Separation flow in a curved duct is a common phenomenon in engineering applications, and it highly contributes to the performance of fluid machinery. Accurate prediction of curved duct flows using the computational fluid dynamics method remains a challenge due to the limitations of turbulence modeling. Hence, the high-fidelity method of the delayed detached eddy simulation (DDES) approach is employed to simulate the U-duct flow with a Reynolds number of 10^{5}. The DDES results are compared with experimental data from the study by Monson *et al.* (1990) and analyzed in detail. The *Q*-criterion is defined to analyze the vortex structures and study the mechanism in the flow separation region. Discussions are made on turbulence characteristics, including turbulence energy spectra, helicity density, and turbulence anisotropy in the U-duct flow. Results indicate that the regions near the wall and within flow separation are highly anisotropic. The turbulence near the wall region is in a two-dimensional state, and the turbulence within the flow separation region is in a “rod-like” state.

## NOMENCLATURE

*A*experimental constant

*b*_{ij}non-dimensional Reynolds stress

*C*_{f}skin friction coefficient

- CC
rotation-curvature correction

- CFD
computational fluid dynamics

- DDES
delayed detached eddy simulation

- DES
detached eddy simulation

- DNS
direct numerical simulation

*E*energy spectrum

- FFT
fast Fourier transformation

*f*_{0}frequency (Hz)

*H*height of the U-duct (cm)

*h*helicity density

*I*_{1},*I*_{2}, and*I*_{3}the first, second, and third invariants of the Reynolds stress tensor, respectively

*k*wave number

- LDV
laser Doppler velocimeter

- LES
large eddy simulation

*LT*_{cri}local trace criterion

- Ma
Mach number

- NASA
National Aeronautics and Space Administration

*p*_{t}total pressure (Pa)

*Q**Q*-criterion (s^{−2})- RANS
Reynolds-averaged Navier–Stokes

- Re
Reynolds number

- S
position along the wall of the U-duct

- SST
shear stress transport (turbulence model)

*T*periodicity of the shedding vortex (s)

*t*time (s)

- $u\u20d7$
velocity vector (m/s)

- URANS
unsteady Reynolds-averaged Navier–Stokes

*V*velocity (m/s)

*V*_{x}velocity in the X direction (m/s)

*x*,*y*Cartesian coordinates of the U-duct grid (cm)

*y*^{+}dimensionless wall distance

### Greek symbols

## I. INTRODUCTION

Internal flow with a high Reynolds number is commonly seen in aero engine components, such as inlet-ducts, compressors, combustors, etc. This flow is accompanied by complex flow features,^{1} such as high turbulence intensity levels, instability, and three-dimensional separation under strong curvature geometry. The three-dimensional separation under strong curvature geometry dramatically affects the performance of the inlet-duct, blade passages, and combustor, which in turn affects the overall performance of the engine. Gaining a deep insight into the mechanism of the separation flow under strong curvature geometry is of great importance, and it could assist in improving the prediction accuracy of the computational fluid dynamics (CFD) method.

Nowadays, the CFD technique is widely used in engineering design or flow mechanism investigation of complex flows.^{2–5} The fast growth of computing capacities makes advanced numerical methods such as direct numerical simulation (DNS), large eddy simulation (LES), and hybrid Reynolds-averaged Navier–Stokes (RANS)/LES available on simulating complex flow fields, such as corner separation flow^{6–8} and tip leakage flow^{9–14} in turbomachinery. However, the RANS method is still the most widely used approach in industrial applications.^{15–19} According to a previous study,^{20,21} it has been assumed that the limitless increases in computing power will someday remove the need for turbulence modeling; however, the estimate for reaching this milestone is 2080, which is a long time away. Furthermore, some novel methods using machine learning have shown good accuracy and efficiency in predicting fluid–structure interactions^{22} and active flow control,^{23} making it a potential approach in the future. For engineering applications, the RANS method offers fast computational speed, but its accuracy is insufficient. The DNS entails excessive computational cost due to its requirement to resolve all flow scales. LES is used for simulating relatively simple flows at low Reynolds numbers. In contrast, hybrid RANS/LES methods present a viable alternative, allowing for faster simulations while maintaining requisite accuracy.

