Procedia Engineering 154 ( 2016 ) 420 – 425
1877-7058 © 2016 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
Peer-review under responsibility of the organizing committee of HIC 2016 doi: 10.1016/j.proeng.2016.07.508
ScienceDirect
12th International Conference on Hydroinformatics, HIC 2016
Numerical Investigation of Backward Facing Step Flow over
Various Step Angles
Hoi Hyun Choi
a, Van Thinh Nguyen
b0F0 F0 F*, John Nguyen
ca,bDept. of Civil and Environmental Engineering, Seoul National University, 01 Gwanak-ro, Gwanak-gu, Seoul 151-744, South-Korea.
cJohn H. Daniels Faculty of Architecture, Landscape and Design, University of Toronto, Toronto, ON M5T 1R2, Canada
Abstract
In this study, based on an open source CFD package OpenFOAM we carried out a number of numerical simulations using Reynolds Averaged Navier-Stokes (RANS) and Large Eddy Simulation (LES) approaches. The numerical investigation has been implemented for various step angles (10o, 15 o, 20 o, 25 o, 30 o, 45 o, and 90 o), different expansion ratios (1.48, 2.00 and 3.27), and Reynolds numbers (5000, 8000, 11000, 15000, 47000 and 64000). The comparisons of the flow structures, separation flows and reattachment lengths between the numerical results and the observations show very good agreement. The results obtained from LES show a better agreement with the observations than the results obtained from RANS model.
© 2016 The Authors. Published by Elsevier Ltd.
Peer-review under responsibility of the organizing committee of HIC 2016.
Keywords: Computational Fluid Dynamics; Numerical analysis; Backward facing step; OpenFOAM; RANS; LES
1. Introduction
The study on the flow over a backward-facing step is very important in hydraulic structures and HVAC systems. The most important characteristics of such flows are the internal flows, flow separation and reattachment caused by sudden changes in cross-section geometries. Experimental study for 90o backward facing step flows was intensified for a wide range of Reynolds numbers and at different expansion ratios, such as Armaly et al. (1983), Ruck and
* Corresponding author. Tel.: +82-2-880-7355; fax: +82-2-873-2684.
E-mail address: [email protected]
© 2016 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
Makiola (1993), Lee and Mateescu (1998), Auburn et al. (2000), etc. A number of significant numerical studies of turbulent flows over 90o backward facing step have been intensively carried out using Reynolds Averaged Navier-Stokes (RANS) Equations; e.g. Armaly et al. (2003), Nallasamy (1987), Speziale and Ngo (1988), Thangam and Speziale (1992), Lasher and Taulbee (1992). However there are still limited numerical results obtained from higher accuracy turbulence calculation methods, such as Large Eddy Simulation (LES), Direct Numerical Simulation; e.g. Le et al. (1997), Fureby (1999), Meri et al. (2000), etc., particularly DNS is mostly applied for a low-Reynolds number, and still very expensive.
Several experiments and numerical studies have been conducted for 90 degree step angle with different expansion ratios, however a detailed study on the turbulent structures, separation flows and reattachment lengths for various step angles is still not well documented. In this study, we have carried out a number of numerical simulations following the experimental cases of Ruck and Makiola for the geometry shown in Fig. 1, with various step angles (10o, 15 o, 20 o, 25 o, 30 o, 45 o, and 90 o), different expansion ratios (1.48, 2.00 and 3.27), and Reynolds numbers (5000, 8000, 11000, 15000, 47000 and 64000), as shown in Table 1.
Fig. 1. Geometry of the experimental channel Table 1. Parameters of Ruck & Makiola’s experimental cases
Reynolds number Expansion ratio Step angle(D) 5000 1.48 10, 15, 20, 25, 30, 45, 90 8000 1.48 10, 15, 20, 25, 30, 90 11000 1.48 10, 15, 20, 25, 30, 90 15000 1.48 10, 15, 20, 25, 30, 45, 90 47000 1.48 10, 15, 20, 25, 30, 45, 90 64000 1.48 10, 15, 20, 25, 30, 90 15000 2.00 10, 15, 20, 25, 30, 45, 90 64000 2.00 10, 15, 20, 25, 45, 90 5000 3.27 15, 20, 25, 90 8000 3.27 15, 20, 25, 90 11000 3.27 15, 20, 25, 90 15000 3.27 15, 20, 25, 90 Nomenclature ER Expansion ratio Xr Reattachment length α Step angle H Step height
To precisely capture the turbulence structures and separation flows, we applied RANS with two-equation k-ω (Wilcox, 1988) and LES with Smagorinsky-Lilly for sub-grid-scale (SGS) viscosity. A comparison between two methods, it shows that the LES approach can produce a better results of the flow behind the step.
