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Drag equation

The drag equation is a quadratic formula in fluid dynamics that calculates the drag force F_d acting on an object moving through a fluid, such as air or water, opposing the direction of motion and proportional to the square of the object's speed.^[1] It is expressed as F_d = \frac{1}{2} \rho v^2 C_d A, where \rho is the mass density of the fluid, v is the speed of the object relative to the fluid, C_d is the dimensionless drag coefficient that accounts for the object's shape, surface roughness, and flow conditions (such as the Reynolds number), and A is the reference area (typically the projected frontal area perpendicular to the flow).^[2] This equation applies primarily to high-speed flows where inertial forces dominate over viscous forces, distinguishing it from linear drag models like Stokes' law used for low Reynolds numbers.^[3] The drag force arises from two main components: form drag (or pressure drag), due to differences in pressure between the front and rear of the object, and skin friction drag, resulting from shear stresses on the object's surface as the fluid flows over it.^[1] The drag coefficient C_d is determined experimentally, often through wind tunnel testing, and varies widely—for example, approximately 0.47 for a sphere, 1.17 for a square flat plate perpendicular to the flow, and as low as 0.04 for streamlined airfoils at optimal angles of attack.^[1]^[4] In engineering applications, the equation is essential for predicting aerodynamic performance in aviation, automotive design, ballistics, and parachuting, enabling optimizations like minimizing fuel consumption in aircraft by maximizing the lift-to-drag ratio.^[2] Historically, the quadratic dependence on velocity was first proposed by Isaac Newton in his Philosophiæ Naturalis Principia Mathematica (1687), where he modeled fluid resistance as proportional to v^2 based on assumptions of non-viscous fluids, laying the groundwork for later developments.^[5] The modern form, incorporating the drag coefficient and reference area, emerged empirically in the late 19th and early 20th centuries; the Wright brothers refined it between 1900 and 1905 using wind tunnel tests on over 200 models to determine accurate coefficients, achieving a Smeaton coefficient of about 0.0033 (equivalent to \frac{1}{2} \rho for air at standard conditions).^[3] Further advancements came with the Navier-Stokes equations in the 19th century, which provided a theoretical basis for viscous effects, though the drag equation remains largely empirical for practical use.^[5]

Fundamentals

Definition and Formula

The drag equation quantifies the aerodynamic drag force acting on an object moving through a fluid, such as air or water, opposing the direction of motion. This force arises from the interaction between the object and the surrounding fluid, primarily due to pressure differences and shear stresses on the surface. The equation is fundamental in aerodynamics for predicting resistance in applications ranging from aircraft design to projectile motion. The standard form of the drag equation is

F_d = \frac{1}{2} \rho v^2 C_d A

where F_d is the drag force, \rho is the density of the fluid, v is the relative velocity of the object with respect to the fluid, C_d is the dimensionless drag coefficient that accounts for the object's shape and surface roughness, and A is the reference area (typically the projected frontal cross-sectional area for blunt bodies).^[6] This equation applies primarily to high Reynolds number flows, where inertial forces dominate over viscous forces, leading to a drag that scales quadratically with velocity; at low Reynolds numbers, linear drag laws like Stokes' law become more appropriate. It specifically describes the component of the aerodynamic force parallel to the oncoming flow, distinguishing it from perpendicular forces like lift or propulsive forces like thrust.^[7] In the International System of Units (SI), F_d is expressed in newtons (N), \rho in kilograms per cubic meter (kg/m³), v in meters per second (m/s), C_d as a unitless quantity, and A in square meters (m²), ensuring the equation's dimensional homogeneity as force (kg·m/s²). The drag equation has empirical origins in experiments on fluid resistance beginning with Isaac Newton's work in the late 17th century and continuing through the 18th and 19th centuries by subsequent researchers, and was formalized within 20th-century aerodynamics through dimensional analysis and empirical correlations.^[7]^[8]

Components of the Drag Force

The drag force in the drag equation arises from the interaction between a moving object and the surrounding fluid, and it is composed of several key terms that quantify the physical effects contributing to this opposition to motion. These components include the fluid density, the relative velocity of the object, the drag coefficient, and the reference area. Together, they form the scalar magnitude of the force, while the direction is inherently vectorial, opposing the relative motion.^[9] Fluid density, denoted as \rho, represents the mass per unit volume of the surrounding medium and plays a crucial role in determining the inertial resistance encountered by the object. It reflects the amount of fluid mass displaced or accelerated by the object's motion, directly scaling the drag force linearly. Density varies significantly with environmental conditions such as altitude, temperature, and pressure; for instance, standard air density at sea level is approximately 1.225 kg/m³ under International Standard Atmosphere conditions. In denser fluids like water, which has a density of about 1000 kg/m³, drag forces are substantially higher for the same object and speed, explaining why objects experience greater resistance in aquatic environments compared to atmospheric ones.^[9]^[10]^[11] The relative velocity, v, is the speed of the object with respect to the fluid medium, and its square in the equation underscores the quadratic dependence of drag on speed, meaning drag increases rapidly at higher velocities. This term captures the kinetic energy transfer from the object to the fluid, where even small increases in speed can lead to disproportionately larger forces due to enhanced momentum exchange and turbulence. Importantly, v is not the object's absolute ground speed but the velocity relative to the undisturbed fluid, accounting for scenarios like wind or currents that alter the effective motion. For example, an aircraft flying into a headwind experiences higher relative velocity and thus greater drag than in still air at the same groundspeed.^[9]^[12] The drag coefficient, C_d, is a dimensionless empirical parameter that encapsulates the aerodynamic or hydrodynamic efficiency of the object's shape in generating drag, primarily through effects like flow separation, pressure differences, and surface friction. It quantifies how streamlines detach from the body, creating wake regions that contribute to form drag, versus smooth attachment that minimizes resistance. Typical values range widely based on geometry; for a sphere, C_d \approx 0.47 in the subcritical Reynolds number regime, reflecting significant flow separation around its blunt form, while a streamlined airfoil might achieve C_d \approx 0.04 at low angles of attack due to reduced separation and favorable pressure recovery. This coefficient is determined experimentally and remains central to comparing drag across diverse shapes without dimensional analysis.^[13]^[14]^[15] The reference area, A, provides the characteristic scale for the interaction surface and is chosen based on the object's geometry and the type of drag being assessed, ensuring consistent normalization in the equation. For blunt bodies or general drag calculations, the frontal projected area perpendicular to the flow is typically used, as it directly influences the initial momentum deflection. In aviation, the planform area of the wing (top-view projection) is often selected when evaluating total aircraft drag to align with lift computations, whereas in marine applications, the wetted surface area (total submerged surface) is common for friction-dominated drag on hulls. The choice of A affects the interpretation of C_d, as a larger area implies a smaller coefficient for the same force.^[13]^[16]^[17] As a vector quantity, the drag force \vec{F_d} acts in the direction opposite to the relative velocity vector \vec{v}, formally expressed as \vec{F_d} = -\frac{1}{2} \rho v^2 C_d A \hat{v}, where \hat{v} is the unit vector in the direction of \vec{v}. This ensures drag always retards the object's motion through the fluid, aligning with the second law of Newton's principles for resistive forces. The dynamic pressure q = \frac{1}{2} \rho v^2 conveniently combines density and velocity into a single term representing the fluid's kinetic energy density.^[12]^[9]

