Large Eddy Simulation of a Compressible Boundary Layer

A large eddy simulation of a compressible boundary layer is performed. To generate an appropriate inflow distribution the rescaling technique for compressible flows is discribed. In this method Morkovin's hypothesis in which the total temperature fluctuations are neglected compared with the static temperature fluctuations is applied to rescale and generate the temperature profile at inlet. This new technique is used for various large eddy simulations of subsonic and supersonic three‐dimensional boundary layers of a flat plate. Simulation results for the time‐averaged mean flow and Reynolds stresses are compared with numerical and analytical data to demonstrate the high quality of the method.     

Large eddy simulations of the turbulent flow between a rotating and a stationary disk

Large eddy simulations of the flow between a rotating and a stationary disk have been performed using a dynamic and a mixed dynamic subgrid-scale model. The simulations were compared to direct numerical simulation results. The mixed dynamic model gave better overall predictions than the dynamic model. Modifications of the near-wall structures caused by the mean flow three-dimensionality were also investigated. Conditional averages near strong stress-producing events led to the same conclusions regarding these modifications as studies of the flow generated by direct numerical simulation, namely a distinct asymmetry of the vortices producing sweeps and ejections.     

A RANS 3D model with unbounded eddy viscosities

We consider the Reynolds Averaged Navier–Stokes (RANS) model of order one (u,p,k)$(\mathbf{u},p,k)$ set in R³${\mathbb{R}}^3$ which couples the Stokes Problem to the equation for the turbulent kinetic energy by k-dependent eddy viscosities in both equations and a quadratic term in the k  -equation. We study the case where the velocity and the pressure satisfy periodic boundary conditions while the turbulent kinetic energy is defined on a cell with Dirichlet boundary conditions. The corresponding eddy viscosity in the fluid equation is extended to R³${\mathbb{R}}^3$ by periodicity. Our contribution is to prove that this system has a solution when the eddy viscosities are nondecreasing, smooth, unbounded functions of k, and the eddy viscosity in the fluid equation is a concave function.     

Investigation of physiological pulsatile flow in a model arterial stenosis using large-eddy and direct numerical simulations

Physiological pulsatile flow in a 3D model of arterial stenosis is investigated by using large eddy simulation (LES) technique. The computational domain chosen is a simple channel with a biological type stenosis formed eccentrically on the top wall. The physiological pulsation is generated at the inlet using the first harmonic of the Fourier series of pressure pulse. In LES, the large scale flows are resolved fully while the unresolved subgrid scale (SGS) motions are modelled using a localized dynamic model. Due to the narrowing of artery the pulsatile flow becomes transition-to-turbulent in the downstream region of the stenosis, where a high level of turbulent fluctuations is achieved, and some detailed information about the nature of these fluctuations are revealed through the investigation of the turbulent energy spectra. Transition-to-turbulent of the pulsatile flow in the post stenosis is examined through the various numerical results such as velocity, streamlines, velocity vectors, vortices, wall pressure and shear stresses, turbulent kinetic energy, and pressure gradient. A comparison of the LES results with the coarse DNS are given for the Reynolds number of 2000 in terms of the mean pressure, wall shear stress as well as the turbulent characteristics. The results show that the shear stress at the upper wall is low just prior to the centre of the stenosis, while it is maximum in the throat of the stenosis. But, at the immediate post stenotic region, the wall shear stress takes the oscillating form which is quite harmful to the blood cells and vessels. In addition, the pressure drops at the throat of the stenosis where the re-circulated flow region is created due to the adverse pressure gradient. The maximum turbulent kinetic energy is located at the post stenosis with the presence of the inertial sub-range region of slope −5/3.     

Subgrid scale eddy viscosity finite element method on optimal control of system governed by unsteady Oseen equations

In this paper we focus on numerical analysis of finite element methods with stabilizations for the optimal control of system governed by unsteady Oseen equations. Using continuous equal-order finite elements for both velocities and pressure, two fully discrete schemes are proposed. Convective effects and pressure are stabilized by adding a subgrid scale eddy viscosity term and a pressure stabilized term. Convergence of the approximate solution is proved. A-Priori error estimates are obtained uniformly with Reynolds number, especially the \(L^2\) -error estimates of numerical solution are independent of Reynolds number. The numerical experiments are shown to be consistent with our theoretical analysis.     

