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The prediction of coherent vortices with standard RANS solvers suffers especially from discretisation and modelling errors which both introduce numerical diffusion. The adaptive Vorticity Confinement (VC) method targets to counteract one part of the discretisation error: the one due to the discretisation of the convection term. This method is applied in conjunction with a hybrid RANS-LES turbulence model to overcome the overprediction of turbulence intensity inside vortex cores which is a typical deficiency of common RANS solvers. The third main source for numerical diffusion originates from the spatial discretisation of the solution domain in the vicinity of the vortex core. The corresponding error is analysed within a grid convergence study. A modification of the adaptive VC method used in conjunction with a high-order discretisation of the convection term is presented and proves to be superior. The simulations of a wing tip vortex flow are validated in terms of vortex velocity profiles using the results of a wind tunnel experiment performed by Devenport and colleagues (1996). Besides, the results are compared with another numerical study by Wells (2009) who uses a Reynolds Stress turbulence model. It turns out that the application of the modified adaptive VC method on the one hand reinforces the tip vortex, and on the other hand accelerates the axial flow which leads to a slight degradation compared to the experimental results. The result of Wells is more accurate close to the wing, but the result obtained here is superior further downstream as no excessive diffusion of the tip vortex occurs.

Coherent tip vortices occur e.g. in the wake of lifting wings like aircraft foils, propeller blades or helicopter rotor blades and extend far downstream showing a slow decay. Considering the cases mentioned above, examples are wake encounters of aircrafts, cavitating propeller tip vortices that may damage the rudder and blade vortex in- teractions for helicopters.

Using Reynolds-Averaged Navier-Stokes (RANS) methods in conjunction with standard one- or two-equation turbulence models for the prediction of tip vortices leads to high errors, as the vortices smear out considerably faster in the simulation compared to the real flow. The increased dissipation results from artificial dissipation effects which are mainly dominated by two numerical errors. The first one is the discretisation error and the second one is the error due to insufficient turbulence modelling. Assuming an appropriate fine mesh in the vicinity of the vortex core, the discretisation error is dominated by the error introduced by the convection discre- tisation scheme. In conjunction with a second-order accurate flow solver, like Open- FOAM, most discretisation schemes, like e.g. linear upwind, introduce diffusive error contributions. These errors create numerical diffusion and lead to a strong dissipation of vortices. As mentioned above, the second numerical error that leads to artificial dissipation refers to turbulence modelling. Due to high velocity gradients inside the tip vortex core, linear isotropic Eddy Viscosity Models (EVMs) create a huge turbulent viscosity inside the core (compared with the physical viscosity). But the available experimental data show that tip vortex cores mostly contain almost laminar flow (see e.g. Devenport and colleagues [

In this paper the potential of an advanced Vorticity Confinement (VC) method applied to a wing tip vortex flow is evaluated. The VC technique was proposed to improve the prediction of vortical flows where the accuracy suffers from numerical diffusion. The basic VC formulation and all modified ones require a user-defined proportionality factor that prescribes the strength of the reinforcement for the vortex. The main disadvantage of the these methods is the lack of a universal procedure to determine the appropriate factor. Even if the vortex was reinforced by VC, this would not necessarily provide a more accurate solution. The adaptive VC method, proposed by Hahn and Iaccarino (see [

The overprediction of the turbulence intensity in the vicinity of the vortex core can be overcome using improved turbulence models. Some modified isotropic eddy viscosity models address the deficiency concerning coherent cortices introducing a curvature correction (see e.g. Spalart and Shur [

The adaptive VC method is employed in the present study in conjunction with a hybrid RANS-LES model. The first study analysing both methods in combination was published recently by the authors of this paper, see [

Presenting the adaptive VC method Hahn and Iaccarino used the upwind dis- cretisation scheme for the convection term successfully. Pierson and Povitsky [

In this study, the adaptive VC method will be assessed in comparison with exper- imental data from Devenport et al. [

This section gives an overview over the numerical errors that dominate the artificial dissipation of coherent vortices. Referring to Lilek and Perić [

Modelling errors are defined as the difference between the actual flow and the exact solution of the mathematical model, which is represented by the Navier-Stokes equations and the mass conservation (continuity) equation in this case. In case flows with coherent vortices are simulated, linear isotropic Eddy Viscosity Models (EVM) fail to predict the laminar vortex core and predict high values of turbulent kinetic energy instead. Common turbulence models based on linear, isotropic EVMs are e.g. the one- equation Spalart-Allmaras model [

In Large Eddy Simulations (LES) large (in relation to the discretisation in space and time) turbulent eddies are resolved and small ones are modelled. The advantage referring to coherent vortices is, that there is no overprediction of the turbulence intensity due to high velocity gradients around the vortex core. In conjunction with a hybrid RANS-LES approach, the near-field of the wing is modelled with a RANS approach saving computational effort.

