# Turbulent Flow over Confined Backward-Facing Step: PIV vs. DNS

^{*}

## Abstract

**:**

## Featured Application

**The paper answers the question about an expected agreement between the very accurate numerical simulations and very accurate experiments in 3D time-averaged turbulent flow with separation.**

## Abstract

^{2}cross-section upstream and 45 × 45 mm

^{2}downstream, while a domain that was three times shorter was used in the DNS. A 2D-2C PIV system with a single high-speed camera and a pulse laser was used for a series of two-dimensional measurements of the velocity field at several cross-sections from two different perspectives. Variables analyzed in the experiment are time-averaged fluid velocities, velocity RMS fluctuations and two components of the Reynolds stress tensor. The key novelty is the comparison of two very accurate approaches, PIV and DNS, in the same cross-section geometry. Comparison of the similarities, and especially the differences between the two approaches, elucidates uncertainties of both studies and answers the question on what kind of agreement is expected when two very accurate approaches are compared.

## 1. Introduction

## 2. Experimental and Computational Geometry

^{2}cross-section, a step height of 25 mm, and a square duct of 45 × 45 mm

^{2}downstream the step. The geometry has an expansion ratio of 2.25.

^{2}in the experiment. The velocities are scaled with the bulk velocity in the upstream part of the test section, which was fixed to a unity value in the DNS, and corresponds to roughly 0.233 m/s inlet velocity in the experiment. Exact velocity used to convert measurements into dimensionless units took into account slight temperature and Reynolds number dependence separately for each measurement (~1% variations). Thus, the basic conversion between the dimensionless units and experiment are as follows:

- (a)
- 12.5 mm = 1 dimensionless length unit,
- (b)
- 0.233 m/s~1 dimensionless velocity unit.

## 3. Computational Set-Up of Direct Numerical Simulation

## 4. PIV Experiment and Data Processing

^{3}they are close to neutrally buoyant in water. PIV measurements of 2D velocity fields were performed in several planes, as described in Table 1. Two of these planes are also shown in the top-right drawing of Figure 1. The positions of the measured planes in Table 1 are given with respect to the origin of the coordinate system shown in Figure 1.

## 5. Results and Discussion

#### 5.1. Inlet Boundary Conditions

^{2}cross-section entrance with 15 holes of 1 mm diameter distributed over the inlet surface. The preliminary analysis performed by Zajec et al. [28], according to the guidelines of White [34], resulted in the inlet part of the test section, which is 68 cm long (~25 hydraulic diameters).

- -
- Measured mean streamwise velocity profile is slightly flatter in the central region of the inlet section (Figure 4, top-left) than in the DNS. Maximum velocity in the axis of the inlet section DNS is higher than in the experiment, despite having the same Reynolds number and mass flow rate. On the other hand, measured mean velocities are higher near the walls (Figure 4, both left graphs: ~0.6 length units of the near-wall regions). We can see that uncertainty bands of both profiles overlap in most of the central region; however, the measurement and DNS seem to be consistent, and they both show very smooth profiles with clearly distinguishable difference. The same behavior is observed in the z direction and in y direction measurements. Thus, we believe that the flat measured profile, which is somewhat different than in the DNS, comes as a consequence of a slightly too short inlet section of around 25 hydraulic diameters based on the guidelines of [34]. In a recent review, most of the pipe entrance length studies analyzed by Düz [35] recommended 25–50 hydraulic diameters to achieve constant pressure gradient, but an even longer section, up to 100 hydraulic diameters, is needed to achieve fully developed values of the mean turbulent statistics, which are studied in the present paper. A longer inlet section gives more length and time for complete development of large-scale turbulent vortices [35]. Nevertheless, using the same argument, we cannot completely exclude the error on the side of DNS. There is a possibility that the periodic domain for the generation of inlet boundary conditions in DNS was too short and did not generate a sufficient amount of large-scale structures, despite being comparable with similar simulations in the literature.
- -
- Unlike in the DNS, the velocity profile in the experiment is weakly asymmetric in the z = 0 plane. Flow in the z > 0 half of the domain is slightly faster than in the z < 0 region. Weak asymmetry, which is visible in some measurement and is almost absent in other cases, is further shown in some of the figures below. The asymmetry was too weak to find the cause: the inlet diffusor has a symmetric arrangement of holes over the z = 0 plane and the dimensions of the test section were found to be accurate within the measurement accuracy of our length measuring tools.
- -
- Due to the proximity of the camera to the laser sheet plane (~20 cm), and the camera view that is centered on the details around x-axis, the near wall measurements are less accurate. Ideally, the walls of the section should be parallel to the line-of-sight; however, walls that are actually seen under the non-negligible angle and out-of-plane motions of particles might contribute to the error. Consequently, the boundary layers in the region of about 0.1 dimensionless units from the wall are poorly resolved in the measurements taken in z = const planes (bottom drawings of Figure 4). Due to the manufacturing method, unresolved near-wall layer is even larger in y = const. measurements (around 0.4 dimensionless length units). Since the focus of our study is not on the boundary layers but on the recirculation zone downstream the BFS, these non-accuracies in the near-wall regions were considered to be acceptable.

