3. Numerical Results and Discussion for the Curved Hill
Turning into the curved hill scenario, let us begin by setting our bases by contrasting our numerical predictions with experimental data from the paper “A turbulent flow over a curved hill Part 1”, [
1].
Figure 11,
Figure 12 and
Figure 13 show contour results of kinematic pressure gauge (
), streamwise velocity (
) and static temperature (
T), respectively; via the SA turbulence model and fine mesh. Zero values of the kinematic pressure gauge were assigned to the outflow plane (reference or atmospheric pressure). It is very important to note that in this work the surface streamline distance (
s) was matched to the presented in [
1], also the wall-normal distance is represented with
n. In the path of the shear-layer region (i.e., turbulent boundary layer), the following aspects can be mentioned. As the flow approaches the hill or obstacle, the pressure increases, the flow decelerates and the pressure gradient becomes strongly adverse. Zones with intense red color can be seen along the first concave region (987 mm <
s < 1191.5 mm). The fluid velocity in the viscous and buffer layer decelerates, inducing a decrease in the skin friction coefficient, as will be shown later in the manuscript. However, no strong backflow or reverse flow is seen due to the moderate APG infringed. Whereas, the temperature generally shows a small gradient with just a small temperature drop in a very limited location at the peak of this pressure strong adverse gradient. In the geometry change from concave to convex, the pressure gradient switches to favorable (FPG), this is translated to streamwise velocity acceleration. This flow acceleration or FPG continues until the flow reaches the hill’s top. Clearly, the flow decelerates downhill, with a significant recovery of the pressure coefficient (as seen in
Figure 14a). The presence of this strong APG (the pressure increase equals the dynamic pressure, since the change in
−1) ends up in flow separation, with a posterior flow reattachment due to the presence of zero-pressure gradient (ZPG) zone, again. Temperature shows no change during this section with the exception of the very near wall. At the start of this separation zone, the temperature begins to drop. At the end of the hill, the separation “bubble” is noticeable due to the presence of a quasi-isothermal zone (in blue). The high level of mixing inside the separation bubble balances the static temperature. This is consistent with observations in the thermal boundary layer downstream of crossflow jet problems via DNS [
36].
Figure 14a depicts the pressure coefficient along the computational domain. The pressure coefficient is defined as
. Here,
is the wall static pressure,
is the freestream static or reference pressure, and
is the freestream dynamic pressure. In general, a fair agreement is observed with experimental data by [
1]. As the flow approaches the obstacle or hill, it decelerates due to the presence of an increasing pressure or APG. The maximum
(≈0.375) is located at the hill feet. Interestingly, good performance of both turbulence models in reproducing the wall pressure coefficient was observed in the vicinity of the hilltop, where the streamwise pressure gradient abruptly switches from FPG to APG, passing through a very short ZPG-zone. It is expected that boundary layer flow experiences a severe distortion in that zone with combined pressure gradients. Major discrepancies occur by the end of the strong APG zone (second half of the hill) where back or recirculating flow can be found. This is consistent with the deficient performance of RANS-eddy viscosity models in capturing boundary layer detachment. While both turbulence models have predicted constant wall static pressures in the separation bubble (ZPG zone), which is physically sound; however, smaller pressure gauges were obtained by SA and SST, e.g.,
−0.0625 and −0.125, respectively, as compared to the measured valued of −0.25. In
Figure 14b, the skin friction coefficient,
, is depicted. The skin friction coefficient is defined as follows,
, where
is the wall shear stress. One can observe an opposite trend of
as compared with the pressure coefficient
. As the flow decelerates due to the presence of moderate APG nearby the hill feet (concave surface), it is seen a decreasing behavior of
just downstream of the ZPG region where almost constant skin friction coefficient values are seen, as expected. However, it never reaches negative values, indicating that the mean flow does not separate. The wall shear stress then recovers as the flow starts to accelerate in the FPG region (convex surface). At roughly one-quarter of the curved hill (where the surface changes geometry from concave to convex) a meaningful increase of the wall shear stress and
is observed since the flow strongly accelerates (FPG), and approximately a 100% increase can be seen with respect to the incoming
under ZPG-flow conditions. Downhill, the pressure coefficient
recovers (presence of APG), inducing a reduction in
, to finally reach slightly negative values in the separation bubble (
2100 mm). Obviously, the boundary layer flow should “pass-through” the laminar skin friction coefficient value before separating (i.e.,
< 0). This may indicate the presence of quasi-laminarized flow within 2000 mm <
s < 2100 mm and in the near wall region [
37] (extension of the viscous sub-layer). This supposition would be better addressed when discussing mean streamwise velocity and Reynolds shear stress profiles in the next pages. In summary, the SA and SST turbulence models have estimated similar and consistent values for
regarding the experimental data from [
1], perhaps the SA model has shown moderate supremacy, overall. The skin friction coefficient in the incoming ZPG zone is slightly under-predicted by both turbulence models (∼15% lower than in [
1]). As previously mentioned, the most “challenging” situation for turbulence models has undoubtedly been the hilltop and vicinity since the flow goes through acceleration and deceleration in a very short distance. Major differences were computed as roughly 35% in that zone. According to [
1], the location of the boundary layer detachment point was found to be situated at
s = 2095 mm by extrapolation. The SA and SST models have predicted a separation point around
2100 mm, in very close agreement with experiments.
