Spectral Element Code Research Articles

Analysing flow structures around a blade using spectral/hp method and HPIV

A still difficult, yet pressing task for blade manufacturers and turbine producers is the correct prediction of the effects of turbulent winds on the blade. Reynolds Averaged Numerical Simulations (RANS) are a limited tool for calculating the effects. For large eddy simulations (LES) boundary layer calculation are still difficult therefore the spectral element method seems to be an approach to improve numerical calculations of flow separation. The flow field around an fx79-w151a airfoil has been calculated by the spectral element code \U0001d4a9ϵκ\U0001d4afαrusing a direct numerical simulation (DNS) solver. In a first step a laminar inflow on the airfoil at angle of attack of α = 12° and a Reynolds number of Re= 33000 was simulated using the 2D Version of the code. The flow pattern was compared to measurements using holographic particle induced velocimetry (HPIV) in a wind tunnel.

The limiting hole pressure in three dimensions

The three-dimensional analog of the Giesekus–Tanner (G–T) theorem [B. Caswell, Non-Newtonian flow at lowest order: the role of the Reiner-Rivlin stress, J. Non-Newtonian Fluid Mech. 133 (1) (2006) 1–13] allows the flow in the vicinity of a pressure tap to be analyzed with the solution of the inertialess flow of a Reiner-Rivlin fluid just as G–T was used [R.I. Tanner, A. Pipkin, Intrinsic errors in pressure-hole measurements, Trans. Soc. Rheol. 14 (1969) 471] to analyze the corresponding two-dimensional flow with the Stokes solution. Later Kearsley [E.A. Kearsley, Intrinsic errors for pressure measurements in a slot along a flow, Trans. Soc. Rheol. 14 (3) (1970) 419–424] extended the Tanner–Pipkin analysis to the rectilinear flow in a channel with a parallel slot. In the two-dimensional case the Tanner–Pipkin term is the hole pressure relative to the Stokes value (usually neglected), and is determined solely by the first normal stress function evaluated at the wall shear rate of the undisturbed flow. It is thus “intrinsic” since dimensional analysis suggests the hole pressure should also depend on the hole geometry. Kearsley’s result is independent of slot geometry only in the limit that the slot is deep and very narrow relative to the size of the main channel. In three-dimensions the Tanner–Pipkin term again contributes an intrinsic pressure independent of hole geometry, but now it is now relative to the Reiner–Rivlin value which itself depends on the hole geometry and is not negligible. Hence the complete hole pressure in three-dimensions requires the solution of the Reiner–Rivlin problem which in general will be obtained numerically. The latter is formulated for numerical simulation so that the hole pressure can be read off from values of the pressure field without any post-processing calculation of velocity gradients. The numerical simulations were performed with a three-dimensional spectral element code [G.E. Karniadakis, S.J. Sherwin, Spectral/ hp Element Methods for CFD, Oxford University Press, 1999] well suited to the efficient solution of flow problems in complex geometries. The limiting values of the Reiner–Rivlin hole pressure have been obtained for several hole/channel ratios. Numerical results for the larger holes are not in agreement with the H–P theory [K. Higashitani, W.G. Pritchard, A kinematic calculation of intrinsic errors in pressure measurements made with holes, Trans. Soc. Rheol. 16 (4) (1972) 687–696], and for such holes the Stokes hole pressure was found to be in the measurable range.

Simulation of binary mixtures with the lattice Boltzman method

A lattice Boltzman model for the simulation of binary mixtures is presented. Contrary to previous models, the present formulation is able to simulate mixtures with different Schmidt numbers and arbitrary molecular mass ratio of the components. In the hydrodynamic limit, the Navier-Stokes and the Stefan-Maxwell binary diffusion equations are recovered. The model is used for the simulation of binary diffusion and mixing layers. The results are found to be in good agreement with a derived similarity solution and with the predictions of a transient spectral element code.