Theoretical Foundations

Relation to Dynamic Pressure

Dynamic pressure, denoted as q, is defined as q = \frac{1}{2} \rho v^2, where \rho is the fluid density and v is the flow velocity relative to the object.^[18] This quantity arises from Bernoulli's principle, representing the difference between the stagnation pressure (total pressure at a point where the flow is brought to rest) and the static pressure (pressure in the undisturbed flow).^[18] In this framework, dynamic pressure captures the contribution of the fluid's motion to the overall pressure field, derived from the conservation of energy along a streamline.^[19] Physically, dynamic pressure quantifies the "ram" effect experienced by an object moving through a fluid, equivalent to the kinetic energy per unit volume of the flow.^[20] It measures the momentum flux of the fluid impacting the object's surface, influencing the magnitude of aerodynamic forces.^[18] This interpretation underscores its role in fluid mechanics, where higher velocities amplify the effective pressure due to the quadratic dependence on v. In the context of the drag equation, dynamic pressure provides a compact way to express the drag force as F_d = q C_d A, where C_d is the drag coefficient and A is the reference area.^[18] This rewritten form is widely used in engineering applications, such as aviation, to simplify calculations of aerodynamic loads by isolating the velocity-dependent term.^[18] The concept of dynamic pressure was developed in the early 20th century by aeronautical engineers, building on 18th-century inventions like the Pitot tube for measuring flow velocity through pressure differences.^[21] It became integral to early flight instrumentation, enabling airspeed determination via the relation between dynamic and static pressures.^[22] Distinct from static pressure, which is the pressure exerted by the fluid at rest, dynamic pressure highlights the effects of motion; in compressible flows, the total pressure is approximated as p_t = p_s + q, where p_s is static pressure, though exact relations involve additional thermodynamic corrections for high speeds.^[18] This distinction is crucial for understanding how flow velocity modulates pressure in aerodynamic scenarios.^[18]

Derivation from Fluid Dynamics

The derivation of the drag equation begins with the application of the conservation of linear momentum to a fluid control volume surrounding a body immersed in a flow. Consider a steady, incompressible flow where the body experiences a drag force F_d aligned with the free-stream velocity v. By Newton's second law, the net force on the fluid within the control volume equals the rate of change of momentum flux across its surface. For a control surface far upstream and downstream of the body, the pressure forces balance, leaving the drag force to balance the difference in momentum flux: the incoming momentum is \rho v^2 A, while the outgoing momentum in the wake is reduced due to velocity deficits, yielding F_d = \int (\rho v^2 - \rho u^2) \, dA, where u is the local wake velocity and the integral is over the cross-sectional area A.^[23]^[24] In a simplified stream-tube model, assume the flow is divided into infinitesimal stream tubes that pass around the body, with each tube experiencing a momentum change due to deflection or deceleration. Applying the momentum equation to such a tube, the drag contribution from a single tube is dF_d = \dot{m} (v - u \cos \theta), where \dot{m} = \rho v \, dA is the mass flow rate, u is the exit velocity, and \theta is the deflection angle. Integrating over all stream tubes gives the total drag as F_d = \rho A v^2 (1 - \cos \theta) for a uniform wake approximation, but this form is idealized and leads to an empirical coefficient to account for complex wake structures.^[23]^[25] This momentum-based approach assumes high Reynolds number (Re \gg 1) flows, where viscous effects are confined to thin boundary layers, allowing an inviscid approximation outside them; viscosity is neglected in the basic momentum balance, though it influences the wake indirectly. For low Re, such as creeping flows, the derivation shifts to Stokes' law (F_d = 6 \pi \mu r v), derived from solving the Navier-Stokes equations, but this is distinct from the quadratic form.^[24]^[26] To obtain the standard dimensionless form, dimensional analysis via the Buckingham \Pi theorem is applied. The drag force F_d depends on fluid density \rho (dimensions: [M L^{-3}]), velocity v ([L T^{-1}]), dynamic viscosity \mu ([M L^{-1} T^{-1}]), and a characteristic length L ([L]), with area A \sim L^2. There are five variables and three fundamental dimensions (M, L, T), yielding two dimensionless \Pi groups: \Pi_1 = \frac{F_d}{\rho v^2 L^2} and the Reynolds number Re = \frac{\rho v L}{\mu}. Thus, \Pi_1 = f(Re). The standard drag coefficient is defined as C_d = \frac{F_d}{\frac{1}{2} \rho v^2 A} = 2 \Pi_1 (assuming A \sim L^2), so conventionally F_d = \frac{1}{2} \rho v^2 C_d A.^[26]^[27] The factor of \frac{1}{2} arises from historical convention in dynamic pressure definitions, but the analysis confirms the quadratic velocity dependence for inertial-dominated flows. At high [Re](/page/Re), C_d approaches a constant, reflecting the empirical nature of the coefficient, which cannot be derived solely from first principles without experimental or computational input to determine its value for specific geometries.^[26]^[27]

Drag Coefficient

Definition and Physical Meaning

The drag coefficient, denoted C_d, is a dimensionless quantity that quantifies the aerodynamic resistance experienced by an object moving through a fluid. It is formally defined by the relation

