2D Periodic Hill

Underlying Flow Regime 3-30

Cross-Comparison of Experiments

The 2D PIV results were checked with 1D LDA measurements. A satisfactory compliance was found as can be seen in the following figures (Re=10,600) [Rapp et al. (2009), Rapp (2009)]. The LDA measurement accuracy depends on the frequency, the number of samples and the window functions. It can be quantfied to <0.01 u. The assesment of the measurement accuracy for PIV measurements is rather complicated. The laser sheet and camera adjustment, the subpixel interpolation and the validation algorithm have got an impact on the measurement. In this case two independently working measurement techniques were compared with each other and only minor deviations were identified (see Fig. 4.1).

Fig. 4.1 Mean streamwise velocity component

\langle u\rangle /u_{b}

(left) and mean normal stress

\langle u\prime u\prime \rangle /u_{b}^{2}

The PIV field data and the point by point LDA measurements show an excellent accordance. The mean velocity component in the x-direction and the streamwise Reynolds stress show only minor deviations that can be traced back to the measurement procedure. As the LDA data could only be acquired by measuring the particular points periodically and at different instants of time, the profiles turn out to be rougher than the PIV data.

The periodicity and the homogeneity of the flow could be shown for Re≥10,600 [Rapp et al. (2009), Rapp (2009)]. The following plots (Fig. 4.2) show the mean streamwise and vertical velocity component at x/h=4.0 between hills six and seven and between hills seven and eight for Re=10,600. Only minor deviations between the two periods can be detected. Within the whole flow field the greatest difference of the mean streamwise velocity component between the two consecutive periods were found at x/h=4.0, y/h=1.1 and can be quantified to $\Delta u=0.012u_{b}$ . The corresponding point and magnitude for the vertical component is: x/h=3.0, y/h=0.55, $\Delta v=0.008u_{b}$ .

Fig. 4.2 Mean streamwise velocity component

\langle u\rangle /u_{b}

(left) and mean vertical velocity component

\langle v\rangle /u_{b}

The mean lateral component w was found to be smaller than |0.01| u_b within $-5\leq z/h\leq 0$ on the left channel half for Re≥10600. The greatest variations of the rms values remain below 0.005 u_b from Re≥10600 on. The contribution of these lateral variations on the momentum balance were investigated in Rapp (2009). $\partial \langle u\prime w\prime \rangle /\partial z$ has got the greatest impact on the momentum balance in the x-direction in the center plane at y/h=1.53 (Re=10,600); it oscillates around zero with magnitudes between $-1,66\cdot 10^{-3}u_{b}^{2}/h$ and $1,64\cdot 10^{-3}u_{b}^{2}/h$ . Insufficient statistics were suspected to be the reason for these oscillations in the velocity/rms gradients. However, $\partial \langle u\prime v\prime \rangle /\partial y$ is one order of magnitude greater but still smaller than the dominating terms $u\partial \langle u\rangle /\partial x$ and $v\partial \langle u\rangle /\partial y$ . Further investigations on the periodicity of the flow in the experiment as well as a detailed analysis of the homogeneity can be found in Rapp et al. (2009) and Rapp (2009).

Cross-Comparison of CFD Calculations

A cross-comparison of the results achieved by the different methods was carried out. For that purpose two different Reynolds numbers were considered. According to Table 1 two DNS predictions are available at Re = 2800. Case 4 was performed by LESOCC using a curvilinear grid and case 5 is based on MGLET using a Cartesian grid with about 3.5 times more grid points. At Re = 5600 case 7 is taken into account which denotes a wall-resolved LES prediction on the curvilinear grid. These results are compared with case 8 providing numerical data on an extremely fine Cartesian grid (231 million grid points) which is beyond all doubt a well-resolved DNS. The cross-comparison was also conducted for Re = 1400. However, since no new findings compared to the outcome presented in the following sections resulted, for the sake of brevity this Reynolds number is omitted here.