2. Numerical Equations
Numerical simulations have been based on an open source CFD package OpenFOAM (http://www.openfoam.org) using Reynolds Averaged Navier-Stokes (RANS) and Large Eddy Simulation (LES) approaches. The main difference of two methods is the way to take the average of the instantaneous velocity and pressure terms from the Navier-Stokes (NS) Equations. While RANS is obtained from averaging the NS over a time period߂ܶ, LES is obtained from a space filtering of the NS which passes the large eddies and rejects the small eddies. The RANS method aimed at obtaining stationary numerical solutions, however the ambiguity in choosing the parameters of the model may lead to non-stationary flows after the step, so the non-stationary turbulent flow regimes may be obtained by LES approach. A summary of both approaches has been described below:
2.1. RANS (k-ω) Model
RANS based on two equation k-ω model (Wilcox, 1998), whose equations are presented as follows:
2 1 1 *
)
3
2
2
(
)]
)
[(
)
(
)]
)
[(
)
(
)
(
1
)
(
Z
E
G
Z
J
Z
V
Q
Q
Z
Z
Z
E
U
V
Q
Q
W
Q
U
w
w
w
w
w
w
w
w
w
w
w
w
ij j i ij ij k t k k t j ij i i i ix
U
S
S
U
t
k
P
k
kU
t
k
x
U
x
P
U
U
t
U
(1)Where: U(U1, U2, U3), ݑൌ ܷ ݑᇱ , ui is instantaneous velocity, Ui is mean velocity over a time period ߂ܶ, ݑᇱ is
fluctuating velocity; i=1,2 and 3 are corresponding to x, y, and z directions in Cartesian coordinates, ൌ ܲ Ԣ, p is pressure, P and p’ are mean and fluctuating components of pressure.
The Reynolds stresses is modeled by ߬ൌ െݑതതതതതത ൌ ߥపᇱݑఫᇱ ௧൬ డ డ௫ೕ డೕ డ௫൰ െ ଶ ଷ݇ߜ, ൌ ଵ ଶݑప ᇱଶ തതതത , ߥ௧ൌ ఠ
and the empirical coefficients are given by: ߪൌ ʹǤͲǡ ߪఠൌ ʹǤͲǡ ߛଵൌ ͲǤͷͷ͵ǡ ߚଵൌ ͲǤͷͷ͵ǡ ߚכൌ ͲǤͲͻ
2.2. LES Model
LES momentum equation equations using the Smagorinsky-Lilly model for Sub-grid-scale (SGS) viscosity are rooted as follows: j ij i i i i
x
U
x
P
U
U
t
U
w
w
w
w
w
w
Q
W
U
(
)
1
)
(
(2) The Sub-grid-scale stresses is modeled as following manner:߬ൌ ݑതതതതത െ ݑపݑఫ ഥ ݑపഥ ൌ െߥఫ ௌீௌቆ߲ܷ ߲ݔ ߲ܷ ߲ݔ ቇ ͳ ͵ ߬ ߩ ߜ ߥௌீௌൌ ሺܥௌீௌȟሻଶටʹܵതതതതܵపఫതതതത, ܵపఫ തതതത ൌపఫ ଵ ଶ൬ డ డ௫ೕ డೕ డ௫൰
3. Numerical Results
As shown in Fig. 2, the numerical results of the reattachment length depended on the inclined step angles show a good agreement with the observations of Ruck and Makiola for the Reynolds number of 64,000 and the expansion ratios of 1.48 and 2.00. The numerical results obtained from other cases for different Reynolds number and expansion ratio have also shown the same tendency. Fig. 3 shows the same results obtained the observations of Ruck & Makiola; i.e. for Re > 15000 the attachment length is no longer dependent on the Reynolds number, and it tends to be constant once the step angle larger than 30o. Fig. 4 shows a difference of numerical results obtained from RANS and LES in comparison with the observation, whereby the velocity profile obtained from LES method shows a best fit to the observation in comparison with the result obtained from RANS method. Fig. 5 shows a very good agreement between the numerical results (LES) and the observation of the velocity profile over the step of 10o and 30o with Reynolds number of 47,000 and the expansion ratio of 1.47.