Iterative versus direct parallel substructuring methods in semiconductor device modelling

The numerical simulation of semiconductor devices is extremely demanding in term of computational time because it involves complex embedded numerical schemes. At the kernel of these schemes is the solution of very ill‐conditioned large linear systems. In this paper, we present the various ingredients of some hybrid iterative schemes that play a central role in the robustness of these solvers when they are embedded in other numerical procedures. On a set of two‐dimensional unstructured mixed finite element problems representative of semiconductor simulation, we perform a fair and detailed comparison between parallel iterative and direct linear solution techniques. We show that iterative solvers can be robust enough to solve the very challenging linear systems that arise in those simulations. Copyright © 2004 John Wiley & Sons, Ltd.     

A numerical study on statistical temporal scales in inertia particle dispersion

The statistical temporal scales involved in inertia particle dispersion are analyzed numerically. The numerical method of large eddy simulation, solving a filtered Navier-Stokes equation, is utilized to calculate fully developed turbulent channel flows with Reynolds numbers of 180 and 640, and the particle Lagrangian trajectory method is employed to track inertia particles released into the flow fields. The Lagrangian and Eulerian temporal scales are obtained statistically for fluid tracer particles and three different inertia particles with Stokes numbers of 1, 10 and 100. The Eulerian temporal scales, decreasing with the velocity of advection from the wall to the channel central plane, are smaller than the Lagrangian ones. The Lagrangian temporal scales of inertia particles increase with the particle Stokes number. The Lagrangian temporal scales of the fluid phase ‘seen’ by inertia particles are separate from those of the fluid phase, where inertia particles travel in turbulent vortices, due to the particle inertia and particle trajectory crossing effects. The effects of the Reynolds number on the integral temporal scales are also discussed. The results are worthy of use in examining and developing engineering prediction models of particle dispersion.     

Smoke flow phenomena and turbulence characteristics of tunnel fires

Gravity currents are similar in behavior with smoke flows. This work aims to provide evidence justifying the use of gravity current approach to model smoke flows downstream of the fire source. The turbulence solver available in almost all commercial CFD codes solves RANS for the flow field. To find out how well the nature of smoke flow be accurately modeled using RANS that is widely used for incompressible flows. The feasibility of using both Reynolds- and Favre-averaging schemes was numerically compared and examined in this paper. In this work, numerical simulations of a fire occurred in a 400-m longitudinally ventilated tunnel have been successfully performed using FDS version 4. Large eddy simulation is employed in this study. Although the ranges of fire size and ventilation velocity vary respectively from 0 MW to 100 MW and 0 m/s to 10 m/s, this paper focuses on the general flow and temperature fields and the turbulence characteristics. Furthermore, the turbulence kinetic energy levels of the flow in the tunnel at several locations were investigated. Since the flow field is generally induced by mechanical ventilation and combustion, the main contribution to the turbulence kinetic energy comes from its longitudinal, vertical, or their combination.     

One-step Taylor–Galerkin methods for convection–diffusion problems

Third and fourth order Taylor–Galerkin schemes have shown to be efficient finite element schemes for the numerical simulation of time-dependent convective transport problems. By contrast, the application of higher-order Taylor–Galerkin schemes to mixed problems describing transient transport by both convection and diffusion appears to be much more difficult. In this paper we develop two new Taylor–Galerkin schemes maintaining the accuracy properties and improving the stability restrictions in convection–diffusion. We also present an efficient algorithm for solving the resulting system of the finite element method. Finally we present two numerical simulations that confirm the properties of the methods.     

Discretization effects on approximate deconvolution modeling for LES

We study the applicability of low‐order schemes with the approximate deconvolution model (ADM) for large‐eddy simulation. As a test case compressible decaying isotropic turbulence is considered. Results obtained with low‐order finite difference schemes and a pseudospectral scheme are compared with filtered well‐resolved direct numerical simulation (DNS) data. It is found that even for low‐order schemes very good results can be obtained if the cutoff wavenumber of the filter is adjusted to the modified wavenumber of the differentiation scheme.     

