On the significance of high spatial resolution to capture all relevant scales in the turbulent flow over periodic hills

10TH INTERNATIONAL SYMPOSIUM ON PARTICLE IMAGE VELOCIMETRY - PIV13 Delft, The Netherlands, July 1-3, 2013

On the significance of high spatial resolution to capture all relevant scales in the

turbulent flow over periodic hills

Christian Cierpka1, Sven Scharnowski1, Michael Manhart2and Christian J. K ¨ahler1

1_{Institute of Fluid Mechanics and Aerodynamics, Bundeswehr University Munich, Werner-Heisenberg-Weg 39, 85577} Neubiberg, Germany, Christian.Cierpka@unibw.de

2_{Fachgebiet Hydromechanik, Technische Universi ¨at M ¨unchen, Arcisstr. 21, 80333 M ¨unchen, Germany,} Michael.Manhart@bv.tum.de

ABSTRACT

Due to the complex nature of turbulence, the simulation of turbulent flows is still challenging and numerical models have to be further improved. For the validation of these numerical flow simulation methods reliable experimental data is essential as the simulation can only be more precise than the validation data but never be more accurate. However, for the correct numerical prediction of flows, the accuracy is the essential quantity. A typical test case is the flow over periodic hills. The numerical prediction is difficult, since flow separation and reattachment are not fixed in space and time due to the smooth geometry [10, 2]. Furthermore, the separated and fully three-dimensional flow from the previous hill impinges on the next hill, which will result in very complex turbulent flow features as shown in Fig. 1 on the left side. With the increasing computer performance available, it becomes possible to examine larger Reynolds numbers with DNS and LES. Typical grid sizes are in the order of several (3-10) Kolmogorov length scales η for LES and approach η for DNS [1]. The resolution of currently available measurements is in the order of 30 η (Re = 8,000) and above which is not sufficient to resolve the large gradients in the shear layer at the hill crest for instance. Even more severe, the contribution of the small eddies is averaged over a region associated with the measurement resolution. Thus an important part of the turbulent energy cannot be measured at all and is lost for the validation of turbulence models. Since these models are supposed to simulate the contribution of these small eddies it is of inherent interest to increase the resolution in the experiment. The aim of the current measurement campaign was therefore to increase the spatial resolution in order to study the resolution effect systematically and to provide an additional data set for the validation of numerical tools.

1. Experimental setup

The particle image velocimetry experiments were performed in a water tunnel at TU Munich [6]. The geometry was derived from the ERCOFTAC test case Nr. 81 [7] and scales with the hill height h. The hill height was set to h = 50 mm and the spacing between the hills, i.e. the periodicity, was 9h. The channel has a total cross-section of 3.035h × 18h and was driven by a hydrostatic water reservoir which is fed by a pump. This construction allows to dynamically decouple the pump from the test section and thus assures that fluctuations from the pump do not influence the flow of interest. The water is homogenized by using honey combs and screens installed upstream of the test section. As opposed to numerical simulations, it is rather difficult in reality to establish periodic boundary conditions. For the current experiment ten hills were arranged in a row and the measurements were performed at the seventh hill to avoid inlet and outlet effects. In order to measure the velocity close to the wall, the flow was seeded with Rhodamin B doped polyamide particles with a diameter range of 1-20 µm and the camera was equipped with a low pass light filter. The reflections of the wall were almost completely suppressed by this procedure. To avoid errors due to misalignment of the light sheet from two cavities, the light sheet in the mid-span of the channel was generated by a single cavity INOLASS laser in double pulse mode. For the image recording a sCMOS camera was operated at 5 Hz for a total measurement time of about 1.2 h for each Reynolds number. The Reynolds number based on the averaged velocity above the hill crest, in the middle of the channel, uband the hill height was Re = ubh/ν = 8, 000 and

Re= 33, 000.

