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

## Abstract

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.

## Keywords

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 \({\varvec{x}}\)- \(A_i\)
Physical quantity of the fixed particle

- \(A_j\)
Physical quantity of the neighbouring particle

*B*Term related to the density fluctuations of the fluid

- \({c}_{\mathrm{v}}\)
Specific heat at constant volume

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

- \(\dfrac{\mathrm{D}}{\mathrm{D}t}\)
Lagrangian (or material) derivative

*d*Side length of the square cavity

- \(\mathrm{d}x\)
Lateral distance between the centres of mass of two adjacent particles

- \(\mathrm{d}{\varvec{x}}^{\prime }\)
Infinitesimal element of volume

*e*Specific internal energy

- \(e_i\)
Specific internal energy of the fixed particle

- \({\varvec{F}}_{ab}\)
Repulsive force exerted by the virtual particle on the fluid particle

- \(g_k\)
Acceleration due to gravity

*h*Support radius

*k*Index of coordinate

*kh*Influence domain

*m*Index of coordinate

- \(m_j\)
Mass of the neighbouring particle

*n*Number of neighbouring particles inside the influence domain

*P*Absolute pressure

- \(P_i\)
Absolute pressure on the fixed particle

- \(P_j\)
Absolute pressure on the neighbouring particle

- \(P_{\mathrm{dyn}(i)}\)
Dynamic pressure on the fixed particle

*Re*Reynolds number

- \(r_o\)
Cut-off distance

*S*Boundary

*t*Time

*T*Temperature

- \(U_o\)
Magnitude of the velocity of the box cover

- \(u_k\)
Velocity vector

- \(u_i\)
Velocity of the fixed particle

- \(u_j\)
Velocity of the neighbouring particle

- \(u_j^m\)
Velocity component of the neighbouring particle, in the direction

*m*- \(u_j^k\)
Velocity component of the neighbouring particle, in the direction

*k*- \({\varvec{x}}\)
Fixed point

- \({\varvec{x}}^{\prime }\)
Variable point

- \({\varvec{x}}_k\)
Cartesian direction

- \({\varvec{x}}_i\)
Point where is located the fixed particle

- \({\varvec{x}}_j\)
Point where is located the neighbouring particle

- \({\varvec{x}}_{ab}\)
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

## Notes

### Compliance with ethical standards

### Conflict of interest

The authors declare that they have no conflict of interest.

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