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摘要
The simulation of transient compressible single-fluid flows, modeled by Euler's classical equation, requires numerical schemes that comply with various stringent physical constraints such as thermodynamic consistency, robustness, and stability when shocks, large volume changes, and transport effects are present. For extreme conditions as encountered in defense applications, the present work introduces a novel generic GEEC (Geometry, Energy, and Entropy Compatible) procedure to derive physics-compatible numerical schemes with the following built-in properties: (i) geometric compatibility whereby volume variations (compressions and expansions) are taken into account consistently with advection relative to the grid; (ii) energetic compatibility whereby total energy is exactly conserved at discrete level for an accurate capture of shock levels and shock velocities, even if solving with primitive variables; and (iii) entropic compatibility whereby compliance with the second law of thermodynamics is ensured (here at least to second-order) - entropy must increase in general and must be conserved for isentropic flows. This novel generic GEEC procedure consists of a three-step mimicking derivation: (i) for a prescribed mass transport equation, a variational least action principle is used to generate the proper pressure forces in the discrete momentum evolution equation - thus ensuring a compatible exchange between kinetic and internal energies;- (ii) corrections are systematically performed on numerical residues in order to force conservation at discrete level up to round-off errors; and (iii) an artificial viscosity term is added as a pressure-like contribution in the evolution equations in order to capture shocks and to stabilize the scheme. As a proof of concept, this procedure is applied in the present work to produce a novel ALE (Arbitrary Lagrangian-Eulerian) compressible hydro-scheme with the following features: (i) a direct ALE formalism where mass, momentum, and energy transport fluxes are directly taken into account without separation in the discrete equations; (ii) continuity with "historic" Lagrangian solvers where velocity fields are space and-time staggered; (iii) second-order accuracy in the Lagrangian limit; and as a consequence of the least action principle, (iv) a non-standard downwind formulation of the pressure gradient, dual of the upwind transport operator. Results on standard numerical test cases involving shocks and large deformations confirm the expected built-in properties of the scheme. In particular, the entropic compatibility leads to the preservation of entropy to second-order for isentropic flows, regardless of mesh motions and regardless of the first order accuracy of the discrete mass transport equation. Indifference and versatility with respect to strenuous grid motion strategies are demonstrated in various situations - including supersonic shearing for new variants of Sod's shock tube and Sedov's blast wave - highlighting the advantages of the compatibility between velocity fields and volume variations in the presence of strong fluids advection.
- 出版日期2017-10
- 单位中国地震局
