Computational Particle Mechanics

, Volume 5, Issue 4, pp 467–475 | Cite as

Meshless Lagrangian SPH method applied to isothermal lid-driven cavity flow at low-Re numbers

  • C. A. D. Fraga FilhoEmail author
  • J. T. A. Chacaltana
  • W. J. N. Pinto


SPH is a recent particle method applied in the cavities study, without many results available in the literature. The lid-driven cavity flow is a classic problem of the fluid mechanics, extensively explored in the literature and presenting a considerable complexity. The aim of this paper is to present a solution from the Lagrangian viewpoint for this problem. The discretization of the continuum domain is performed using the Lagrangian particles. The physical laws of mass, momentum and energy conservation are presented by the Navier–Stokes equations. A serial numerical code, written in Fortran programming language, has been used to perform the numerical simulations. The application of the SPH and comparison with the literature (mesh methods and a meshless collocation method) have been done. The positions of the primary vortex centre and the non-dimensional velocity profiles passing through the geometric centre of the cavity have been analysed. The numerical Lagrangian results showed a good agreement when compared to the results found in the literature, specifically for \({ Re} < 100.00\). Suggestions for improvements in the SPH model presented are listed, in the search for better results for flows with higher Reynolds numbers.


Lid-driven cavity SPH particle method Incompressible Newtonian fluid Physical laws of conservation Navier–Stokes equations Low Reynolds numbers 

List of symbols

\(A\left( {\varvec{x}} \right) \)

Scalar function

\(\left\langle {A\left( {\varvec{x}} \right) } \right\rangle \)

Approximation for the function A at the position \({\varvec{x}}\)


Physical quantity of the fixed particle


Physical quantity of the neighbouring particle


Term related to the density fluctuations of the fluid


Specific heat at constant volume


Parameter of the same order of magnitude as the square of the highest flow velocity


Lagrangian (or material) derivative


Side length of the square cavity


Lateral distance between the centres of mass of two adjacent particles

\(\mathrm{d}{\varvec{x}}^{\prime }\)

Infinitesimal element of volume


Specific internal energy


Specific internal energy of the fixed particle


Repulsive force exerted by the virtual particle on the fluid particle


Acceleration due to gravity


Support radius


Index of coordinate


Influence domain


Index of coordinate


Mass of the neighbouring particle


Number of neighbouring particles inside the influence domain


Absolute pressure


Absolute pressure on the fixed particle


Absolute pressure on the neighbouring particle


Dynamic pressure on the fixed particle


Reynolds number


Cut-off distance








Magnitude of the velocity of the box cover


Velocity vector


Velocity of the fixed particle


Velocity of the neighbouring particle


Velocity component of the neighbouring particle, in the direction m


Velocity component of the neighbouring particle, in the direction k


Fixed point

\({\varvec{x}}^{\prime }\)

Variable point


Cartesian direction


Point where is located the fixed particle


Point where is located the neighbouring particle


Difference position vector between the fluid and virtual particle (a and b, respectively)

\(\left| {{\varvec{x}}_{ab}} \right| \)

Distance between a fluid and a virtual particle

\(\left| {{\varvec{x}}_i -{\varvec{x}}_j } \right| \)

Distance between a fluid and a virtual particle

\(W({{\varvec{x}}}-{{\varvec{x}}}^{\prime },h)\)

Kernel interpolation function evaluated at the point \(\left( {{{\varvec{x}}}-{{\varvec{x}}}^{\prime }} \right) \)

\(\dfrac{\partial }{\partial x_k }\)

Partial derivative with respect to \(x_k\)

Greek symbols

\(\alpha _D\)

Normalization constant

\(\delta \left( {{{\varvec{x}}}-{{\varvec{x}}}^{\prime }} \right) \)

Dirac delta function

\(\mu _i\)

Absolute viscosity of the fixed particle

\(\mu _j\)

Absolute viscosity of the neighbouring particle

\(\rho \)

Fluid density

\(\rho _i\)

Density of the fixed particle

\(\rho _o\)

Density of the fluid at rest

\(\rho _i^{*}\)

Corrected density of the fixed particle

\(\tau _{mk}\)

Stress tensor

\(\Omega \)

Influence domain


Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.


  1. 1.
    Ertuk E (2009) Discussions on driven cavity flow. Int J Numer Methods Fluids 60:275–294MathSciNetCrossRefGoogle Scholar
  2. 2.
    Ghia U, Ghia KN, Shin CT (1982) High-Re solutions for incompressible flow using the Navier–Stokes equations and a multigrid method. J Comput Phys 48:387–411CrossRefGoogle Scholar
  3. 3.
    Marques ACH, Doricio JL (2006) Numerical investigation of the flow in a two-dimensional cavity: meshless, finite volumes and finite difference methods. Latin Am J Solids Struct 3:301–324Google Scholar
  4. 4.
    Liu GR, Liu MB (2003) Smoothed particle hydrodynamics, a meshfree particle method. World Scientific, SingaporezbMATHGoogle Scholar
  5. 5.
    Liu MB, Liu GR (2010) Smoothed particle hydrodynamics (SPH): an overview and recent developments. Arch Comput Methods Eng 17:25–76MathSciNetCrossRefGoogle Scholar
  6. 6.
    Monaghan JJ (1994) Simulating free surface flow with SPH. J Comput Phys 100:399–406CrossRefGoogle Scholar
  7. 7.
    Chen JK, Beraun JE, Carney TC (1999) A corrective smoothed particle method for boundary value problems in heat conduction. Int J Numer Methods Eng 46:231–252CrossRefGoogle Scholar
  8. 8.
    Rapaport DC (2004) The art of the molecular dynamics simulation, 2nd edn. Cambridge University Press, New YorkGoogle Scholar
  9. 9.
    Groot RD, Warren PB (1997) Dissipative particle dynamics: bridging the gap between atomistic and mesoscopic simulation. J Chem Phys 107(11):4423–4435CrossRefGoogle Scholar
  10. 10.
    Pinto WJN (2013) Application of the Lagrangian smoothed particle hydrodynamics (SPH) method for solution of the shear-driven cavity problem. Master’s Dissertation, Federal University of Espírito Santo, Brazil. Accessed 25 Oct 2017

Copyright information

© OWZ 2018

Authors and Affiliations

  1. 1.Development, Implementation and Application of Computational Tools for Problem Solving in Engineering Research GroupFederal Institute of Espírito SantoVitóriaBrazil
  2. 2.Laboratory of Simulation of Free Surface FlowsFederal University of Espírito SantoVitóriaBrazil

Personalised recommendations