A generalised multi-directional characteristic-based Godunov-type framework for elliptic, parabolic and hyperbolic pressure-based incompressible methods

The objective of the current research is to construct numerical methods based on physical principles to reduce modelling errors in the field of computational fluid dynamics. In order to investigate the non-linearities of the convective flux term, a multi-directional characteristic-based scheme has been developed in this work to capture the anisotropic behaviour of the incompressible Navier{Stokes equations. To avoid the pressure-velocity decoupling and to promote stability at high Reynolds numbers, the Riemann problem has been incorporated into the scheme which creates a multi-directional Godunov-type framework. In order to capture the pressure correctly, which through its coupling to the velocity field is depending on the velocity's non-linear effects, it is postulated that the pressure should have its own transport equation which should have a parabolic type. This is necessary to align the pressure with the mathematical properties of the Navier{Stokes equations. Thus, a novel incompressible method has been developed which features a pressure transport equation which is referred to as the Fractional-Step with Velocity Projection (or FSVP) method. It is further extended through a perturbed continuity equation of the Arti cial Compressibility (AC) method to hyperbolise the first Fractional-Step of the system of equations, while the second Fractional-Step retains the required parabolic behaviour, which is called the FSAC-VP method in turn. Through the hyperbolic Fractional-Step, the multi-directional Godunov-type framework is directly applicable to the newly developed method. Parametric simulations for the lid driven cavity, backward facing step, sudden expan- sion and Taylor{Green vortex problem have been performed using the AC, FSVP, FSAC-VP and the Fractional-Step, Arti cial Compressibility with Pressure Projection, or FSAC-PP, method. The FSVP and FSAC-VP method showed superior convergence properties compared to the AC method for unsteady flows, where a speed up of a factor up to 193.0 times has been observed. Since the parabolic pressure transport equation has a memory of the time history of the flow, smooth error curves have been produced over time while the other methods showed oscillatory profi les. Generally speaking, the most accurate results have been obtained with the FSAC-PP method, closely followed by the FSAC-VP and FSVP method. The inclusion of the multi-directional Godunov-type framework showed generally better or equally well resolved results compared to the benchmark numerical scheme for the FSAC-PP and FSAC-VP / FSVP method. Furthermore, the multi-directional scheme by itself showed its capabilities to predict vortical flows better than a simple numerical reconstruction scheme. The FSAC-VP method has shown a higher degree of scheme independence where velocity and pressure curves showed little variations compared to reference data. This was particularly pronounced for the sudden expansion which had consequences on the prediction of the correct bifurcation behaviour. Finally, it has been argued that what the numerical scheme development is to the non-linear term of the Navier{Stokes equations should be similarly done with incompressible flow method development to capture the correct pressure behaviour. This work shows that differences between elliptic, parabolic and hyperbolic pressure treatments do exist which can have a significant effect on the overall prediction of the flow features

A high-resolution, unified incompressible solver framework for turbulent flows in OpenFOAM

This work introduces the Factional-Step, Artificial Compressibility with Pressure Projection (FSAC-PP) method into OpenFOAM, a fast pressure-velocity coupling algorithm for incompressible flows. It is tested for the lid driven cavity problem and it is shown that the pressure Poisson solver speeds up the solution by up to 27.1% compared to the Pimple algorithm available in OpenFOAM. Comparison against the Pressure Projection method from which the FSAC-PP method is derived, are similar favourably

Aerodynamic performance investigation through different chemistry modelling approaches for space re-entry vehicles using the DSMC method

High-speed flows with Mach numbers well above the hypersonic regime pose significant modelling com-plexities due to increased levels of thermal energy, which in turn result in a variety of chemical reactionsthat become dominant and thus have to be accurately modelled. Within Computational Fluid Dynamics(CFD), the Direct Simulation Monte Carlo (DSMC) method is commonly chosen here as it has shownsuperior performance over traditional Navier-Stokes-based solvers due to a breakdown in the continuumhypothesis. Space re-entering vehicles are commonly exposed to high Mach numbers when entering intoearth’s atmosphere and low density so that the mean free path of particles is comparable to the lengthof the vehicle itself. Thus, these types of applications require challenging modelling approaches which isthe subject of this study. We use the open-source CFD solver OpenFOAM in this study, which comesprebuilt with the dsmcFoam solver. This implementation of the DSMC method lacks, however, the abilityto model chemical reactions and thus is not equipped to predict aerodynamic coefficients for high-speedflows. Recently, the dsmcFoam+ solver has been proposed [1] and implemented into OpenFOAM whichfeatures, among other things, the ability to model chemical reactions through the Quantum-Kinetic (QK)model. The aim of this study, then, is threefold; 1) Validate the new dsmcFoam+ solver against availablereference data from the literature and compare it to the default dsmcFoam solver, highlighting the im-portance of chemical modelling, 2) Publish all simulation and setup files through an online repository tofacilitate an easy case setup for researchers wishing to evaluate or adopt the new dsmcFoam+ solver, 3)Provide documentation for the new dsmcFoam+ solver in the context of OpenFOAM where there is littledocumentation available. We investigate the flow of a re-entry vehicle with a freestream Mach number of25.6 at different angle of attacks and find that the chemical modelling approach taken has a significantinfluence over the aerodynamic coefficients which are up to 24% apart. Similar results are obtained forthe heat transfer coefficient, which shows differences of up to 28%. Based on our findings, we advocatethat the dsmcFoam+ solver should be used for aero-thermodynamic calculations as its ability to predictchemical reactions and thus changes in the flow field will significantly affect the overall solution accuracycompared to a non-reacting modelling approach

