Unsteady Test Cases Research Articles

On the Use of Different Sets of Variables for Solving Unsteady Inviscid Flows with an Implicit Discontinuous Galerkin Method

This article presents a comparison between the performance obtained by using a spatial discretization of the Euler equations based on a high-order discontinuous Galerkin (dG) method and different sets of variables. The sets of variables investigated are as follows: (1) conservative variables; (2) primitive variables based on pressure and temperature; (3) primitive variables based on the logarithms of pressure and temperature. The solution is advanced in time by using a linearly implicit high-order Rosenbrock-type scheme. The results obtained using the different sets are assessed across several canonical unsteady test cases, focusing on the accuracy, conservation properties and robustness of each discretization. In order to cover a wide range of physical flow conditions, the test-cases considered here are (1) the isentropic vortex convection, (2) the Kelvin–Helmholtz instability and (3) the Richtmyer–Meshkov instability.

On the use of high order central difference schemes for differential equation based wall distance computations

A computationally efficient high-order solver is developed to compute the wall distances by solving the relevant partial differential equations, namely: Eikonal, Hamilton–Jacobi (HJ) and Poisson equations. In contrast to the upwind schemes widely used in the literature, we explore the suitability of high-order central difference schemes (explicit/compact) for the wall-distance computation. While solving the Hamilton–Jacobi equation, the high-order central difference schemes performed approximately 1.4–2.8 times faster than the upwind schemes with a marginal improvement in the solution accuracy. A new pseudo HJ formulation based on the localized artificial diffusivity (LAD) approach has been proposed. It is demonstrated to predict results with an accuracy comparable to that of the Eikonal equation and the simulations are ≈ 1.5 times faster than the baseline HJ solver using upwind schemes. A curvature correction is also incorporated in the HJ equation to correct for the near-wall errors due to concave/convex wall curvatures. We demonstrate the efficacy of the proposed methods on both the steady and unsteady test cases and exploit the unsteady wall-distance solver to estimate the instantaneous shape and burning surface area of a dendrite propellant grain in a solid propellant rocket motor.

An updated Gappy-POD to capture non-parameterized geometrical variation in fluid dynamics problems

In this work, we propose a new method to fill the gap within an incomplete turbulent and incompressible data field in such a way to satisfy the topological and intensity changes of the fluid flow after a non-parameterized geometrical variation in the fluid domain. This work extends the one that has been published as a conference proceeding to the 2018 AIAA Scitech Forum and Exposition (Akkari et al. in Geometrical reduced order modeling (ROM) by proper orthogonal decomposition (POD) for the incompressible navier stokes equations. In: 2018 AIAA Aerospace Sciences Meeting, AIAA SciTech Forum, (AIAA 2018-1827), 2018). A single baseline large eddy simulation (LES) is assumed to be performed prior geometrical variations. The proposed method is an enhancement of the Gappy-POD method proposed by Everson and Sirovich in 1995, in the case where the given set of empirical eigenfunctions is not sufficient and is not interpolant for the recovering of the modal coefficients for each Gappy snapshot by a least squares procedure. This happens when the available data cannot be written as an interpolation of the baseline POD modes. This is typically the case when we introduce non-parameterized geometrical modifications in the fluid domain. Here, after the baseline simulation, additional solutions of the incompressible Navier–Stokes equations are solely performed over a restricted fluid domain, that contains the geometrical modifications. These local LESs that we will call hybrid simulations are performed by using the immersed boundary technique, which uses of a fluid boundary condition and the baseline velocity field. Then, we propose to update the POD modes using a local modification of the baseline POD modes in the restricted fluid domain. Furthermore, we will propose a physical correction of the latter enhanced Gappy-POD modal coefficients thanks to a Galerkin projection of the Navier–Stokes equations upon the new modes of the available data. This enhancement procedure on the global velocity reconstruction by the physical constraint was tested on a 3D semi-industrial test case of a typical aeronautical injection system and, a 2D laminar and unsteady incompressible test case. The speed-up relative to this new technique is equal to 100, which allows us to perform an exploration of two new designs of the aeronautical injection system.

