Fig. 1 Schematic description of the plane turbulent Couette flow subjected to spanwise rotation
System rotation is known to substantially affect the turbulence structure in rotating channel flows. The fully developed rotating Couette flow is particularly well suited for the study of the Coriolis instability in turbulent channel flow. The constant sign of the mean vorticity distribution makes the stabilising or destabilising effect symmetric with respect to the centreline. This simplifies the interpretation of the observed changes in turbulence structure due to system rotation, as compared to the pressure driven Poisseuille flow which was studied in the 7th workshop.
This flow can be defined by two parameters: Reynolds number Re and Rotation number Ro. Reynolds number is Uw h / v = 1300, where Uw is half of the velocity difference between the walls, h is the channel half height and v is the kinematic viscosity. Results from direct numerical simulation (DNS) are available for six different rotation numbers: 0.0, -0.01, 0.01, 0.1, 0.2, and 0.5. The rotation number Ro is defined as Ro=2 Omega h / Uw, where Omega is the angular velocity of the coordinate system [rad/s].
Fig. 1 Schematic description of the plane turbulent Couette flow subjected to
spanwise rotation
At certain regime of positive rotation numbers the flow is characterised by longitudinal roll cells, see Fig. 2. These roll cells are secondary motion originating from the unstable balance between the Coriolis acceleration and the pressure gradient in the cross stream direction (y). If the secondary flow is omitted, the flow field can be modelled as one-dimensional (changes in y-direction only). However, the kinetic energy of the roll cells may be of the same order as the kinetic energy of the real turbulence. In one-dimensional computation, turbulence and secondary motion are modelled together by the turbulence model. Because the secondary motion is not turbulence by nature, turbulence models may fail badly in predicting the effects of the secondary motion on the mean flow. This is why a two dimensional numerical model, that is able to capture the roll cells, may be needed in order to obtain realistic and accurate solutions. The decision whether to use a 1- or a 2-dimensional numerical model, or even both, will be left to each participant.
Fig. 2 Illustration of the secondary flow at Ro=0.01 after Bech and Anderson [2]
(used with permission).
If a two-dimensional numerical model is employed, the velocity field should be decomposed following Ref. 2. In this decomposition the velocity field is splitted in three parts: the spanwise averaged one-componential mean velocity that is only a function of y, the three-componential secondary flow that is a function of y, and z and the turbulent motion which is a function of x, y, z, and t. The two first parts of this triple-decomposition will be represented by the mean flow in the RANS approach. This mean flow is steady and two-dimensional (y,z), but has three components of the (mean) velocity field. Then only the real turbulence is modelled by the turbulence closure, and all turbulence variables will be independent of x and t. The results should be averaged in the spanwise direction and the secondary flow correlations should be evaluated. The total stresses i.e. the sums of real turbulent stresses and secondary flow stresses will be compared with the DNS-results. The secondary flow stresses and the turbulent Reynolds stresses from DNS are also given separately for the case Ro=0.01. This data is useful for investigating if the secondary flow is correctly solved. If the time schedule of the workshop permits, the predicted secondary flow and turbulent stress components can be also separately compared with the DNS-data in this particular case.
The stabilising or destabilising effect of the system rotation on the turbulence can be illustrated by plotting for instance the friction Reynolds number Retau = utau h / v as a function of the Ro-number with Ro ranging from a high negative rotation up to a high positive rotation. With sufficiently high negative rotation, the flow remains laminar due to the stabilising effect of the rotation on turbulence. When the Ro-number is gradually increased the flow becomes turbulent at certain Ro-value. When Ro is further increased, the turbulence intensity, and hence Retau, increase until at certain Ro-number Retau reaches its maximum and a stabilising effect on the turbulence again takes place. Eventually the flow laminarises again, see Fig. 3.
In Fig. 3 Retau is plotted as a function of Ro based on the DNS-results as well as on three turbulence models. These models are: an explicit algebraic Reynolds stress models with the pressure-strain model based on the one of Launder Reece and Rodi (EARSM-1) and with the linearised SSG pressure-strain model (EARSM-2). Both of these EARSM models are built upon the k-w two-equation model. In addition results obtained with a linear k-w-model with a rotation and curvature correction (RCSST) are also presented.
Fig. 3 The friction Reynolds number Retau as a function of the
Ro-number according to the DNS and a few turbulence models (based on one-dimensional
computations).
The mean velocity as well as the Reynolds stress profiles for all of the six Ro-numbers: 0, -0.01, 0.01, 0.1, 0.2, and 0.5 are available. Note, that the shear stress component is given as -uv and the shear-stress budget is for -uv as well. The mean velocity has been scaled by the wall velocity Uw and the Reynolds stresses by the square of the friction velocity utau2. The given stresses actually contain the contribution of the secondary flow in addition to the contribution of the real turbulence. However, in one case, Ro=0.01, the turbulent and secondary flow stresses are also given separately. The friction Reynolds number Retau as a function of Ro is also given. In the non-rotating case, the budget data of k as well as each stress component are given. This includes: Production and viscous dissipation of k, turbulent diffusion + pressure diffusion, turbulent and pressure diffusion terms separately and finally viscous diffusion. The same terms in addition the pressure-strain term are also given for each stress component. The budget terms are scaled by utau4 / v .
The file names are organised so that the first five letters define the variable in question. The sixth letter which is either "%" or "+" indicates the scaling so that "%" means scaling by a reference quantity such as Uw and "+" stands for the wall units i.e. scaling by utau and possibly v. The seventh and eighth character in the file names stand for the sense of rotation so that "nr" means negative Ro while "pr" means positive Ro and "_r" represents the non-rotating case. The three-letter extension tells us the absolute value of Ro. There is one exception to this naming system, in case of the file "g_ret%" the 7th and 8th letters as well as the extension have been dropped.
The variable codes i.e. the first five letters of the data-file names are given with their explanations in Table 1. All the files are of two-column format and the first column always contains the y-coordinate scaled by h except for the file g_ret% in which the first column contains the Ro-number.
Table 1. The variable codes and their explanations.
| First five letters of the file name | Variable |
| m___u | mean velocity |
| f__?? | total stress, ?? means: uu, vv, ww, or uv |
| f_t?? | separated turbulent stress |
| f_s?? | separated secondary stress |
| f___k | total k |
| b??_p | production term of each stress component |
| b??ep | dissipation term of each stress component |
| b??fi | pressure-strain term of each stress component |
| b??ds | turbulent+pressure diffusion of each stress component |
| b??dt | turbulent diffusion of each stress component |
| b??dp | pressure diffusion of each stress component |
| b??dv | viscous diffusion of each stress component |
| b_k_p | production term of k |
| b_kep | dissipation term of k |
| b_kds | turbulent+pressure diffusion of k |
| b_kdt | turbulent diffusion of k |
| b_kdp | pressure diffusion of k |
| b_kdv | viscous diffusion of k |
| g_ret | Retau as a function of Ro |