Secondary Flow Effects Research Articles

An actuator-disk model augmentation for low-pressure axial fan simulation

Abstract Actuator-disk rotor models are important simulation tools for cost-effective industrial axial fan system analysis. Actuator-disk fan model performance, however, is constrained by the conventional use of 2D airfoil coefficient input data which limits the accuracy of the models to a narrow operating range free from significant secondary radial blade flow effects. Radial blade flow is characteristic of off-design fan operation which is often unavoidable within typical industrial fan system environments, so the enhancement of actuator-disk model performance for these conditions is desired. This paper accordingly presents a new means of robustly determining actuator-disk model coefficient inputs that are suitable for a wide range of fan operating conditions. The proposed Augmented Actuator-disk Method (AADM) capitalizes on new insights on the unique aerodynamic behavior of low-pressure axial fan rotors. The performance of the AADM is evaluated for two different industrial cooling fans and is shown to outperform existing actuator-disk coefficient formulations through computational fluid dynamics (CFD) simulations. The AADM is shown to better predict key fan performance metrics, spanwise blade force distributions, and to produce flow fields that are more physically representative (an important feature for industrial heat exchanger studies where the AADM is anticipated to be commonly applied). The AADM has been developed to be easily adopted in generic industrial fan analyses and is expected to serve as a valuable springboard for future actuator-disk fan model developments.

A method for modeling the lateral distributions of depth-averaged velocities behind an emergent vegetation patch

Aquatic vegetation provides ecological, hydrological, and aesthetic functions for rivers, and measuring velocity in vegetated channels is essential for river management. The paper presents a method for modeling the lateral distributions of depth-averaged velocities behind an emergent vegetation patch. Based on SKM, this approach divides the channel behind an emergent vegetation patch into pseudo-vegetation and free-flow regions, offering analytical solutions for the depth-averaged velocities in these two regions. The model incorporates several critical parameters, including the Darcy–Weisbach coefficient, lateral dimensionless eddy viscosity, drag force coefficient, and secondary flow coefficients. These coefficients are associated with bed friction, vegetation-induced resistance, secondary flow effects, and lateral momentum exchange, respectively, affecting the depth-averaged velocities. A comparison with published experimental data validates that the proposed model can predict the lateral distributions of depth-averaged velocities behind an emergent vegetation patch. A steady wake section exists behind an emergent vegetation patch for low-flow blockage. In the steady wake section, the secondary flow coefficients in pseudo-vegetation and free-flow regions remain relatively constant. Upon exiting the stable wake section, the secondary flow coefficient in the pseudo-vegetation region decreases with increasing distance from an emergent vegetation patch, and it in the free-flow region increases with increasing distance from an emergent vegetation patch. A sensitivity analysis of drag coefficient and secondary flow coefficients suggests that secondary flow coefficient in the free-flow region has a more significant effect on the lateral distributions of depth-averaged velocities compared to drag coefficient and secondary flow coefficient in the pseudo-vegetation region.

Experimental Study of Single and Two-Phase Flow and Heat Transfer Characteristics in Helical Tube under Motion Condition

The unique structural characteristic of the helical coiled pipe causes some special phenomena such as the secondary flow effect inside the tube, and the heat transfer characteristics are more complicated than those in circular pipe. There have been few published papers about experimental research of flow and heat transfer characteristics of helical tube under motion conditions. An experimental loop was built based on the background above. An experimental study of single-phase and saturated boiling two-phase heat transfer characteristics and flow instability in helical tubes under multiple motion conditions was carried out. In which the maximum system pressure was 4 MPa, mass flux was 550 kg/(m2·s), and heat flux of the test section was 360 kW/m2. The results show that the heat transfer capacity of the side close to the central axis of the helical tube is significantly lower than that of the side away from the central axis under rocking and inclination conditions. An empirical correlation of the heat transfer coefficient in helical tube under static and motion conditions was drafted based on the experiment data of single-phase conditions. The influence of motion conditions on two-phase heat transfer characteristics was similar to that of a single phase but much weaker.

Deciphering the Evolution of Inertial Migration in Serpentine Channels.

