_l varies with camber and thickness; for instance, symmetric NACA 0012 (12% thick, 0% camber) exhibits \alpha_{L=0} \approx 0^\circ and a maximum c_l around 1.6 at stall (Reynolds number ≈6×10^6), while the cambered NACA 2412 (12% thick, 2% camber) has \alpha_{L=0} \approx -2^\circ and produces c_l \approx 0.25 at \alpha = 0^\circ, with a higher maximum c_l of about 1.7 due to the camber-induced lift (Reynolds number ≈6×10^6).^[16] Thickness has a lesser influence on the lift curve slope, which remains near 2π per radian across 6% to 15% thickness ratios in the NACA 6A-series, though increasing camber typically lowers \alpha_{L=0} by about 1° per percent camber for NACA 4-digit series.^[16]

Finite Wing Lift Coefficient

The lift coefficient for a finite wing accounts for three-dimensional flow effects that reduce the overall lift generation compared to an infinite two-dimensional airfoil, primarily due to the formation of trailing vortices from the wingtips. These effects are characterized by the aspect ratio of the wing, defined as AR = b²/S, where b is the wing span and S is the reference wing area. Higher aspect ratios generally yield lift coefficients closer to two-dimensional values by minimizing relative tip losses.^[17] Prandtl's lifting-line theory models the finite wing as a horseshoe vortex system, with a bound vortex along the span and trailing sheet vortices that induce a downwash velocity across the wing. This downwash creates an induced angle of attack α_i that effectively reduces the geometric angle of attack α, leading to lower lift than predicted by two-dimensional theory. For an elliptical wing planform, which produces a uniform downwash, the theory yields the lift coefficient C_L = \frac{2\pi \alpha}{1 + \frac{2}{\mathrm{AR}}}, where α is in radians; this formula shows that as AR increases, C_L approaches the two-dimensional value of 2\pi \alpha.^[18]^[19] For non-elliptical planforms, the spanwise lift distribution deviates from ideal, requiring corrections such as the Oswald efficiency factor e, which quantifies the aerodynamic efficiency relative to an elliptical wing. The general expression for the wing lift curve slope is C_{L\alpha} = \frac{a_0}{1 + \frac{a_0}{\pi e \mathrm{AR}}}, where a_0 \approx 2\pi is the two-dimensional lift curve slope per radian, leading to C_L = C_{L\alpha} \alpha; typical e values range from 0.7 to 0.9 for conventional unswept wings, with higher values for planforms closer to elliptical loading.^[19]^[20] In some approximations, planform corrections are incorporated via a factor τ (often 0.05 to 0.25 for rectangular or tapered wings), modifying the denominator to 1 + \frac{a_0 (1 + \tau)}{\pi \mathrm{AR}} to account for non-uniform downwash.^[21] Wingtip effects arise from the pressure difference between the upper and lower surfaces, causing high-pressure air to roll up into counter-rotating tip vortices that trail downstream and further induce downwash, resulting in local lift loss near the tips. Rectangular wings exhibit higher tip loading and stronger vortices, amplifying this loss and reducing overall efficiency, whereas tapered wings promote a more elliptical lift distribution, mitigating vortex strength and preserving lift closer to the tips.^[22] For high-aspect-ratio wings, the finite wing lift coefficient approximates the section lift coefficient as tip effects become negligible.^[18]

