Three-dimensional Separated Flow Research Articles

Wing design optimization and stall analysis with Co-flow Jet active control

Coupled with Co-flow Jet (CFJ) technology, the Non-dominated Sorting Genetic Algorithm II was utilized for the multi-objective combination optimization of an optimized Co-flow Jet wing, based on National Advisory Committee for Aeronautics (NACA) 6421. A high-precision numerical simulation using the delayed detached eddy simulation model was performed on the optimized wing to investigate the three-dimensional flow separation characteristics after static stall. The stall improvement was investigated by adjusting the momentum coefficient of the injection. The results show that the optimized wing exhibits significant improvements in aerodynamic performance and corrected aerodynamic efficiency. At an angle of attack of 10°, the average lift increased by 16.25% and the drag decreased by 27.23% compared to the CFJ6421 wing, while effectively addressing the problem of low modified aerodynamic efficiency of the CFJ wing at lower angles of attack. By utilizing higher momentum and improving the boundary layer control capability, flow separation is effectively suppressed, thus achieving the goal of stall recovery of the CFJ wing.

Water tunnel testing of downwind yacht sails

Downwind yacht sails, such as spinnakers, are low-aspect-ratio highly cambered wings with a sharp leading edge. They are characterised by substantial three-dimensional flow separation and are thus modelled with difficulty with numerical simulations. Furthermore, accurate full-scale validation data are not available. The first quantitative flow measurements have only recently been achieved in water tunnels. In this study, we aim to provide guidelines on this emerging sail testing methodology. We consider six model-scale rigid models at average-chord-based Reynolds numbers ranging from 5 870 to 61 870. A critical Reynolds number is identified, below which relaminarisation of the reattached boundary layer downstream of the leading edge separation bubble occurs. Both lift and drag increase monotonically at subcritical Reynolds numbers while remaining about constant at transcritical Reynolds numbers. The critical Reynolds number decreases with increasing incidence and is insensitive to the blockage ratio. Spinnakers are normally sailed in front of the mainsail, whose circulation is found to generate an approximately 3∘\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$3^{\\circ }$$\\end{document} upwash on the spinnaker and higher flow velocity on both sides of it. These findings provide guidelines for the experimental testing of spinnaker-like wings in water tunnels and provide new insights into the flow and experimental testing of highly cambered wings with massive flow separation at low Reynolds numbers.

Assimilating experimental data of a mean three-dimensional separated flow using physics-informed neural networks

In this article, we address the capabilities of physics-informed neural networks (PINNs) in assimilating the experimentally acquired mean flow of a turbulent separation bubble occurring in a diffuser test section. The training database contains discrete mean pressure and wall shear-stress fields measured on the diffuser surface as well as three-component velocity vectors obtained with particle image velocimetry throughout the volumetric flow domain. Imperfections arise from the measurement uncertainty and the inability to acquire velocity data in the near-wall region. We show that the PINN methodology is suited to handle both of these issues thanks to the incorporation of the underlying physics that, in the present study, are taken into account by minimizing residuals of the three-dimensional incompressible Reynolds-averaged Navier–Stokes equations. As a result, measurement errors are rectified and near-wall velocity profiles are predicted reliably. The latter benefits from the incorporation of wall shear-stress data into the PINN training, which has not been attempted so far to the best of our knowledge. In addition to demonstrating the influence of this novel loss term, we provide a three-dimensional, highly resolved, and differentiable model of a separating and reattaching flow that can be readily used in future studies.

Suppression of flow separation around a finite wall-mounted square cylinder by suction at the side leading edge

We investigate the control of three-dimensional flow separation around a finite wall-mounted square cylinder by applying suction at the side leading edge. Direct numerical simulations are conducted at a Reynolds number of 250, with suction ratios Γ of 0–2 (where Γ is the absolute value of the suction velocity divided by the free stream velocity). The effect of Γ on the aerodynamic forces acting on the cylinder is studied. The results show that suction reduces the aerodynamic forces, with the best control effect for the fluctuating lift coefficient (corresponding to a reduction of over 70%) achieved at Γ = 0.375. As the suction ratio increases, the pressure drag experienced by the square cylinder decreases. Simultaneously, the mean frictional drag force exerted on the square cylinder increases. The optimal mean drag coefficient (corresponding to a reduction of nearly 20%) is achieved at Γ = 1. The effect of the suction ratio on the flow topology in the wake is also investigated. Suction significantly suppresses the flow separation. As the suction ratio increases, the spanwise counter-rotating vortices in the streamwise and transverse directions decreases in size, and the downwash vortex shrinks, and shifts toward the free end of the square cylinder. The far-wake streamwise base vortex disappears when active suction is applied to the side leading edge. However, a new pair of base vortices splits from the original base vortex and persists into the far wake flow field, forming a quadrupole vortex structure with the tip vortex.