The disretisation error introduces numerical diffusion in two different ways. The first one is insufficient grid resolution around the vortex core and can be overcome by mesh refinement. The second one is the discretisation of the governing equations. In the context of industrial flows with high Reynolds numbers, the convection term dominates the flow and hence its discretisation is important.

Usually, polynomial functions are used to describe the gradient of flow variables between grid points. In conjunction with the second-order accurate FVM the use of second-order accurate discretisation schemes would be reasonable as they preserve the overall (second-order) accuracy of the approach. A basic second-order accurate discre- tisation scheme is the central differencing scheme (termed CDS). But the use of second- order schemes for the convection discretisation may lead to numerical oscillations during the iterative solution process of the equation system. Following, schemes introducing diffusive truncation errors are usually used to assure stability. The stability is increased at the cost of accuracy. The total diffusion is then composed of the (anisotropic) numerical and the (isotropic) physical diffusion.

Vorticity Confinement describes a technique which reinforces vortices and therewith acts against the effects of artificial dissipation discussed in the previous section. As part of the VC approach an artificial source term

with velocity

The proportionality factor

where

The influence of the VC source term was described smartly by Hahn and Iaccarino (see [

The original VC technique was proposed by Steinhoff and colleagues in 1992 ( [

The adaptive VC method introduces a momentum source term counteracting the error of the convection discretisation in an adaptive manner. The magnitude of the source term is proportional to the estimated numerical diffusion defined with the difference between the central scheme (introducing no diffusive error) and the utilised scheme.

Hahn and Iaccarino [

The derivation of the adaptive VC method is based on the assumption that the numerical diffusion is known a priori, hence the total viscosity

The target of the adaptive approach is to minimize the influence of

which has the unit

for more information on the derivation see [

In this study, the effect of restricting the adaptive VC source term to the vicinity of the vortex will be analysed. Therefor the local vortex identification criterion

In addition to the adaptive VC method presented above, there are further for- mulations of the VC method which lack the necessity for a user-defined forcing coefficient

In this study, the adaptive method is used in conjunction with the high-order con- vection (HOC) scheme LUDS. Following, the difference in Equation (5) is calculated as the difference between CDS and LUDS, which is consistent to the target of cancelling out the effect of

This still scales with the magnitude of the numerical diffusion and always points along

In the present study, simulations are carried out for the experiments conducted by Devenport and colleagues [

consists of a rectangular NACA0012 profile with a span of

There are two reasons why this experimental data is especially valuable for a comparison with numerical methods that target an improved resolution of tip vortices. The first one is the large distance downstream

frequency

Devenport et al. suppose the reason of wandering to be a slight unsteady behaviour of the flow in the wind tunnel ( [

Measurements are carried out for several Reynolds numbers

For the setup chosen for comparison here, the tangential and axial velocity are measured in five times chord length distances until 30 times the chord length (behind the leading edge of the foil). Besides, the mean velocity {redcomponents and turbulence properties are measured at certain positions. A main outcome of the experiment is that the vortex core is laminar. The observed velocity fluctuations are attributed to the interaction of the vortex with the turbulent wake ( [

The simulations are conducted with OpenFOAM 3.0.1 on a hexahedral mesh. Firstly, the computational domain and the mesh are presented, afterwards, the solver settings will be given.

The origin of the used coordinate system is located at the leading edge on the tip end of the foil. The x-axis points downstream and the y-axis points along the wing. This coordinate system is applied also in [

The mesh is generated with Hexpress (version 5.1),

All meshes consist of eight base cells along the cross section length of the domain. For the medium mesh, the cells around the tip vortex are refined by seven steps in the y- and z-direction, hence the measured mean vortex core diameter

The bounding walls (normals y and z) are modelled by a slip boundary condition. Thus, the slight pressure gradient along the wind tunnel due to the boundary layer growth observed in the experiments cannot be modelled. It can be assumed that this effect is negligible for the evolution of the tip vortex. At the inlet the inflow velocity is

prescribed and a pressure boundary condition is applied at the outlet. The surface of the wing is represented by a no-slip condition and layer cells are used for the resolution of the boundary layer. At the inlet, the eddy viscosity is prescribed with

Devenport et al. do not indicate the free-stream velocity

termed Detached-Eddy Simulation, DES) is based on the Spalart-Allmaras delayed DES model. The reference for the SpalartAllmarasDDES model can be found in [

The simulations are conducted with the pimpleFoam solver, a large time-step transient solver using the PISO and the SIMPLE algorithms. Both algorithms are described e.g. by Ferziger and Perić [

A further remark concerns the source domain, where

In the following section the simulation results will be analysed and compared to the experimental data. The simulation results are evaluated after

In this study the tangential and axial velocity profiles are gained from velocity evaluations on lines through the vortex core.

