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Showing papers on "Drag coefficient published in 2022"


Journal ArticleDOI
TL;DR: In this article , the authors considered the simple case of two identical spherical particles with different relative orientations and particle distances and discussed the flow and fluid-particle heat-transfer characteristics by numerical simulation of supercritical water flowing around the fixed twoparticle system.

39 citations


Journal ArticleDOI
TL;DR: In this article , the Lattice Spring Method (LSM) is combined with the Immersed Boundary-Lattice Boltzmann Method (IB-LBM) to investigate fluid-elastic body interaction problems in unsteady conditions.

16 citations


Journal ArticleDOI
Michael Hanson1
TL;DR: In this article , the authors focused on the terminal velocity and drag coefficient models for disc-shaped particles and proposed models fitted by the LM + UGO optimization algorithm, have more accuracy and stability than the published models.

15 citations


Journal ArticleDOI
TL;DR: In this article , a sweeping jet (SWJ) actuator is integrated into the base of a flat-back Ahmed body such that its nozzle width corresponds to about 2/3 of the bluff body base width.
Abstract: The dynamics of the wake behind a flat-back Ahmed body are modified using a large-scale fluidic oscillator, aiming at drag reduction and the reflectional symmetry breaking (RSB) mode suppression. In the present experiment, the sweeping jet (SWJ) actuator is integrated into the base of the bluff body such that its nozzle width corresponds to about 2/3 of the bluff body base width. The jet is sweeping in the horizontal plane, which coincides with the plane of the wake flow state switching due to the RSB mode. The impact of the SWJ actuator is evaluated for three different locations along the base's height, and for a range of blowing coefficients. The base suction coefficient is recorded from four pressure measurements at the base, while the drag coefficient is obtained from direct force and moment measurements. Particle image velocimetry of the near wake reveals the modifications of the mean flow, which elucidate on the changes in the base suction and drag coefficients. Both drag reduction and RSB mode suppression are achieved, however, not for the same blowing coefficient. The symmetrized wake yields a near Gaussian distribution of the base pressure gradients around zero in both gy and gz directions. This result is corroborated by the joint probability distributions of pitching and yawing moment fluctuations from force balance measurements.

15 citations


Journal ArticleDOI
TL;DR: In this article , the influence of a wide range of particle size distributions (PSDs) on the soil permeability coefficient is quantified and a prediction equation for k is proposed based on the PSD of the particle system.

14 citations


Journal ArticleDOI
Chunxiang Li1
TL;DR: In this paper , a hybrid algorithm combining the BP neural network (BPNN) and particle swarm optimization (PSO) is employed to optimize the parameters in the model to estimate the vehicle drag coefficient in a platoon.

11 citations


Journal ArticleDOI
TL;DR: In this paper , the authors numerically investigated the time-resolved laminar flow over rectangular cylinders with cross-sectional aspect ratio Ar varying from square cylinder (Ar = 1) to normal flat plate (ar = 0.1) for Reynolds number Re = 40-100.

11 citations


Journal ArticleDOI
TL;DR: In this article , a new screen force model based on numerical simulations of a portion of knotless net panels is proposed, which is validated through a computational grid and domain independence study for the mean hydrodynamic loads on full size net panels.

10 citations



Journal ArticleDOI
TL;DR: In this article , a numerical analysis of the two-dimensional flow behavior of power-law fluids past a rotating elliptic cylinder in unconfined domain has been performed using an ANSYS (FLUENT 15) method.
Abstract: In this paper, a numerical investigation is done to understand the two-dimensional flow behaviorof power-law fluids past a rotating elliptic cylinder in unconfined domain. The flow field around the cylinder is calculated using ANSYS (FLUENT 15) The sliding mesh method (SMM) is used to treat moving solid–fluid boundary. The engineering parameters like drag coefficient, lift coefficient and complete kinematic variables such as stream function and vorticity have been calculated for the range of dimensionless parameters namely aspect ratio of the cylinder e=0.1 and 0.7, rotational speed (1≤α≤4), Reynolds number (5≤Re≤40) and power-law index (0.4≤n≤1.8). The emphasis has been given to see the effect of the power-law index (n) and Reynolds number (Re), rotational speed α and aspect ratio of cylinder e on flow characteristics around the cylinder in laminar flow regime. Before presenting the results, a qualitative as well as a quantitative comparison has been done between the present result and experimental and numerical result presented in the literature. Further, a detailed streamline and vorticity patterns in the vicinity of the cylinder is shown to demonstrate the wake formation phenomenon around the cylinder. The increase in rotational speed and power-law index exhibit a similar effect on wake formation phenomena. Due to rotational motion of the cylinder drag and lift coefficient vary in periodic manner with time/orientation of cylinder. The average value of drag is seen to decrease with Re and α and a complete trend is displayed with power-law index, n. The negative lift coefficient is observed due to anticlockwise motion of the cylinder for all parameters considered here. The results of this work clearly show the strong dependency of flow phenomenon around the rotating elliptic cylinder on power-law index, rotational speed, aspect ratio, and Reynolds number.

