^{1,a)}, S. J. Peltier

^{1}and R. D. W. Bowersox

^{1}

### Abstract

The effects of convex curvature on the outer structure of a Mach 4.9 turbulent boundary layer (*Re* _{θ} = 4.7 × 10^{4}) are investigated using condensate Rayleigh scattering and analyzed using spatial correlations, intermittency, and fractal theory. It is found that the post-expansion boundary layer structure morphology appears subtle, but certain features exhibit a more obvious response. The large-scale flow structures survive the initial expansion, appearing to maintain the same physical size. However, due to the nature of the expansion fan, a differential acceleration effect takes place across the flow structures, causing them to be reoriented, leaning farther away from the wall. The onset of intermittency moves closer towards the boundary layer edge and the region of intermittent flow decreases. It is likely that this reflects the less frequent penetration of outer irrotational fluid into the boundary layer, consistent with a boundary layer that is losing its ability to entrain freestream fluid. The fractal dimension of the turbulent/nonturbulent interface decreases with increasing favorable pressure gradient, indicating that the interface's irregularity decreases. Because fractal scale similarity does not encompass the largest scales, this suggests that the change in fractal dimension is due to the action of the smaller-scales, consistent with the idea that the small-scale flow structures are quenched during the expansion in response to bulk dilatation.

This work was sponsored by the AFOSR/NASA National Center for Hypersonic Laminar-Turbulent Transition Research (Grant No. FA9550-09-1-0341). The authors wish to thank Dr. B. S. Thurow for the provision of the high-speed laser system and for his helpful suggestions on an earlier draft of this manuscript, as well as K. P. Lynch for his role in the high-speed laser system experiments.

I. INTRODUCTION

II. EXPERIMENTAL SETUP

A. Experimental flow facility and conditions

B. Streamline pressure gradients

III. FLOW VISUALIZATION METHOD

A. Condensate light scattering

B. Particle size and dynamics

IV. DATA ANALYSIS METHODS

A. Convective velocity and spatial correlations

B. Intermittency

C. Fractalanalysis

V. RESULTS AND DISCUSSION

A. Flow visualizations

B. Convective velocity structure

C. Two-point spatial correlations

D. Structure angles

E. Intermittency

F. Fractalanalysis

VI. SUMMARY AND CONCLUSIONS

### Key Topics

- Fractals
- 49.0
- Flow visualization
- 43.0
- Turbulent flows
- 37.0
- Boundary layer turbulence
- 24.0
- Condensation
- 22.0

## Figures

Rendering of the experimental flow facility. The flow direction is from left to right.

Rendering of the experimental flow facility. The flow direction is from left to right.

Schematic representation of the streamline pressure gradients. The curved walls are given by *Y* = *AX* ^{3} + *BX* ^{2} (see text for further details). The dashed-dotted line indicates the center of the measurement domains. The origin (*X*, *Y*) = (0, 0) is located where the curvature begins. The coordinates *x* and *y* are the body intrinsic coordinates of the tangential and wall-normal directions, respectively. Note the axes are not shown on the same scale. Inset shows three-dimensional renderings of the curvature-driven pressure-gradient models.

Schematic representation of the streamline pressure gradients. The curved walls are given by *Y* = *AX* ^{3} + *BX* ^{2} (see text for further details). The dashed-dotted line indicates the center of the measurement domains. The origin (*X*, *Y*) = (0, 0) is located where the curvature begins. The coordinates *x* and *y* are the body intrinsic coordinates of the tangential and wall-normal directions, respectively. Note the axes are not shown on the same scale. Inset shows three-dimensional renderings of the curvature-driven pressure-gradient models.

Rendering of the experimental arrangement. The flow is from left to right. Laser light can be seen to enter from the top. The flow is imaged with a CCD camera. Note that the knife-edge filter has been omitted for clarity.

Rendering of the experimental arrangement. The flow is from left to right. Laser light can be seen to enter from the top. The flow is imaged with a CCD camera. Note that the knife-edge filter has been omitted for clarity.

