^{1}, Jens H. M. Fransson

^{1,a)}and P. Henrik Alfredsson

^{1,b)}

### Abstract

An experimental study of the receptivity of disturbances and their subsequent development into a three-dimensional boundary layer has been carried out. The three-dimensional boundary layer was set up using a flat plate with a swept leading edge and a pressure gradient using a displacement body at the ceiling of the test section. Low level free-stream turbulence was generated with five different screens and was shown to generate traveling crossflow modes for all but the lowest turbulence level, i.e., for , where instead a stationary crossflow disturbance dominated. Stationary crossflow disturbances were triggered by small cylindrical roughness elements arranged in an array. For high enough roughnessReynolds number () stationary disturbances growing exponentially were seen and their amplitude seems to scale with .

This work is supported by the European Commission through the FP6 project “Telfona” (Contract No. AST4-CT-2005-516109) and the Linné Flow Centre. We would like to additionally thank David Tempelmann and Professor Ardeshir Hanifi for valuable inputs.

I. INTRODUCTION

A. Receptivity and transition of 3D flows

B. Motivation of present work

II. EXPERIMENTAL SETUP

A. Facility and measurement technique

B. Experimental Design

III. TRIGGERING OF CROSSFLOW MODES

A. Traveling modes

B. Stationary modes

IV. RECEPTIVITY TO FREE-STREAM TURBULENCE

A. Base flow

B. Disturbance structure

C. Growth of disturbances

V. RECEPTIVITY TO SURFACE ROUGHNESS

A. Base flow

B. Detection of the stationary mode

C. Initial receptivity and development

D. Structure of stationary modes

VI. SUMMARY AND CONCLUSIONS

A. Traveling disturbances

B. Stationary disturbances

C. Final statement

### Key Topics

- Turbulent flows
- 62.0
- Boundary layer turbulence
- 15.0
- Velocity measurement
- 10.0
- Wind tunnels
- 9.0
- Reynolds stress modeling
- 7.0

## Figures

(a) Coordinate system of the test section with its corresponding velocity components. (b) Configuration of X-probe allowing for all three velocity component to be measured.

(a) Coordinate system of the test section with its corresponding velocity components. (b) Configuration of X-probe allowing for all three velocity component to be measured.

(a) Sketch of the leading edge and plate along with coordinate systems. (b) Photo of the leading edge. (c) Side view of the setup at the test section centerline with the displacement body placed on the ceiling in black and the plate in gray. (d) Top view of the setup with side walls in black and the plate in gray. The lengths and are used to scale the streamwise and spanwise coordinates, respectively.

(a) Sketch of the leading edge and plate along with coordinate systems. (b) Photo of the leading edge. (c) Side view of the setup at the test section centerline with the displacement body placed on the ceiling in black and the plate in gray. (d) Top view of the setup with side walls in black and the plate in gray. The lengths and are used to scale the streamwise and spanwise coordinates, respectively.

Free-stream velocity distributions for the streamwise and spanwise velocity components. (a) and (b) correspond to and velocity distributions, respectively. .

Free-stream velocity distributions for the streamwise and spanwise velocity components. (a) and (b) correspond to and velocity distributions, respectively. .

(a) Streamwise and (b) spanwise velocity in normal coordinates. The solid line in (a) corresponds to the power-law relation, Eq. (1), fitted to the data. .

(a) Streamwise and (b) spanwise velocity in normal coordinates. The solid line in (a) corresponds to the power-law relation, Eq. (1), fitted to the data. .

Spanwise variation of the (a) streamwise and (b) spanwise velocity in normal coordinates for increasing .

Spanwise variation of the (a) streamwise and (b) spanwise velocity in normal coordinates for increasing .

Spanwise variation of the Hartree parameter, .

Spanwise variation of the Hartree parameter, .

Parameter values of integral length scale and turbulence intensity measured at the leading edge for the grids (cf. Table I for symbols).

