Smoothed Particle Hydrodynamics - Mathematical Basis
Smoothed particle hydrodynamics (SPH) is a Lagrangian method for
modelling heat and mass flows. Materials are approximated by particles
that are free to move rather than by fixed grids or meshes. The governing
PDEs are converted into equations of motion for these particles. The SPH
method has been developed over the past two decades for astrophysical
applications [1]. More recently the method has been extended to
incompressible enclosed flows [2]. SPH has several advantages that make it
particularly well suited to this type of problem:
- It handles momentum dominated flows well
- Complex free surfaces, including break-up into fragments are
modelled naturally
- Complicated physics such as multi-phase, realistic equations of
state, compressibility, radiation and solidification can be added
easily
- It is easily able to handle complex geometries in two and three
dimensions
SPH version of the Navier-Stokes Equations
In SPH, the interpolated value of a function A at
position r is given by:
where the sum is over all particles b within a
radius 2h of r. Here W = W(r,h) is a
spline based interpolation kernel of radius 2h. It is a C2
function that approximates the shape of a Gaussian function and has
compact support. The gradient of the function A is then given by:
The best choice of SPH continuity equation for free
surface flows is:
It is Galilean invariant, has good numerical conservation
properties and is not affected by free surfaces. The SPH momentum equation
derived in [1] is:
It automatically ensures continuous of stress across
material interfaces and allows multiple materials with viscosities varying
by up to five orders of magnitude to be accurately simulated. This is
particularly important for solidifying metal simulation. SPH is actually a
quasi-compressible method and the pressure is given by the stiff equation
of state:
where P0 is the magnitude of the
pressure and r 0 is the reference
density. For water or liquid metals the exponent g
= 7 is used and
where V is the characteristic or maximum fluid velocity.
Typically a is between 10 and 40 which ensures
that the density variation is less than 0.25% to 1% and the flow can be
regarded as incompressible. The boundaries are modelled in the normal
direction as Leonard-Jones forces applied at the boundary particles and
interpolated to produce arbitrary, smoothly-varying boundaries. In the
tangential direction, the particles are included in the summation for the
shear force to give non-slip boundary conditions for the walls. Details
are to be found in [3].
SPH Heat Equation
A new form of the SPH heat equation based on internal
energy is described in [4]. For solidifying metals, it is more appropriate
to use an enthalpy formulation, giving an SPH energy equation for particle
a:
where H =  is the
enthalpy per unit mass, cp is the specific heat, L
is the latent heat and fs(T) is the volume
fraction of the metal that is solid at temperature T. Also for each
particle b, kb is the conductivity, r
b is the density and mb is the mass.
Here rab is the position vector from particle b
to particle a, Tab = Ta – Tb
and Wab = W(rab,h) is
the interpolation kernel with smoothing length h. The above heat
equation has an explicit conductivity which can be temperature dependent
and ensures that heat flux is automatically continuous across material
interfaces. This allows multiple materials with substantially different
conductivities, specific heats and densities to be accurately simulated.
References
[1] J.J. Monaghan, Smoothed particle hydrodynamics,
Annual Review of Astronomy and Astrophysics, 30, 543-574 (1992).
[2] J.J. Monaghan, Simulating free surface flows
with SPH, Journal of Computational Physics, 110, 399-406
(1994).
[3] P.W. Cleary and J.J. Monaghan, Boundary
interactions and transition to turbulence for standard CFD problems using
SPH, Proceedings of the 6th International Computational Techniques
and Applications Conference, Canberra ACT, pp. 157 (1993).
[4] P.W. Cleary and J.J. Monaghan, Conduction
modelling using Smoothed Particle Hydrodynamics, Journal of
Computational Physics, 148, 227–264 (1999).
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