Dual-time Stepping Research Articles

Influence of ribs on train aerodynamic performances

The influence of ribs on the train aerodynamic performance was computed using detached eddy simulation (DES), and the transient iteration was solved by the dual-time step lower-upper symmetric Gauss-Seidel (LU-SGS) method. The results show that the ribs installed on the roof have a great effect on the train aerodynamic performance. Compared with trains without ribs, the lift force coefficient of the train with convex ribs changes from negative to positive, while the side force coefficient increases by 110% and 88%, respectively. Due to the combined effect of the lift force and side force, the overturning moment of the train with convex ribs and cutting ribs increases by 140% and 106%, respectively. There is larger negative pressure on the roof of the train without ribs than that with ribs. The ribs on the train would disturb the flow structure and contribute to the air separation, so the separation starts from the roof, while there is no air separation on the roof of the train without ribs. The ribs can also slow down the flow speed above the roof and make the air easily sucked back to the train surface. The vortices at the leeward side of the train without ribs are small and messy compared with those of the train with convex or cutting ribs.

Investigation of Unsteady Flow in an Axial Compression Based on Double Times Dual Time Stepping

A numerical investigate of unsteady (rotor 37) based on dual time step is carried out .Results were used to identify the existence of an interface between the approach fluid and the tip-leakage flow. This paper discusses mechanism of spike stall.

A central compact scheme for numerical solution of two-phase incompressible flow using Allen–Cahn phase field model

This article presents a numerical method for the Allen–Cahn phase field model coupled with the incompressible Navier–Stokes equations for simulating two-phase incompressible flow. The numerical method is based on the artificial compressibility method with dual-time stepping technique applied to the governing equations written in conservative form. The convective terms are approximated by using an eighth-order central compact scheme with filtering while the viscous terms are approximated by a sixth-order central compact scheme. Approximate factorization-based alternating direction implicit algorithm is used to solve the discretized linear system. The effectiveness of the method is demonstrated by computing several benchmark two-phase incompressible flow problems.

A High-Order Central ENO Finite-Volume Scheme for Three-Dimensional Low-Speed Viscous Flows on Unstructured Mesh

AbstractHigh-order discretization techniques offer the potential to significantly reduce the computational costs necessary to obtain accurate predictions when compared to lower-order methods. However, efficient and universally-applicable high-order discretizations remain somewhat illusive, especially for more arbitrary unstructured meshes and for incompressible/low-speed flows. A novel, high-order, central essentially non-oscillatory (CENO), cell-centered, finite-volume scheme is proposed for the solution of the conservation equations of viscous, incompressible flows on three-dimensional unstructured meshes. Similar to finite element methods, coordinate transformations are used to maintain the scheme’s order of accuracy even when dealing with arbitrarily-shaped cells having non-planar faces. The proposed scheme is applied to the pseudo-compressibility formulation of the steady and unsteady Navier-Stokes equations and the resulting discretized equations are solved with a parallel implicit Newton-Krylov algorithm. For unsteady flows, a dual-time stepping approach is adopted and the resulting temporal derivatives are discretized using the family of high-order backward difference formulas (BDF). The proposed finite-volume scheme for fully unstructured mesh is demonstrated to provide both fast and accurate solutions for steady and unsteady viscous flows.

High-performance Computing of Periodic Unsteady Flow Based on Time Spectral Method

The engineering application requires the solver of periodic unsteady flows to be of high efficiency and precision. Based on the developing Frequency-Domain methods, a fully implicit version of the Time Spectral Method (TSM) is established, the stability problem that occurs as the sampling number grows is improved. The TSM is extended toSST turbulence model, which improves the numerical accuracy and the ability of simulating complex flows. The periodic unsteady flows over a forced oscillating NACA0015 and Hypersonic Ballistic Shape (HBS) are numerically simulated. The computational results agree well with the experiment data at subsonic and hypersonic regime, validating the application of the TSM solver. Influences of the angle of attack and Ma on dynamic derivatives of the HBS are analyzed. Comparing with the Dual-Time-Step (DTS) method, the relative efficiency of the TSM improves as the Ma number increases. The TSM provides an order of magnitude improvement in computational efficiency over DTS without loss of generality at hypersonic speed regime.

