Unsteady Incompressible Flow Problems Research Articles

CFD Simulation of Low Speed Turbulent Flow Problems Using Unsteady Rans and Large Eddy Simulation Approach

The present paper focuses on the recent development of an implicit pressure-based finite volume algorithm for numerical solution of Navier Stokes equation in an inertial frame of reference for prediction of unsteady incompressible flow problems. The algorithm uses boundary-conforming, multiblock structured grid with moving boundaries, collocated variable arrangement with momentum equations resolved along cartesian directions, second order accurate spatial and temporal discretisation schemes for the convective fluxes and a pressurevelocity solution strategy. Effect of turbulence is simulated using one of the two different approaches. In the Unsteady Reynolds Averaged (URANS) approach coupled to appropriate eddy viscosity based turbulence models, the Navier Stokes (NS) equations, averaged over the whole range of turbulent length scales of the flow, are solved numerically. On the other hand, in the Large Eddy Simulation (LES) approach, the model filtered 3D NS equations are directly solved for the flow variables to resolve the large scale turbulent motions whereas the transport processes at the fine subgrid scale level only are simulated using simple algebraic turbulence model. The capabilities and limitations of both the cost-effective URANS approach and the relatively expensive but rich in physics LES approach have been demonstrated for few application problems of engineering interest.

Improved simplified and highly stable lattice Boltzmann methods for incompressible flows

In this work, improved simplified and highly stable lattice Boltzmann methods (SHSLBMs) are developed for incompressible flows. The SHSLBM is a newly developed scheme within the lattice Boltzmann method (LBM) framework, which utilizes the fractional step technology to resolve the governing equations recovered from lattice Boltzmann equation (LBE) and reconstructs the equations in the Lattice Boltzmann frame. By this treatment, the SHSLBM directly tracks the macroscopic variables in the evolution process rather than the distribution functions of each grid node, which greatly saves virtual memories and simplifies the implementation of physical boundary conditions. However, the Chapman–Enskog expansion analysis reveals that the SHSLBM recover the weakly compressible Navier–Stokes equations with the low Mach number assumption. Therefore, the original SHSLBM can be regarded as an artificial compressible method and may cause some undesired errors. By modifying the evolution equation for the density distribution function, the improved SHSLBMs can eliminate the compressible effects. The incompressible SHSLBMs are compared with the original SHSLBM in terms of accuracy and stability by simulating several two-dimensional steady and unsteady incompressible flow problems, and the results demonstrate that the present SHSLBMs ensure the second order of accuracy and can reduce the compressible effects efficiently, especially for the incompressible flows with large pressure gradients. We then extended the present SHSLBMs to study the more complicated two-dimensional lid-driven flow and found that the present results are in good agreement with available benchmark results.

An implicit dual-time stepping high-order nodal discontinuous Galerkin method for solving incompressible flows on triangle elements

In this work, a high-order nodal discontinuous Galerkin method (NDGM) is developed and assessed for the simulation of 2D incompressible flows on triangle elements. The governing equations are the 2D incompressible Navier–Stokes equations with the artificial compressibility method. The discretization of the spatial derivatives in the resulting system of equations is made by the NDGM and the time integration is performed by applying the implicit dual-time stepping method. Three numerical fluxes, namely, the local Lax–Friedrich, Roe and AUSM+-up are formulated and applied to assess and compare their accuracy and performance in the simulation of incompressible flows using the NDGM. Several steady and unsteady incompressible flow problems are simulated to examine the accuracy and robustness of the proposed solution methodology and they are the Kovasznay, backward facing step, NACA0012 airfoil, circular cylinder and two side-by-side circular cylinders. Indications are that the NDGM applied for solving the incompressible Navier–Stokes equations with the artificial compressibility approach and the implicit dual-time stepping method is accurate and robust for the simulation of steady and unsteady incompressible flow problems.

Stabilized dimensional factorization preconditioner for solving incompressible Navier-Stokes equations

In this paper, we propose a stabilized dimensional factorization (SDF) preconditioner for saddle point problems arising from the discretization of Navier-Stokes equations. The idea is based on regularization, block factorization and selective approximation. The spectral properties of the preconditioned matrix are analyzed in details. Based on the analysis, we prescribe a reasonable choice of the regularization matrix W in the preconditioner. By using the connection with the RDF preconditioner, we determine the relaxation parameter α for the problems discretized by uniform grids and stretched grids, respectively. Finally, numerical experiments on the finite element discretizations of both steady and unsteady incompressible flow problems show that the SDF preconditioner is more efficient and robust than the RDF preconditioner, which has been illustrated very competitive with some existing preconditioners.

