Turbulent Plane Couette Flow

The turbulent structure in plane Couette flow at low Reynolds number is studied using data obtained from Direct Numerical Simulation. The geometry is shown in figure 1, and the Reynolds number \(Re=U_wh/\nu\) is approximately 1300.

 Flow configuration Fig. 1: Flow configuration

The Computer Code

The present direct numerical simulation (DNS) was carried out utilising a finite-difference computer code.

In this code the incompressible momentum equations are discretized by the second-order- accurate central-difference scheme. The Poisson equation for the pressure is Fourier transformed with respect to the streamwise and spanwise homogeneous directions, and the resulting three-diagonal equations are solved directly for each time step. The flow field is advanced in time using a fractional-step method and with the second-order Adams-Bashforth time discretization scheme. At the walls, no-slip boundary conditions are imposed on the velocity components. Periodicity is assumed in the streamwise and spanwise directions.

The Present Computation

The case-specific information of the present computation is listed in the table below. The effect of the streamwise and spanwise geometrical length scales \(L_x\) and \(L_z\) on numerically generated turbulent Couette flow was investigated through a series of large-eddy simulations (LES) with different computational boxes at the present Reynolds number. These simulations were performed using the sub-grid model due to Mom and Kim (1982). The LES indicated that by choosing \(L_z/h = 4\pi\), the turbulent field was allowed to decorrelate in the spanwise direction.

Sampling time \(Re_{\tau}\) \(N_1\times N_2 \times N_3\) \(L_x/h\) \(L_z/h\) \(\Delta x\) \(\Delta z\) \(\Delta y\) (min, max)
1350 82.2 \( 256 \times 70 \times 256\) \( 10\pi\) \(4\pi\) 10.1 4.0 (0.7, 3.9)

The initial field for the present DNS was generated by interpolating an LES flow field from a \(64 \times 32 \times 64\) grid to a \(256 \times 70 \times 256\) grid. The LES flow field had been run for \(3500 t_*\). The DNS was then run for \(1300 t_*\), before the sampling process was started, during the first 400 of these 1300 time units, a transient behaviour occurred during which the volume-averaged turbulent kinetic energy \(k\) decreased. (The too high initial level of \(k\) was caused by the crude resolution and the sub-grid model in the LES.) During the remainder of the 1300 time units, the planar averages of the turbulence quantities were fluctuating about approximately statistically steady mean values. Because the expected large-scale vortical cells with a high degree of spanwise periodicity were not present in the flow field, the time advancement was considerable before starting the sampling in order to ensure that the large-scale field really was statistically steady. The flow field did not enter into the state with substantial periodicity during the run.

In order to satisfy \(-\overline{uv} + dU/dy = 1\) (all quantities normalized with viscous scales) and to get sufficiently smooth time-averaged two-point correlations for calculating the one-dimensional energy spectra, the calculation was run for another \(1300 t_*\) during which a data-base of flow fields was generated and time-averaged statistics were calculated. The data-base consisted of 12 fields of the primitive variables \(u\), \(v\), \(w\) and \(p\) recorded with \(12t_*\) intervals. The computation was performed with a time step of 0.03, keeping the Courant number below 0.3. The maximum fluctuation of the volume-averaged kinetic energy during sampling was below 10% of the mean value.

The marginal computer memory available restricted the conditional sampling to one half-channel, i.e. only flow fields contained between the centreplane and one of the walls were considered.

Numerical Resolution

One-dimensional spectra will not provide definite information regarding the resolution of the smallest turbulence scales in a finite-difference DNS like they do in spectral-method DNS. However, experience shows that too poor a resolution corrupts the spectra. The spectra give an impression of the range of length scales present in the simulation. The decay rate at high wavenumbers provides information regarding the amount of energy associated with the smallest length scales.

For the streamwise spectra , the energy decays at least 6 decades. For intermediate streamwise wavenumbers, the spectra contained more energy than found in Poiseuille flow. The pile up of energy around \(k_xh = 10\) caused the spectra to drop sharply at high wavenumbers. The spanwise spectra are also monotonically decreasing in the high-wavenumber range. \(E_{vv}\) displays the shortest span in energy, especially near the wall where the drop is slightly more than 3 decades. The amount of energy contained in the small-scale wall-normal motions close to the wall was, however, considered to be relatively small. Because the spectra indicated that the wall-normal velocity was most likely to be inadequately resolved very close to the wall, some further studies concerning the numerical resolution of this velocity component were carried out by studying the higher-order moments in the immediate vicinity of the wall.

The problem of numerical resolution in the case of Couette flow DNS is associated with the largest scales. The two-point correlations indicate that the turbulent velocity fluctuations decorrelated at separations corresponding to half the computational box with some exceptions: \(R_{uu}(L_x/2h)\) was approximately 0.01 at \(y = 13\) and 0.03 in the channel centre, this being the absolute maximum of the correlation at this separation. A weak periodicity in \(R_{uu}(\Delta z)\) resulted in a 2% maximum correlation in the centreplane for \(\Delta z = L_z/2\). These minor shortcomings of \(R_{uu}\), which ideally should decay to zero at separations corresponding to the half-box size, may be caused by the finite computational box.

Data available includes profiles across the half-channel of:

  • Mean velocity
  • Rms fluctuating velocities \(u'\), \(v'\), \(w'\) and Reynolds shear stress \(uv\)
  • Flatness and skewness parameters

Sample plots of selected quantities are available.

The data can be downloaded as compressed archives from the links below, or as individual files.

Quantities File
Mean velocity umean.dat
\(u\), \(w\) rms, flatness and skewness uw.dat
\(v\) rms, flatness and skewness; \(\overline{uv}\) v.dat
  1. Bech, K.H., Tillmark, N., Alfredsson, P.H., Andersson, H.I. (1995). An investigation of turbulent plane Couette flow at low Reynolds numbers. J. Fluid Mech., Vol. 304, pp. 285-319.
  2. Andersson, H.I., Pettersson, B.A. (1994). Modelling plane turbulent Couette flow at low Reynolds numbers. AIAA Paper No. 94-2342.

Indexed data:

case071 (dbcase, confined_flow)
titleTurbulent plane Couette flow
authorBech, Tillmark, Alfredsson, Andersson
flow_tagconstant_cross_section, channel_flow