High Reynolds Number Incompressible Flow Research Articles

Efficient SVV stabilized triangular spectral element methods for incompressible flows of high Reynolds numbers

In this paper, we propose a spectral vanishing viscosity method for the triangular spectral element computation of high Reynolds number incompressible flows. This can be regarded as an extension of a similar stabilization technique for the standard spectral element method. The difficulty of this extension lies in the fact that a suitable definition of spectral vanishing viscosity operator in non-structured elements does not exist, and it is not clear that if a suitably defined spectral vanishing viscosity provides desirable dissipation for the artificially accumulated energy. The main contribution of the paper includes: 1) a well-defined spectral vanishing viscosity operator is proposed for non-standard spectral element methods for the Navier-Stokes equations based on triangular or tetrahedron partitions; 2) an evaluation technique is introduced to efficiently implement the stabilization term without extra computational cost; 3) the accuracy and efficiency of the proposed method is carefully examined through several numerical examples. Our numerical results show that the proposed method not only preserves the exponential convergence, but also produces improved accuracy when applied to the unsteady Navier-Stokes equations having smooth solutions. Especially, the stabilized triangular spectral element method efficiently stabilizes the simulation of high Reynolds incompressible flows.

On the convergence order of the finite element error in the kinetic energy for high Reynolds number incompressible flows

The kinetic energy of a flow is proportional to the square of the L2(Ω) norm of the velocity. Given a sufficient regular velocity field and a velocity finite element space with polynomials of degree r, then the best approximation error in L2(Ω) is of order r+1. In this survey, the available finite element error analysis for the velocity error in L∞(0,T;L2(Ω)) is reviewed, where T is a final time. Since in practice the case of small viscosity coefficients or dominant convection is of particular interest, which may result in turbulent flows, robust error estimates are considered, i.e., estimates where the constant in the error bound does not depend on inverse powers of the viscosity coefficient. Methods for which robust estimates can be derived enable stable flow simulations for small viscosity coefficients on comparatively coarse grids, which is often the situation encountered in practice. To introduce stabilization techniques for the convection-dominated regime and tools used in the error analysis, evolutionary linear convection–diffusion equations are studied at the beginning. The main part of this survey considers robust finite element methods for the incompressible Navier–Stokes equations of order r−1, r, and r+1/2 for the velocity error in L∞(0,T;L2(Ω)). All these methods are discussed in detail. In particular, a sketch of the proof for the error bound is given that explains the estimate of important terms which determine finally the order of convergence. Among them, there are methods for inf–sup stable pairs of finite element spaces as well as for pressure-stabilized discretizations. Numerical studies support the analytic results for several of these methods. In addition, methods are surveyed that behave in a robust way but for which only a non-robust error analysis is available. The conclusion of this survey is that the problem of whether or not there is a robust method with optimal convergence order for the kinetic energy is still open.

An efficient parallel algorithm for DNS of buoyancy-driven turbulent flows

In this paper, an explicit low-storage simplified M — stage Runge-Kutta (SRK) scheme for high Reynolds-number incompressible flows is presented. In the SRK scheme, the Poisson equation is solved only once in the final substage of each time step. By taking advantage of the SRK scheme and the advanced hybrid MPI+MPI model, we have developed an efficient parallel solver for buoyancy-driven turbulent flow. The spatial and temporal accuracies of the solver are validated with Taylor-Green vortex flow. Both the RK and SRK schemes are implemented for the simulation of turbulent Rayleigh-Benard convection as well as Rayleigh-Taylor flow. The results show that the SRK scheme can save approximately 20% of the computation time.

A new high-order particle method for solving high Reynolds number incompressible flows

In this study, a new high-order particle method is proposed to solve the incompressible Navier–Stokes equations. The proposed method combines the advantages of particle and mesh methods to approximate the total and the spatial derivative terms under the Lagrangian and the Eulerian frameworks. Our aim is to avoid convective instability and increase solution accuracy at the same time. Data transfer from Lagrangian particles to Eulerian grids is realized by moving least squares interpolation. In contrast to the previously proposed method, there is no need to interpolate diffusion terms from Eulerian grids to Lagrangian particles. Therefore, the accuracy of the present solution will not be deteriorated by interpolation error. Additionally, no extra work is required to manage particles for searching procedure. Because no convection term needs to be discretized by upwinding schemes, false diffusion and dispersion errors will not be introduced, thereby increasing the solution accuracy. To verify the proposed particle method, several benchmark problems are solved to show that the present simulation is more stable, accurate, and efficient. The proposed particle method renders fourth- and second-order accurate solutions in space for velocity and pressure, respectively.