The large variance of structures and events on different scales in this highly turbulent flow are visible on the right side of Fig. 1 where the stream-wise velocity component at Re = 8,000 is shown. Due to the confinement of the channel the flow accelerates towards the hill. Behind the hill a separated region can be clearly seen in the instantaneous stream-wise velocity field with a very irregular interface towards the free stream region. For a detailed analysis and comparison with numerical results it is therefore important to make sure that the mean values for this highly turbulent flow reached statistical convergence.

2. Temporal sampling

In general, it is useful to define an integral time scale treffor the investigation to analyze the convergence. This reference time refers

to the time, required by an event or structure to travel with the mean velocity over the hill crest ubthrough one periodic part of the

channel, i.e. 9h, and is also often denoted as ’flow through time’ tref= 9h/ub[6]. This reference time is tref= 2.63 s for Re = 8,000

and tref= 0.64 s for Re = 33,000. Since the acquisition rate of the double frame images was 5 Hz, a structure traveling with the mean

33,000, respectively. However, for the convergence especially the low frequency events are critical. As a criterion for convergence a deviation of the mean value of 0.05 pixel was chosen for the mean displacement and 0.01 pixel for the turbulence intensity. Since the sampling rate for the 22,000 instantaneous velocity fields was identical for both Reynolds numbers, a total measurement time of T = 1, 680 · trefwas achieved for Re = 8,000 and T = 6, 840 · treffor Re = 33,000. Therefore, the convergence of the mean values

within the total measurement time for the lower Reynolds number are more critical. To prove the statistical convergence, the evolution of the mean values for the stream-wise velocity hui /uband the turbulence level Tu/ubvs. t/tref, with Tu =

q

u02_{+ v}02_{/2, is}

shown in Fig. 2 for three different wall-normal positions at x/h = 1.8, indicated by the circles in Fig. 3. As expected, the time to reach the convergence criteria is much higher for the turbulence level compared to the time necessary to reach convergence for the velocity distribution. However, the time span also differs considerably for the different positions within the flow field.

10

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1.2

t / t

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u

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t / t

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Figure 2: Evolution of the mean value for the stream-wise velocity u/ub (top) and the turbulence level Tu/ub vs. t/treffor three

different wall-normal positions at x/h = 1.8. The crosses indicate the time, necessary to reach statistical convergence.

The spatial distribution of the time necessary to reach the convergence criteria of the stream-wise mean velocity hui and the turbulence level Tu for Re = 8,000 is shown in Fig 3. The three different positions from Fig 2 are indicated as circles. As can be seen in the figure, the convergence for the upper half of the channel can be reached for fairly low time spans. A much larger time span is necessary in the region of the developing shear layer where the flow shows higher turbulence and vortex interaction. However, in the uphill region a large time span is also required due to the impinging of the vortices on the hill. These findings are especially interesting for numerical simulations where it is usually difficult to acquire as many time steps for reliable mean values.

3. Spatial sampling

For the spatial resolution, the Kolmogorov length scale η can be estimated by taking the value of Breuer et al. [1] obtained for DNS data at Re = 5, 600 and scaling it with Re3/4[6]. This results in a mean Kolmogorov length scale of ηRe=8,000≈ 100µm and ηRe=33,000≈

35µm. The instantaneous velocity field shown in Fig. 1 was obtained by standard window correlation-based image evaluation with a final interrogation window size of 16×16 pixels and 50% overlap. For a comparison different interrogation window sizes were evaluated. Interrogation windows of 32×32 pixels provide a resolution of 6 mm, 16×16 pixels of 3 mm and 8×8 pixels of 1.5 mm, respectively. To increase the spatial resolution for the mean values, the single-pixel ensemble-correlation is required [5]. This technique provides a velocity vector at each pixel location. Nevertheless, the final resolution, i.e. the distance of independent velocity vectors, does strongly depend on the particle image size [3]. The mean particle image diameter was estimated to be 2.4 pixels, which gives a final resolution of 0.45 mm. The velocity field is therefore 5 times better resolved than in previous measurements [6]. For Re = 8,000 a physical spatial resolution results of 4.4·η and of 12.7·η for Re = 33,000 was reached. These values are comparable to the direct numerical simulations performed by Breuer et al. [1].

x / h y / h 〈u〉 t c / tref −4 −2 0 2 4 0 1 2 3 0 500 1000 1500 x / h y / h Tu t c / tref −4 −2 0 2 4 0 1 2 3 0 500 1000 1500

Figure 3: Spatial distribution of the duration to reach the convergence criteria of the stream-wise mean velocity hui and the turbulence level Tu for Re = 8,000. The circles mark the positions for the temporal evolution shown in Fig. 2.

