Drag Formula Research Articles

Optimizing the methodology for accurate and accessible slip length measurement with atomic force microscopy

The remarkable performance of cooling devices employing nano- and microscale channels has drawn considerable interest, highlighting the need for surfaces with large slip lengths to improve their efficiency. However, the large errors in slip length associated with existing techniques hinder a clear understanding of slip phenomena. In this paper, we evaluate existing analytical methods for slip length measurement with atomic force microscopy (AFM) and propose a new reliable method. We performed force curve measurements on mica, silica and highly ordered pyrolytic graphite surfaces in water using an AFM equipped with a colloidal probe. The obtained force curves were analyzed through three methods: two commonly utilized procedures, namely the recursive and intercept methods, and a novel one called the two-parameter method which we developed. Our analyses showed that the recursive method yielded slip lengths with relatively large errors, fluctuation of ±5.8 nm, which were due to inaccuracies in the cantilever's spring constant and the fluid viscosity. On the other hand, it was found that the intercept method leads to restrictions on the choice of fitting range because of the simplified formula for viscous drag. As a result, by altering the data range, the calculated slip length shows significant variations within the ranges of ±27.5 nm. The two-parameter method, unlike the standard ones, overcome these drawbacks. This method requires no pre-measured parameters, and the slip length fluctuation is independent of the fitting range and only ±3.6 nm, which is around 2/3 of that observed in the recursive method and 1/8 of that in the intercept method. Our study optimizes existing analytical protocols and offers a new way for accessible and reliable calculations of slip lengths.

Model-based manoeuvre analysis: a path to a new paradigm in aircraft flight dynamics

We propose a closed-form system of nonlinear equations for the pitch plane or longitudinal motions of a fixed-wing aircraft and use it to demonstrate a possible path to the unification of theoretical flight dynamics and practical analysis of aircraft manoeuvres. The derivation of an explicit model free of data tables and interpolated functions is enabled by our use of empirical formulae for lift and drag which agree with experiments. We validate the model by recovering the well-known short period and phugoid modes, and the regions of normal and reversed command. We then use the model to present detailed simulations of two acrobatic manoeuvres, an Immelmann turn and a vertical dive. Providing new quantitative insights into the dynamics of aviation, our model-based manoeuvre analysis has the potential to impact both the academic flight dynamics curriculum and the ground training program for pilots of manned and unmanned aircraft. Possible consequences of future model-centric pilot training may include improved safety standards in general and commercial aviation as well as expedited theoretical course completion in air transport.

Uniform Flow Of Inviscid Fluid Past A Permeable Sphere Filled With Viscous Fluid

The investigativestudy of continuousflow of micro-polar fluid past a spherical porous ballsaturated with viscous fluidis considered. The velocity field within as well as outside the sphere is referred in terms of stream function. The flow pattern is obtained under regularity condition at infinity and normal velocity continuity on the sphere's surface. The formula for Drag acting on the sphere is derived and the results of physical parameters like porosity parameter and micro-polarity parameter on the flow and on the Dragon the sphere are presented in the form of diagrams.

A floating system integrating a wind turbine with a steel fish farming cage: Experimental validation of the hydrodynamic model

This paper is dedicated to comparing the numerical results of the dynamic responses of a floating offshore wind turbine integrated with a steel fish farming cage (FOWT-SFFC) under wind and waves against the experimental data. FOWT-SFFC's numerical model is built with OrcaFlex where the potential flow theory and quadratic drag formula are used to calculate the hydrodynamic loads on the structural components of the floater, while Morison-type formulation is employed to model the force on fish nets. The numerical model is tuned in terms of the measured data. The simulated motion response amplitude operators and stochastic responses under irregular waves are compared with the experimental results to fathom the influence of drag's modeling on the simulation outcome. The effects of additional damping and 2nd-order wave forces are also investigated to determine the optimal simulation method. With the selected numerical model, the responses of FOWT-SFFC with and without nets under 100-year extreme waves, combined wind and irregular waves are calculated. The simulated response time histories, spectra and statistics are in good agreement with the experimental data. Besides, it is revealed that the effects of 2nd-order wave forces are important for the accurate simulation of low-frequency pitch motion, but unfavorable to the prediction of other responses.