Investigation on the behavior of flow and aerodynamic noise generated around the tandem seal-vibrissa-shaped cylinder

Through comparing with the tandem circular and elliptic cylinders with the same characteristic dimensions, the behavior of flow and flow-induced noise generated around the tandem seal-vibrissa-shaped cylinder is studied based on delayed detached-eddy simulation model and acoustic analogy approach. The co-shedding pattern of flow developed around the tandem cylindrical-like bars is investigated. The spatial modes, mode energy, and mode coefficients of turbulent flow around the geometries are analyzed through spectral proper orthogonal decomposition. Results show that the lift fluctuations of downstream bar are stronger than those of upstream bar, and more aerodynamic noise is radiated from the downstream bar than from the upstream bar. The alternative arrangement of nodal and saddle planes of seal-vibrissa-shaped cylinder introduces three-dimensional flow separations and suppresses the shear layer interactions, inhibiting the regular vortex shedding of Kármán vortex street occurring in the tandem cylinder wake. The reversed vortex shedding generated by two adjacent saddle surfaces in the wake of seal-vibrissa-shaped cylinder balances the lateral force and mitigates the lift fluctuations greatly, thereafter reduces the aerodynamic noise generated by wall pressure fluctuations introduced by unsteady fluctuating forces exerting on the surfaces of geometries. Compared to the tandem circular and elliptic cylinders, the good noise reduction effect with sound pressure level reduced at main frequency range has been achieved from the tandem seal-vibrissa-shaped cylinder. The calculated spectra and amplitude levels of aerodynamic noise agree well with the experimental measurements from the anechoic wind tunnel, verifying the accuracy of the numerical simulations.

Flow-induced noise from a seal-vibrissa-shaped cylinder

Inspired by the special structure of seal vibrissae and their ability to effectively suppress vortex-induced vibrations, this paper studies the characteristics of flow and aerodynamic noise generated around a seal-vibrissa-shaped cylinder at a Reynolds number (based on the hydrodynamic diameter and freestream velocity) of 6 × 104 using the delayed detached-eddy simulation model combined with the acoustic analogy approach. The far-field aerodynamic noise has also been measured at Reynolds numbers of 6 × 104∼1.2 × 105 in an anechoic wind tunnel to compare the seal-vibrissa-shaped cylinder with cylindrical and elliptical bars of the same characteristic dimensions. The calculated spectra of aerodynamic noise agree well with the wind tunnel measurements, verifying the accuracy of the numerical simulations. It is found that the three-dimensional flow separations introduced by the alternative saddle and nodal planes of the seal-vibrissa-shaped cylinder inhibit shear layer interactions and reduce the spanwise coherence, resulting in a wake with no dominant coherent structures and thus suppressing pressure fluctuations on the solid surfaces. The configuration of seal-vibrissa-shaped cylinder eliminates the regular Kármán vortex street which generally occurs in a cylinder wake, and consequently inhibits the tonal peak in the aerodynamic noise. The sound pressure level is also reduced in most frequencies. The consistency of the normalized spectra of the far-field aerodynamic noise and the vortex shedding frequency at different inflow speeds demonstrates the high accuracy of the anechoic wind tunnel measurements. Compared with cylinders with circular or elliptical cross sections, the seal-vibrissa-shaped cylinder effectively reduces the flow-induced noise by, respectively, 13 dB or 11 dB for a far-field receiver of 90° at Reynolds number of 6 × 104, showing great potential in reducing aerodynamic noise.