10 citations


Journal ArticleDOI
TL;DR: In this article , a new formula for Cd was developed by introducing the dimensionless particle diameter (d) and two shape descriptors, and the absolute error of the new formula is 15.2%, smaller than those (42.5-72.8%) of other existing formulas.

Journal ArticleDOI
TL;DR: In this article , the effect of canards' geometric design on the rear-body wake using the application of CFD-based optimization is investigated, and 7 design variables based on the geometry of the canard are considered, and the objective functions are set to be drag and lift coefficients that are aimed to be minimized.
Abstract: Canard is a cutting-edge aerodynamic attachment for lowering the vehicle's drag coefficient by efficiently directing the airflow as well as reducing the lift coefficient by enhancing down-force. This paper aims to simulate the airflow crossing over the car to investigate the effect of canards' geometric design on the rear-body wake using the application of CFD-based optimization. Hence, 7 design variables based on the geometry of the canard are considered, and the objective functions are set to be drag and lift coefficients that are aimed to be minimized. Firstly, ANSYS Fluent is utilized to generate CFD calculations for a series of Design of Experiment (DOE) points. Then, the GMDH-ANN processes the results to elicit the polynomials that demonstrate the relation of design variables and objective functions. A genetic algorithm is next implemented for multi-objective optimization using polynomials as its input, and consequently, Pareto optimal points are achieved. The numerical results show that an appropriate design for canards on the rear bumper causes a potential drag and lifts reduction of 9.62% and 9.6% in comparison to the car without canards. Moreover, the size of the wake behind the car decreases and the differences in the pressure distribution between car fore-body and rear-body is reduced. Finally, the fuel efficiency is potentially enhanced due the changes in car drag coefficient and frontal area.

Journal ArticleDOI
TL;DR: In this article , particle-resolved direct numerical simulation is used to calculate the drag and lift forces acting on linear and irregular agglomerates formed by spherical particles, and a new correlation is also proposed to predict the lift coefficient CL for linear agglomers.
Abstract: Nano- and micrometer particles tend to stick together to form agglomerates in the presence of attractions. An accurate calculation of the drag and lift forces on an agglomerate is a key step for predicting the sedimentation rate, the coagulation rate, the diffusion coefficient, and the mobility of the agglomerate. In this work, particle-resolved direct numerical simulation is used to calculate the drag and lift forces acting on linear and irregular agglomerates formed by spherical particles. For linear agglomerates, the drag coefficient CD follows the sine squared function of the incident angle. The ratio between CD of a linear agglomerate and that for a sphere increases with the agglomerate size, and the increasing rate is a function of the Reynolds number and the incident angle. Based on this observation, explicit expressions are proposed for CD of linear agglomerates at two reference incident angles, 60° and 90°, from which CD at any incident angle can be predicted. A new correlation is also proposed to predict the lift coefficient CL for linear agglomerates. The relative errors for the drag and lift correlations are ∼2.3% and ∼4.3%, respectively. The drag coefficient for irregular agglomerates of arbitrary shape is then formulated based on the sphericity and the crosswise sphericity of agglomerates with a relative error of ∼4.0%. Finally, the distribution of the lift coefficient for irregular agglomerates is presented, which is non-Gaussian and strongly depends on the structure. The mean values and the standard deviations of CL can be well correlated with the Reynolds number.

