forward-facing step on the instability of crossflow vortices
Guillaume Chauvat
1, Francesco Tocci
2, Alberto Rius-Vidales
3, Marios Kotsonis
3, Stefan Hein
2and Ardeshir Hanifi
11 KTH Royal Institute of Technology, Engineering Mechanics, FLOW Centre and Swedish e-Science Research Centre (SeRC)
2DLR German Aerospace Center, 37073 Göttingen, Germany
3 Delft University of Technology, Delft, The Netherlands, 2629HS
To be submitted
We reproduce three variants of an experimental setup consisting of a swept wing in a wind tunnel through direct numerical simulation (DNS). The effect of a forward-facing step of two different heights at location x/cx= 0.2on a steady crossflow instability is analysed. The effect of discrete roughness elements used to condition the instability modes in the experiments is accurately reproduced via perturbations computed through nonlinear parabolised stability equations.
Our results demonstrate that the setup can be reproduced accurately numerically as far as the steady flows are concerned, with a good agreement between the experimental and numerical profiles upstream and downstream of the step. Our results shed light on the flow structure in the vicinity of the step.
1. Introduction
Improving aviation’s ecological footprint is a current drive for the aerospace industry. In the quest for greater aircraft efficiency, reduction of skin-friction drag has shown a great potential as it constitutes a significant part of the total drag budget (Schrauf 2005).
The skin friction drag associated with a laminar boundary layer is signifi- cantly lower than in a turbulent one and therefore maintaining laminar flow on wings at a chord Reynolds number beyond what is usually transitional or turbu- lent is very appealing. Therefore, several active or passive laminar flow control techniques (LFC) (Joslin 1998; Arnal & Archambaud 2008; Saric et al. 2011) have been developed for the delay of the laminar-turbulent transition. However, their success depends on the smoothness of the wing surface. Two-dimensional surface irregularities, such as gaps or steps, which may cause a premature boundary-layer transition arise from junction defaults, manufacturing processes or operational constraints. Consequently the practical implementation of a
99
laminar-flow wing on operational aircraft requires the knowledge of the permit- ted tolerances for the above-mentioned irregularities such that the breakdown of laminar flow is not promoted.
The laminar-turbulent transition on highly swept-wings, typical of contem- porary transport aircraft, is mainly due to the growth and breakdown of the so-called crossflow instability (CFI). This instability leads to the growth of co-rotating vortices, oriented within a few degrees of the local inviscid stream- lines, depending on the boundary-layer receptivity to the external disturbance environment and surface roughness (Bippes 1999; Saric et al. 2003), they can be of stationary or travelling nature. The majority of the literature focuses on the studies on crossflow instability on otherwise idealised and smooth geometries (Högberg & Henningson 1998; Wassermann & Kloker 2003; White & Saric 2005;
Hosseini et al. 2013; Serpieri & Kotsonis 2016) and only recently the effect of two-dimensional steps on such conditions has been investigated.
Tufts et al. (2017) investigated the interaction of forward-facing (FFS) and backward facing steps (BFS) with stationary crossflow vortices numerically.
Regarding FFS they proposed that when the core-height of the crossflow vortices is lower than the height of the step a constructive interaction occurs between the CF vortices and the recirculation region downstream of the FFS edge which leads to the amplification of the stationary crossflow vortices. However, experimental investigations by Eppink (2020) on the mechanisms of the FFS- CFI interaction casts doubts on the above mentioned constructive interaction.
Moreover, Eppink (2020) suggest that the interaction is largely dominated by the initial amplitude of the CF vortices. In addition, experimental investigations by Rius-Vidales & Kotsonis (2020b) indicate that the exclusive use of local one-parameter correlations (i.e. CF vortices core-height or relative step height h/δ∗) might not fully capture the influence of the FFS on the laminar-turbulent transition.
The unresolved aspects of the development of the crossflow instability with FFS surface irregularities lead to a recent experimental investigation conducted by Rius-Vidales & Kotsonis (2020a). They have found that effects of the step height on the transition location is not monotonic. In certain parameter region, it is observed that steps with small height can delay transition while higher steps cause the transition line move upstream. The present work aims to numerically investigate these recent experiments in order to gain a better understanding of the transition process on a swept-wing in the presence of a forward-facing step.