A high‐order discontinuous Galerkin method for the incompressible Navier‐Stokes equations on arbitrary grids

SummaryA high‐order discontinuous Galerkin (DG) method is proposed in this work for solving the two‐dimensional steady and unsteady incompressible Navier‐Stokes (INS) equations written in conservative form on arbitrary grids. In order to construct the interface inviscid fluxes both in the continuity and in the momentum equations, an artificial compressibility term has been added to the continuity equation for relaxing the incompressibility constraint. Then, as the hyperbolic nature of the INS equations has been recovered, the local Lax‐Friedrichs (LLF) flux, which was previously developed in the context of hyperbolic conservation laws, is applied to discretize the inviscid term. Unlike the traditional artificial compressibility method, in this work, the artificial compressibility is introduced only for the construction of the inviscid numerical fluxes; therefore, a consistent discretization of the INS equations is obtained, irrespective of the amount of artificial compressibility used. What is more, as the LLF flux can be obtained directly and straightforward, no numerical iteration for solving an exact Riemann problem is entailed in our method. The viscous term is discretized by the direct DG method, which was developed based on the weak formulation of the scalar diffusion problems on structured grids. The performance and the accuracy of the method are demonstrated by computing a number of benchmark test cases, including both steady and unsteady incompressible flow problems. Due to its simplicity in implementation, our method provides an attractive alternative for solving the INS equations on arbitrary grids.

A Comparison between the Reduced Differential Transform Method and Perturbation-Iteration Algorithm for Solving Two-Dimensional Unsteady Incompressible Navier-Stokes Equations

In this work, approximate analytical solutions to the lid-driven square cavity flow problem, which satisfied two-dimensional unsteady incompressible Navier-Stokes equations, are presented using the kinetically reduced local Navier-Stokes equations. Reduced differential transform method and perturbation-iteration algorithm are applied to solve this problem. The convergence analysis was discussed for both methods. The numerical results of both methods are given at some Reynolds numbers and low Mach numbers, and compared with results of earlier studies in the review of the literatures. These two methods are easy and fast to implement, and the results are close to each other and other numerical results, so it can be said that these methods are useful in finding approximate analytical solutions to the unsteady incompressible flow problems at low Mach numbers.

A truly incompressible smoothed particle hydrodynamics based on artificial compressibility method

In the present study, a truly incompressible smoothed particle hydrodynamics based on the artificial compressibility method for simulating steady and unsteady incompressible flows is proposed and assessed. The incompressible Navier–Stokes equations in the primitive variables formulation using the artificial compressibility method proposed by Chorin in the Eulerian reference frame are written in a Lagrangian reference frame to provide an appropriate incompressible SPH algorithm. The proposed SPH formulation implemented here is based on an implicit dual-time stepping scheme to be capable of time-accurate analysis of unsteady flows. The advantage of the Artificial Compressibility-based Incompressible SPH (ACISPH) method proposed over the Weakly Compressible SPH (WCSPH) and the pseudo-compressibility SPH methods is that the ACISPH formulation is a truly incompressible SPH algorithm and it does not involve any approximate enforcement of the incompressibility condition that usually causes to use a large magnitude speed of sound implying a small time step in the computations. The approximate enforcement of the incompressibility condition used in the WCSPH method also causes spurious oscillations in the density and pressure fields which is not the case for the ACISPH formulation proposed. Unlike the projection SPH algorithm and the moving-particle semi-implicit (MPS) method formulation, the ACISPH formulation presented herein does not involve the iterative solution of a Poisson equation for the pressure field. The accuracy and robustness of the proposed incompressible SPH (ACISPH) are demonstrated by solving different incompressible flow problems. A sensitivity study is also conducted to evaluate the effects of particle resolution and the value of artificial compressibility parameter on the accuracy and convergence rate of the solution. To validate the results, different test cases are considered herein that are incompressible flows in a 2-D simple channel, a 2-D locally expanded channel, a 2-D cavity, the unsteady Couette flow with pressure gradient, the Taylor vortex problem and a dam break problem with dry bed. Results obtained for these cases are in good agreement with the available analytical and numerical results. The study shows the artificial compressibility-based ISPH (ACISPH) method proposed is accurate and robust for simulating steady and unsteady incompressible flow problems.

A high-order solver for unsteady incompressible Navier–Stokes equations using the flux reconstruction method on unstructured grids with implicit dual time stepping

We report development of a high-order compact flux reconstruction method for solving unsteady incompressible flow on unstructured grids with implicit dual time stepping. The method falls under the class of methods now referred to as flux reconstruction/correction procedure via reconstruction. The governing equations employ Chorin's classic artificial compressibility formulation with dual time stepping to solve unsteady flow problems. An implicit non-linear lower–upper symmetric Gauss–Seidel scheme with backward Euler discretization is used to efficiently march the solution in pseudo time, while a second-order backward Euler discretization is used to march in physical time. We verify and validate implementation of the high-order method coupled with our implicit time stepping scheme using both steady and unsteady incompressible flow problems. The current implicit time stepping scheme is proven effective in satisfying the divergence-free constraint on the velocity field in the artificial compressibility formulation within the context of the high-order flux reconstruction method. This compact high-order method is very suitable for parallel computing and can easily be extended to moving and deforming grids.

