UFR 2-11 Evaluation
High Reynolds Number Flow around Airfoil in Deep Stall
Flows Around Bodies
Underlying Flow Regime 2-11
Evaluation
Comparison of CFD Calculations with Experiments
A dramatic improvement in solution fidelity for DES compared to URANS, first reported by Shur et al. [22], was observed in the extensive cross-validation exercise carried out in the EU FLOMANIA project [4]. Figure 4 depicts the relative deviation from experimental drag achieved by DES and URANS within this work.
Figure 4: Comparison of URANS and DES for the prediction of mean drag coefficient for the NACA0012 airfoil at α = 60°. Results of 11 different simulations conducted by different partners with different codes and turbulence models within the EU FLOMANIA project [4]. Experimental data cited by Hoerner [6] are used as reference. |
The effect of spatial and temporal numerical schemes on DES was
investigated for the NACA0012 case at α = 45° by
Shur et al. (2004)
[23]. Using a localised "hybrid" convection
scheme [29] (in which 4^{th}
order central differences are applied within the vortical wake region)
and a 2^{nd} order temporal integration was seen to resolve fine turbulent
structures to a scale near to that of the local grid spacing. Switching
the convection scheme to 3^{rd} order upwind or, to a lesser extent, the
temporal scheme to 1st order was seen to strongly damp the fine
vortices in the wake (Figure 5). Correspondingly, a strong under-
prediction of the Power Spectral Density (PSD) of the drag and lift
forces at higher frequencies was observed. The effect on the mean
forces and pressure distributions was however comparatively mild for
this case.
Figure 5: Effect of different spatial and temporal numerical schemes on the resolved wake structures of the NACA0012 at α = 45° [23]. "Hybrid" refers to the localized blending between 4^{th} order central and 3^{rd} or 5^{th} order upwind convection schemes proposed by Travin et al. [29] |
Having clearly demonstrated the benefits of DES compared to URANS [4, 22]
(Figure 4),
no further URANS computations were carried out in the
successor EU project DESider [5], and the
focus shifted to cross-comparison of different turbulence-resolving approaches.
Figure 6
compares flow visualizations from 3 simulations carried out with the
use of different approaches (k – ω
SST SAS and DES based on SA and CEASM
RANS models) in the form of instantaneous fields of the vorticity
magnitude. They reveal quite similar flow and turbulent structures thus
supporting a marginal sensitivity of the simulations to the turbulence
modelling approach and numerics used.
Figure 6: Comparison of snapshots of vorticity magnitude from three simulations |
The same is to a major extent correct regarding the PSD of the lift
coefficient and mean pressure distributions over the airfoil shown in
Figure 7.
Figure 7(a) also suggests that all
the simulations are
capable of predicting the experimental spectra, particularly the main
shedding frequency and its harmonic, fairly well, whereas
Figure 7(b)
reveals a systematic difference between the predicted and measured
pressure on the suction side. Note also that SAS results somewhat
deviate from those of SA DES and are closer to the experiment. The same
trend is observed for the integral lift and drag forces
(Table 5).
A concrete reason for the difference between the SAS and DES predictions
is not clear but, in any case, it is not significant when compared to
e.g. the differences between DES and URANS or between different URANS
approaches (see Figure 4).
This justifies the above conclusion on the
weak sensitivity of the predictions to the turbulence modelling
approach and numerics used in different turbulence-resolving
simulations.
(a) | (b) |
Figure 7: Comparison of PSD of the lift coefficient (left) and mean pressure distributions over the airfoil (right) from three simulations with experimental data [27, 28] |
Partner and approach | μ [C_{l}] | Statistical 95% CI^{*} | μ [C_{d}] | Statistical 95% CI^{*} |
ANSYS (k – ω SST SAS, L_{z}=4) | 0.915 | ±0.017 | 1.484 | ±0.030 |
EADS-M (SA DES, in wind-tunnel) | 0.889 | ±0.016 | 1.425 | ±0.029 |
NTS (SA DES, L_{z}=4) | 0.879 | ±0.007 | 1.381 | ±0.012 |
TUB (SA DES, L_{z}=4) | 0.875 | ±0.007 | 1.33 | ±0.012 |
Experiment [24, 25] | 0.931 | ±0.004 | 1.517 | ±0.008 |
The parameters to which the results of the simulations turned out to be
very sensitive include the span-size of the computational domain and
time sample the turbulence statistics is calculated for. These
parameters are therefore the most important in the sense of formulation
the BPG for the considered UFR, and for this reason below we focus
exactly on those studies which analyse their effect. These studies
along with some original results related to this issue were summarized
in [2], so the further presentation of
the material closely follows this work.