2. Governing equations
We perform numerical simulations for flow condition in experiments performed at low free-stream velocity. Therefor, we neglect here the compressibility effects and consider the three-dimensional incompressible Navier-Stokes equations:
∂u
∂t + (u · ∇)u +∇p
ρ = ν∇2u (1)
∇ · u = 0 (2)
Z Y
IR-A b
Wind tunnel Bottom
IR-B
0.24b
Wind tunnel Top
X y x z
PIV Laser Unit PIV Imaging
U∞
0.3b 0.4b
0.76b
�
dDDREx\cx,Dz,D
λ
FFS Upper Pressure Taps
v u
w Outboard Side
Inboard Side Lower Pressure Taps
Laser-plane
Figure 1: Schematic of the experimental setup (flow direction is from left to right) showing the forward-facing step location, Infrared Themography analysis regions (IR-A,IR-B) and the planar particle image velocimetry camera and laser arrangement (reproduced from Rius-Vidales & Kotsonis (2020a))
where ν is the kinematic viscosity, ρ is the density, p is the pressure and u = (u, v, w)is the velocity vector.
Following an infinite-swept wing approximation, the flow can be considered spanwise invariant and as such, computation of the spanwise velocity component wcan be decoupled from that of the u and v components of the velocity and considered as a passive scalar transported by a two-dimensional flow. The equations become
∂u
∂t + (u · ∇)u +∇p
ρ = ν∇2u, (3)
∇ · u = 0, (4)
∂w
∂t + (u · ∇)w = ν∇2w. (5)
Here, u = (u, v) is the velocity in the plane normal to the spanwise direction.
This approach is referred to as 2.5D in the following and has been used to compute the steady base flow for the considered problem.
3. Geometry and flow parameters
In this work, we numerically reproduce the experiments conducted by Rius- Vidales & Kotsonis (2020a) in the Low Turbulence Tunnel (LTT) at Delft University of Technology. The geometry presented here in figure 1, consists of a 45 degree swept-wing model which features a chord length normal to the leading edge of cx= 0.9 m(cX= 1.27 min the streamwise direction) and the measured surface roughness standard deviation is 0.2 µm. The model is placed vertically in the wind tunnel test-section and set at a α = 3◦ angle of attack. The airfoil
cx 0.9 m
cX 1.27 m
Sweep 45◦
AoA 3◦
U∞ 27.542 m s−1
ρ 1.2069 kg m−3
µ 1.7829 × 10−5kg m−1s−1 RecX 2.37 × 106
hS1 368µm
hS2 474µm
λ0 7.5 mm
Table 1: Summary of the flow and geometry parameters.
shape is symmetric, thus the angle of attack can be considered negative, as measurements are conducted on the pressure side.
The airfoil shape, which consists of a modified version of the NACA 66018 (Serpieri & Kotsonis 2015), at mild angles of attack enhances the growth of the crossflow instability by promoting a favorable pressure gradient up to x/cx≈ 0.65. Moreover, the lack of concave surfaces and the small leading-edge radius of approximately 1% of the chord avoid the amplification of Görtler vortices and attachment line instabilities (Poll 1985; Bippes 1999).
At the reference conditions of the experiments the free-stream turbulence intensity is as low as T u = (1/U∞)p(u02+ v02)/2 ≤ 0.03% (hot wire mea- surements, bandpass filtered between 2 and 5000 Hz), ensuring dominance of stationary crossflow vortices in the laminar-turbulent transition(Bippes 1999;
Downs & White 2013). This is confirmed in the experiments through thermal IR measurements of the model surface which show a jagged transition front char- acteristic of the breakdown of stationary crossflow vortices. Moreover, during the experiment discrete roughness elements are used near the leading edge to force a CFI mode featuring a spanwise wavelength λ0= 7.5 mm corresponding to a wavenumber of β0= 837.758 m−1
In order to replicate as well as possible the flow in the near-field of the airfoil, we choose the following far-field flow parameters: free-stream velocity U∞= 27.542 m s−1 (measured just upstream of the wind tunnel model using a pitot-static tube), free-stream pressure p∞ = 9.941 × 104Pa, density ρ = 1.2069 kg m−3 and dynamic viscosity µ = 1.7829 × 10−5kg m−1s−1. These parameters, summarised in table 1, correspond to a streamwise chord Reynolds number of RecX = 2.37 × 106.
During the experiments, the height h of the FFS on the surface of the swept-wing model was measured in-situ employing a laser line profilometer.
Based on this information we selected three different cases for their numerical study and comparison. A clean (i.e. no step) reference case (S0), and two
Case h/δ h/δ∗
S0 0 0
S1 0.23 0.77 S2 0.30 0.99
Table 2: Normalised step heights.
−1.5 −1 −0.5 0 0.5 1 1.5 2
−1
−0.5 0 0.5 1
x/cx
y/cx
Figure 2: The geometry of the different domains used in the present simulations projected on a plane normal to the leading edge. Dashed lines indicate extend of domain in y-direction for RANS simulations, corresponding to the width of the wind tunnel. The streamwise extend of the RANS mesh is about eight chord lengths. The domain for the 2.5D simulations is shown in light blue, and the one used for the 3D simulations is shown in red.