C_d = \frac{F_d}{\frac{1}{2} \rho v^2 A},

where F_d is the drag force, \rho is the fluid density, v is the relative velocity of the object to the fluid, and A is the reference area (typically the frontal projected area for bluff bodies). This normalization expresses the drag in terms of dynamic pressure \frac{1}{2} \rho v^2, allowing C_d to encapsulate the effects of geometry and flow conditions independently of size or speed scales.^[13] Physically, C_d represents the combined influence of form drag, arising from pressure imbalances due to flow separation on the object's surface, and skin friction drag, resulting from viscous shear stresses in the boundary layer. For bluff bodies like spheres or cylinders, form drag dominates, often accounting for approximately 90% of the total drag, as the large wake created by early separation leads to significant pressure differences between the front and rear. In contrast, streamlined shapes minimize form drag by delaying separation, shifting the balance toward skin friction.^[28] The value of C_d varies with flow regime: in subcritical conditions (laminar boundary layer), it remains high due to extensive separation and a wide wake; it drops in supercritical flow following transition to a turbulent boundary layer, which energizes the flow and delays separation. For instance, a flat plate oriented perpendicular to the flow exhibits C_d \approx 1.28, reflecting nearly complete momentum loss in the wake. High C_d values signify inefficient streamlining, increasing the power required to maintain motion and thus elevating fuel consumption in applications like aircraft or vehicles.^[29]^[30] As a dimensionless parameter, C_d facilitates scaling laws in aerodynamic testing, enabling predictions for full-scale objects based on wind tunnel models by matching similarity criteria such as the Reynolds number. This universality supports design optimization across diverse scales without repeated full-size experiments.^[31]

Factors Affecting the Drag Coefficient

The drag coefficient C_d is profoundly influenced by the Reynolds number (Re), which characterizes the ratio of inertial to viscous forces in the fluid flow around an object. At low Re (typically below 10^3), flows are laminar, resulting in higher C_d due to dominant viscous drag and early boundary layer separation. As Re increases into the transitional regime (around 10^3 to 10^5), the flow begins to exhibit unsteadiness, with C_d gradually decreasing for streamlined shapes. A critical phenomenon occurs in bluff bodies like spheres at Re ≈ 3 × 10^5, known as the drag crisis, where the boundary layer transitions to turbulent, delaying separation and causing a sharp drop in C_d by up to 50% as pressure drag diminishes.^[32]^[33] Beyond this, in fully turbulent regimes (Re > 10^6), C_d stabilizes at lower values, though surface roughness can trigger the crisis earlier.^[34] Compressibility effects, governed by the Mach number (Ma), become significant at higher speeds, altering C_d through the onset of shock waves and wave drag. For subsonic flows (Ma < 0.3), C_d is relatively insensitive to Ma, dominated by incompressible viscous and pressure components. As Ma approaches 0.8, local supersonic regions form on the object, leading to drag divergence where C_d rises rapidly due to shock-induced separation and increased pressure drag.^[35] In transonic regimes (Ma ≈ 0.8–1.2), wave drag can increase C_d by factors of 2–3 compared to subsonic values, while supersonic flows (Ma > 1) exhibit further elevation from oblique shocks, though C_d may decrease slightly post-shock stabilization before hypersonic rarefaction effects intervene.^[36]/03:_Aerodynamics/3.02:_Airfoils_shapes/3.2.04:_Compressibility_and_drag-divergence_Mach_number) Object geometry and surface characteristics play a pivotal role in determining C_d, with bluff bodies exhibiting higher values due to large wakes and dominant form drag, while streamlined shapes minimize separation for lower C_d via predominant skin friction. For instance, a sphere (bluff) has C_d ≈ 0.47 in subcritical flow, compared to ≈ 0.04 for an airfoil at zero incidence (streamlined). Surface roughness exacerbates skin friction drag in laminar layers but can reduce overall C_d by promoting early turbulence and delaying separation; dimples on a golf ball, for example, lower C_d by nearly 50% at relevant Re (≈ 10^5), extending the attached flow region and shrinking the wake.^[37]^[38]^[39] The angle of attack, defined as the angle between the oncoming flow and the object's reference line, significantly impacts C_d by altering pressure distribution and separation patterns. At zero angle, C_d is minimized for symmetric bodies; as the angle increases (e.g., up to 10°–15°), induced drag rises quadratically due to lift-related components, elevating total C_d by 20%–50% or more. Beyond the critical angle (near stall, ≈ 15°–20°), massive separation causes C_d to surge dramatically, often doubling or tripling. Surface tension effects on C_d are negligible in macroscopic flows but can influence microscale phenomena like droplet drag.^[40]^[41] Fluid properties indirectly affect C_d through their influence on Re and flow behavior; for Newtonian fluids, temperature variations alter viscosity (μ), thereby changing Re = ρVL/μ and shifting the flow regime. Higher temperatures reduce μ, increasing Re and typically lowering C_d in transitional flows by promoting turbulence. Density (ρ) and speed (V) also scale Re linearly, with analogous effects. In non-Newtonian fluids, such as shear-thinning suspensions, the standard drag equation assumes constant viscosity, leading to inaccuracies; here, effective viscosity varies with shear rate, often reducing C_d at high Re compared to Newtonian counterparts, though predictive models require rheological corrections.^[42]^[43] The conventional drag equation provides limited coverage for nanoscale or rarefied flows, where the Knudsen number (Kn = λ/L, with λ as mean free path) exceeds 0.01, invalidating continuum assumptions. In such regimes, encountered during space re-entry or vacuum conditions, slip flow and free-molecular effects increase C_d by 20%–100% or more as Kn rises, due to reduced momentum transfer and non-equilibrium kinetics, necessitating kinetic theory or DSMC simulations for accurate prediction.^[44]^[45]

