cases:case017

# Laterally Strained Boundary Layers

Turbulent boundary layers on a flat surface laterally strained by converging or diverging walls. 3D flow with constant temperature.

#### Geometry of the Computational Domain

The cross-section of the duct remains nearly constant along the $x$ axis for both the laterally converging and diverging wall geometry configurations. The geometry of the duct is designed to obtain a potential flow with uniform velocity far upstream, far downstream and a prescribed $\partial W/\partial z(x,0,0)$ gradient along the line $y=0$ and $z=0$.

The flat plate surface, over which the boundary layer studied develops, is on the $y=0$ plane shown in figure 1. In the laterally converging wall case the duct has a wide rectangular cross-section at the inlet, and which contracts symmetrically in the lateral direction with downstream distance, whilst expanding in the positive $y$ direction. The laterally diverging case is obtained by simply reversing the duct geometry.

Cross-sections in both cases:

laterally converging walls laterally diverging walls
inlet 0.8 m wide by 0.25 m height 0.4 m wide by 0.5 m height
outlet 0.4 m wide by 0.5 m height 0.8 m wide by 0.25 m height

Fig. 1: Duct geometries (dimensions are in mm)

#### Flow Characteristics

The test-section is placed 500 mm downstream from a 3.5 m long contraction with a contraction ratio of 15 so that the entry flow is mainly two-dimensional. The geometry of the duct is designed to generate a strong convergence or divergence of the flow on the flat floor of the test-section without risk of flow separation under the roof. The secondary flow is generated by the lateral streamline curvature and the development of the three-dimensional boundary layers on the floor is mainly pressure-driven.

After being smoothly distorted along the transition, the flow finally relaxes towards two-dimensionality at outlet. The boundary layer is thickened in the case of convergence and thinned in case of divergence.

#### Flow Parameters

• Air at temperature: $T = 28$oC.
• Inlet free-stream velocity: $U_o = 42$ m/s.
• Barometric pressure: $p = 967$ hPa.
• Kinematic viscosity: $\nu = 1.7 \times 10^5$ m2/s.
• Density: $\rho = 1.11$ kg/m3.

#### Inflow Conditions

At station $x = 0$, the flow is largely two-dimensional, with boundary layer features:

• boundary-layer thickness: $\delta = 16$ mm
• boundary-layer displacement thickness: $\delta_1 = 2.3$ mm
• boundary-layer momentum thickness: $\delta_2 = 1.7$ mm
• free stream turbulence level: 0.1%

The following measurements are available for the turbulent boundary layer on the bottom wall:

#### Measured Quantities:

Wall static pressures (file names containing pst) have been measured through 0.8 mm tappings. The pressure coefficient is defined as $C_p = (p - p(2500,0,0))/(0.5\rho U(0,80,0)^2)$.

Wall shear stresses (file names containing cf) have been measured using either Preston tubes with diameters of 0.5 mm and 1 mm or a floating-element balance.

Velocities have been measured using either a rotatable crossed-hot-wire or a simple crossed-hot-wire probe (file names containing hw), a three-hole pressure probe (file names containing p3) and a five-hole (file names containing p5) pressure probe.

#### Measurement Errors:

 $\delta(\text{probe positions})$ $\pm 0.05$ mm $\delta(\text{angles})$ $\pm 0.3^o$ $\delta(p)$ $\pm 1$ Pa $\delta(C_f)$ $\pm 2\%$ $\delta(\text{mean velocities})$ $\pm 2\%$ $\delta(\text{Reynolds stresses})$ $\pm 10\%$

Data available include:

• Wall pressure and skin friction coefficients for the 2D and laterally converging and diverging cases
• Mean velocity profiles across the ducts at a number of streamwise and spanwise locations
• Reynolds stress profiles at a selection of streamwise locations.

Sample plots of selected quantities are available.

The data are contained in a number of files detailed in the table below. The file info.dat contains some further description of the files and content.

Apart from the geometry files, those with names starting “2_” correspond to the 2D boundary layer case, those starting with “3c_” correspond to the laterally converging walls case, and those starting with “3d_” correspond to the laterally diverging walls case.