Spike formation theory in three-dimensional flow separation

We develop a frame-invariant theory of material spike formation during flow separation over a no-slip boundary in three-dimensional flows with arbitrary time dependence. Based on the exact evolution of the largest principal curvature on near-wall material surfaces, our theory identifies fixed and moving separation. Our approach is effective over short time intervals and admits an instantaneous limit. As a byproduct, we derive explicit formulas for the evolution of the Weingarten map and the principal curvatures of any surface advected by general three-dimensional flows. The material backbone we identify acts first as a precursor and later as the centrepiece of Lagrangian flow separation. We discover previously undetected spiking points and curves where the separation backbones connect to the boundary and provide wall-based analytical formulas for their locations. We illustrate our results on several steady and unsteady flows.

Control of bow shock induced three-dimensional separation using bleed through holes

The unsteady three-dimensional separated flow on a wall induced by a square protrusion (approximately twice the local boundary layer thickness in width and height), and its control by means of passive suction through holes, is investigated using wind tunnel experiments at Mach 2.87. The baseline flow without any control was characterized and compared against the cases with bleed. A bow-shaped separation line on the wall with a mid-span separation length of 5.57δ from protrusion face was traced from the oil-flow visualization. The averaged pressure distribution surveyed using static pressure ports placed on the wall has mapped plateau, high-pressure, and low-pressure regions in the separated flow, distinctive to three-dimensional interactions. Ten control configurations were tested with suction holes placed along mid-span in the different pressure regions. Significant spanwise “Mean Reduction in Separation Length” of up to 0.93δ was observed from oil-flow visualization. A comparison of observations from various control configurations suggested that bleeding the flow from the high-pressure region could delay the separation and reduce the bubble size in general. Furthermore, time-resolved Schlieren visualizations have confirmed reduction in both “mid-span separation length” and “shock-intermittent-region” with the introduction of suction in high-pressure region. Fourier and Proper Orthogonal Decomposition analysis done on the Schlieren data has confirmed the presence of low-frequency separation-shock oscillations at Strouhal Numbers of order 10−2, both with and without control. Furthermore, the amplitudes of separation-shock oscillations in the spectrum were reduced with the introduction of suction simultaneously from two holes placed in high- and low-pressure regions.

Research on the flow characteristics of the bulk carrier wake field based on particle image velocimetry

The wake field has three-dimensional flow separation characteristics, affecting various ship performances. The research on the characteristics of the wake field can provide theoretical guidance for the formation of new ship design methods and provide an understanding of flow mechanisms for improving ship performance. To explore the flow mechanism of the wake field, particle image velocimetry was used to carry out the detailed flow measurement of bulk carriers. First, the experimental uncertainty and convergence are analyzed. Then, the spatial distribution characteristics of the time-averaged field, instantaneous field, and turbulent flow statistics are discussed in detail, and a criterion for discriminating turbulent anisotropy is proposed. The results show that the vortex structure significantly affects flow characteristics, and the axial velocity contours present a U-form distribution with prominent “hook-like” features. Compared with the time-averaged velocity field, the instantaneous velocity field is chaotic and has multiple additional vortex structures, and the velocity contours and streamlines have prominent non-smooth characteristics. The wake field has an apparent double vortex structure, and the aggregation of many small vortices forms the bilge vortex. The instantaneous rotation characteristics of vortices in the wake field are highly time-dependent and fluctuate with time. The turbulent kinetic energy, the root mean square of fluctuation velocity, and the Reynolds stress have a U-form distribution. The U-form region is concentrated in the area with a large gradient. The wake distribution is in a state of turbulent anisotropy, and the kinetic energy change layer and low kinetic energy region have a low turbulent anisotropy.

Numerical modeling of the dynamic unsteadiness with global mode decomposition approach in a high-speed compressor with abnormal diffuser vane