Influences and Variations

Angle of Attack Effects

The lift coefficient C_L of an airfoil varies with the angle of attack \alpha, typically plotted as a lift curve that exhibits a linear region followed by nonlinear behavior leading to stall. In the linear regime, which extends up to approximately 12–16 degrees for most conventional airfoils, C_L increases proportionally with \alpha, characterized by a slope C_{L\alpha} of about 5.7 per radian (or 0.1 per degree), slightly below the theoretical value of $2\pi from thin airfoil theory due to real viscous effects. The curve intersects the \alpha-axis at the zero-lift angle \alpha_0, which is 0 degrees for symmetric airfoils like the NACA 0012 but shifts to negative values (e.g., -2 to -4 degrees) for cambered airfoils such as the NACA 2412, reflecting the inherent lift from camber at zero geometric angle.^[23]^[24] Camber significantly influences the shape of the lift curve by enhancing lift across a range of angles, particularly at low \alpha, while also affecting the stall characteristics. For cambered airfoils, the curve is vertically shifted upward compared to symmetric ones, allowing higher C_L at a given \alpha, but it may introduce earlier stall on the upper surface due to adverse pressure gradients. Beyond the linear region, nonlinear effects dominate as flow separation begins near the trailing edge, causing C_L to plateau and reach a maximum of approximately 1.2–1.5 for conventional NACA airfoils before a sharp post-stall drop, where C_L decreases rapidly due to massive separation. This maximum C_{L,\max} is influenced by airfoil thickness and camber, with thicker sections generally achieving higher values before stall at 15–18 degrees.^[23]^[25] Historical wind tunnel tests conducted by the National Advisory Committee for Aeronautics (NACA) from the late 1920s through the 1940s established these standard lift curve behaviors through systematic investigations in facilities like the Langley Variable-Density Tunnel, providing foundational data for airfoil design. Reports from this era, such as NACA Report No. 824 (1944), compiled lift curves for series like the four-digit (e.g., NACA 2412) and five-digit (e.g., NACA 23012) airfoils, confirming consistent linear slopes around 0.1 per degree and C_{L,\max} values in the 1.2–1.5 range across Reynolds numbers of 3×10^6 to 9×10^6, with camber enabling better low-speed performance. These results underscored the role of camber in optimizing the curve for practical applications while highlighting nonlinear stall drops as a critical limitation.^[24]^[25] The variation of lift with \alpha is coupled with the pitching moment coefficient C_m about the aerodynamic center, which is constant with respect to \alpha, typically ranging from -0.05 to -0.1 for conventional cambered airfoil sections, providing a nose-down moment.^[26] The lift curve slope can be slightly modulated by Reynolds number, with lower values yielding marginally reduced C_{L\alpha}.^[23]

Other Influencing Factors

The Reynolds number, defined as Re = \frac{\rho V c}{\mu}, where \rho is the fluid density, V is the freestream velocity, c is the chord length, and \mu is the dynamic viscosity, significantly influences the lift coefficient through its impact on boundary layer development and transition from laminar to turbulent flow.^[27] At higher Reynolds numbers, the boundary layer transitions earlier, delaying separation and allowing for a higher maximum lift coefficient, typically increasing it by 10-20% relative to lower Re conditions.^[27] For example, experimental data on airfoils demonstrate that maximum C_L rises from 1.04 at Re = 2.7 \times 10^6 to 1.23 at Re = 9.7 \times 10^6, highlighting the role of Re in enhancing stall resistance.^[27] In subsonic high-speed flows, the Mach number introduces compressibility effects that modify the lift coefficient, necessitating corrections for accurate prediction. The Prandtl-Glauert factor provides this adjustment, scaling the incompressible lift coefficient as C_L = \frac{C_{L,\infty}}{\sqrt{1 - M^2}}, where M is the freestream Mach number.^[28] This correction accounts for the increase in lift due to local density variations as M approaches unity, with the factor diverging near transonic conditions to reflect amplified pressure differences.^[29] Such effects are critical for aircraft operating near their critical Mach number, where uncorrected low-speed data would underestimate lift. Surface conditions exert a profound influence on the lift coefficient by altering flow attachment and boundary layer behavior. Roughness from manufacturing tolerances or environmental exposure can trigger premature transition, modestly boosting lift at moderate angles of attack but often reducing maximum C_L through induced separation. Ice accretion, particularly on leading edges, disrupts smooth airflow, decreasing maximum C_L by up to 50%; for instance, tests on an NACA 23012 airfoil showed C_L max falling from approximately 1.8 in clean conditions to 0.5 when iced.^[30] High-lift devices like flaps counteract such losses by increasing camber and effective wing area, potentially elevating C_L by 50% or more during takeoff and landing, though excessive roughness or contamination on flapped configurations can still degrade performance by promoting early stall. Comparisons between clean and contaminated wings underscore these sensitivities, with even thin ice layers causing substantial lift penalties.^[31] Fluid properties such as viscosity and temperature have a minor, indirect effect on the lift coefficient, primarily manifesting in non-standard environments like high-altitude flight. Viscosity, which varies with temperature according to \mu \propto T^{0.7} for air, influences boundary layer thickness; at high altitudes, lower temperatures reduce \mu slightly, but the overriding factor is decreased density \rho, which lowers Re and can subtly diminish maximum C_L by advancing separation.^[32] These variations are secondary to the primary impacts of density on dynamic pressure, with temperature-driven changes in \mu contributing less than 5% to overall C_L shifts in typical stratospheric conditions.^[32]