FFS cases (S1 h = 368 µm and S2 h = 474 µm) positioned at a location of x/cx = 0.2. The normalised step heights with respect to the boundary layer thickness δ and the displacement thickness δ∗are given in table 2. Here, δ and δ∗ are based on the velocity in direction of the local outer streamline. The case S1 corresponds to the step height which has caused a delay of transition in the experiments, while in the case S2 transition line has moved upstream compared to the smooth surface.
4. Numerical methods and computational setup
4.1. Computational toolDirect numerical simulations are performed with the spectral element code Nek5000 (Fischer et al. 2008). For the 3D cases, one spanwise wavelength is simulated using eight spectral elements in the spanwise direction. The choice of spanwise extension of domain is based on the fact that no subharmonics of the forced crossflow modes were observed in the experiments. The number of elements for the cases S0, S1 and S2 is respectively 103488, 103904 and 104880.
The polynomial order is chosen to N = 9. For the sake of convergence analysis, a simulation for the S1 case is also performed with N = 7.
The airfoil is parametrised using cubic splines as a function of curvilinear abscissa. This allows the curvature to be approximated in a consistent way at each location.
4.2. Computational approach
The three-dimensional simulations being expensive, they are run on a relatively small domain that requires proper boundary conditions. Those boundary condi- tions are obtained in two steps. First, a 2.5D RANS calculation is performed in a section of the wind tunnel in which the full airfoil section and the walls of the wind tunnel are modeled.
The RANS calculations have been performed with the M-Edge code (Elias- son 2002) using Spalart-Allmaras one-equation model (Spalart & Allmaras 1992). In order to avoid unsteady RANS solution due to the laminar separation, the laminar-turbulent transition was triggered at x/c = 0.17 on the lower side of the wing (identical to the location of a zig-zag tape, used in the wind tunnel experiment) and at x/c = 0.6, close to the minimum pressure, on the upper side by turning on the turbulence production. The resulting field is used to set the far-field conditions for a 2.5D DNS that computes an accurate approximation of the steady flow field in a domain of intermediate size. This DNS is performed once for each step height, including the clean case. Note that the RANS simu- lation was performed for the wing with smooth surface, considering that the step height is significantly smaller than the chord, and thus does not affect the far-field pressure. In addition, the domain for 2.5D calculations is so large that presence of the steps is thought not affecting the flow at the boundaries. The three different domains are shown together in figure 2.
4.3. Boundary conditions
Surface of the wing is subjected to a no-slip condition. We impose Dirichlet conditions on the inflow boundary. The inflow velocity values for the 2.5D DNS are taken from the RANS calculations. In the case of the three-dimensional DNS the inflow velocity is extracted from the 2.5D DNS.
The 2.5D DNS possess two outflow boundaries on which the pressure is specified according to
µ∂u
∂n− pn = −p0n (6)
where p0 is the target pressure interpolated from the RANS solution and n is the normal to the surface.
For the top boundary, Dirichlet conditions would not be satisfactory given the closeness of the boundary to the boundary-layer where the flow can be turbulent or contain large amplitudes of crossflow vortices which make imposing the vertical velocity too constraining. Instead, a modification of the condition used in previous related works (Brynjell-Rahkola et al. 2017) is derived. The top boundary is both horizontal and roughly parallel to the wall surface, which is achieved by rotating the domain by 8◦ clockwise (see figure 2 for an overview of the domains before rotation). The original condition for the vertical velocity was similar to the outflow condition (6) and was observed to be unstable in our case due to the significant downwards velocity present on that boundary, making it act as an inflow rather than an outflow. The modified condition reads
u = u0 (7)
µ∂v
∂y − ρv0(v − v0) − p = µ∂v0
∂y − p0 (8)
w = w0 (9)
where y denotes the vertical direction in the rotated domain, where it is normal to the boundary, and (u0, v0, w0)is the baseflow velocity in the rotated domain.
The boundary condition 8 is obtained by subtracting the condition µ∂v∂y = µ∂v∂y0 from the linearised Bernoulli equation p + v0(v − v0) = p0+ v20.
4.4. Fringe
In order to avoid back flow or reflection at the outflow boundary, which could result in numerical instabilities, we force the flow to the unperturbed state determined from the 2.5D simulations upstream of the outflow. The forcing function used is
f (x, y, z, t) = −ρkS x − xstart
∆xrise
(u(x, y, z, t) − u0) (10) with an amplitude k = 2000 s−1. The spatial distribution of the fringe is given by
S(x) =
0 if x ≤ 0,
1/ (1 + exp(1/(x − 1) + 1/x)) if 0 < x < 1,
1 if x ≥ 1. (11)
(Chevalier et al. 2007) where xstart= 0.54 mand ∆xrise= 25 mm.