Many numerical investigations on the U-duct flow are based on two-dimensional (2D) simulations because they can save computation time. Monson *et al.*^{24,25} conducted experimental and numerical studies with two-equation turbulence models for a 2D U-duct. The performance predicted by six *k*–*ɛ* turbulence models is compared, and it is concluded that the extended *k*–*ɛ* turbulence model by Chen and Kim^{26} performs best in predicting the flow of U-ducts. Smirnov and Menter^{27} modified the shear stress transport (SST) model with the rotation-curvature correction (SST-CC), and this modified SST-CC model is applied to simulate Monson’s 2D U-duct flow.^{27} The simulation results indicated that the SST-CC model offers significant improvements in predicting complex three-dimensional separation flows involving a strong curvature while maintaining computational efficiency and robustness. In engineering, the curvature of the U-duct causes a strong pressure gradient to the internal flow, leading to a large amount of backflow and secondary flow, which is three-dimensional. The study from the Helsinki University of Technology,^{28} Finland, in 1999, confirmed that RANS models cannot accurately predict three-dimensional flow in U-ducts. To study the effects of rotation on the flow with a strong curvature and separation, Laskowski and Felten^{29,30} used DNS and LES to investigate the flow in a strong curvature duct with a low Reynolds number of 5600. The results show that rotation has a significant impact on turbulence statistics, such as intensity and Reynolds stress.

Detached eddy simulation (DES) is one of the hybrid RANS/LES methods. When the turbulent length scale is smaller than the maximum grid dimension, the simulation operates in the RANS mode, and this mode typically occurs near solid boundaries. When the turbulent length scale exceeds the grid dimension, the simulation transitions to the LES mode. This approach combines the advantages of both RANS and LES in the simulation, thereby achieving better performance between the turbulent region and the boundary layer. Previous studies^{31–35} have qualitatively demonstrated that the hybrid RANS/LES method produces higher quality data than the RANS/unsteady RANS (URANS) method for a wide range of flows. Based on the two-equation SST *k*–*ω* turbulence model, Travin *et al.*^{36} proposed DES based on the SST (DES-SST) method. Subsequently, Davidson^{37} improved the DES-SST method and named it the delayed DES-SST (DDES-SST) method. In comparison to the DES-SST method, the DDES-SST method yields more accurate numerical simulations, capturing a greater amount of flow and turbulence information. RANS methods have been predominantly utilized for the prediction of strong curvature flow.^{24,25,27} However, RANS methods exhibit limitations in simulation accuracy, particularly in capturing accurate turbulent information. Alternatively, DNS and LES methods can also be applied to U-duct simulations.^{29,30} Nevertheless, these approaches demand extensive computational resources, especially for high Reynolds number flows. Hence, DNS and LES methods are generally employed for lower Reynolds number flows. The DDES method allows for computationally feasible simulations of high Reynolds number U-duct flows, and it contributes to capture rich flow data and turbulence information of the flow field. This method is well-suited for conducting comprehensive multidimensional analyses of flow field mechanisms.

In this study, a U-duct flow is investigated using DDES methods. The performance of the DDES method is tested using a three-dimensional (3D) simulation and compared with the experimental data. To investigate the mechanism of turbulent flow with a strong curvature, detailed analyses are made based on the results obtained from DDES. The results of DDES are analyzed from multiple aspects, including the flow separation properties, the vortex structure based on the *Q*-criterion method, turbulence energy spectra, and turbulence anisotropy.

## II. CONFIGURATIONS AND NUMERICAL METHODS

### A. U-duct experiment apparatus

The rectangular U-duct investigated in this study was tested by Monson *et al.*^{25} in National Aeronautics and Space Administration (NASA) Ames High Reynolds Number Channel I. Figure 1 shows the sketch of the geometry of the U-duct test section. This U-duct is a rectangular channel with an aspect ratio of 10. The height of the U-duct is *H* = 3.81 cm, and it remains constant throughout the whole duct. A flow condition with a high Reynolds number of Re = 10^{5} is tested. The inlet condition is a nominal average velocity, *V* = 32 m/s (Ma ≈ 0.1), and the total pressure is set to *p*_{t} = 1.2 atm. In this test case, a two-component laser Doppler velocimeter (LDV) was used to measure turbulent and mean velocity distributions. An electronically scanned pressure system is used to measure static pressures. More details about the test cases can be found in Ref. 24.

