SPH-ALE Scheme for Weakly Compressible Viscous Flow with a Posteriori Stabilization

A highly accurate SPH method with a new stabilization paradigm has been introduced by the authors in a recent paper aimed to solve Euler equations for ideal gases. We present here the extension of the method to viscous incompressible flow. Incompressibility is tackled assuming a weakly compressible approach. The method adopts the SPH-ALE framework and improves accuracy by taking high-order variable reconstruction of the Riemann states at the midpoints between interacting particles. The moving least squares technique is used to estimate the derivatives required for the Taylor approximations for convective fluxes, and also provides the derivatives needed to discretize the viscous flux terms. Stability is preserved by implementing the a posteriori Multi-dimensional Optimal Order Detection (MOOD) method procedure thus avoiding the utilization of any slope/flux limiter or artificial viscosity. The capabilities of the method are illustrated by solving one- and two-dimensional Riemann problems and benchmark cases. The proposed methodology shows improvements in accuracy in the Riemann problems and does not require any parameter calibration. In addition, the method is extended to the solution of viscous flow and results are validated with the analytical Taylor–Green, Couette and Poiseuille flows, and lid-driven cavity test cases.

Implementation and assessment of a flux-limiter based wetting and drying scheme

<p>Wetting and drying processes in shallow water systems by surges, tides and seiches have important societal, physical and biological impacts. Operational regional models are now of sufficient resolution, O(1 km), that the processes of wetting and drying need to be included. Here we describe a flux limiter based approach that allows a numerical ocean model with a flux formulation of tracer advection to wet and dry. Following Warner et al. (2013), the flux limiter approach limits the outflow from a cell whose depth is below a critical value defined by the user. The limiter can be a step function or a smooth function of the water depth flux limiter, the latter increases model stability and avoids rapid alternation between dry and wet states on long slopes as the critical depth is approached. Furthermore, the user may proportionally limit the baroclinic fluxes as a cell transitions from wet to dry over the course of the large baroclinic time step. The simplicity of the flux limiter approach lends itself to its application within existing numerical models without significant intrusion into the code base. Here we explore the scheme's effectiveness, sensitivities and limitations within the 3D NEMO ocean model by assessing it using test cases of increasing complexity. It is shown to perform well in classic channel test cases and 2D parabolic test cases with analytic solutions. Its performance against analytical 1D dam break experiments is explored and used to interpret its performance against laboratory measurements of a 2D dam break. The scheme is also shown to run stably for a realistic 3D regional domain of the North West European shelf and to improve some aspects of the model's performance against tide gauges.</p>

Reacting flow analysis of a cavity-based scramjet combustor using a Jacobian-free Newton–Krylov method

ABSTRACTThe scramjet is a rather a new technology and there are many issues related to their operation, especially when it comes to the combustion processes. Combustion in high-speed flows causes various problems such as flame instability and poor fuel–air mixing efficiency. One of the methods used to overcome these problems is to recess a cavity in the combustor wall where a secondary flow is generated. In this study, a computational fluid dynamics (CFD) code is developed to analyse the reacting flow passing through the cavity-based scramjet combustor. The developed code is based on three-dimensional coupled Navier–Stokes and finite rate chemistry equations. An ethylene-air reduced chemical reaction model is used as a fuel–air combination. The Spalart–Allmaras model is utilised for turbulence closure. The non-dimensional form of the flow and chemical reaction equations are discretised using a finite volume method. The Jacobian-Free Newton–Krylov (JFNK) method is used to solve the coupled system of non-linear equations. The JFNK is a matrix-free solution method which improves the computational cost of Newton’s method. The parameters that affect the performance of the JFNK method are studied in the analysis of a scramjet combustor. The influence of the forcing term on the convergence of the JFNK method is studied in the analysis of scramjet combustor. Different upwind flux vector splitting methods are utilised. Various flux limiter techniques are employed for the calculations of higher order flux vectors. The effects of flux vector splitting and flux limiter methods on the convergence and accuracy of the JFNK method are evaluated. Moreover, the variations of the mixing efficiency with fuel injection angles are studied.

A Weno-Tvd Implementation for Solving Some Problems of Hyperbolic Conservation Laws

This work deals with a numerical implementation of a fifth order CENTRAL WENO-TVD (\textit{Weighted Essentially Non-Oscillatory-Total Variation Dimimishing}) of Haschem (2006) scheme applied to the convective terms of some hyperbolic conservation laws problems, in a volume finite framework. The WENO-TVD scheme is used to solve the 1D advection and Burgers equations. For this case is implemented two different numerical fluxes: The Lax-Friedrichs and TVD fluxes. In the TVD fluxes the schemes applied are in flux-limiter form. The schemes implemented for this flux are: Van Albada-1 (van Albada et al.,1982), van Albada-2 (Kermani et al., 2003), van Leer (Hassanzadeh, 2009) and MINMOD (Hirsch, 2007). The WENO type schemes are characterized for their high order approximation, and do not produce spurious oscilations near discontinuities, shocks and higher gradients. A third order Runge-Kutta TVD for the temporal variable is used. Qualitative and quantitative comparison are presented. The numerical solutions are computed with an in-house computer code developed in MATLAB software. In future works, it will develope a paralelization of computer code for solving systems of conservation laws, e.g. Euler equations of gas dynamics.

A 4D-Var inversion system based on the icosahedral grid model (NICAM-TM 4D-Var v1.0) – Part 1: Offline forward and adjoint transport models

A four-dimensional variational (4D-Var) method is a popular algorithm for inverting atmospheric greenhouse gas (GHG) measurements. In order to meet the computationally intense 4D-Var iterative calculation, offline forward and adjoint transport models are developed based on the Nonhydrostatic ICosahedral Atmospheric Model (NICAM). By introducing flexibility into the temporal resolution of the input meteorological data, the forward model developed in this study is not only computationally efficient, it is also found to nearly match the transport performance of the online model. In a transport simulation of atmospheric carbon dioxide (CO2), the data-thinning error (error resulting from reduction in the time resolution of the meteorological data used to drive the offline transport model) is minimized by employing high temporal resolution data of the vertical diffusion coefficient; with a low 6-hourly temporal resolution, significant concentration biases near the surface are introduced. The new adjoint model can be run in discrete or continuous adjoint mode for the advection process. The discrete adjoint is characterized by perfect adjoint relationship with the forward model that switches off the flux limiter, while the continuous adjoint is characterized by an imperfect but reasonable adjoint relationship with its corresponding forward model. In the latter case, both the forward and adjoint models use the flux limiter to ensure the monotonicity of tracer concentrations and sensitivities. Trajectory analysis for high CO2 concentration events are performed to test adjoint sensitivities. We also demonstrate the potential usefulness of our adjoint model for diagnosing tracer transport. Both the offline forward and adjoint models have computational efficiency about 10 times higher than the online model. A description of our new 4D-Var system that includes an optimization method, along with its application in an atmospheric CO2 inversion and the effects of using either the discrete or continuous adjoint method, is presented in an accompanying paper Niwa et al.(2016).