Session R29: CFD: Algorithms II

Comparison of Quasi-Conservative Pressure-Based and Fully-Conservative Formulations for the Simulation of Transcritical Flows

High-pressure flows are known to be challenging to simulate due to thermodynamic non-linearities occurring in the vicinity of the pseudo-boiling line. This study investigates the origin of this issue by analyzing the behavior of thermodynamic processes at elevated pressure and low temperature. We show that under transcritical conditions, non-linearities significantly amplify numerical errors associated with construction of fluxes. These errors affect the local density and energy balances, which in turn creates pressure oscillations. For that reason, solvers based on a conservative system of equations that transport density and total energy are subject to unphysical pressure variations in gradient regions. These perturbations hinder numerical stability and degrade the accuracy of predictions. To circumvent this problem, the governing system can be reformulated to a pressure-based treatment of energy. We present comparisons between the pressure-based and fully conservative formulations using a progressive set of canonical cases, including a cryogenic turbulent mixing layer at rocket engine conditions.   

A GPU-based High-order Multi-resolution Framework for Compressible Flows at All Mach Numbers

The Wavelet Adaptive Multiresolution Representation (WAMR) method is a general and robust technique for providing grid adaptivity around the evolution of features in the solutions of partial differential equations and is capable of resolving length scales spanning 6 orders of magnitude. A new flow solver based on the WAMR method and specifically parallelized for the GPU computing architecture has been developed. The compressible formulation of the Navier-Stokes equations is solved using a preconditioned dual-time stepping method that provides accurate solutions for flows at all Mach numbers. The dual-time stepping method allows for control over the residuals of the governing equations and is used to complement the spatial error control provided by the WAMR method. An analytical inverse preconditioning matrix has been derived for an arbitrary number of species that allows preconditioning to be efficiently implemented on the GPU architecture. Additional modifications required for the combination of wavelet-adaptive grids and preconditioned dual-time stepping on the GPU architecture will be discussed. Verification using the Taylor-Green vortex to demonstrate the accuracy of the method will be presented.   

High order accurate finite difference schemes based on symmetry preservation

A new algorithm for development of high order accurate finite difference schemes for numerical solution of partial differential equations using Lie symmetries is presented. Considering applicable symmetry groups (such as those relevant to space/time translations, Galilean transformation, scaling, rotation and projection) of a partial differential equation, invariant numerical schemes are constructed based on the notions of moving frames and modified equations. Several strategies for construction of invariant numerical schemes with a desired order of accuracy are analyzed. Performance of the proposed algorithm is demonstrated using analysis of one-dimensional partial differential equations, such as linear advection diffusion equations inviscid Burgers equation and viscous Burgers equation, as our test cases. Through numerical simulations based on these examples, the expected improvement in accuracy of invariant numerical schemes (up to fourth order) is demonstrated. Advantages due to implementation and enhanced computational efficiency inherent in our proposed algorithm are presented. Extension of the basic framework to multidimensional partial differential equations is also discussed.   

The Space-Time CESE Method Applied to Viscous Flow Computations with High-Aspect Ratio Triangular or Tetrahedral Meshes

Flow physics near the viscous wall is intrinsically anisotropic in nature, namely, the gradient along the wall normal direction is much larger than that along the other two orthogonal directions parallel to the surface. Accordingly, high aspect ratio meshes are employed near the viscous wall to capture the physics and maintain low grid count. While such arrangement works fine for structured-grid based methods with dimensional splitting that handles derivatives in each direction separately, similar treatments often lead to numerical instability for unstructured-mesh based methods when triangular or tetrahedral meshes are used. The non-splitting treatment of near-wall gradients for high-aspect ratio triangular or tetrahedral elements results in an ill-conditioned linear system of equations that is closely related to the numerical instability. Altering the side lengths of the near wall tetrahedrons in the gradient calculations would make the system less unstable but more dissipative. This research presents recent progress in applying numerical dissipation control in the space-time conservation element solution element (CESE) method to reduce or alleviate the above-mentioned instability while maintaining reasonable solution accuracy.   

Framework for a Robust General Purpose Navier-Stokes Solver on Unstructured Meshes

A numerical framework for a pressure-based all-speeds flow solver operating on unstructured meshes, which is robust for a broad range of flow configurations, is proposed. The distinct features of our framework are the full coupling of the momentum and continuity equations as well as the use of an energy equation in conservation form to relate the thermal quantities with the flow field. In order to overcome the well-documented instability occurring while coupling the thermal energy to the remaining flow variables, a multistage iteration cycle has been devised which exhibits excellent convergence behavior without requiring any numerical relaxation parameters. Different spatial schemes for accurate shock resolution as well as complex thermodynamic gas models are also seamlessly incorporated into the framework. The solver is directly applicable to stationary and transient flows in all Mach number regimes (sub-, trans-, supersonic), exhibits strong robustness and accurately predicts flow and thermal variables at all speeds across shocks of different strengths. We present a wide range of results for both steady and transient compressible flows with vastly different Mach numbers and thermodynamic conditions in complex geometries represented by different types of unstructured meshes.   