### B. Computational grid

The computational grid of the U-duct is shown in Fig. 2. The computational domain used in this study is identical to that used in Ref. 38. A series of grid numbers and distributions have been compared to validate the grid independence of the solution. For this 3D U-duct grid, the computational domain extends to 2.5 *H* in the spanwise direction. There are 702 nodes in the flow direction, 149 in the vertical wall direction, and 60 in the spanwise direction. Hence, the total mesh account is about 6.28 × 10^{6}. The grid provides accurate near-wall resolution, with the value of *y*^{+} being less than 1. Periodic boundary conditions are enforced for the upper and lower faces of the computational domain, as shown in Fig. 2(a).

### C. Numerical settings and boundary conditions

The DDES method based on the SST *k*–*ω* turbulence model (DDES-SST) is adopted to predict the three-dimensional flow in a U-duct. Grid independence has been analyzed to mitigate the influence of other factors. To reduce numerical diffusion, second-order upwind and central differencing schemes are employed for the convection and viscous terms, respectively. The SIMPLE algorithm is utilized to handle pressure-velocity coupling. For the case with Re = 10^{5}, the air density is 1.47 kg/m^{3} and the total pressure is 1.2 atm. The inlet boundary condition is defined using the velocity and turbulent kinetic energy profiles measured by the experiment at *x*/*H* = −4. A constant static pressure is applied at the outlet boundary. The spanwise plane is treated as periodic, while the inner and outer walls are set as non-slip boundaries.

## III. RESULTS AND DISCUSSION

### A. Comparison with experimental data

Numerical predictions of the flow in the U-duct using the DDES-SST method are compared with experimental data. In Fig. 3, velocity profiles at different positions along the U-duct are compared. At *x*/*H* = −4, the simulation results from DDES-SST closely match the experimental data, with only a small difference observed near the outer wall. At *θ* = 0°, the velocity profile predicted by DDES-SST shows closer agreement with experimental results near both the inner and outer walls than other turbulence models. Similarly, at *θ* = 90°, the DDES-SST results exhibit good agreement, particularly near the inner wall, although there is significant deviation near the outer wall. At the outlet, noticeable discrepancies exist between the velocity profile predicted by DDES-SST and the experimental results, particularly near the outer wall. Conversely, near the inner wall, the simulation results of DDES-SST align better with the experimental data. Overall, a comprehensive comparison of velocity profiles at different positions reveals that the accuracy of DDES-SST is higher near the inner wall than its accuracy near the outer wall.

Figure 4 illustrates the distribution of the skin friction coefficient *C*_{f} along the inner and outer walls of the U-duct for Re = 10^{5}. In the inlet section (*S*/*H* < 0), the *C*_{f} distribution predicted by the DDES-SST method closely matches the experimental results for both the inner and outer walls. However, at the bend inlet (*S*/*H* = 0), there is a slight overprediction of the peak *C*_{f} value on both walls. Despite this, flow separation occurs near the end of the U-duct bend (*S*/*H* = π), yet the *C*_{f} results predicted by DDES-SST align well with experimental data. Downstream of the flow separation, the DDES-SST method accurately predicts the distribution of *C*_{f} values on both walls. By combining the velocity and *C*_{f} predictions, it can be concluded that the DDES method accurately captures the separation flow under a strong curvature. However, there is still room for improvement in the predictive accuracy of the DDES method.

