Diffusive terms applied in smoothed particle hydrodynamics simulations of incompressible and isothermal Newtonian fluid flows

Abstract

Currently, there is not an extensive literature dedicated to the presentation of the diffusive modeling applied to the momentum equation of incompressible and isothermal Newtonian fluid flows solved by the smoothed-particle hydrodynamics (SPH) method. This paper aims to present the most common viscosity modelings and the LES (Large Eddy Simulation) method applied to the solution of the Navier–Stokes (N–S) equations. A comparative study showing different modelings of the diffusive terms has been carried out. Two incompressible free surface flows were simulated: the generation and propagation of waves on a flat beach and the collapse of a water column. In the first case study, the SPH results were compared to the results provided by the Eulerian modeling (Boussinesq-type nonlinear wave equations solved by the finite difference method and validated from laboratory data). It was verified that the laminar shear stress modeling is the most adequate in the wave period and wave amplitude simulated, although great divergences have not been noticed in relation to the other models used. In the collapse of the water column study, the SPH results obtained after implementation of different approaches for the diffusive terms of the N–S equations presented good agreement with experimental or literature data.

Appendices

Appendix

Boussinesq-type nonlinear wave equations

These equations are obtained after integration of mass and momentum conservation equations in the vertical direction [49].

Continuity equation:

$$\frac{\partial \eta }{{\partial {\text{t}}}} + \frac{{\partial M_{i} }}{{\partial x_{i} }} = 0$$

(A1)

where the flux $M_{i}$ is:

$$\begin{aligned} M_{i} & = \left( {{\text{h}} + \delta \eta } \right)\left\{ {u_{i} + \chi^{2} \left[ {\frac{{1}}{{2}}z^{{2}} - \frac{{1}}{{6}}\left( {{\text{h}}^{{2}} - {\text{h}}\delta \eta + \delta^{{2}} \eta^{{2}} } \right)} \right]\frac{\partial }{{\partial x_{i} }}\left( {\frac{{\partial u_{j} }}{{\partial x_{j} }}} \right)} \right\} \\ & \quad + \left. {\chi^{2} \left[ {z + \frac{{1}}{2}\left( {{\text{h}} - \delta \eta } \right)} \right]\,\frac{\partial }{{\partial x_{i} }}\left[ {\frac{{\partial \left( {{\text{h}}u_{j} } \right)}}{{\partial x_{j} }}} \right]} \right\} + O\left( {\chi^{{4}} } \right) \\ \end{aligned}$$

(A2)

Momentum equation:

$$\frac{{\partial u_{i} }}{{\partial {\text{t}}}} + \delta \left( {u_{j} \frac{{\partial u_{i} }}{{\partial x_{j} }}} \right) + \frac{\partial \eta }{{\partial x_{i} }} + \chi^{{2}} J_{i} + \delta \chi^{2} K_{i} = O\left( {\chi^{{4}} } \right)$$

(A3)

where the nonlinear terms $J_{i}$ and $K_{i}$ are given as

$${\begin{aligned} J_{i} & = \frac{{1}}{{2}}z^{{2}} \frac{\partial }{{\partial x_{i} }}\left[ {\frac{\partial }{{\partial x_{j} }}\left( {\frac{{\partial u_{j} }}{\partial t}} \right)} \right] + z\frac{\partial }{{\partial x_{i} }}\left[ {\frac{\partial }{{\partial x_{j} }}\left( {{\text{h}}\frac{{\partial u_{j} }}{{\partial {\text{t}}}}} \right)} \right] \\ & \quad - \frac{\partial }{{\partial x_{i} }}\left[ {\frac{{1}}{{2}}\left( {\delta \eta } \right)^{{2}} \frac{\partial }{{\partial x_{j} }}\left( {\frac{{\partial u_{j} }}{\partial t}} \right) + \delta \eta \frac{\partial }{{\partial x_{j} }}\left( {{\text{h}}\frac{{\partial u_{j} }}{\partial t}} \right)} \right] \\ \end{aligned} }$$

(A4)

$${\begin{gathered} {K_i} = \frac{\partial }{{\partial {x_i}}}\left\{ {\left( {z - \delta \eta } \right)\left( {{u_j}\frac{\partial }{{\partial {x_j}}}} \right)\left[ {\frac{{\partial \left( {{\text{h}}{u_j}} \right)}}{{\partial {x_j}}}} \right] + \frac{{\text{1}}}{{\text{2}}}\left( {{z^{\text{2}}} - {\delta ^{\text{2}}}{\eta ^{\text{2}}}} \right)\left( {{u_j}\frac{\partial }{{\partial {x_j}}}} \right)\left( {\frac{{\partial {u_j}}}{{\partial {x_j}}}} \right)} \right\} + \hfill \\ {\text{ }} \hfill \\ \end{gathered} }{\text{ + }}\frac{{\text{1}}}{{\text{2}}}\frac{\partial }{{\partial {x_i}}}\left\{ {{{\left[ {\frac{{\partial \left( {{\text{h}}{u_j}} \right)}}{{\partial {x_j}}} + \delta \eta \frac{{\partial {u_j}}}{{\partial {x_j}}}} \right]}^{\text{2}}}} \right\}$$

(A5)

where the level $z$ is the reference depth for calculating the velocities, which may be given as

$$z \approx 0.531\,{\text{h}}_{0}$$

(A6)

where:

$\eta$ is the free surface elevation.

${\text{h}}$ is the total water depth.

$\text t$ is the time.

$\delta = {{a_{0} } \mathord{\left/ {\vphantom {{a_{0} } {{\text{h}}_{0} }}} \right. \kern-\nulldelimiterspace} {{\text{h}}_{0} }}$ and $\chi = k_{0} {\text{h}}_{0}$ are scales of nonlinearity and dispersion, respectively.

$a_{0} ,{\text{h}}_{0} ,k_{0}$ are, in sequence, the typical wave amplitude, the depth of water at rest and the number of waves.

$u_{i} = (u,v)$ is the velocity at depth in the coordinated $z:u_{i} = \left( {\frac{\partial \phi }{{\partial x_{i} }}} \right)$, $\phi$ being the velocity potential.