Unsteady Drag Coefficient Research Articles

Influence of boundary layer and pressure lag on unsteady aerodynamics of airfoil based on a simple semi‐empirical dynamic stall engineering model

AbstractIn view of the fact that dynamic stall models in the wind energy industry such as ONERA model, Beddoes–Leishman model, and Snel model are mostly semi‐empirical models, and the determination of empirical time constants has a great influence on the model accuracy. To optimize the time constant in dynamic stall model and improve the prediction accuracy of unsteady aerodynamics, the influence of boundary layer and pressure lag on the unsteady performance of the S809 airfoil under 2D flow conditions is explored using a simple semi‐empirical dynamic stall engineering model. The proposed model consists of four first‐order differential equations accounting for attached flow and dynamic separation flow of trailing edge based on the Theodorsen theory. A validation is carried out by the wind tunnel experiment in the Key Laboratory of Wind and Solar Energy Utilization Technology of the Ministry of Education at Inner Mongolia University of Technology. The main conclusions are as follows. The time constants for lag in pressure and boundary layer both have a great influence on the unsteady lift coefficient. When the mean angle of attack is relatively small and the airflow is between the attached flow and the separated flow, appropriately reducing the time constant can make the prediction results closer to the experimental values. When the mean angle of attack is relatively large and the airflow is in condition of fully separated flow, the time constant value can be appropriately increased. The influence of pressure lag and boundary layer lag on the unsteady drag coefficient is not significant.

Experimental Study of Unsteady Drag Coefficient of Droplets in a Liquid–Liquid System

In liquid–liquid contact process, the motion of droplets relative to the surrounding fluid always involves accelerating and decelerating, which affects the mass, heat, and momentum transfer. The lack of experimental data of the unsteady drag coefficient has been one of the limitations on the prediction of the unsteady flow field. In this study, the accelerated and decelerated water droplets in an organic phase were measured by a high-speed camera. The results show that with a decrease in droplet diameter, the acceleration becomes more significant, while the absolute relative velocity decreases, causing a lower Reynolds number. The Basset force and add mass force were solved numerically and compared with drag force. The unsteady drag coefficient is always smaller than the corresponding steady drag coefficient in the case of accelerating relative flow and larger than that in the case of decelerating relative flow. A new unsteady drag coefficient model has been established, which has an acceptable agreement with the experimental data.

Analysis of tidal turbine blade loading due to blade scale flow

A blade element model is developed to include unsteady lift and drag coefficients and it is shown that this provides improved prediction of the spectrum of root bending moment of a tidal turbine blade for two definitions of onset turbulence. Computational Fluid Dynamics (CFD) is used to model the lift and drag forces on a 2D aerofoil using the relative onset flow from the different turbulence generation methods. Inclusion of the magnitude of high frequency fluctuations within the blade load spectra, results in an increased number of load cycles and improves prediction of Damage Equivalent Load (DEL). Discrepancy between prediction and existing experimental data is improved from 15% to within 2%. The signal-to-noise ratio (SNR) of relative velocity has been used to characterise the onset flow, and a relationship is established between onset flow conditions and the resultant fatigue loads (DEL). This has been applied to a number of onset conditions, typical of channel shear flows and upstream turbine wakes. For a given SNR, there is a 3.3 factor of variation in the DEL across the range of tip-speed-ratio (TSR) shown and over a single TSR value there is a 6.5 factor of variation across the DEL, over the largest range of SNR.

Flow over a hydrofoil at incidence immersed within the wake of a propeller

The flow over a hydrofoil in the wake of a marine propeller is studied using large-eddy simulation on a cylindrical grid composed of 3.8 billion points. Four angles of incidence of the downstream hydrofoil are considered, ranging from 0° to 15°. The impact of the propeller wake on the flow within the boundary layer of the hydrofoil is substantial, increasing the skin-friction and producing significant spanwise flows, associated especially with the deflection of the tip and hub vortices. This deflection is strongly influenced by the incidence angle of the hydrofoil, producing an overall expansion of the propeller wake on its pressure side and a contraction on its suction side. The tip and hub vortices are also the major source of pressure fluctuations on the surface of the hydrofoil, affecting this way its unsteady lift and drag coefficients. On the pressure side, the most significant pressure fluctuations are due to the hub vortex, while on the suction side, their maxima originate from the overlapping effects by the tip vortices and the adverse streamwise pressure gradient, promoting the instability of the boundary layer. Pressure fluctuations are an increasing function of the incidence of the hydrofoil on both its pressure and suction sides.