The coordinate system

For the determination of the tangential and axial velocity components, the vortex axis is assumed to be parallel to the x-axis for the sake of simplicity. The angle between the vorticity vector at the vortex centre and the x-axis is less than 2˚ downstream of

is simply

The vortex core parameters (the radius, the peak tangential velocity and the axial velocity deficit) will be plotted at the x-coordinates of the measurement sections

The use of the VC method does not significantly increase the computational time. The time which a time step needs fluctuates by about

In the following subsections, first the restriction of the VC source term to the zone in the vicinity of the vortex core will be analysed. Secondly the effect of the modified formulation according to Equation (7) will be discussed. Afterwards, the simulation results will be compared to the experimental data.

The influence of the restriction of the VC source term to the vicinity of the vortex core will be analysed in the following. As will be shown in the next section, the modified formulation presented in Section 3.3 is superior compared to the original one, so the modified VC source term according to Equation (7) will be used here. The reason for the restriction is the acceleration of the axial flow due to the source term outside of the vortex core in some places. In these places, the magnitude of the source term is comparable to the maximum value in the plane perpendicular to the current vortex axis and the source vector is quasi parallel to the vortex axis (with a deviation of a few degrees). Although

the vortex core parameters. The comparison to the experimental results is evaluated in subsection 6.4. In both cases, the application of VC reduces the vortex core size and increases the peak tangential velocity which corresponds to a stronger vortex. The disadvantage of the unrestricted source term reveals in the large overprediction of the axial velocity at the vortex centre: Further downstream, the axial velocity deficit nearly vanishes. Furthermore, the unrestricted source term induces velocity fluctuations close to the trailing edge. This effect manifests in the sudden drop of

The magnitude of the adaptive VC source term is large in the vicinity of the vortex core radius (where

First, the necessity of the modification for the adaptive VC formulation, presented in Section 3.3, will be discussed. Afterwards, the effect of the modification on the vortex flow will be shown.

arrows (which are not projected to the plane). The grey cells indicate cells where

The vortex core has an approximately cicular shape in both cases. For LUDS, the vortex core is smaller and the vector

conducted also with a negative source term

The original VC method leads to a slightly increased core radius and slightly reduced tangential velocity compared to the result without VC. This corresponds to the source term acting in the wrong direction in many cells. If the original source term is multiplied with

In the following subsection, the velocity profiles through the vortex core and the core parameters from the experimental results and the simulation results will be compared. Therefor, the modified formulation of the adaptive VC source term will be used and restricted to zones with

Further downstream at

results to the experimental one (at

The vortex core parameters are shown in

size monotonically increases with distance downstream. Upstream of

The peak tangential velocity is shown in

Applying VC reduces the axial velocity deficit in each case. The relative reduction of the axial velocity deficit downstream of

In the following, the results presented above will be compared to the results obtained in the study by Wells [

In this subsection, the best result obtained by Wells will be compared to the results obtained in this study. The best result in [

The velocity profiles Wells evaluated were extracted on a line parallel to the z-axis through the vortex centre. The vortex core parameters were determined from the resulting profiles. The influence of the different procedures to extract core parameters that were used by Wells [

Wells evaluated the simulation results only until

The original adaptive VC method presented by Hahn and Iaccarino mitigated the vortex, because the source term reversed in conjunction with the linear upwind dis- cretisation of the convection term. The modified formulation is based on an evaluation of the magnitude of the estimated numerical diffusion and neglects its fluctuating direction. This modification turned out to be superior compared to the original for- mulation.

The modified, adaptive VC method improved the prediction of vortices in some aspects. It led to a reduction of about

Compared to results obtained with a Reynolds Stress turbulence Model (RSM) (by Wells), the results obtained with VC and the underlying hybrid RANS-LES approach lead to less accurate results at the most upstream measurement section but to superior results further downstream. In this study, the diffusion of the vortex core could be limited to an acceptable level unlike in the other numerical study using the RSM. A possible reason for the difference of further downstream is a coarsening of the mesh resolution in the RSM case. It would be interesting to further study the potential of the (modified) adaptive VC method in conjunction with other turbulence models (e.g. RSM), especially with respect to the relation of the boundary layer flow and the axial velocity near the vortex. Subsequently, the acceleration of the axial flow due to VC should be analysed further.

The used mesh resolutions with a total cell count of up to 20 M don’t show grid convergence. To assess the influence of numerical diffusion due to domain discre- tisation, Adaptive Mesh Refinement would be necessary to keep the total cell count to an acceptable level.

The investigations presented in this paper have been obtained within the research project “Entwicklung von numerischen Methoden zur Auflösung konzentrierter Wirbelstrukturen” (Development of Numerical Methods for the Resolution of Coherent Vortices) funded by the Deutsche Forschungsgemeinschaft (DFG). This support is greatly appreciated.

Feder, D.-F. and Abdel-Maksoud, M. (2016) Tracking a Tip Vortex with Adaptive Vorticity Confinement and Hybrid RANS-LES. Open Journal of Fluid Dynamics, 6, 406-429. http://dx.doi.org/10.4236/ojfd.2016.64030