Theoretical Drag Research Articles

An Economical Open‐Source Lagrangian Drifter Design to Measure Deep Currents in Lakes

AbstractThe objective was to construct and test an economical, accurate, and open‐source Lagrangian drifter design suitable for lakes <200 km2. Lagrangian drifters are used to trace water currents in marine and freshwater settings and comprise of a low‐friction surface float containing instrumentation for location and environmental measurement, tethered to a high‐friction drogue at the depth of interest. Oceanic drifters are robust but expensive, and this design tailored to inland lake waterbodies fills a durability and cost gap for lake environments. Water‐following characteristics were tested using theoretical drag coefficient calculations, practical drag measurements, and comparison of wind and drifter vectors while deployed on two deep inland lakes (maximum area 175 km2) in the Finger Lakes region of New York, USA. The ratio of drag between float and drogue met or exceeded the minimum value of 40 recommended in the literature, and the vectors of wind and drifter during deployment were independent of one another, meaning the device accurately traced the movement of water currents at depth without undue influence of wind and waves. Each device cost USD $265 in 2021 and was built from materials readily available at hardware and sporting goods stores, allowing their use by research institutions and communities with smaller budgets. This design reliably measured lake currents at sampling depths that ranged from 2 to 30 m. We anticipate that this design will have application to a wide range of hydrodynamic and ecological research where empirical insights to physical processes like lake currents are sought by scientists and managers.

Airfoil Design Framework for Optimized Boundary-Layer Integral Parameters

An airfoil design framework is introduced in which boundary-layer integral parameters serve as the driving design mechanism. The method consists of generating a parameterized pressure distribution capable of producing the desired boundary-layer characteristics for inverse design use. Additionally, by deduction from the Squire–Young theory, the method allows for the determination of the pressure distribution that results in the minimum theoretical drag. To assess this design framework, several airfoils were developed based on the mission requirements of the RQ-4B Global Hawk aircraft. Numerical results obtained using a viscous-inviscid solver of the integral boundary layer and Euler equations showed that the optimized airfoils achieved profile drag reductions of 9.06 and 6.00%, respectively, for α=0° and L/Dmax design points. A validation experimental campaign was also performed using the optimized CA5427-72 airfoil. The acquired data produced the expected pressure distribution characteristics and aerodynamic performance improvements, typifying the efficacy of the design framework.

Theoretical and experimental study on drag torque of wet clutch considering surface tension and shrinkage of oil film

The drag torque produced by the viscous shearing action of lubricating oil is an important component of the power loss of wet clutches. Studying the prediction model and its influencing factors will provide important theoretical support for reducing the drag torque. Taking a single friction pair wet clutch as the research object, this research obtained the oil film shrinkage under different working conditions based on the VOF model. On this basis, a drag torque calculation model considering the surface tension and shrinkage of oil film is established, in which the drag torque is composed of the torque transmitted by oil film in the oil film continuous zone and the torque transmitted by oil-liquid and gas-liquid in the oil film rupture area, respectively. Comparing the theoretical drag torque with the simulation and experimental results, the variation law of drag torque with the speed difference under different oil film thicknesses and different supply flow is obtained, and the validity of the theoretical model is also verified. Results show that increasing the oil film thickness and decreasing the supply flow will lead to a decrease in drag torque, which provides a useful reference for reducing the drag torque.

Theoretical and numerical analysis of drag force at the interface between the dilute and dense phases

Resolving the sharp interface between the dilute and dense phases in gas–solid flows is a bottleneck in fine-grid simulations. This study addresses this issue through the theoretical analysis of an infinite gas–structure interface with arbitrary flow directions. The drag force at the interface is decomposed into three parts: the homogeneous drag forces of the dilute and dense regions and a stress divergence difference term. All the three parts are expressed as the functions of the solid volume fractions, particle Reynolds numbers, and stress divergences of the interface grid and its adjacent grids. The developed theoretical drag models at the interface are verified and improved based on particle-resolved direct numerical simulations (PR-DNSs) of flows past plug-like structures. The models are then tested against PR-DNSs of flows past bubble-containing, spherical, ellipsoidal structures. They yield significantly better performance than the traditional Beetstra et al.'s model [Beetstra et al., “Drag force of intermediate Reynolds number flow past mono- and bidisperse arrays of spheres,” AIChE J. 53(2), 489–501 (2007)].