Unsteady shock wave dynamics in accelerating and decelerating flight

AbstractIncreasingly agile manoeuvre is an advantage in the flight of aircraft, missiles and aerial vehicles, but the principles of accelerating aerodynamics in the transonic regime are only now being fully investigated. This study contributes to the understanding of shock and separation effects on drag during axial acceleration, using a simple geometric configuration. Unsteady shock wave behavior was numerically investigated for an axisymmetric cone-cylinder using a commercial solver and the Moving Reference Frame acceleration technique. This acceleration technique was validated using unsteady numerical and experimental methods. The cone-cylinder was accelerated from Mach number 0.6 to Mach number 1.2 at 100g constant and deceleration was from Mach number 1.2 until Mach number 0.6 at –100g constant. Three cone angles were tested for the cone-cylinder with uniform cylinder diameter. Acceleration through the transonic Mach regime was characterised by a delayed and gradual shock wave development when compared to steady state, demonstrating a clear flow history effect. Deceleration through the transonic Mach regime was characterised by shock wave propagation from the base to the nose. New flow structures appeared during deceleration that do not have counterparts in the steady state, including shock interactions and propagating expansion-compression features. Gross changes in the unsteady drag coefficient curves for each cone-angle are explained with reference to unsteady shock wave behaviour for accelerating and decelerating motion.

Influence of baroclinic vorticity production on unsteady drag coefficient in shock–particle interaction

The influence of baroclinic vorticity production on the unsteady drag coefficient in shock–particle interaction is numerically studied. Numerical simulations are performed for shock–particle interaction utilizing a high–resolution axisymmetric solver for the Euler equations that allows for multi-material interface and shock propagation in both the particle and surrounding medium. We consider an aluminum particle in nitromethane and allow for particle deformation. We compute the vorticity production and unsteady drag coefficient as a function of time to explain the complex physical mechanisms that occur during shock–particle interaction. We observe baroclinic vorticity production as the shock propagates over the particle and find that the vorticity is primarily generated at the surface of the particle. After the passage of the shock over the particle, the generated vortex traverses downstream, thus creating a sharpened particle edge and low pressure on the downstream side of the particle, followed by the trapping of the vortex at the particle edge. These mechanisms lead to the generation of a quasi-steady drag force even after the passage of the shock, thus suggesting the importance of baroclinic vorticity production to the unsteady drag coefficient. Finally, we compute the unsteady drag coefficient for various shock Mach numbers and particle ellipticities.

A new method to calculate unsteady particle kinematics and drag coefficient in a subsonic post-shock flow

We introduce a new method (piecewise integrated dynamics equation fit, PIDEF) that uses the particle dynamics equation to determine unsteady kinematics and drag coefficient (CD) for a particle in subsonic post-shock flow. The uncertainty of this method is assessed based on simulated trajectories for both quasi-steady and unsteady flow conditions. Traditional piecewise polynomial fitting (PPF) shows high sensitivity to measurement error and the function used to describe CD, creating high levels of relative error (1) when applied to unsteady shock-accelerated flows. The PIDEF method provides reduced uncertainty in calculations of unsteady acceleration and drag coefficient for both quasi-steady and unsteady flows. This makes PIDEF a preferable method over PPF for complex flows where the temporal response of CD is unknown. We apply PIDEF to experimental measurements of particle trajectories from 8-pulse particle tracking and determine the effect of incident Mach number on relaxation kinematics and drag coefficient of micron-sized particles.

Experimental study and modelling of unsteady aerodynamic forces and moment on flat plate in high amplitude pitch ramp motion