Journal ArticleDOI
TL;DR: In this article , the authors investigate the characteristics of a dynamic wake and of flow separation for a square cylinder with steady suction at its leading-edge corners, and the measurements were analyzed by applying a proper orthogonal decomposition (POD) to study the control effectiveness.
Abstract: We experimentally investigate the characteristics of a dynamic wake and of flow separation for a square cylinder with steady suction at its leading-edge corners. The wind tunnel experiments were conducted at a Reynolds number of 5946, and suction slots were manufactured symmetrically at the leading corners of the square cylinder. Steady suction was characterized with a suction momentum coefficient Cμ varying from 0.0227 to 0.3182. A time-resolved particle image velocimetry system was used to evaluate the control of leading-edge suction at different Cμ. Next, the measurements were analyzed by applying a proper orthogonal decomposition (POD) to study the control effectiveness. The POD results suggest that the first four modes of wake vortex shedding are transformed in controlled cases and that periodic Karman vortex shedding is suppressed. The results also show that, even with a very small momentum coefficient, the steady suction at the leading-edge corners stabilizes the cylinder wake. The wake region becomes longer and narrower in comparison with the baseline case. In addition, modifications of separation flow were visualized. At quite small Cμ, flow separation at the leading-edge corners is considerably suppressed. Upon increasing the suction momentum coefficient to 0.1364, flow separation at the leading edges is almost eliminated. Finally, we estimate the effect of drag reduction due to the leading-edge suction.

Journal ArticleDOI
TL;DR: In this article , a 3D printing structured porous surface was introduced and experimentally investigated to manipulate the dynamic wake evolution and flow characteristics around a circular cylinder at a Reynolds number of 2.02×104.

Journal ArticleDOI
21 Jan 2022-Polymers
TL;DR: A meta-model was developed using stacking technique to ensemble RF, XGBoost and DNN models and output a prediction based on the individual learner’s predictions and a DNN meta-regressor, which consistently outperformed the individual models on all datasets.
Abstract: This study presents a framework based on Machine Learning (ML) models to predict the drag coefficient of a spherical particle translating in viscoelastic fluids. For the purpose of training and testing the ML models, two datasets were generated using direct numerical simulations (DNSs) for the viscoelastic unbounded flow of Oldroyd-B (OB-set containing 12,120 data points) and Giesekus (GI-set containing 4950 data points) fluids past a spherical particle. The kinematic input features were selected to be Reynolds number, 0

Journal ArticleDOI
TL;DR: In this paper , a flexible plate was used at the free-end leading edge of a wall-mounted finite square cylinder to modulate the aerodynamic forces on a wall mounted finite-square cylinder.
Abstract: A flexible plate vertically clamped at the free-end leading edge was used to modulate the aerodynamic forces on a wall-mounted finite square cylinder. The side width (d) of the cylinder was 40 mm and the aspect ratio (H/d) was 5. The flexible plate was made of low-density polyethylene, with a width of d and thickness of 0.04 mm. The length of the flexible plate ranged from d/8 to d. All measurements were carried out in a low-speed wind tunnel with the free-stream velocity (U∞) ranging from 4 to 20 m/s, corresponding to a Reynolds number ranging from 10 960 to 54 800. It was found that the flexible plate behaves distinctly depending on its length and has significant effects on the aerodynamic forces on the finite square cylinder. When U∞ is smaller than the critical velocity Ucr, which is closely related to the length of the plate, the plate statically deforms, having a negligible influence on the aerodynamic forces on the cylinder. When U∞ exceeds Ucr, the plate flaps periodically, resulting in a significant reduction in the aerodynamic forces. The maximum reduction in the mean drag, fluctuating drag, and fluctuating lateral force reaches approximately 5%, 25%, and 60%, respectively. The reduction in the aerodynamic forces is insensitive to both the plate length and flapping frequency. Flow visualization and particle image velocimetry results point out that the flapping plate induces large-scale vortices in the free-end shear flow, which suppress the formation of spanwise vortex shedding and make the upper part of the near wake symmetrical. The flapping configuration of the flexible plate and the corresponding pressure fluctuation on the free end were also addressed.