Parameter values of integral length scale and turbulence intensity measured at the leading edge for the grids (cf. Table I for symbols).

(a) Sketch of the location of the roughness elements. , , and denote the wavelength or spacing, the height and the diameter of the cylindrical roughness elements. (b) The parameters investigated in the experiments. Filled squares indicate traversing was done in full planes, whereas circles indicate wider spans at one wall-normal position.

(a) Sketch of the location of the roughness elements. , , and denote the wavelength or spacing, the height and the diameter of the cylindrical roughness elements. (b) The parameters investigated in the experiments. Filled squares indicate traversing was done in full planes, whereas circles indicate wider spans at one wall-normal position.

(a) Free-stream velocity variation in normal coordinates along the centerline for all grids. The gray line represents a fit to the data with . (b) All wall-normal profiles plotted together with the theoretical (gray line) Falkner–Skan–Cooke profile in tunnel coordinates with .

(a) Free-stream velocity variation in normal coordinates along the centerline for all grids. The gray line represents a fit to the data with . (b) All wall-normal profiles plotted together with the theoretical (gray line) Falkner–Skan–Cooke profile in tunnel coordinates with .

(a) Two-point velocity correlation functions in the spanwise direction around the disturbance peak for all grids at . (b) The spanwise variation of the mean velocity around the disturbance peak at . (c) Amplitude of the stationary mode taken from (b). (d) Averaged spanwise length scale for all grids for both the traveling and stationary modes in open and filled symbols respectively. See Table I for symbols.

(a) Two-point velocity correlation functions in the spanwise direction around the disturbance peak for all grids at . (b) The spanwise variation of the mean velocity around the disturbance peak at . (c) Amplitude of the stationary mode taken from (b). (d) Averaged spanwise length scale for all grids for both the traveling and stationary modes in open and filled symbols respectively. See Table I for symbols.

Evolution of wall-normal disturbance profiles in the downstream direction for grids (a) , (b) , (c) , (d) , (e) , and (f) all profiles with subtracted free-stream disturbance levels and normalized with its maximum value. .

Evolution of wall-normal disturbance profiles in the downstream direction for grids (a) , (b) , (c) , (d) , (e) , and (f) all profiles with subtracted free-stream disturbance levels and normalized with its maximum value. .

(a) Phase shift at different spanwise distances for . (b) Resulting phase speed, for all the grids.

(a) Phase shift at different spanwise distances for . (b) Resulting phase speed, for all the grids.

(a) Growth of the integrated disturbance profiles for each grid, in absolute terms. (b) Growth of the integrated disturbance profiles for each grid, normalized with the value at . (c) Derivative of N using central difference. (d) Same as (b) plotted with spline curves through the data points to emphasize the locations of regions I, II, and III. See Table I for symbols.

(a) Growth of the integrated disturbance profiles for each grid, in absolute terms. (b) Growth of the integrated disturbance profiles for each grid, normalized with the value at . (c) Derivative of N using central difference. (d) Same as (b) plotted with spline curves through the data points to emphasize the locations of regions I, II, and III. See Table I for symbols.

The spanwise variation of the fluctuating velocity for all grids. See Table I for symbols.

The spanwise variation of the fluctuating velocity for all grids. See Table I for symbols.

(a) Energy levels for all grids at , 0.50, and 0.55 as a function of . Solid line represents a quadratic fit to the data. (b) Growth of the integrated disturbance profiles for each grid, normalized with the square of the free-stream turbulence level measured at the leading edge. See Table I for symbols.

(a) Energy levels for all grids at , 0.50, and 0.55 as a function of . Solid line represents a quadratic fit to the data. (b) Growth of the integrated disturbance profiles for each grid, normalized with the square of the free-stream turbulence level measured at the leading edge. See Table I for symbols.