Nozzle Geometry
Profile Case File
Computed lateral wall streamline on $y=0$ 3Dc bott_c01.dat
Computed upper wall streamline 3Dc roof_c01.dat
Measured lateral wall streamline on $y=80$ mm 3Dc bott_m01.dat
Measured lateral wall streamline on $y=40$ mm 3Dc bott_m02.dat
Measured lateral wall streamline on $y=8$ mm 3Dc bott_m03.dat
Measured lateral wall streamline on $y=40$ mm 3Dd bott_m04.dat
Measurements
2D Case Laterally Converging Case Laterally Diverging Case
Wall Pressure Measurements
Static wall pressure 2_pst01.dat 3c_pst01.dat 3d_pst01.dat Traverses across the duct, at a number of streamwise, $x$, locations
Friction Coefficient $C_f$ ( mostly from Preston tube, $D=1$ mm)
$C_f$ Spanwise at $x=0$ mm 2_cfc01.dat 3c_cfc01.dat 3d_cfc01.dat
$C_f$ Spanwise, $x=1000$ mm 2_cfc02.dat
$C_f$ Spanwise, $x=2000$ mm 2_cfc03.dat
$C_f$ Spanwise profiles 3c_cfc02.dat, 3c_cfc03.dat 3d_cfc02.dat Traverses across the duct, at a number of streamwise, $x$, locations
$C_f$ Streamwise, $z=0$ 2_cfs01.dat 3c_cfs01.dat 3d_cfs01.dat Along duct centreline, from Preston tube
$C_f$ Streamwise, $z=0$ 2_cfs02.dat 3c_cfs02.dat 3d_cfs02.dat Along duct centreline, from skin friction balance
Boundary Layer Integral Properties
Integral Properties, $z=0$ 2_p1s02.dat 3c_p1s02.dat 3d_p1s02.dat $U_e$, $\delta$, $\delta^*$, $\delta_{\theta}$, $H_{12}$, $H_{23}$ along duct centreline
Integral Properties, $x=0$ 2_p3c03.dat Values at selected $z$ locations on the inlet plane
Mean Velocity Profiles
Vel. Profiles, $z=0$ 2_p1s01.dat 3c_p1s01.dat 3d_p1s01.dat Profiles at selected $x$ locations on centreline, from Preston tube
Vel profiles, $x=0$ 2_p3c01.dat 3c_p3c01.dat 3d_p3c01.dat Profiles at selected $z$ locations, from 3-hole probe
Vel profiles, $x=50$ mm 3c_p3c02.dat 3d_p3c02.dat Profiles at selected $z$ locations, from 3-hole probe
Vel profiles, $x=900$ mm 3c_p3c03.dat 3d_p3c03.dat Profiles at selected $z$ locations, from 3-hole probe
Vel profiles, $x=1000$ mm 3c_p3c04.dat 3d_p3c04.dat Profiles at selected $z$ locations, from 3-hole probe
Vel profiles, $x=1100$ mm 3c_p3c05.dat 3d_p3c05.dat Profiles at selected $z$ locations, from 3-hole probe
Vel profiles, $x=2000$ mm 2_p3c02.dat Profiles at selected $z$ locations, from 3-hole probe
Vel profiles, $x=2500$ mm 3c_p3c06.dat 3d_p3c06.dat Profiles at selected $z$ locations, from 3-hole probe
Vel profiles, $z=-200$ mm 3c_p3s01.dat, 3c_p3s11.dat 3d_p3s01.dat Profiles at selected $x$ locations, from 3-hole probe
Vel profiles, $z=-150$ mm 3c_p3s02.dat 3d_p3s02.dat Profiles at selected $x$ locations, from 3-hole probe
Vel profiles, $z=-100$ mm 3c_p3s03.dat 3d_p3s03.dat Profiles at selected $x$ locations, from 3-hole probe
Vel profiles, $z=-0$ mm 3c_p3s04.dat 3d_p3s04.dat Profiles at selected $x$ locations, from 3-hole probe
Vel profiles, $z=100$ mm 3c_p3s05.dat 3d_p3s05.dat Profiles at selected $x$ locations, from 3-hole probe
Vel profiles, $z=150$ mm 3c_p3s06.dat 3d_p3s06.dat Profiles at selected $x$ locations, from 3-hole probe
Vel profiles, $z=200$ mm 3c_p3s07.dat, 3c_p3s71.dat 3d_p3s07.dat Profiles at selected $x$ locations, from 3-hole probe
Vel profiles, $y=0.2$ mm 3c_p3y03.dat 3d_p3y03.dat Profiles across the duct at selected $x$ locations, from 3-hole probe
Vel profiles, $y=10$ mm 3c_p3y02.dat Profiles across the duct at selected $x$ locations, from 3-hole probe
Vel profiles, $y=100$ mm 3c_p3y01.dat 3d_p3y01.dat Profiles across the duct at selected $x$ locations, from 3-hole probe
Vel profiles, $x=0$ mm 3c_p5x01.dat 3d_p5x01.dat Profiles at selected $z$ locations, from 5-hole probe
Vel profiles, $x=500$ mm 3c_p5x02.dat 3d_p5x02.dat Profiles at selected $z$ locations, from 5-hole probe
Vel profiles, $x=1000$ mm 3c_p5x03.dat 3d_p5x03.dat Profiles at selected $z$ locations, from 5-hole probe
Vel profiles, $x=1500$ mm 3c_p5x04.dat 3d_p5x04.dat Profiles at selected $z$ locations, from 5-hole probe
Vel profiles, $x=2000$ mm 3c_p5x05.dat 3d_p5x05.dat Profiles at selected $z$ locations, from 5-hole probe
Vel profiles, $x=2500$ mm 3c_p5x06.dat 3d_p5x06.dat Profiles at selected $z$ locations, from 5-hole probe
Vel profiles, $y=100$ mm 3c_p5y01.dat 3d_p5y01.dat Profiles across the duct at selected $x$ locations, from 5-hole probe
$U_e(x,z)$, $z>0$ 3c_p5e01.dat 3d_p5e01.dat Boundary layer edge velocity, from 5-hole probe
$U_e(x,z)$, $z<0$ 3c_p5e02.dat 3d_p5e02.dat Boundary layer edge velocity, from 5-hole probe
Hot Wire Measurements of Mean Velocity and Turbulence Statistics (Reynolds, stresses, triple moments, flatness, skewness)
Centreline profiles 2_hws01.dat 3c_hws01.dat 3d_hws01.dat From X-wire on the $(U,V)$ plane
Centreline profiles 2_hws02.dat 3c_hws02.dat 3d_hws02.dat From X-wire on the $(U,W)$ plane
Profiles at $x=1000$ mm 3c_hwc01.dat 3d_hwc01.dat From rotatable X-wire