Serpentine channels coupling inertial and secondary flows enable effective particle focusing and separation, showing great potential in clinical diagnostics and drug screening. However, the nonsteady secondary flows in the serpentine channel make the evolution of inertial migration unclear, hindering the development and application of the serpentine channel. Herein, to refine the inertial migration mechanism, we established a model with varying curvature ratios to study the effect of secondary flow on particle migration in the serpentine channel. This method used direct numerical simulation (DNS) to calculate inertial lift, mapped the inertial lift to cross sections of the serpentine channel, and deciphered the inertial migration by using the Lagrangian particle tracking (LPT) method. The inertial migration of microparticles is experimentally investigated to validate the established numerical model. The results indicate that particle migration in serpentine channels follows a two-stage migration. An increase in secondary flow accelerates the second stage of the migration process while slowing the first stage process. Subsequently, we investigated the effects of different parameters, including Reynolds number, aspect ratio, and blockage ratio, on the equilibrium positions of particles, providing guidelines for the high-resolution separation of particles. Taking flow resistance into account, the dimensionless study makes the separation of arbitrary-sized particles possible. This work reveals the migration mechanism in serpentine channels, paving the way for the inertial separation of the particles.

Effect of secondary flow and wall collisions on particle-laden flows in 90° pipe bends

Fully developed, statistically stationary particle-laden turbulent flows in a 90° pipe bend at a moderate Reynolds number are studied using direct numerical simulation coupled to a Lagrangian particle tracking technique. Three populations of particles are studied which are two-way coupled with the carrier phase. The focus is the investigation of the effect of strong curvature on particle transport dynamics across a range of particle inertias. A validation of the carrier phase predictions is performed, with good agreement found with available experimental data. It is demonstrated that the presence of particles notably affects the turbulence statistics of the carrier phase, decreasing the mean fluid velocity in the outer bend while slightly increasing it in the inner bend. Two intriguing phenomena emerge: the presence of a particle void near the inner bend and reflection layers near the outer bend. Centrifugal forces are responsible for driving particles to the outer wall, with some rebounding back into the main body of the flow, and others, influenced by the secondary flow in the plane of the pipe cross-section, converging towards the inner wall before re-entering the central pipe region. These effects dictate the size and shape of the particle void region and the formation of reflection layers due to particle-wall collisions. These two phenomena have a significant influence on the particle distribution in the pipe bend, as well as on the first- and second-order moments of the velocity and acceleration of the particulate phase. Finally, a heat map of particle-wall collision statistics indicates that the effect of the secondary flow on particle distribution and particle-wall collisions is not negligible.

Tip air injection and plasma DBD actuator as active flow control in axial compressors - Review

Abstract Nowadays, flow control is extensively studied in fluid dynamic domain, and the principal aim is to manipulate the flow path that passes over a surface to reduce losses generated by turbulent flow, generated by detachment. The Flow control involves passive or active methods, and for approximately thirty years, active flow control methods have been studied and developed in axial compressors more intensively, active flow control methods are those in which energy is added to the fluid through an external source. The flow control reduces losses generated by secondary flow effects, such as flow detachment at rotor tip, flow detachment at casing and flow detachment at stator vanes hub. The present paper provides a summary of investigation of active flow control methods tip air injection and plasma actuators.

A total three-dimensional evaluation method to analyze the loss mechanism of film cooling on turbine guide vane

In view of the situation that the secondary flow in the passage deflects the coolant from film cooling holes and changes heat transfer, a total three-dimensional evaluation method is developed based on the HS1A high-pressure turbine guide vane to take a new insight into the loss mechanism. Film cooling holes are placed on the suction surface with compound angles to obtain different blade models. After satisfying the requirement of film cooling effectiveness, TTDM, a new total three-dimensional evaluation method where both mainstream and coolant are treated in three dimensions is established. Then, loss mechanism is analyzed by controlling the temperature ratio, changing the exit Mach number and blowing ratio. The entropy creation due to mixing is used as the evaluation basis to analyze different models within the high film cooling effectiveness range. It is believed that the mixing temperature and velocity need to be elaborated and distinguished in high subsonic environments. Compared with the original model within a certain range, the increase of the film cooling effectiveness can be found under the application of different compound angles in a short distance on suction surface. A slight increase of entropy is also noticed. By lessen the loss of secondary flow within a certain range of blowing ratio and exit Mach number conditions, the total loss can be reduced with reasonable setting of compound angle. Finally, the loss with film cooling is evaluated more accurately by the total three-dimensional evaluation method considering the effects of secondary flow.