0 0.1 0.2 0.3 0.4 0.5 0.6
−0.6
−0.4
−0.2 0 0.2 0.4 0.6
x/cx
−Cp
Figure 3: Pressure coefficient distribution for the smooth case in the DNS (plain line) and in the experiment in the lower (crosses) and upper (circles) pressure taps.
5. Results: Steady flow
We start comparison of the DNS results with the experimental data by looking at the distribution of the pressure coefficient Cp. In figure 3, both numerical and experimental values of Cp are plotted. Experimental data includes values obtained from two rows of pressure taps located at 24 and 76% of the model span. As it can be seen there, a close agreement between data from simulations and experiment is found.
In order to illustrate the stability characteristics of the CF vortices pertain- ing to the present flow conditions, linear PSE calculations have been performed for vortices with wavenumbers β ∈ [500, 2100] m−1 (spanwise wavelengths of approximately 3 to 12 mm) and frequencies f ∈ [0, 1000] Hz. Results of these calculations are presented in figure 4 where amplification of the perturbations (N-factor) based on the maximum of their streamwise velocity component is plotted. At the location of the step (x/cx= 0.2), the dominating modes have a wavenumber around β = 1000 m−1 (wavelength of about 6.2mm) and a fre- quency around f = 400 Hz. Nevertheless, as mentioned earlier, the dominance of steady crossflow instabilities is well established at low freestream environments, such as the one established in the present experiment. This is particularly important in cases of upstream conditioning using DREs.
-0
0.1 0.2 0.3 0.4 0.5 0.6
x/cx 500
1000 1500 2000
- [rad/m]
0 1 2 3 4 5 6 7 8
N-factor
0 200 400 600 800 1000
f [Hz]
500 1000 1500 2000
- [rad/m]
0 1 2 3 4 5 6 7 8
N-factor
Figure 4: N-factors based on the maximum streamwise disturbance velocity for the wing without step. (top) steady cross-flow vortices, (bottom) unsteady vortices at x/c = 0.2. Wavenumber β0 corresponds to the spacing of the roughness elements. Isolines have an step of 1 and start from N = 1.
5.1. Generation of steady perturbations
In order to reduce the size of the computational domain and thereby the computational cost, the receptivity stage, namely generation of the CF vortices by the roughness elements, is not considered in the DNS performed here.
Instead, the stationary CF vortices are imposed as a boundary condition on the inflow boundary of the domain, superposed to the homogeneous base flow
calculated in section 4.2. Their shape and amplitude is determined using nonlinear Parabolised Stability Equations (PSE) (Bertolotti et al. 1992). The baseflow for the PSE is calculated with a boundary-layer code using the wall pressure distribution obtained by the 2.5D baseflow solution (section 4.2).
The PSE are integrated from the location x/c = 0.05 and the amplitude of the fundamental mode, corresponding to spacing of the roughness elements, is chosen to match the experimental data as closely as possible. Calculations are performed using the nonlinear version of the NoLoT code (Hanifi et al. 1994;
Hein et al. 1994). The proper matching is demonstrated in figure 5 by the close agreement of the PSE solutions with experimental data around the location of the step. Here, the time-average and the r.m.s. values of the spanwise velocity component are plotted. Note that due to uncertainty in the wall position in the experiments, the experimental data has slightly been shifted in the wall-normal direction such that the location of the maximum of wrmsmatches those from PSE.
5.2. Smooth geometry
In order to demonstrate the validity of our approach for the generation of steady crossflow vortices, their amplitude from PSE is compared with experimental data, see figure 5. As it can be seen there, a close agreement between results of the nonlinear PSE computations and the measured data is found at the streamwise locations 0.173 ≤ x/cx≤ 0.227.
The pressure distribution in the absence of perturbations is in good agree- ment with the experiments (figure 3). The pressure coefficient in the experiments is determined from static pressure measurements on the surface of the model along the streamwise direction at two different span locations as shown in figure.1 and reference to the static and dynamic pressure measured by a Pitot-Static tube located just upstream of the swept-wing model. The velocity profiles (figures 9 and 10) agree very well with the experiments.
5.3. Geometries with step
Here, we discuss effects of the steps on the baseflow and development of the stationary cross flow vortices. The accuracy of our results was checked by performing simulations with a lower polynomial order. It was found that the amplitude of the crossflow vortices in the S1 case remain the same within 0.3%
when the order is decreased to N = 7, which indicates that the spatial resolution is sufficient.
Even though the steps are small, they have a visible effect on the pressure coefficients, especially in the upstream region, as shown in figure 6. The flow experiences an increase of pressure upstream of the step and a rapid acceleration down stream of the step. In figure 7 the flow filed around the steps is plotted.