#### 5.2. Recirculation Zone, Planes z = Const

_{RMS}are almost two times higher and the maximum of v

_{RMS}is roughly three times higher.

_{RMS}, are shown in Figure 6, with the other three processed quantities showing a similar agreement. Dotted contours in Figure 6 show that positions of the violet contours increased and decreased for 0.03 and 0.01 for u and v

_{RMS}, respectively. These are values of the estimated uncertainty in the mean streamwise velocity and velocity fluctuations given in Section 5.5; however, they conveniently show the sensitivity of the measurements. DNS contours in Figure 6 are black and serve as an independent measure of the uncertainty.

#### 5.3. Recirculation Zone, Planes y = Const

#### 5.4. Results at the Cross-Section of the Planes z = Const and y = Const

_{RMS}is larger than in other cases. Higher uncertainties are observed also for the near-wall measurement ‘y − 1.92’, where only mean velocity u is show, while the u

_{RMS}exhibits fluctuations of around 0.04 dimensionless velocity units, which is not shown in the right graph.

#### 5.5. Uncertainties of the Measurements

_{RMS}fluctuations ~0.25. These integral uncertainties obtained from various properties of our measurements are based on the fact that we performed 22 independent (or weakly dependent) measurements, where 10 of them in y = const. planes exhibit an additional internal symmetry. Uncertainty quantification of each 2D measurement case was consequently performed on around 10 profiles of each particular variable taken at various streamwise or spanwise line segments on the 2D plane, and crosswise compared with results of other independent measurements of the same variable in the same line segments. As such, this set of independent measurements provides a reasonable estimate of the measurement uncertainties.

- -
- Repeatability of the measurement performed in plane z = 0 is discussed in Section 5.2. Next to the graphs shown for u and v
_{RMS}in Figure 6, graphs of other variables, v, u_{RMS}and v’u’, were considered in the uncertainty analysis. - -
- Comparison of the measurements in symmetry planes z = ±0.8 and z = ±1.36 is discussed in Section 5.2. Beside the comparison of the mean streamwise velocities shown in Figure 7, a symmetry check was also done for other variables. This comparison encompassed all measurement uncertainties, since the measurements in different z = const. planes are independent of each other; some of them were made on different days.
- -
- Quantification of the uncertainties in y = const. measurements were slightly less precise and were performed through mirroring of the measurements over the z = 0 plane, as shown in Figure 10 for plane y = 0.8. All other planes were analyzed in a similar fashion. The y = const. measurements do not encompass all systematic errors. Uncertainties due to the camera and laser sheet position are not available in a single measurement and only the discontinuities due to the merging of measurement pairs into a single dataset provided an insight into this type of uncertainties.
- -
- Measurements in z = const. and y = const. planes were analyzed in the line segments that represent intersections of these planes. This analysis gave us 25 line segments, where the streamwise velocity u and u
_{RMS}fluctuations can be compared from entirely independent measurements. An example of such analysis is discussed in Section 5.4 and in Figure 11 for 5 out of 25 available line segments. This approach allows more precise uncertainty quantification of u and u_{RMS}, but not for other components of velocity and velocity fluctuations.