A semi-Lagrangian Vortex-In-Cell method and its application to high-Re lid-driven cavity flow

PurposeThe purpose of this paper is to develop a Vortex-In-Cell (VIC) method with the semi-Lagrangian scheme and apply it to the high-Re lid-driven cavity flow.Design/methodology/approachThe VIC method is developed for simulating high Reynolds number incompressible flow. A semi-Lagrangian scheme is incorporated in the convection term to produce unconditional stability, which gets rid of the constraint of the convection Courant-Friedrichs-Lewy (CFL) condition; the adaptive time step is used to maintain the numerical stability of the diffusion term; and the velocity boundary condition is readily converted to the vorticity formulation to suit discontinuous boundary treatment. The VIC simulation results are compared with those produced by other gird methods reported in open literature studies.FindingsThe lid-driven cavity flow is simulated from Re = 100 to 100,000. Similar vortex birth mechanisms are exhibited though, but distinct flow characteristics are revealed. At Re = 100 to 7,500, the cavity flow is confirmed steady. At Re = 10,000, 15,000 and 20,000, the cavity flow is periodical with a primary vortex held spatially at the center. In particular, at Re = 100,000 highly turbulent characteristics is first revealed and an analogous primary vortex is formed but in motion rather than stationary, which is caused by the considerable flow separation at all the boundaries.Originality/valueIn the lid-driven cavity, the flow becomes extremely complex and highly turbulent at Re = 100,000, and the analogous primary vortex structure is observed. Boundary layer separation is observed at all walls, producing small vortices and causing the displacement of the analogous primary vortex. Such a finding original and has not yet been reported by other investigators. It may provide a basis for conducting in-depth studies of the lid-driven cavity flow.

A hybrid collocation/Galerkin scheme for convective heat transfer problems with stochastic boundary conditions

AbstractWe present a numerical method to study convective heat transfer in a high Reynolds number incompressible flow around a cylinder subject to uncertain boundary conditions. We exploit the one‐way coupling of the energy and momentum transport to derive a semi‐intrusive uncertainty propagation scheme, which combines Galerkin and collocation approaches for computing approximate statistics of the stochastic temperature field. The hybrid scheme converges rapidly and dramatically reduces the overall computational cost compared with the conventional uncertainty propagation schemes. Copyright © 2009 John Wiley & Sons, Ltd.

Near-Wake Turbulence Properties in the High Reynolds Number Incompressible Flow Around a Circular Cylinder Measured by Two- and Three-Component PIV

The main objective of the present experimental study is to analyse the turbulence properties in unsteady flows around bluff body wakes and to provide a database for improvement and validation of turbulence models, concerning the present class of non-equilibrium flows. The flow around a circular cylinder with a low aspect ratio ( $L/D=4.8$ ) and a high blockage coefficient ( $D/H=0.208$ ) is investigated. This confined environment is used in order to allow direct comparisons with realisable 3D Navier–Stokes computations avoiding ‘infinite’ conditions. The flow is investigated in the critical regime at Reynolds number 140,000. A cartography of the velocity fields in the near wake of the cylinder is obtained by PIV and Stereoscopic PIV techniques. Statistical means and phase-averaged quantities are determined. Furthermore, POD analysis is performed on the data set in order to extract coherent structures of the flow and to compare the results with those obtained by the conditional sampling technique. The Reynolds stresses, the strain-rate and vorticity fields as well as the turbulence production terms are determined.

Stabilized spectral element computations of high Reynolds number incompressible flows

The possibility of using the spectral vanishing viscosity method for the spectral element computation of high Reynolds number incompressible flows is investigated. An exponentially accurate stabilized formulation is proposed and then applied to the computation of the 2D wake of a cylinder. Such a formulation can be easily implemented in existing spectral element solvers, since only modifying the computation of the viscous term while preserving the symmetry of the corresponding bilinear form.

Simulation of rotating stall by the vortex method

The vortex method, coupled to a boundary-layer solver, is applied to the numerical simulation of high Reynolds number incompressible flow in two-dimensional cascades. Periodic conditions are imposed along the plane of the cascade, with several blades per period. Good agreement is found with two finite-difference methods for a single-blade case. When a staggered cascade is treated with five independent blades, the simulation predicts rotating stall, for a range of angles of attack and stagger, and the essential features of the flow are correct. The stall cell steadily propagates along the cascade. The sensitivity of this phenomenon to two parameters is studied, and the stall boundary is found. Quantitative results and visualizations are presented.