Transonic Compressor Darmstadt Open Test Case: Experimental Investigation of Stator Secondary Flows and Hub Leakage

Abstract The future trend of clean sky aviation has led to a smaller core engine, resulting in highly loaded stages with increased relative tip gaps and leakage flows, which causes a reinforced influence of secondary flow phenomena. This article experimentally investigates the secondary flow effects in the 3D-optimized shrouded stator of the TUDa-GLR-OpenStage, representative of a transonic high-pressure compressor front stage, and its susceptibility to typical hub leakage flows. The impact of stator hub leakage on efficiency and total pressure ratio is investigated for several operating speeds, from subsonic to nominal transonic operating conditions. Steady five-hole probe and time-resolved virtual multi-hole probe measurements at different operating conditions at nominal speed are conducted to investigate the underlying loss mechanisms. The leakage mass flow is determined using the measured pressure difference within the fin seal. Steady results show that the optimized 3D stator effectively suppresses the hub corner separation compared to the predecessor 2D stator, but its performance is further limited by a near-tip separation. With the increased stator hub leakage (at a leakage-to-main flow ratio of about 0.4%), the compressor isentropic efficiency generally drops (by about −0.5%) at the full operating range due to an enhanced hub corner separation effect. However, the increased leakage also helps alleviate the tip-corner separation when operating near stall, leading to a smaller efficiency penalty under these conditions. Unsteady results reveal the sensitivity of transient compressor performance to the passing rotor wake, but limited interaction between this phenomenon and the increased leakage is observed. These findings provide insight into the stator hub leakage-related penalties on the compressor aerodynamics efficiency.

Heat transfer enhancement in a triple-layered turbine blade internal cooling channel

An effective measure of enhancing the coupled convection and radiation in the turbine blade internal cooling channel is developed. A triple-layered channel is presented for the first time to further improve the flow organization to suppress the negative effect of secondary flows induced by rotation as well as extend cold surfaces for radiation. Explorations of flow and coupled heat transfer are executed numerically for various channel models. The results manifest that, heat transfer is markedly improved with the triple-layered channel. The highest comprehensive thermal performance is achieved when employing the triple-layered channel with smooth surfaces and ribs (Model 3SW), which can increase the total Nusselt number ratio by 105.78 % and performance evaluation criterion by 18.75 %, comparing with the existing Model 1 N. Besides, the average and maximum wall temperature drop by 54.26 K and 293.02 K, respectively. In addition, the triple-layered channel is demonstrated to have the ability to reduce rotational effects on heat transfer uniformity. The heat transfer difference between the leading and trailing walls induced by the Coriolis force is restricted. It also has the self-adaptive ability to increase the degree of heat transfer enhancement when thermal load increases.

Secondary flows in the actuator-disk simulation of wind-turbine wakes

This study explores the generation of secondary flows of Prandtl's second kind in the actuator-disk simulation of wind-turbine wakes. Leveraging large-eddy simulation data and conducting an analysis of the mean streamwise vorticity budget, we uncover the physical mechanisms contributing to this phenomenon. Our investigations attribute the emergence of such flows to the spatial gradients of the Reynolds stresses in the wake downstream of the turbines, which are, in turn, influenced by ground effects. To further investigate the phenomenon, we specifically isolate the impact of secondary flows on the wake by employing a model recognized for its incapacity to capture such dynamics: a two-equation Reynolds-averaged Navier–Stokes (RANS) model founded on the linear eddy-viscosity hypothesis. By comparing the predictions of the RANS model with those of large-eddy simulations and wind-tunnel experiments, we highlight the effect of secondary flows on the wake structure and, in particular, the upward shift of the wake. Motivated by the obtained results, we then enhance the baseline RANS model by introducing a non-linear term within the Reynolds stress tensor. This modification leads to a more accurate representation of Reynolds stresses, enabling the RANS model to capture the secondary flows in the wake. Our analysis emphasizes the importance of employing advanced RANS models in the simulation of wind farms.

