page_title: Smoothed-particle hydrodynamics
section_title: Weakly compressible approach
hierarchical_section_title: Smoothed-particle hydrodynamics / Modelling Physics / Hydrodynamics / Weakly compressible approach
caption_attribution_description: English: Smoothed Particle Hydrodynamics simulation: pressure fiels of a dam-break flow using standard δ-SPH formulation
context_page_description: Smoothed-particle hydrodynamics is a computational method used for simulating the mechanics of continuum media, such as solid mechanics and fluid flows. It was developed by Gingold and Monaghan and Lucy in 1977, initially for astrophysical problems. It has been used in many fields of research, including astrophysics, ballistics, volcanology, and oceanography. It is a meshfree Lagrangian method, and the resolution of the method can easily be adjusted with respect to variables such as density.
context_section_description: Another way to determine the density is based on the SPH smoothing operator itself. Therefore, the density is estimated from the particle distribution utilizing the SPH  interpolation. To overcome undesired errors at the free surface through kernel truncation, the density formulation can again be integrated in time.
  The weakly compressible SPH in fluid dynamics is based on the discretization of the Navier–Stokes equations or Euler equations for compressible fluids. To close the system, an appropriate equation of state is utilized to link pressure  and density . Generally, the so-called Cole equation
  (sometimes mistakenly referred to as the "Tait equation") is used in SPH. It reads
  where  is the reference density and  the speed of sound. For water,  is commonly used.  
  The background pressure  is added to avoid negative pressure values.
  Real nearly incompressible fluids such as water are characterised by very high speed of sounds of the order . Hence, pressure information travels fast compared to the actual bulk flow, which leads to very small Mach numbers . The momentum equation leads to the following relation:
  where  is the density change and  the velocity vector.
  In practice a value of c smaller than the real one is adopted to avoid time steps too small in the time integration scheme. 
  Generally a numerical speed of sound is adopted such that density variation smaller than 1% are allowed. This is the so-called weak-compressibility assumption.
  This corresponds to a Mach number smaller than 0.1, which implies:
  where the maximum velocity  needs to be estimated, for e.g. by Torricelli's law or an educated guess. Since only small density variations occur, a linear equation of state can be adopted:
  Usually the weakly-compressible schemes are affected by a high-frequency spurious noise on the pressure and density fields.
  
  This phenomenon is caused by the nonlinear interaction of acoustic waves and by fact that the scheme is explicit in time and centred in space
  .
  Through the years, several techniques have been proposed to get rid of this problem. 
  They can be classified in three different groups: 
  the schemes that adopt density filters,
  the models that add a diffusive term in the continuity equation,
  the schemes that employ Riemann Solvers to model the particle interaction.
  The schemes of the first group apply a filter directly on the density field to remove the spurious numerical noise. 
  The most used filters are the MLS (Moving Least Squares) and the Shepard filter 
  which can be applied at each time step or every n time steps. 
  The more frequent is the use of the filtering procedure, the more regular density and pressure fields are obtained.
  On the other hand, this leads to an increase of the computational costs. 
  In long time simulations, the use of the filtering procedure may lead to the disruption of the hydrostatic pressure component and 
  to an inconsistency between the global volume of fluid and the density field.
  Further, it does not ensure the enforcement of the dynamic free-surface boundary condition.
  A different way to smooth out the density and pressure field is to add a diffusive term inside the continuity equation (group 2) :
  The first schemes that adopted such an approach were described in Ferrari
  
  and in Molteni
  where the diffusive term was modelled as a Laplacian of the density field. A similar approach was also used in 
  
  .
  In 
  
  a correction to the diffusive term of Molteni was proposed to remove some inconsistencies close to the free-surface. 
  In this case the adopted diffusive term is equivalent to a high-order differential operator on the density field 
  .
  The scheme is called δ-SPH and preserves all the conservations properties of the SPH without diffusion (e.g., linear and angular momenta, total energy,
  see 
  
  ) along with a smooth and regular representation of the density and pressure fields.
  In the third group there are those SPH schemes which employ numerical fluxes obtained through Riemann solvers to model the particle interactions 
  
  .
caption: SPH simulation: pressure distribution of a dam-break flow using standard δ-SPH formulation
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