Effect of time sample
Obtaining accurate estimates of statistical quantities naturally requires a sufficient statistical sample. This raises fundamental questions concerning the engineering application of computationally expensive turbulence-resolving methods: how many time steps must be computed to estimate the statistical quantities of interest to the desired accuracy? The inverse perspective is just as important: what is the error bar on the statistical quantities when only a given number of time steps are affordable? The implications of this issue for the cost planning of industrial computational studies and the avoidance of false conclusions are clear.
An arbitrary section of the time series for the lift and drag coefficients from the experiment is shown in Figure 8. The quasi-periodic von Kármán vortex shedding cycles are seen (with a time period of roughly 5 convective time unites, (c / |U_{∞} |) together with a strong low- frequency intermittency. This represents an apparently multi-modal behaviour of the flow, in which alternating weak and strong vortex shedding modes occur. The duration of each such mode as well as its initiation are random. As a result, it can intuitively be recognised that very long time samples will be required for accurate mean values of this flow.
Figure 8: Sample lift and drag coefficients time traces from the experimental data of Swalwell [27, 28], showing occurrence of weak and strong shedding behaviour. |
A common method to obtain an impression of the time sample effect on
the mean values is the computation of running averages. Indeed, this
approach was applied in the DESider project study of this
test case [5].
Figure 9 demonstrates such a running average for the experimental
C_{d} trace, from which an indication is given that the
mean C_{d} is still
not reliable to the second decimal place after 4000 non-dimensional
time units. Although useful in this respect, the running average
technique does not provide any quantitative estimate of the statistical
error. A potential peril therefore arises: were the data acquisition to
have been ceased at = 800, the strong and sudden drift that occurs
shortly thereafter would not be anticipated^{[1]}.
In order to get a
quantitative estimate of the statistical error, a novel signal
processing algorithm has been developed. The details of this technique
have been summarised in [2]
and published in more detail in [15]. For a
given input signal, the magnitude of the statistical error as a
function of sample length is estimated. The 95% confidence intervals
for the experimental drag coefficient have been obtained in this way
and plotted over the running average in Figure 9.
Figure 9: Running average of experimental drag coefficient [27, 28], giving an impression of the level of drift that occurs even after long averaging times (red curve). Envelope of 95% confidence interval arising from statistical error (black curves). |
Span-size effects
The spanwise separation between the periodic boundaries to the computational domain has been varied between L_{z} = 1c and 4c in the calculations of NTS and TUB, allowing the influence of this user parameter to be studied. Figure 10 and Figure 11 draw upon all DES results employing a periodic spanwise domain and all applicable experimental data to provide an overview. The 95% confidence intervals due to the statistical error estimated using the technique mentioned in the preceding section [2, 15][2]^{[2]} are portrayed in the figures by error brackets.
Figure 11: Trends of standard deviation of the lift and drag coefficients against L_{z} for the NACA0021 at α = 60° and Re = 270000, bars representing the statistical 95% confidence interval. |
The effect of a narrow span size is to increase all quantities
depicted. The effect is indeed pronounced, far outweighing the
statistical error and the model/code dependency. When sufficient
statistical samples are computed, L_{z} hence constitutes the most
influential source of error. It is to be expected that this finding
would apply to nominally 2D bluff bodies in general.
The plots of Figure 10
and Figure 11
suggest that the L_{z} influence
becomes negligible at values of around 4c for this test case, although
further computations at still larger L_{z} values would strictly be
required to demonstrate this. An indication that the required L_{z} is
strongly case-specific is given by comparison with the results of
Guenot [3] for the lower α = 45° case.
In that case, the L_{z} effect was
seen to settle at lower values of L_{z} ≈ 2c.
It is believed that the
wider wake is responsible for the larger Lz required for
our α = 60° case.
Figure 12: The effect of L_{z} on the mean C_{p} distribution. Comparison of NTS SA-DES results (left) and TUB CEASM and SA-DES results (right) with the experimental data of Swalwell [27,28] |
The mean surface pressure coefficient, C_{p},
plotted in Figure 12,
corroborates the trends shown by Fig. 10
and shows that the L_{z} effect
is manifested in a strong variation of the base pressure on the
downstream (upper) surface of the airfoil. The high comparability
between the left-hand and right-hand plots demonstrates the negligible
influence of different solvers and underlying RANS models. The paradox
that the agreement with experiment degrades with increasing L_{z} was
initially a source of concern. However, the observation of a strong
scatter between the experiments (particularly visible in
Fig. 10)
reduces this. Furthermore, the conclusion to be reached from
Fig. 10 is
that increasing L_{z} brings the mean C_{l}
inside the experimental range and
in excellent agreement with the Sheldahl & Klimas data. For the mean C_{d}
however, the agreement is less positive. This could in part be because
the C_{d} value from the Raghunathan et al., experiment is unknown:
Assuming that the lift-to-drag ratio of 0.60 observed in the other
experiments applies, this could be estimated to lie in the region of
C_{d} ≈ 1.37.