Heat transfer characteristics of the air-oil heat exchanger with spiral tube

The cooling problem of hot-end components has become an important factor restricting the aero-engine performance, life and reliability. In order to improve the cooling quality of the cooling air, the supercritical aviation kerosene was used as a heat sink to cool the cooling air in a spiral heat exchanger. The flow and heat transfer process of the air-oil heat exchanger with spiral tube were numerically simulated using Fluent software. The flow and heat transfer situation of air and aviation kerosene were analyzed in detail. The optimized spiral heat exchanger with the diaphragm was presented to enhance the heat transfer capacity of the air side. It is found that the heat transfer performance of the air-oil heat exchanger is greatly enhanced by the spiral tube due to the secondary flow effect. With the increase in the turning angle of the spiral tube, the effect of the secondary flow for the heat transfer enhancement is strengthened. The PEC index of the spiral heat exchanger increases with the increase in the air flow rate. The spiral heat exchanger with the diaphragm can effectively improve the heat transfer performance of the heat exchanger, and increase the PEC index of the air side by more than 20%.

Spur Dike Applications for the Sustainability of Channels in Incised Steep Bend Streams

Japan’s rivers are shaped by distinctive topography and abundant rainfall, and they face flooding and sediment supply escalation concerns under climate change. Small- and medium-sized rivers tend to catch unprecedented forces that exceed planned levels, leading to substantial widening and excavation. Thus, there is a demand for a method that is capable of managing significant flood flows over an extended period. The spur dike can maintain channel clearance by promoting erosion as well as providing bank protection. However, the effectiveness of this spur dike function has not been well studied in small- and medium-sized rivers and curved reaches. In this study, we evaluate the function of spur dikes in improving channel sustainability based on examples of small- and medium-sized rivers that have maintained their channel for more than ten years after spur dike installation. First, the applicability of the empirical rule was evaluated by comparing it with actual cases of erosion depths in curved sections in Japan. Next, one-dimensional simulations were performed to evaluate the sustainability of the section over a long period. Finally, a depth-averaged morphodynamic simulation, including the secondary flow effect, was applied to evaluate the location of the flow core and elevation changes due to the spur dike. The results showed that a slight difference in the ratio of river curvature radius to river width (r/B) caused the river channel to be erosive and sedimentary. The reasons for the difference were the cross-sectional expansion caused by the excavation of the bend and the difference in the plane flow regime caused by the shift of the flow core to the inside of the bend. Although it is structurally challenging to reproduce localized scour around a spur dike in a depth-averaged simulation, it is essential for designing to apply the simulation model and combine empirical knowledge.

Effect of secondary flow and secondary reactions on pyrolysis and heat transfer of supercritical hydrocarbon aviation fuel in a U-bend tube

The regenerative cooling channels generally consists of straight channel sections and curved sections for the scramjet engines. In this study, pyrolysis and heat transfer of supercritical hydrocarbon aviation fuel HF-1 in a U-bend tube (UBT) were numerically simulated. Discussion was held regarding how the secondary flow and secondary reactions affected the temperature distribution, heat transfer and pyrolysis. Results indicated that the secondary reactions have obvious effects on temperature distribution and pyrolysis at high pyrolysis conversion rate. Besides, the temperature stratification in the UBT is obvious. The existence of secondary flow inside the bending section of the UBT increases non-uniformity of HF-1 mass fraction distribution inside the cross sections. Compared with a straight tube (ST), the radial distribution of DamkÖhler number (Da) inside the cross-section of the UBT is asymmetrical, while Da near the inner side of the UBT is larger. When the r increases from −1 to 0, Da in the ST decreases from 24.4 to 0.3 while Da in the UBT decreases from 102.4 to 0.46. Therefore, the flow of HF-1 has a diminishing effect on the pyrolysis from the main flow area to the inner side wall to a larger extent.

A Numerical Design Space Investigation of Low-Speed Axial Compressor Stages Using Single-Row and Tandem Bladings

Abstract The present article numerically investigates the design space of an axial, low-speed compressor stage using tandem and single aerofoil configurations. The investigated rotor and stator designs follow a coupled throughflow and blade-to-blade design approach and will yield quasi-3D designed compressor stages with either single-rotor single-stator or tandem-rotor tandem-stator configurations. In past studies, tandem aerofoil configurations have shown a potential for increasing stage working coefficients, while maintaining their efficiency compared to their single aerofoil reference configuration. However, it was also shown that tandem configurations are able to reduce efficiency due to an increase of secondary flow caused by the higher loading and introduction in the second aerofoil. Currently, there is no clear design guideline on when tandem aerofoil configurations should be preferred over single ones, and under which circumstances they should be avoided. To answer this question, the present article will attempt to create a “Smith chart” for tandem aerofoil stages and compare them to their single aerofoil counterparts. Originally intended for use with turbine stages, the Smith chart maps out the potential efficiencies of turbomachinery designs. It plots the working coefficient (Ψ) over the throughflow coefficient (φ), which gives a direct indication of the shape of velocity triangles in the stage. However, the Smith chart has been well investigated for single aerofoil stages, both numerically and experimentally, as well as for low- and high-speed applications, and this is not the case so far for tandem configurations. The current work presents a fast Q3D design and analysis procedure used on a low-speed axial compressor stage and will yield quasi-3D efficiencies for a large number of different design coefficients. The findings of this work will enable future research to identify preferred combinations of design coefficients for tandem aerofoil configurations, which can then be further analyzed in 3D investigations at a higher fidelity level.