_{RMS}is used to plot uncertainty band of u

_{RMS}in DNS in Figure 4 top-right). The same statistical uncertainty applies to the experiment due to the similar length of the time-averaging interval; however, this uncertainty is added to the systematic errors, which are actually a dominant source of uncertainty in our experiment.

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Terekhov, V.I. Heat Transfer in Highly Turbulent Separated Flows: A Review. Energies
**2021**, 14, 1005. [Google Scholar] [CrossRef] - Oder, J.; Shams, A.; Cizelj, L.; Tiselj, I. Direct numerical simulation of low-Prandtl fluid flow over a confined backward facing step. Int. J. Heat Mass Transfer.
**2019**, 142, 118436. [Google Scholar] [CrossRef] - Da Vià, R.; Giovacchini, V.; Manservisi, S. A Logarithmic Turbulent Heat Transfer Model in Applications with Liquid Metals for Pr = 0.01–0.025. Appl. Sci.
**2020**, 10, 4337. [Google Scholar] [CrossRef] - Schaub, T.; Arbeiter, F.; Hering, W.; Stieglitz, R. Forced and mixed convection experiments in a confined vertical backward facing step at low-Prandtl number, to appear in Exp. Fluids
**2021**. [Google Scholar] [CrossRef] - Kim, J.; Kline, S.J.; Johnston, J.P. Investigation of a reattaching turbulent shear layer: Flow over a backward-facing step. J. Fluids Eng.
**1980**, 102, 302. [Google Scholar] [CrossRef] - Abbott, D.E.; Kline, S.J. Experimental investigation of subsonic turbulent flow over single and double backward facing steps. J. Basic Eng.
**1962**, 84, 317. [Google Scholar] [CrossRef] - De Brederode, V.; Bradshaw, P. Three-Dimensional Flow in Nominally Two Dimensional Separation Bubbles. I, Flow behind a Rearward-Facing Step; I.C. Aero Report 1972; Department of Aeronautics, Imperial College of Science and Technology: London, UK, 1972. [Google Scholar]
- Armaly, B.F.; Durst, F.; Pereira, J.C.F.; Schönung, B. Experimental and theoretical investigation of backward-facing step flow. J. Fluid Mech.
**1988**, 127, 473–496. [Google Scholar] [CrossRef] - Kasagi, N.; Matsunaga, A. Three-dimensional particle-tracking velocimetry measurement of turbulence statistics and energy budget in a backward-facing step flow. Int. J. Heat Fluid Flow
**1995**, 16, 477–485. [Google Scholar] [CrossRef] - Beaudoin, J.-F.; Cadot, O.; Aider, J.-L.; Wesfreid, J.E. Three-dimensional stationary flow over a backward-facing step. Eur. J. Mech.—B/Fluids
**2004**, 23, 147–155. [Google Scholar] [CrossRef] [Green Version] - Liakos, A.; Malamataris, N.A. Topological study of steady state, three dimensional flow over a backward facing step. Comput. Fluids
**2015**, 118, 1–18. [Google Scholar] [CrossRef] - Buckingham, S. Prandtl Number Effects in Abruptly Separated Flows: LES and Experiments on an Unconfined Backward Facing Step. Ph.D. Thesis, Université Catholique de Louvain, Ottignies-Louvain-la-Neuve, Belgium, 2018. [Google Scholar]
- Chen, L.; Asai, K.; Nonomura, T.; Xi, G.; Liu, T. A review of backward-facing step (BFS) flow mechanisms, heat transfer and control. Thermal Sci. Eng. Prog.
**2018**, 6, 194–216. [Google Scholar] [CrossRef] - Polewski, M.D.; Cizmas, P.G.A. Several Cases for the Validation of Turbulence Models Implementation. Appl. Sci.
**2021**, 11, 3377. [Google Scholar] [CrossRef] - Pont-Vílchez, A.; Trias, F.; Gorobets, A.; Oliva, A. Direct numerical simulation of backward-facing step flow at ${\mathrm{Re}}_{\tau}=395$ and expansion ratio 2. J. Fluid Mech.
**2019**, 863, 341–363. [Google Scholar] [CrossRef] [Green Version] - Schumm, T.; Frohnapfel, B.; Marocco, L. Investigation of a turbulent convective buoyant flow of sodium over a backward-facing step flow. Heat Mass Transf.