Impinging laminar jets at moderate Reynolds numbers and separation distances

An experimental and numerical study of impinging, incompressible, axisymmetric, laminar jets is described, where the jet axis of symmetry is aligned normal to the wall. Particle streak velocimetry (PSV) is used to measure axial velocities along the centerline of the flow field. The jet-nozzle pressure drop is measured simultaneously and determines the Bernoulli velocity. The flow field is simulated numerically by an axisymmetric Navier-Stokes spectral-element code, an axisymmetric potential-flow model, and an axisymmetric one-dimensional stream-function approximation. The axisymmetric viscous and potential-flow simulations include the nozzle in the solution domain, allowing nozzle-wall proximity effects to be investigated. Scaling the centerline axial velocity by the Bernoulli velocity collapses the experimental velocity profiles onto a single curve that is independent of the nozzle-to-plate separation distance. Axisymmetric direct numerical simulations yield good agreement with experiment and confirm the velocity profile scaling. Potential-flow simulations reproduce the collapse of the data; however, viscous effects result in disagreement with experiment. Axisymmetric one-dimensional stream-function simulations can predict the flow in the stagnation region if the boundary conditions are correctly specified. The scaled axial velocity profiles are well characterized by an error function with one Reynolds-number-dependent parameter. Rescaling the wall-normal distance by the boundary-layer displacement-thickness-corrected diameter yields a collapse of the data onto a single curve that is independent of the Reynolds number. These scalings allow the specification of an analytical expression for the velocity profile of an impinging laminar jet over the Reynolds number range investigated of .

Hydrodynamic forces acting on a rigid fixed sphere in early transitional regimes

A spectral – spectral-element code is used to investigate the hydrodynamic forces acting on a fixed sphere placed in a uniform flow in the Reynolds number interval [10–320] covering the early stages of transition, i.e. the steady axisymmetric regime with detached flow, the steady non-axisymmetric and the unsteady periodic regimes of the sphere wake. The mentioned changes of regimes, shown by several authors to be related to a regular and a Hopf bifurcations in the wake, result in significant changes of hydrodynamic action of the flow on the sphere. In the present paper, we show that the loss of axisymmetry is accompanied not only by an onset of lift but also of a torque and we give accurate values of drag, lift and torque in the whole interval of investigated Reynolds numbers. Among other results show, moreover, that each bifurcation is accompanied also by a change of the trend of the drag versus Reynolds number dependence, the overall qualitative effect of instabilities being an increase of drag.

The Newmark scheme as velocity-stress time-staggering: an efficient PML implementation for spectral element simulations of elastodynamics

Perfectly matched layers (PMLs) provide an exponential decay, independent of the frequency, of any propagating field along an assigned direction without producing spurious reflections at the interface with the elastic volume. For this reason PMLs have been applied as absorbing boundary conditions (ABCs) and their efficiency in attenuating outgoing wave fields on the outskirts of numerical grids is more and more recognized. However, PMLs are designed for first-order differential equations and a natural extension to second-order Partial Differential Equations (PDEs) involves either additional variables in the time evolution scheme or convolutional operations. Both techniques are computationally expensive when implemented in a spectral element (SE) code and other ABCs (e.g. paraxial or standard sponge methods) still remain more attractive than PMLs. Here, an efficient second-order implementation of PMLs for SE is developed from interpreting the Newmark scheme as a time-staggered velocity–stress algorithm. The discrete equivalence with the standard scheme is based on an L2 approximation of the stress field, with the same polynomial order as the velocity. In this case, PMLs can be introduced as for first-order equations preserving the natural second-order time stepping. Subsequently, a non-classical frequency-dependent perfectly matched layer is introduced by moving the pole of the stretching along the imaginary axis. In this case, the absorption depends on the frequency and the layer switches from a transparent behaviour at low frequencies to a uniform absorption as the frequency goes to infinity. It turns out to be more efficient than classical PMLs in the absorption of the incident waves, as the grazing incidence approaches, at the cost of a memory variable for any split component. Finally, an extension of PMLs to general curvilinear coordinates is proposed.

Large eddy simulation of turbulent channel flows by the rational large eddy simulation model

The rational large eddy simulation (RLES) model is applied to turbulent channel flows. This approximate deconvolution model is based on a rational (subdiagonal Padé) approximation of the Fourier transform of the Gaussian filter and is proposed as an alternative to the gradient (also known as the nonlinear or tensor-diffusivity) model. We used a spectral element code to perform large eddy simulations of incompressible channel flows at Reynolds numbers based on the friction velocity and the channel half-width Reτ=180 and Reτ=395. We compared the RLES model with the gradient model and the Smagorinsky model with Van Driest damping. The RLES model was much more stable than the gradient model and yielded improved results. Both the RLES model and the gradient model predicted the off-diagonal Reynolds stresses better than the Smagorinsky model with Van Driest damping. The latter, however, yielded better results for the diagonal Reynolds stresses.

