Navier-Stokes-ω model with slip and friction boundary conditions at high Reynolds numbers
DOI:
https://doi.org/10.15282/jmes.18.2.2024.7.0794Keywords:
Slip With Friction Boundary Conditions, Reattachment Point, Friction Coefficient, Turbulence, Navier-Stokes-ω Model (NS-ω)Abstract
The no-slip boundary condition is indeed a fundamental concept in fluid dynamics, especially for flows at lower Reynolds numbers where viscous effects dominate. However, inertial effects become more significant at higher Reynolds numbers, and the no-slip condition might not accurately represent the behavior of the fluid near the boundary. In such cases, partial slip or slip boundary conditions become more relevant as they take into account the slip between the fluid and the boundary. This study offers the presentation of numerical experiments for a 2-dimensional channel flow, through a step Navier-Stokes-ω model at high Reynolds numbers. The slip boundary conditions with friction is used in these numerical tests, namely along the step and on the lower and upper walls. The impact of the friction coefficient on the flow characteristics is illustrated. Especially for large Reynolds numbers, the effect of the friction coefficient on the flow region is examined. In the numerical tests, the Crank-Nicolson method is used for time discretization, while the Galerkin finite element method is applied for space discretization. It can be observed that as the coefficient of friction decreased, the eddies are further away from the step and moved towards the outer flow. In addition, the size of the eddies are larger for small coefficients of friction. For
Re = 5000, the reattachment length calculated on a fine mesh at time T = 50 is close to the step. For Re = 10000, the reattachment lengths determined for different friction coefficients on both meshes are very similar, with eddies forming just behind the step. Similarly, for Re = 15000 and β = 0.0001, the reattachment lengths calculated on the fine mesh are farther from the step. Conversely, for other values of β, the reattachment lengths are close to the step. The results are explained according to flow physics.
References
G.P. Galdi and W. J. Layton, “Approximation of the larger eddies in fluid motion II: A model for space filtered flow,” Mathematical Models and Methods in Applied Sciences, vol. 10, no. 3, pp. 343-350, 2000.
C. M. L. H. Navier, “Sur les lois de l’equilibre et du mouvement des corps elastiques,” Memoires de l’Academie des Sciences de l’Institut de France, vol. 6, pp. 389-441, 1827.
J. Serrin, “Mathematical principles of classical fluid mechanics,” Encyclopedia of Physics, vol. 3/8/1, pp. 125-263, 1959.
P. Duhem, “On the boundary conditions in hydrodynamics,” Comptes Rendus, vol. 134, pp. 149-151, 1902.
P. Duhem, “On certain cases of adherence of a viscous liquid to solids immersed in it,” Comptes Rendus, vol. 134, pp. 265-267, 1902.
C. W. Oseen, “Contributions to hydrodynamics,” Annalen der Physik, vol. 46, pp. 231-252, 1915.
P. Noaillon, “Attempt at a new theory of the resistance of fluids,” Comptes Rendus, vol. 188, pp. 441-443, 1929.
T. Clopeau, A. Mikelić and R. Robert, “On the vanishing viscosity limit for the 2D incompressible Navier-Stokes equations with the friction type boundary conditions,” Nonlinearity, vol. 11, no. 6, pp. 1625-1636, 1998.
J. M. Coron, “On the controllability of the 2-D incompressible Navier- Stokes equations with the Navier slip boundary conditions,” ESAIM Controle Optimisation and Calculus of Variations, vol. 1, pp. 35-75, 1996.
R. R. Hwang, Y. C. Chow and Y. F. Peng, “Numerical study of turbulent flow over two-dimensional surface-mounted ribs in a channel,” International Journal for Numerical Methods in Fluids, vol. 31, pp. 767-785, 1999.
J. Fan and Y. Zhou, “Global well-posedness of the Navier- Stokes- equations,” Applied Mathematics Letter, vol. 11, no. 24, pp. 1915-1918, 2011.
W. J. Layton, ‘Existence of smooth attractors for the Navier-Stokes- model of turbulence’ Journal of Mathematical Analysis and Applications, vol. 366, no. 1, pp. 81-89, 2010.
J. Fan, F. Li and G. Nakamura, “Global solutions to the Navier-Stokes- and related models with rough initial data,” Zeitschrift für angewandte Mathematik und Physik, vol. 65, pp. 301-314, 2014.
M. Aggul, A. E. Labovsky and K. J. Schwiebert, “NS- model for fluid-fluid interaction problems at high Reynolds numbers,” Computer Methods in Applied Mechanics and Engineering, vol. 395, p. 115052, 2022.
S. T. Bose and P. Moin, “A dynamic slip boundary condition for wall-modeled large-eddy simulation,” Physics of Fluids, vol. 26, no. 1, p. 015104, 2014.