A high‐order compact finite‐difference lattice Boltzmann method for simulation of steady and unsteady incompressible flows

SUMMARYA high‐order compact finite‐difference lattice Boltzmann method (CFDLBM) is proposed and applied to accurately compute steady and unsteady incompressible flows. Herein, the spatial derivatives in the lattice Boltzmann equation are discretized by using the fourth‐order compact FD scheme, and the temporal term is discretized with the fourth‐order Runge–Kutta scheme to provide an accurate and efficient incompressible flow solver. A high‐order spectral‐type low‐pass compact filter is used to stabilize the numerical solution. An iterative initialization procedure is presented and applied to generate consistent initial conditions for the simulation of unsteady flows. A sensitivity study is also conducted to evaluate the effects of grid size, filtering, and procedure of boundary conditions implementation on accuracy and convergence rate of the solution. The accuracy and efficiency of the proposed solution procedure based on the CFDLBM method are also examined by comparison with the classical LBM for different flow conditions. Two test cases considered herein for validating the results of the incompressible steady flows are a two‐dimensional (2‐D) backward‐facing step and a 2‐D cavity at different Reynolds numbers. Results of these steady solutions computed by the CFDLBM are thoroughly compared with those of a compact FD Navier–Stokes flow solver. Three other test cases, namely, a 2‐D Couette flow, the Taylor's vortex problem, and the doubly periodic shear layers, are simulated to investigate the accuracy of the proposed scheme in solving unsteady incompressible flows. Results obtained for these test cases are in good agreement with the analytical solutions and also with the available numerical and experimental results. The study shows that the present solution methodology is robust, efficient, and accurate for solving steady and unsteady incompressible flow problems even at high Reynolds numbers. Copyright © 2014 John Wiley & Sons, Ltd.

Implementation of a high-order compact finite-difference lattice Boltzmann method in generalized curvilinear coordinates

In this work, the implementation of a high-order compact finite-difference lattice Boltzmann method (CFDLBM) is performed in the generalized curvilinear coordinates to improve the computational efficiency of the solution algorithm to handle curved geometries with non-uniform grids. The incompressible form of the discrete Boltzmann equation with the Bhatnagar–Gross–Krook (BGK) approximation with the pressure as the independent dynamic variable is transformed into the generalized curvilinear coordinates. Herein, the spatial derivatives in the resulting lattice Boltzmann (LB) equation in the computational plane are discretized by using the fourth-order compact finite-difference scheme and the temporal term is discretized with the fourth-order Runge–Kutta scheme to provide an accurate and efficient incompressible flow solver. A high-order spectral-type low-pass compact filter is used to regularize the numerical solution and remove spurious waves generated by boundary conditions, flow non-linearities and grid non-uniformity. All boundary conditions are implemented based on the solution of governing equations in the generalized curvilinear coordinates. The accuracy and efficiency of the solution methodology presented are demonstrated by computing different benchmark steady and unsteady incompressible flow problems. A sensitivity study is also conducted to evaluate the effects of grid size and filtering on the accuracy and convergence rate of the solution. Four test cases considered herein for validating the present computations and demonstrating the accuracy and robustness of the solution algorithm are: unsteady Couette flow and steady flow in a 2-D cavity with non-uniform grid and steady and unsteady flows over a circular cylinder and the NACA0012 hydrofoil at different flow conditions. Results obtained for the above test cases are in good agreement with the existing numerical and experimental results. The study shows the present solution methodology based on the implementation of the high-order compact finite-difference Lattice Boltzmann method (CFDLBM) in the generalized curvilinear coordinates is robust, efficient and accurate for solving steady and unsteady incompressible flows over practical geometries.

A Relaxed Splitting Preconditioner for the Incompressible Navier‐Stokes Equations

A relaxed splitting preconditioner based on matrix splitting is introduced in this paper for linear systems of saddle point problem arising from numerical solution of the incompressible Navier‐Stokes equations. Spectral analysis of the preconditioned matrix is presented, and numerical experiments are carried out to illustrate the convergence behavior of the preconditioner for solving both steady and unsteady incompressible flow problems.

A Relaxed Dimensional Factorization preconditioner for the incompressible Navier–Stokes equations

In this paper we introduce a Relaxed Dimensional Factorization (RDF) preconditioner for saddle point problems. Properties of the preconditioned matrix are analyzed and compared with those of the closely related Dimensional Splitting (DS) preconditioner recently introduced by Benzi and Guo [7]. Numerical results for a variety of finite element discretizations of both steady and unsteady incompressible flow problems indicate very good behavior of the RDF preconditioner with respect to both mesh size and viscosity.