Journal ArticleDOI
TL;DR: In this article , the effect of the shape and size of the grooves on flow dynamics over a circular cylinder was investigated numerically for low freestream Reynolds number ranging from 50 to 300, and the predicted flow-field for the grooved cylinder cases were analyzed and compared with that for the smooth cylinder.
Abstract: Laminar cross-flow over a circular cylinder with smooth as well as longitudinally grooved surfaces was investigated numerically for low freestream Reynolds number ranging from 50 to 300. Most of the currently published works on similar flow configurations considered high Reynolds number in the turbulent regime. The computations were performed by the ANSYS Fluent code. The numerical results were validated against the experimental results. The triangular V-shaped, the U-shaped, and Rectangular shaped grooves, with unit aspect ratio; along with the smooth cylinder, are considered. The predicted flow-field for the grooved cylinder cases are analyzed and compared with that for the smooth cylinder. The aerodynamic loads, time-averaged velocities, rms velocity, vorticity, and pressure coefficient plots around the smooth and grooved cylinders were compared to explore the effect of the shape of the grooves on flow dynamics over the cylinder. The flow physics around the cylinder is significantly affected by the shape and size of the grooves, and their circumferential distribution. It is found that the U-grooves significantly decrease the mean drag coefficient (around 13% for Re = 200 and 10% for Re = 300 compared to the smooth cylinder case). The grooves decreased viscous drag significantly (up to 30%) compared to pressure drag. The grooves on the cylinder also delayed the separation and reduced the extent of the recirculation zone in the cylinder wake due to the presence of recirculating flow within the individual grooves. The conclusion is that the properly designed longitudinal grooves on the cylinder in a cross-flow can be used advantageously for reducing aerodynamic loads like drag and lift force in laminar flow as well.

Journal ArticleDOI
TL;DR: In this article, the authors compared the separation angles, drag coefficients and average Sherwood numbers for both clean and fully contaminated spherical bubbles and proposed a correlation giving the separation angle versus Re and θ cap.

Journal ArticleDOI
TL;DR: In this article, the effect of the shape and size of the grooves on flow dynamics over a circular cylinder was investigated numerically for low freestream Reynolds number ranging from 50 to 300, and the predicted flow-field for the grooved cylinder cases were analyzed and compared with that for the smooth cylinder.
Abstract: Laminar cross-flow over a circular cylinder with smooth as well as longitudinally grooved surfaces was investigated numerically for low freestream Reynolds number ranging from 50 to 300. Most of the currently published works on similar flow configurations considered high Reynolds number in the turbulent regime. The computations were performed by the ANSYS Fluent code. The numerical results were validated against the experimental results. The triangular V-shaped, the U-shaped, and Rectangular shaped grooves, with unit aspect ratio; along with the smooth cylinder, are considered. The predicted flow-field for the grooved cylinder cases are analyzed and compared with that for the smooth cylinder. The aerodynamic loads, time-averaged velocities, rms velocity, vorticity, and pressure coefficient plots around the smooth and grooved cylinders were compared to explore the effect of the shape of the grooves on flow dynamics over the cylinder. The flow physics around the cylinder is significantly affected by the shape and size of the grooves, and their circumferential distribution. It is found that the U-grooves significantly decrease the mean drag coefficient (around 13% for Re = 200 and 10% for Re = 300 compared to the smooth cylinder case). The grooves decreased viscous drag significantly (up to 30%) compared to pressure drag. The grooves on the cylinder also delayed the separation and reduced the extent of the recirculation zone in the cylinder wake due to the presence of recirculating flow within the individual grooves. The conclusion is that the properly designed longitudinal grooves on the cylinder in a cross-flow can be used advantageously for reducing aerodynamic loads like drag and lift force in laminar flow as well.


Journal ArticleDOI
TL;DR: In this paper , Wang et al. measured the drag forces acting on a circular cylinder model using wind tunnel balance when porous materials with different permeability are applied within different intersection angles on the trailing edge and leading edge, and the flow fields were visualized with a particle image velocimetry system with high time resolution.
Abstract: Due to its unique pore structure, porous materials have the potential to be used in the fields of acoustic noise reduction and flow drag reduction control. In order to study their effects and mechanism of drag reduction on the flow around a circular cylinder, experiments are conducted in a low-speed wind tunnel with low turbulence intensity. The drag forces acting on a circular cylinder model are measured using wind tunnel balance when porous materials with different permeability are applied within different intersection angles on the trailing-edge and leading edge, and the flow fields are visualized with a particle image velocimetry system with high time resolution. The method of dynamic mode decomposition (DMD) is also used for reduced-order analysis of the vorticity field in the wake of the cylinder. The measured drag forces and wake flow fields are then compared with those of a smooth cylinder, and the results show that porous materials laid on the trailing-edge can reduce drag, when a porous material with 20 pores per inch is laid within 270° on the leeward side, the best effect of the drag reduction ratio of 10.21% is reached. The results of flow visualization indicate that after the porous material is applied, the vortex region in the wake of the cylinder is expanded; both the frequency of vortex shedding and the magnitude of vorticity fluctuation decrease; the Reynolds-shear-stress decreases significantly, and both indicate that vorticity is dissipated earlier. The results of DMD analysis show that porous materials can effectively relax the energy of vortices in different modes.