Theoretical drag analyses of laminar channel and pipe flows with wall roughness

In this paper, an exact expression for the drag coefficient of a streamwise-periodic steady incompressible laminar channel and pipe flow with micro- or macro-scale wall roughness is derived, whereby the drag coefficient is decomposed into contributions from different components of the velocity gradient tensor in the flow field. It is shown through our theoretical analysis that drag reduction cannot be achieved by adding micro- or macro-scale spanwise-periodic/-symmetry wall roughness structures to the smooth inner walls of streamwise-periodic steady incompressible laminar channel/pipe flows while maintaining the same volumetric flow rate. It is also shown that wall roughness produces a higher drag due to two factors: (a) wall roughness induces other non-zero velocity gradient terms apart from the wall-normal/radial gradient of streamwise velocity that exists in a smooth channel/pipe flow; (b) the profile of streamwise velocity in the wall-normal/radial direction deviates from the parabolic profile that produces the minimum kinetic energy loss for a given volumetric flow rate. Finally, numerical simulations of laminar channel flow with longitudinal and transverse bars are conducted, and the numerical results confirm the theoretical finding.

Interspecific differences in the flow regimes and drag of North Pacific skate egg cases.

Skates are a diverse group of dorso-ventrally compressed cartilaginous fishes found primarily in high-latitude seas. These slow-growing oviparous fishes deposit their fertilized eggs into cases, which then rest on the seafloor. Developing skates remain in their cases for 1-4 years after they are deposited, meaning the abiotic characteristics of the deposition sites, such as current and substrate type, must interact with the capsule in a way to promote long residency. Egg cases are morphologically variable and can be identified to species. Both the gross morphology and the microstructures of the egg case interact with substrate to determine how well a case stays in place on a current-swept seafloor. Our study investigated the egg case hydrodynamics of eight North Pacific skate species to understand how their morphology affects their ability to stay in place. We used a flume to measure maximum current velocity, or "break-away velocity," each egg case could withstand before being swept off the substrate and a tilt table to measure the coefficient of static friction between each case and the substrate. We also used the programming software R to calculate theoretical drag on the egg cases of each species. For all flume trials, we found the morphology of egg cases and their orientation to flow to be significantly correlated with break-away velocity. In certain species, the morphology of the egg case was correlated with flow rate required to dislodge a case from the substrate in addition to the drag experienced in both the theoretical and flume experiments. These results effectively measure how well the egg cases of different species remain stationary in a similar habitat. Parsing out attachment biases and discrepancies in flow regimes of egg cases allows us to identify where we are likely to find other elusive species nursery sites. These results will aid predictive models for locating new nursery habitats and protective policies for avoiding the destruction of these nursery sites.

Transition delay using biomimetic fish scale arrays

Aquatic animals have developed effective strategies to reduce their body drag over a long period of time. In this work, the influence of the scales of fish on the laminar-to-turbulent transition in the boundary layer is investigated. Arrays of biomimetic fish scales in typical overlapping arrangements are placed on a flat plate in a low-turbulence laminar water channel. Transition to turbulence is triggered by controlled excitation of a Tollmien–Schlichting (TS) wave. It was found that the TS wave can be attenuated with scales on the plate which generate streamwise streaks. As a consequence, the transition location was substantially delayed in the downstream direction by 55% with respect to the uncontrolled reference case. This corresponds to a theoretical drag reduction of about 27%. We thus hypothesize that fish scales can stabilize the laminar boundary layer and prevent it from early transition, reducing friction drag. This technique can possibly be used for bio-inspired surfaces as a laminar flow control means.