As it can be observed there, a small recirculation bubble is present upstream of the step attached to it. The streamwise extend of the separation bubble in the S2 case is almost twice of that in the S1 case. Time signals of the velocity
0 1 2 3 4 5 7
w=we
0 1 2 3 4
y[mm]
S2 S1
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
wrms=we
0 1 2 3 4
y[mm]
S2S1
Figure 5: Comparison of the experimental measurements (circles) with the PSE solutions (black lines) for (top) the spanwise average velocity ¯w/weand (bottom) its spanwise standard deviation wrms at the streamwise locations x/cx= 0.173, 0.180, 0.188, 0.196, 0.204, 0.211, 0.219 and 0.227. The plots for consecutive locations are shifted by (top) 0.5 and (bottom) 0.2 for legibility.
and pressure at locations close to the steps verify that these separation bubbles are steady. There is no recirculation downstream of the steps in the baseflows for the step heights considered here. Further, it is observed that the boundary layer is thicker upstream of the steps and a new boundary layer starts to grow after the step.
In the experiments, the flow velocity components in the spanwise and wall- normal directions are measured. Since the wall-normal component is in general small and more difficult to measure with high precision, we choose to compare our results in terms of the spanwise velocity component. The amplitude of the vortices is determined using the maximum of the spanwise standard deviation of the spanwise velocity. The computed amplitudes along with the measured ones are presented in figure 8. We observe that the presence of the step increases the amplitude of the vortices at the step location, but decreases it downstream.
0.05 0.1 0.15 0.2 0.25 0.3
−0.2
−0.1 0 0.1 0.2
x/cx
−Cp
Figure 6: Pressure coefficient around the step location for S0 (solid line), S1 (dashed line) and S2 (dash-dotted line). The red vertical line marks the location of the step, x/cx= 0.2.
This could be related to the fast variation of the pressure gradient around the step (figure 6). The flow is first decelerated upstream of the step, then decelerated again after the corner, then accelerated again. The observed trends in amplitude variation in the simulations and the experiments are the same.
The amplitudes of perturbations upstream of the steps are in a good agreement.
However, downstream of the steps experimental data show a larger decay of the amplitudes.
Velocity profiles for the three cases in the presence of crossflow vortices are extracted at selected positions around the step location and compared with the experimental profiles. Note that the exact location of the plane around the step in those figures is slightly modified to compensate for uncertainty in the location of the experimental measurement planes and the fact that the laser sheet in the experiments has a finite thickness. In figures 9, the spanwise component of the velocity is plotted. These data are the averaged in the spanwise direction.
As it can be seen in this figure, a good agreement between the data from the simulations and the experiments is find. The corresponding perturbation profiles in terms of their spanwise-averaged values are presented in figure 10, showing a close agreement between DNS and measurements. In general, the perturbations in all three cases have a similar shape. Increasing the step height cause a an increase of the amplitude of the crossflow vortices at location of the steps. At a location slightly upstream of the steps (x/cx≈ 0.23) amplitude of the perturbations decays slightly in the presence of the step compared to the smooth case. This amplitude reduction is smaller for the lowest step, S1. This
variation of perturbation amplitude is more significant in the experimental data.
Moreover, data show appearance of a second peak in the perturbation profile at outer part of the boundary layer. This second peak is more pronounced in the experimental data for the highest step, S2.
Details of the 3D flow around the step are shown in figure 11. The recircula- tion bubble observed in the 2D flow is significantly reshaped by the presence of the CF vortices. It is not present at all spanwise locations, but forms separate periodic bubbles instead. New recirculation regions also appear just downstream of the steps, especially in the S2 case, where the boundary-layer is being lifted from the wall by the CF vortices. In this simulation without unsteady forcing the flow appears completely steady, including in the region around the step.
6. Results: Unsteady flow
6.1. Generation of unsteady perturbations
If steady crossflow vortices alone are imposed at the inflow, no transition is observed due to the low amount of numerical noise in the simulations. Unsteady noise is therefore added as a forcing in the boundary-layer just downstream of the outflow. The approach for the forcing is identical to the trip forcing described in Hosseini et al. (2016). The forcing acts in the momentum equation on the component normal to the wall surface. It is centred at the location x/c = 0.142, 2.1 × 10−4m above the wall, and acts on 16 Fourier modes in the spanwise direction. In the time domain, these perturbations have a cut-off frequency of 6500 Hz. This value is chosen based on the earlier measurements of disturbance spectrum in the same facility at similar flow conditions (Serpieri
& Kotsonis 2016).