Adjustable vane diffuser is addressed as one of the main techniques in centrifugal compressor in seeking of wider stability margin. Flow instabilities, however, may be driven by the three-dimensional flow separation induced by the abnormal vane in the diffuser. In the present work, a numerical investigation of the radial vaned diffuser with abnormal vane was applied to identify the correlation between stall inception and asymmetric flow distribution. Unsteady numerical simulations of the centrifugal compressor with vaned diffuser was carried out and validated against experimental data. Then, the aerodynamic characteristics and evolution process of the flow instabilities in vaned diffuser was analyzed with the compressed dynamic mode decomposition in term of perturbation frequency and dominating flow mode. Finally, concerning the relative position of the volute tongue and abnormal vane, the influence of the abnormal vane on the matching between the diffuser and volute regarding the stall mechanism was summarized. The results indicate that the abnormal vane needs to be altered large enough, which is above 10°, to trigger the flow instability of vaned diffuser before the spike stall in impeller. With the increase of stagger angle alternation, the stall mechanism turns from a global flow separation in multiple passages into a regional and abrupt phenomenon in limited passages. With the mode composition approach, the flow separation near diffuser outlet and the tornado-type vortices at the leading edge of the abnormal vane are responsible for the flow instability of the case with 15° vane alternation. When combined with the volute, the case with abnormal vane close to the volute tongue tends to be more unstable than that 180° away from volute tongue. The tornado-type vortices are also moved to the trailing edge rather than the leading edge under the influence of the asymmetric volute.

On the scaling of three-dimensional shock-induced separated flow due to protuberances

Supersonic flow over three-dimensional bodies protruding out of the turbulent boundary layer was investigated by means of experiments and numerical computations. A parametric study was performed by varying the shape and dimensions of the protuberance, as well as the freestream Mach numbers (1.5, 2, 2.5, 2.89, and 3.5). Surface streak line visualization, surface pressure measurements, and time-resolved Schlieren visualization were employed along with Reynolds-averaged Navier–Stokes computations to elicit the complex flow features such as the separation line, shock pattern, and the horseshoe vortex, which greatly influence the flow dynamics in the separated region. The rise in surface pressure at mid-span due to separation (plateau pressure) was dependent only on the incoming flow parameters and independent of protuberance geometry. The two-dimensional free interaction theory, applied for normal shock-induced separation, closely predicts the mid-span plateau pressure. Although protuberances are of varying shapes and dimensions, the inviscid bow shock (obtained from Euler computations) provided generalized scales, whose effects on shock boundary layer interactions are analyzed. The radius of curvature of the inviscid shock on the wall plane at the nose, which is theoretically related to the local second derivative (along the shock) of pressure jump, was found to be a determining parameter of mid-span separation length (Lsep). Since the spanwise distance of the sonic point on the inviscid shock was found to be strongly correlated with its nose radius of curvature, it follows that the “strong” portion of the inviscid bow shock fixes the mid-span separation location. These observations concerning mid-span plateau pressure, and the role of strong shock portion in fixing mid-span separation, suggest that the Lsep shall be predicted from a modification of the scaling laws for the length of plateau pressure region in two-dimensional shock boundary layer interaction, with the inclusion of a spanwise relieving effect. A correlation is obtained relating the Lsep with various incoming flow parameters and inviscid shock nose radius. The mid-span vortex core position was found to be linearly related to the Lsep. The radius of curvature of the separation shock is, however, found to be influenced by the entire inviscid shock, including the “weak” portion.

Data augmented turbulence modeling for three-dimensional separation flows

Field inversion and machine learning are implemented in this study to describe three-dimensional (3D) separation flow around an axisymmetric hill and augment the Spalart–Allmaras (SA) model. The discrete adjoint method is used to solve the field inversion problem, and an artificial neural network is used as the machine learning model. A validation process for field inversion is proposed to adjust the hyperparameters and obtain a physically acceptable solution. The field inversion result shows that the non-equilibrium turbulence effects in the boundary layer upstream of the mean separation line and in the separating shear layer dominate the flow structure in the 3D separating flow, which agrees with prior physical knowledge. However, the effect of turbulence anisotropy on the mean flow appears to be limited. Two approaches are proposed and implemented in the machine learning stage to overcome the problem of sample imbalance while reducing the computational cost during training. The results are all satisfactory, which proves the effectiveness of the proposed approaches.

Active flow control in an S-shaped duct at Mach 0.4 using sweeping jet actuators

An experimental study was conducted on the flow control capability of sweeping jet actuators (SJAs) in an aggressive offset S-shaped duct with an inlet Mach number of 0.4. The SJAs were placed in two different spacing arrangements to suppress flow separation in the duct and flow distortion at the aerodynamic interface plane (AIP). Multiple measurements were taken, including surface oil flow visualization, steady pressure on the wall surface, five-hole probe pressure, and velocity at the AIP. For the baseline case without control, the surface oil flow visualization and pressure measurement results clearly reflected the three-dimensional flow separation characteristics in the duct. In the controlled cases, the streamwise surface pressure coefficient distribution showed that SJAs can effectively suppress the flow separation. At a spacing ratio of 8.13 and an excitation pressure of 250 kPa, the inflection point of the pressure coefficient Cp was completely eliminated. As a result, the absolute pressure recovery at the upper region of the AIP increased by nearly 5%, and the mean value and RMSE of the cross-flow velocity declined by 54.9% and 48.1%, respectively.