Unlike the mean quantities, the effect of increasing L_{z} on the
prediction of unsteady quantities is definitely positive: as seen in
Figure 11,
a dramatic improvement of agreement with the data on the
standard deviation of the integral forces is achieved at longer L_{z}
values. More detail is offered by the PSD of C_{l}, plotted in
Figure 13.
Particularly regarding the lowest frequencies, which are associated
with the long-wavelength intermittency, the agreement of the broadband
energy is strongly improved at L_{z} = 4c.
The sharp spectral peak at
St = 0.2 represents the vortex shedding frequency which well agrees with
the experiment. However a tangible disagreement with the data in the
energy of the peak which is more clearly seen in plots of the energy
pre-multiplied by frequency (not shown) is somewhat troublesome.
Figure 13: PSD of C_{l} compared between NTS and TUB DES results with varying L_{z} and the Swalwell experimental data [27,28]. |
A natural statistical quantity to examine in such an investigation is
the two-point correlation of velocity in the spanwise direction,
, where i represents the u, v and w
velocity components. These
are plotted in Figure 14
at four probe positions in the near wake for
TUB's computations at L_{z} = 1c
and L_{z} = 4c. For the u and v components,
the correlations settle to lower values in the wider domain, however
they clearly remain above zero. Such long spanwise correlation lengths
naturally arise due to the strong spanwise coherence of the large-scale
vortex shedding. For the w component a different behaviour is seen: in
the narrow domain the correlation consistently reaches a negative
value, implying an anti-phase behaviour. For the longer domain however,
the w correlation settles around zero. It is therefore suggested that
could provide a generic test for sufficient span size in
practice. This would however require more testing to confirm.
Figure 14: Two-point spanwise correlations of u, v and w compared for L_{z} = 1c (TUB CEASM-DES) and L_{z} = 4c (TUB SA-DES) at the probe locations shown in the left-hand figure. |
Finally, further insight into the span size effect can be obtained from
instantaneous visualisations of the resolved vortex structures
(Figure 15).
The streamwise rib vortices and spanwise von Kármán vortices
maintain a high degree of orthogonality in the narrow spanwise domain.
In contrast, these exhibit less orthogonality with spanwise
dislocations and higher degrees of slant apparent in the wider domain.
Figure 15: Instantaneous snapshots of resolved vortical structures compared between L_{z} = 1c (TUB CEASM-DES) and L_{z} = 4c (TUB SA-DES) calculations. Isosurfaces of λ_{2} = -4 [7] shaded with velocity magnitude, view from below. |
Effect of wind tunnel walls
It was suspected that the surrounding wind tunnel walls, neglected in the spanwise-periodic computations, may contribute to the discrepancy with the Swalwell experiments. This was the motivation for the EADS simulation including the wind tunnel walls. The mean surface C_{p} and PSD of C_{l} plotted in Figure 16 indicate a negligible impact of the tunnel walls. The time-averaged force coefficients give a similar message: The C_{l} = 0.89 ± 0.016 and C_{d} = 1.43 ± 0.029 from the computation with wind tunnel walls do not differ significantly from the NTS periodic computation (C_{l} = 0.88 ± 0.007 and C_{d} = 1.38 ± 0.012). Although the comparison is unfortunately not direct (employing different L_{z}, solvers, grid resolution, and time step), the experience of various related investigations implies that these factors have a negligible influence for this test case^{[3]}.
Figure 16: Comparison of PSD of C_{l} (left) and mean surface C_{p} (right) between the spanwise-periodic NTS computation (L_{z} = 4c) and the EADS computation with side walls (L_{z} = 7.2c) and with the experimental data of Swalwell [27,28]. |
- ↑ This drift arises as the flow switches to a protracted bout of strong shedding behaviour at around t* = 800.
- ↑ A software implementing this algorithm for the estimation of statistical error and the detection of initial transient is commercially available: Contact info@cfd-berlin.com for details.
- ↑ L_{z} is not expected to have a large effect for values above around 4c
Contributed by: Charles Mockett; Michael Strelets — Upstream CFD GmbH, Berlin; New Technologies and Services LLC (NTS) and St.-Petersburg State Polytechnic University
© copyright ERCOFTAC 2023