6.2. Transition
As mentioned above, unsteady noise was artificially added to the filed close to the inflow boundary in order to replicate noise in the wind tunnel. Their initial amplitude is chosen such that the laminar-turbulent transition in the simulations is close to that in the experiments in the smooth case. The transition location is determined as the location where the friction coefficient reaches a value halfway between those of laminar and turbulent regimes. It must pointed out that if the type and spectrum of the unsteady perturbations present in the DNS is not close to that in experimental environment, the transition scenario can be different and the location of the transition could prove unreliable in different cases. In figure 12 result of a set of our simulations in terms of the time- and spanwise-averaged skin friction is presented.
In order to assess the properties of the generated unsteady perturbations, the standard deviation in time of the flow is computed for the case S0 and shown in figure 13. Here, also the experimental data are presented for the sake of comparison. As expected, the unsteady perturbations are concentrated to the region of high shear generated by the steady vortices (Högberg & Henningson 1998; Malik et al. 1999, see e.g.).
(a) (b)
(c) (d)
(e) (f)
Figure 7: Velocity (streamwise, wall-normal and spanwise relative to the wall coordinates at the step location, from left to right) around the step in the spanwise-homogeneous base flows for ((a),(c),(e)) the S1 case and ((b),(d),(f)) the S2 case, showing the regions of negative streamwise velocity upstream of the step in orange.
The experimental data show that the time r.m.s values of perturbations has similar maximum values (around 2 m s−1) at x/cx= 0.179and 0.239 but their structures are different. At x/cx = 0.239, the shape of the unsteady perturbations resemble that of the so-called z-type of the secondary instability modes. The simulations data show a different behaviour. The structure of the
0.15 0.2 0.25 0.3 0.35 0.4 0
1 2 3
x/c
|w|
Figure 8: Amplitude of the crossflow vortices in the DNS for S0 (plain line), S1 (dashed line) and S2 (dash-dotted line), measured as the maximum of the spanwise standard deviation of w at different chordwise locations. The same data for the experimental data is plotted with red circles (S1) and triangles (S2).
unsteady perturbations does not have any similarity with the z- or y-type of secondary instability modes (Malik et al. 1999). Further, the amplitude of the observed structure in those planes is one to two orders of magnitude smaller and strongly increases downstream. The structure and growth of this perturbation indicates that they may consist of travelling CF vortices, which have a low frequency with the most unstable ones at 400 Hz, whereas the experimental data in the same facility and at similar flow conditions (Serpieri & Kotsonis 2016) shows high frequency noise around 6.5 kHz that is damped as it travels downstream. This is also consistent with the large extent of the transition region (figure 12) which indicates a large region of intermittent turbulence. To exactly identify the type of the observed perturbation in our simulations further analysis are required.
7. Summary and conclusions
We have performed direct numerical simulations corresponding to the recent experiments by Rius-Vidales & Kotsonis (2020a) studying effects of the forward- facing steps on the laminar-turbulent transition over a swept wing model.
The objective of the present work is to through detailed analysis of the flow understand the flow physics behind the observed behaviour in the experiments.
Three flow cases are considered here: a smooth case (S0) and two cases with step heights h/δ∗= 0.77 (S1) and 0.99 (S2) at x/cx= 0.2. The case S1 corresponds to the step height which has caused a delay of transition in the experiments, while in the case S2 transition line has moved upstream compared to the smooth surface. Results of the baseflow simulations revealed existence of steady small recirculation bubbles upstream of the steps. The streamwise
0 5 10 15 20 0
1 2
w
y(mm)
(a)
0 5 10 15 20
0 1 2
w
y(mm)
(b)
0 5 10 15 20
0 1 2
w
y(mm)
(c)
0 5 10 15 20
0 1 2
w
y(mm)
(d)
0 5 10 15 20
0 1 2
w
y(mm)
(e)
0 5 10 15 20
0 1 2
w
y(mm)
(f)
0 5 10 15 20
0 1 2
w
y(mm)
(g)
0 5 10 15 20
0 1 2
w
y(mm)
(h)
0 5 10 15 20
0 1 2
w
y(mm)
(i)
Figure 9: Mean spanwise velocity (in m/s) in selected planes ((a), (d), (g)) upstream, ((b), (e), (h)) around and ((c), (f), (i)) downstream of the step, for (a)–(c) S0, (d)–(f) S1 and (g)–(i) S2. The exact locations are x/c = (a) 0.173, ((d), (g)) 0.176, ((b), (e), (h)) 0.200, (c) 0.227, and ((f), (i)) 0.231.
extend of the bubble in S2 case was almost twice that for S1 case. No flow separation was observed downstream of the steps in the baseflow simulations.
In the experimental setup discrete roughness elements generate CF vortices, that are replicated in our simulations by profiles obtained through nonlinear PSE calculations. The pressure distribution and velocity profiles in the three cases agree very well with the experimental measurements in the laminar regimes.