A free–energy stable p–adaptive nodal discontinuous Galerkin for the Cahn–Hilliard equation

A novel free–energy stable discontinuous Galerkin method is developed for the Cahn–Hilliard equation with non–conforming elements. This work focuses on dynamic polynomial adaptation (p–refinement) and constitutes an extension of the method developed by Manzanero et al. (2020) [8], which makes use of the summation–by–parts simultaneous–approximation term technique along with Gauss–Lobatto points and the Bassi–Rebay 1 (BR1) scheme. The BR1 numerical flux accommodates non–conforming elements, which are connected through the mortar method. The scheme has been analytically proven to retain its free–energy stability when transitioning to non-conforming elements. Furthermore, a methodology to perform the adaptation is introduced based on the knowledge of the location of the interface between phases. The adaptation methodology is tested for its accuracy and effectiveness through a series of steady and unsteady test cases. We test the scheme for freestream preservation and primary quantity conservation on non–conforming curvilinear meshes. We solve a steady one–dimensional interface test case to initially examine the accuracy of the adaptation. Furthermore, we study the formation of a static bubble in two dimensions and verify that the accuracy of the solver is maintained while the degrees of freedom decrease to less than half compared to the uniform solution. Lastly, we examine an unsteady case such as the spinodal decomposition and show that the same results for the free–energy are recovered, with a 35% reduction of the degrees of freedom for the two–dimensional case considered and a 48% reduction for the three–dimensional case.

Sensitivity equation method for the Navier‐Stokes equations applied to uncertainty propagation

SummaryThis works deals with sensitivity analysis (SA) for the Navier‐Stokes equations. The aim is to provide an estimate of the variance of the velocity field when some of the parameters are uncertain and then to use the variance to compute confidence intervals for the output of the model. First, we introduce the physical model and analyze its stability. The sensitivity equations are derived, and their stability analyzed as well. We propose a finite element‐volume numerical scheme for the state and the sensitivity, which is integrated into the open‐source industrial code TrioCFD. Finally, we present some numerical results: a steady and an unsteady test case for the channel flow problem are investigated. For the steady case, we compare the results to the Monte Carlo method and show how the SA technique succeeds in providing very accurate estimates of the variance. For the unsteady case, a new filtering procedure is proposed to deal with a sensitivity that grows in time. The filtered sensitivity is then used to compute the variance of the output and to provide confidence intervals.

Arbitrary Lagrangian-Eulerian unstructured finite-volume lattice-Boltzmann method for computing two-dimensional compressible inviscid flows over moving bodies.

The objective of this study is to develop and apply an arbitrary Lagrangian-Eulerian unstructured finite-volume lattice-Boltzmann method (ALE-FVLBM) for solving two-dimensional compressible inviscid flows around moving bodies. The two-dimensional compressible form of the LB equation is considered and the resulting LB equation is formulated in the ALE framework on an unstructured body-fitted mesh to correctly model the body shape and properly incorporate the mesh movement due to the body motion. The spatial discretization of the resulting system of equations is performed by a second-order cell-centered finite-volume method on arbitrary quadrilateral meshes and an implicit dual-time stepping method is utilized for the time integration. To stabilize the numerical solution, appropriate numerical dissipation terms are added to the formulation. At first, the shock tube problem is computed to examine the accuracy of the solution obtained by applying the proposed FVLBM for this unsteady test case which includes shock, expansion wave, and contact discontinuity in the flow domain. Then, the stationary isentropic vortex is simulated on both the stationary and moving meshes to assess the implementation of the geometric conservation law in enhancing the solution accuracy of the ALE-FVLBM. The compressible inviscid flow in the transonic regime is then computed around the stationary NACA0012 airfoil in order to further study the sensitivity of the solution method to the user defined parameters. Now, the transonic inviscid flow is simulated over the pitching or plunging NACA0012 airfoil to investigate the accuracy and capability of the proposed solution method (ALE-FVLBM) for the computation of the compressible flows over moving bodies. Finally, the pitching or plunging NACA0012 airfoil near the ground in the transonic inviscid flow is simulated as a practical and challenging problem to study the ground effect on the aerodynamic characteristics of the airfoil. It is indicated that the solution methodology proposed based on the finite-volume LBM formulated in the arbitrary Lagrangian-Eulerian framework (ALE-FVLBM) is capable of accurately computing the compressible inviscid flows around the moving bodies with and without the ground effect.