The effects of threshold level on the intermittency. Results are shown for the ZPG boundary layer case but are also representative of the MPG and SPG cases. Open circles show boundary layer intermittency within a ZPG Mach 2.84 boundary layer using the hot-wire anemometry mass flux signal from Selig *et al.* ^{ 45 } Open triangles show boundary layer intermittency within a ZPG Mach 4.95 boundary layer using fluctuating Pitot pressures from McClure ^{ 46 } (the latter data are taken from nalmis ^{ 47 } ).

The effects of threshold level on the intermittency. Results are shown for the ZPG boundary layer case but are also representative of the MPG and SPG cases. Open circles show boundary layer intermittency within a ZPG Mach 2.84 boundary layer using the hot-wire anemometry mass flux signal from Selig *et al.* ^{ 45 } Open triangles show boundary layer intermittency within a ZPG Mach 4.95 boundary layer using fluctuating Pitot pressures from McClure ^{ 46 } (the latter data are taken from nalmis ^{ 47 } ).

Schematic illustration of the fractal (box-counting) dimension determination procedure. The object (the boundary layer interface) is covered with area elements of decreasing size, and we observe how the number of required elements varies. If the object exhibits scale similarity, i.e., it looks the same at different levels of magnification (in a statistical sense), then the increase in the number of elements will follow a power law, which is symptomatic of fractal behavior.

Schematic illustration of the fractal (box-counting) dimension determination procedure. The object (the boundary layer interface) is covered with area elements of decreasing size, and we observe how the number of required elements varies. If the object exhibits scale similarity, i.e., it looks the same at different levels of magnification (in a statistical sense), then the increase in the number of elements will follow a power law, which is symptomatic of fractal behavior.

A series of temporally uncorrelated instantaneous flow visualization images. (Top row) ZPG, (middle row) MPG, (bottom row) SPG. In all cases the flow direction is from left to right. Note that the camera has been rotated in the pressure gradient cases to be parallel with the wall. The exposure time is 500 ns, although the laser pulse duration is 4 ns. The mean boundary layer thicknesses are δ = 9.0, 9.9, 12.9 mm, respectively. Lettered regions are discussed in the text.

A series of temporally uncorrelated instantaneous flow visualization images. (Top row) ZPG, (middle row) MPG, (bottom row) SPG. In all cases the flow direction is from left to right. Note that the camera has been rotated in the pressure gradient cases to be parallel with the wall. The exposure time is 500 ns, although the laser pulse duration is 4 ns. The mean boundary layer thicknesses are δ = 9.0, 9.9, 12.9 mm, respectively. Lettered regions are discussed in the text.

Three-dimensional rendering of the ZPG boundary layer's instantaneous structure. Results show nucleated isosurfaces corresponding to the interface between the boundary layer and freestream flow. The field-of-view of the volume is (*x*, *y*, *z*) ∈ [93.0 mm (12.1δ) (Taylor's hypothesis), 28.6 mm (3.7δ), 28.6 mm (3.7δ)]. The view is looking upstream.

Three-dimensional rendering of the ZPG boundary layer's instantaneous structure. Results show nucleated isosurfaces corresponding to the interface between the boundary layer and freestream flow. The field-of-view of the volume is (*x*, *y*, *z*) ∈ [93.0 mm (12.1δ) (Taylor's hypothesis), 28.6 mm (3.7δ), 28.6 mm (3.7δ)]. The view is looking upstream.

Examples of temporally uncorrelated instantaneous convective velocity vector fields. (a) ZPG, (b) SPG. Results show the instantaneous convective velocity vector field determined from the cross-correlation of double-pulsed image pairs (0.5 μs time separation). The results are shown in a streamwise convective reference frame of 0.92*U* _{∞}, with a 0.1*U* _{∞} wall-normal subtraction in the SPG case due to the diverging nature of the expansion. The background shows the corresponding second image. Camera exposure time is 500 ns, although the laser exposure time is 4 ns.

Examples of temporally uncorrelated instantaneous convective velocity vector fields. (a) ZPG, (b) SPG. Results show the instantaneous convective velocity vector field determined from the cross-correlation of double-pulsed image pairs (0.5 μs time separation). The results are shown in a streamwise convective reference frame of 0.92*U* _{∞}, with a 0.1*U* _{∞} wall-normal subtraction in the SPG case due to the diverging nature of the expansion. The background shows the corresponding second image. Camera exposure time is 500 ns, although the laser exposure time is 4 ns.