Formation of drops and rings in double-diffusive sedimentation

We conduct numerical simulations to investigate the formation and evolution of drops and vortex rings of particle-laden fingers in double-diffusive convection in stably stratified environments. We show that the temporal evolution can be divided into double diffusion, acceleration and deceleration phases. The acceleration phase is a result of the vanishing temperature perturbation in the drop during the descent in the layer of uniform temperature. The drop decelerates because it transforms into a vortex ring. A theoretical drag model is presented to predict the speed of the spherical drop with the low drop Reynolds number. By formulating the boundary condition based on the vorticity, our drag model gives a more general form of the drag coefficient for small spherical drops and shows good agreement in predicting the drag coefficient. Drops with five particle sizes are compared, and it is found that although the greater vertical settling enhances vertical transport, the final state differs little among the various sizes. Comparison of our drag model with the simulation results under various bulk conditions and previous experimental results shows good model predictability. Finally, a comparison with the salt-finger case shows that the diffusive nature of the dissolved scalar field, along with the wake effect, can result in an apparent loss of mass from the drop and a permanent presence of the connection between the drop and its parent finger. This makes the observed detachment of the particle-laden drop much less likely in the salt-finger case.

Predicting Current-Induced Drag in Emergent and Submerged Aquatic Vegetation Canopies

Canopies formed by aquatic vegetation, such as mangroves, seagrass and kelp, play a crucial role in altering the local hydrodynamics in rivers, estuaries and coastal regions, and thereby influence a range of morphodynamic and biophysical processes. Prediction of the influence of canopies on these hydrodynamic processes requires a fundamental understanding of canopy drag, which varies significantly with both flow conditions and canopy properties (such as density and submergence). Although our knowledge on canopy drag has increased significantly in recent decades, a conclusive, physics-based description for canopy drag that can be applied to both emergent and submerged canopies is currently lacking. Here, we extend a new theoretical canopy drag model (that employs the velocity between canopy elements as the reference velocity) to submerged aquatic canopies. The model is validated for the first time with direct measurements of drag forces exerted by canopies across broad ranges of flow conditions and canopy density and submergence. The skill and broader applicability of the model are further assessed using a comprehensive set of existing experimental data, covering a broad range of natural conditions (including flexible vegetation). The resulting model provides a simple tool to estimate canopy drag forces, which govern hydraulic resistance, sediment transport and biophysical processes within aquatic ecosystems.

Sensitivity of surface drag predictions using Monin–Obukhov similarity theory in non-ideal flow conditions

Turbulent flux measurements in the atmospheric surface layer (ASL), acquired from a vertical tower of sonic anemometers, are used to investigate the transient behavior of the drag coefficient over a relatively flat, homogeneous, desert playa surface. The goal of the study is to quantify the extent to which Monin–Obukhov similarity theory (MOST) successfully predicts the empirical drag coefficient when the data are not necessarily restricted to idealized conditions. Here, “idealized” refers to the conditions under which MOST is assumed to be valid, namely that the ASL exhibits statistically stationary behavior and stresses are constant with height. In order to quantify the agreement between the empirical and theoretical drag coefficients, uncertainty bands as a function of time of day were calculated for each. These uncertainty bands account for random errors and nonstationarity effects in the momentum flux, horizontal wind velocity, and stability parameter, as well as uncertainties associated with the model parameters appearing in MOST, i.e., the von Karman constant and aerodynamic roughness length. Results show that the average uncertainties in the empirical and theoretical drag coefficients are about $$\pm \,30\%$$ and $$\pm \,20\%$$ , respectively. In the case of the former, the major contribution lies from uncertainty in the momentum flux measurements; while, the latter is dominated by uncertainty in the von Karman constant. Joint probability density functions reveal that large differences between the empirical and theoretical drag coefficient have a high probability of occurrence when the ASL is nonstationary and/or exhibits a non-constant stress layer.