### B. Vortex identification

The DDES-SST method offers detailed flow field data, and it is helpful for understanding the mechanics of separated flow under a large curvature through various analytical approaches applied to this dataset. Figure 5 shows the streamlines at the separation region in different transient cases, with color indicating the magnitude of vorticity. The flow period *T* in Fig. 5 represents the duration from the generation to the disappearance of the shedding vortex, and it is calculated by averaging the durations of multiple shedding vortex generation and disappearance processes. Two main vortices are observed downstream of the bend. As circled in Fig. 5, the one residing at the center of the separation region is caused by recirculation flow; hence, it is termed “recirculation vortex.” The other, termed “shedding vortex,” is positioned close to the inner wall after the recirculation vortex. The recirculation vortex remains stable at the center of the separation, as depicted in Fig. 5(a). In the subsequent transient demonstrated in Fig. 5(b), the right recirculation vortex intensifies, while the left vortex diminishes, and the shedding vortex is divided into two smaller ones. Moving to the transient depicted in Fig. 5(c), the right recirculation vortex strengthens further, while the left recirculation vortex weakens due to shedding from the separation region and downstream flow. The shedding vortex dissipates rapidly due to turbulence viscosity. In the subsequent transient shown in Fig. 5(d), the shedding vortex diminishes until disappearing downstream, while the right recirculation vortex continues to intensify until merging with the weak left recirculation vortex. Finally, in the last transient shown in Fig. 5(e), the merged vortex replaces the left recirculation vortex, and a small vortex appears in the position formerly occupied by the right recirculation vortex. The shedding vortex completely disappeared, while a new small shedding vortex generates from the left recirculation vortex. This process repeats periodically.

The streamlines can help show the separation region of the flow field and indicate the macro-recirculation vortex and shedding vortex, but they cannot figure out the small-scale vortices in the U-duct. In reality, many small-scale vortices exist within both the separation region and downstream of it. Some vortex identification methods, such as the *Q*-criterion,^{39} Liutex,^{40–42} and local trace criterion (*LT*_{cri}),^{43,44} help to find these micro-turbulence coherent structures. The *Q*-criterion adopted in this study presents the balance of the local strain rate and vorticity. This criterion is described and analyzed in detail in Ref. 39. The vortices can be figured out when *Q* > 0; different thresholds of *Q* can show various scales of the vortex.

Figure 6 shows the vortex structures identified by the *Q*-criterion at various time steps, and the value for *Q* is 10^{5} s^{−2}. The color denotes the magnitude of the *x*-velocity (*V*_{x}). Due to the periodic boundary condition being in the spanwise direction, this U-duct behaves as an infinitely extended duct. Prior to separation, the flow field exhibits strong two-dimensional characteristics. However, upon separation, numerous large-scale strip vortices are generated. These vortices are turbulent coherent structures with higher velocities near the outer wall. The separation vortex near the inner wall reduces the effective flow area at the end of the bend, leading to an acceleration near the outer wall. The blue contour of vortices indicates *V*_{x} < 0 m/s near the inner wall at the end of the bend. The region demarcated by the red circle residing within the separation vortex constitutes a recirculation zone. With the flow development, these large-scale strip vortices migrate downstream and gradually dissipate into smaller scales. Without the influence of the separation vortex, the velocity of small-scale vortices remains relatively low and maintains consistency with the mainstream velocity in the streamwise direction. Near the outlet, the longitudinal strip vortex progressively merges with the mainstream strip vortex. Given sufficient duct length, the small-scale vortices will dissipate entirely.

Figure 7 shows the vortex identification result at *Q* = 10^{6} s^{−2}, aimed at capturing smaller vortex structures in the downstream and outlet sections. For example, the vortices indicated by the red circles are high-velocity strip vortices that gradually stretch and bend as turbulence flows downstream. The bent vortex forms a hairpin vortex and eventually dissipates due to viscosity.

### C. Turbulence energy spectra and backscatter

^{45}The energy spectrum is solely dependent on the dissipation rate and wave number, and it can be written as follows:

*A*is an experimental constant,

*ɛ*is the dissipation rate, and

*k*is the wave number. Furthermore,

*E*(

*k*) can be calculated by the velocity data using the Fast Fourier Transformation (FFT) method,

*t*and $ui\u0302(k)$ is the velocity component in the frequency domain. Then, the energy spectrum

*E*(

*k*) can be calculated by

*f*

_{0}is the frequency.

