======Couette Flow with Fixed Wavy Wall======
=====Experiments by Nakabayashi, Kitoh and Iwata=====

----

====Description====

Experimental work has been performed to investigate turbulent Couette-type flow with
repeated longditudinal pressure gradients that are realized in a channel consisting of a
wavy fixed wall on one side and a moving wall on the other.

This type of flow simulates the flow in high speed journal bearings operating in 
turbulent conditions. 2D flow with constant temperature.

===Geometry===

The geometry of the channel, and coordinate system used, are shown in <imgref cas10-geom>.
The channel has its lower wall moving and its upper wall wavy and fixed.

<imgcaption cas10-geom| Channel geometry>{{ figs:case010:cfww-geom1.png | Channel geometry}}</imgcaption>

The test section is \(L = 1.4\) m long (one wavelength) with a mean height (distance
between the two walls) of \(2h = 15\) mm.

The fixed wavy wall is approximated by \(y = 2h - e\,cos(2\pi x/L)\) with \(e = 4.5\) mm,
although an accurate set of spline fitting functions is provided in the file {{cdata:case010:geom1c.dat|geom1c.dat}}.

The measurements have been carried out in the middle part of the channel (total length of 3.5
times the wavelength) in order to minimize upstream and downstream effects. The pressure
has been found to be the same at the inlet and outlet of the considered one
wavelength-long portion. Moreover the longitudinal pressure gradient is also found to be
periodic.

The mean velocity distributions and turbulent intensities are measured at various
streamwise positions of \(x/l\) for a Reynolds number \(Re=2hU_b/\nu=8000\). Non-dimensional
pressure gradient parameters \(u^{-1} = (\nu dp/dx)/(\rho (U^*)^3)\)
can be seen in the table below.

^ \(x/l\) |  0  |  0.125  |  0.250  |  0.375  |  0.500  |  0.625  |  0.750  |  0.875  |
^ \(u^{-1}\) |  -0.009  |  0.017  |  0.025  |  0.026  |  0.018  |  -0.016  |  -0.015  |  -0.023  |

===Inflow Conditions===

Profiles from the fixed wall to moving wall at \(x=0\) are in provided in the file
{{cdata:case010:couette.dat|couette.dat}} of:

  * First order moments: \(U\), \(V\)
  * Second order moments: \(\overline{u^2}\), \(\overline{v^2}\)

====Experimental Details====

All velocity measurements were made using hot-wire anemometer.

===Measurement Errors===

Measurement errors are not published in any English language paper.

====Available Measurements====

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

  * {{cdata:case010:cfww-allfiles.zip|cfww-allfiles.zip}}
  * {{cdata:case010:cfww-allfiles.tar.gz|cfww-allfiles.tar.gz}}

[[case010-plots|Sample plots]] of selected quantities are available.


The file {{cdata:case010:couette.dat|couette.dat}} contains 

  * Profiles at \(x = 0.0\),  \(0.175\), \(0.35\), \(0.525\), \(0.7\), \(0.875\), \(1.05\) and \(1.225\) m of:
      * First order moments: \(U/U_b\), \(V/U_b\), \(U/U^*\)
      * Second order moments: \(\overline{u^2}\), \(\overline{v^2}\), \(\overline{u^2}/(U^*)^2\)

  * Friction velocity (\(U^*\)) along the wavy wall at the above eight \(x\) locations.

====Recommendations for Calculations====

The following suggestions were given to contributors to a collaborative workshop in which the case was studied.

**Inlet and Outlet Conditions**

Ideally the calculation of the channel flow should be performed using periodicity 
conditions for both velocities and pressure gradient (\(dp(x+L)/dx = dp(x)/dx\)). 
However, given the short length of the experimental set-up, this may not be fully
periodic, and it is therefore also possible to start at \(x = 0\), prescribing the experimental 
values provided as inlet conditions, and assume zero gradients for the flow variables 
at outlet. 


====References====

  - Nakabayashi, K., Kitoh, O., Iwata, H. (1991). Turbulent Couette type flow with an alternating pressure gradient. //Proc. 8th Int. Symp. on Turbulent Shear Flows//, Munich.
  - El Telbany, M.M.M., Reynolds, A.J. (1980). [[https://doi.org/10.1017/S0022112080000973|Velocity distributions in plane turbulent channel flows]]. //J. Fluid Mech.//, Vol. 100, pp. 1-29.
  - Huffman, G.D., Bradshaw P. (1972). [[https://doi.org/10.1017/S0022112072000035|A note on von Kármán's constant in low Reynolds number turbulent flows]]. //J. Fluid Mech.//, Vol. 53, pp. 45-60.
  - Townsend, A.A. (1961). [[https://doi.org/10.1017/S0022112061000883|Equilibrium layers and wall turbulence]]. //J. Fluid Mech.//, Vol. 11, pp. 97-120.

----

Indexed data:
<data dbcase confined_flow>
case       : 010
title      : Couette Flow with Fixed Wavy Wall
author*    : Nakabayashi, Kitoh, Iwata
year       : 1989
type       : EXP
flow_tag*  : 2d, varying_cross_section
</data>