On the use of thermally perfect gas model for heating prediction of laminar and turbulent SWBLI

The model of thermally perfect gas has been investigated in the heating prediction of laminar and turbulent shock wave boundary layer interaction (SWBLI) flows. The method of the polynomial curve fit coupling to thermodynamics equation is examined for higher temperature range. The characteristic weighted essentially non-oscillatory (WENO) scheme is utilized to capture the complex flow details and meantime stabilize the strong shocks in the SWBLI flows. Both the laminar and turbulent, steady and unsteady test cases are carried out to assess the performance of the designed method. Numerical results have substantiated the thermally perfect gas model as an effective model in the challenging SWBLI heating predictions in comparison with the existing model of calorically perfect gas.

Multiple-relaxation time color-gradient lattice Boltzmann model for simulating contact angle in two-phase flows with high density ratio

The contact line dynamics is an important phenomenon in natural and industrial processes and is still a challenging problem. The complexity of this problem in computational simulations is much more than theoretical and experimental works. In this paper, a color-gradient (CG) lattice Boltzmann method (LBM) for simulating wetting phenomena and the dynamics of contact line in two-phase flows with very high density-ratio (∼ 1000 was used. This method applies a multi-relaxation time (MRT) collision operator to enhance the stability of numerical scheme. Both static and dynamic contact angles were enforced at the wall through geometrical wetting boundary condition. Note, the main nobility of this paper is the supplementation of geometrical wetting boundary condition to the color gradient LBM and applying the present computational methodology for two-phase flow physics with a very high density ratio. This method is first validated by simulating a stationary drop and static contact angles. Also, the dynamical behavior of a drop on an ideal surface in shear flow was computationally validated. Finally, simulation of a drop motion subjected to gravitational force was the fourth test case studied. According to the results of these test cases, small values of spurious velocities and equilibrium contact angle errors were found in the simulations of steady cases studied. While, in our unsteady test cases, the behavior of the interface shapes and contact angles were in good agreements with previous reliable studies.

A Developed Numerical Method for Turbulent Unsteady Fluid Flow in Two-Phase Systems with Moving Interface

This paper presents the development of an accurate and robust numerical modeling of instability of an interface separating two-phase system, such as liquid–gas and/or solid–gas systems. The instability of the interface can be refereed to the buoyancy and capillary effects in liquid–gas system. The governing unsteady Navier–Stokes along with the stress balance and kinematic conditions at the interface are solved separately in each fluid using the finite-volume approach for the liquid–gas system and the Hamilton–Jacobi equation for the solid–gas phase. The developed numerical model represents the surface and the body forces as boundary value conditions on the interface. The adapted approaches enable accurate modeling of fluid flows driven by either body or surface forces. The moving interface is tracked and captured using the level set function that initially defined for both fluids in the computational domain. To asses the developed numerical model and its versatility, a selection of different unsteady test cases including oscillation of a capillary wave, sloshing in a rectangular tank, the broken-dam problem involving different density fluids, simulation of air/water flow, and finally the moving interface between the solid and gas phases of solid rocket propellant combustion were examined. The latter case model allowed for the complete coupling between the gas-phase physics, the condensed-phase physics, and the unsteady nonuniform regression of either liquid or the propellant solid surfaces. The propagation of the unsteady nonplanar regression surface is described, using the Essentially-Non-Oscillatory (ENO) scheme with the aid of the level set strategy. The computational results demonstrate a remarkable capability of the developed numerical model to predict the dynamical characteristics of the liquid–gas and solid–gas flows, which is of great importance in many civilian and military industrial and engineering applications.

Wind turbine control in computational fluid dynamics with OpenFOAM

In this article, the implementation of the pitch-control option is described within the OpenFOAM computational fluid dynamics library supported by the preliminary results for a few representative test cases. The proposed pitch-control logic represents a simplified version of the typically model available for the load simulation based on blade element momentum theory codes. The control variable for the blade pitch regulation is the rotor rotational speed, for which an additional torque-speed controller determining the equilibrium between the aerodynamic and the electrical torque is required. The simulations are performed on the 80-m-diameter turbine used as a reference design within the SmartBlades project. The unsteady test cases are based on the assumption of rigid, not pre-bend blade, and refer to an emergency stop and an extreme operating gust. Results verified the correctness of the implemented control logic and its sensitivity to the control parameters, that is, the key factors for a successful regulation of power production and load mitigation in wind turbine design. The implemented simulation methodology constitutes the first step toward the reproduction of very complex operating conditions for wind turbines by means of accurate computations.