The streamwise variation of the Stanton number and the Reynolds analogy ratio (i.e.,
) are shown in
Figure 15. The Stanton number is defined as follows,
where
is the wall heat flux defined as:
Here,
k is the fluid thermal conductivity,
is the fluid’s specific heat at constant pressure, and
is the thermal gradient at the wall in the wall-normal direction. The Stanton number is a dimensionless number that relates the heat interchanged between the surface and the fluid to the thermal capacity of the fluid. From
Figure 15a, one can observe nearly constant
in the incoming flow (ZPG), which is typical in canonical or flat-plate boundary layers. A very good agreement was obtained with the empirical correlation by Kays and Crawford [
35], who proposed a variation of
as a function of the local momentum thickness Reynolds number,
, for ZPG turbulent flows (adapted to
= 0.71). The Stanton number (and heat transferred) peaks by
1125 mm, where (
)
is located, demonstrating a high similarity between maximum viscous shear force and total heat transferred at the wall. This peak in the heat transfer (representing about an 80% increase regarding the incoming baseline
) is situated approximately by the end of the first concave bend. Downstream, the Stanton number decreases much faster than
does, suggesting a non-similarity between these two boundary layer parameters. The strong changes of streamwise pressure gradients in this zone, which are sources of dissimilarity between the momentum and thermal boundary layer transport [
38], are the reasons for that behavior. Beyond
s = 1500 mm, a “plateau” is observed in
values, and an abrupt reduction of the heat transfer is achieved by
2100 mm, caused by the presence of the flow recirculation zone. This separation bubble is characterized by a quasi-adiabatic process, since no heat transfer occurs between the surface and the fluid (
0). Furthermore, the Stanton number can also be related to the skin friction coefficient via the Reynolds analogy (similarity between the viscous drag to the heat interchanged). We introduce the
ratio in
Figure 15b. An excellent agreement with the
empirical correlation by [
35] is seen in the ZPG zone. This ratio significantly departs from the unitary value by the hill feet (beginning of the concave bend,
987 mm). It is worth highlighting that
remains very close to one in most of the curved hill for
s > 1100 mm, getting large negative values in the vicinity of the separation bubble due to the very small (and negative) values of
. In essence, both turbulence models have generated very similar Stanton numbers.
To the best of our knowledge, the implemented approach, based on a potential flow-based scheme, has shown robustness and accuracy in the detection of boundary layer edge parameters as well as its integral values in comparison to the classical
criterion. This made the overall calculation of the parameters shown in
Figure 16 and
Figure 17 consistent in the presence of strong pressure gradients, and subsequently, severe boundary layer distortion. The boundary layer thickness is slightly underestimated, although it follows the experimental results’ behavior. Baskaran et al. have limited data in the separation bubble, and we focus on the sections highlighted by them [
1]. The overestimated shape factor is likely due to an overestimation in the boundary layer’s edge velocity incurred when comparing the potential and RANS flow fields. Nonetheless, the comparisons along the hill are extremely favorable to our approach compared to the experimental baseline. Overall, the SA model has superior performance when compared to the
SST [
2]. This is particularly noticeable within the portion 800 mm <
s < 1200 mm seen in the integral parameters prediction. Both turbulence models significantly over-predict the maximum shape factor,
H, located at
1000. At this point, the meaningful thickening of the turbulent boundary layer is consistent with the presence of strong APG and flow deceleration (note the
peak in
Figure 14a). Consequently, the shape factor increases (up to ∼15% increases with respect to the incoming flow), and discrepancies in numerical results are within 25% regarding experimental values. Since the shape factor,
H, is the ratio of the displacement thickness to the momentum thickness, the previously mentioned discrepancies on
are caused by over-predictions on the displacement thickness,
, as confirmed from
Figure 17. In addition, the momentum thickness,
, has been almost faithfully replicated by RANS as compared to experiments. This is consistent with the good agreement on the
variation (see
Figure 14b), since the momentum thickness is proportional to the drag force over the surface.