Energy spectra for all grids at the location of the disturbance peak at (a*)* and (b) . The legend identifying the different grids is below figure (b). Some amount of aliasing error can be seen for at but does not affect the behavior of the peak. (c) Evolution of the premultiplied energy spectra at the disturbance peak in the streamwise direction for grid .

Energy spectra for all grids at the location of the disturbance peak at (a*)* and (b) . The legend identifying the different grids is below figure (b). Some amount of aliasing error can be seen for at but does not affect the behavior of the peak. (c) Evolution of the premultiplied energy spectra at the disturbance peak in the streamwise direction for grid .

Base flow for the stationary disturbance experiments along the fit to the power law. .

Base flow for the stationary disturbance experiments along the fit to the power law. .

Base flow mean velocity profiles, relative to the streamline, without any disturbance for the stationary disturbance experiments at all streamwise and spanwise locations. .

Base flow mean velocity profiles, relative to the streamline, without any disturbance for the stationary disturbance experiments at all streamwise and spanwise locations. .

Spanwise variation of the velocity along with their energy spectra for and . (a), (b), and (c) correspond to the downstream positions , 0.5, and 1.0 respectively.

Spanwise variation of the velocity along with their energy spectra for and . (a), (b), and (c) correspond to the downstream positions , 0.5, and 1.0 respectively.

Spanwise variation of the velocity along with their FFTs for and . (a), (b), and (c) correspond to the downstream positions , 0.5, and 1.0 respectively.

Spanwise variation of the velocity along with their FFTs for and . (a), (b), and (c) correspond to the downstream positions , 0.5, and 1.0 respectively.

The disturbance development of the stationary modes, , for two roughness heights, normalized with the for (gray) and (black). —–: mode 1, – – –: mode 2, : mode 3. (a) With linear axes. (b) In a semilog plot to show the exponential behavior.

The disturbance development of the stationary modes, , for two roughness heights, normalized with the for (gray) and (black). —–: mode 1, – – –: mode 2, : mode 3. (a) With linear axes. (b) In a semilog plot to show the exponential behavior.

Contour plots of the streamwise disturbances for and in the -plane taken at the -position with the maximum disturbance level. Low speed areas are in white, high-speed areas are in black.

Contour plots of the streamwise disturbances for and in the -plane taken at the -position with the maximum disturbance level. Low speed areas are in white, high-speed areas are in black.

Contour plots of the streamwise disturbances for and at (a) , (b) , (c) , (d) , (e) , (f) , (g) , (h) , (i) , and (j) . The values at the wall are extrapolated to 0. Low speed areas are in white, high-speed areas are in black.

Contour plots of the streamwise disturbances for and at (a) , (b) , (c) , (d) , (e) , (f) , (g) , (h) , (i) , and (j) . The values at the wall are extrapolated to 0. Low speed areas are in white, high-speed areas are in black.

Contour plots of the spanwise disturbances for and . The -locations are identical to Fig. 23. The values at the wall are extrapolated to 0. Negative velocities are in white, positive velocities are in black.

Contour plots of the spanwise disturbances for and . The -locations are identical to Fig. 23. The values at the wall are extrapolated to 0. Negative velocities are in white, positive velocities are in black.

## Tables

The five turbulence generating grids used in this investigation. Bar width, mesh width and grid solidity are denoted , , and , respectively. The streamwise integral length scale, , and the turbulence intensity, , are measured at the leading edge in the center of the wind tunnel.

The five turbulence generating grids used in this investigation. Bar width, mesh width and grid solidity are denoted , , and , respectively. The streamwise integral length scale, , and the turbulence intensity, , are measured at the leading edge in the center of the wind tunnel.

Cylindrical surface roughness parameters studied during the experiment.

Cylindrical surface roughness parameters studied during the experiment.

Free-stream variation of parameters [cf. Eq. (1)] determined in a least square fit sense to the data for each grid.

Free-stream variation of parameters [cf. Eq. (1)] determined in a least square fit sense to the data for each grid.

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