Journal ArticleDOI
TL;DR: In this article , Wang et al. studied the gas-liquid mass transfer from spherical bubbles for various Reynolds numbers (1⩽Re ⩽300), Schmidt numbers and bubble surface contamination degrees.

Journal ArticleDOI
TL;DR: In this paper , the effects of aspect ratios on the wake flow of a simplified ground transportation system (GTS) model using improved delayed detached-eddy simulation (IDDES) at a Reynolds number of 2.7
Abstract: This study numerically investigates the effects of aspect ratios on the wake flow of a simplified ground transportation system (GTS) model using improved delayed detached-eddy simulation (IDDES) at a Reynolds number of 2.7 × 104. The aspect ratio Ra* ∈ [1.0, 2.0] is defined as the ratio of the height ( H, variable) to the width ( W, constant) of the GTS. The primary purpose of this work is to identify the relationship between the aspect ratio and the wake flow topology. The accuracy of the IDDES method has been validated by comparing the recirculation bubble configuration, vortex core position, velocity profiles, and aerodynamic drag of the baseline model ( Ra* = 1.41) with those obtained from the previous large-eddy simulation study and the wind tunnel experiment. The results show that three typical flow states are observed in the near-wake region for various aspect-ratio cases. The aerodynamic drag increases by 4.60% and 2.06% for the aspect-ratio value equal to Ra* = 2.0 and Ra* = 1.8 (flow state II) and reduces by 6.75%, 7.37%, and 7.98% for the models with the aspect-ratio value of Ra* = 1.15, Ra* = 1.05, and Ra* = 1.0 (flow state III) compared to the aerodynamic drag of the baseline model with the aspect-ratio value of Ra* = 1.41 (flow state I). The dominant shedding frequency of the turbulent wake flow is identical for the aspect-ratio cases when the corresponding wake topology stays in the same flow state. The flow state acts as the substantial factor, which has an essential influence on the GTS's wake flow and its inducing aerodynamic response.

Journal ArticleDOI
TL;DR: In this article , the authors proposed a novel study on finding the best possible combination of bluff body shape and flag configuration to enhance the low-speed wind energy harvesting, in which several bluff body shapes in different cross sections and flag configurations were considered for finding an appropriate combination.
Abstract: Vortex-induced vibration (VIV) is an appropriate mechanism to harvest energy from the low-speed wind energy by flexible piezoelectric flags as transducers. To enhance the low-speed wind energy harvesting, this work proposes a novel study on finding the best possible combination of bluff body shape and flag configuration. This study considered several bluff body shapes in different cross sections and flag configurations as two crucial parameters for finding an appropriate combination. In high flexible piezoelectric flag, zero strain or electrical canceling point along the length of the flag is an important parameter that could be considered in energy harvester design. To this purpose, wind tunnel experiments were conducted to investigate the combination of proper bluff body shape and flag configuration to improve the harvester performance. The proposed bluff body shapes are classified by drag and lift coefficients which are calculated by a computational fluid dynamics (CFD) analysis. Then, several flag configurations in different active area length were clamped to these bluff bodies and tested in the wind tunnel in low wind speed range. The analysis in time and frequency domain of the acquired voltage lead to the conclusion that in low wind speed the bluff body with higher drag coefficients can excite more the longer full active piezoelectric flags, consequently generating more energy. However, for the same bluff body shapes, short full active flags generate more energy in higher wind speed. This study could develop a new experimental approach on finding the most favorable combination of bluff body shape and flag configuration which is as important as just considering bluff body shape to improve the efficiency of the low-speed wind piezoelectric energy harvesting system.

Journal ArticleDOI
TL;DR: In this article , the flow passing through three cylinders in staggered and equilateral-triangular arrangement were numerical calculated using a momentum exchange-based immersed boundary-lattice Boltzmann method.