The mean velocity distributions using a interrogation window size of 16×16 pixels and single-pixel ensemble-correlation are shown in Fig. 4. Qualitatively, the results look quite similar. However, quantitatively there are significant differences visible. In Fig. 5 the velocity profiles for the different evaluation window sizes are shown for x/h = −0.8, which corresponds to the region of acceleration and x/h = 0.23, which was determined to be the starting point for the flow separation. The position of both flow profiles are also indicated by the lines in Fig. 4.

16 x 16 pixel

Ensemble corr.

Figure 4: Mean velocity in x-direction for Re = 8,000 using 32×32 window correlation (top) and single pixel ensemble correlation (bottom).

For the measurement in front of the hill at x/h = −0.8 the strong gradients are fully resolved for the single-pixel ensemble-correlation. Only the last point at the wall does not reach zero velocity but the strong non-linear gradient is well captured. However, using 8×8 pixel interrogation windows, the first data point at the wall is by far not zero due to the spatial averaging effects. The strong non-linear gradient cannot be resolved as well, due to the inhomogeneous seeding at the wall, which causes velocity biases. Furthermore, the position of the wall cannot be determined by means of the velocity profiles [4]. The effect of the finite size of the interrogation windows would also strongly bias the estimation of the Reynolds stresses, since vortices smaller than the interrogation window size are filtered out and do not contribute to the total Reynolds stresses.

The demand for the high resolution becomes even more evident looking at the profiles at x/h = 0.23. The single-pixel ensemble-correlation results clearly indicate that the flow starts to separate whereas all window ensemble-correlation results show a rather large velocity in the x-direction. This would clearly lead to an erroneous determination of the geometry of the recirculation zone and makes a comparison with numerical results difficult. On the basis of the single-pixel ensemble- correlation results the point for flow separation and reattachment and the position of the center of the recirculation region were determined and are given in Tab. 1.

Table 1: Main experimental parameters and geometry of the recirculation zone. Re ≈8,000 ≈33,000 trefin s 2.62 0.645 T/tref 1,680 6,840 ubin m/s 0.171 0.698 h~uimax/ub 1.106 ± 0.003 1.178 ± 0.004 at x/h -0.29 ± 0.02 -0.33 ± 0.02 at y/h 1.03 ± 0.02 0.99 ± 0.02 flow separation at x/h 0.23 ± 0.05 0.31 ± 0.05 flow reattachment at x/h 4.29 ± 0.02 3.73 ± 0.03 recirculation center at x/h 2.07 ± 0.03 2.09 ± 0.04 at y/h 0.51 ± 0.01 0.46 ± 0.02

In order to compare the flow at both Reynolds numbers, profiles of the mean stream-wise velocity, the turbulence intensity and the Reynolds shear stresses are shown in Fig. 6. The high spatial resolution of the measurements allows for the determination of the large gradient of the velocity overshoot at the hill top as can clearly be seen in the upper plot. The velocity overshoot at x/h = 0.05 is about 0.11 · ubfor Re = 8,000 and 0.18 · ubfor Re = 33,000 in the time averaged sense. Due to the confinement of the flow the velocity in the

upper half of the channel is slightly larger than the bulk velocity for Re = 8,000 which is not the case for the higher Reynolds number. For the higher Reynolds number a larger momentum exchange is expected, which can clearly be seen at x/h = 1 for instance, where the momentum deficit in the recirculation zone is already substantially smaller than for the lower Reynolds number. However, the flow finally reattaches in a time averaged sense. Due to the high spatial resolution the small differences for the different Reynolds numbers can clearly be measured, such as the very thin remaining region of reversed flow for the lower Reynolds number seen in the profile at x/h = 4 for example. In general, the momentum transfer from the free stream to the recirculation zone is lower for the lower Reynolds number and thus the profiles in the upper half of the channel show higher velocities compared to Re = 33,000 and vice versa in the lower half of the channel. This results in a smaller recirculation zone for the higher Reynolds number which was determined to be 3.42 x/h and thus 84% of the length of the recirculation zone for Re = 8,000 (4.06 x/h).