Numerical investigation on the generation mechanism of aero-heating of rudder shaft from three-dimensional flow separation and vortices

The hypersonic flow past a controlled rudder mounted at a gap to the aircraft fuselage is numerically investigated by solving three-dimensional (3D) Reynolds-averaged Navier–Stokes equations. This paper aims to explain the mechanism of production of extreme thermal environment faced by a rudder shaft from the view of physics of flow in the mounting gap. Simulations are conducted at Mach number of 10, and the gap ratio h/δ varies from 0 to 1.42, where h denotes the gap height and δ represents the thickness of the incoming turbulent boundary layer. Topological theory is utilized to identify the separation types. The formation of vortices is traced by extracting volume streamlines in the 3D space. The results indicate that the types of 3D separation appear in the gap shifts from the coexistence of horseshoe-type and tornado-type separations to only horseshoe-type separation that persists with the increase of h/δ. It is found that high heat flux is generated by the high-momentum fluid transported toward the surface by the horseshoe vortices. The tornado-type vortex prevents the incoming flow from arriving at the rudder shaft, which avoids the generation of high heat flux at the center of the rudder shaft. The rate of local heat transfer increases with h/δ as a result of the shrink and disappearance of the tornado-type vortices, which means that the region of low-speed backflow in front of the rudder is reduced and vanished. This study contributes to a clearer understanding of the flow physics in the complex disturbance area.

Interaction of synthetic jets with a massively separated three-dimensional flow field

The interaction of an array of finite span synthetic jet actuators with a massively separated flow was explored experimentally over a cantilevered, swept, and tapered model having a deflected control surface. The interaction of one synthetic jet with the local flow led to a flow field associated with formation and advection of vortex rings that bent into the flow, resulting in flow reattachment in their trajectory. Under specific conditions, when multiple jets were actuated, their interaction was either constructive or distractive, depending on the separation severity and spanwise flow magnitude. This work improves control of three-dimensional flow separation using distributed actuators.

Influence of the inlet boundary layer on non-axisymmetric endwall contouring effects in a linear compressor cascade

The modern aero-engine compressors are commonly designed to be highly loaded which can introduce strong three-dimensional flow separation, and effective flow control is of vital importance in improving compressor performance. Non-axisymmetric endwall contouring nowadays is conducted on axial compressors to control the flow separation. In this paper, a contoured endwall is designed based on a linear CDA compressor cascade. The genetic algorithm and ANN surrogate model are used in the optimization process of the hub contouring, based on RANS simulations. Three inlet boundary layers which follow the fully turbulent boundary layer assumption are generated with different thickness, and the influence mechanism is numerically investigated. The optimal endwall with streamwise concave near the suction side provides great performance improvement at all inlet boundary layers at the aerodynamic design point. The endwall flow near the suction side is accelerated by the contoured endwall with cross-flow pressure gradient altered, suppressing the corner separation. The main pressure loss is closely related to one streamwise vortex that blocks the endwall flow. A thicker inlet boundary layer can induce early forming and increased strength on the streamwise vortex, enhancing the cross-flow movement and its interaction with the blade surface flow. In addition, the pressure loss near endwall in the inlet boundary layer is found more dominant on the endwall contouring effect than the boundary layer thickness.

