# High Reynolds Number Flow around Airfoil in Deep Stall

## Key Physics

The key physics of this UFR is predominantly characterised by the unsteady, three-dimensional, massively-separated wake region. This takes the form of a nominally periodic shedding of large scale, coherent vortices in a vortex sheet pattern, which is overlaid with finer random turbulent fluctuations at higher frequencies and random modulation and intermittency at frequencies lower than the vortex shedding frequency. It has been found that it is necessary to capture these key physical features in a simulation in order to reliably predict the assessment parameters. Whether this is achieved or not in a simulation should be checked by:

• Obtaining a visual impression of the range of spatial scales present in the wake using e.g. a snapshot of the vorticity magnitude.
• Confirming the mixed tonal and broadband nature of the force coefficient time histories by e.g. visual inspection or spectral analysis of time histories.

## Numerical Modelling

### Discretisation method

• Use a numerical scheme with low numerical dissipation (e.g. pure CDS for convective fluxes) in the region of resolved turbulence near the airfoil.
• Check for low numerical dissipation by examining instantaneous snapshots (e.g. of vorticity magnitude) in the turbulent wake region: the smallest resolved turbulent scales should have nearly the same size as the local grid spacing (if they are noticeably larger, this is an indication of excessive numerical dissipation).
• Use a numerical scheme with sufficient numerical dissipation to prevent grid oscillations or wiggles in the coarse grid and/or irrotational flow regions.
• Use a minimum of second order accurate spatial discretisation. No evident benefit of higher spatial accuracy has been proven for this UFR.
• Use a minimum of second order accurate temporal integration scheme.
• Use a time step sufficiently fine to capture the motion of the turbulent eddies resolved by the grid in the region of resolved turbulence near the airfoil. This corresponds to the approximate guideline CFLmax ≈ 1 in this region.

### Grids and grid resolution

Note: These BPA items regarding grid & resolution are not backed up by any specific studies on grid sensitivity, but describe recommendations from experience.

• Use a minimum grid spacing of around 1/32 of the chord length in the near wake region.
• Use roughly isotropic cells (as far as possible) in the near wake region.
• Expand the grid cell size gradually towards the outflow boundary, avoiding sudden jumps where possible and beginning the expansion roughly two chord lengths downstream of the airfoil.

### Boundary conditions and computational domain

• Employ a minimum size of domain of 15c in XY-plane and of 4c in the spanwise direction for the NACA0021 at 60° angle of attack case.
• Conduct a sensitivity study to determine the required spanwise domain size for different geometries or angles of attack. The required spanwise domain is smaller for lower angles of attack (e.g. around 2 chords seen for NACA0012 case at 45°) and is expected to be larger for larger angles up to 90°.

## Physical Modelling

### Turbulence modelling

• Use a turbulence-resolving approach (e.g. DES, SAS or Wall-Modelled LES).
• Do not use a pure turbulence-modelling approach, i.e. steady RANS and "classical" unsteady RANS because of their insufficient accuracy and wall-resolved LES because of its too high computational cost for the considered high Re number.
• The choice of RANS model in a hybrid RANS-LES approach is non-critical.

### Transition modelling

• Although no data on the transition location in the experiment are available, based on the simulation results no specific transition modelling is required (no effect on results seen comparing simulations with turbulent and laminar boundary layer separation).

### Near-wall modelling

All the simulations have been carried out with the use of low-Re versions of background RANS models. However a general very low sensitivity of the flow to the choice of RANS model and transition treatment implies that wall-functions may be probably employed with no tangible damage to prediction accuracy.

N/A

## Application Uncertainties

### Time sample / statistical processing

• Ensure that the simulation has bridged the initial transient and entered a statistically steady state before commencing the collection of statistics. This must either be checked by visual inspection of a monitor signal (e.g. force coefficients) or using a suitable statistical algorithm (e.g. [‌15]). As a guideline, roughly 80 convective time units of initial transient can be expected.
• Compute sufficiently long time samples for reliable statistical quantities:
• As a general guideline, a minimum time sample of 400 convective units are recommended, for which the 95% confidence intervals are below 5% for the mean forces and are around 30% for the standard deviation of the forces.
• Include statistical confidence intervals on the mean and standard deviation of the forces before drawing conclusions:
• e.g.: If the 95% confidence intervals from two simulations do not overlap, it can be stated with 95% confidence that the results differ.
• The confidence intervals can be computed for the mean and standard deviation of the lift and drag using the following formulae:

• Relative statistical error on mean: ${\displaystyle \varepsilon [\mu ]\approx {\sqrt {\frac {1}{2BT}}}{\frac {\sigma }{\mu }}}$

• Relative statistical error on standard deviation:  ${\displaystyle \varepsilon [\sigma ]\approx {\sqrt {\frac {1}{4BT}}}}$

• 95% confidence interval on statistical quantity ${\displaystyle \left.\phi \right.}$ (${\displaystyle \left.\phi =\mu \right.}$ for mean, ${\displaystyle \left.\phi =\sigma \right.}$ for standard deviation):

${\displaystyle {\frac {\phi }{1+2\varepsilon [\phi ]}}\leq \phi \leq {\frac {\phi }{1-2\varepsilon [\phi ]}}}$ with 95% confidence.

• In these formulae, ${\displaystyle \left.T\right.}$ is the statistical sample length in dimensionless convective time units (i.e., ${\displaystyle {T=tU_{\infty }/c}}$) and ${\displaystyle \left.B\right.}$ is the error scaling bandwidth, for which the following values apply to the lift and drag coefficient signals of the NACA0021 at 60° case:

${\displaystyle \left.B=0.149\right.}$ for ${\displaystyle \left.C_{l}\right.}$ and ${\displaystyle \left.B=0.0898\right.}$ for ${\displaystyle \left.C_{d}\right.}$.

### Comparability of force spectra and standard deviation between CFD and experiment

• Integrate lift and drag over individual spanwise grid slice(s) around the chord, rather than over the entire span, in order to ensure comparability with experimental force spectra and standard deviation, which were integrated at single spanwise locations. Averaging of spectra obtained from multiple spanwise slices is helpful to reduce statistical noise.
• If the force time traces have been integrated over the entire span, only the value of the vortex shedding peak frequency can be considered comparable with experiment: the spanwise averaging reduces the amplitude of broadband content in the simulation time traces.

### Uncertainty in experimental mean force coefficients

• Do not pay excessive attention to the agreement with experimental mean force coefficients due to the high level of scatter between experimental facilities.
• Check the level of agreement in the mean and standard deviation of the force coefficients with existing CFD data, provided that the same spanwise domain size was computed and that statistical uncertainty is taken into account.

## Recommendations for Future Work

• It would be very worthwhile to investigate the difference seen between the DES and SAS approaches (Figure 7). To this end, direct comparison of the two methods using identical grid, CFD solver and numerical settings would be most beneficial.
• The high scatter in mean forces between different experimental data sets represents the most unsatisfactory aspect of this test case, since it eliminates the possibility to validate the mean force predictions from CFD with any confidence. Efforts to remedy this situation and, also, to get experimental data on the mean and unsteady near wake characteristics of the flow would be most welcome.

Contributed by: Charles Mockett; Michael Strelets — CFD Software GmbH and Technische Universitaet Berlin; New Technologies and Services LLC (NTS) and St.-Petersburg State Polytechnic University