Time-derivative Preconditioning Research Articles

Parametric and V&V study in a fundamental CFD process: revisiting the lid-driven cavity flow

PurposeThe openings on aircraft structures can be modeled from an aerodynamical point of view as lid-driven cavities (LDC). This paper aims to show the primary verification and validation (V&V) process in computational fluid dynamics (CFD, and to investigate the influences of numerical settings on the efficiency and accuracy for solving the LDC problem.Design/methodology/approachTo dig into the details of CFD approaches, this paper outlines the design, implementation, V&V and results of an efficient explicit algorithm. The parametric study is performed thoroughly focusing on various iteration methods, grid density discretization terms and Reynolds number effects.FindingsThis study parameterized the numerical implementation which provides empirical insights into how computational accuracy and efficiency are affected by changing numerical settings. At a low Reynolds number (not over 1,000), the time-derivative preconditioning is necessary, and k = 0.1 can be the optimal value to guarantee the efficiency, as well as the stability. A larger artificial viscosity (c = 1/16) would relieve the calculating oscillation issue but proportionally increase the discretization error. Furthermore, the iteration method and the mesh quality are two key factors that affect the convergence efficiency, thus need to be selected “wisely”.Practical implicationsThe study shows how numerical implementation can enhance an accurate and efficient solution. This workflow can be used to determine the best parameter settings whenever CFD researchers applying this LDC problem as a complementary design tool for testing newly developed solvers.Originality/valueThe studied LDC problem is representative of CFD analysis in real aircraft structures. These numerical simulations provide a cost-effective and convenient tool to understand the parameter sensitivity, solution receptivity and physics of the CFD process.

New approaches for computation of low Mach number flows

By using all speed numerical flux schemes, such as SLAU [Simple Low Dissipation AUSM (Advection Upstream Splitting Method)], in MUSCL (Monotone Upwind Scheme for Conservation Laws) approach for compressible CFD, low Mach number flows can be computed without loss of accuracy nor parameter tuning. For an efficient computation, this paper deals with new approaches of implicit time integration method. In this approach, the large sparse matrix system, which consists of flux Jacobian of numerical flux function, has to be solved in each time step. Firstly, a simple Gauss–Seidel iteration method named TC-PGS1(Time Consistent Preconditioned Gauss–Seidel 1) which has flavor of the time derivative preconditioning is introduced. Secondary, we tried to use FGMRES (k) (Flexible Generalized Minimum Residual Method) to solve the non-diagonal dominant linear system arising from Jacobian of flux function SLAU. TC-PGS1 is also used as the matrix preconditioner for FGMRES (k). Optimal parameters for FGMRES (k) is investigated numerically and the performances on computational efficiency of the new methods are compared. It is indicated that FGMRES (k) has apparent advantage on computation of low Mach number flows with sound propagation, however, simpler TC-PGS1 has comparable performance if only flow fields are of interest.

Implicit methods for viscous incompressible flows

Numerical methods and simulation tools for incompressible flows have been advanced largely as a subset of the computational fluid dynamics (CFD) discipline. Especially within the aerospace community, simulation of compressible flows has driven most of the development of computational algorithms and tools. This is due to the high level of accuracy desired for predicting aerodynamic performance of flight vehicles. Conversely, low-speed incompressible flow encountered in a wide range of fluid engineering problems has not typically required the same level of numerical accuracy. This practice of tolerating relatively low-fidelity solutions in engineering applications for incompressible flow has changed. As the design of flow devices becomes more sophisticated, a narrower margin of error is required. Accurate and robust CFD tools have become increasingly important in fluid engineering for incompressible and low-speed flow. Accuracy depends not only on numerical methods but also on flow physics and geometry modeling. For high-accuracy solutions, geometry modeling has to be very inclusive to capture the elliptic nature of incompressible flow resulting in large grid sizes. Therefore, in this article, implicit schemes or efficient time integration schemes for incompressible flow are reviewed from a CFD tool development point of view. Extension of the efficient solution procedures to arbitrary Mach number flows through a unified time-derivative preconditioning approach is also discussed. The unified implicit solution procedure is capable of solving low-speed compressible flows, transonic, as well as supersonic flows accurately and efficiently. Test cases demonstrating Mach-independent convergence are presented.