Measurement and Determination

Experimental Methods

Experimental methods for determining the drag force and coefficient have evolved from rudimentary drop tests to sophisticated laboratory setups, enabling empirical validation of the drag equation by measuring forces on scaled models under controlled flow conditions. In 1687, Isaac Newton conducted early experiments on air resistance using pendulum decay tests to quantify drag in air and water, laying foundational insights into fluid resistance proportional to velocity squared for blunt bodies.^[46] Modern techniques trace back to the early 1900s, when the Wright brothers constructed a wind tunnel in 1901 to test over 200 airfoil shapes, measuring lift and drag to refine wing designs for their glider experiments.^[47] These historical efforts established the importance of controlled environments for accurate drag assessment, achieving precisions that have improved to within ±0.5% uncertainty in contemporary setups.^[48] Wind tunnel testing remains the primary laboratory method for measuring drag in aerodynamic applications, applicable across subsonic and supersonic regimes. In subsonic tunnels, airflow speeds typically range from low velocities to near Mach 0.3, while supersonic facilities achieve Mach numbers above 1 using nozzles and diffusers to simulate high-speed flows. The drag force F_d is directly measured using internal or external force balances mounted on the model, which capture axial components as the model is exposed to varying airspeeds v. By recording F_d at multiple velocities and using the drag equation F_d = \frac{1}{2} \rho v^2 A C_d to solve for the drag coefficient C_d, researchers plot C_d versus the Reynolds number \text{Re} = \frac{\rho v L}{\mu} to characterize flow regimes from laminar to turbulent.^[49] Force balances employ strain gauge or piezoelectric transducers for precise force detection. Strain gauge balances, utilizing Wheatstone bridge configurations, excel in steady-state measurements of both static and dynamic loads, offering high sensitivity for low-drag configurations like airfoils. Piezoelectric balances, based on quartz crystal deformation, are preferred for transient forces in unsteady flows, providing rapid response times under high-frequency vibrations. These systems typically resolve forces to millinewton levels, ensuring reliable C_d determination across a wide dynamic range.^[50]^[51]^[52] Testing procedures emphasize similitude to extrapolate model results to full-scale prototypes. Geometric similarity requires proportional scaling of all linear dimensions, while kinematic similarity ensures velocity ratios match between model and prototype flows. Dynamic similarity is achieved by equating dimensionless numbers like Reynolds and Mach, often necessitating pressurized tunnels or variable-density air to replicate full-scale conditions. Models are typically scaled at 1:10 to 1:50 ratios, with surface finishes mimicking prototype roughness to avoid discrepancies in boundary layer development. Post-test data undergo corrections for wall effects and blockage: solid blockage from the model's volume accelerates flow, increasing apparent C_d by up to 10% for blockage ratios exceeding 5%, while wake blockage and streamline curvature demand iterative adjustments based on potential flow theory. These corrections, standardized in facilities like NASA's tunnels, restore equivalence to free-air conditions.^[53]^[31]^[54]^[55] For marine applications, towing tank experiments measure hydrodynamic drag on submerged or surface-piercing hull models, following Froude's pioneering work in the 1870s. William Froude developed scaling laws in 1867–1871, constructing the first experimental tank at Torquay to tow models of varying lengths (3 ft, 6 ft, 12 ft) and derive full-scale resistance via Froude number scaling, \text{Fr} = \frac{v}{\sqrt{gL}}, which preserves wave patterns by matching gravitational effects. Modern towing tanks, up to 300 m long, use carriage systems to propel models at speeds of 0.1–10 m/s, with dynamometers recording drag via tension in towing wires. These tests separate frictional and wave-making components, applying geometric and dynamic similarity to predict total resistance, though viscous scaling requires additional corrections for Reynolds effects.^[56]^[57]^[58] Despite advancements, experimental methods face limitations, particularly scale effects and challenges in replicating unsteady flows. Scale effects arise from incomplete Reynolds number matching, leading to premature transition or altered separation on small models, which can inflate C_d by 5–20% compared to full-scale. Unsteady flows, such as gusts or vortex shedding, are difficult to simulate accurately due to facility constraints on oscillation frequencies and amplitudes, often requiring specialized dynamic rigs. While traditional intrusive balances provide direct force data, they overlook flow field details; modern non-intrusive techniques like particle image velocimetry (PIV) address this gap by mapping velocity fields in the wake to infer drag via momentum deficits, offering spatial resolution without physical contact.^[59]^[60]^[61]^[62]

Computational Approaches

Computational fluid dynamics (CFD) provides a numerical framework for predicting drag forces by solving the Navier-Stokes equations, which govern fluid motion, through discretization methods such as the finite volume method (FVM) and finite element method (FEM). In FVM, the domain is divided into control volumes where conservation laws are applied, ensuring flux balance across boundaries, while FEM approximates solutions using basis functions over elements. For low Reynolds number (Re) flows, direct numerical simulation (DNS) resolves all turbulent scales without modeling, offering high fidelity but at prohibitive computational expense. In contrast, large eddy simulation (LES) models only small-scale turbulence while resolving larger eddies, balancing accuracy and cost for transitional regimes.^[63]^[64] Drag prediction in CFD involves post-processing the simulated flow field to compute the drag force F_d by integrating the pressure distribution and viscous shear stresses over the body's surface, typically projected onto the flow direction:

F_d = \int_S (-p \mathbf{n} + \boldsymbol{\tau} \cdot \mathbf{n}) \cdot \mathbf{e}_x \, dS

where p is pressure, \boldsymbol{\tau} is the viscous stress tensor, \mathbf{n} is the surface normal, and \mathbf{e}_x is the unit vector in the streamwise direction. The resulting F_d and drag coefficient C_d are validated against empirical data from experiments to assess simulation reliability.^[65]^[66]^[67] Widely used software tools for drag prediction include ANSYS Fluent, which employs Reynolds-averaged Navier-Stokes (RANS) solvers for efficient engineering approximations of turbulent flows via turbulence models like k-ε or k-ω, and the open-source OpenFOAM, which supports customizable RANS, LES, and DNS implementations. These tools enable simulations of complex geometries where analytical solutions are infeasible. RANS, in particular, averages turbulent fluctuations to reduce computational demands, making it suitable for high-Re industrial applications.^[68]^[69]^[70] CFD offers significant advantages over physical experiments for drag prediction, including cost-effectiveness for iterating on intricate geometries and unsteady flows without building prototypes or scaling issues inherent to wind tunnels. It provides comprehensive flow field data, such as velocity profiles and pressure gradients, unattainable through surface measurements alone in experiments. Additionally, post-2020 advancements in machine learning surrogates accelerate CFD by training neural networks on simulation data to approximate drag responses, reducing evaluation times from hours to seconds.^[71]^[72] Despite these benefits, CFD faces challenges, particularly in turbulence modeling, where RANS approximations can introduce errors up to 20% in C_d predictions due to inadequate capture of separation or transition phenomena. High-Re simulations also demand immense computational resources, often requiring supercomputers for LES or DNS to achieve grid resolutions of billions of cells.^[73]^[74]^[75] As of 2025, recent developments integrate artificial intelligence (AI) with CFD to enable real-time drag optimization, such as physics-informed neural networks that surrogate Navier-Stokes solutions for rapid design iterations in aerodynamics. These AI-enhanced methods, often combining multi-fidelity data and machine learning, achieve up to 50-fold speedups in surrogate modeling while maintaining predictive accuracy for drag minimization in applications like wing design.^[76]^[77]^[78]

Applications and Variations

In Aerodynamics and Engineering

In aerodynamics and engineering, the drag equation is pivotal for designing vehicles and structures to minimize resistance and enhance efficiency. In aviation, engineers apply the equation to optimize aircraft configurations, distinguishing between parasite drag—which arises from skin friction, form, and interference—and induced drag, which results from lift generation via wingtip vortices. Parasite drag dominates at high speeds, while induced drag is more significant at lower speeds, guiding designs like high-aspect-ratio wings to reduce the latter. For instance, the Boeing 787 Dreamliner achieves a cruise drag coefficient of approximately 0.026 through advanced composites and laminar flow control, enabling fuel savings of up to 20% compared to predecessors.^[79]^[80] This optimization directly impacts fuel efficiency, as overcoming aerodynamic drag consumes a major portion of an aircraft's fuel during cruise. The power required to counter drag is given by