Figures 4.3 to 4.6 show shows exemplarily the distribution of the averaged streamwise velocity $\langle U\rangle /U_{B}$ at four different vertical positions in the flow field, i.e. x / h = 0.5, 2, 4, and 6. The first position is located shortly after the separation line and crosses the strong shear layer; the second profile is at the beginning of the flat floor and hence within the main recirculation region. The third one is located near the end of the recirculation bubble and finally the last is positioned behind the main separation region in the reattached flow. At both Re the agreement observed between the results obtained by the different numerical methods is excellent. Only marginal deviations are found. For example, MGLET predicts slightly larger back-flow velocities at x / h = 4 for Re = 2800 compared with the data of LESOCC. This marginal deviation is still visible further downstream at x / h = 6 close to the lower wall. Besides that the profiles resulting from the two independent simulation techniques lie on top of each other at both Re.

Fig. 4.3 Comparison of predictions by LESOCC and MGLET based on the streamwise velocity profiles at x / h = 0.5 and two different Reynolds numbers, i.e. Re = 2800 (case 4 and 5) and 5600 (case 7 and 8, see Table 1)

Fig. 4.4 Comparison of predictions by LESOCC and MGLET based on the streamwise velocity profiles at x / h = 2. and two different Reynolds numbers, i.e. Re = 2800 (case 4 and 5) and 5600 (case 7 and 8, see Table 1)

Fig. 4.5 Comparison of predictions by LESOCC and MGLET based on the streamwise velocity profiles at x / h = 4. and two different Reynolds numbers, i.e. Re = 2800 (case 4 and 5) and 5600 (case 7 and 8, see Table 1)

Fig. 4.6 Comparison of predictions by LESOCC and MGLET based on the streamwise velocity profiles at x / h = 6. and two different Reynolds numbers, i.e. Re = 2800 (case 4 and 5) and 5600 (case 7 and 8, see Table 1)

Furthermore, Figs. 4.7 to 4.12 display the resolved Reynolds stresses ${\langle u'u'\rangle /U_{B}^{2}}$ , ${\langle v'v'\rangle /U_{B}^{2}}$ , and ${\langle u'v'\rangle /U_{B}^{2}}$ at two locations. The two main observations are as follows. First, a clear trend is obvious concerning the variation of the Reynolds stresses with varying Reynolds numbers. Second, a reasonable agreement is found between the predictions based on the two different codes at both Re. Similar to the normal velocity component $\langle V\rangle$ small deviations are visible near the extrema occurring in the free shear layer. Except in this narrow region the normal stresses and the shear stress belonging to a certain Re agree satisfactorily. The extremely well-resolved prediction by MGLET at Re = 5600 (case 8 in Table 1) shows the largest peak values for ${\langle u'u'\rangle /U_{B}^{2}}$ . The extrema of the normal stress ${\langle v'v'\rangle /U_{B}^{2}}$ as well as the shear stress ${\langle u'v'\rangle /U_{B}^{2}}$ in the shear layer (x/h = 2) are also slightly larger than the results predicted by LESOCC (case 7). The deviations found at this Reynolds number can be explained by the fact that the last is an LES prediction whereas MGLET provides DNS data. The modeled stress contributions, although known to be small, are not taken into account in the LES data at Re = 5600 in Figs. 4.7 to 4.12.

Fig. 4.7 Comparison of predictions by LESOCC and MGLET based on the streamwise normal Reynolds stress at x/h = 2.

Fig. 4.8 Comparison of predictions by LESOCC and MGLET based on the streamwise normal Reynolds stress at x/h = 4.

Fig. 4.9 Comparison of predictions by LESOCC and MGLET based on the wall-normal Reynolds stress at x/h = 2.

Fig. 4.10 Comparison of predictions by LESOCC and MGLET based on the wall-normal Reynolds stress at x/h = 4.

Fig. 4.11 Comparison of predictions by LESOCC and MGLET based on the Reynolds shear stress at x/h = 2.

Fig. 4.12 Comparison of predictions by LESOCC and MGLET based on the Reynolds shear stress at x/h = 4.

More comparisons including wall shear stresses and pressure distributions can be found in Breuer et al. (2009).

