PAMM · Proc. Appl. Math. Mech. 5, 569–570 (2005) / DOI 10.1002/pamm.200510261
DNS and Experiment of a Turbulent Channel Flow with Streamwise Rotation - Study of the Cross Flow Phenomena Tanja Weller∗1 , Martin Oberlack1 , Ingo Recktenwald2 , and Wolfgang Schr¨oder 2 1 2
Fluid- and Hydromechanics Group, Technische Universit¨at Darmstadt, Petersenstraße 13, 64287 Darmstadt, Germany Chair of Fluid Mechanics and Institute of Aerodynamics, RWTH Aachen, W¨ullnerstr. zw. 5 u. 7, 52062 Aachen, Germany
Modelling of rotating turbulent flows is a major issue in engineering applications. Intensive research has been dedicated to rotating channel flows in spanwise direction such as by [1], [2] to name only two. In this work a turbulent channel flow rotating about the streamwise direction is presented. The theory is based on the investigations of [4] employing the symmetry theory. It was found that a cross flow in the spanwise direction is induced. A series of direct numerical simulations (DNS) at different rotation numbers is carried out to examine these effects. Further, the results of the DNS are compared to the measuremets of a corresponding experiment. © 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
1
Introduction
The flow has several common features with the classical rotating channel flow [1] but also has some different characteristics. The induction of a mean velocity in x3 -direction is the most obvious difference compared to the classical case. This cross flow can be deduced by investigating the mean momentum equation and the Reynolds stress transport equation. The analysis of the mean momentum equation and the two-point velocity correlation equation in a rotating frame of reference [5] has shown in particular that analogically to the classical case selfsimilar mean velocity profiles exist which are linear functions of the form u ¯1 = C1 Ω1 x2 + C2
and u ¯3 = C3 Ω1 x2 + C4 .
(1)
Because the reflection symmetry about the center line is not broken the mean velocity stays symmetrical. Near the center two linear regions may appear according to an observation in [4]. Oberlack states that, except for the log-law, the highest degree of symmetry is usually obtained in the flow regions with the least wall influence. The cross flow is closely coupled with the rotation of the system and a linear profile can emerge also for small rotation rates.
2
Direct Numerical Simulation
To verify the theoretical results two DNS have been conducted at rotation numbers Ro=3.2 and Ro=10. The numerical technique which was chosen is a standard spectral method with a Fourier decomposition in streamwise and spanwise direction as well as a Chebyshev decomposition in wall-normal direction. The numerical code was developed at KTH [3]. The size of the domain used in the x1 , x2 , and x3 directions is respectively 4π, 2, and 2π on a 128 × 129 × 128 grid at rotation number Ro=3.2. At rotation number Ro=10 the length of the domain was increased to 8π and the number of grid points to 256 in x1 -direction. The boundary conditions are non-slip at x2 = ±1 and periodic in x1 - and x3 -direction. The pressure-gradient is held constant. The Reynolds number and the rotation number are defined by Reτ =
huτ = 180 2ν
and Roτ =
Ωh . 2uτ
(2)
In figure 1 the mean velocity profiles are presented. The velocity profile decreases at higher rotation numbers. Each mean velocity profile has a linear region (e.g. see figure 2) on each side of the centerline in the core of the flow. This has been expected from the global time scale analysis [5]. The near-wall regions up to x2 = ±0.9 are only marginally perturbed. In the DNS the cross flow could be proved. The secondary profiles are skew-symmetric about the centerline and the predicted linear profiles are at least marginally visible. Figure 3 shows the cross flow directly after the initialization phase averaged over a time period of 500 time steps. In this period the velocity profile increases at higher rotation number. In contrast the opposite effect can be noticed in figure 4 for a time period of 3000 time steps. This effect could not be explained immediately. Conjecturable is that the metastable large scale structures which are established are responsible for this effect. Particularly it is to mention that at Reynolds number Re = 180 there is concern about a viscosity dominated turbulence which makes the rotation induced large scale structures even more persisting. It is essential to conduct further computations at higher Reynolds numbers. ∗
Corresponding author: e-mail:
[email protected], Phone: +49 6151 16 5249, Fax: +49 6151 16 7061
© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
Section 13
570
18
20 Ro = 3.2 Ro = 10
Ro = 3.2
15 16
u ¯1 10
u ¯1 14
5 0 -1
0
-0,5
0,5
12 -1
1
0
-0,5
x2
0,5
1
x2
Fig. 1 Streamwise mean velocity profiles.
Fig. 2 Streamwise mean velocity profile at core region.
3
1,5
Ro = 3.2 Ro = 10
2
Ro = 3.2 Ro = 10
1
1
0,5
u ¯3 0
u ¯3
0
-1
-0,5
-2
-1
-3 -1
0
-0,5
0,5
x2 Fig. 3 Spanwise mean velocity profiles, tstatistics = 500.
3
1
-1,5 -1
0
-0,5
0,5
1
x2 Fig. 4 Spanwise mean velocity profiles, tstatistics = 3000.
Experiment
Experiments have been carried out in a test facility that has been configured to provide the same boundary conditions, i.e. the same Reynolds number and rotation number, as the numerical simulation. Several rotation numbers have been investigated, and the secondary flow that is generated by the interaction of the burst events and coriolis forces has been observed [6].
4
Conclusions and Summary
With the DNS the induced phenomena of a cross flow in spanwise direction has been computed. Furthermore, it has been confirmed by DNS that there are linear regions in both the streamwise and spanwise mean velocity. The experiments indicate a qualitative agreement as far as the secondary flow is concerned. Symmetries and scaling laws provide the theoretical results for turbulence modeling which can quantitatively be adjusted to employing the DNS and experimental data. Future investigations are planned, especially simulations at higher Reynolds numbers to examine this mentioned effect.
References [1] Johnston, J. P. and Halleen, R. M. and Lazius, D. K. (1972): Effects of spanwise rotation on the structure of two-dimensional fully developed turbulent channel flow, J. Fluid Mech. , 56, 533–557. [2] Lamballais, E. and Metais, O. (1996): Effect of spanwise rotation on the vorticity stretching in transitional and turbulent channel flow. Intern. J. of Heat and Fluid Flow, 17-3, 325-332. [3] Lundbladh, A., Henningson, D., Johanson, A. (1992): An efficient spectral integration method for the solution of the Navier-Stokes eqautions. FFA-TN 1992-28, Aeronautical Research Institute of Sweden, Bromma. [4] Oberlack, M. , Cabot, W. , Rogers, M. M. (1998): Group analysis, DNS and modeling of a turbulent channel flow with streamwise rotation. Studying Turbulence Using Numerical Datebase - VII, Center for Turbulence Research, Stanford University/NASA Ames, 221–242. [5] Oberlack, M. (2001): A unified approach for symmetries in plane parallel turbulent shear flows. J. Fluid Mech., 427, 299–328. [6] Recktenwald, I., Br¨ucker, Ch., Schr¨oder, W. (2004): PIV investigations of a turbulent channel flow rotating about the streamwise axis. Proceedings of the 10th European Turbulence Conference, Trondheim, Norway, 561–564. © 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
