Abstract The discontinuous Galerkin method (DGM) has been widely used since the 1980s due to its ability to provide accurate high-order (larger than 2) numerical solutions of partial differential equations (PDEs) modeling convection-dominated problems and nonlinear ordinary differential equations (ODEs). This work aims to study two effects that can break the high-order numerical accuracy of the DGM: (i) the use of finite precision arithmetic, and (ii) the existence of discontinuous right-hand sides (source terms). These types of discontinuities occur in many applications in computational science and engineering. To this end, a time-dependent optimal control problem (OCP) (an ODE system) and the shallow water equations (a convection-dominated PDE problem) have been studied. The results show that when the discontinuity lies within an element, the numerical accuracy of the solution decreases significantly and is independent of the degree of the polynomial approximation used in the computations. The results also show that this conclusion is independent of the arithmetic precision used in the computations. It is also shown that the number of digits used in the computations, for example, double precision, limits the achievable order of mesh convergence as the polynomial order increases. The paper also presents an efficient matrix-based implementation of the proposed algorithms.

Tidal turbines are a renewable energy source on the rise. The exceptional predictability of tidal currents contributes to a high reliability of this technology, which represents a key advantage in the endeavor to become a major contributor to the energy mix. To foster the development and to support the design process of tidal turbines, reliable numerical modeling techniques are required. This paper presents verification and validation work performed within the framework of the Supergen ORE Tidal Turbine Benchmarking Study. Viscous-flow CFD code ReFRESCO is used to conduct blade-resolved simulations of the towing tank experiments. In a first approximation, a steady-state frozen-rotor approach is chosen. A transition model, Gamma-ReTheta, is employed to predict the flow state transition on the turbine blades. In the process, the sensitivity to input turbulence quantities is highlighted. The numerical uncertainty is estimated based on mesh refinements. Finally, a conclusion is drawn to which accuracy the presented numerical models can predict the outcome of the experiments.

This paper reports on the recent NATO Advanced Vehicle Technology (AVT) effort associated with smooth-wall two-dimensional turbulent boundary layer flows subjected to streamwise pressure gradients. The effort considered experiments, Reynolds Averaged Navier-Stokes (RANS) simulations, modelling, scaling and flow physics relative to the subject flows. Special attention is given to the current predictive capabilities and deficiencies of RANS simulations, and the interplay between experiments and RANS validation and development. In addition, the efficacy of the prediction of velocity field response and wall pressure statistics are respectively demonstrated via Resolvent and Gene Expression Programming based models. The persistence of a logarithmic mean velocity profile is evaluated and measures of non-equilibrium are described and discussed. A number of open issues are described and recommendations for future research are suggested.

In the present paper, we focus on the simulation of viscous flows at high Reynolds numbers using the Reynolds-averaged Navier-Stokes (RANS) equations. Time-averaging is used to define the mean flow properties and only deterministic simulations are considered. Therefore, numerical errors are a consequence of round-off, iterative and discretization errors. In carefully performed simulations, round-off and iterative errors are reduced to negligible levels when compared to the discretization error and so the numerical error is dominated by the contribution of the discretization error. The use of grid refinement studies is one of the most flexible and popular techniques for the estimation of discretization errors for steady simulations. Several methods have been proposed in the open literature and most of them share common features. The discretization error of a quantity of interest is described as a function of the typical cell size by power series expansions. The estimation of the exact solution requires numerical solutions in more than one grid and so a family of (nearly) geometrical similar grids needs to be generated. The requirement of grid similarity is a consequence of the definition of the typical cell size. In the numerical solution of the RANS equations, the determination of the shear-stress at the wall tau(w) can be performed in two alternative ways: directly from its definition, or using wall functions. The grid refinement strategy required by each case is significantly different. In the first option, the near-wall cell must be systematically refined as all the remaining grid cells. When wall functions are used, the size of the near-wall cell size should remain fixed. In this paper, we present the consequences of using the wrong refinement strategy, i.e., by keeping the size of the near-wall cell fixed when tau(w) is calculated from its definition and by refining the near-wall cell when tau(w) is determined from wall functions. The selected test case is the flow over a flat plate at Reynolds numbers of 10(7) and 10(9). The results show that using the wrong grid refinement strategy can lead to misleading results that exhibit reasonable orders of grid convergence.

