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Modelling discontinuities and Kelvin-Helmholtz instabilities in SPH
Daniel J. Price
TL;DR
SPH has difficulty treating contact discontinuities, contributing to problems in Kelvin–Helmholtz simulations across entropy gradients. The paper analyzes integral and differential formulations and proposes artificial thermal conductivity that resolves these interfaces while limiting dissipation elsewhere, producing excellent Kelvin–Helmholtz results.
Problem
Standard SPH formulations inadequately treat contact discontinuities, creating pressure errors in Kelvin–Helmholtz simulations across entropy gradients.
Method
The paper analyzes integral versus differential SPH formulations and develops artificial thermal conductivity that activates at discontinuities while limiting dissipation elsewhere.
Results
The proposed conductivity equalises pressure at contact discontinuities, minimises unwanted dissipation, and gives excellent Kelvin–Helmholtz results consistent with analytic and grid-based calculations.
Takeaways & Limitations
Kelvin–Helmholtz instabilities require no special treatment in isothermal flows because those flows lack the problematic entropy gradient.
Takeaways & Limitations
The SPH derivation assumes differentiability and vanishing surface terms, limiting how discontinuities are represented by the resulting equations.
Abstract
from arXiv · showhide
In this paper we discuss the treatment of discontinuities in Smoothed Particle Hydrodynamics (SPH) simulations. In particular we discuss the difference between integral and differential representations of the fluid equations in an SPH context and how this relates to the formulation of dissip ative terms for the capture of shocks and other discontinuities. This has important implications for many problems, in particular related to recently highlighted problems in treating Kelvin-Helmholtz instabilities across entropy gradients in SPH. The specific problems pointed out by Agertz et al. (2007) are shown to be related in particular to the (lack of) treatment of contact discontinuities in standard SPH formulations which can be cured by the simple application of an artificial thermal conductivity term. We propose a new formulation of artificial thermal conductivity in SPH which minimises dissipation away from discontinuities and can therefore be applied quite generally in SPH calculations.
1 Introduction
The paper addresses reported difficulties simulating Kelvin–Helmholtz instabilities and density discontinuities in SPH by analyzing discontinuity treatment and proposing improved artificial thermal conductivity. It distinguishes integral and differential fluid-equation representations and motivates discontinuity-capturing terms for differential formulations.
- SPH is presented as a Lagrangian particle method for solving fluid-dynamics equations and is widely used in astrophysics, geophysics, and engineering.
- The paper targets difficulties in SPH simulations of Kelvin–Helmholtz instabilities between fluids with different densities, highlighted by Agertz et al. (2007).
- Its central aim is to explain how integral versus differential SPH representations, particularly for continuity, determine the need for discontinuity-capturing terms.
- The paper proposes artificial thermal conductivity that resolves contact discontinuities while minimizing dissipation of thermal-energy gradients elsewhere.
- The broader discontinuity treatment is illustrated using shock-tube tests, alongside the paper’s basic SPH formulation and general discussion.
2 Standard variable−h SPH equations
The standard variable−h SPH equations are derived from a discretized fluid Lagrangian, yielding conservative particle dynamics with smoothing lengths coupled to density. The formulation supports thermal-energy, total-energy, or entropy evolution, although exact entropy advection is a differential assumption and dissipative terms govern entropy production.
- Lagrangian derivation: Discretizing the perfect-fluid Lagrangian over SPH particles and applying Euler–Lagrange equations produces the particle equations of motion.The derivation assumes thermal energy depends on density and entropy, with constant entropy used for the spatial derivatives.
- Conservation: The variational formulation guarantees exact conservation of momentum, angular momentum, and energy through translational, rotational, and temporal symmetries of the Lagrangian.
- Variable smoothing length: In variable−h SPH, density is computed by summation while h depends on density, requiring a nonlinear iterative solution; its time derivative introduces the Ω correction to the continuity equation.The smoothing-length relation uses η = 1.2 throughout the paper and is solved with Newton–Raphson iteration.
- Energy evolution: SPH can evolve thermal energy, conserved total energy, or the entropy function, with the energy formulations equivalent to timestepping accuracy when smoothing-length-gradient terms are handled properly.This equivalence differs from Eulerian schemes, where advection terms introduce additional differences between formulations.
- Entropy evolution: Entropy-function evolution strictly controls entropy sources because entropy is purely advected without dissipation, but this exact advection relies on a differential relation with no corresponding integral representation.
3 Discontinuities in SPH
SPH’s integral and differential formulations treat discontinuities differently: density summation retains them, whereas differentiated momentum and energy equations assume differentiability and require dissipation to resolve jumps. The proposed pressure-difference-based thermal conductivity targets contact discontinuities while limiting diffusion away from discontinuities.