Aeroelastic Simulations of High Reynolds Number Aero Structural Dynamics Wind Tunnel Model

This article presents results of numerical simulations of the High Reynolds Number Aero Structural Dynamics wind tunnel configuration. The activities consist of the selection of relevant transonic test cases, the setup and simulation of the numerical models, and the comparison of computational and experimental results. Two steady (static coupling) and three unsteady (forced motion) test cases were selected, all in the transonic flow regime at Mach numbers up to 0.85 and Reynolds numbers up to 23 million. Good agreement between numerical and experimental data was obtained for both the steady and the unsteady test cases. It is shown that the consideration of the static elastic deformation of the wing is indispensable for both the steady and unsteady simulations. The unsteady simulations were executed in such a way that the elastic motion of the wing in a selected structural eigenmode was prescribed in a forced motion computational fluid dynamics simulation using grid deformation. This approach yields very satisfactory results in terms of the unsteady pressure distribution.

Validation of a vortex ring wake model suited for aeroelastic simulations of floating wind turbines

In order to evaluate aerodynamic loads on floating offshore wind turbines, advanced dynamic analysis tools are required. As a unified model that can represent both dynamic inflow and skewed inflow effects in it basic formulation, a wake model based on a vortex ring formulation is discussed. Such a model presents a good intermediate solution between computationally efficient but simple momentum balance methods and computationally expensive but complete computational fluid dynamics models. The model introduced is shown to be capable of modelling typical steady and unsteady test cases with reasonable accuracy.

Modified extended BDF scheme for the discontinuous Galerkin solution of unsteady compressible flows

SUMMARYIn this paper, a high‐order DG method coupled with a modified extended backward differentiation formulae (MEBDF) time integration scheme is proposed for the solution of unsteady compressible flows. The objective is to assess the performance and the potential of the temporal scheme and to investigate its advantages with respect to the second‐order BDF. Furthermore, a strategy to adapt the time step and the order of the temporal scheme based on the local truncation error is considered. The proposed DG‐MEBDF method has been evaluated for three unsteady test cases: (i) the convection of an inviscid isentropic vortex; (ii) the laminar flow around a cylinder; and (iii) the subsonic turbulent flow through a turbine cascade. Copyright © 2014 John Wiley & Sons, Ltd.

Numerical Investigation of Sub-Iteration Criterions for Unsteady Flow Computations with Dual-Time Method

The convergence of sub-iteration with the dual-time method is very important for the prediction of unsteady flow field. The influence of sub-iteration step number, criterion of sub-iteration convergence and the choice of physical time step size on the calculation results are discussed by solving of the two-dimensional unsteady Euler equations. A new convergence criterion (named residual criterion) of sub-iteration for unsteady flows is proposed, and the unsteady flow test case AGARD-CT5 is calculated to verify the new criterion. The results show that, with the same criterion of sub-iteration, the results from different physical time step sizes are in agreement with each other. The difference between the experiment data and the numerical results are small, and if the sub-iteration criterion used is reasonable and small enough, the dependence of numerical results of unsteady flows on the physical time step will be decreased as possible. The new criterion of sub-iteration for dual-time step unsteady calculations can be used for engineering problem.

Implementation of a Numerical Scheme Based on the Dual Time Stepping in COSMO LM: Idealized Test Cases

This paper is part of research on the development and testing of a modified version of COSMO LM, that is aimed to assess the feasibility of a time integration technique called Dual Time Stepping (DTS). In a previous paper it was described the implementation of the proposed time integration core with an application for a steady mountain ideal flow. In the present work the DTS scheme has been applied to an unsteady idealized test case, the non-hydrostatic inertia-gravity wave, that involves the evolution of a potential temperature perturbation in a channel. This study on idealized cases will be followed, in future research, by an assessment of the proposed numerical scheme on a realistic test case for Meteorology and Climatology.