Measurement and Analysis

Experimental Determination

The experimental determination of the lift coefficient has evolved significantly since the early 20th century, beginning with rudimentary kite and glider tests conducted by the Wright brothers in the late 1890s and early 1900s. These initial experiments involved manned glider flights and kite setups at Kitty Hawk to measure lift and drag forces qualitatively, followed by the construction of a small wind tunnel in 1901 equipped with custom balances to quantify lift coefficients on scaled airfoil models, achieving measurements that informed their 1903 Flyer design.^[33]^[34] By the mid-20th century, advancements led to standardized wind tunnel facilities, and modern cryogenic wind tunnels, operational since the 1970s, enable high-Reynolds-number simulations by cooling air with liquid nitrogen to replicate full-scale flight conditions without excessive model scaling issues.^[35]^[36] Wind tunnel testing remains the primary method for experimentally determining lift coefficients, involving scaled aerodynamic models mounted in controlled airflow environments to measure forces directly. The setup typically features a test section where the model is supported by a force balance system that records lift via transducers, with airflow speed, density, and model angle of attack varied systematically to generate lift curves.^[37] Scaling laws ensure dynamic similarity, preserving the aspect ratio (AR) of the full-scale wing—such as maintaining AR=6 for a NACA 0015 airfoil model—to minimize distortions in induced drag and tip effects, while limiting model span to about 80% of the tunnel width to keep blockage below 5-10%.^[37] Blockage corrections are essential post-measurement, accounting for solid blockage from the model's volume (e.g., ε_s ≈ V_model / V_test_section) and wake blockage from flow displacement, using methods like Maskell's formulation to adjust observed coefficients to free-air equivalents and ensure accuracy within tunnel constraints.^[37]^[38] Two main force balance types are employed in wind tunnels: strain gauge balances and pressure integration systems, each offering distinct advantages for lift measurement. Strain gauge balances, cemented to flexure elements, detect deformations from lift forces with high sensitivity, providing direct three- to six-component measurements (lift, drag, moments) and resolutions up to 1 part in 20,000, often achieving lift coefficient accuracy of ±0.01 through digital processing and repeated calibrations.^[37]^[39] In contrast, pressure integration derives lift by summing surface pressure distributions from numerous taps or pressure-sensitive paint (PSP) across the model, integrating via the formula L = \int (p_l - p_u) \, dA (where p_l and p_u are lower and upper pressures, A is area), which is particularly useful for detailed flow mapping but requires more taps (400-800) and is less suited for dynamic loads compared to strain gauges.^[37] Both methods undergo tare and interference calibrations to isolate aerodynamic lift from support and buoyancy effects. Flight testing provides in-situ validation of lift coefficients on full-scale aircraft, using embedded strain gauges on wing spars and fuselage to measure structural deformations correlated to aerodynamic loads, often supplemented by accelerometers for inertial corrections.^[40]^[41] These sensors capture data during maneuvers at varying angles of attack, with lift inferred from strain-load calibrations performed via ground loading rigs or in-flight maneuvers, though challenges include environmental factors like temperature-induced gauge drift (up to 0.1-0.2% error) and the need for real-time data reduction to account for aircraft weight and thrust contributions.^[40] Calibration typically involves applying known loads to replicate flight spectra, ensuring uncertainties below 5% for critical maneuvers, and these empirical results are often cross-checked against theoretical lift curves for consistency.^[42]