Splitting based scheme for three-dimensional MHD with dual time stepping

A new hybrid numerical scheme of combining an E-CUSP (Energy-Convective Upwind and Split Pressure) method for the fluid part and the Constrained Transport (CT) for the magnetic induction part is proposed. In order to avoid the occurrence of negative pressure in the reconstructed profiles and its updated value, a positivity preserving method is provided. Furthermore, the MHD equations are solved at each physical time step by advancing in pseudo time. The use of dual time stepping is beneficial in the computation since the use of dual time stepping allows the physical time step not to be limited by the corresponding values in the smallest cell and to be selected based on the numerical accuracy criterion. This newly established hybrid scheme combined with positivity preserving method and dual time technique has demonstrated the accurateness and robustness through numerical experiments of benchmark problems such as the 2D Orszag-Tang vortex problem and the 3D shock-cloud interaction problem.

A Unified Fractional-Step, Artificial Compressibility and Pressure-Projection Formulation for Solving the Incompressible Navier-Stokes Equations

AbstractThis paper introduces a unified concept and algorithm for the fractional-step (FS), artificial compressibility (AC) and pressure-projection (PP) methods for solving the incompressible Navier-Stokes equations. The proposed FSAC-PP approach falls into the group of pseudo-time splitting high-resolution methods incorporating the characteristics-based (CB) Godunov-type treatment of convective terms with PP methods. Due to the fact that the CB Godunov-type methods are applicable directly to the hyperbolic AC formulation and not to the elliptical FS-PP (split) methods, thus the straightforward coupling of CB Godunov-type schemes with PP methods is not possible. Therefore, the proposed FSAC-PP approach unifies the fully-explicit AC and semi-implicit FS-PP methods of Chorin including a PP step in the dual-time stepping procedure to a) overcome the numerical stiffness of the classical AC approach at (very) low and moderate Reynolds numbers, b) incorporate the accuracy and convergence properties of CB Godunov-type schemes with PP methods, and c) further improve the stability and efficiency of the AC method for steady and unsteady flow problems. The FSAC-PPmethod has also been coupled with a non-linear, full-multigrid and fullapproximation storage (FMG-FAS) technique to further increase the efficiency of the solution. For validating the proposed FSAC-PP method, computational examples are presented for benchmark problems. The overall results show that the unified FSAC-PP approach is an efficient algorithm for solving incompressible flow problems.

Delayed Detached Eddy Simulation of Airfoil Stall Flows Using High-Order Schemes

An advanced hybrid Reynolds-Averaged Navier–Stokes/large eddy simulation (RANS/LES) turbulence model delayed detached eddy simulation (DDES) is conducted in thispaper to investigate the dynamic stall flows over 3D NACA0012 airfoil at 17 deg, 26 deg, 45 deg, and 60 deg angle of attack (AOA). The spatially filtered unsteady 3D Navier–Stokes equations are solved using a fifth-order weighted essentially nonoscillatory (WENO) reconstruction with a low diffusion E-CUSP (LDE) scheme for the inviscid fluxes and a conservative fourth-order central differencing for the viscous terms. An implicit second-order time marching scheme with dual time stepping is employed to achieve high stability and convergency rate. A 3D flat plate is validated for the DDES model. For quantitative prediction of lift and drag of the stalled NACA0012 airfoil flows, the detached eddy simulation (DES) and DDES achieve much more accurate results than the Unsteady Reynolds-Averaged Navier–Stokes (URANS) simulation. In addition to the quantitative difference, the DES/DDES and URANS also obtain qualitatively very different unsteady stalled flows of NACA0012 airfoil with different vortical structures and frequencies. This may bring a significantly different prediction if those methods are used for fluid–structural interaction. For comparison purpose, a third-order WENO scheme with a second-order central differencing is also employed for the DDES stalled NACA0012 airfoil flows. Both the third- and fifth-order WENO schemes predict the stalled flow similarly for lift and drag at AOA less than 45 deg, while at AOA of 60 deg, the fifth-order WENO scheme shows better agreement with the experiment than the third-order WENO scheme. The high-order scheme of WENO 5 also resolves more small scales of flow structures than the second-order scheme. The prediction of the stalled airfoil flow using DDES with both the high-order scheme and second-order scheme is overall significantly more accurate than the URANS simulation.