Navier-Stokes equations with several independent pressure laws and explicit predictor-corrector schemes

This work is concerned with the numerical capture of stiff viscous shock solutions of Navier-Stokes equations for complex compressible materials, in the regime of large Reynolds numbers. After [2] and [6], a relevant numerical capture is known to require the satisfaction of an extended set of non classical Rankine-Hugoniot conditions due to the non conservation form of the governing PDE model. Here, we show how to enforce their validity at the discrete level without the need for solving local non linear algebraic problems. Non linearities are bypassed when introducing new averaging techniques which are proved to satisfy all the desirable stability properties when invoking suitable approximate Riemann solutions. A relaxation procedure is proposed to that purpose with the benefit of a fairly simple overall numerical method.     

Dynamical characteristics of decaying Lamb couples

A multiple-scale adiabatic asymptotic theory is developed to describe the dissipation of the solitary Lamb couple or modon solutions of the two-dimensional Navier-Stokes equations. The transport equations describing the evolution of the Lamb couple are obtained as solvability conditions for a direct asymptotic expansion assuming a relatively large but finite Reynolds number and are equivalent to globally-integrated leading-order enstrophy and energy balances. The asymptotic theory predicts that the spectral or spatial characteristics of the decaying Lamb couple are temporally invariant and that there is a simple exponential decay in the amplitude and translation speed. We compare the predictions of the theory with a high-resolution numerical simulation. The global and local predictions of the theory and the results of the numerical simulation are in very good agreement. As well, we present a time-series of vorticity-stream function scatter diagrams as derived from the numerical simulation to show that thenon-analytic linear vorticity-stream function relationship is being continuously maintained during the perturbed evolution of the Lamb couple.     

Numerical simulation of airfoil vibrations induced by turbulent flow

The subject of this paper is the numerical simulation of the interaction between two-dimensional incompressible viscous flow and a vibrating airfoil. A solid elastically supported airfoil with two degrees of freedom, which can rotate around the elastic axis and oscillate in the vertical direction, is considered. The numerical simulation consists of the stabilized finite element solution of the Reynolds averaged Navier–Stokes equations with algebraic models of turbulence, coupled with the system of ordinary differential equations describing the airfoil motion. Since the computational domain is time dependent and the grid is moving, the Arbitrary Lagrangian–Eulerian (ALE) method is used. The developed method was applied to the simulation of flow-induced airfoil vibrations.     

Direct numerical simulation of bi-disperse particle-laden gravity currents in the channel configuration

We present a numerical investigation of bi-disperse particle-laden gravity currents in the lock-exchange configuration. Previous results, based on numerical simulation and laboratory experiments, are used to establish comparisons. Our discussion focuses on explaining how the presence of more than one particle diameter influences the main features of the flow, such as deposit profile, the evolution of the front location and suspended mass. We develop the complete energy budget equation for bi-disperse flows. A set of two and three-dimensional direct numerical simulations (DNS), with different initial compositions of coarse and fine particles, are carried out for Reynolds number equal to 4000. Such simulations show that the energy terms are strongly affected by varying the initial particle fractions. The addition of a small amount of fine particles into a current predominantly composed of coarse particles increases its run-out distance. In particular, it is shown that higher amounts of coarse particles have a dumping effect on the current development. Comparisons show that the two-dimensional simulation does not reproduce the intense turbulence generated in 3D cases accurately, which results in a significant difference in the suspended mass, front position as well as the dissipation term due to the advective motion.     

MHD turbulence in a channel with spanwise field

We study turbulent channel flow of an electrically conducting liquid with a homogeneous magnetic field imposed in the spanwise direction. The Lorentz force is modelled using the quasistatic approximation. Direct and large–eddy simulations are performed for hydrodynamic Reynolds numbers Re=10000 and Re=20000 and the Hartmann number varying in a wide range. The main effect of the magnetic field is the suppression of turbulent velocity fluctuations and momentum transfer in the wall–normal direction. Comparing the results from direct and large–edddy simulations we show that the dynamic Smagorinsky model accurately reproduces the flow transformation. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Semi-implicit Runge-Kutta schemes for the Navier-Stokes equations

The stationary Navier-Stokes equations are solved in 2D with semi-implicit Runge-Kutta schemes, where explicit time-integration in the streamwise direction is combined with implicit integration in the body-normal direction. For model problems stability restrictions and convergence properties are studied. Numerical experiments for the flow over a flat plate show that the number of iterations for the semi-implicit schemes is almost independent of the Reynolds number.     