P = F_d v,

where F_d is the drag force from the drag equation and v is the flight velocity, underscoring how even small reductions in drag coefficient translate to substantial energy savings over long distances. Techniques such as fairings—streamlined covers for protrusions like landing gear—and vortex generators, which energize boundary layers to delay flow separation, are employed to lower overall drag by 5-15% in commercial jets.^[81]^[82]^[83] In automotive engineering, the drag equation informs vehicle shaping to combat wind resistance, particularly for electric vehicles where range is limited by battery capacity. Tesla models exemplify this, with the Model S achieving a drag coefficient of 0.208 through sleek profiling and active grille shutters, targeting values below 0.20 to extend highway range. Ground effect, generated by underbody diffusers, enhances downforce for stability but must balance against increased drag from proximity to the road surface, while wheel drag—stemming from rotating hubs and spokes—contributes up to 25% of total aerodynamic losses and is mitigated via enclosed designs or optimized spoke patterns. By 2025, trends in electric vehicle development emphasize ultra-low drag shapes, with leading models attaining coefficients of 0.20-0.25 to prioritize range extension amid growing adoption.^[84]^[85]^[86] For civil engineering applications, the drag equation evaluates wind loads on structures like bridges and buildings, ensuring stability against aerodynamic forces. The 1940 Tacoma Narrows Bridge collapse highlighted the risks of aeroelasticity, where torsional flutter amplified by wind-induced drag led to structural failure at moderate gusts of 42 mph, prompting modern designs to incorporate open lattices and dampers. Drag coefficients for bluff bodies such as buildings typically range from 1.2 to 2.0, depending on shape and orientation, with rectangular forms experiencing higher values due to flow separation at edges; optimization involves rounded corners or sloped facades to reduce these by up to 20%. In bridge design, fairings and streamlined girders apply the same principles to cut wind-induced drag, linking directly to lower maintenance costs and safer spans.^[87]^[88]

Special Cases and Extensions

In regimes where the Reynolds number is low (Re ≪ 1), viscous forces dominate over inertial effects, rendering the standard drag equation inadequate as the drag coefficient C_d becomes Reynolds-number dependent and approaches infinity. Instead, the drag force on a sphere is given by Stokes' law:

F_d = 6\pi \mu r v,

where \mu is the dynamic viscosity, r is the sphere radius, and v is the relative velocity; this expression replaces the quadratic form and arises from solving the Stokes equations for creeping flow around a sphere.^[89] For intermediate Reynolds numbers (Re ≈ 0.1–1), where inertial effects begin to matter but remain small, the Oseen approximation corrects Stokes' law by incorporating a linearized inertial term, yielding a drag force of approximately F_d = 6\pi \mu r v \left(1 + \frac{3}{8} \mathrm{Re}\right), which better matches experimental data for slightly higher Re flows. For rotating bodies, the standard drag equation must account for spin-induced asymmetries in the boundary layer, often via the Magnus effect, which generates a lateral force perpendicular to the velocity and spin axis while also altering drag. On a spinning sphere, the drag coefficient C_d typically increases with the spin parameter \alpha = \frac{\omega r}{v}, where \omega is the angular velocity; experimental studies at intermediate Re (e.g., 10^4–10^5) show C_d rising by up to 20–30% for \alpha > 0.5, as rotation delays boundary layer separation on one side but advances it on the other.^[90] This modification explains phenomena like the curving trajectory of a spinning baseball, where the increased drag combines with the Magnus lift to amplify path deviation under air viscosity influences.^[90] At hypersonic speeds (Mach > 5), the drag equation extends to account for strong shock waves and molecular dissociation, causing C_d to rise beyond the subsonic plateau (typically from ~0.5 to values approaching 1–2 for blunt bodies) due to elevated post-shock temperatures and real-gas effects. Newtonian impact theory provides a simplified model for the pressure coefficient on windward surfaces: C_p = 2 \sin^2 \theta, where \theta is the angle between the surface normal and flow direction; integrating this over the body yields the wave drag component, which dominates total drag in hypersonic flows and assumes particle-like momentum transfer from the inviscid flow.^[91] In multiphase flows, such as suspensions of particles in a carrier fluid, the drag equation modifies to include inter-particle interactions and hindered settling, with the Schiller-Naumann correlation providing an empirical drag coefficient for spheres: C_d = \frac{24}{\mathrm{Re}} (1 + 0.15 \mathrm{Re}^{0.687}), valid for Re < 800 and volume fractions up to 0.2, which accounts for increased effective viscosity and wake interference in dense suspensions.^[92] In bio-fluids like blood, where red blood cells form deformable aggregates, drag on microparticles deviates further due to non-Newtonian rheology and cell-fluid partitioning, reducing effective drag by 10–50% compared to plasma alone as cells shield particles from viscous shear.^[93] Recent extensions to the drag equation for nano-scale flows in micro-electro-mechanical systems (MEMS) incorporate velocity slip at solid-liquid interfaces, where the no-slip condition fails due to molecular mean free path effects. Molecular dynamics simulations show that slip length b modifies the drag force to F_d \approx 6\pi \mu r v \left(1 - \frac{b}{r}\right) for Re ≪ 1, reducing drag by up to 30% for nanoparticles (r < 10 nm) in liquids, as slip allows partial momentum transfer across the interface; this is critical for 2024–2025 MEMS designs in biomedical sensors and nanofluidic devices.^[94]