Inlet Conditions: The calculation can be started at $x = 0$ using the experimental values provided as inlet conditions. These values correspond to a 2D turbulent boundary layer. For the region close to the wall ($0 \le y \le 3$ mm), the values of the turbulent quantities are not measured and the value of $k$ may need to be approximated. An alternative is to perform a separate flat plate boundary layer calculation and extract profiles from that at an appropriate stage of development.

Symmetry: Due to geometric symmetry with respect to the $z=0$ plane one can use a computational domain of only one half of the test-section.

Outlet Conditions: The rectangular duct has a constant cross-section beyond the section $x = 2500$ mm. The outlet section may be moved sufficiently far downstream in order to assume zero streamwise gradients for the flow variables.

The author of the experiments and his coworkers have also performed calculations of the two different flows using a finite-difference boundary layer code (Pompeo et al, 1993). First-order boundary layer equations have been used with the turbulent viscosity equations of Cebeci and Smith (1968). In the case of the diverging walls, good agreement with the experimental results has been found without any particular problem. In the case of the converging walls, the choice of the boundary conditions at the outer edge of the boundary layer has been found to have a considerable influence on the computed results and good agreement has only been obtained after several adjustments of the constants of the model.

1. Pompeo, L.P. (1992). An experimental study of three-dimensional turbulent boundary layers. Diss. ETH, Zurich No. 9780.
2. Pompeo, L.P., Bettelini, M.S.G., Thomann, H. (1993). Laterally strained turbulent boundary layers near a plane of symmetry. J. Fluid Mech., Vol. 257, pp. 507-532.

Indexed data:

case017 (dbcase, semi_confined_flow, confined_flow)
case017
titleLaterally Strained Boundary Layers
authorPompeo, Bettelini, Thomann
year1992
typeEXP
flow_tag3d, 3dbl, varying_cross_section