Direct numerical simulation of turbulent flows in eccentric pipes

A numerical algorithm was developed for solving the incompressible Navier-Stokes equations in curvilinear orthogonal coordinates. The algorithm is based on a central-difference discretization in space and on a third-order accurate semi-implicit Runge-Kutta scheme for time integration. The discrete equations inherit some properties of the original differential equations, in particular, the neutrality of the convective terms and the pressure gradient in the kinetic energy production. The method was applied to the direct numerical simulation of turbulent flows between two eccentric cylinders. Numerical computations were performed at Re = 4000 (where the Reynolds number Re was defined in terms of the mean velocity and the hydraulic diameter). It was found that two types of flow develop depending on the geometric parameters. In the flow of one type, turbulent fluctuations were observed over the entire cross section of the pipe, including the narrowest gap, where the local Reynolds number was only about 500. The flow of the other type was divided into turbulent and laminar regions (in the wide and narrow parts of the gap, respectively).     

Application of finite element method in aeroelasticity

The subject of this paper is the numerical simulation of the interaction of two-dimensional incompressible viscous flow and a vibrating airfoil, which can rotate around the elastic axis and oscillate in the vertical direction. The numerical simulation consists of the finite element approximation of the Navier–Stokes equations coupled with the system of ordinary differential equations describing the airfoil motion. The arbitrary Lagrangian–Eulerian (ALE) formulation of the Navier–Stokes equations, stabilization the finite element discretization and coupling of both models is discussed. Moreover, the Reynolds averaged Navier–Stokes (RANS) system of equations together with the Spallart–Almaras turbulence model is also discussed. The computational results of aeroelastic calculations are presented and compared with the NASTRAN code solutions.     

Temporally regularized direct numerical simulation

Experience with fluid-flow simulation suggests that, in some instances, under-resolved direct numerical simulation (DNS), without a residual-stress model per se but with artificial damping of small scales to account for energy lost in the cascade from resolved to unresolved scales, may be as reliable as simulations based on more complex models of turbulence. One efficient and versatile manner to selectively damp under-resolved spatial scales is by a relaxation regularization, e.g. Stolz and Adams [S. Stolz, N.A. Adams, An approximate deconvolution procedure for large eddy simulation, Phys. Fluids II (1999) 1699-1701]. We consider the analogous approach based on time scales, time filtering and damping of under-resolved temporal features. The paper explores theoretical and practical aspects of temporally damped fluid-flow simulations. We prove existence of solutions to the resulting continuum model. We also establish the effect of the damping of under-resolved temporal features as the energy balance and dissipation and prove that the time fluctuations → 0 in a precise sense. The method is then demonstrated to obtain both steady-state and time-dependent coarse-grid solutions of the Navier-Stokes equations.     

Two Way Coupled Micro/Meso Scale Method for Wind Farms

Wind turbines extract energy from the approaching flow field resulting in reduced wind speeds, increased turbulence and a wake downstream of the wind turbine. The wake has a multitude of negative effects on downstream wind turbines. This includes reduced efficiency and increased unsteadiness resulting in vibrations and potentially in material fatigue. Moreover, the maintenance can increase compared to non-interfering wind turbines. The simulation of these effects is challenging. Computational fluid dynamics (CFD) simulations of these large and complex geometries requires exceedingly large computational resources. With present Reynolds Averaged Navier-Stokes (RANS) or Large Eddy Simulation (LES) based CFD methods it is virtually impossible to perform such simulations of the interaction between individual wind turbines in a complete wind turbine farm. Coupling to the mesoscale accounting for local weather situations becomes yet more challenging. This is due to the wide range of length and time scales that have to be considered for these simulations and therefore the tremendous computational power needed to perform such simulations. To investigate these effects we propose to combine ideas from existing methods, the Coarse-Grid-CFD (CGCFD) ( [1]) developed at the KIT and the meso-/ micro scale method developed at the University of Thessaloniki ( [2]). Goal of the proposed methodology is to provide a numerical method that allows to implement a wind farm in a meso-scale weather simulation which includes two-way coupling. Thus both the micro and the meso scale wind and energy production of wind farms can be addressed. This proposed multi scale coupling strategy can also be applied in two hierarchies reducing the numerical effort of the global approach yet more. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)