Global Optimization of Wing Aerodynamic Optimization Case Exhibiting Multimodality

An investigation into an aerodynamic optimization benchmark case that exhibits multimodality is presented using a global optimization approach. The recently suggested case 6 of the AIAA Aerodynamic Design Optimization Discussion Group involves the drag minimization of a rectangular NACA0012 wing subject to lift and the root bending moment, as well as other geometric constraints. In this paper, optimization of that case using a state-of-the-art constrained population-based global optimization framework is presented. Shape control is achieved via a hierarchical application of the radial basis function domain element approach. Three different fidelities of optimization cases are performed to investigate the multimodality of various aspects of the full problem, with four independent runs of each case being performed. When considering chord optimization, the optimizer converges to two independently identifiable minima with very similar performances. When adding dihedral and sweep variation, further multimodality is introduced, although this multimodality is reasonably well defined, with minima including forward and rearward sweeps, and an upward and a downward winglet. Introducing thickness variables leads to a highly multimodal problem due to the coupling between the chord and thickness variables. The theoretical minimum drag for this case is 24.7 counts, which a number of the runs gets close to achieving.

Numerical investigation of V-shaped riblets and an improved model of riblet effects

Symmetric V-shaped riblets are simulated by using the computational fluid dynamic method to understand the riblet effects on the turbulent boundary layer and the skin friction reduction. Three classical turbulence models, namely Spalart–Allmaras, shear stress transport, and re-normalization group k-epsilon models, are investigated under different grid densities. The re-normalization group model produces good results consistent with the experiment, as compared with the existing theoretical and experimental drag results of the flat plate and the V-shaped riblets with different sizes. Simulating V-shaped riblets yield the unexpected discovery that the shear stress transport model produces large errors, and the Spalart–Allmaras model even produces results of qualitative errors. Another finding is that von Kármán’s constants can no longer meet the requirement of describing velocity profiles in the logarithmic law layer. Aside from the traditional shift of the logarithmic law’s intercept, the slope is also changed by riblet height and spacing. Therefore, an improved model of riblet effects is proposed by redefining von Kármán’s constants.

Investigation and Analysis on Racing Car Front Wings

A great number of studies have been done on aerodynamic test for the racing car. Due to critical competition among the racing car teams, the results of the studies will only been published when they are obsolete. This study was conducted by using the Computational Fluid Dynamics (CFD) utilising SolidWorksTM to perform simulations about the airflow on the front and rear wings of a racing car with different angles of attack. The analysis generates the theoretical Lift force, Down force, and Drag force for a racing car wing. Consequently, at higher velocity, the down force and interrelated induced drag increases. The maximum speed on a straight part is thus reduced due to the increase in induced drag. Finally, tests from the C15 wind tunnel also show similar trends to those derived from the simulations. Compared with results of Model geometry and wind tunnel tested, it is shown that an angle between -10° and -20° below the horizontal indicates the stalling conditions.

A Numerical Investigation of Cumulus Thermals

Abstract Although the steady, entraining, updraft plume is widely taken as the foundational concept of cumulus convection, past studies show that convection is typically dominated by thermals that are transient, more isotropic in shape, and possess interior vortical circulations. Here, several thousand such thermals are tracked in cloud-resolving simulations of transient growing convective events. Most tracked thermals are small (with radius R < 300 m), ascend at moderate rates (~ 2–4 m s−1), maintain an approximately constant size as they rise, and have brief (4–5 min) lifetimes, although a few are much larger, faster, and/or longer lived. They show slight vertical elongation, but few, if any, would be described as plumes. As convection deepens, thermals originate higher up, are larger, and rise faster, although radius and ascent rate are only weakly correlated among individual thermals. The main force opposing buoyancy is a nonhydrostatic pressure drag, not mixing of momentum. This drag can be expressed in terms of a drag coefficient cd that decreases as convection intensifies: deep convective thermals are less damped, with cd ~ 0.2, while shallow convective thermals are more damped, with cd ~ 0.6. The expected dependence of cd based on theoretical form and wave drag coefficients for a solid sphere is inconsistent with these results, since it predicts the opposite dependence on the Froude number. Thus, a theory for drag on cumulus thermals is not straightforward. Overall, it is argued that thermals are a more realistic prototype for atmospheric deep convection than plumes, at least for the less organized convection types simulated here.