The aerodynamic performance analysis of Formula Student racecars has been mostly done by teams with CFD tools, for time and cost savings, that often lack proper validation. To address this, the FST Lisboa team performed a detailed wind tunnel (WT) test campaign, using a one-third scale model, under different configurations, including variable ride heights, bullhorn appendix, and rear wing flap settings, also replicated in CFD. The simulations used RANS with the SST k-omega turbulence model, with a 13.7 million polyhedral mesh for the test chamber region domain. Both experimental and numerical errors were estimated from the instrumentation and mesh convergence analysis, respectively. Comparisons were made between WT and CFD both in terms of local flow, using tufts for flow visualization, and global flow, using lift, drag, and pitching moment coefficients. Overall, the numerical streamlines agreed very well with the orientations of the tufts in experiments, but some discrepancies were found in regions of cross-flow and high-frequency unsteadiness, mainly caused by limitations of the visualization technique. The gamma transition model in CFD was abandoned as it could not replicate the WT observations. In terms of aerodynamic coefficients, a strong correlation was found between WT and CFD. The parametric studies revealed that the simulations captured the experimental sensitivity to each car setting parameter studied but the uncertainties did not enable a full quantitative evaluation of the aerodynamic performance. The drag reduction system significantly impacted the aerodynamic balance of the racecar, while the current bullhorn design proved to be ineffective. The ride height increase led to higher downforce, mostly due to the higher pitch angle of the vehicle, with negligible variation of the aerodynamic balance. This work validated the team CFD studies, building confidence in that trends estimated in numerical parametric studies are likely to be translated to the real prototype performance.

The effects of roughness were considered as part of a NATO Advanced Vehicle Technology effort titled 'Non-Equilibrium Turbulent Boundary Layers at High Reynolds Numbers' (NATO AVT-349). This paper comments on the current state of understanding of the flow physics and modelling efforts to predict rough-wall boundary layer behaviour. Outer layer similarity to smooth wall flows and Reynolds number effects are discussed for zero, favourable, and adverse pressure gradients based on the results of experiments and numerical simulations. Various types of modelling are considered including Reynolds averaged Navier-Stokes (RANS) models with different roughness and turbulence models, wall-modelled large eddy simulations (WMLES), and resolvent models. Current needs and gaps in present understanding are discussed along with recommendations for future experiments and computations.

This paper presents Solution Verification exercises with the pressure-based compressible flow solver ReFRESCO for five test cases available in the NASA Turbulence Modeling Resource: the two-dimensional flows over a flat plate, a bump-in-channel, a DSMA661 airfoil and a multi-element airfoil and the three-dimensional flow of a bump-in-channel. Simulations are performed with the Favre-averaged continuity and Navier-Stokes equations using the Spalart & Allmaras turbulence model. ReFRESCO results are compared with reference data from density-based compressible flow solvers (CFL3D and FUN3D). two aspects of the implementation of the turbulence model are addressed: the calculation of the distance to the wall and the discretization scheme used in the convective terms of the turbulence model transport equation. Results of this study show perfect consistency with the reference data for the test cases that are not affected by the determination of the distance to the wall.

This work investigates the importance of verification and validation (V&V) to achieve predictive scale-resolving simulations (SRS) of turbulence, i.e., computations capable of resolving a fraction of the turbulent flow scales. Toward this end, we propose a novel but simple V&V strategy based on grid and physical resolution refinement studies that can be used even when the exact initial flow conditions are unknown, or reference data are unavailable. This is particularly relevant for transient and transitional flow problems, as well as for the improvement of turbulence models. We start by presenting a literature survey of results obtained with distinct SRS models for flows past circular cylinders. It confirms the importance of V&V by illustrating a large variability of results, which is independent of the selected mathematical model and Reynolds number. The proposed V&V strategy is then used on three representative problems of practical interest. The results illustrate that it is possible to conduct reliable verification and validation exercises with SRS models, and evidence the importance of V&V to predictive SRS of turbulence. Most notably, the data also confirm the advantages and potential of the proposed V&V strategy: separate assessment of numerical and modeling errors, enhanced flow physics analysis, identification of key flow phenomena, and ability to operate when the exact flow conditions are unknown or reference data are unavailable.

