Numerical Investigation of the Flow Field about the VFE2 deltawing
The industrial application of deltawings is manifold and reaches from the classical aerospace engineering, e.g. highly agile aircraft, aerodynamic devices or control surfaces, to unique environmental technologies, such as devices for snow clearance. In all cases the development of leading edge vortices is exploited.
However, steadiness and stability of these leading edge vortices is essential for controllability, particularly for highly agile aircraft. It is well known that vortices can undergo a sudden expansion often related to vortex breakdown [5]. The occurrence of unsteady vortex breakdown is critical for aircraft. It is physically not fully understood, thus further investigation is required. The ongoing investigation is funded by the DFG project “Numerische Untersuchung der instationären Strömung um generische schlanke Deltaflügel“ (DFGB506/2).

Figure 1: Isosurface of streamwise vorticity colored by streamwise velocity for the VFE2 Delta Wing at an angle of attack of 13° and a Reynolds number of 0.5 million.

The international Vortex Flow Experiment 2 (VFE2) deltawing is taken as a generic aerodynamic configuration for which small angles of attack already lead to the development of leading edge vortices.
Underlying Theory
A profound understanding of vortex formation and breakdown requires a comprehensive insight into the complete unsteady flow field. This insight can only be obtained from timeaccurate simulations accompanied by experiments.
The most complete description of flows in continuum mechanics is given by the NavierStokes equations, which describe the exchange of momentum in the fluid considering friction.
Solving the Navier Stokes equations requires very high spatial and temporal resolution. A Direct Numerical Simulation (DNS) is still not feasible for complex turbulent flows in industrial applications due to the required tremendous computational resources. For simulating the turbulent flow about the VFE2 DeltaWing several hundred CPU years using 1 Terra flops would be needed, making a simplification of the Navier Stokes equation inevitable.
Commonly used in industry is the RANS (ReynoldsAveraged Navier Stokes) simulation with appropriate statistical turbulence models. This simplified approach often fails to accurately predict separated and reattached flows. Also in the case of the VFE2 deltawing results do not compare well with the existing experimental results [9].
Better results are expected from LargeEddy Simulation (LES). In LES only the large flow structures are resolved while small, stochastic structures are modelled with the help of SubGrid Scale (SGS) models. In Implicit Large Eddy Simulation (ILES) the truncation error of the discretization of the convective terms is deliberately tailored to act as a SGSmodel which is therefore implicit to the discretization. One implementation of ILES is the Adaptive Local Deconvolution Method (ALDM) [1,4] which has shown considerable potential for the efficient representation of physically complex flows in generic configurations, such as isotropic turbulence, turbulent channel flow with periodic constrictions, turbulent boundary layer separation and turbulent cylinder flow [7]. In this project the ILES approach will be used for the essential numerical investigations.
Numerical Method
The Adaptive Local Deconvolution Method (ALDM) turbulence model is incorporated in a solver for the incompressible NavierStokes equations with constant density. Continuity is ensured by the pressurePoisson equation. The equations are discretized on a staggered Cartesian mesh allowing for an easy control of the truncation error and offering superior computational efficiency compared to bodyfitted grids [6]. Bounding surfaces of the flow that are not aligned with the grid are accounted for by the Conservative Immersed Interface (CIIM) approach [8].
For time advancement an explicit thirdorder RungeKutta scheme is used. The pressurePoisson equation and diffusive terms are discretized by secondorder centred differences.
Further improvement of efficiency is achieved by modelling the turbulent boundary layer using a wall model [2] and by locally adapting the mesh resolution with Local Mesh Refinement, such as used in [7]. In this context new criteria for the Local Mesh Refinement algorithm based on physical criteria are applied. A further considerable reduction of the number of computational cells is achieved, rendering the simulation much more effective.
Results
The results in this reporting period have been obtained for a Reynolds number Re = 0.5 million based on root chord length and an angle of attack of 13°. The objective is to reach a qualitative agreement with the respective experiments of Furman and Breitsamter [3].


Figure 2: Cp distribution on the upper (top) and front (bottom) surface of the VFE2 Delta Wing at an angle of attack of 13° and a Reynolds number of 0.5 million. Qualitative results for this specific configuration look very promising. A profound understanding is expected after results have been quantitatively validated with the experimental data. The necessary simulations are ongoing.

For this angle of attack both, experiment [3] and the current simulation, show vortex formation over half chord length, see Fig. 1. Close to the apex the boundary layer flow accelerates over the leading edge and undergoes laminarturbulent transition. This is also reflected by the high suction levels visible in Fig. 2. Severe pressure gradients in lateral direction provoke boundary layer separation further downstream. The separation region is indicated by the low pressure regions on the upper surface of the wing (see Fig. 2).
Computational Details
For this test case again a SGIALTIX with ItaniumII processors have been used. CIIM and the wall modelling consume 2% of the overall computational time, the flux calculation with ALDM accounts for 7% and the Poisson solver uses 83%.
Table 1 summarizes further computational details for the present and planned simulations.
Ongoing Research/Outlook
Since the objective is to investigate vortex bursting process, further investigation at the angles of attack of 18° and 23° is with varying leading edge geometries, i.e. MR= Medium Round and S= Sharp, are planned. As a next step an investigation is planned for actively controlling and preventing the vortex burst with leading edge devices, i.e. oscillating control surfaces.

Table 1: Survey of conducted simulations (black) and planned simulations (grey). (AoA = Angle of Attack, Re= Reynolds number, LE=Leading Edge, number of cells, number of CPUs, number CPU hours)

• Michael Meyer
• Stefan Hinkel
• Nikolaus A. Adams
Institute of Aerodynamics and Fluidmechanics, Technische Universität München
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