Predicting non-linear flow phenomena through different characteristics-based schemes

The present work investigates the bifurcation properties of the Navier–Stokes equations using characteristics-based schemes and Riemann solvers to test their suitability to predict non-linear flow phenomena encountered in aerospace applications. We make use of a single- and multi-directional characteristics-based scheme and Rusanov’s Riemann solver to treat the convective term through a Godunov-type method. We use the Artificial Compressibility (AC) method and a unified Fractional-Step, Artificial Compressibility with Pressure-Projection (FSAC-PP) method for all considered schemes in a channel with a sudden expansion which provides highly non-linear flow features at low Reynolds numbers that produces a non-symmetrical flow field. Using the AC method, our results show that the multi-directional characteristics-based scheme is capable of predicting these phenomena while the single-directional counterpart does not predict the correct flow field. Both schemes and also Riemann solver approaches produce accurate results when the FSAC-PP method is used, showing that the incompressible method plays a dominant role in determining the behaviour of the flow. This also means that it is not just the numerical interpolation scheme which is responsible for the overall accuracy. Furthermore, we show that the FSAC-PP method provides faster convergence and higher level of accuracy, making it a prime candidate for aerospace applications

Progress in particle-based multiscale and hybrid methods for flow applications

This work focuses on the review of particle-based multiscale and hybrid methods that have surfaced in the field of fluid mechanics over the last 20 years. We consider five established particle methods: molecular dynamics, direct simulation Monte Carlo, lattice Boltzmann method, dissipative particle dynamics and smoothed-particle hydrodynamics. A general description is given on each particle method in conjunction with multiscale and hybrid applications. An analysis on the length scale separation revealed that current multiscale methods only bridge across scales which are of the order of O(102)−O(103) and that further work on complex geometries and parallel code optimisation is needed to increase the separation. Similarities between methods are highlighted and combinations discussed. Advantages, disadvantages and applications of each particle method have been tabulated as a reference

A generalised and low-dissipative multi-directional characteristics-based scheme with inclusion of the local Riemann problem investigating incompressible flows without free-surfaces,

In the present study, we develop a generalised Godunov-type multi-directional characteristics-based (MCB) scheme which is applicable to any hyperbolic system for modelling incompressible flows. We further extend the MCB scheme to include the solution of the local Riemann problem which leads to a hybrid mathematical treatment of the system of equations. We employ the proposed scheme to hyperbolic-type incompressible flow solvers and apply it to the Artificial Compressibility (AC) and Fractional-Step, Artificial Compressibility with Pressure Projection (FSAC-PP) method. In this work, we show that the MCB scheme may improve the accuracy and convergence properties over the classical single-directional characteristics-based (SCB) and non-characteristic treatments. The inclusion of a Riemann solver in conjunction with the MCB scheme is capable of reducing the number of iterations up to a factor of 4.7 times compared to a solution when a Riemann solver is not included. Furthermore, we found that both the AC and FSAC-PP method showed similar levels of accuracy while the FSAC-PP method converged up to 5.8 times faster than the AC method for steady state flows. Independent of the characteristics- and Riemann solver-based treatment of all primitive variables, we found that the FSAC-PP method is 7–200 times faster than the AC method per pseudo-time step for unsteady flows. We investigate low- and high-Reynolds number problems for well-established validation benchmark test cases focusing on a flow inside of a lid driven cavity, evolution of the Taylor–Green vortex and forced separated flow over a backward-facing step. In addition to this, comparisons between a central difference scheme with artificial dissipation and a low-dissipative interpolation scheme have been performed. The results show that the latter approach may not provide enough numerical dissipation to develop the flow at high-Reynolds numbers. We found that the inclusion of a Riemann solver is able to overcome this shortcoming. Overall, the proposed generalised Godunov-type MCB scheme provides an accurate numerical treatment with improved convergence properties for hyperbolic-type incompressible flow solvers

Validation and verification of a 2D lattice Boltzmann solver for incompressible fluid flow