The simulations were performed using a polynomial order N = 9. The accuracy of our results was checked by performing simulations with a lower polynomial
0 1 2 3 4 0
1 2
wrms
y(mm)
(a)
0 1 2 3 4
0 1 2
wrms
y(mm)
(b)
0 1 2 3 4
0 1 2
wrms
y(mm)
(c)
0 1 2 3 4
0 1 2
wrms
y(mm)
(d)
0 1 2 3 4
0 1 2
wrms
y(mm)
(e)
0 1 2 3 4
0 1 2
wrms
y(mm)
(f)
0 1 2 3 4
0 1 2
wrms
y(mm)
(g)
0 1 2 3 4
0 1 2
wrms
y(mm)
(h)
0 1 2 3 4
0 1 2
wrms
y(mm)
(i)
Figure 10: Spanwise standard deviation of the spanwise velocity (in m/s) in selected planes ((a), (d), (g)) upstream, ((b), (e), (h)) around and ((c), (f), (i)) downstream of the step, for (a)–(c) S0, (d)–(f) S1 and (g)–(i) S2. The x/c locations are identical as for figure 9.
order. It was found that the amplitude of the crossflow vortices in the S1 case remain the same within 0.3% when the order is decreased to N = 7, which indicates that the spatial resolution is sufficient.
Without any artificial noise in the simulation, the stationary crossflow vortices appear in all cases and the flows remain fully laminar. The effects of the step on the stationary vortices seems to be mostly concentrated around location of step. At the location of the steps the amplitude of these vortices
(a)
(b)
Figure 11: Visualisation of the steady flow around the step for (a) S1 and (b) S2. The red surfaces delimit the recirculation regions (negative streamwise velocity), and the spanwise velocity is plotted on the plane in the background.
For clarity, several identical spanwise wavelengths are shown together.
increases with increasing step height, while a short distance after the steps their amplitude is slightly lower compared to the smooth case.
The evolution of the laminar-turbulent transition location aimed to be studied by forcing the flow with random disturbances at frequencies similar than what is observed in the experiments. We used a randomly generated noise in a frequency range up to 6500 Hz. This noise has a flat temporal amplitude
0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0
2 4 6
·10−3
x/cx
cf
Figure 12: Average local skin friction coefficient as a function of chordwise location for S0 showing the average location of the transition. The experimentally observed transition region is marked by vertical lines.
for all temporal frequencies. However the exact shape of the noise may need to be more realistic in order to reproduce the transition location consistently in the three cases as significant differences are observed in the unsteady perturbations between the experiments and simulations. A comparison between the structures of the unsteady perturbation observed in experiments and those found in our simulations shows a significant difference. The structure, strong growth of the unsteady perturbations long upstream of the location of transition and the extended region of intermittent turbulence in transition region in the simulation point to a transition triggered by the interaction between the steady and travelling CF. While the frequency range of flow unsteadiness along with the observed structure of the measured r.m.s perturbations in the experiment indicate a transition based on secondary instability of the steady CF. This indicates that the spectrum of the noise used in simulations should be tailored more closely to that existing in the wind tunnel. Further simulations will be performed to achieve a closer agreement with the conditions in the experiments in order to be able explaining the observed effects of the step height variation on the transition in the three-dimensional flows.
Acknowledgement
This project has been funded through the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No 675008 and the European Research Council under grant agree- ment 694452-TRANSEP-ERC-2015-AdG. Computer resources provided by the Swedish National Infrastructure for Computing (SNIC) at the Center for High
(a) (b)
(c) (d)
Figure 13: Standard deviation in time of the spanwise velocity (in m/s) at x/cx= 0.176((a), (c)) and 0.239 ((b) and (d)), in the experiments ((a), (b)) and DNS ((c), (d)). Isocontours of the time-averaged spanwise velocity are shown superposed in yellow. Note the different colour scales for the perturbation amplitudes.
Performance Computing (PDC) at the Royal Institute of Technology (Stock- holm), the High Performance Computing Center North (HPC2N) at Umeå University, and the National Supercomputer Centre at Linköping University are gratefully acknowledged.
References
Arnal, D. & Archambaud, J. P. 2008 Laminar–turbulent transition control: NLF, LFC, HLFC. In Advances in Laminar–Turbulent Transition Modeling. VKI Lecture Series.
Bertolotti, F. P., Herbert, T. & Spalart, P. R. 1992 Linear and nonlinear stability of the Blasius boundary layer. Journal of Fluid Mechanics 242, 441–474.
Bippes, H. 1999 Basic experiments on transition in three-dimensional boundary layers dominated by crossflow instability. Progress in Aerospace Sciences 35 (4), 363 – Brynjell-Rahkola, M., Shahriari, N., Schlatter, P., Hanifi, A. & Hen-412.
ningson, D. S. 2017 Stability and sensitivity of a cross-flow-dominated Falkner–Skan–Cooke boundary layer with discrete surface roughness. Journal of Fluid Mechanics 826, 830–850.