- Integral versus differential formulations: Density summation preserves discontinuities through an integral continuity formulation, whereas differentiated SPH momentum and energy equations assume differentiability and omit surface terms at discontinuities.The missing surface contribution generally vanishes except at boundaries or flow discontinuities, explaining the mismatch between the continuity, momentum, and energy formulations.
- Dissipative terms: Dissipation terms recover discontinuities by diffusing them across the smoothing scale, and artificial thermal conductivity is specifically required to smooth jumps in thermal energy.The thermal-conductivity contribution is widely omitted in SPH formulations despite its role in resolving thermal-energy discontinuities.
- Signal velocities: Sound-speed-based signal velocities suit shocks but are inappropriate for contact discontinuities, which lack compression and move at the post-shock velocity.This distinction motivates a separate treatment of thermal-energy jumps rather than applying one signal velocity to all variables.
- Artificial thermal conductivity: The proposed conductivity signal velocity vanishes at pressure equilibrium, eliminating spurious pressure gradients across contacts while avoiding unwanted diffusion elsewhere.The method is reported to be particularly effective in the Kelvin–Helmholtz tests.
- Dissipation control: Dissipative terms can erase smooth gradients permanently, so switches must turn conductivity off away from discontinuities; the new signal velocity makes the switch nearly unnecessary.The paper identifies minimizing dissipation away from discontinuities as the central design requirement for artificial thermal conductivity.
4 Tests
The tests show that density summation resolves contact-discontinuity density gradients but requires artificial thermal conductivity to remove pressure errors. Applied to Kelvin–Helmholtz flows, the proposed conductivity enables classical instability growth and agrees well with analytic and grid-based results.
- Shock tube: Density summation well resolves the shock-tube contact-discontinuity density gradient, but leaves a spurious pressure blip caused by thermal-energy overshoot.Evolving the continuity equation instead produces a significant density-gradient problem.
- Shock tube: Artificial thermal conductivity is required in the energy equation to treat discontinuities, regardless of the specific discontinuity-capturing formulation.This conclusion is supported by both artificial-viscosity and first-order Godunov-SPH shock-tube tests.
- Kelvin–Helmholtz instability: Without conductivity, standard SPH inhibits Kelvin–Helmholtz mixing through artificial surface tension, while artificial viscosity further suppresses early velocity-perturbation growth.The RT01 formulation improves instability-like features but introduces substantial interface particle noise.
- Kelvin–Helmholtz instability: Adding artificial thermal conductivity produces classical Kelvin–Helmholtz growth, with the λ = 1/6 mode visible at τKH ∼2 and overtaken by λ = 1/2 at τKH ∼6.The relative mode growth agrees well with the stated comparison, although the overall evolution remains slower than analytic theory.
- Kelvin–Helmholtz instability: At higher resolution, both λ = 1/6 and λ = 1/2 modes develop well-resolved vortex rolls, while the generic formulation is cleaner than RT01 and performs well on shock-tube tests.For a 10:1 density ratio, the method agrees very well with the analytic growth timescale and grid-based calculations of Agertz et al. (2007).
5 Discussion and Conclusions
The paper attributes SPH’s Kelvin–Helmholtz problems across density jumps to improperly treated entropy discontinuities, which create an artificial surface-tension effect. Its proposed thermal-conductivity formulation equalises pressure at contacts while minimising unnecessary dissipation and performs well on the instability test.
- 5 Discussion and Conclusions: Because SPH momentum and energy equations assume differentiability, discontinuities require diffusion in both momentum and thermal energy, with artificial viscosity treating momentum jumps.The continuity equation can instead be represented integrally through the density sum.
- 5 Discussion and Conclusions: Standard SPH produces an artificial surface-tension effect because a pressure ridge at the fluid interface creates the particle-distribution gap identified by Agertz et al. (2007).The pressure discontinuity explains why the interface energetically favours an unmixed configuration under exact entropy advection.
- 5 Discussion and Conclusions: The proposed dissipation formulation gives excellent Kelvin–Helmholtz results, agreeing with analytic estimates and grid-based calculations from Agertz et al. (2007).It equalises pressure at contact discontinuities while minimising dissipation elsewhere.
- 5 Discussion and Conclusions: Kelvin–Helmholtz suppression in standard SPH arises from entropy gradients, not density gradients; isothermal flows therefore require no special treatment.The conclusion applies specifically to the absence of entropy gradients in isothermal flows.
- 5 Discussion and Conclusions: The paper recommends its generic dissipation formulation as standard practice because standard SPH inhibits Kelvin–Helmholtz growth across density jumps while the signal velocity suppresses unnecessary thermal conductivity.This recommendation is stated for at least non-self-gravitating calculations.