Numerical Calculations for Curved Open Channel Flows with Advanced Depth-Integrated Models

This paper examines the effect of coupling several equations for vertical velocity profiles in a depth-integrated model to clarify the roles of three-dimensional flow structures minimizing the errors from the two-dimensional calculation model. In addition, this paper develops a numerical discretization method for the dispersion terms in the horizontal momentum equations. It is revealed that the use of the 2DC model underestimates the water surface elevation through the comparisons with the experimental datasets, advanced 2DC and 3DC results. The prediction of water surface elevation is considerably improved by taking into account of secondary flow effect with vorticity equations. It is clarified that the accuracy in predicting the water surface elevation is increased with increasing the degree of the function for vertical velocity profiles in advanced 2DC models or the number of vertical grids in 3DC model, while non-hydrostatic pressure and variation in vertical velocity have a second importance.

Adaptation of von Karman Institute for Fluid Dynamics’ Isentropic Compression Tube Facility for High-Speed Low-Pressure Turbines Testing

Abstract This paper presents the commissioning of a newly designed High-Speed Low-Pressure Turbine (LPT) stage, operating at transonic exit Mach numbers and low Reynolds numbers, typical of modern geared turbofan (GTF) applications. The work falls within the scope of the Clean Sky 2 project SPLEEN (Secondary and Leakage Flow Effects in High-speed Low-pressure Turbines), which consists of an extensive experimental characterization of LPTs for GTFs. Geometries and measurements will constitute a valuable open-access database for the validation of simulation methods and data analysis tools. At first, the characteristics of the research turbine are illustrated. The LPT stage is designed with an equal number of stationary vanes and shrouded rotor blades. The upstream and downstream hub cavities are purged and feature engine-realistic rim seals. The nominal operating condition of the stage is reported, along with a set of off-design conditions, obtained by varying rotor speed and purge mass flows. The second part of the paper describes the substantial revamping of the CT3 large-scale compression tube at Von Karman Institute for Fluid Dynamics, traditionally employed for high-pressure turbine testing, now adapted to host a high-speed LPT stage. The third section is an overview of the time-averaged and fast-response instrumentation. The test article is heavily instrumented, to maximize the amount of acquired data, while minimizing the number of blowdown tests. The dataset for each operating conditions includes aerothermal measurements in the annulus flow, in the hub cavities and in the shroud labyrinth seal. In addition, this section presents the design of an in-house traversing system for continuous probe traversing during the test. Finally, the outcome of the commissioning phase is discussed, with particular emphasis on the operating conditions stability, as well as on the inlet and cavity injection uniformity. The commissioning of a traversing system for continuous probe traversing during the short-duration test is also discussed.

Experimental investigation on thermochemical heat storage using CaO/CaCO3 in a cyclone reactor

In this study, a cyclone reactor with a secondary flow effect was applied to the CaCO3/CaO thermochemical energy storage system. An experimental platform for thermochemical energy storage with a cyclone reactor was built. A cold-state experiment was performed to study the feeding and conveying performance of CaCO3 and CaO particles. The experimental results reveal that the average feeding speeds of CaCO3 and CaO were 25.7–29.6 g/min and 3.5–11.0 g/min, respectively. The effects of different heating temperatures (750–950 °C) and inlet air velocities (4–14 m/s) on the reaction performance were studied. The results indicate that CaCO3 decomposes above 750 °C. The reaction degree increased with increased reaction temperature, and the peak value of the outlet CO2 concentration peaked (4.72 %) when the heating temperature was 950 °C, where the reaction degrees under different air velocities were similar. In addition, the inlet air velocity and the outlet relative humidity decreased owing to the formation of CO2 and CaO. In the energy release experiment, it was difficult to smoothly feed CaO into the cyclone reactor for synthesis owing to its characteristics, and no obvious change in the CO2 concentration was observed at the outlet. The energy storage reaction and flow characteristics were experimentally studied to provide a reference for the design of thermochemical energy storage reactors.