Averaged Propulsive Body Acceleration (APBA) Can Be Calculated from Biologging Tags That Incorporate Gyroscopes and Accelerometers to Estimate Swimming Speed, Hydrodynamic Drag and Energy Expenditure for Steller Sea Lions.

Forces due to propulsion should approximate forces due to hydrodynamic drag for animals horizontally swimming at a constant speed with negligible buoyancy forces. Propulsive forces should also correlate with energy expenditures associated with locomotion—an important cost of foraging. As such, biologging tags containing accelerometers are being used to generate proxies for animal energy expenditures despite being unable to distinguish rotational movements from linear movements. However, recent miniaturizations of gyroscopes offer the possibility of resolving this shortcoming and obtaining better estimates of body accelerations of swimming animals. We derived accelerations using gyroscope data for swimming Steller sea lions (Eumetopias jubatus), and determined how well the measured accelerations correlated with actual swimming speeds and with theoretical drag. We also compared dive averaged dynamic body acceleration estimates that incorporate gyroscope data, with the widely used Overall Dynamic Body Acceleration (ODBA) metric, which does not use gyroscope data. Four Steller sea lions equipped with biologging tags were trained to swim alongside a boat cruising at steady speeds in the range of 4 to 10 kph. At each speed, and for each dive, we computed a measure called Gyro-Informed Dynamic Acceleration (GIDA) using a method incorporating gyroscope data with accelerometer data. We derived a new metric—Averaged Propulsive Body Acceleration (APBA), which is the average gain in speed per flipper stroke divided by mean stroke cycle duration. Our results show that the gyro-based measure (APBA) is a better predictor of speed than ODBA. We also found that APBA can estimate average thrust production during a single stroke-glide cycle, and can be used to estimate energy expended during swimming. The gyroscope-derived methods we describe should be generally applicable in swimming animals where propulsive accelerations can be clearly identified in the signal—and they should also prove useful for dead-reckoning and improving estimates of energy expenditures from locomotion.

Evaluation of the Drag Force on Single-Walled Carbon Nanotubes in Rarefied Gases

The drag force on carbon nanotubes (CNTs) in dilute gases has been previously derived. However, the drag force formulae involve collision integrals, which are complex functions of the gas-CNT interaction potential. The unavailability of the collision integrals and interaction potential makes the application of the theoretical drag force laws impossible. In this work, we develop a potential model for the interaction between a gas and single-walled CNT. The collision integrals are then calculated based on the potential and empirical expressions are proposed. Finally, the drag force is computed directly through molecular dynamics simulations and compared with the theoretical predictions.

Modeling of the flue gas desulfurization in a CFB riser using the Eulerian approach with heterogeneous drag coefficient

Flue gas desulfurization (FGD) in a CFB at moderate temperatures with the T–T sorbent, a hydrated calcium-based sorbent was numerically simulated using the Eulerian approach with two drag models for heterogeneous gas–solid flow, one is an EMMS-based theoretical model (QL-EMMS), the other is an experimental model (O-S). The QL-EMMS model was analyzed in terms of the cluster parameters, the drag reduction and CFB modeling by comparing with the experimental data and results of the O-S model. Results using a heterogeneity index show that the QL-EMMS model is more accurate than other theoretical drag models, but there are still some differences from experimental data due to an inaccurate description of the solid volume fractions in clusters. The factors influencing the moderate temperature desulfurization were then investigated based on the accurate O-S drag model. Predictions of the desulfurization efficiency and reaction rate indicate that higher solid volume fractions and more homogeneous solid distributions lead to higher desulfurization efficiencies. The reaction time also has an important effect on the desulfurization.