Cross-Comparison of CFD Calculations with Literature Data at Re = 10,595

Furthermore, the wall-resolved LES prediction by LESOCC (case 9) is compared with literature data by Fröhlich et al. (2005). For this comparison one should bear in mind that both simulations are based on similar numerical techniques (second-order accurate in space and time using curvilinear grids), but the main issues from section about the resolution should be recapitulated. The present grid consists of about three times more grid points than that used in Fröhlich et al. (2005) whereby the increased number of points is especially applied in the wall-normal direction to resolve also the upper wall and thus to avoid wall functions.

Figure 4.xx depicts a comparison at three different locations (x/h = 0.5, 2, and 6) showing averaged velocity profiles for $\langle U\rangle /U_{B}$ and $\langle V\rangle /U_{B}$ as well as turbulent quantities such as the resolved turbulent kinetic energy $\langle k\rangle /U_{B}^{2}=0.5(\langle u'u'\rangle +\langle v'v'\rangle +\langle w'w'\rangle )/U_{B}^{2}$ and the shear stress $\langle u'v'\rangle /U_{B}^{2}$ . For the streamwise velocity component $\langle U\rangle /U_{B}$ it is obvious that a very good agreement can be observed in the lower part of the computational domain close to the lower wall. Solely in the vicinity of the upper wall deviations are visible which can be assigned to the application of wall functions in \cite{froehlich_05}. The streamwise velocity component Failed to parse (syntax error): {\displaystyle \langle U \rangle / U_B<math> is underpredicted here and shows a non-smooth behavior. The resolution was quite coarse in that case in Fröhlich et al. (2005) partially leading to wiggles in the solution as obvious in Fig. 4.xxx(b) for the wall-normal velocity component <math>\langle V \rangle / U_B } . The deficits resulting from the usage of wall functions at the upper wall are also detectable in the turbulent quantities, e.g. the location of the peak value of $\langle k\rangle /U_{B}^{2}$ is shifted away from the upper wall and the shear stress profile $\langle u'v'\rangle /U_{B}^{2}$ strongly deviates from the wall-resolved LES prediction (case 9) in this region. Since the total mass flux through the channel was fixed, the underprediction of $\langle U\rangle /U_{B}$ in the vicinity of the upper wall has a certain influence on the overall flow development. Nevertheless, the agreement between the present wall-resolved LES prediction (case 9) and the literature data (Fröhlich et al. 2005) is satisfactory and further validates the former.

Comparison of CFD Calculations with Experiments at Re = 10,595

The experimental data at Reynolds number 10,595 are compared with the highly resolved LES results by LESOCC [Breuer et al. (2009), Rapp et al. (2009)]. The following figures show averaged velocity profiles for <u>/u_b at four streamwise positions.

Fig. 4.4 Mean streamwise velocity component

<U>

Fig. 4.5 Mean streamwise velocity component

<V>

The agreement between the predicted and the measured mean streamwise velocity component in the x-direction is very good. Minor deviations can be found in the shear layer (slightly higher velocities in the experiment) and in the post-reattachment zone. The vertical velocity component <v>/u_b is about one order of magnitude smaller than the streamwise component and yet more sensitive. However, the measurements comply with the simulation results in a fully satisfactory manner. The largest deviations are found at x/h=2.

The measured and the predicted Reynolds shear stress show a close agreement at the different x/h-positions. The location of the experimental peak values and their distributions are in close accordance with the predictions. Solely at x/h=6 a slightly higher stress was found in the experimental data.

Fig. 4.6 Mean Reynolds normal stresses

<uu>

Fig. 4.7 Mean Reynolds shear stresses

<uv>

Fig. 4.8 Mean Reynolds normal stresses

<vv>

However, the agreement between the PIV measurement and the LES prediction is highly satisfactory.

In conclusion, the variety of cross-comparisons carried out have demonstrated that the predicted data are reliable.

Separation and Reattachment Lengths

The flow separates shortly behind the hill crest at all Re. As shown in Fig.~4.9 the separation point moves upstream from $x_{S}/h\approx 0.45$ at Re = 100 to $x_{S}/h\approx 0.3$ at Re = 700 and finally to $x_{S}/h\approx 0.18$ at Re = 5600 but then settles down at a slightly larger value of $x_{S}/h\approx 0.19$ at Re = 10,595. The undulations in the wall shear stress at the beginning of the main recirculation bubble seem to be geometry related and not much affected by Reynolds number effects. This statement is supported by the observation that the undulations are visible in all simulations using LESOCC but also for the predictions based on MGLET.