Numerical investigation on separated turbulent flow in a three-dimensional U-turn duct with spanwise diverging

The flow in a U-turn duct with a 180-degree curved section is a typical model of various applications such as the bend and return channel in multi-stage radial turbo-machines. The flow exhibits complex patterns including the separation on the inner wall, highly sheared flow in the curved and downstream sections, and recovery of deficit flow to uniform flow. Most of current researches focus on the simplified models of two-dimensional plane duct or three-dimensional duct with constant cross-section, while in realistic circumstances the duct can be varying in the shape and area of the cross-section to meet the design requirement. In this work, we performed a numerical investigation on the turbulent flow in a U-turn duct with spanwise diverging at Reynolds number Re = 106. The effect of diverging, as represented by the height of the duct, on the formation of secondary flow, flow separation and recovery is explored, and the three-dimensionality of the separated flow is quantitatively assessed. Numerical results reveal that with stronger spanwise diverging, the flow separates earlier from the inner wall and the size of the separated vortex is larger; the primary characteristics of the secondary flow also change significantly. The three-dimensional separated flow substantially affects the flow in the downstream straight duct that both streamwise and radial velocities exhibit fluctuating patterns as a result of the spatially non-uniform separation. The characteristics of boundary layer flow on all solid walls are also analyzed and discussed regarding the pressure field and local shearing.

Influence of Surface Roughness on a Highly Loaded Axial Compressor Stage Performance at Low Reynolds Number

In the present study, a numerical simulation was conducted to investigate the influence of surface roughness on the aerodynamic performance of a 1.5-stage highly loaded axial compressor at low Reynolds number. It was especially considered how the roughness Reynolds number ( k + ) affected the change of the inlet and outlet conditions, the growth of the separation bubble (LSB), the status of the limiting streamline, and the patterns of the wake. Regarding the roughness settings, five roughness magnitudes and four roughness locations were mainly studied. The results showed that at low Re, surface roughness mainly improved the stage performance by reducing the length and width of the LSB as well as the rotor tip vorticity, delaying the occurrence of three-dimensional flow separation and increasing the turbulence level near the wall. However, it also aggravated the incoordination between the subsequent stages to a certain extent, which limited further improvement of the overall aerodynamic performance. Generally, with k + increasing, the compressor aerodynamic performance improved and achieved the best at k + = 137.8 . The maximum increases in the total pressure ratio, peak efficiency, and chocked mass flow were approximately 4.01%, 5.34%, and 2.24%, respectively. In addition, for all the four roughness locations, the roughness covering from the leading edge to 50% of the axial chord length on the suction surface had a relatively evident advantage in improving the compressor peak efficiency because of the better control of the LSB and wall shear stress.

Investigation of the three-dimensional flow past a flatback wind turbine airfoil at high angles of attack

Flatback airfoils are airfoils with a blunt trailing edge. They are currently commonly used in the inboard part of large wind turbine blades, as they offer a number of aerodynamic, structural, and aeroelastic benefits. However, the flow past them at high angles of attack (AoA) has received relatively little attention until now. This is important because they usually operate at high AoA at the inboard part of Wind Turbine blades. The present investigation uses Reynolds averaged Navier–Stokes (RANS) and hybrid RANS + large eddy simulation predictions to analyze the flow in question. The numerical results are validated against previously published wind tunnel experiments. The analysis reveals that to successfully simulate this flow, the spanwise extent of the computational domain is crucial, more so than the selection of the modeling approach. Additionally, a low-drag regime observed at angles of attack before stall is identified and analyzed in detail. Finally, the complex interaction between the three-dimensional separated flow beyond maximum lift (stall cells) with the vortex shedding from the blunt trailing edge is revealed.

A 3D flow separation model for open propellers with blunt trailing edge

The panel method is widely used in the design and analysis of marine propellers. However, this method cannot handle propellers with a blunt trailing edge due to a lack of proper models for the flow separation downstream. In this study, a numerical scheme, i.e., the three-dimensional flow separation model, is proposed to enable the panel method to be applied to propellers with a blunt trailing edge. In this model, the flow separation zone is represented as an extension of the trailing edge. The criteria of zero local lifts and zero local moments over the extension of all blade sections are used to establish the equations that determine the shape of the flow separation zone. Those equations are solved either by the Newton–Raphson method or as an optimization problem. A low-order panel method coupled with a boundary layer solver (the viscous/inviscid interaction method) is used to predict the pressure distributions and forces for the extended geometry. To validate this model, the pressure distributions of the propeller blade are compared with those from solving the Reynolds-Averaged Navier–Stokes equations. The predicted open water characteristics of a controllable pitch propeller are compared with the experimental measurements at a wide range of advance ratios with different pitches. This model requires much less computational effort while preserving high accuracy, and thus can be used reliably in designing and analyzing propellers with a blunt trailing edge.