Neutrally buoyant particle dynamics in fluid flows: Comparison of experiments with Lagrangian stochastic models

We study the validity of various models for the dynamics of finite-sized particles in fluids by means of a direct comparison between theory and experimentally measured trajectories and velocities of large numbers of particles in chaotic two-dimensional flow. Our analysis indicates that finite-sized particles follow the predicted particle dynamics given by the Maxey-Riley equation, except for random correlated fluctuations that are not captured by deterministic terms in the equations of motion, such as the Basset-Boussinesq term or the lift force. We describe the fluctuations via spectral methods and we propose three different Lagrangian stochastic models to account for them. These Lagrangian models are stochastic generalizations of the Maxey-Riley equation with coefficients calibrated to the experimental data. We find that one of them is capable of describing the observed fluctuations fairly well, while it also predicts a drag coefficient in near agreement with the theoretical Stokes drag.

Thin Film Nitinol Microstent for Aneurysm Occlusion

Thin film nitinol produced by sputter deposition was used in the design of microstents intended to treat small vessel aneurysms. Thin film microstents were fabricated by "hot-target" dc sputter deposition. Both stress-strain curves and differential scanning calorimetry curves were generated for the film used to fabricate stents. The films used for stents had an A(f) temperature of approximately 36 degrees C allowing for body activated response from a microcatheter. The 10 microm film was only slightly radio-opaque; thus, a Td marker was attached to the stents to guide fluoroscopic delivery. Thin film microstents were tested in a flow loop with and without nitinol support skeletons to give additional radial support. Stents could be compressed into and easily delivered with <3 Fr catheters. Theoretical frictional and wall drag forces on a thin film nitinol small vessel vascular stent were calculated, and the radial force exerted by thin film stents was evaluated theoretically and experimentally. In vivo studies in swine confirmed that thin film nitinol microstents could be deployed accurately and consistently in the swine cranial vasculature.

Lee waves from a sphere in a stratified flow

Two asymptotic analyses of the generation of lee waves by horizontal flow at velocityUof a stratified fluid of buoyancy frequencyNpast a sphere of radiusaare presented, for either weak or strong stratification, corresponding to either large or small internal Froude numberF=U/(Na), respectively. ForF⋙1, the fluid separates into two regions radially: an inner region of scaleawith three-dimensional irrotational flow unaffected by the stratification, and an outer region of scaleU/Nwith small-amplitude lee waves generated by theO(1) vertical motion in the inner region. ForF⋘1, the fluid separates into five layers vertically: from the lower dividing streamsurface situated at a distanceU/Nabove the bottom of the sphere to the upper dividing streamsurface situated at a distanceU/Nbelow the top, there is a middle layer with two-dimensional horizontal irrotational flow; from the upper dividing streamsurface to the top of the sphere, and from the lower dividing streamsurface to the bottom, there are top and bottom transition layers, respectively, with three-dimensional flow; above the top and below the bottom, there are upper and lower layers, respectively, with small-amplitude lee waves generated by theO(F) vertical motion in the transition layers.The waves are calculated where they have small amplitudes. The forcing is represented by a source of mass: forF⋙1, the surface distribution of singularities equivalent to the sphere in three-dimensional irrotational flow; forF⋘1, the horizontal distribution of singularities equivalent, in the upper (resp. lower) layer, to the flat cut-off obstacle made up of the top (resp. bottom) portion of the sphere protruding above (resp. below) the upper (resp. lower) dividing streamsurface. The analysis is validated by comparison of the theoretical wave drag with existing experimental determinations. ForF⋙1, the drag coefficient decreases as (lnF+7/4-γ)/(4F4), with γ the Euler constant; forF⋘1, it increases as$(32\surd2)/(15\upi)F^{3/2}$. The waves have the crescent shape of the three-dimensional lee waves from a dipole, modulated by interferences associated with the finite size of the forcing. For strong stratification, the hydrostatic approximation is seen to produce correct leading-order drag, but incorrect waves.