Compressible turbulence and shock-capturing using a variational multiscale method

We have previously developed a dynamic extension of Hughes' variational multiscale method which is implemented in an entropy-stable Discontinuous-Galerkin spectral-element solver\footnote{Murman {\em{et al.}}, AIAA 2016-1059}. This solver and sub-grid model have been examined on standard low-speed benchmark flows, {\em{e.g.}} homogeneous turbulence, channel flow, {\em{etc.}} Here we extend the approach to higher speeds where compressibility effects are no longer insignificant, and the flowfields develop unsteady shocklets and shock waves. Homogeneous isotropic turbulence at high turbulent Mach number is tested for two cases - decaying and passing through a normal shock. Numerical simulations using the multiscale sub-grid model, no sub-grid model, and a variation of Barter and Darmofal's shock-capturing scheme\footnote{Barter and Darmofal, J. Comp. Physics, 229(5)} are examined in isolation and combination. The computed results are compared against theoretical observations and previous computational results.   

Cost and accuracy comparison between the diffuse interface method and the geometric volume of fluid method for simulating two-phase flows

The diffuse interface(DI) and volume of fluid(VOF) methods are mass conserving front capturing schemes which can handle large interfacial topology changes in realistic two phase flows. The DI method is a conservative phase field method that tracks an interface with finite thickness spread over a few cells and does not require reinitialization. In addition to having the desirable properties of level set methods for naturally capturing curvature and surface tension forces, the model conserves mass continuously and discretely. The VOF method, which tracks the fractional tagged volume in a cell, is discretely conservative by requiring costly geometric reconstructions of the interface and the fluxes. Both methods however, suffer from inaccuracies in calculation of curvature and surface tension forces. We present a quantitative comparison of these methods in terms of their accuracy, convergence rate, memory, and computational cost using canonical 2D two-phase test cases: damped surface wave, oscillating drop, equilibrium static drop, and dense moving drop. We further compared the models in their ability to handle thin films by looking at the impact of a water drop onto a deep water pool. Considering these results, we suggest qualitative guidelines for using the DI and VOF methods.   

Impact Detection for Characterization of Complex Multiphase Flows

Multiphase flows often involve a wide range of impact events, such as liquid droplets impinging on a liquid pool or gas bubbles coalescing in a liquid medium. These events contribute to a myriad of large-scale phenomena, including breaking waves on ocean surfaces. As impacts between surfaces necessarily occur at isolated points, numerical simulations of impact events will require the resolution of molecular scales near the impact points for accurate modeling. This can be prohibitively expensive unless subgrid impact and breakup models are formulated to capture the effects of the interactions. The first step in a large-eddy simulation (LES) based computational methodology for complex multiphase flows like air-sea interactions requires effective detection of these impact events. The starting point of this work is a collision detection algorithm for structured grids on a coupled level set / volume of fluid (CLSVOF) solver (Mortazavi \textit{et al}., JFM 2016) adapted from an earlier algorithm for cloth animations (Bridson \textit{et al}., 2002) that triangulates the interface with the marching cubes method (Lorensen and Cline, Comp. Graphics 1987). We explore the extension of collision detection to a geometric VOF solver and to unstructured grids (Ivey and Moin, JCP 2015). Supported by ONR/A*STAR.   

An implicit control-volume finite element method for well-reservoir modelling

Here a novel implicit approach (embodied within the IC-Ferst) is presented for modelling wells with potentially a large number of laterals within reservoirs. IC-Ferst is a conservative and consistent, control-volume finite element method (CV-FEM) model and fully unstructured/geology conforming meshes with anisotropic mesh adaptivity. As far as the wells are concerned, a multi-phase/multi-well approach, where well systems are represented as phases, is taken here. Phase volume fraction conservation equations are solved for in both the reservoir and the wells, in addition, the field within wells is also solved for. A second novel aspect of the work is the combination of modelling and resolving of the motherbore and laterals. In this case wells do not have to be explicitly discretised in space. This combination proves to be accurate (in many situations) as well as computationally efficient. The method is applied to a number of multi-phase reservoir problems in order to gain an insight into the effectiveness, in terms of production rate, of perforated laterals. Model results are compared with semi-analytical solutions for simple cases and industry-standard codes for more complicated cases.