Past the hill the main recirculation region follows with a nearly constant pressure plateau in a large part of the separation bubble. The reattachment position $x_{R}/h$ displayed decreases from Re = 100 to 1400. Between Re = 1400 and Re = 2800 a sudden change can be observed in Fig.~4.9. The reattachment length $x_{R}/h$ increases again in this narrow Re-range. Afterwards, a clear trend concerning the reattachment length is visible for $Re\geq 2800$ where $x_{R}/h$ decreases again with increasing Re. Since this phenomenon is consistently predicted by both independent codes, it is assumed to be no numerical artifact. The corresponding values of the reattachment lengths are ${x_{R}/h=5.24,5.19,5.41,5.09}$ , and ${4.69}$ for Re = 700 to 10,595, respectively. A similar behavior of the reattachment length with increasing Re has been observed for the backward-facing step flow by Armaly et al. (1983). This observation also explains why with increasing Re there is no continuous trend visible for the averaged pressure distribution.

Fig. 4.9 Separation and reattachment lengths versus Reynolds number; comparison of predictions by LESOCC and MGLET

(100\leq Re\leq 10,595)

In the experiment only the reattachment lengths are determined.

Fig. 4.10 despicts the separation and reatachment lengths for a wider Reynolds number range. In the experiment it was not possible to determine the separation point reliably. Thus only the reattachment length is shown. The general trend of a decreasing reattachment length with increasing Re found in the simulations is confirmed by the experimental data.

Fig. 4.10 Separation and reattachment lengths versus Reynolds number; comparison of predictions by LESOCC and MGLET

(100\leq Re\leq 37,000)

In the experiment only the reattachment lengths are determined.

Flow Development with increasing Reynolds number

The figures shows a series of prediction between Re = 100 and 10,595. At the lowest Reynolds number considered, the flow is found to reach steady-state after about 100 dimensionless time units which is equivalent to about 11 flow-through times. The flow separates past the hill crest at about x/h = 0.45 and forms a large recirculation region which fills the lower portion of the entire constriction completely. Thus reattachment is observed in this case at the rising edge of the second hill at about x/h = 7.73. No three-dimensional structures can be detected. The situation changes completely when the Reynolds number is increased to Re = 200. Now and for all other higher Re the flow remains unsteady all the time. Up to about 140 dimensionless time units the flow field at Re = 200 is two-dimensional. However, after this initial phase first three-dimensional flow structures develop which are detected by monitoring the spanwise velocity component W. With increasing Re the initial two-dimensional stage is shortened so that at Re = 700 first three-dimensional phenomena can be found at about 50 dimensionless time units past the initial stage. As visible with increasing Re the spatial structures observed become smaller and smaller.

Fig. 4.11 Comparison of (instantaneous) vorticity component

\omega _{z}

normal to the cross-section at eleven different Reynolds numbers (Re = 100 to 10,595) at an arbitrarily chosen instant in time

Contributed by: (*) Christoph Rapp, (**) Michael Breuer, (*) Michael Manhart, (*) Nikolaus Peller — (*) Technische Universität München, (**) Helmut-Schmidt Universität Hamburg

UFR 3-30 Evaluation

Contents

2D Periodic Hill

Cross-Comparison of Experiments

Cross-Comparison of CFD Calculations

Cross-Comparison of CFD Calculations with Literature Data at Re = 10,595

Comparison of CFD Calculations with Experiments at Re = 10,595

Separation and Reattachment Lengths

Flow Development with increasing Reynolds number

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UFR 3-30 Evaluation

2D Periodic Hill

Cross-Comparison of Experiments

Cross-Comparison of CFD Calculations

Cross-Comparison of CFD Calculations with Literature Data at Re = 10,595

Comparison of CFD Calculations with Experiments at Re = 10,595

Separation and Reattachment Lengths

Flow Development with increasing Reynolds number

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