References

[1]
5.2 Drag Forces – College Physics - UCF Pressbooks
Drag force is a force on an object moving in a fluid, opposing motion, and proportional to the square of the object's speed.
[2]
C L = L/qS - mechanics -
Similarly, the drag coefficient is written as: CD = D/qS , where D is the drag force and the other symbols have the same meaning. The numerator and denominator ...
[3]
Drag Equation of the 1900's | Glenn Research Center - NASA
Jul 15, 2024 · Between 1900 and 1905, the Wright brothers designed and built three unpowered gliders and three powered aircraft. In the design of each ...Missing: fluid | Show results with:fluid
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A History of Fluid Drag – Joseph Henry Project - McGraw Commons
Jul 18, 2025 · Fluid drag research in particular focuses on shear stress, the force of a fluid against the structure that can slow down or damage equipment over time.
[5]
Drag Equation | Glenn Research Center - NASA
Jul 1, 2025 · The drag equation states that drag D is equal to the drag coefficient Cd times the density rho (ρ) times half of the velocity V squared times the reference ...
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6.4 Drag Force and Terminal Speed - University Physics Volume 1
Sep 19, 2016 · Using the equation of drag force, we find m g = 1 2 ρ C A v 2 . m g = 1 2 ρ C A v 2 .
[7]
Aerodynamic Drag - The Physics Hypertextbook
Thus, if drag is proportional to the square of speed, then the power needed to overcome that drag is proportional to the cube of speed (P ∝ v3).
[8]
The Drag Equation
The drag equation states that drag D is equal to the drag coefficient Cd times the density r times half of the velocity V squared times the reference area A.
[9]
[PDF] The Standard Atmosphere
– R = gas constant for air = 287.053 joules/kg-deg K = 1716.551 ft-lb/sl ... = standard sea-level density 1,22500 kg/m3 = 0.00237691 sl/ft3. 9. Page 10 ...
[10]
[PDF] Drag Reduction in Wave-Swept Macroalgae
where Fdrag is drag force (N), ρ is density of water (1000 kg · m − 3), U is water ... Water motion, marine macroalgal physiology, and pro- duction.
[11]
What is Drag? | Glenn Research Center - NASA
Jul 21, 2022 · Drag is the aerodynamic force that opposes an aircraft's motion through the air. Drag is generated by every part of the airplane (even the engines!).
[12]
The Drag Coefficient
The drag coefficient Cd is equal to the drag D divided by the quantity: density r times half the velocity V squared times the reference area A. Cd = D / (A * .5 ...Missing: applicability | Show results with:applicability
[13]
[PDF] Experiment on Drag - Air Resistance
Sphere. 0.47. Typical Car. 0.5. Cylinder. 0.7-1.3. Cyclist. 0.9. Truck. 0.8-1.0. Motorcyclist. 1.8. FD. W = mg. Figure 1-2: Free Body Diagram of a Free Falling ...Missing: source | Show results with:source
[14]
Chapter 1. Introduction to Aerodynamics - Pressbooks at Virginia Tech
On an airplane, the dominant area for lift and drag is the wing, and ... It can also be expressed in terms of the square of the span and the planform area.Missing: marine | Show results with:marine
[15]
75. Worked Examples: Bluff Body Flows - Eagle Pubs
This is because the definition of the drag coefficient depends on the reference area. In the case of a wing, the reference area is the planform area, but in ...Missing: marine | Show results with:marine
[16]
Drag Coefficients of Swimming Animals: Effects of Using Different ...
However, CD is calculated using an arbitrary reference area for which there is no uniform convention; both total surface area ("wetted area") and maximum cross- ...
[17]
Dynamic Pressure | Glenn Research Center - NASA
Jun 28, 2024 · The dynamic pressure is a defined property of a moving flow of gas. We have performed this simple derivation to determine the form of the dynamic pressure.
[18]
Bernoulli's Equation | Glenn Research Center - NASA
Jul 19, 2024 · We call this pressure the dynamic pressure. The form of the dynamic pressure is the density times the square of the velocity divided by two.
[19]
Dynamic Pressure - The Engineering ToolBox
Dynamic pressure is the kinetic energy per unit volume of a fluid in movement, calculated as 1/2 * fluid density * velocity^2.
[20]
https://www.engineeringtoolbox.com/dynamic-pressure-d_1037.html
[21]
Bernoulli's Principle | SKYbrary Aviation Safety
Dynamic pressure is the difference between stagnation pressure and static pressure. Bernoulli's principle is used to calibrate the airspeed indicator so that it ...
[22]
[PDF] Fluids – Lecture 9 Notes - MIT
This is simply the summation of the momentum flow “lost” by all the infinitesimal stream- tubes in the flow. The lost momentum flow appears as drag on the body.
[23]
Momentum Equation – Introduction to Aerospace Flight Vehicles
The objective is to apply the conservation of linear momentum principle to a flow to find a mathematical expression for the forces produced.
[24]
[PDF] DRAG FORCE CALCULATION - UTRGV Faculty Web
The drag force of an object can also be calculated by applying the conservation of momentum equation for your stationary object. F⃗ = 𝜕. 𝜕t. ∫ V⃗⃗. CV. 𝜌d∀ + ∫ V⃗⃗𝜌V⃗⃗ ...
[25]
[PDF] 5.5 Buckingham Pi theorem - MIT
May 13, 2010 · The drag force depends on four quan- tities: two parameters of the cone and two parameters of the fluid (air). Any dimensionless form can be ...
[26]
Dimensional Analysis – Introduction to Aerospace Flight Vehicles
The Buckingham \Pi Theorem (usually just called the Buckingham \Pi method or just the \Pi method) is a generalized formalization of the method of dimensions ...
[27]
Measured and Predicted Turbulent Kinetic Energy in Flow Through ...
Nov 9, 2020 · ... form drag constitutes 90% of the total drag for bluff objects. Therefore, it is reasonable to approximate urn:x-wiley:00431397:media ...
[28]
Drag of a Sphere | Glenn Research Center - NASA
Jun 30, 2025 · Where D is equal to the drag, ρ is the air density, V is the velocity, A is a reference area, and Cd is the drag coefficient.Missing: components relative
[29]
Drag Equations of the 1900's
For the modern drag equation the drag coefficient of a flat plate moving perpendicular to the flow is 1.28; for the 1900's drag equation, the drag coefficient ...
[30]
Lift to Drag Ratio | Glenn Research Center - NASA
Jul 19, 2024 · Because lift and drag are both aerodynamic forces, the ratio of lift to drag is an indication of the aerodynamic efficiency of the airplane.
[31]
[PDF] Similitude requirements and scaling relationships as applied to ...
The aerodynamic coefficients are functions of the fourteen dimensionless param- eters , which represent the requirements for complete static and dynamic ...
[32]
Drag Crisis - an overview | ScienceDirect Topics
The law of drag for large Reynolds numbers, ie the relation between the drag force acting on the body and the value of R when the latter is large.
[33]
[PDF] Drag on Spheres - Penn State Mechanical Engineering