This paper examines the unsteady lift, drag and moment coefficients experienced by a thin airfoil in high-amplitude pitch ramp motion. Experiments have been carried out in a wind tunnel at moderate Reynolds number ($Re\approx 1.45\times 10^{4}$), using a rigid flat-plate model. Forces and moments have been measured for reduced pitch rates ranging from 0.01 to 0.18, four maximum pitch angles ($30^{\circ },45^{\circ },60^{\circ },90^{\circ }$) and different pivot axis locations between the leading and the trailing edge. Results confirm that for reduced pitch rates lower than 0.03, the unsteady aerodynamics is limited to a stall delay effect. For higher pitch rates, the unsteady response is dominated by a buildup of the circulation, which increases with the pitch rate and the absolute distance between the pivot axis and the $3/4$-chord location. This circulatory effect induces an overshoot in the normal force and moment coefficients, which is slightly reduced for a flat plate with a finite aspect ratio close to 8 in comparison with the two-dimensional configuration. A new time-dependent model has been tested for both the normal force and moment coefficients. It is mainly based on the superposition of step responses, using the Wagner function and a time-varying input that accounts for the nonlinear variation of the steady aerodynamics, the pivot point location and an additional circulation which depends on the pitch rate. When compared with experiments, it gives satisfactory results for $0^{\circ }$ to $90^{\circ }$ pitch ramp motion and captures the main effect of reduced pitch rate and pivot point location.

Unsteady Stokes flow near boundaries: the point-particle approximation and the method of reflections

Problems of particle dynamics involving unsteady Stokes flows in confined geometries are typically harder to solve than their steady counterparts. Approximation techniques are often the only resort. Felderhof (see e.g. J. Phys. Chem. B, vol. 109 (45), 2005, pp. 21406–21412; J. Fluid Mech., vol. 637, 2009, pp. 285–303) has developed a point-particle approximation framework to solve such problems, especially in the context of Brownian motion. Despite excellent agreement with past experiments, this framework produces unsteady drag coefficients that depend on particle density. This is inconsistent, since the problem can be formulated mathematically without any reference to the particle’s density. We address this inconsistency in our work. Upon implementing our modifications, the framework passes consistency checks that it previously failed. Further, it is not obvious that such an approximation should work for short-time-scale motion. We investigate its validity by deriving it from a general formalism based on integral equations through a series of systematic approximations. We also compare results from the point-particle framework against a calculation performed using the method of reflections, for the specific case of a sphere near a full-slip plane boundary. We find from our analysis that the reasons for the success of the point-particle approximation are subtle and have to do with the nature of the unsteady Oseen tensor. Finally, we provide numerical predictions for Brownian motion near a full-slip and a no-slip plane wall based on the point-particle approximation as used by Felderhof, our modified point-particle approximation and the method of reflections. We show that our modifications to Felderhof’s framework would become significant for systems of metallic nanoparticles in liquids.

Detailed numerical simulation of unsteady drag coefficient of deformable droplet

Abstract The detailed numerical simulation of the unsteady drag coefficient of deformable droplet is investigated with a mass conserving level set (LS) method. The simulation results indicate that the unsteady drag coefficient is always larger than the steady standard drag coefficient for the present decelerating relative flow. The effects of Weber number, density ratio and viscosity ratio on the unsteady drag coefficient of drop deformation are studied. It is found that the unsteady term (including density ratio) has the most effect on the unsteady drag coefficient and Weber number secondly. The viscosity ratio has little effect on the unsteady drag coefficient due to low Ohnesorge number. The numerical results confirm that the difference between drag coefficient and standard steady drag coefficient has an approximately linear curve fit with the unsteady parameter for low Weber number. These results will lay the foundation for the modeling of unsteady drag coefficient in our further work.

Estimation of unsteady aerodynamic forces using pointwise velocity data

A novel method to estimate unsteady aerodynamic force coefficients from pointwise velocity measurements is presented. As opposed to other existing methodologies, time-resolved full velocity fields are not required. The methodology is based on a resolvent-based reduced-order model which requires the mean flow to obtain physical flow structures and pointwise measurement to calibrate their amplitudes. A computationally affordable time-stepping methodology to obtain resolvent modes in non-trivial flow domains is introduced and compared with previous existing matrix-free and matrix-forming strategies. The technique is applied to the unsteady flow around an inclined square cylinder at low Reynolds number. The potential of the methodology is demonstrated through good agreement between the fluctuating pressure distribution on the cylinder and the temporal evolution of the unsteady lift and drag coefficients predicted by the model and those computed by direct numerical simulation.

Numerical investigation of shock interaction with one-dimensional transverse array of particles in air

In this paper, we present numerical simulations of shock propagation in air over a one-dimensional transverse array of particles. Simulations are carried out by varying the particle spacing and shock Mach number. We compute the unsteady inviscid drag coefficient as a function of time and make relevant comparisons to that for a single particle. We find that deviations in the drag coefficient in time from that of a single particle can be correlated to the acoustic-particle interaction time. Finally, we investigate and classify the interaction of the bow shocks in front of the transverse array of particles.

