Three-dimensional receptivity of hypersonic sharp and blunt cones to free-stream planar waves using hierarchical input-output analysis (2024)

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Three-dimensional receptivity of hypersonic sharp and blunt cones to free-stream planar waves using hierarchical input-output analysis

David A. Cook and Joseph W. Nichols

Phys. Rev. Fluids 9, 063901 – Published 3 June 2024

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Abstract

Understanding the receptivity of hypersonic flows to free-stream disturbances is crucial for predicting laminar to turbulent boundary layer transition. Input-output analysis as a receptivity tool considers which free-stream disturbances lead to the largest response from the boundary layer using the global linear dynamics. Two technical challenges are addressed. First, we restrict the allowable forcing to physically realizable inputs via a free-stream boundary modification to the classic input-output formulation. Second, we develop a hierarchical input-output (H-IO) analysis which allows us to solve the three-dimensional problem at a fraction of the computational cost otherwise associated with directly inverting the fully three-dimensional resolvent operator. Next, we consider Mach 5.8 flows over a sharp cone and two blunt cones with $3.6 mm$ and $7.2 mm$ spherically blunt tips. H-IO correctly predicts that the sharp cone boundary layer is most receptive to slow acoustic waves at an optimal incidence angle of $10^{\circ}$ , validating the method. We then investigate the effect of free-stream disturbances on the blunt cone boundary layer and identify two distinct vorticity-dominated receptivity mechanisms for the oblique first-mode instability at $10 kHz$ and an entropy layer instability at 40 and $70 kHz$ . Our results reveal these receptivity processes to be highly three-dimensional in nature, involving both the nose tip and excitation along narrow bands at certain azimuthal angles along the oblique shock downstream. We interpret these processes in terms of critical angles from linear shock/perturbation interaction theory. Finally, we show how these receptivity processes vary with frequency and nose tip bluntness, and demonstrate how this methodology might be applied to transition prediction from first principles.

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Received 7 June 2023
Accepted 13 February 2024

DOI:https://doi.org/10.1103/PhysRevFluids.9.063901

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Compressible boundary layersFlow instabilityHigh-speed flowShock waves

Fluid Dynamics

Authors & Affiliations

David A. Cook^* and Joseph W. Nichols

Department of Aerospace Engineering and Mechanics, University of Minnesota, Minneapolis, Minnesota 55414, USA

^*Corresponding author: cookx894@umn.edu

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Vol. 9, Iss. 6 — June 2024

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Three-dimensional receptivity of hypersonic sharp and blunt cones to free-stream planar waves using hierarchical input-output analysis (12)