This article presents a high-order accurate method to solve the two-phase incompressible Stokes flow problem. The method has three main features; First, the divergence-free condition of the incompressible Stokes/Navier-Stokes equations are satisfied numerically everywhere in the computational domain; secondly, it satisfies zero jumps in the normal velocities across inter-elements faces and the interface between the two fluids; those two features are essential for the method to be energy-stable. Finally, local discontinuities arising in the two-phase flow fields are captured accurately on a fixed unfitted mesh.In this work, a novel Hybridizable Discontinuous Galerkin (HDG) method is used for the spatial discretization. This hybrid method that belongs to the family of Discontinuous Galerkin (DG) methods satisfies the energy-stability conditions by featuring the following options: the velocity and pressure trace variables are of the same order on both inter-elements faces and on the interface between the two-fluids within each element; compatible velocity and pressure approximations are used inside the elements. Furthermore, the concepts of eXtended Finite Element Method (X-FEM) are used to approximate discontinuities in the flow fields without the need of fitting the computational mesh to the material interface.

View Video Presentation: https://doi.org/10.2514/6.2022-0696.vid This paper describes a collaborative experimental and computational study of smooth wall boundary layers in a systematic family of favorable and adverse pressure gradients. The objective is to advance turbulence modeling of these flows, in particular the effects of pressure gradients that can be classified as non-equilibrium. This collaboration is a component of the larger NATO AVT-349 Research Task Group. Experiments under this effort are conducted at Virginia Tech and computational efforts are presented from Virginia Tech, the German Aerospace Center (DLR), the University of Melbourne, Chalmers University of Technology, the Maritime Research Institute Netherlands (MARIN) in conjunction with the University of Lisbon Instituto Superior Técnico (IST) (MARIN/IST), and the Sirehna Naval Group. This paper describes some of the key elements of the experimental and computational approaches, the efforts made for cross discipline collaboration, verification, and validation, and reports on some initial results and findings. The agreement between various RANS solutions and RANS turbulence models and between RANS solutions and experiment are generally good, but questions remain as to the efficacy of RANS modeling for non-equilibrium boundary layer flows and some potential directions for future investigations are suggested.

The incorporation of the effects of laminar-to-turbulent transition in CFD has become a topic of growing importance, due to the inability of well established turbulence models in accurately predicting transition. This work details simulations done for two of the three-dimensional test cases of the first AIAA Transition Modeling and Prediction Workshop. These are the NASA 3D Bump-in-Channel, and the 6:1 prolate spheroid. The results for the first case are used to verify the implementation of the underlying turbulence model. On the other hand, the calculations performed for the prolate spheroid at three different angles of attack, using two distinct transition models, highlight some of the physical and numerical challenges that accompany the use of this type of model. These challenges consist of the influence of the boundary conditions at the inlet for the variables of the turbulence model, and the modeling of crossflow transition, which is unaccounted for in the original formulation of the two selected transition models.

This paper presents the results obtained with an incompressible flow solver for the two-dimensional test cases proposed for the first AIAA CFD Transition Modeling and Prediction Workshop: a zero-pressure-gradient flow over a flat plate and the flow around the NLF(1)-0416 airfoil. Simulations were performed with the Reynolds-averaged (time averaging) Navier-Stokes equations using the Shear-Stress Transport (SST) $k-\omega$ two-equation, eddy-viscosity model combined with three transition models: $\gamma-Re_{\theta}$, $\gamma$ and Amplification Factor Transport (AFT). The flat plate flow is used to illustrate the dependence of the predicted location of transition from laminar to turbulent flow on the domain size and turbulence quantities inlet boundary conditions. To this end, simulations are performed for the settings adopted by the AIAA CFD Transition Modeling and Prediction and Applied Vehicle Technology Activity 313 Workshops. This simple geometry is also used to illustrate the consequences of specifying large values of inlet eddy-viscosity on the laminar part of the flow calculated with the SST $k-\omega$ model with and without transition models. The flow around the NLF(1)-0416 airfoil is simulated in a domain with smaller dimensions than those recommended by the AIAA Workshop instructions. Uncertainties due to the size of the calculation domain are quantified and compared with numerical uncertainties and input uncertainties due to inexact values of the turbulence quantities inlet boundary conditions. Finally, the results obtained with the SST $k-\omega$ model with and without transition models are compared with experimental data to illustrate the significant modeling improvement achieved by transition models in low Reynolds numbers flows.