Water displacement drag decomposition of aircraft elastic tires rolling on contaminated runways

During the taxiing of an aircraft on a water-contaminated runway, the rolling tires impact water film resulting in a considerable water displacement drag and a large amount of spray, which will affect the safety of take-off and landing. The characteristics and flow mechanism of the water displacement drag generated by an elastic tire are numerically studied based on the coupling method of Smoothed Particle Hydrodynamics and Finite Element Method. The water displacement drag is divided into wave-generated drag and film-generated drag based on flow mechanism for the first time, and the former is much larger than the latter. The water displacement drag satisfies the Gaussian distribution along the rotation axis. The longitudinal grooves cause a significant increase in the wave-generated drag and a reduction in the film-generated drag. A simple potential flow model around a semi-infinite bluff body is proposed to reveal the flow physics of film-generated drag. Two drag formulas are proposed to quickly predict the wave-generated drag and film-generated drag.

On deducing the drag coefficient formula of cylindrical vegetation in non-uniform channel flow

Research on water flow resistance characteristics in a vegetation environment is a hotspot in environmental fluid research, which is primarily concentrated on the calculation of the vegetation drag coefficient Cd. At present, relatively few studies exist on the resistance characteristics of vegetation under non-uniform flow conditions, resulting in few general expressions for the research of Cd for this type of condition. In response to these scientific problems, this study selects shrub vegetation as the research object and generalised it as cylinders for the simulation study. This study adopts quadraticand Gaussian functions to change the coordinate expression of cylindrical vegetation Cd and then proposes the drag formulas of cylindrical vegetation in non-uniform flow for non-rainfall and heavy rainfall conditions based on regression analysis. Finally, this study substitutes the proposed Cd formula into the Saint-Venant equation to calculate the depth of channel flow. The newly proposed equations are verified by comparing the measured flow depth data with the calculation results. This study provides technical support for refined hydrodynamic simulations of vegetated flow regions.

Hydrodynamics of a slip-stick sphere with a non-axisymmetric patch

We study the low Reynolds number hydrodynamics of a slip-stick sphere suspended in an arbitrary ambient Stokes flow, whose surface is partitioned into two regions with different slip lengths. The fore-aft symmetry of the sphere breaks due to the varied slip length over the surface, which causes translational and rotational motion of the slip-stick sphere. An analytical solution is developed using the double curl method to evaluate Faxén's formulae for the hydrodynamic drag and torque exerted on the slip-stick sphere for the sub-cases, namely, (a) cap/strip model and (b) patch model. Subsequently, we compute the flow field, velocity, and rotation rate, which strongly depend on the slip lengths and configuration of the patch. As a specific example, we consider the slip-stick sphere immersed in a Poiseuille flow. For the cap/strip model, we find an optimal configuration for which the velocity of the slip-stick sphere is maximum compared to the slip-stick sphere with uniform slip. We also find configurations for which the velocity is independent of the slip lengths. Subsequently, in the patch model, we obtain the optimal azimuthal angles for the maximum rotation rate of the slip-stick sphere. We observe near-field deviations in streamlines due to the heterogeneous nature of the surface of the slip-stick sphere. These findings help design efficient artificial passive swimmers with prescribed slip lengths.