Jan 11, 2012 · Observe the “drag crisis” associated with a sphere at high Reynolds numbers, and the effect of surface roughness on sphere drag. 4. Measure the ...
[34]
Mach Number
This compressibility effect will alter the amount of resulting force on the object. The effect becomes more important as speed increases. Near and beyond the ...
[35]
Critical Mach Number - an overview | ScienceDirect Topics
At a slightly higher Mach number a shock wave appears on the upper surface and the drag rises rapidly. The Mach number at which the no-lift drag coefficient ...
[36]
Drag of Blunt Bodies and Streamlined Bodies
For streamlined bodies, frictional drag is the dominant source of air resistance. For a bluff body, the dominant source of drag is pressure drag. For a given ...Missing: typical | Show results with:typical
[37]
Bluff Body Flows – Introduction to Aerospace Flight Vehicles
Understand the use of reference areas in defining bluff body drag coefficients and the concept of equivalent drag area. Appreciate why the drag of bluff bodies ...<|control11|><|separator|>
[38]
(PDF) A Study of Dimple Characteristics on Golf Ball Drag
Aug 6, 2025 · ... The drag coefficient can be reduced by almost 50% in comparison with the smooth sphere in the critical region [14]. For dimpled particles, ...
[39]
Inclination Effects on Drag | Glenn Research Center - NASA
Jul 28, 2022 · The angle between the chord line and the flight direction is called the angle of attack and has a large effect on the drag generated by the wing.
[40]
Form Drag - an overview | ScienceDirect Topics
Streamlined bodies will have a relatively small wake and the drag will be mainly skin friction, whilst a bluff body with a large wake will have a higher ...
[41]
Drag Coefficient - an overview | ScienceDirect Topics
Drag coefficient is due to friction, pressure, and induced drag. The friction component is associated to the boundary layer development.<|separator|>
[42]
A Note on the Drag Coefficient of Steady Flow of Non-Newtonian ...
At low Reynolds numbers, shear-thinning fluids' drag coefficient follows cD = A/Re. At high power-law index, drag coefficients tend to coincide for all bodies.
[43]
Kinetic comparative study on aerodynamic characteristics of ...
May 7, 2021 · The drag coefficient increases significantly as the Knudsen number increases. The drag coefficient at the limit of free molecular flow is ...
[44]
Supersonic and Hypersonic Drag Coefficients for a Sphere
The rarefaction and compressibility flow regimes can be characterized by the Mach and Knudsen numbers as indicated Tables 1 and 2, where the quantitative ...
[45]
Newton's Philosophiae Naturalis Principia Mathematica
Dec 20, 2007 · From the resistance measured in the pendulum-decay experiments, Newton could conclude that the forces in air and water are dominated by a ...
[46]
Wind Tunnel Tests, 1901 - NPS Historical Handbook: Wright Brothers
Sep 28, 2002 · The Wright brothers used a wind tunnel to test model airfoils, measuring lift and drag, and testing over 200 surfaces, including monoplane, ...
[47]
Development of an experimental apparatus for flat plate drag ...
Feb 23, 2022 · A wind tunnel system was developed to measure aerodynamic drag, achieving less than 0.5% uncertainty, and a correction for pressure forces was ...
[48]
Drag Measurement | Glenn Research Center - NASA
Jul 18, 2024 · Aerodynamicists use wind tunnels to test models of proposed aircraft and engine components. During a test, the model is placed in the test ...Force Balance · Wright Memorial Tunnel (WMT)
[49]
[PDF] Review of Potential Wind Tunnel Balance Technologies
Piezoelectric based strain gages have been used successfully in dynamic wind tunnel balances [33] [34].
[50]
Force measurement using strain-gauge balance in a shock tunnel ...
May 24, 2016 · These strain-gauge sensors are arranged in a Wheatstone bridge to measure the strain produced by the loads. The output voltage of a balance ...
[51]
Piezoelectric or Strain Gauge Based Force Transducers? - HBK
Strain gauges use a spring and are good for long-term monitoring and large forces. Piezoelectric sensors use crystals, are good for dynamic and small forces, ...
[52]
Geometric Similarity - an overview | ScienceDirect Topics
3.1 Similarity. Wind tunnel tests are usually performed on a scale model of the real geometry. In order for the results of the wind tunnel test to be ...
[53]
[PDF] wind tunnel wall interference corrections - NASA
This approach makes it possible to correct dynamic pressure, Mach number, and angle of attack using three single numerical values. The dynamic pressure and the ...
[54]
Blockage Effect - an overview | ScienceDirect Topics
However, if the blockage ratio is larger than 10%, the tunnel walls would interfere with wake development and ultimately impair the accuracy of experiments.
[55]
History and Technology - Towing Tank Tests - NavWeaps
Dec 20, 2017 · In 1867, Froude did his model tests using models of two different hull shapes (called Swan and Raven). Each was built in 3ft, 6ft and 12 ft ...
[56]
William Froude: the father of hydrodynamics
Jan 20, 2024 · He built a sequence of 3, 6 and 12 foot scale models and used them in towing trials. These took place on the River Dart and enabled him to ...
[57]
Towing Experiments on Models to Determine the Resistance of Full ...
Froude's dynamometric apparatus used in towing models, and a description of the method of obtaining from such experiments the resistance of a full-sized ship, ...
[58]
[PDF] A Review of Scale Effects in Unsteady Aerodynamics
Unsteady experiments of all types are more difficult and often inherently less accurate than steady measurements.Missing: limitations | Show results with:limitations
[59]
A review of scale effects in unsteady aerodynamics - ScienceDirect
The review suggests that for the usual recommended model conditions with fixed transition, scale effects are small for fully-attached or well-separated flows, ...Missing: limitations | Show results with:limitations
[60]
Wind Tunnel Measurement Systems for Unsteady Aerodynamic ...
Aug 17, 2020 · The main problem of the test techniques is that the “unsteady” effect that has been proved to have a significant effect on response predictions ...
[61]
3-dimensional particle image velocimetry based evaluation of ...
Aug 26, 2020 · A high-resolution planar PIV measurement is employed to determine the drag reduction directly from wall shear measurements and to analyze ...
[62]
[PDF] An Introduction to Computational Fluid Dynamics - ResearchGate
Page 1. An Introduction to Computational. Fluid Dynamics. THE FINITE VOLUME METHOD ... Navier–Stokes equations for a Newtonian fluid. 21. 2.4. Conservative form ...
[63]
[PDF] Theoretical Foundations and Applications of Computational Fluid ...
The course covered the basics of computational fluid dynamics in the field of nuclear engineering, from mathematical description to numerical representation and ...
[64]
[PDF] Drag Forces in External Flows