For the creation of
Figure 17, our potential flow-based scheme used Equations (
30) and (
31) to calculate displacement and momentum thickness parameters. Meanwhile, for
Figure 16, the parameter
used a reference velocity,
, where after following the results of [
1], it was found to be
s = 596 mm, i.e., the reference station.
To assess the distortion of the momentum boundary layer due to the hill, we present wall normal, inner scaled profiles at several streamwise stations in
Figure 18. Comparison is performed with experiments by [
1] as well as against two DNS baselines by Schlatter and Orlu (
) [
10] and Lagares and Araya (
) [
9]. For the ZPG zone in
Figure 18a, the inner scaled velocity profiles of both closure models collapse near perfectly with a very slight variation observed in the wake region where the
SST variant predicts a slightly higher streamwise velocity,
. Both DNS databases exhibit a high level of consistency (almost overlap), and a long log region can be seen due to the high Reynolds numbers considered. The flow starts to decelerate by the first concave bend beginning (
710 mm), which causes a decrease in the skin friction coefficient and in the friction velocity as well. This is the reason for the slight upward movement of the wake in
Figure 18b, which is over-predicted by the turbulence models. The effects of flow acceleration and FPG can be observed in
Figure 18c,d, inducing a “hump” in velocity profiles over the log region, also reported by [
39] in sink flows. The boundary layer suffers a significant distortion around the hilltop where a “plateau” in
can be observed between
and 1000 from experimental data. In this FPG zone, the SA model predictions are in better agreement with experiments by [
1]. As discussed in
Figure 14a, the pressure coefficient recovers from
mm and the streamwise pressure gradient becomes more adverse. Therefore, one can describe the APG influence as steeper slopes of the velocity profiles within the log-wake region. Finally, the sudden deceleration (strong APG) of the boundary layer produces a recirculation bubble above the wall, i.e., between 2 to 1000 wall units, as can be seen in
Figure 19, where streamwise velocity profiles can be seen at three different stations. While the SA model predicts a slightly stronger reversal flow, the SST model predicts a larger (in terms of wall-normal distance) bubble. This recirculation zone is characterized by a low level of velocity fluctuations (this will be confirmed later in the Reynolds shear stress profiles) and back (negative) flow, where almost constant values of
can be observed (within 5 to 10 in wall units, according to the streamwise station). Beyond the separation zone in the wall-normal direction, the streamwise velocity exhibits very sharp increases towards the boundary layer edge, resembling Blasius velocity profiles. The substantial discrepancies observed in both turbulence models confirm our previous statement regarding the limitations of eddy viscosity models to accurately predict boundary layer detachment.
We apply the Boussinesq hypothesis to estimate Reynolds shear stresses from the RANS output as per the equation:
In
Figure 20, profiles of Reynolds shear stresses are plotted in inner units and several streamwise stations by considering the friction velocity at the reference station (
596 mm or ZPG zone). This choice is based on the isolated assessment of the baseline (incoming) Reynolds shear stresses under combined streamwise pressure gradients caused by the curved hill. Moreover, we remove any scaling effect according to the local values of the friction velocity. At the ZPG station (
596 mm), it can be seen that both turbulence models tend to capture the inner portion of the boundary layer with very good agreement with DNS from [
9,
10]. The comparison breaks down in the outer region where both models predict larger values, perhaps, caused by the higher Reynolds numbers considered in RANS predictions. The APG effect at
s = 1139 mm is manifested as a clear secondary peak on
around
800. We can infer that the flow is subject to a very strong deceleration or APG since the outer peak is larger (almost twice as large) than the inner peak, around
15. The SST model predicts a more intense outer peak, addressing one of the original research questions of this study. These outer secondary peaks of
have also been reported by [
38] in DNS of turbulent spatially developing boundary layers subject to strong streamwise APG. Moreover, outer streaks are enhanced by APG, which in turn cause local increases of streamwise velocity fluctuations and Reynolds shear stresses, according to DNS studies by Skote et al. [
40]. Interestingly, a much stronger APG effect can be seen at
s = 1990 mm, just upstream of the separation bubble, given by the inclined shear layer or “plateau” in the zone 10 <
< 200. The larger the APG, the more inclined the shear layer [
38]. At the flow recirculation zone, i.e., at
s = 2500 mm, there is an appreciable attenuation of the Reynolds shear stresses in the near wall and buffer region (
< 100).