Prediction of Wind Loadings on Offshore Wind Turbines Using CFD

Analysing the structural response of offshore wind turbines (OWTs) requires a robust method to estimate environmental loadings associated with extreme weather conditions. This study aimed to accurately predict wind loadings on a non-rotating OWT using a Computational Fluid Dynamics (CFD) code, Ansys Fluent package. The inlet velocity boundary condition was defined as a uniform inlet wind speed and then as a Normal Wind Profile (NWP). Three large wind speeds of 25 m/s, 40 m/s and 50 m/s were tested with the aid of the RANS equations and the Shear Stress Transport (SST) turbulence model. A thorough mesh convergence study was conducted for both 2D and 3D simulations, and their results were assessed using the Richardson extrapolation method. Overall, the NWP method was found to produce larger wind forces and moments in comparison with the uniform wind speed conditions (on average 52% and 63% higher, respectively), and its predictions were consistent with the estimations obtained using the simplified drag formula recommended by offshore standards (within approx. 7% difference for wind forces). This paper provided the preliminary steps towards investigating the structural integrity of OWTs under extreme weather conditions.

Evaluation of Urban Canopy Models against Near-Surface Measurements in Houston during a Strong Frontal Passage

Urban canopy models (UCMs) in mesoscale numerical weather prediction models need evaluation to understand biases in urban environments under a range of conditions. The authors evaluate a new drag formula in the Weather Research and Forecasting (WRF) model’s multilayer UCM, the Building Effect Parameterization combined with the Building Energy Model (BEP+BEM), against both in-situ measurements in the urban environment as well as simulations with a simple bulk scheme and BEP+BEM using the old drag formula. The new drag formula varies with building packing density, while the old drag formula is constant. The case study is a strong cold frontal passage that occurred in Houston during the winter of 2017, causing high winds. It is found that both BEP+BEM simulations have lower peak wind speeds, consistent with near-surface measurements, while the bulk simulation has winds that are too strong. The constant-drag BEP+BEM simulation has a near-zero wind speed bias, while the new-drag simulation has a negative bias. Although the focus is on the impact of drag on the urban wind speeds, both BEP+BEM simulations have larger negative biases in the near-surface temperature than the bulk-scheme simulation. Reasons for the different performances are discussed.

Invariant Vortex-Force Theory Extending Classical Aerodynamic Theories to Transonic Flows

Recent studies about the Lamb vector have led to the development of the vortex-force theory: a formulation able to predict the aerodynamic force in compressible flows and decompose it into lift, lift-induced drag, and profile drag. Here, a revised formulation of the vortex-force theory developed at ONERA in collaboration with the University of Naples is presented and applied to steady transonic flows. In the mathematical developments, special care is given to the presence of shock wave discontinuities within the flowfield. The equivalence between the new definition, the Kutta–Joukowski lift theorem, Maskell’s lift-induced drag formula (“Progress Towards a Method for the Measurement of the Components of the Drag of a Wing of Finite Span,” Procurement Executive, Ministry of Defence, Royal Aircraft Establishment TR 72232, Farnborough, England, U.K., 1972) and Betz’s profile drag formula (“A Method for the Direct Determination of Wing-Section Drag,” NACA TM 337, 1925) are emphasized. The revised formulation also presents several practical advantages: the decomposition is naturally invariant to the reference point chosen for the calculation of moment transformations, and the computation can be performed in a fine part of the grid, close to the body skin. The formulation is finally tested on the NASA Common Research Model wing–fuselage configuration in cruise flight conditions.

The Drag Forces on a Taylor Bubble Rising Steadily in Vertical Pipes

By the adoption of a drag-buoyancy equality model, analytical solutions were obtained for the drag coefficients (CD) of Taylor bubbles rising steadily in pipes. The obtained solutions were functions of the geometry of the Taylor bubble and the gas volume fraction. The solutions were applicable at a wide range of Capillary numbers. The solution was validated by comparison with experimental data of other investigators. All derived drag formulas were subject to the condition that Bond number >4, for air-water systems.