**2018**, 54, 2533–2543. [Google Scholar] [CrossRef] - Oder, J.; Tiselj, I.; Jaeger, W.; Schaub, T.; Hering, W.; Otic, I.; Shams, A. Thermal fluctuations in low-Prandtl number fluid flows over a backward facing step. Nucl. Eng. Des.
**2020**, 359, 110460. [Google Scholar] [CrossRef] - Terekhov, V.I.; Yarygina, N.I.; Zhdanov, R.F. Heat transfer in turbulent separated flows in the presence of high free-stream turbulence. Int. J. Heat Mass Transf.
**2003**, 46, 4535–4551. [Google Scholar] [CrossRef] - Isomoto, K.; Honami, S. The effect of inlet turbulence intensity on the reattachment process over a backward-facing step. JSME
**1988**, B54, 51–58. [Google Scholar] [CrossRef] [Green Version] - Jaňour, Z.; Jonás, P. On the flow in a channel with a backward-facing step on one wall. Eng. Mech.
**1994**, 1, 313–320. [Google Scholar] - Wang, F.; Gao, A.; Wu, S.; Zhu, S.; Dai, J.; Liao, Q. Experimental Investigation of Coherent Vortex Structures in a Backward-Facing Step Flow. Water
**2019**, 11, 2629. [Google Scholar] [CrossRef] [Green Version] - Le, H.; Moin, P.; Kim, J. Direct numerical simulation of turbulent flow over a backward-facing step. J. Fluid Mech.
**1997**, 330, 349–374. [Google Scholar] [CrossRef] [Green Version] - Zhao, P.; Wang, C.; Ge, Z.; Zhu, J.; Liu, J.; Ye, M. DNS of turbulent mixed convection over a vertical backward-facing step for lead-bismuth eutectic. Int. J. Heat Mass Transf.
**2018**, 127, 1215–1229. [Google Scholar] [CrossRef] - Lim, K.S.; Park, S.O.; Shim, H.S. A low aspect ratio backward-facing step flow. Exp. Therm. Fluid Sci.
**1990**, 3, 508–514. [Google Scholar] [CrossRef] - Nie, J.H.; Armaly, B.F. Reverse flow regions in three-dimensional backward-facing step flow. Int. J. Heat Mass Transf.
**2004**, 47, 4713–4720. [Google Scholar] [CrossRef] - Piirto, M.; Karvinen, A.; Ahlstedt, H.; Saarenrinne, P.; Karvinen, R. PIV Measurements in Square Backward-Facing Step. J. Fluids Eng. ASME
**2007**, 129, 984–990. [Google Scholar] [CrossRef] - Barri, M.; El Khoury, G.K.; Andersson, H.I.; Pettersen, B. DNS of backward-facing step flow with fully turbulent inflow. Int. J. Numer. Meth. Fluids
**2010**, 64, 777–792. [Google Scholar] [CrossRef] [Green Version] - Zajec, B.; Oder, J.; Matkovič, M. Two-dimensional PIV Measurements of Water Flow Over a Backward-facing Step. In Proceedings of the NENE 2018, 27th Conference Nuclear Energy for New Europe, Portorož, Slovenia, 10–13 September 2018. [Google Scholar]
- Argyropoulos, C.D.; Markatos, N.C. Recent advances on the numerical modelling of turbulent flows. Appl. Math. Model.
**2015**, 39, 693–732. [Google Scholar] [CrossRef] - Weaver, D.; Miškovic, S. A Study of RANS Turbulence Models in Fully Turbulent Jets: A Perspective for CFD-DEM Simulations. Fluids
**2021**, 6, 271. [Google Scholar] [CrossRef] - Fisher, P.F.; Lottes, J.W.; Kerkemeier, S.G. nek5000 Web Page. Available online: http://nek5000.mcs.anl.gov (accessed on 15 June 2021).
- Oder, J.; Flageul, C.; Tiselj, I. Statistical Uncertainty of DNS in Geometries without Homogeneous Directions. Appl. Sci.
**2021**, 11, 1399. [Google Scholar] [CrossRef] - Sciacchitano, A. Uncertainty quantification in particle image velocimetry. Meas. Sci. Technol.
**2019**, 30, 092001. [Google Scholar] [CrossRef] [Green Version] - White, F.M. Fluid Mechanics, 7th ed.; McGraw-Hill Education Ltd.: New York, NY, USA, 2011. [Google Scholar]
- Düz, H. Numerical and experimental study to predict the entrance length in pipe flows. J. Appl. Fluid Mech.
**2019**, 12, 155–164. [Google Scholar] [CrossRef]