Fluid forces acting on a cylinder undergoing streamwise vortex-induced vibrations

This brief communication examines the fluid forces acting on a cylinder free to move in the streamwise direction throughout its response regime. The amplitude and phase of the unsteady drag coefficient are estimated from the displacement signals and a simple harmonic oscillator model. We examine the counter-intuitive reduction in vibration amplitude observed in streamwise vortex-induced vibrations (VIV) at resonance, which has remained one of the most poorly understood aspects of VIV. Our results show that it is not caused by a change in the phase of the fluid forcing with respect to the cylinder displacement, as suggested by previous researchers; instead, we show that there is a sudden decrease in the amplitude of the unsteady drag coefficient in this region. The possible cause of this result, relating to three-dimensionality in the wake, is briefly discussed.

Shock interaction with deformable particles using a constrained interface reinitialization scheme

In this paper, we present axisymmetric numerical simulations of shock propagation in nitromethane over an aluminum particle for post-shock pressures up to 10 GPa. We use the Mie-Gruneisen equation of state to describe both the medium and the particle. The numerical method is a finite-volume based solver on a Cartesian grid, that allows for multi-material interfaces and shocks, and uses a novel constrained reinitialization scheme to precisely preserve particle mass and volume. We compute the unsteady inviscid drag coefficient as a function of time, and show that when normalized by post-shock conditions, the maximum drag coefficient decreases with increasing post-shock pressure. We also compute the mass-averaged particle pressure and show that the observed oscillations inside the particle are on the particle-acoustic time scale. Finally, we present simplified point-particle models that can be used for macroscale simulations. In the Appendix, we extend the isothermal or isentropic assumption concerning the point-force models to non-ideal equations of state, thus justifying their use for the current problem.

On the flow past a circular cylinder from critical to super-critical Reynolds numbers: Wake topology and vortex shedding

Large-eddy simulations (LES) of the flow past a circular cylinder are used to investigate the flow topology and the vortex shedding process at Reynolds numbers Re=2.5×105-8.5×105. This range encompasses both the critical and super-critical regimes. As the flow enters the critical regime, major changes occur which affect the flow configuration. Asymmetries in the flow are found in the critical regime, whereas the wake recovers its symmetry and stabilizes in the super-critical regime. Wake characteristic lengths are measured and compared between the different Reynolds numbers. It is shown that the super-critical regime is characterised by a plateau in the drag coefficient at about CD≈0.22, and a quasi-stable wake which has a non-dimensional width of dw/D≈0.4. The periodic nature of the flow is analysed by means of measurements of the unsteady drag and lift coefficients. Power spectra of the lift fluctuations are computed. Wake vortex shedding is found to occur for both regimes investigated, although a jump in frequencies is observed when the flow enters the super-critical regime. In this regime, non-dimensional vortex-shedding frequency is almost constant and equal to St=fvsD/Uref≈0.44. The analysis also shows a steep decrease in the fluctuating lift when entering the super-critical regime. The combined analysis of both wake topology and vortex shedding complements the physical picture of a stable and highly coherent flow in the super-critical regime.

Shock interaction with one-dimensional array of particles in air

In this paper, we present axisymmetric numerical simulations of shock propagation in air over an aluminum particle for Mach numbers up to 10. The numerical method is a finite-volume based solver on a Cartesian grid that allows for multi-material interfaces and shocks. Validation of the solver is demonstrated by comparing to existing experimental data. We compute the unsteady inviscid drag coefficient as a function of time, and show that when normalized by post-shock conditions, the maximum drag coefficient decreases with Mach number. Furthermore, for supercritical Mach numbers, we show that the inviscid steady-state drag asymptotes to a non-zero value due to the presence of a bow shock formed just upstream of the particle. Using this information, we also present a simplified point-particle force model that can be used for mesoscale simulations. Finally, we investigate the dynamics of a shock propagating over a 1-D array of particles aligned in the flow direction. We show that the maximum drag coefficient increases as the shock travels deep into the array and then asymptotes to a final value, which can be as high as 50% more than that of the first particle, depending on Mach number and particle spacing.