Drag acts in the direction of the upstream velocity. It includes friction drag (viscous) and pressure drag (form drag), influenced by object shape.Missing: flux | Show results with:flux
[65]
How Do I Compute Lift and Drag? | COMSOL Blog
Jun 16, 2015 · We can integrate the local wall shear stress in the direction of drag (the y-direction) with the following expression: spf. rho*spf. u_tau*spf.
[66]
Verification of CFD analysis methods for predicting the drag force ...
The pressure (form) drag force is obtained by calculating the integral of the pressure on the surface of the vehicle. It was found that the pressure drag force ...
[67]
Best Practice: RANS Turbulence Modeling in Ansys CFD
This paper guides you through the process of optimal RANS turbulence model selection within the Ansys CFD codes, especially Ansys Fluent and Ansys CFX.
[68]
[PDF] RANS Simulations using OpenFOAM Software - DTIC
The purpose of this report is to illustrate and test the use of the steady-state Reynolds Averaged Navier-Stokes (RANS) solver in OpenFOAM. (simpleFoam), as ...
[69]
Development and Prediction of Vehicle Drag Coefficient Using ...
Jan 9, 2019 · This papers summarizes CFD simulation results for three standard MIRA body configurations (Notchback, Fastback and Estate back) using OpenFoam ...Missing: RANS | Show results with:RANS
[70]
Aerodynamic study of different cyclist positions: CFD analysis and ...
May 7, 2010 · A strong advantage of CFD is that detailed flow field information is obtained, which cannot easily be obtained from wind-tunnel tests. This ...
[71]
CFD analysis and full-scale wind-tunnel tests - PubMed
May 7, 2010 · A strong advantage of CFD is that detailed flow field information is obtained, which cannot easily be obtained from wind-tunnel tests. This ...
[72]
High‐order CFD methods: current status and perspective
Jan 24, 2013 · For example, a 5% error in velocity may translate into a 20% or higher error in skin friction depending on many factors such as Reynolds number, ...
[73]
Turbulence Modeling Effects on the CFD Predictions of Flow over a ...
It is even true that more often, the RANS approach fails to predict correct integral aerodynamic quantities like lift, drag, or moment coefficients, and as such ...
[74]
[PDF] Statistical Analysis of CFD Solutions from the Fifth AIAA Drag ...
A graphical framework is used for statistical analysis of the results from an extensive N- version test of a collection of Reynolds-averaged Navier-Stokes ...<|control11|><|separator|>
[75]
AI-Assisted CFD Optimisation of Multi-Element Wing Angle of Attack ...
Oct 3, 2025 · This study presents a multi-level aerodynamic optimisation framework that integrates Computational Fluid Dynamics (CFD) with Artificial ...
[76]
Physics-Enhanced Neural Networks for Aerodynamic Shape ...
Jul 16, 2025 · A multi-fidelity neural network architecture is developed, combining large amounts of low-fidelity data with fewer high-fidelity evaluations.
[77]
Advancing CFD with AI: Surrogate Modeling Approaches in the ...
Aug 19, 2025 · Once trained, the surrogate could deliver results in near real time, enabling up to 50X more design iterations without increasing compute costs.
[78]
[PDF] Aircraft Drag Polar Estimation Based on a Stochastic Hierarchical ...
Dec 7, 2018 · These two parameters, CD0 and k are the zero-lift drag coefficient and lift-induced drag coefficient factor respectively. ... Boeing 787-9. 0.026.
[79]
Induced Drag Explained - Pilot Institute
Jun 5, 2023 · Unlike induced drag, which decreases as your speed increases, parasite drag does the exact opposite. It grows with speed, becoming more of a ...Induced Drag and Angle of... · Parasite vs. Induced Drag
[80]
Chapter 4. Performance in Straight and Level Flight – Aerodynamics ...
The drag coefficient relationship shown above is termed a parabolic drag “polar” because of its mathematical form. It is actually only valid for inviscid wing ...Missing: flux | Show results with:flux
[81]
Fairings - an overview | ScienceDirect Topics
A fairing is defined as a streamlined structure designed to reduce drag caused by the underlying geometry. It is characterized by parameters such as chord, ...
[82]
An overview of flow control in aerodynamic surfaces using vortex ...
Mar 27, 2025 · This review paper aims to provide an overview of the principle of vortex generator design and its chronological development and optimization ...
[83]
New Model 3 Has "Lowest Absolute Drag Of Any Tesla" With Cd Of ...
Sep 1, 2023 · The new Model 3 is "the most aerodynamic Tesla ever," with a drag coefficient of 0.219. However, the Model S has a drag coefficient of 0.208, so what does ...
[84]
[PDF] Ground Effect Aerodynamics of Race Cars - ePrints Soton
We review the progress made during the last 30 years on ground effect aerodynamics associated with race cars, in particular open wheel race cars.
[85]
Longest Range EVs 2025: Real-World Tested Rankings & Guide
Oct 2, 2025 · Ultra-low drag coefficients: Leading EVs achieve 0.20-0.25 Cd compared to 0.30+ for most gasoline vehicles; Active aerodynamics: Adjustable ...
[86]
Why the Tacoma Narrows Bridge Collapsed: An Engineering Analysis
Dec 12, 2023 · The Tacoma Narrows Bridge collapsed primarily due to the aeroelastic flutter. In ordinary bridge design, the wind is allowed to pass through the ...
[87]
[PDF] “Drag Coefficient of Tall Building by CFD Method using ANSYS”
Checking the effect of shape of building on wind analysis, drag coefficients are found out using CFD analysis for different shapes of building in plan and ...
[88]
Terminal fall velocity: the legacy of Stokes from the perspective of ...
Aug 28, 2019 · Equation (2.17) is known as Stokes' law, where the viscous shear force and pressure force contribute two-thirds and one-third, respectively. (ii) ...
[89]
Varying Magnus effect on a rotating sphere at intermediate ...
In this paper, we have experimentally investigated the effect of imposed transverse rotation on flow past a sphere in the intermediate Reynolds number range.
[90]
Theory and Implementation of a Rapid Hypersonic Impact Method ...
The original Newtonian method, also referred to as the straight Newtonian method, utilizes the following equation for pressure calculations [1]. CP = 2 sin2 𝜃.
[91]
Particle Distribution Studies in Highly Concentrated Solid-liquid ...
Amongst the drag correlations implemented the Schiller & Naumann showed the best agreement for the highest particle concentrations at intermediate velocities, ...
[92]
Drag force on a microrobot propelled through blood - Nature
Jul 13, 2024 · The drag is reduced presumably because the microrobot is propelled through the fluid among RBCs, and the fluid has a viscosity lower than the ...
[93]
Drag on nanoparticles in a liquid: from slip to stick boundary conditions
In this work, the drag force acting on nanoparticles suspended in a liquid and the hydrodynamic boundary coefficient were calculated by using molecular ...Missing: MEMS 2025