A Liquid drop Falling in Another Fluid: A Two-Phase Flow Phenomenon

Liquid drops exist everywhere in daily life. However, such a common phenomenon involves complex mechanisms, and therefore has never been investigated thoroughly—the achievements of the studies are limited and unsatisfactory. In this article, the phenomenon of a liquid drop falling in another immiscible fluid (mostly its settling speed) was investigated. A series of experiments, in which liquid drops emerging from a syringe falls inside a measuring cylinder filled another fluid, was performed. Fluids used in the experiment include ethanol solutions and rice bran oil; for rice bran oil, its temperature-dependent viscosity was approximated by a function, which was proved effective on other similar oils. The dependence of the settling speed of the liquid drop on the density difference between the liquid drop and the surrounding fluid, as well as on the temperature of the surrounding fluid, was measured. Finite-Element Modeling (FEM) simulations were then carried out to model these situations, and lead to results that agreed with the experiments. It also visualized the flow field and revealed more details of the two-phase flow that were not detectable by our devices. Furthermore, inspired by some previous formulae, quantitative equations were derived semi-analytically by modifying an approximate drag formula originally developed for a rigid object at moderate Reynolds numbers. The form of the modification was never introduced in any other previous works. The equations were validated by simulations under ideal conditions.

Turbulent Brownian motion

An outstanding result of classical physics is (Einstein A., Ann. Phys. (Berlin), 17 (1905) 549; Smoluchowski M., Ann. Phys. (Berlin), 21 (1996) 756; Pomeau Y. and Piasecki J., C. R. Phys., 18 (2017) 570 and references therein) Stokes-Einstein's formula for the diffusion coefficient of tiny spheres driven by pressure fluctuations of a fluid at equilibrium. This formula includes both the macroscopic Stokes formula for the viscous drag and the fluctuations described by Gibbs-Boltzmann statistics. In the present work we consider a somewhat related problem, namely the diffusion (in space) induced by the turbulent fluctuations of the wake behind a body settling at constant speed under the effect of gravity —because of a difference between its mass density and the one of the fluid. We assume that the body is so big and heavy that the Reynolds number is large enough to make valid scaling relations independent of the fluid viscosity in a regime of fully developed turbulence of the wake.

Couple stress fluid flow due to slow steady oscillations of a permeable sphere

AbstractThe study of oscillating flow of a Couple Stress fluid past a permeable sphere is considered. Analytical solution for the flow field in terms of stream function is obtained using modified Bessel functions. The formula for Drag acting on the sphere due external flow is evaluated. Pressure field for the flow region past and inside the sphere is obtained. Effects of physical parameters like couple stress parameter, permeability, frequency and geometric parameters on the drag due to internal and external flows are represented graphically. It is observed that the drag for viscous fluid flow will be less than the case of couple-stress fluid flow and hence couple stress fluids offer resistance for flow.

Wave Attenuation by Spartina alterniflora under Macro-Tidal and Storm Surge Conditions

Salt marshes are coastal wetlands that provide many ecosystem services. They protect coasts from erosion, storm surge, and small tsunami by dissipating wave energy. Plant flexibility has been reported to reduce drag force on and wave attenuation of marsh plants. In this study, we proposed a new drag coefficient formula to account plant flexibility so that the classical theory based on rigid cylinders can be extended to predict wave decay by flexible plants. To validate this new formula, the field survey was conducted in the North Hangzhou Bay, China. The wave attenuation was measured in a Spartina alterniflora marsh under a combined macro-tidal and storm surge condition, during which strong bending motions of the plants were observed. The new formula provided better predictions of wave decay than that by the traditional drag formula when compared to the field data measured in both the present study and the earlier studies.