A Numerical Test Rig for Turbomachinery Flows Based on Large Eddy Simulations With a High-Order Discontinuous Galerkin Scheme—Part III: Secondary Flow Effects

Abstract In this final paper of a three-part series, we apply the numerical test rig based on a high-order discontinuous Galerkin scheme to the MTU T161 low-pressure turbine with diverging end walls at off-design Reynolds number of 90,000, Mach number of 0.6, and inflow angle of 41 deg. The inflow end wall boundary layers are prescribed in accordance with the experiment. Validation of the setup is shown against recent numerical references and the corresponding experimental data. Additionally, we propose and conduct a purely numerical experiment with upstream bar wake generators at a Strouhal number of 1.25, which is well above what was possible in the experiment. We discuss the flow physics at midspan and in the end wall region and highlight the influence of the wakes from the upstream row on the complex secondary flow system using instantaneous flow visualization, phase averages, and modal decomposition techniques.

Modelling secondary circulations in stratified bi-directional flows: A potential mechanism for flux transfer from lower to upper fluid layers

Results are presented for new laboratory-scale numerical simulations of bi-directional stratified flows generated within an idealized trapezoidal sill-channel topography, focussing on the lateral (or secondary) flow structure across the channel both in non-rotating and rotating frames of reference. The simulations utilise the non-hydrostatic Bergen Ocean Model (BOM), a three-dimensional general ocean circulation model. The results from the BOM simulations are verified by large-scale experimental data obtained in the LEGI Coriolis rotating platform in Grenoble, within which velocity and density fields for bi-directional stratified flows were measured through particle image velocimetry (PIV) and micro-conductivity density probes, respectively. The BOM simulations reproduce the main dynamic flow patterns and structure of the large-scale exchange flows generated through the trapezoidal sill-channel, with the lower layer saline intrusion flux shown to reduce, due to partial blockage, as the upper freshwater flow is increased (i.e. net barotropic forcing). Non-rotational BOM simulations show upward (and downward) flow along the centreline of the trapezoidal sill-channel in the upper (and lower) layers, with symmetrical counter-rotating circulation cells forming both above and below the density interface. By contrast, equivalent BOM simulations within a rotating frame of reference show the secondary flow circulation in the lower dense water layer to be dominated by Ekman dynamics, while two co-rotating cells are formed in the upper freshwater layer above the inclined density interface. In these rotational cases, significant upward and downward velocities (i.e. strong upwelling and downwelling) are generated adjacent to the inclined sill-channel side walls (in comparison to non-rotating cases). This secondary flow effect may be especially relevant for exchange flows in estuaries, fjords, deep-ocean channels and sea straits, where the channel width exceeds the Rossby radius of deformation and, thus, Earth rotation effects become important dynamically. In particular, any potential vertical transport pathway associated with this upwelling and downwelling at the inclined channel side slopes may have strong managerial implications in coastal areas, whereby tracers (e.g. contaminants, salinity, reduced DO waters, fine sediments) can be transferred from deep waters to the near surface.

A numerical study of the slurry erosion in 90° horizontal elbows

The slurry erosion processes of horizontal elbows are investigated numerically using an Eulerian-Lagrangian method coupled with particle rebound and erosion model. First, a mechanical model for the internal multiphase flow involving slurry erosion in the 90° horizontal elbow is established and the corresponding governing equations are presented. Then, a two-way coupled numerical solution procedure is developed and its accuracy and stability are examined carefully using benchmark pipe flow models with experimental results. Using the validated numerical methods, the effects of flow velocity, fluid density and fluid viscosity on the flow and slurry erosion in the horizontal elbow are analyzed in detail. Finally, a prediction method of erosion rate distribution based on the pipe Froude number, particle Stokes number and the Dean number is proposed. In this method the pipe Froude number is employed to qualify the effect of ununiform distribution of particles, particle Stokes number and Dean number are combined to qualify the effects of inertia force, drag force and secondary flow. Using this approach, the location of the maximum erosion rate in horizontal elbows under different operating conditions can be predicted more conveniently.