Abstract
An alternative formulation is presented for the incompressible form of the Navier–Stokes equations. This is done through the introduction of a new vector variable which represents the sum of the velocity vector and the pressure gradient vector. The introduction of the new variable modifies the continuity equation to a Poisson equation for pressure, with the forcing function on the right-hand side representing the divergence of the new vector variable. Hence, the proposed formulation utilizes pressure to satisfy the divergence-free condition on the differential level in a simple and straightforward manner. Moreover, with pressure introduced to the continuity equation on the differential level, the present formulation allows direct computation of the pressure and velocity fields from the continuity and momentum equations, respectively, and eliminates the need for computing pressure and velocity corrections to satisfy mass conservation as in SIMPLE-type pressure correction methods. The proposed procedure is numerically tested against several two-dimensional and three-dimensional steady flow problems. Discretization is carried out using a second-order accurate central difference scheme on a staggered grid and the discretized equations are solved in a segregated manner. Numerical results from the present formulation are in good agreement with conventional incompressible flow solvers and with experimentation.
Similar content being viewed by others
References
Patankar SV, Spalding DB (1972) A calculation procedure for heat, mass and momentum transfer in three-dimensional parabolic flows. Int J Heat Mass Transf 15(10):1787–1806
Patankar S (1980) Numerical heat transfer and fluid flow. CRC Press, Boca Raton
Chorin AJ (1968) Numerical solution of the Navier–Stokes equations. Math Comput 22(104):745–762
Temam R (1969) Sur l’approximation de la solution des équations de Navier–Stokes par la méthode des pas fractionnaires (II). Arch Ration Mech Anal 33(5):377–385
Guermond JL, Minev P, Shen J (2006) An overview of projection methods for incompressible flows. Comput Methods Appl Mech Eng 195(44):6011–6045
Chorin AJ (1967) A numerical method for solving incompressible viscous flow problems. J Comput Phys 2(1):12–26
Turkel E (1987) Preconditioned methods for solving the incompressible and low speed compressible equations. J Comput Phys 72(2):277–298
Kwak D, Kiris C, Kim CS (2005) Computational challenges of viscous incompressible flows. Comput Fluids 34(3):283–299
Weinan E, Liu JG (2003) Gauge method for viscous incompressible flows. Commun Math Sci 1(2):317–332
Hafez M, Shatalov A, Wahba E (2006) Numerical simulations of incompressible aerodynamic flows using viscous/inviscid interaction procedures. Comput Methods Appl Mech Eng 195(23):3110–3127
Edmund DO, Maki KJ, Beck RF (2013) A velocity-decomposition formulation for the incompressible Navier–Stokes equations. Comput Mech 52(3):669–680
Perić M, Kessler R, Scheuerer G (1988) Comparison of finite-volume numerical methods with staggered and colocated grids. Comput Fluids 16(4):389–403
Rauwoens P, Vierendeels J, Merci B (2002) A solution for the odd–even decoupling problem in pressure-correction algorithms for variable density flows. J Comput Phys 227(1):79–99
Johnston H, Liu JG (2002) Finite difference schemes for incompressible flow based on local pressure boundary conditions. J Comput Phys 180(1):120–154
Wahba EM (2012) Steady flow simulations inside a driven cavity up to Reynolds number 35,000. Comput Fluids 66:85–97
Turkel E (1999) Preconditioning techniques in computational fluid dynamics. Annu Rev Fluid Mech 31(1):385–416
Ghia UKNG, Ghia KN, Shin CT (1982) High-Re solutions for incompressible flow using the Navier–Stokes equations and a multigrid method. J Comput Phys 48(3):387–411
Beaudan P, Moin P (1994) Numerical experiments on the flow past a circular cylinder at sub-critical Reynolds number, (Report no. TF-62). Thermo-sciences Division, Stanford University, CA
Coutanceau M, Bouard R (1977) Experimental determination of the main features of the viscous flow in the wake of a circular cylinder in uniform translation. Part 1. Steady flow. J Fluid Mech 79(02):231–256
Lanzerstorfer D, Kuhlmann HC (2012) Global stability of multiple solutions in plane sudden-expansion flow. J Fluid Mech 702:378–402
Battaglia F, Tavener SJ, Kulkarni AK, Merkle CL (1997) Bifurcation of low Reynolds number flows in symmetric channels. AIAA J 35(1):99–105
Drikakis D (1997) Bifurcation phenomena in incompressible sudden expansion flows. Phys Fluids 9(1):76–87
Wahba EM (2007) Iterative solvers and inflow boundary conditions for plane sudden expansion flows. Appl Math Model 31(11):2553–2563
Albensoeder S, Kuhlmann HC (2005) Accurate three-dimensional lid-driven cavity flow. J Comput Phys 206(2):536–558
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Wahba, E.M. A pressure-based velocity decomposition procedure for the incompressible Navier–Stokes equations. Meccanica 51, 1675–1684 (2016). https://doi.org/10.1007/s11012-015-0326-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11012-015-0326-6