Furthermore, the SST model predicts large positive values of the cross-correlation
inside the bubble (not seen in the SA model), which is consistent with previous DNS studies in flow separation [
36,
40]. Clearly, the very low values of the Reynolds shear stresses suggest that the flow is quasi-laminarized or on the verge of relaminarization [
41] (viscous sublayer extended). On the other hand, the very large values of
in the outer region (1000 <
< 3000) plus the non-negligible wall normal gradients of the streamwise velocity indicate the presence of significant turbulence production (i.e.,
/
) well above the separation bubble, and thus, the flow is highly turbulent in that zone, as will be shown in the next figure. The published data by Baskaran et al. [
1] do not contain much information beyond the early portions of the separation bubble.
The turbulent kinetic energy (
K) production,
, inside the boundary layer can be evaluated by computing the term with the highest contribution to
K and mean-flow kinetic energy equations. The time-averaged velocity gradients act against the Reynolds stresses, removing kinetic energy from the mean flow and transferring it to the fluctuating velocity field, Pope [
42]. The turbulent kinetic energy is defined as in Equation (
6); whereas the Boussinesq hypothesis leads to the following definition of
in Equation (
33), where the Reynolds shear stress definition in Equation (
32) is employed.
Figure 21 shows the principal term of
K production in wall units at different streamwise stations, as done with the RANS-modeled Reynolds shear stresses. It was observed that the term
in the mean flow gradient was negligible, even in zones with large wall curvatures. For the incoming flow (ZPG zone) at
596 mm, the SA turbulence model reproduces properly the turbulence production in the turbulent boundary layer, as contrasted to DNS from [
9,
10] at lower Reynolds numbers. Peak values are approximately 0.25 to 0.29 in the buffer layer at
10. While an increase in the term
would suggest a mean flow deceleration; whereas, an enhancement of the fluctuating component of the velocity field and Reynolds shear stresses. Previous flow physics descriptions can be clearly seen at stations
1469 mm and 1665 mm. Down the hilltop, turbulence production begins to recover the inflow features (attenuation process) as the Reynolds shear stresses decrease in the viscous sub-layer and buffer region. In addition, flow deceleration by APG tends to destabilize the boundary layer, inducing turbulence intensification in the outer portion. Based on DNS studies by Skote et al. [
40], the outer streaks are intensified by strong APG and can be related to local increases in turbulence production and <
> (outer peaks). As seen in
Figure 21, SA and SST models predict outer peaks of turbulence production around
1500–2500 at
2500 mm, where the separation bubble is thicker. Furthermore, the production of
K inside the bubble is almost negligible, suggesting that the flow is locally quasi-laminar.
In general, the thermal boundary layer profiles presented in
Figure 22 follow similar tendencies (i.e., Reynolds analogy) to those presented for the momentum boundary layer. This is expected since the temperature is modeled as a passive scalar. Particularly, a high level of similarity has been observed in ZPG zones since streamwise pressure gradient is a source of dissimilarity between momentum and thermal fields. For instance, nearby the hilltop, strong FPG effects were described by the presence of “humps” in velocity profiles over the log region. However, thermal profiles look very different at
1345 mm and 1596 mm, indicating Reynolds analogy breakdown. Actually, a significant portion of the thermal boundary could be represented by a logarithmic curve fitting, given by the observed “linear” behavior when plotted on a semi-log scale. From
Figure 23, the separation bubble seems to elongate the thermal “plateau”, although the distortion effects are less “violent" than those seen in the momentum boundary layer. As expected, the predictions at the separation bubble seem to disagree more than on any other station between turbulence models. Interestingly, we can now visualize two distinctive logarithmic regions of the thermal profile: one log slope inside the bubble and the other (steeper) outside the separation zone up to the thermal boundary layer edge. This log behavior may open up the opportunity for future better turbulence modeling of passive scalar transport in flow separation. Furthermore, a deeper analysis must be performed in this sense, which falls outside the scope of the present manuscript and may be published elsewhere.
It is worth highlighting that the 2D results presented in this work assume spanwise homogeneity. The main expected deviations when considering a full 3D domain and unsteady simulations can be summarized as follows: (i) the appearance of G
rtler-like vortices due to the strong concave curvatures (
−0.13 to −0.15, where
is the local boundary layer thickness and
R is the local radius of curvature) present in the complex geometry and (ii) flow separation bubble. Our previous experience on supersonic turbulent boundary layers subject to strong concave surfaces via DNS [
43] dictated the existence of G
rtler-like vortices caused by centrifugal forces, which in turn, enhanced spanwise flow fluctuations. However, no spanwise inhomogeneity has been observed in time-averaged flow statistics of turbulent flows. On the other hand, flow separation at the second concave surface should be ruled by unsteadiness and three-dimensionality effects. Even when the assumption of spanwise homogeneity may not be perfect, our results have shown reasonable agreement with wind-tunnel experiments.