Chevalier, M., Lundbladh, A. & Henningson, D. S. 2007 SIMSON–a pseudo- spectral solver for incompressible boundary layer flow. Tech. Rep.. KTH Royal Institute of Technology.
Downs, R. S. & White, E. B. 2013 Free-stream turbulence and the development of cross-flow disturbances. Journal of Fluid Mechanics 735, 347–380.
Eliasson, P. 2002 Edge, a navier–stokes solver for unstructured grids. In Proc. to Finite Volumes for Complex Applications III, pp. 527–534.
Eppink, J. L. 2020 Mechanisms of stationary cross-flow instability growth and breakdown induced by forward-facing steps. Journal of Fluid Mechanics 897, A15.
Fischer, P. F., Lottes, J. W. & Kerkemeier, S. G. 2008 Nek5000 Web page.
http://nek5000.mcs.anl.gov.
Hanifi, A., Henningson, D. S., Hein, S., Bertolotti, F. P. & Simen, M. 1994 Linear nonlocal instability analysis - the linear nolot code -. Tech. Rep. FFA TN 1994-54. FFA.
Hein, S., Bertolotti, F. P., Simen, M., Hanifi, A. & Henningson, D. S. 1994 Linear nonlocal instability analysis - the linear nolot code -. Tech. Rep. DLR-IB 223-94 A43. DLR.
Högberg, M. & Henningson, D. 1998 Secondary instability of cross-flow vortices in Falkner-Skan-Cooke boundary layers. J. Fluid Mech. 368, 339–357.
Hosseini, S. M., Tempelmann, D., Hanifi, A. & Henningson, D. S. 2013 Stabilization of a swept-wing boundary layer by distributed roughness elements.
Journal of Fluid Mechanics 718, R1.
Hosseini, S. M., Vinuesa, R., Schlatter, P., Hanifi, A. & Henningson, D. S.
2016 Direct numerical simulation of the flow around a wing section at moderate Reynolds number. International Journal of Heat and Fluid Flow 61, 117–128.
Joslin, R. D. 1998 Aircraft laminar flow control 1. Ann. Rev. Fluid Mech. 30 (1), 1–29.
Malik, M. R., Li, F., Choudhari, M. M. & Chang, C.-L. 1999 Secondary instability of crossflow vortices and swept-wing boundary-layer transition. J.
Fluid Mech. 399, 85–115.
Poll, D. I. A. 1985 Some observations of the transition process on the windward face of a long yawed cylinder. Journal of Fluid Mechanics 150, 329–356.
Rius-Vidales, A. F. & Kotsonis, M. 2020a Impact of a forward-facing step on the development of crossflow instability in swept wing. Manuscript in preparation . Rius-Vidales, A. F. & Kotsonis, M. 2020b Influence of a forward-facing step surface
irregularity on swept wing transition. AIAA Journal pp. 1–11.
Saric, W. S., Carpenter, Andrew, L. & Reed, H. L. 2011 Passive control of transition in three-dimensional boundary layers, with emphasis on discrete roughness elements. Philosophical transactions. Series A, Mathematical, physical, and engineering sciences 369 (1940), 1352–1364.
Saric, W. S., Reed, H. L. & White, E. B. 2003 Stability and transition of three-dimensional boundary-layers. Annual Review of Fluid Mechanics 35 (1), 413–440.
Schrauf, G. 2005 Status and perspectives of laminar flow. Aeronautical J. 109 (1102), 639–644.
Serpieri, J. & Kotsonis, M. 2015 Design of a swept wing wind tunnel model for study of cross-flow instability. AIAA paper 2015-2576.
Serpieri, J. & Kotsonis, M. 2016 Three-dimensional organisation of primary and secondary crossflow instability. Journal of Fluid Mechanics 799, 200–245.
Spalart, P. & Allmaras, S. 1992 A one-equation turbulence model for aerodynamic flows. AIAA paper 1992-0439.
Tufts, M. W., Reed, H. L., Crawford, B. K., Duncan, G. T. & Saric, W. S.
2017 Computational investigation of step excrescence sensitivity in a swept-wing boundary layer. Journal of Aircraft 54 (2), 602–626.
Wassermann, P. & Kloker, M. 2003 Transition mechanisms induced by travel- ling crossflow vortices in a three-dimensional boundary layer. Journal of Fluid Mechanics 483, 67–89.
White, E. b. & Saric, W. S. 2005 Secondary instability of crossflow vortices. Journal of Fluid Mechanics 525, 275–308.
