A Low Dissipative and Stable Cell-Centered Finite Volume Method with the Simultaneous Approximation Term for Compressible Turbulent Flows

A simultaneous-approximation term is a non-reflecting boundary condition that is usually accompanied by summation-by-parts schemes for provable time stability. While a high-order convective flux based on reconstruction is often employed in a finite-volume method for compressible turbulent flow, finite-volume methods with the summation-by-parts property involve either equally weighted averaging or the second-order central flux for convective fluxes. In the present study, a cell-centered finite-volume method for compressible Naiver–Stokes equations was developed by combining a simultaneous-approximation term based on extrapolation and a low-dissipative discretization method without the summation-by-parts property. Direct numerical simulations and a large eddy simulation show that the resultant combination leads to comparable non-reflecting performance to that of the summation-by-parts scheme combined with the simultaneous-approximation term reported in the literature. Furthermore, a characteristic boundary condition was implemented for the present method, and its performance was compared with that of the simultaneous-approximation term for a direct numerical simulation and a large eddy simulation to show that the simultaneous-approximation term better maintained the average target pressure at the compressible flow outlet, which is useful for turbomachinery and aerodynamic applications, while the characteristic boundary condition better preserved the flow field near the outlet.

Boundary Friction of ZDDP Tribofilms

The frictional properties of ZDDP tribofilms at low entrainment speeds in boundary lubrication conditions have been studied in both rolling/sliding and pure sliding contacts. It has been found that the boundary friction coefficients of these tribofilms depend on the alkyl structure of the ZDDPs. For primary ZDDPs, those with linear alkyl chains give lower friction those with branched alkyl chain ZDDPs, and a cyclohexylmethyl-based ZDDP gives markedly higher friction than non-cyclic ones. Depending on alkyl structure, boundary friction coefficient in rolling-sliding conditions can range from 0.09 to 0.14. These differences persist over long duration tests lasting up to 120 h. For secondary ZDDPs, boundary friction appears to depend less strongly on alkyl structure and in rolling-sliding conditions stabilises at ca 0.115 for the three ZDDPs studied. Experiments in which the ZDDP-containing lubricant is changed after tribofilm formation by a different ZDDP solution or a base oil indicate that the characteristic friction of the initial ZDDP tribofilm is lost almost as soon as rubbing commences in the new lubricant. The boundary friction rapidly stabilises at the characteristic boundary friction of the replacement ZDDP, or in the case of base oil, a value of ca 0.115 which is believed to represent the shear strength of the bare polyphosphate surface. The single exception is when a solution containing a cyclohexylethyl-based ZDDP is replaced by base oil, where the boundary friction coefficient remains at the high value characteristic of this ZDDP despite the fact that rubbing in base oil removes about 20 nm of the tribofilm. XPS analysis of the residual tribofilm reveals that this originates from presence of a considerable proportion of C-O bonds at the exposed tribofilm surface, indicating that not all of the alkoxy groups are lost from the polyphosphate during tribofilm formation. Very slow speed rubbing tests at low temperature show that the ZDDP solutions give boundary friction values that vary with alkyl group structure in a similar fashion to rolling-sliding MTM tests. These variations in friction occur immediately on rubbing, before any measurable tribofilm can develop. This study suggest that ZDDPs control boundary friction by adsorbing on rubbing steel or tribofilm surfaces in a fashion similar to organic friction modifiers. However it is believed that, for primary ZDDPs, residual alkoxy groups still chemically bonded to the phosphorus atoms of newly-formed polyphosphate/phosphate tribofilm may also contribute to boundary friction. This understanding will contribute to the design of low friction, fuel efficient crankcase engine oils. Graphical Abstract

Type II critical collapse on a single fixed grid: a gauge-driven ingoing boundary method

We develop a numerical method suitable for gravitational collapse based on Cauchy evolution with an ingoing characteristic boundary. Unlike similar methods proposed recently (Ripley; Bieri et al. in Class Quantum Grav 37:045015, 2020), the numerical grid remains fixed during the evolution and no points need to be removed or added. Increasing coordinate refinement of the central region as the field collapses is achieved solely through the choice of spatial gauge and particularly its boundary condition. We apply this method to study critical collapse of a massless scalar field in spherical symmetry using maximal slicing and isotropic coordinates. Known results on mass scaling, discrete self-similarity and universality of the critical solution (Choptuik in Phys Rev Lett 70:9, 1993) are reproduced using this considerably simpler numerical method.

Stable Finite-Difference Methods for Elastic Wave Modeling with Characteristic Boundary Conditions

In this paper, a new stable finite-difference (FD) method for solving elastodynamic equations is presented and applied on the Biot and Biot/squirt (BISQ) models. This method is based on the operator splitting theory and makes use of the characteristic boundary conditions to confirm the overall stability which is demonstrated with the energy method. Through the stability analysis, it is showed that the stability conditions are more generous than that of the traditional algorithms. It allows us to use the larger time step τ in the procedures for the elastic wave field solutions. This context also provides and compares the computational results from the stable Biot and unstable BISQ models. The comparisons show that this FD method can apply a new numerical technique to detect the stability of the seismic wave propagation theories. The rigorous theoretical stability analysis with the energy method is presented and the stable/unstable performance with the numerical solutions is also revealed. The truncation errors and the detailed stability conditions of the FD methods with different characteristic boundary conditions have also been evaluated. Several applications of the constructed FD methods are presented. When the stable FD methods to the elastic wave models are applied, an initial stability test can be established. Further work is still necessary to improve the accuracy of the method.