**Figure 1.**

**Left:**Test loop configuration.

**Top-right:**Schematic of the test-section, coordinate system and measurement planes z = 0 (green) and y = 0 (blue). In parentheses: dimensionless lengths. X-length of the measurement planes is around 25 cm.

**Bottom-right**: raw image for PIV analysis in plane z = 0.

**Figure 2.**

**Top**: DNS spectral element mesh.

**Bottom**: reference mesh dimension divided by the local Kolmogorov scale in the z = 0 plane.

**Figure 3.**Time history of the streamwise velocity component u in point #46. PIV: (5.83, −0.16, 0), DNS: (6, −0.2, 0).

**Top-left**: experiment, case ‘z − 0’,

**Bottom-left**: DNS.

**Bottom-right**: frequency spectra of PIV and DNS time histories in point #46.

**Figure 4.**

**Top**: profiles of u (

**left**), u

_{RMS}, w

_{RMS}(

**right**) at line segment x = −0.8, case ‘y + 0.8’ in Table 1 (green line on top inset drawing).

**Bottom**: profiles of u (

**left**), u

_{RMS}(

**right**) at line segments x = −0.8 in cases ‘z − 0’, ‘z − 1.36’, ‘z + 1.36’ (green lines on bottom inset drawing). Symbols: PIV, lines: DNS.

**Figure 5.**Time-averaged velocity field of case ‘z − 0’: components u and v (

**top pair**), u

_{RMS},v

_{RMS}, u’v’ (

**bottom triplet**) in plane z = 0. PIV experiment: color-scale and blue contours. DNS—black contours.

**Figure 6.**Streamwise velocity u contours (

**top**) and v

_{RMS}(

**bottom**) in plane z = 0. Solid contours: case ‘z − 0’—violet, case ‘z − 0a’—blue, DNS—black. Violet dotted contours: sensitivity contours of ‘z − 0’ case; ±0.03 and ±0.01 for u and v

_{RMS}, respectively.

**Figure 7.**Flow symmetry: time-averaged u velocity contours.

**Top**: measurements ‘z − 0.8’ (violet) and ‘z + 0.8’ (blue).

**Bottom**: measurements ‘z − 1.36’ (violet), ‘z + 1.36’ (blue). DNS—black contours. Dotted violet contours: sensitivity contours at ±0.03 units of ‘z − 0.8’ (

**top**) and ‘z − 1.36’ (

**bottom**) cases.

**Figure 8.**Measurements ‘z − 0’ and ‘z − 0a’. Profiles at x = 3.2, 9.6, 16 mm. Variables: u, v, u

_{RMS}, v

_{RMS}, and u′v′. Measurements—solid lines. DNS—dotted lines. Uncertainties are given for ‘z − 0’ measurement in a single point per profile.

**Figure 9.**Time-averaged field of velocity component u in all y = const. planes. Measurements—color scale and blue contours. DNS—black contours.

**Figure 10.**Measurement ‘y + 0.8’, profiles at x = 3.2, 9.6, 16. Variables: u, w, u

_{RMS}, w

_{RMS}, and u′w′. Solid lines: original measurements and measurements mirrored over z = 0. DNS—dotted lines. Uncertainties are given in a single point per profile.