For the validation of numerical methods the turbulence intensity and the Reynolds stresses are of particular interest, as they are determined from turbulence models, which are always uncertain. Scharnowski et al. [8] developed a method to determine the Reynolds stresses directly from the correlation function with the same spatial resolution. The main benefit of estimating the Reynolds stresses directly from the correlation function is that structures, much smaller than a typical interrogation window in PIV contribute to the velocity probability distribution function that is encoded in the shape of the correlation function. Therefore the Reynolds stresses are not underestimated by the spatial filtering effect of the finite interrogation windows. On the other hand if outliers are present in the velocity field and the Reynolds stresses are determined by velocity vector fields, these outliers would increase the value for the final Reynolds stresses. The estimation of the Reynolds stresses from the correlation function is free of both of these biases and lead to very accurate results for a large number of images. Recently, the method was extended to allow for the estimation of higher order moments [9]. As already mentioned, the Reynolds stresses were directly determined from the correlation functions obtained by the

−5 −4 −3 −2 −1 0 1 2 3 4 5 0 0.5 1 1.5 2 2.5 3 〈u〉 / u b + x / h y / h Re_h = 8,000 Re_h = 33,000 −5 −4 −3 −2 −1 0 1 2 3 4 5 0 0.5 1 1.5 2 2.5 3 2 ⋅ Tu / u b + x / h y / h −5 −4 −3 −2 −1 0 1 2 3 4 5 0 0.5 1 1.5 2 2.5 3 −10 _⋅〈v_′⋅u_′〉 / u b 2 _{+ x / h} y / h

Figure 6: Profiles for the mean stream-wise velocity (top), the turbulence intensity (middle) and the Reynolds shear stress (bottom) for Re = 8,000 (blue) and Re = 33,000 (red).

The profiles for the turbulence intensity show the largest values in the evolving shear layer at the hill top. With the wall-normal growth of this shear layer, the turbulence profiles become smoother. However, the peak at y/h ≈ 0.75 can still be seen for the flow, impinging on the hill at x/h = −3 . . . − 2. Due to the confinement and the acceleration of the flow by the geometry of the hill, the turbulence levels decrease strongly in the uphill region. The Reynolds shear stresses for both Reynolds numbers are shown in the lower part of Fig. 6. It is obvious that the Reynolds shear stresses for the larger Reynolds number reach a much higher level than for Re = 8,000 as expected. The difference is particularly pronounced within the recirculation region. Also the wall-normal position of the maximum is lower for the higher Reynolds number. This effect is due to the larger momentum exchange which effects the position of the shear layer. The width of the layer with high Reynolds shear stresses is also much smaller for the higher Reynolds number. However, the highest values are to be observed in the thin shear layer evolving from the hill top and smooth out for further downstream positions especially when the flow starts to accelerate due to the next hill.

Conclusion and outlook

Single-pixel ensemble-correlation provides the necessary spatial resolution to characterize the flow in a time averaged sense and to elaborate on the evolution of the large scale structures and regions of high phenomenological relevance. The first examination of the mean velocity, turbulence levels and Reynolds stress distributions already indicated that with advanced measurement equipment and sophisticated evaluation techniques many new flow features can be resolved. This gives new insight into the complex flow phenomena and allows to prove the numerical predictions, which could not be validated in the past due to a lack in spatial resolution for experimental data.

Acknowledgments

The financial support from the European Community’s Seventh Framework program (FP7/2007-2013) under grant agreement No. 265695 and from the German research foundation (DFG) under the individual grants program KA 1808/8 and Transregio 40 is gratefully acknowledged.

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