Study on the transports in transient flow over impulsively started circular cylinder using Lagrangian coherent structures

The material transport in transient flow over impulsively started circular cylinder is studied from Lagrangian point of view, in order to understand the nature of some complex transient phenomena. The transient flow is simulated numerically by CBS scheme with dual time stepping. Then, the transport processes during the development of separation bubbles and vortices are analyzed in terms of Lagrangian coherent structures (LCSs), in order to understand the transport and mixture in the transient phenomena and their contributions to the aerodynamic performance of circular cylinder. Results show that the transport processes in the transition flow are well described by the LCSs. In particular, at the beginning stage of formations of flow separation and vortex, the transports between main flow and separated flow are strong, and the aerodynamic performance of circular cylinder can be affected significantly. During the transition of flow states from symmetric flow to periodic vortex shedding, the dynamical behaviors of flow are different from that of periodic flow. Compared with Eulerian description, it can be seen that the Lagrangian description is much more available for the analysis of transient flow. More, the results are also instructive to the flow control from the viewpoint of nonlinear dynamics.

Coupled CFD/GNC Modeling for the Basic Finner Projectile with Closed Loop

The virtual flight simulation of the Basic Finner projectile has been investigated through coupling solving unsteady Navier-Stokes equations, rigid-body six degree-of-freedom motion equations, guidance, navigation and control law. The flow solver uses a finite-volume method based on structure grid with dual time stepping, the chimera method is used to simulate relative motions, the fourth-order Runger-Kutta method is used to solve the motion equations. A closed loop feedback control law with PID control is required for the elevator to attain the commanded deflection. The predictions show that the PID parameters for the controller must be correctly selected to obtain the proper response. Simulation results show that the virtual flight simulation platform that we developed is capable of solving the complicated unsteady flows with moving boundary, has a strong applicability to engineering application.

Lagrangian coherent structure and transport in unsteady transient flow

Unsteady transient phenomena in flow over impulsively started circular cylinder, such as the generation of separation, burst of separation bubble, vortex shedding, etc., are studied from Lagrangian viewpoint. The transient flow is solved numerically by using characteristic-based split scheme with dual time stepping. Then Lagrangian coherent structures (LCSs) are extracted to study the transport and mixing of these transient phenomena. Results show that the variation of drag is closely related to the evolutions of separation bubbles and vortex shedding. The evolutions of the symmetric bubbles in streamwise induce high pressure distribution at rear of cylinder and result in drag reduction of the circular cylinder. As separation bubbles become asymmetric, the transport between separation bubbles and main flow is enhanced and thus can reduce the separation region and suppress flow separation as well. The results also show that the shedding vortices are induced by the transpor between separation bubble and main flow. Compared with streamline patterns, LCSs have huge advantages in describing the dynamic features of the unsteady phenomena.

Formula omitted] Continuous finite elements on tetrahedra with local mass matrix inversion to solve the preconditioned compressible Navier–Stokes equations

For unsteady simulations a global mass matrix must be inverted when using a continuous finite element method. To avoid this, a local approximate mass matrix inversion procedure is given for a p=2 basis. This inversion process when combined with a low Mach Turkel preconditioner, dual time stepping, and multigrid gives an efficient and accurate iterative scheme to solve the compressible Navier–Stokes equations. The performance of the preconditioner for both p=1 and p=2 is examined on a perturbed free stream flow under a wide range of flow conditions. Both preconditioning and multigrid improved the convergence rates for all cases and for both polynomial degrees the performance is similar in most cases. The best performance is seen for unsteady and viscous flow problems. In these regimes the iterative convergence is almost independent of the polynomial degree and it takes around 1–3 multigrid cycles to reduce the residual one order of magnitude. A large scale simulation of flow around a NACA 0012 airfoil is also presented. For the flow conditions of M=0.1,Re=500, and CFL=55 the convergence rate averaged 2.5 cycles per decade. Also, the method showed good linear speedup when solved in parallel, as a speedup of 160 was achieved with 192 processors.

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.