Two-point spatial correlation functions at various heights within the boundary layer *y* _{0}/δ = 0.7, 0.8, and 0.9. (Top row) ZPG, (middle row) MPG, (bottom row) SPG. Contours show 0.5 to 1.0 in increments of 0.05. The results are based on an ensemble size *N* = 1000 images for each test case.

Two-point spatial correlation functions at various heights within the boundary layer *y* _{0}/δ = 0.7, 0.8, and 0.9. (Top row) ZPG, (middle row) MPG, (bottom row) SPG. Contours show 0.5 to 1.0 in increments of 0.05. The results are based on an ensemble size *N* = 1000 images for each test case.

Variation of the structure angle θ throughout the boundary layer for the ZPG, MPG, and SPG cases. Inset shows the definition of θ. Results are based on an ensemble size *N* = 1000 images for each test case. The compressible ZPG boundary layer results of Poggie *et al.* ^{ 21 } and Ringuette *et al.* ^{ 44 } are shown for comparison. Note that the results have been corrected for the reorientation of the reference frame in order to isolate the effects of the pressure gradient.

Variation of the structure angle θ throughout the boundary layer for the ZPG, MPG, and SPG cases. Inset shows the definition of θ. Results are based on an ensemble size *N* = 1000 images for each test case. The compressible ZPG boundary layer results of Poggie *et al.* ^{ 21 } and Ringuette *et al.* ^{ 44 } are shown for comparison. Note that the results have been corrected for the reorientation of the reference frame in order to isolate the effects of the pressure gradient.

Intermittency profiles within the boundary layer for the ZPG, MPG, and SPG cases. The incompressible (curve-fit) results of Klebanoff ^{ 64 } are included for comparison. The inset shows a zoomed view of the data.

Intermittency profiles within the boundary layer for the ZPG, MPG, and SPG cases. The incompressible (curve-fit) results of Klebanoff ^{ 64 } are included for comparison. The inset shows a zoomed view of the data.

Fractal analysis. (a) ZPG, (b) MPG, (c) SPG. The main figure shows an example of the variation of the number of boxes *N* _{ɛ} against the box size ɛ for a single image. The inner cutoff is the laser sheet thickness, *l* _{ z }, and the outer cutoff is the integral length scale, *L* _{ x }. The fractal dimension *d* is extracted from the power-law region in between. The upper inset shows the probability density distribution for *d*. The lower inset shows the results for 100 images. The average (mean) fractal dimension is shown in each figure part along with the corresponding standard deviation, σ_{ d }.

Fractal analysis. (a) ZPG, (b) MPG, (c) SPG. The main figure shows an example of the variation of the number of boxes *N* _{ɛ} against the box size ɛ for a single image. The inner cutoff is the laser sheet thickness, *l* _{ z }, and the outer cutoff is the integral length scale, *L* _{ x }. The fractal dimension *d* is extracted from the power-law region in between. The upper inset shows the probability density distribution for *d*. The lower inset shows the results for 100 images. The average (mean) fractal dimension is shown in each figure part along with the corresponding standard deviation, σ_{ d }.

Schematic representation of an idealized conceptual model summarizing the main features observed when the hypersonic boundary layer negotiates convex curvature. Note this model only applies during the initial expansion process and that the sketch is not shown to scale. Annotation arrows represent either motion in the convective reference frame of the coherent motions or to emphasize a qualitative behavior.

Schematic representation of an idealized conceptual model summarizing the main features observed when the hypersonic boundary layer negotiates convex curvature. Note this model only applies during the initial expansion process and that the sketch is not shown to scale. Annotation arrows represent either motion in the convective reference frame of the coherent motions or to emphasize a qualitative behavior.

## Tables

Experimental conditions.

Experimental conditions.

Summary of impulse parameters in convex curvature-driven pressure-gradient studies. MPG and SPG refer to where more than one pressure-gradient strength was considered.

Summary of impulse parameters in convex curvature-driven pressure-gradient studies. MPG and SPG refer to where more than one pressure-gradient strength was considered.

Summary of the fractal dimension of some classical turbulent flows. Note that the rule of codimension has been used where appropriate.

Summary of the fractal dimension of some classical turbulent flows. Note that the rule of codimension has been used where appropriate.

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