Fig. 2. Comparison of reattachment length between Ruck’s experiment and numerical simulation
Fig. 3. Reattachment length as a function of step angle for Re > 15,000, ER=2.0
4. Conclusion
The numerical investigation has tried to simulate the turbulent flow regimes over various step angles, different expansion ratios and Reynolds numbers following the experiments of Ruck and Makiola. The results have been verified that the reattachment length remains almost constant once Re > 15,000, even increasing of the step angle larger than 30o. The LES approach shows higher accuracy to capture the flow structures behind the step. A study on this characteristic will be carried out more on various Reynolds numbers and expansion ratios to give a further conclusion. 0 2 4 6 8 10 0 20 40 60 80 100 Xr/H α
Re=64000, ER=1.48 Exp. Sim
0 2 4 6 8 10 0 20 40 60 80 100 Xr/H α Re=64000, ER=2.00
Fig. 4. Comparisons of velocity profile between RANS, LES and experiment for Re=5000 (up) and Re=8000 (down), α = 90R, ER = 1.48.
Fig. 5. Comparisons of velocity profile between numerical results and experiment for Re=47000 and ER = 1.48.
Acknowledgements
This work was supported by the National Research Foundation of Korea grand funded by the Korean Government (NRF-2015R1D1A1A01060880).
References
[1] B.F. Armaly., F. Durst, J. C. F. Pereira, and B. Schönung. "Experimental and theoretical investigation of backward-facing step flow." Journal of Fluid Mechanics 127 (1983): 473-496.
[2] B.F. Armaly, A. Li, and J. H. Nie. "Measurements in three-dimensional laminar separated flow." International Journal of Heat and Mass Transfer 46, no. 19 (2003): 3573-3582.
[3] S. Aubrun, P. L. Kao, and Henri-Claude Boisson. "Experimental coherent structures extraction and numerical semi-deterministic modelling in the turbulent flow behind a backward-facing step." Experimental thermal and fluid science 22, no. 1 (2000): 93-101.
[4] C.Fureby. "Large eddy simulation of rearward-facing step flow." AIAA journal 37, no. 11 (1999): 1401-1410.
[5] B. Ruck, and B. Makiola. "Flow separation over the inclined step." In Physics of Separated Flows—Numerical, Experimental, and Theoretical Aspects, Vieweg+ Teubner Verlag (1993): 47-55.
[6] T. Lee and D. Mateescu. "Experimental and numerical investigation of 2-D backward-facing step flow." Journal of Fluids and Structures 12, no. 6 (1998): 703-716.
[7] W.C. Lasher and D. B. Taulbee. "On the computation of turbulent backstep flow." International journal of heat and fluid flow 13, no. 1 (1992): 30-40.
[8] H. Le, P. Moin, and J. Kim. "Direct numerical simulation of turbulent flow over a backward-facing step." Journal of fluid mechanics 330 (1997): 349-374. -2 0 2 4 6 8 10 12 14 16 0 0.5 1 1.5 2 2.5 3
angle 45 ER 1.48 Reynlods number 47000
U/Umax
y(
m
[9] A. Meri, H. Wengle, and R. Schiestel. "DNS and LES of a backward-facing step flow using 2nd-and 4th-order spatial discretization and LES of the spatial development of mixing of turbulent streams with non-equilibrium inflow conditions." In Numerical Flow Simulation II,Springer Berlin Heidelberg (2001): 268-287.
[10] M. Nallasamy, "Turbulence models and their applications to the prediction of internal flows: a review." Computers & Fluids 15, no. 2 (1987): 151-194.
[11] C.G. Speziale and Tuan Ngo. "Numerical solution of turbulent flow past a backward facing step using a nonlinear K-ε model." International Journal of engineering science 26, no. 10 (1988): 1099-1112.
[12] S. Thangam and C. G. Speziale. "Turbulent flow past a backward-facing step-A critical evaluation of two-equation models." AIAA journal 30, no. 5 (1992): 1314-1320.