Numerical Simulation of Instantaneous Heat Transfer in Reciprocating Internal Combustion Engines

This study provides a detailed instantaneous heat transfer solution in a cylinder during a motored cycle. The piston-cylinder assembly is modeled by a two-dimensional cavity with a sinusoidal moving wall. A high order differential algorithm is used to solve the unsteady compressible Navier-Stokes equations. The order of differencing is raised progressively from the walls toward the center of the cylinder, enabling oscillation free and accurate calculation on a very coarse grid. The numerical fluxes are integrated based on the dual-time stepping of the preconditioned matrix with a third-order Runge-Kutta scheme. The absolute pressure is determined by enforcing the global mass conservation for each grid. The predicted results of the absolute pressure, temperature, and velocity components of the fluid inside the cylinder at any instant during the start-up and the periodically stable periods are compared well with the results given in the literature.

2D Unstructured Mesh Finite Volume Method for Simulating Structural Dynamics

A novel procedure for calculating the dynamic response of elastic solid structures is presented. The ultimate aim of this study is to develop a consistent set of finite volume (FV) methods on unstructured meshes for the analysis of dynamic fluidstructure interaction (FSI). This paper describes a two-dimensional (2D) FV cell-vertex based method for dynamic solid mechanics. A novel matrix-free implicit scheme was developed using the Newmark method and dual time step algorithm and the model is validated with a 2D cantilever test case as well as a 2D plate one.

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.

Numerical Computation of Unsteady Compressible Flow past a Plunging Aerofoil

Unsteady Reynolds-averaged Navier-Stokes (RANS) computations are presented for subsonic and transonic flow past a plunging NACA 64A010 aerofoil. The Implicit RANS solver used for obtaining the time-accurate solution is based on finite volume nodal point spatial discretization scheme with dual time stepping. Results for the subsonic and transonic cases compare well with the experimental data, thus demonstrating the capability of the solver to provide useful unsteady pressure data for aeroelastic analysis.

Numerical simulation of rotor-airframe aerodynamic interaction based on unstructured dynamic overset grids

A method of unstructured dynamic overset grids is developed for the numerical simulation of helicopter unsteady rotorairframe aerodynamic interaction. For the effective treatment of the relative motion between the rotor and the airframe, the domain of flowfield is divided into two overset subzones, namely, a rotational subzone containing the blades and a stationary subzone containing the airframe. The overset part of two subzones is used to convect the flow variables of the two zones. The Taylor series expansion is used to obtain a second-order spatial accuracy, and dual-time stepping is adopted to improve the solution accuracy. Mesh deformation from the blade motion in forward flight is treated by using a spring analogy. Validation is made by numerically simulating the flows around a wind tunnel configuration and comparing the predicted time-averaged and instantaneous inflow and airframe surface pressure distributions with the experimental data. It shows that the present method is efficient and robust for the prediction of complicated unsteady rotor-airframe aerodynamic interaction phenomena. unstructured grid, overset grids, rotor-airframe interaction, spring analogy, relative motion

Theoretical Study and Software Design for the Implementation of a Numerical Scheme Based on the Dual Time Stepping in COSMO LM

This research paper describes a theoretical study for the re-design of the time integration core of COSMO based on a time integration technique called Dual Time Stepping. This technique has been successfully adopted in Aerodynamics for the resolution of the unsteady Navier-Stokes equations, but no previous attempt has been found in Metereology and Climatology. This preliminary theoretical study aims to assess the feasibility of implementing the Dual Time Stepping in the time integration core of COSMO. After the description of the Dual Time Stepping, the procedure to be implemented is described in comparison with the current time integration core. The typical convergence acceleration techniques (i.e. local time stepping, preconditioning, residual smoothing) that can be combined with the Dual Time Stepping are presented. A simple test case with a mountain flow is used to obtain a preliminary comparison between the re-designed and the existing software. This theoretical study will be followed by an assessment of the proposed numerical scheme that will space from stability considerations and implementation issues to the analysis of ideal and real test case for Meteorology and Climatology.

와류 셀을 이용한 풍력블레이드 에어포일 주위 유동 제어

A flow control on airfoil installed a vortex cell for high efficiency wind turbine blade in stationary and dynamic stall conditions have been numerically investigated by solving the compressible Navier-Stokes equations. The numerical scheme is based on a node-based finite-volume method with Roe`s flux-difference splitting and an implicit time-integration method coupled with dual time step sub-iteration. The computed result for the airfoil in the stationary showed that lift-drag ratio increases due to low pressure by the vortex cell. The oscillating airfoil with the vortex cell showed that the magnitude of hysteresis loop is reduced due to the enhanced vortex in the cell.