Theoretical and Computational Methods

Analytical methods for predicting the lift coefficient rely on potential flow theory, which assumes inviscid, irrotational flow to solve for pressure distributions and resulting aerodynamic forces. For two-dimensional airfoils, solutions to the thin airfoil equation provide the foundational approach, yielding a lift coefficient proportional to the angle of attack with a slope of approximately 2π per radian for symmetric sections. This theory, developed in the early 20th century, enables rapid estimation of lift curves for simple geometries without viscous effects. For finite wings and more complex three-dimensional configurations, the vortex lattice method extends potential flow theory by discretizing the lifting surface into panels with bound vortices, solving for circulation distribution to satisfy the no-penetration boundary condition. This method, building on Prandtl's lifting-line theory, accurately captures three-dimensional effects like induced drag and lift distribution for subsonic flows. Panel methods further generalize this by representing surfaces with source and doublet distributions, allowing computation of potential flow around arbitrary lifting bodies, including fuselages and nacelles. Pioneered in the 1960s, these methods provide efficient predictions for preliminary design, with lift coefficients computed via integration of surface pressures.^[43] Computational fluid dynamics (CFD) advances these inviscid approaches by solving the Navier-Stokes equations to include viscous effects critical for realistic lift prediction. Reynolds-averaged Navier-Stokes (RANS) solvers dominate for steady-state simulations, employing turbulence models such as the k-ε model to close the equations for high-Reynolds-number flows. Grid convergence studies ensure numerical accuracy, typically requiring refinement until changes in lift coefficient are below 1%. These methods incorporate Reynolds number and Mach number effects through compressible formulations and wall modeling. Validation of theoretical and computational predictions involves direct comparison with experimental lift coefficients, where RANS methods achieve errors typically less than 5% for attached flows at low angles of attack.^[44] Post-2000 advancements integrate machine learning for surrogate models, training neural networks on CFD datasets to accelerate lift coefficient predictions by orders of magnitude while maintaining fidelity, particularly in design optimization loops.^[45]