E. Bostrom, “Boundary conditions for spectral simulations of atmospheric boundary layers,” Ph.D. Thesis, Royal Institute of Technology, Stockholm, Sweden, 2017.
Y. Cao, “Global classical solutions to the compressible Navier-Stokes equations with Navier-type slip boundary condition in 2D bounded domains’ Journal of Mathematical Physics, vol. 64, p. 111506, 2023.
Ö. Ilhan and N. Şahin, “Testing of logarithmic law for the slip with friction boundary condition,” International Journal of Nonlinear Sciences and Simulation, vol. 24, no. 8, pp. 2823-2837, 2024.
I. Bermúdez, C. I. Correa, G. N. Gatica and J. P. Silva, “A perturbed twofold saddle point-based mixed finite element method for the Navier-Stokes equations with variable viscosity,” Applied Numerical Mathematics, vol. 201,
pp. 465-487, 2024.
F. Gamar, M. D. Shamshuddin, M. S. Ram, S. O. Salawu and E. O. Fatunmbi, “Exploration of thermal radiation and stagnation point in MHD micropolar nanofluid flow over a stretching sheet with Navier slip,” Numerical Heat Transfer, Part A: Applications, vol. 85, no. 15. pp. 1–13, 2024.
S. G. Cai, J. Jacob and P. Sagaut, “Immersed boundary based near-wall modeling for large eddy simulation of turbulent wall-bounded flow,” Computers & Fluids, vol. 259, p. 105893, 2023.
S. Lyu and S. Utyuzhnikov, “A computational slip boundary condition for near-wall turbulence modelling,” Computers & Fluids, vol. 246, pp. 105628, 2022.
H. Maeyama, T. Imamura, J. Osaka and N. Kurimoto, “Turbulent channel flow simulations using the lattice Boltzmann method with near-wall modeling on a non-body-fitted Cartesian grid,” Computers & Mathematics with Applications, vol. 93, pp. 20-31, 2021.
Z. Li, X. Pan and J. Yang, ‘Constrained large solutions to Leray's problem in a distorted strip with the Navier-slip boundary condition,” Journal of Differential Equations, vol. 377, pp. 221-270, 2023.
F. Wang, M. Ling, W. Han and F. Jing, “Adaptive discontinuous Galerkin methods for solving an incompressible Stokes flow problem with slip boundary condition of frictional type,” Journal of Computational and Applied Mathematics, vol. 371, p. 112700, 2020.
J. K. Djoko, J. Koko, M. Mbehou and T. Sayah, “Stokes and Navier-Stokes equations under power law slip boundary condition: Numerical analysis,” Computers & Mathematics with Applications, vol. 128, pp. 198-213, 2022.
V. John, W. Layton and N. Sahin, “Derivation and analysis of near-wall models for channel and recirculating flows,” Computers & Mathematics with Applications, vol. 28, pp. 1135-1151, 2004.
N. Sahin, “Derivation, analysis and testing of new near wall models for large eddy simulation,” Ph.D. Thesis, Pittsburgh University, United States, 2003.
Ö. Ilhan, “Slip with friction boundary conditions for the Navier–Stokes-alpha turbulence model and the effects of the friction on the reattachment point,” International Journal of Non-Linear Mechanics, vol. 159, p. 104614, 2024.
R. G. Hill, “Benchmark testing the α-models of turbulence,” Master Thesis, Clemson University, United States, 2010.
V. John, “Slip with friction and penetration with resistance boundary conditions for the Navier-Stokes equations: Numerical tests and aspects of the implementation,” Journal of Computational and Applied Mathematics, vol. 147, pp. 287-300, 2002.
V. John and A. Liakos, “Time-dependent flow across a step: the slip with friction boundary condition,” International Journal for Numerical Methods in Fluids, vol. 50, 713-731, 2006.
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, vol. 127, pp. 473-496, 1983.
A. Halim and M. Hafez, ‘Calculation of separation bubbles using boundary layer type equations,” Computational Methods in Viscous Flows, vol. 3, pp. 395-415, 1984.
M. D. Gunzburger. Finite Element Methods for Viscous Incompressible Flows: A Guide to Theory, Practice, and Algorithms, 1st ed. New York: Academic Press, 1989.
I. Moulinos, C. Manopoulos and S. Tsangaris, “Computational analysis of active and passive flow control for backward facing step,” Computation, vol. 10, no. 1, pp.12, 2022.
F. Hecht, FreeFEM Documentation (Release 4.13), pp. 1-740, 2024.
Ö. Ilhan, “Application of a near wall model to Navier-Stokes equations with nonlinear time-relaxation model,” Journal of Applied Mathematics and Computational Mechanics, vol. 21, no. 2, pp. 39-50 2022.
Downloads
Published
How to Cite
Issue
Section
License
Copyright (c) 2024 The Author(s)
This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.