The use of transition models in Reynolds-averaged Navier-Stokes solvers is usually accompanied by high values of the eddy viscosity at the inlet, due to the role of this quantity in controlling the decay of freestream turbulence, a critical factor in the behavior of these models. This paper presents a new approach to control the decay of freestream turbulence that avoids setting high values for the inlet eddy viscosity. The proposed method is based on the modification of the dissipation term of the turbulence model only in the freestream, ensuring that the original formulation of the model is retained in regions where viscous effects are significant. The new technique is tested on the flow over a flat plate and around a NACA 0012 airfoil and a 6:1 prolate spheroid, using the gamma-Re-theta model. The results show that the method allows for reductions of up to three orders of magnitude of the inlet eddy viscosity, without disturbing the solution of the original model for cases with low or moderate turbulence intensity. The method requires the Kato-Launder stagnation point correction to be used in the underlying turbulence model, due to the strong influence of the turbulence production limiter on the location of transition.

Numerous hybridizable discontinuous Galerkin methods for numerically solving convection-diffusion, and Navier-Stokes equations have emerged. Nevertheless, their stability, well-posedness, and order of convergence are highly dependent on the proposed stabilization parameter. Conventionally, the stabilization parameter can be defined in two terms as, the advection term and the diffusion term. The advection stabilization term had been studied extensively for discontinuous Galerkin methods on hyperbolic problems. These studies can be extended to the hybridizable discontinuous Galerkin method for convection-diffusion problems, in which the advection stabilization term can be chosen based on mathematical derivations to guarantee the stability of the method. Notwithstanding that the diffusion stabilization term is studied for purely diffusive as well as convection-diffusion problems in hybridizable discontinuous Galerkin framework, its significance in the convection dominated flows is not clear. Referring to literature, the diffusion stabilization term is dependent on a characteristic length and the diffusivity or the viscosity for the Navier-Stokes equations.In this paper, the role of this stabilization parameter of the hybridizable discontinuous Galerkin method is reviewed in detail for the convection-diffusion and the incompressible Navier-Stokes. A new expression for the diffusion stabilization term is mathematically derived, where the term is studied with respect to the Reynolds number. Its importance for convection dominated flows is emphasized and supported by numerical examples.

When dealing with flows at moderate Reynolds numbers, the laminar and transition regions are a key component of the flow solution. In applications such as wind turbines and unmanned aerial vehicles, an accurate prediction of the aerodynamic forces requires accounting for these effects. In Reynolds-averaged Navier-Stokes simulations, this is done by incorporating transition models because commonly used turbulence models are unable to predict a significant extent of laminar flow. In this paper, we present and study the coupling of the local correlation-based gamma -Re theta and gamma transition models with the k-kL (KSKL) turbulence model, and we compare it to the original formulation using the k-omega shear-stress transport (SST) turbulence model. The coupling of the models is calibrated for the flow over a flat plate and subsequently tested for the flow around the S809 and NLF(1)-0416 airfoils, as well as around a 6:1 prolate spheroid. The results show that the combination of the gamma -Re theta transition model with the KSKL turbulence model leads to a reduction of the numerical uncertainty (discretization errors) when compared to its application with the SST model. On the other hand, the combination of the KSKL model with the gamma transition model leads to the sharpest transition of the four combinations tested.