References

[1]
Lift Equation | Glenn Research Center - NASA
Nov 20, 2023 · For lift, this variable is called the lift coefficient, designated “Cl.” This allows us to collect all the effects, simple and complex, into a ...
[2]
The Lift Coefficient
The lift coefficient is a number that aerodynamicists use to model all of the complex dependencies of shape, inclination, and some flow conditions on lift.
[3]
Aerodynamics of Airfoil Sections – Introduction to Aerospace Flight ...
In fact, for most airfoils, it is found that the lift coefficient varies linearly to within about 10% of that given in Eq. 7 up to an angle of about 10o to 12o, ...<|control11|><|separator|>
[4]
Inclination Effects on Lift
(For thin airfoils at subsonic speed, the lift coefficient is 2 x pi x angle, where pi is 3.1415926..., and the angle is given in radians {pi radians = 180 ...Missing: definition | Show results with:definition<|control11|><|separator|>
[5]
Stalling & Spinning – Introduction to Aerospace Flight Vehicles
An airplane has a wing with a maximum lift coefficient, C_{L_{\rm max}} , of 1.7. The in-flight weight, W , is 20,000 lb, and the wing area, S , is 340 ft2 ...
[6]
Lift Equation of the 1900's | Glenn Research Center - NASA
Jul 9, 2025 · When the Wrights began to design the 1900 aircraft, they used values for the lift coefficient based on the work by Lilienthal so they too used ...
[7]
What is Lift? | Glenn Research Center | NASA
### Definition of Lift Force in Aerodynamics
[8]
Lifting Line Theory – Introduction to Aerospace Flight Vehicles
Lifting line theory, initially developed by Ludwig Prandtl and his students in the early 1900s, revolutionized the understanding of finite wing aerodynamics, ...
[9]
Lift Coefficient - an overview | ScienceDirect Topics
The lift coefficient relates the AOA to the lift force. If the lift force is known at a specific airspeed the lift coefficient can be calculated from:
[10]
[PDF] Fluids – Lecture 4 Notes - MIT
The Buckingham Pi Theorem states that this functional statement can be rescaled into an ... lift coefficient. Π2. = α. = α angle of attack. Π3. = ρ∞V∞ c. µ ...
[11]
[PDF] Fluids – Lecture 3 Notes - MIT
The local coefficients are then defined as follows. Local Lift coefficient: cℓ ≡. L′ q∞ c. Local Drag coefficient: cd ≡. D′ q∞ c. Local Moment coefficient: cm ≡.
[12]
Aerodynamic Forces - Glenn Research Center - NASA
Sep 3, 2025 · The component of the net force perpendicular (or normal) to the flow direction is called the lift; the component of the net force along the flow ...
[13]
[PDF] 1 Performance 6. Airfoils and Wings The primary lifting surface of an ...
the lift coefficient varies with angle-of-attack. To simplify the resulting ... cosine of the angle is approximately 1 and the sine of the angle equals ...
[14]
Chapter 3. Additional Aerodynamics Tools
The two dimensional lift coefficient will now be CL = 2π(α – αL0), where αL0 is called the “zero lift angle of attack” and is a negative angle for a positively ...
[15]
[PDF] 5.7.2.1. Thin Airfoil Theory Derivation
Thin airfoil theory starts with a thin cambered plate, using vortexes to approximate velocity, and uses the Biot-Savart Law to compute induced velocity.Missing: definition | Show results with:definition
[16]
Appendix A: Airfoil Data – Aerodynamics and Aircraft Performance ...
Note that the lift coefficient at zero angle of attack is no longer zero but is approximately 0.25 and the zero lift angle of attack is now minus two degrees, ...
[17]
[PDF] naca 6a-series airfoil sections
Fi~ure 25.- Variation of section minimum drag coefficient with airfoil thickness for some. NACA bl.j-series and NACA 64A-series airfoil sections of various ...
[18]
Wing Shapes & Nomenclature – Introduction to Aerospace Flight ...
The aspect ratio of a wing on an airplane can be increased by increasing the wing span and decreasing the wing chord, as shown in the figure below; this example ...
[19]
[PDF] CHAPTER 12 WINGS OF FINITE SPAN
is the slope of the lift coefficient versus angle of attack curve of a two-dimensional. (infinite) wing with the same airfoil section. Combine (12.43) ...
[20]
[PDF] Incompressible Flow Over Finite wings - UTRGV Faculty Web
Prandtl's Lifting Line Theory Equation. Page 18. Prandtl's Lifting Line Theory - Elliptic Lift Dist. ... The local section lift coefficient is cl = a0(αeff - αL=0).Missing: formula | Show results with:formula
[21]
Aerodynamics of Finite Wings – Introduction to Aerospace Flight ...
The lifting part is referred to as the induced drag coefficient, or more generally, “drag due to lift,” and it can be observed that this value depends ...
[22]
3D Prandtl Lifting Line Theory - Aerodynamics for Students
A simple solution for unswept three-dimensional wings can be obtained by using Prandtl's lifting line model. For incompressible, inviscid flow, the wing is ...Missing: 1910s | Show results with:1910s
[23]
Vortex Drag - for How Things Fly
A rectangular wing creates more wingtip vortices than a tapered or elliptical wing. Few aircraft have elliptical wing shapes, because of the manufacturing ...Missing: loss | Show results with:loss
[24]
[PDF] Summary of Low-Speed Airfoil Data
This document summarizes low-speed airfoil data, showing pressure distribution around airfoils with varying thickness and camber, and is the second book in the ...
[25]
None