The lattice Boltzmann method (LBM) is becoming increasingly popular in the fluid mechanics society because it provides a relatively easy implementation for an incompressible fluid flow solver. Furthermore the particle based LBM can be applied in microscale flows where the continuum based Navier-Stokes solvers fail. Here we present the validation and verification of a two-dimensional in-house lattice Boltzmann solver with two different collision models, namely the BGKW and the MRT models [1]. Five different cases were studied, namely: (i) a channel flow was investigated, the results were compared to the analytical solution, and the convergence properties of the collision models were determined; (ii) the lid-driven cavity problem was examined [2] and the flow features and the velocity profiles were compared to existing simulation results at three different Reynolds number; (iii) the flow in a backward-facing step geometry was validated against experimental data [3]; (iv) the flow in a sudden expansion geometry was compared to experimental data at two different Reynolds numbers [4]; and finally (v) the flow around a cylinder was studied at higher Reynolds number in the turbulent regime. The first four test cases showed that both the BGKW and the MRT models were capable of giving qualitatively and quantitatively good results for these laminar flow cases. The simulations around a cylinder highlighted that the BGKW model becomes unstable for high Reynolds numbers but the MRT model still remains suitable to capture the turbulent von Karman vortex street. The in-house LBM code has been developed in C and has also been parallelised for GPU architectures using CUDA [5] and for CPU architectures using the Partitioned Global Address Space model with UPC [6

Simulate cavitation bubble with single component multi-phase Lattice Boltzmann method

Cavitation occurs when the pressure drops below a critical value at which point it can cause great damage to the machines such as propellers. In this study, a two-dimensional single bubble with different pressure differences between the boundary and the bubble will be studied based on the single component Shan-Chen model with the Carnahan-Starling (C-S) Equation of State (EOS) incorporated, which is similar to the model in [1-2]. Firstly, the model with the C-S EOS will be validated based on Maxwell’s equal area construction. The equilibrium density of liquid and vapor is obtained using a flat interface simulation according to [3]. It was demonstrated that the model has great thermal consistency according to this validation. Furthermore, we show results for a single bubble case for which its growth and collapse can be validated against the RayleighPlesset (R-P) equation with various pressure differences. Results show good agreement with the R-P equation and literature

Validation and verification of a 2D lattice Boltzmann solver for incompressible fluid flow

The lattice Boltzmann method (LBM) is becoming increasingly popular in the fluid mechanics society because it provides a relatively easy implementation for an incompressible fluid flow solver. Furthermore the particle based LBM can be applied in microscale flows where the continuum based Navier-Stokes solvers fail. Here we present the validation and verification of a two-dimensional in-house lattice Boltzmann solver with two different collision models, namely the BGKW and the MRT models [1]. Five different cases were studied, namely: (i) a channel flow was investigated, the results were compared to the analytical solution, and the convergence properties of the collision models were determined; (ii) the lid-driven cavity problem was examined [2] and the flow features and the velocity profiles were compared to existing simulation results at three different Reynolds number; (iii) the flow in a backward-facing step geometry was validated against experimental data [3]; (iv) the flow in a sudden expansion geometry was compared to experimental data at two different Reynolds numbers [4]; and finally (v) the flow around a cylinder was studied at higher Reynolds number in the turbulent regime. The first four test cases showed that both the BGKW and the MRT models were capable of giving qualitatively and quantitatively good results for these laminar flow cases. The simulations around a cylinder highlighted that the BGKW model becomes unstable for high Reynolds numbers but the MRT model still remains suitable to capture the turbulent von Karman vortex street. The in-house LBM code has been developed in C and has also been parallelised for GPU architectures using CUDA [5] and for CPU architectures using the Partitioned Global Address Space model with UPC [6

Effects of cornering conditions on the aerodynamic characteristics of a high-performance vehicle and its rear wing

This study investigates the aerodynamic behavior of a high-performance vehicle and the interaction with its rear wing in straight-line and steady-state cornering conditions. Analyses are performed with Reynolds-averaged Navier–Stokes based computational fluid dynamics simulations using a moving reference frame and overset mesh technique, validated against moving ground wind tunnel experiments. The results indicate a significant 20% decrease in downforce and 35% increase in drag compared to straight-line conditions at the smallest considered corner radius of 2.9 car-lengths. Downforce losses primarily stem from performance deficits on the underbody and rear wing, alongside elevated upper body lift. Drag penalties mainly result from additional pressure drag induced by a recirculation wake vortex generated behind the vehicle's inboard side. The vehicle's lateral pressure distribution is also affected, introducing a centripetal force that increases with smaller corner radii. Additionally, analyses of the rear wing reveal alternations of its aerodynamic characteristics in cornering, particularly impacting vortical flow and suction on the lower surface. Throughout the operating conditions, the rear wing's individual downforce contribution falls off beyond its stall angle. At higher angles of attack, the rear wing primarily generates downforce by pressurizing the vehicle's upper surfaces, but its interaction with the near-wake leads to a substantially increased pressure drag. Overall, these findings provide crucial insights into the intricate aerodynamic interactions of high-performance vehicles in diverse operating conditions as well as form an essential foundation for future research on static and active aerodynamic designs in the pursuit to optimize vehicle performance in dynamic driving conditions