Figure 8 illustrates the distribution of monitor points in the separation region. From the *Q*-criterion, as shown in Fig. 7, there are many turbulence vortices in the separation region, which make the turbulence kinetic energy transportation active within different turbulence length scales. Point 3, which is set in the reattachment region, shows higher turbulence energy spectra as many vortices are generated and dissipated continuously, leading to active turbulence energy transportation. This result agrees with the analysis of the vortex in the separation region shown in Fig. 5.

The turbulence energy spectra results are shown in Fig. 9. It is evident that the energy spectra in the separation region are significantly higher than those observed prior to separation. Furthermore, as shown in Fig. 9, the slope of the energy spectra conforms to the −5/3 power law for monitoring points located outside the separation region. This indicates that the turbulence flow in the pre-separation region (point 1) is closer to an equilibrium state. However, in contrast to the monitoring points prior to the separation region, the point within the separation region (point 2) does not strictly adhere to the −5/3 power law, suggesting that the turbulence in the separation region is in a non-equilibrium state. Through the analysis of turbulence vortices in the separation region, it becomes apparent that numerous coherent structures exist within this region, causing the turbulence flow downstream to become unsteady and be non-equilibrium.

*et al.*,

^{46}the correlation between helicity and energy backscatter allows the usage of helicity to represent energy backscatter. If the helicity density exceeds 0.7, the energy backscatter surpasses forward dissipation. The definition of helicity density can be written as follows:

Figure 10 shows the contour of helicity density and vorticity. High helicity density is predominantly concentrated in the reattachment region and the downstream flow field after separation. It indicates active turbulence backscatter following the separation region. In the separation region, high helicity density is observed near the outlet of the bend. In the center of the separation region, the helicity density is not as strong as that downstream, while abundant vortices exist at the outlet region of the bend, suggesting a close relationship between energy backscatter and vortices. Comparison of helicity density and vorticity shows that regions of high vorticity correspond to regions of high helicity density, and it is consistent with the findings by Yan *et al.*^{8}

Figure 11 shows the iso-surface of helicity density (*h* = 0.8), which is colored by vorticity magnitude. As is shown in the figure, the red region mainly appears downstream of the separation region, which indicates that the energy backscatter mainly happens at the exit of the bend and the reattachment region, where the vorticity magnitude is high. The contour of the helicity density region (*h* > 0.6) and the mainstream vorticity (X-vorticity) are shown in Fig. 12. The colored region in the contour of helicity density is the place where turbulence energy backscatter appears. The X-vorticity can represent the strength of the secondary vortex. The two contours reveal that turbulence backscatter is accompanied by the evolution of the secondary vortex in the separation flow region.

### D. Anisotropy of turbulence

Turbulence in the three-dimensional flow separation has strong anisotropy. In order to study the mechanism, the invariants of the dimensionless Reynolds partial stress are used by Lumley to express the natural characteristics of Reynolds stress.^{47} This method is called the Lumley triangle (shown in Fig. 13).

*b*

_{ij}has three invariants, expressed as

*I*

_{2}and

*I*

_{3}represent the second and third invariants of the Reynolds stress tensor, respectively. For an incompressible flow,

*I*

_{1}= 0. By cross plotting

*ξ*and

*η*as an abscissa and ordinate, as shown in Fig. 13, the

*ξ*and

*η*of

*b*

_{ij}in a real flow should fall within a triangle. The original point, namely, the bottom vertex of the triangle, where

*ξ*= 0 and

*η*= 0, represents the isotropic turbulence. The double edges from the original point represent the axial symmetry turbulence; the left edge represents “disk-like” turbulence, which means that the turbulent velocity fluctuations of one direction are obviously weaker than those of the other two directions, and the right edge represents “rod-like” turbulence, which means that the turbulent velocity fluctuations of one direction are obviously stronger than those in the other two directions. Finally, the top curved edge represents two-dimensional turbulence, which indicates one component of the turbulent velocity fluctuations close to zero. For the points in the triangle, the closer they are to the bottom vertex, the more isotropic the turbulence is, and the closer they are to the top curved edge, the more anisotropic the turbulence is.