Below is a merged summary of all provided segments on airfoil data, focusing on NACA airfoils and their lift characteristics versus angle of attack (alpha). The information is consolidated into a dense, structured format, including text summaries and tables where appropriate to retain all details. Due to the complexity and volume of data, I’ll use tables in CSV-like format to summarize key airfoil data (e.g., lift coefficients vs. alpha) and provide a narrative overview for historical testing, sources, and additional notes.
[26]
None
### Summary of NACA Airfoil Development and Characteristics
[27]
[PDF] Aerodynamic Characteristics of NACA 0012 Airfoil Section at Angles ...
The small dis- continuity in lift coefficient at a = 182° is probably due to small angle-of-attack errors in alining the model previous to one or both of the ...
[28]
[PDF] Independent Effects of Reynolds Number and Mach Number on ...
Maximum lift coefficient was again increased by approximately 20 percent from 1.04 at Re = 2.7×106 to 1.23 at. Re = 9.7×106, with a corresponding increase in ...
[29]
[PDF] Subsonic Compressible Flow over Airfoils - Aerostudents
The lift coefficient and moment coefficient for compressible flow can be derived similarly, using cl = cl,0 p1 − M2. ∞ and cm = cm,0 p1 − M2. ∞ . (3.2).
[30]
[PDF] Revisiting the Transonic Similarity Rule: Critical Mach Number ...
That is, the pressure coefficients at high speeds increase inversely proportional to the Prandtl-Glauert scaling parameter, β; the lift coefficient at any given ...
[31]
The Impact of Icing on the Airfoil on the Lift‐Drag Characteristics and ...
Jun 14, 2021 · The maximum lift coefficient is reduced from 1.8 to 0.5, and the stall angle of attack is reduced from 18° to 6°. The resistance has increased ...Abstract · Introduction · Materials and Methods · Results and Discussion
[32]
Experimental study of ice accretion effects on aerodynamic ...
Oct 19, 2025 · In this case, the stall angle drops about 10° and the maximum lift coefficient reduces about 50% which is hazardous for an airplane. While horn ...
[33]
Variation in aerodynamic coefficients with altitude - ScienceDirect.com
At higher altitude Reynolds number is lower if Mach number remains same. It is due to the rapid decrease in density while temperature decreases slowly with ...
[34]
Wright 1901 Wind Tunnel Tests
The brothers decided to measure the lift and drag coefficients themselves. They first built some small models of a wing and a flat plate and attached them ...
[35]
Wind Tunnel Tests, 1901 - NPS Historical Handbook: Wright Brothers
Sep 28, 2002 · The most ingenious parts of the Wright wind tunnel were the two balances they designed for measuring the lift and drag of the model air-foils.
[36]
[PDF] THE CRYOGENIC WIND-TUNNEL CONCEPT FOR HIGH ...
The results of this work indicate that cryogenic subsonic, transonic, and super- sonic wind tunnels offer significant increases of test Reynolds number without ...
[37]
[PDF] The Cryogenic Wind Tunnel Cologne (KKK) - DNW Aero
The findings of airfoil and half model tests are confirmed: not only does the drag depend on the Reynolds number; even more important are the Reynolds number ...
[38]
[PDF] Low speed wind tunnel testing - Portland State University
This book covers low-speed wind tunnel testing, including types of wind tunnels, design, and measurements of pressure, flow, and shear stress.
[39]
[PDF] Investigation of blockage correction methods for full-scale wind ...
This study investigates wall interference effects in truck wind tunnel testing, using simulations and various blockage correction methods to correct drag ...Missing: laws | Show results with:laws<|control11|><|separator|>
[40]
Force (Wind Tunnel Fundamentals) - Amrita Virtual Lab
Most aerodynamic performance measurement techniques for airfoils rely on using balance systems or pressure systems, or a combination of both. The approach ...Missing: setup | Show results with:setup
[41]
[PDF] A Summary of Numerous Strain-Gage Load Calibrations on Aircraft ...
Fifteen aircraft structures that were calibrated for flight loads using strain gages are examined. The primary purpose of this paper is to document ...Missing: accelerometers | Show results with:accelerometers
[42]
[PDF] Flight loads measurements obtained from calibrated strain-gage ...
Calibrated strain-gage bridges have been used extensively to determine flight loads. (shear, bending moment, and torque) on a variety of aircraft structures.Missing: accelerometers | Show results with:accelerometers
[43]
[PDF] AGARD Flight Test Techniques Series. Volume 1. Calibration of Air ...
The pressure error of a sensor thus trailed outside the influence of the aircraft will be independent of aircraft parameters such as lift coefficient and should.
[44]
[PDF] Calculation of Potential Flow About Arbitrary Three-Dimensional ...
This report presents a complete discussion of a method for calculating potential flow about arbitrary lifting three-dimensional bodies without the ...
[45]
Prediction of high lift: review of present CFD capability - ScienceDirect
The focus of the current paper is to provide an overview of present CFD capability applied to the prediction of high-lift flow fields. To accomplish this ...
[46]
End-to-End Deep-Learning-Based Surrogate Modeling for ... - MDPI
A novel end-to-end multitask Convolutional Neural Network is proposed to predict the aerodynamic coefficients of an airfoil shape.Missing: post- | Show results with:post-