We investigate the main challenges to prediction of turbulent external flows of practical interest with Reynolds-Averaged Navier–Stokes equations (RANS) and Scale-Resolving Simulation (SRS) models. This represents a crucial step toward further developing and establishing these formulations so they can be confidently utilized in engineering problems without reference data. The study initiates by identifying the major challenges to prediction. A literature review is performed to illustrate their effects in RANS and SRS computations. Afterward, we evaluate the impact of the challenges to prediction by analyzing representative statistically steady and unsteady flows with prominent RANS and SRS methods. These include multiple turbulent viscosity and second-moment RANS closures, and hybrid and bridging SRS models. The results demonstrate the potential of the selected SRS models to predict engineering flows. Yet, they also show the importance of considering the challenges to prediction during the setup and conduction of numerical experiments. These can suppress the advantages of using SRS formulations. The data also indicate that only SRS models can confidently predict statistically unsteady flows. In contrast, the results demonstrate that mean-flow quantities of statistically steady flows can be efficiently calculated with RANS closures, especially second-moment closures. Among the selected SRS methods, bridging models reveal better suited for prediction due to their ability to prevent commutation errors and enable the robust evaluation of numerical and modeling errors. This last property allows the use of a new validation technique that does not require reference data.

This paper presents a solution verification exercise for the simulation of subsonic, transonic and supersonic flows of an inviscid fluid over a circular arc (bump). Numerical simulations are performed with a pressure-based, single-phase compressible flow solver. Sets of geometrically similar grids covering a wide range of refinement ratios have been generated. The goal of these grids is twofold: obtain a reference solution from power series expansion fits applied to the finest grids; check the numerical uncertainties obtained from coarse grids that do not guarantee monotonic convergence of the quantities of interest. The results show that even with very fine grids it is not straightforward to define a reference solution from power series expansions. The level of discretization errors required to obtain reliable reference solutions implies iterative errors reduced to machine accuracy, which may be extremely time consuming even in two-dimensional inviscid flows. Quantitative assessment of the estimated uncertainties for coarse grids depends on the selected reference solution.

This paper addresses the influence of the scheme used in the convective terms of turbulence and transition transport equations on the numerical accuracy of transitional flow simulations. Three transition models are combined with the Shear Stress Transport turbulence model: the model, the gamma model and the Amplification Factor Transport model. Two airfoils and a 6:1 prolate spheroid are used as test cases. Using first-order upwind in the convective terms of the k and omega transport equations has a weak effect on the numerical accuracy of the simulations when there is no transition model. However, when combined with the transition models, there is a significant improvement if second-order schemes are used. The iterative convergence obtained for the first-order approach was similar to that observed with the second-order schemes. Therefore, common 'rules of thumb' from RANS simulations at high Reynolds numbers may fail when transition models are included.

In many Engineering applications, the Reynolds-Averaged Navier-Stokes (RANS) equations are still used to simulate high Reynolds numbers (turbulent) flows around complex geometries. In flows that exhibit significant regions of flow separation leading to vortex shedding, it does not make sense to define the mean flow using time-averaging. Therefore, the use of RANS (even if locally) in statistically unsteady flows requires the application of ensemble-averaging to the flow variables and to the mass and momentum equations, which generates the appearance of the Reynolds stresses. Turbulence models available in the open literature have been developed for the simulation of statistically steady flows (mean flow defined by time-averaging). Nonetheless, the same models are used for the simulation of statistically unsteady flows. Therefore, it is not guaranteed that such models provide sufficient diffusion (damping) to capture only the mean flow. In this paper, we have investigated the modeling and numerical properties of RANS supplemented by the k–ω SST eddy-viscosity model when applied to the classical problem of the flow around captive and moving (imposed motion) cylinders with Reynolds numbers ranging from 102 to 106. Two and three-dimensional simulations are performed and numerical (statistical, iterative and discretization) convergence properties are assessed for moving and deforming grids techniques. The quantities of interest are the drag and lift coefficients, for which we determine the frequency content of the time signal to assess if the numerical results correspond (as intended) to the mean flow. Results obtained at a Reynolds number of 104 are compared with experimental data available in the open literature.