English

The present study focuses on the recent development of an implicit pressure-based finite volume algorithm for numerical solution of Reynolds averaged Navier-Stokes equations (RANS) in an inertial frame of reference for the prediction of unsteady incompressible flow problems. The algorithm uses boundary-conforming, multiblock structured grid with moving boundaries, collocated variable arrangement with momentum equations resolved along cartesian directions, second-order accurate spatial and temporal discretisation schemes for the convective fluxes and a pressure-velocity solution strategy. Effect of turbulence was simulated using appropriate linear eddy viscosity-based turbulence models. The capabilities and limitations of the cost-effective unsteady Reynolds averaged Navier-Stokes (URANS) approach has been demonstrated for few application problems of engineering interest. Defence Science Journal, 2010, 60(6), pp.614-627 , DOI:http://dx.doi.org/10.14429/dsj.60.603

Evaluation of a bounded high order upwind scheme for 3D incompressible free surface flow computations

In the context of normalized variable formulation (NVF) of Leonard and total variation diminishing (TVD) constraints of Harten, this paper presents an extension of a previous work by the authors for solving unsteady incompressible flow problems. The main contributions of the paper are threefold. First, it presents the results of the development and implementation of a bounded high order upwind adaptative QUICKEST scheme in the 3D robust code (Freeflow), for the numerical solution of the full incompressible Navier–Stokes equations. Second, it reports numerical simulation results for 1D shock tube problem, 2D impinging jet and 2D/3D broken dam flows. Furthermore, these results are compared with existing analytical and experimental data. And third, it presents the application of the numerical method for solving 3D free surface flow problems.

Computation for Unsteady Incompressible Flow on Unstructured Grid.

The artificial compressibility algorithm for incompressible flows is applied to an unstructured grid method. It is based on a finite volume cell-vertex upwind technique and uses a time-marching procedure to efficiently solve the continuity equations as well as the momentum equations. The Lower-Upper Symmetric Gauss-Seidel(LU-SGS)approximate factorization scheme is implemented with the Navier-Stokes solver on arbitrary three-dimensional unstructured grid. Numerical simulation is performed for steady and unsteady incompressible flow problems, 2-D wake of cylinder, 3-D stenosed tube and branched tube.

An Exponentially Fitted Finite Volume Method for the Numerical Solution of 2D Unsteady Incompressible Flow Problems

In this paper we develop and test an exponentially fitted finite volume method for the numerical solution of the Navier-Stokes equations describing 2D incompressible flows. The method is based on an unstructured Delaunay mesh and its dual Dirichlet tessellation, combined with a locally constant approximation to the flux. This yields a piecewise exponential approximation to the exact solution. Numerical tests are presented for a linear advection-diffusion problem with boundary layers. The method is then applied to the driven cavity problem with Reynolds numbers up to 10 4. The numerical results indicate that the method is robust for a wide range of values of the Reynolds number. In the case Re = 10 4 unsteady solutions are captured if the mesh is sufficiently fine.

Numerical simulation of flows past periodic arrays of cylinders

We present a detailed numerical investigation of three unsteady incompressible flow problems involving periodic arrays of staggered cylinders. The first problem is a uniperiodic flow with two cylinders in each cell of periodicity. The second problem is a biperiodic flow with two cylinders in each cell, and the last problem is a uniperiodic flow with ten cylinders. Both uniperiodic flows are periodic in the direction perpendicular to the main flow direction. In all three cases, the Reynolds number based on the cylinder diameter is 100, and initially the flow field has local symmetries with respect to the axes of the cylinders parallel to the main flow direction. Later on, these symmetries break, vortex shedding is initiated, and gradually the scale of the shedding increases until a temporally periodic flow field is reached. We furnish extensive flow data, including the vorticity and stream function fields at various instants during the temporal evolution of the flow field, time histories of the drag and lift coefficients, Strouhal number, initial and mean drag coefficients, amplitude of the drag and lift coefficient oscillations, and the phase relationships between the drag and lift oscillations associated with each cylinder. Our data confirms that, at this Reynolds number, there are no stable steady-state solutions with local symmetries. Of course, one can obtain such unphysical solutions by assuming symmetry conditions along the axes of the cylinders parallel to the main flow direction and taking half of the computational domain needed normally. In such cases, the “steady-state” flow fields obtained would be identical to the flow fields observed at the initial stages of our computations. However, we show that such flow fields do not represent the temporally periodic flow fields even in a time-averaged sense, because, in all three cases, the initial drag coefficients are different from the mean drag coefficients. Therefore, we conclude that stability studies involving periodic arrays of cylinders should be carried out, as it is done in this work, with the true implementation of the spatial periodicity.