In Fig. 14, the contour shows the velocity distribution across the entire U-duct field. The contour results of averaged velocity facilitate clear differentiation between low and high velocity regions. The color of the contour corresponds with the color of the scatter point in the Lumley triangle below. By comparing different regions from the velocity contour, the downstream flow field can be distinctly divided into three regions: the separation region, the transition region, and the mainstream region. The separation region, near the inner wall, exhibits low velocity and strong recirculation flow, while the mainstream region, near the outer wall, features high-velocity magnitudes. The transition region lies between the separation and mainstream regions, characterized by a small area with a high-velocity gradient. Detailed analysis of turbulence anisotropy is essential to understand the influence of separation.

Three streamlines extracted from the U-duct flow and their positions are depicted in Fig. 15. The points along these lines are shown in the Lumley triangle in order to study turbulence anisotropy. As shown in Fig. 16(a), streamline 1 is close to the outer wall, and the points are close to the upper curved edge of the Lumley triangle. The points along streamline 1 in the Lumley triangle are distributed close to the top curvature edge, which indicates that the turbulence is in a strong two-dimensional state similar to other near-wall flows. For streamline 2 in the mainstream region, as shown in Fig. 16(b), the points are distributed in the middle of the triangle, which indicates that the turbulence tends to be less anisotropic than in the other region. Most points along streamline 2 are distributed near the right edge of the Lumley triangle, which means the turbulence flow in the mainstream tends to be in a weak “rod-like” state. For the streamline near the separation, the points trend to move to the right corner, which is close to the right edge, so the turbulence is in a strong “rod-like” state with one-component velocity fluctuation predominating over the other components. This turbulence state is obviously different from the mainstream line. The results of streamlines 2 and 3 indicate that the separation flow affects the turbulence anisotropy incredibly.

## IV. CONCLUSIONS

In this study, numerical simulations are conducted on a U-duct using the DDES method. The performance of the DDES method in predicting separation flow in the U-duct under a strong curvature is evaluated. The vortex structures are carefully studied, and the turbulence characteristics are discussed in detail from the abundant turbulence data from the DDES method. The main conclusions are as follows:

The DDES method accurately predicts the velocity distribution and surface friction coefficient distribution within the U-duct, exhibiting good agreement with experimental results. Meanwhile, there is still room for further improvement in its accuracy.

The

*Q*-criterion can demonstrate the development of vortices in the separation region, in which different scale vortices are generated and dissipated in a periodic process. With the development of the flow after the bend, a lot of strip vortices are produced, and the flow gets more complex after separation.Helicity density mainly concentrates in the region of the reattachment and downstream flow field after the separation. The high helicity density region and secondary flow are related, which indicates that the evolution of the secondary vortex in the separation flow region is accompanied with turbulence backscatter.

The results of the Lumley triangle indicate that the regions near the wall and within flow separation are highly anisotropic. The turbulence of the near wall region is in a two-dimensional state, and the turbulence within the flow separation region is in a “rod-like” state.

## ACKNOWLEDGMENTS

This work was supported by the National Natural Science Foundation of China (Grant No. 52106039), the Industry-University-Research Cooperation Project of AECC (Grant Nos. HFZL2022CXY001 and HFZL2023CXY002), the Fundamental Research Funds for the Central Universities, and the Science Center for the Gas Turbine Project (Grant No. P2022-B-II-005-001).

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

**Xiaosong Yong 雍小松**: Conceptualization (equal); Funding acquisition (equal); Investigation (equal); Resources (equal); Software (equal); Supervision (equal); Validation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). **Yangwei Liu 柳阳威**: Conceptualization (equal); Funding acquisition (equal); Investigation (equal); Resources (equal); Supervision (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). **Hao Yan 闫昊**: Investigation (equal); Software (equal); Validation (equal); Visualization (equal); Writing – original draft (equal). **Yumeng Tang 唐雨萌**: Funding acquisition (equal); Investigation (equal); Software (equal); Validation (equal); Visualization (equal); Writing – original draft (equal).

## DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding authors upon reasonable request.

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*Internal Flow-Concepts and Applications*

*k*–ɛ eddy viscosity model for high Reynolds number turbulent flows

*-*ε turbulence closure model

*Advances in LES of Complex Flows: Proceedings of EUROMECH Colloquium*