DERIVE A NEW DRAG FORMULA ON POROUS MEDIA LAMINAR FLOW FROM THE MINOR RESISTANCE ASSUMPTION

DERIVE A NEW DRAG FORMULA ON POROUS MEDIA LAMINAR FLOW FROM THE MINOR RESISTANCE ASSUMPTION

A drag force formula for heterogeneous granular flow systems based on finite average statistical method

The existing drag models are mostly based on the assumption of homogenous fluidization. However, the use of a homogeneous drag model to predict a heterogeneous granular flow system will cause a deviation. In this study, we developed a drag force model based on the assumption of heterogeneous fluidization. To prevent weakening of the heterogeneous characteristics in the drag force formula, we propose a finite average statistical method to filter the information of the heterogeneous granular cluster. The filtered information was used to fit the modified drag formula, which can reflect the heterogeneity of the granular cluster considering different configurations. A comparison shows that the new proposed drag formula filtered by the finite average statistical method fits well with energy minimization multi-scale simulation results.

Effect of the rope system deformation on the working cycle of the mobile crane during interaction of wind pressure

This work concerns the analysis of the load motion, taking into account wind pressure and the deformation of the rope system. Disturbances resulting from the physical properties of the linear system are presented. The Kelvin–Voigt model was adopted as the deformable model of the rope system. The carried load was treated as a rigid body. The formulated initial–value problem allows the analysis of load motion during kinematic forcing or its free movement. The wind pressure was included in the analytical model by using the formula for aerodynamic drag. Numerical tests were carried out in conditions resulting from the crane operating standards (ISO 4302).

The importance of the wind-drag coefficient parameterization for hydrodynamic modeling of a large shallow lake

Wind strongly impacts the hydrodynamic and biogeochemical process of large shallow lakes, therefore wind stress also plays an important role in modeling the hydrodynamics and water quality of shallow lakes. In large shallow lakes, it may be necessary to modify the empirical wind-drag coefficient formula derived from ocean surface experiments because lake current velocities may be seriously underestimated in inland waters. To resolve this limitation, we added a wind-drag multiplier (α) to the wind drag formula in a lake hydrodynamic model. We used the Environmental Fluid Dynamics Code (EFDC) to model the hydrodynamics of Upper Klamath Lake (UKL), Oregon. The moment-independent method for global sensitive analysis (GSA) based on sampling of input parameters was utilized. We found that the original model underestimated lake current velocities when compared to field observations, so we developed a modified model with a wind-drag multiplier. This model was calibrated to observed data from June 21–September 12, 2005, and verified with data from May 24–September 25, 2006. The results showed the calibrated modified model resolved the underestimation problem, e.g., at three sites in UKL the water velocity increased by 59–85%, and the relative error for the model decreased by 15–32%. Sensitivity analysis showed the modeled current velocities were more sensitive to the α coefficient than to the bottom roughness height z0 and the coefficients in the original wind-drag formula. We believe the wind-drag multiplier affects wave propagation in the model and reconciles the mismatch between large shallow lake and open ocean conditions. Our results show that a relatively simple modification can alleviate the fundamental mismatch between modeling the hydrodynamics of the open ocean and large shallow lakes.

A Bound on Ekman Pumping

AbstractMomentum transport by boundary layer turbulence causes a weak synoptic‐scale vertical motion. The classical textbook solution for the strength of this Ekman pumping depends on the curl of the surface momentum flux. A new solution for Ekman pumping is derived in terms of the curl of the geostrophic wind and a term that depends in a nontrivial way on the vertical profile of the turbulent momentum flux. The solution is confined to a boundary layer regime that is vertically well mixed and horizontally homogeneous. The momentum flux is computed from a commonly used bulk surface drag formula and a flux jump relation to capture the entrainment flux of momentum at the top of the boundary layer. It is found that the strength of Ekman pumping is bounded. The weakening of Ekman pumping for enhanced turbulent surface friction can be explained from the fact that it will reduce the magnitude of the horizontal wind. It is demonstrated that entrainment of momentum across the top of the boundary layer tends to diminish the large‐scale divergence of the wind. As momentum transport is parameterized in large‐scale models, the analysis is relevant for the understanding and interpretation of the evolution of synoptic‐scale vertical motions as predicted by such models.