**Figure 11.**Mean streamwise velocity u and velocity fluctuations u

_{RMS}at cross-sections of the measurement planes. Dashed lines: profiles of the case ‘z − 0’ in line segments at y = −1.92, −1.6, −0.8, 0, 0.8. Solid lines: profiles from cases ‘y − 1.92’, ‘y − 1.6’, ‘y − 0.8’, ‘y − 0’, and ‘y + 0.8’ at z = 0. DNS—black dotted lines.

**Table 1.**List of measurements in the planes z = const. and in the planes y = const. Each case contains two measurements: one with camera at position at x = 4 cm and another at x = 16 cm. Reynolds numbers are given for each of them.

Case Name | Re | z (mm) | z Dimensionless |
---|---|---|---|

‘z − 1.36’ | 7120, 7120 | −17 | −1.36 |

‘z − 0.8’ | 7140, 7120 | −10 | −0.8 |

‘z − 0’ | 7120, 7120 | 0 | 0 |

‘z + 0.8’ | 7120, 7120 | 10 | 0.8 |

‘z + 1.36’ | 7120, 7120 | 17 | 1.36 |

‘z − 0a’ | 7130, 7120 | 0 | 0 |

y (mm) | y Dimensionless | ||

‘y + 0.8’ | 7120, 7120 | 10 | 0.8 |

‘y − 0’ | 7120, 7120 | 0 | 0 |

‘y − 0.8’ | 7130, 7130 | −10 | −0.8 |

‘y − 1.6’ | 7110, 7130 | −20 | −1.6 |

‘y − 1.92’ | 7130, 7120 | −24 | −1.92 |

**Table 2.**Absolute uncertainties of measured quantities rounded to the leading digit (* uncertainties up to 0.05 are present in the interval 0 < x < 3 where maximum u

_{RMS}values are measured and calculated).

Case Name | u | u_{RMS} | v | v_{RMS} | u’v’ |
---|---|---|---|---|---|

‘z − 1.36’ | 0.03 | 0.02 * | 0.02 | 0.01 | 0.003 |

‘z − 0.8’ | 0.03 | 0.01 * | 0.01 | 0.01 | 0.003 |

‘z − 0’ | 0.03 | 0.01 * | 0.007 | 0.01 | 0.003 |

‘z + 0.8’ | 0.03 | 0.01 * | 0.01 | 0.01 | 0.003 |

‘z + 1.36’ | 0.03 | 0.02 * | 0.02 | 0.01 | 0.003 |

‘z − 0a’ | 0.03 | 0.01 * | 0.007 | 0.01 | 0.003 |

u | u_{RMS} | w | w_{RMS} | u’w’ | |

‘y + 0.8’ | 0.03 | 0.01 | 0.005 | 0.01 | 0.001 |

‘y − 0’ | x < 8: 0.15 x > 8: 0.03 | x < 8: 0.04 x > 8: 0.01 | 0.007 | 0.02 | 0.002 |

‘y − 0.8’ | x < 8: 0.05 x > 8: 0.03 | x < 8: 0.02 x > 8: 0.01 | 0.01 | 0.01 | 0.004 |

‘y − 1.6’ | 0.03 | 0.01 | 0.02 | 0.01 | 0.002 |

‘y − 1.92’ | 0.08 | 0.04 | / | / | / |

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**MDPI and ACS Style**

Zajec, B.; Matkovič, M.; Kosanič, N.; Oder, J.; Mikuž, B.; Kren, J.; Tiselj, I.
Turbulent Flow over Confined Backward-Facing Step: PIV vs. DNS. *Appl. Sci.* **2021**, *11*, 10582.
https://doi.org/10.3390/app112210582

**AMA Style**

Zajec B, Matkovič M, Kosanič N, Oder J, Mikuž B, Kren J, Tiselj I.
Turbulent Flow over Confined Backward-Facing Step: PIV vs. DNS. *Applied Sciences*. 2021; 11(22):10582.
https://doi.org/10.3390/app112210582

**Chicago/Turabian Style**

Zajec, Boštjan, Marko Matkovič, Nejc Kosanič, Jure Oder, Blaž Mikuž, Jan Kren, and Iztok Tiselj.
2021. "Turbulent Flow over Confined Backward-Facing Step: PIV vs. DNS" *Applied Sciences* 11, no. 22: 10582.
https://doi.org/10.3390/app112210582