Journal ArticleDOI
TL;DR: In this article , the effects of the Bingham number (Bn) on the drag and lift coefficients were investigated in a channel-driven cavity with a square obstacle installed near the inlet.
Abstract: In the current work, an investigation has been carried out for the Bingham fluid flow in a channel-driven cavity with a square obstacle installed near the inlet. A square cavity is placed in a channel to accomplish the desired results. The flow has been induced using a fully developed parabolic velocity at the inlet and Neumann condition at the outlet, with zero no-slip conditions given to the other boundaries. Three computational grids, C1, C2, and C3, are created by altering the position of an obstacle of square shape in the channel. Fundamental conservation and rheological law for viscoplastic Bingham fluids are enforced in mathematical modeling. Due to the complexity of the representative equations, an effective computing strategy based on the finite element approach is used. At an extra-fine level, a hybrid computational grid is created; a very refined level is used to obtain results with higher accuracy. The solution has been approximated using P2 − P1 elements based on the shape functions of the second and first-order polynomial polynomials. The parametric variables are ornamented against graphical trends. In addition, velocity, pressure plots, and line graphs have been provided for a better physical understanding of the situation Furthermore, the hydrodynamic benchmark quantities such as pressure drop, drag, and lift coefficients are assessed in a tabular manner around the external surface of the obstacle. The research predicts the effects of Bingham number (Bn) on the drag and lift coefficients on all three grids C1, C2, and C3, showing that the drag has lower values on the obstacle in the C2 grid compared with C1 and C3 for all values of Bn. Plug zone dominates in the channel downstream of the obstacle with augmentation in Bn, limiting the shear zone in the vicinity of the obstacle.

Journal ArticleDOI
TL;DR: In this paper , a general shape factor was proposed and applied together with 16 previously proposed C D correlations to the 60 reference model shapes over a range of Re from 0.001 to 300,000 to derive a general C D correlation using nonlinear regression analysis.

Journal ArticleDOI
TL;DR: In this paper , the potential of a Rhizophora mangrove forest of moderate cross-shore thickness to attenuate wave heights using an idealized prototype-scale physical model constructed in a 104 m long wave flume was investigated.
Abstract: This study investigates the potential of a Rhizophora mangrove forest of moderate cross-shore thickness to attenuate wave heights using an idealized prototype-scale physical model constructed in a 104 m long wave flume. An 18 m long cross-shore transect of an idealized red mangrove forest based on the trunk-prop root system was constructed in the flume. Two cases with forest densities of 0.75 and 0.375 stems/m2 and a third baseline case with no mangroves were considered. LiDAR was used to quantify the projected area per unit height and to estimate the effective diameter of the system. The methodology was accurate to within 2% of the known stem diameters and 10% of the known prop root diameters. Random and regular wave conditions seaward, throughout, and inland of the forest were measured to determine wave height decay rates and drag coefficients for relative water depths ranging 0.36 to 1.44. Wave height decay rates ranged 0.008–0.021 m–1 for the high-density cases and 0.004–0.010 m–1 for the low-density cases and were found to be a function of water depth. Doubling the forest density increased the decay rate by a factor two, consistent with previous studies for other types of emergent vegetation. Drag coefficients ranged 0.4–3.8, and were found to be dependent on the Reynolds number. Uncertainty in the estimates of the drag coefficient due to the measured projected area and measured wave attenuation was quantified and found to have average combined standard deviations of 0.58 and 0.56 for random and regular waves, respectively. Two previous reduced-scale studies of wave attenuation by mangroves compared well with the present study when their Reynolds numbers were re-scaled by λ3/2 where λ is the prototype-to-model geometric scale ratio. Using the combined data sets, an equation is proposed to estimate the drag coefficient for a Rhizophora mangrove forest: CD = 0.6 + 3e04/ReDBH with an uncertainty of 0.69 over the range 5e03 < ReDBH < 1.9e05, where ReDBH is based on the tree diameter at breast height. These results may improve engineering guidance for the use of mangroves and other emergent vegetation in coastal wave attenuation.

Journal ArticleDOI
TL;DR: In this article, a general shape factor was proposed and applied together with 16 previously proposed CD correlations to the 60 reference model shapes over a range of Re from 0.001 to 300,000.