Multi-dimensional limiting for high-order schemes including turbulence and combustion

In the present paper a fourth/fifth order upwind biased limiting strategy is presented for the simulation of turbulent flows and combustion. Because high order numerical schemes usually suffer from stability problems and TVD approaches often prevent convergence to machine accuracy the multi-dimensional limiting process (MLP) [1] is employed. MLP uses information from diagonal volumes of a discretization stencil. It interacts with the TVD limiter in such a way, that local extrema at the corner points of the volume are avoided. This stabilizes the numerical scheme and enables convergence in cases, where standard limiters fail to converge. Up to now MLP has been used for inviscid and laminar flows only. In the present paper this technique is applied to fully turbulent sub- and supersonic flows simulated with a low Reynolds-number turbulence closure. Additionally, combustion based on finite-rate chemistry is investigated. An improved MLP version (MLP ld , low diffusion) as well as an analysis of its capabilities and limitations are given. It is demonstrated, that the scheme offers high accuracy and robustness while keeping the computational cost low. Both steady and unsteady test cases are investigated.

A space–time adaptation scheme for unsteady shallow water problems

We provide a space–time adaptation procedure for the approximation of the Shallow Water Equations (SWE). This approach relies on a recovery based estimator for the global discretization error, where the space and time contributions are kept separate. In particular we propose an ad hoc procedure for the recovery of the time derivative of the numerical solution and then we employ this reconstruction to define the error estimator in time. Concerning the space adaptation, we move from an anisotropic error estimator able to automatically identify the density, the shape and the orientation of the elements of the computational mesh. The proposed global error estimator turns out to share the good properties of each recovery based error estimator. The whole adaptive procedure is then combined with a suitable stabilized finite element SW solver. Finally the reliability of the coupled solution–adaptation procedure is successfully assessed on two unsteady test cases of interest for hydraulics applications.

A Posteriori Error Estimation for Subgrid Flamelet Models

A dual-weighted residual error estimation strategy is applied to the modeling error associated with a class of subgrid scale models widely used in turbulent combustion. The approach is implemented and validated for an idealized test problem consisting of a system of multiscale advection-reaction-diffusion equations. The regime of interest corresponds to the limit of fast reaction. A fully resolved numerical simulation of the resulting very fine reaction scale would be excessively demanding, so typically a subgrid model is used instead to account for its impact at large scales. The class of subgrid models investigated here is based on the asymptotic flamelet approximation, which leads to the so-called flamelet libraries, i.e., precomputed tables expressing the reactive scalars in terms of the appropriate passive scalar. We show that a similar library can be constructed for the dual-based estimator. The performance of the estimation strategy is validated by comparison with exact results for steady and unsteady test cases. Asymptotic scalings are recovered, and new light is shed on some specific implementations of the flamelet models for unsteady test cases.

Time-Centered Split Method for Implicit Discretization of Unsteady Advection Problems

A new method for implicit solution of unsteady nonlinear advection equations is presented. This time-centered split (TCS) method uses a nested application of the midpoint rule to computationally decouple advection terms in a temporally second-order accurate time-marching discretization. The method requires solution of only two sets of linear equations without an outer iteration, and is theoretically applicable to quadratically nonlinear coupled equations for any number of variables. The TCS algorithm is compared to other nonlinear solution methods (local linearization, Picard iteration, and Newton iteration) and applied to the Crank-Nicolson discretization of the one-dimensional Burgers’ equation. The temporal accuracy and practical stability of the method is confirmed using an unsteady flow test case with an analytical solution. The method is shown to require computational effort similar to local linearization, but does not require discrete computation of a functional Jacobian for solution.

Detailed comparisons of front‐capturing methods for turbulent two‐phase flow simulations

AbstractIn the framework of numerical study of multiphase flows, different front‐capturing methods are compared. Two different approaches dealing with the incompressibility constraint will be compared, too. Then, every technique is tested on several relevant test cases in order to make a comparison. The surface tension treatment is evaluated for each method and every front‐capturing method is tested on a case where interface deformations have something in common with turbulent behaviour. Finally, every method is studied through a phase inversion problem, which is an unsteady test case. Front‐capturing methods are tested on physical cases where Navier–Stokes and advection equations are coupled in a non‐linear way. Copyright © 2008 John Wiley & Sons, Ltd.