Numerical simulations of the early stages of high-speed droplet breakup

Experiments reported in the literature are reproduced using numerical simulations to investigate the early stages of the breakup of water cylinders in the flow behind normal shocks. Qualitative features of breakup observed in the numerical results, such as the initial streamwise flattening of the cylinder and the formation of tips at its periphery, support previous experimental observations of stripping breakup. Additionally, the presence of a transitory recirculation region at the cylinder’s equator and a persistent upstream jet in the wake is noted and discussed. Within the uncertainties inherent to the different methods used to extract measurements from experimental and numerical results, comparisons with experimental data of various cylinder deformation metrics show good agreement. To study the effects of the transition between subsonic and supersonic post-shock flow, we extend the range of incident shock Mach numbers beyond those investigated by the experiments. Supersonic post-shock flow velocities are not observed to significantly alter the cylinder’s behavior, i.e., we are able to effectively collapse the drift, acceleration, and drag curves for all simulated shock Mach numbers. Using a new method that minimizes noise errors, the cylinder’s acceleration is calculated; acceleration curves for all shock Mach numbers are subsequently collapsed by scaling with the pressure ratio across the incident shock. Furthermore, we find that accounting for the cylinder’s deformed diameter in the calculation of its unsteady drag coefficient allows the drag coefficient to be approximated as a constant over the initial breakup period.

Numerical Simulations of Droplet Aerobreakup

The work presented in this thesis aims to bridge an existing gap in the state of droplet aerobreakup knowledge associated with the fundamental flow physics that govern the experimentally observable droplet morphologies. Using direct numerical simulations of the aerobreakup of water cylinders and droplets in the flow behind shock waves in air, we investigate the behavior of the surrounding gas flow to gain insight into the droplet’s deformation and evolution in the stripping breakup regime. The compressible multicomponent Navier-Stokes equations are solved using the Multicomponent Flow Code — a high-order accurate structured finite-volume flow solver with shock- and interface-capturing. Following qualitative descriptions of the aerobreakup process, comparisons are made with available experimental data. In 2D, accurate measurements of the cylinder’s center-of-mass acceleration across a range of incident shock Mach numbers allow characterization of the unsteady drag coefficient. Additionally, mass loss measurements from viscous simulations refute a well-known boundary layer stripping theory. The results of a 3D nonaxisymmetric aerobreakup simulation are presented with an emphasis on describing the intricate flow phenomena observable in the wake region. Subsequent analyses of the surface instabilities and a Fourier decomposition of the flow field reveal asymmetrical azimuthal modulations and broadband instability growth that result in the devolution of the wake region into chaotic flow.

Fluttering flags: An experimental study of fluid forces

Unsteady fluid forces are measured at the onset of flutter and during the post-critical flutter of flags placed in a wind tunnel, focusing on the drag force and the moment around the flagpole. The evolution of these forces during flutter mode switches, induced by varying either the mass ratio or the wind velocity, is discussed by using additional high-speed imaging. For the highest wind velocities, losses of periodicity and snapping events are reported. Time-averaged and unsteady drag coefficients are reported and compared to previous works.

Launch Dynamics of Supersonic Projectiles

The aerodynamics of projectiles launched from barrels of various devices is quite complicated due to their interactions with the unsteady flowfield around them. A computational study using a moving grid method is performed here to analyze various fluid dynamic phenomena in the near field of a gun, such as the projectile–shock wave interactions and interactions between the flow structures and the aerodynamic characteristics of the projectile when it passes through various flow interfaces. Cylindrical and conical projectiles have been employed to study such interactions and the fluid dynamics of the flowfields. The aerodynamic characteristics of the projectile are hardly affected by the projectile configuration during the process of the projectile overtaking the primary blast wave for small Mach numbers. However, it is noticed that the projectile configurations do affect the unsteady flow structures before overtaking and hence, the unsteady drag coefficient for the conical projectile shows considerable variation from that of the cylindrical projectile. The projectile aerodynamic characteristics during the interaction with the secondary shock wave are also analyzed in detail. It is observed that the change in the characteristics of the secondary shock wave during the interaction is fundamentally different for different projectile configurations. Both inviscid and viscous simulations were carried out to study the projectile aerodynamics and the fluid dynamics. Although the effect of the viscosity on the projectile aerodynamic characteristics is not significant, the viscosity greatly affects the unsteady flow structures around the projectile.