Optimal preconditioning of lattice Boltzmann methods

A preconditioning technique to accelerate the simulation of steady-state problems using the single-relaxation-time (SRT) lattice Boltzmann (LB) method was first proposed by Guo et al. [Z. Guo, T. Zhao, Y. Shi, Preconditioned lattice-Boltzmann method for steady flows, Phys. Rev. E 70 (2004) 066706-1]. The key idea in this preconditioner is to modify the equilibrium distribution function in such a way that, by means of a Chapman–Enskog expansion, a time-derivative preconditioner of the Navier–Stokes (NS) equations is obtained. In the present contribution, the optimal values for the free parameter γ of this preconditioner are searched both numerically and theoretically; the later with the aid of linear-stability analysis and with the condition number of the system of NS equations. The influence of the collision operator, single- versus multiple-relaxation-times (MRT), is also studied. Three steady-state laminar test cases are used for validation, namely: the two-dimensional lid-driven cavity, a two-dimensional microchannel and the three-dimensional backward-facing step. Finally, guidelines are suggested for an a priori definition of optimal preconditioning parameters as a function of the Reynolds and Mach numbers. The new optimally preconditioned MRT method derived is shown to improve, simultaneously, the rate of convergence, the stability and the accuracy of the lattice Boltzmann simulations, when compared to the non-preconditioned methods and to the optimally preconditioned SRT one. Additionally, direct time-derivative preconditioning of the LB equation is also studied.

Large eddy simulation of turbulent heat transfer in a rotating square duct

Large eddy simulations of turbulent mixed convection heat transfer in a variable-property thermally developing rotating square duct are presented. A finite volume lower–upper symmetric Gauss–Seidel (LU-SGS) scheme coupled with time derivative preconditioning is used to simulate low Mach number compressible three-dimensional turbulent flow. A localized dynamic subgrid-scale (SGS) model is used to evaluate the unresolved stresses. Characteristic outflow conditions are employed so that the flow can develop further as it responds to the heating as well as rotating conditions. Several isothermal rotating duct cases were calculated and compared with previous DNS results to validate the numerical procedure. Then heated rotating ducts under constant wall heat flux are simulated. The wall heat flux is strong enough to cause the flow to separate in the outward mixed convections. A vanishing inviscid flux derivative method is designed to overcome the difficulty caused by the velocity reversal at the duct outlet when the flow separates. The Reynolds number varies from 4500 to 10,000; The rotation number changes from 0.0133 to 0.176; The Grashof number ranges from −2.2 × 10 6 to 2.2 × 10 6. Simulation of forced and mixed convection cases shows that the flow is strongly influenced by the Coriolis and centrifugal buoyancy forces through complex and delicate mechanisms.

A compressible finite volume formulation for large eddy simulation of turbulent pipe flows at low Mach number in Cartesian coordinates

A finite volume formulation for large eddy simulation (LES) of turbulent pipe flows based on the compressible time-dependent three-dimensional Navier–Stokes equations in Cartesian coordinates with non-Cartesian control volumes is presented. The small scale motions are modeled by a dynamic subgrid-scale (SGS) model. A dual-time stepping approach with time derivative preconditioning is employed to enable the simulations to run efficiently at low Mach numbers. The equations with primitive variables, ( p, u, v, w, T), are solved with an implicit lower-upper-symmetric-Gauss–Seidel (LU-SGS) scheme. An isothermal turbulent pipe flow at two Reynolds numbers, and a turbulent pipe flow with a low heat transfer are simulated to evaluate this compressible LES finite volume formulation. The results agree very well with the experimental data and DNS results, verifying the accuracy of the present scheme.

All Speed and High-Resolution Scheme Applied to Three-Dimensional Multi-Block Complex Flowfield System

ABSTRACTThe improved numerical approach is implemented with preconditioned Navier-Stokes solver on arbitrary three-dimensional (3-D) structured multi-block complex flowfield. With the successful application of time-derivative preconditioning, present hybrid finite volume solver is performed to obtain the steady state solutions in compressible and incompressible flows. This solver which combined the adjective upwind splitting method (AUSM) family of low-diffusion flux-splitting scheme with an optimally smoothing multistage scheme and the time-derivative preconditioning is used to solve both the compressible and incompressible Euler and Navier-Stokes equations. In addition, a smoothing procedure is used to provide a mechanism for controlling the numerical implementation to avoid the instability at stagnation and sonic region. The effects of preconditioning on accuracy and convergence to the steady state of the numerical solutions are presented. There are two validation cases and three complex cases simulated as shown in this study. The numerical results obtained for inviscid and viscous two-dimensional flows over a NACA0012 airfoil at free stream Mach number ranging from 0.1 to 1.0E-7 indicates that efficient computations of flows with very low Mach numbers are now possible, without losing accuracy. And it is effectively to simulate 3-D complex flow phenomenon from compressible flow to incompressible by using the advanced numerical methods.