Spectral Element Methods for Transitional Flows in Complex Geometries

We describe the development and implementation of an efficient spectral element code for simulating transitional flows in complex three-dimensional domains. Critical to this effort is the use of geometrically nonconforming elements that allow localized refinement in regions of interest, coupled with a stabilized high-order time-split formulation of the semi-discrete Navier–Stokes equations. Simulations of transition in a model of an arteriovenous graft illustrate the potential of this approach in biomechanical applications.

Overlapping Schwarz Methods for Unstructured Spectral Elements

A parallel and scalable domain decomposition method for unstructured and hybrid spectral element discretizations of elliptic problems is introduced and studied. The spectral elements are affine images of the reference triangle or square in two dimensions and of the reference tetrahedron, pyramid, prism, or cube in three dimensions. The method is based on overlapping Schwarz techniques applied to the Schur complement of the discrete system and is implemented as a preconditioner for a Krylov space method. Numerical results in two and three dimensions show that the iteration counts of our method are bounded by a constant independent of the spectral degree and the number of subdomains. The resulting elliptic solver can be used in Navier–Stokes simulations using the spectral element code NekTar.

Stability analysis of perturbed plane Couette flow

Plane Couette flow perturbed by a spanwise oriented ribbon, similar to a configuration investigated experimentally at the Centre d’Etudes de Saclay, is investigated numerically using a spectral-element code. Two-dimensional (2-D) steady states are computed for the perturbed configuration; these differ from the unperturbed flows mainly by a region of counter-circulation surrounding the ribbon. The 2-D steady flow loses stability to three-dimensional (3-D) eigenmodes at Rec=230, βc=1.3 for ρ=0.086 and Rec≈550, βc≈1.5 for ρ=0.043, where β is the spanwise wave number and 2ρ is the height of the ribbon. For ρ=0.086, the bifurcation is determined to be subcritical by calculating the cubic term in the normal form equation from the time series of a single nonlinear simulation; steady 3-D flows are found for Re as low as 200. The critical eigenmode and nonlinear 3-D states contain streamwise vortices localized near the ribbon, whose streamwise extent increases with Re. All of these results agree well with experimental observations.

Spectral element simulation of ultrafiltration

A spectral element method for simulating stationary 2-D ultrafiltration is presented. The mathematical model is comprised of the Navier–Stokes equations for the velocity field of the fluid and a transport equation for the concentration of the solute. In addition to the presence of the velocity vector in the transport equation, the system is coupled by the dependency of the fluid viscosity on the solute concentration and by a concentration-dependent boundary condition for the Navier–Stokes equations at the membrane surface. The spectral element discretization yields a nonlinear algebraic system for the unknowns at the mesh nodes. This system is solved via a technique combining the penalty method, Newton–Raphson iterations, static condensation, and a solver for banded linear systems. In addition, a smoothing technique is used to handle a singularity in the boundary condition at the membrane. The performance of the spectral element code when applied to several ultrafiltration problems is reported.

A Spectral Element Simulation of Gravitational Flow During Plastic Pipe Extrusion

When plastic pipe is extruded, it emerges from an annular die and passes through a sizing sleeve to set its outer diameter. The pipe is then solidified in a cooling tank by spraying the outer surface with cold water. For thick walled pipe, material on the inside stays molten for a long time, and flows under its own weight. This gravitational flow (or sag) therefore governs the final pipe wall thickness distribution. Thus the solidification of extruded plastic pipe involves heat transfer with phase-change coupled with gravitational flow. The inside thermal boundary condition is arguably adiabatic. For the outside boundary, one can use either Newton’s law of cooling or an isothermal condition. In this paper, a commercially available spectral element code for simulating unsteady incompressible fluid flow with heat transfer, has been used to simulate sag flow. The model predictions of the solid pipe thickness distribution compared well with process data. Also, the effects of different heat transfer parameters on the thickness profile are evaluated. The extrusion temperature is found to have the greatest effect on the pipe wall profile.

A new wall shear stress gauge

A new technique has been developed to measure the wall shear stress and its direction in a turbulent boundary layer. This technique involves the measurement of torque upon a very small cylindrical body placed above the wall deep in the viscous sublayer, so that the device is operating in the creeping flow regime. The approach has involved calibration tests on a gauge 8 mm long by 0.8 mm in diameter, located in uniform shearing flow of glycerol created in a cone-and-plate apparatus. Our theoretical, computational and experimental results show that the torque has a linear relation with the wall shear stress. We have used a spectral element code, Nekton, for determining the creeping flow about the gauge. There is a very good agreement between the experimental and computational results.