Multigrid solution of the Navier–Stokes equations at low speeds with large temperature variations

Multigrid methods for the Navier–Stokes equations at low speeds and large temperature variations are investigated. The compressible equations with time-derivative preconditioning and preconditioned flux-difference splitting of the inviscid terms are used. Three implicit smoothers have been incorporated into a common multigrid procedure. Both full coarsening and semi-coarsening with directional fine-grid defect correction have been studied. The resulting methods have been tested on four 2D laminar problems over a range of Reynolds numbers on both uniform and highly stretched grids. Two of the three methods show efficient and robust performance over the entire range of conditions. In addition, none of the methods has any difficulty with the large temperature variations.

Low-Diffusion Flux-Splitting Methods for Real Fluid Flows with Phase Transitions

Methods for extending the AUSM+ low-diffusion flux-splitting scheme toward the calculation of real fluid flows at all speeds are presented. The single-phase behavior of the fluid is defined by the Sanchez-Lacombe equation of state, a lattice-fluid description. Liquid-vapor phase transitions are modeled through a homogeneous equilibrium approach. Time-derivative preconditioning is utilized to allow effective integration of the equation system at all flow speeds and all states of compressibility. Modifications to the preconditioned variant of AUSM+ necessary to preserve solution accuracy under such conditions are presented in detail. One-dimensional results are presented for the faucet problem, a classic test case for multifluid algorithms, as well as for liquid octane flow through a converging-diverging nozzle. Two-dimensional calculations are presented for water flow over a hemisphere/cylinder geometry and liquid carbon dioxide flow through a capillary nozzle

A preconditioned Navier–Stokes method for two-phase flows with application to cavitation prediction

An implicit algorithm for the computation of viscous two-phase flows is presented in this paper. The baseline differential equation system is the multi-phase Navier–Stokes equations, comprised of the mixture volume, mixture momentum and constituent volume fraction equations. Though further generalization is straightforward, a three-species formulation is pursued here, which separately accounts for the liquid and vapor (which exchange mass) as well as a non-condensable gas field. The implicit method developed here employs a dual-time, preconditioned, three-dimensional algorithm, with multi-block and parallel execution capabilities. Time-derivative preconditioning is employed to ensure well-conditioned eigenvalues, which is important for the computational efficiency of the method. Special care is taken to ensure that the resulting eigensystem is independent of the density ratio and the local volume fraction, which renders the scheme well-suited to high density ratio, phase-separated two-fluid flows characteristic of many cavitating and boiling systems. To demonstrate the capabilities of the scheme, several two- and three-dimensional examples are presented.

Low-Diffusion Flux-Splitting Methods for Flows at All Speeds

Methods for extending the advective upwind splitting method (AUSM) family of low-diffusion flux-splitting schemes to operate effectively at all flow speeds are developed. The extensions developed are designed for use with time-derivative preconditioning and are based on the idea that the speed of sound should cease to be an important scaling parameter for the diffusive contributions to the interface flux as the Mach number becomes small. Using this criterion, alternative definitions for the interface Mach numbers are developed, and methods for ensuring pressure-velocity coupling at low speeds are formulated. Results are presented for inviscid flows through a channel at various Mach numbers, developing viscous flow in a two-dimensional duct, driven-cavity flows at various Mach and Reynolds numbers, flow over a backward-facing step, and hydrogen-nitrogen mixing layers

Evaluation of multigrid acceleration for preconditioned time-accurate Navier-Stokes algorithms

The development of a two-dimensional time-accurate dual time step Navier-Stokes flow solver with time-derivative preconditioning and multigrid acceleration is described. The governing equations are integrated in time with both an explicit Runge-Kutta scheme and an implicit lower-upper symmetric-Gauss-Seidel scheme in a finite volume framework, yielding second-order accuracy in space and time. Issues concerning the implementation of multigrid for preconditioned, dual time step algorithms are discussed. Steady and unsteady computations were made of lid driven cavity flow, thermally driven cavity flow and pulsatile channel flow for a variety of conditions to validate the schemes and evaluate the effectiveness of multigrid for time-accurate simulations. Significant speedups were observed for steady and unsteady simulations. The speedups for unsteady simulations were problem dependent, a function of how rapidly the flow varied in time and the size of the allowable time step.

Preconditioning applied to variable and constant density flows

A time-derivative preconditioning of the Navier-Stokes equations, suitable for both variable and constant density fluids, is developed. The ideas of low-Mach-number preconditioning and artificial compressibility are combined into a unified approach designed to enhance convergence rates of density-based, time-marching schemes for solving flows of incompressible and variable density fluids at all speeds. The preconditioning is coupled with a dual time-stepping scheme implemented within an explicit, multistage algorithm for solving time-accurate flows. The resultant time integration scheme is used in conjunction with a finite volume discretization designed for unstructured, solution-adaptive mesh topologies. This method is shown to provide accurate steady-state solutions for transonic and low-speed flow of variable density fluids. The time-accurate solution of unsteady, incompressible flow is also demonstrated.