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Figure 1
Schematic of the shock-kinematic boundary condition. The function $X (y, z, t)$ describes the displacement of the shock as a function of space and time.
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Figure 2
Illustration of the main steps in the hierarchical approach to three-dimensional I/O analysis.
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Figure 3
H-IO gain vs wavenumber for several sharp and blunt cone analyses. For each case shown, the gain at $m = 50$ is less than 10% of the maximum gain.
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Figure 4
Gain and input error quantification for H-IO analyses of the sharp cone at (a)–(c) 10kHz and (d)–(f) 70kHz. Error is quantified by varying the number of included Fourier coefficients and then showing (a), (d) the leading gains, (b), (c) the input distributions, and (c), (f) log-scale relative input error with respect to $m_{t} = 50$ . Also shown are $D_{1}$ outputs for sharp cone H-IO analyses at 10kHz and for (g) $m_{t} = 30$ , (h) $m_{t} = 40$ , and (i) $m_{t} = 50$ . As more wavenumbers are included, the output physics converge to a single physical mechanism.
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Figure 5
Low-rank truncation error are quantified for sharp cone H-IO analysis at 60kHz in terms of (a), (d) the input error, (b), (e) the gain error, and (c), (f) the output error. For several truncation numbers, (a)shows good agreement between the $D_{1}$ forcing distributions as a function of $ψ$ , and (d)shows log-scale relative input error with respect to $D_{t} = 25$ . For several truncation numbers, (b)shows good agreement between the first 100 gains for $D_{t} > 1$ , and (e) shows log-scale relative gain error with respect to $D_{t} = 25$ . Output is shown for two truncation numbers: (c) $D_{t} = 1$ and (f) $D_{t} = 10$ . Relative density norm error between (c), (f) and $D_{t} = 25$ are shown in (c)and (f).
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Figure 6
$D_{1}$ gain error quantification for H-IO analysis of the sharp cone across several frequencies and truncation values. Error is relative to case with $D_{t} = 25$ .
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Figure 7
$D_{1}$ gain as a function of frequency from hierarchical input-output analysis of $M = 5.8$ flows over sharp and blunt cones.
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Figure 8
Mean boundary layer profiles of (a)velocity, (b)temperature, and (c)density at several streamwise positions as a function of wall-normal coordinate $η$ for both flows. Shown in (d)are mean boundary layer thicknesses based on edge enthalpy $δ_{h}$ .
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Figure 9
LST $N$ -factors contours for (a–c) the sharp cone, (d–f) the $R_{N} = 3.6 mm$ blunt cone, and (g–i) the $R_{N} = 7.2 mm$ blunt cone. $N$ -factors are shown at $x = 1.0 m$ as functions of frequency $f$ and azimuthal wavenumber $m$ in (a), (d), and (g). $N$ -factors are shown as a functions of $x$ and $f$ at $m = 0$ in (b), (e), and (h), and $m = 17$ in (c), (f), and (i). Panels (a)and (d)both contain a peak at low frequency and high wavenumber corresponding to oblique Mack first-mode instability, and a second peak at high frequency but low wavenumber corresponding to Mack second-mode instability. Comparing (a)–(c), (d)–(f), and (g)–(i), bluntness reduces the $N$ -factors associated with both modes of instability and thus should delay the onset of laminar to turbulent transition.
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Figure 10
Global linear response of the sharp cone to (a)a free-stream slow acoustic wave at $f = 70 kHz$ and $ψ = 0$ , and (b)the leading input direction ( $D_{1}$ ) at $f = 70 kHz$ . Pressure contours on the outermost surface are shown in the free stream. Pressure contours on the inner surfaces show the boundary layer response.
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Figure 11
(a)Wall-normal profiles of fluctuating (i) velocity, (ii) pressure, and (iii) temperature at $x = 1.0 m$ and $θ = 0$ for the sharp cone boundary layer. The profiles from the forced response (solid lines) are compared to LST eigenfunctions (dash-dotted lines) corresponding to axisymmetric Mack second mode instability. Wall-parallel profiles along the sharp cone of Chu energy amplitude in response to (b)five different types of free-stream waves at $ψ = 0$ angle of incidence, (c)slow acoustic waves at three different incidence angles, and (d)H-IO $D_{1}$ forcing. In panels (b)–(d), the wall profiles are plotted with fitted $A_{0} e^{N}$ functions corresponding to LST N-factors.
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Figure 12
H-IO results at 70kHz for (a)–(c) the sharp cone, (d)–(f) the 3.6mm blunt cone, and (g)–(i) the 7.2mm blunt cone. Shown are (a), (d), (e) gains vs I/O direction, (b), (e), (h) $D_{1}$ input directions, and (c), (f), (i) physical realizations of the optimal forcing in the free stream.
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Figure 13
Global response of (a)the $R_{N} = 3.6 mm$ blunt cone and (b)the $R_{N} = 7.2 mm$ blunt cone to the $D_{1}$ forcing direction at $70 kHz$ . Contours on the outermost surface are in the free stream, while density isosurfaces show the downstream response. Reference contours of local shock obliqueness (solid lines) and incidence angles (dashed lines) are included.
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Figure 14
Theoretical acoustic generation from vortical waves impinging on oblique shock waves as a function of obliqueness angle ( $θ_{o b l}$ ) and incidence angle ( $θ_{i}$ ).
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Figure 15
Contours of spatially amplifying (a), (b) velocity, (c), (d) temperature, and (e), (f) pressure for the (a), (c), (e) $R_{N} = 3.6 mm$ blunt cone and (b), (d), (f) $R_{N} = 7.2 mm$ blunt cone at 70kHz. The solid streamlines are extracted at the boundaries of the injected velocity packet, and the dashed lines show the boundary layer edge.
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Figure 16
Streamwise energy amplification envelope of the entropy layer instability taken along several entropy layer streamlines for (a)the $R_{N} = 3.6 mm$ blunt cone and (b)the $R_{N} = 7.2 mm$ blunt cone at 70kHz. Also shown are streamline heights above the wall ( $δ_{str}$ ) and the boundary layer edge height ( $δ_{b l}$ ) as a visualization of where the entropy layer interacts with the boundary layer.
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Figure 17
Profiles of absolute value of fluctuating velocity from the direct response to the $D_{1}$ forcing for the (a) $R_{N} = 3.6 mm$ blunt cone at and (b)the $R_{N} = 7.2 mm$ blunt cone. Profiles are shown along with the F modes from the LST at the same streamwise positions at $m = 0$ .
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Figure 18
Contours of fluctuating temperature near the end of the domain for the (a) $R_{N} = 3.6 mm$ blunt cone and (b) $R_{N} = 7.2 mm$ blunt cone. The entropy layer in (a)undergoes swallowing, whereas no swallowing occurs in (b).
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Figure 19
H-IO results at 10kHz for (a)–(c) the sharp cone and (d)–(f) the 3.6mm blunt cone. Shown are (a), (d) gains vs I/O direction, (b), (e) $D_{1}$ input directions, and (c), (f) physical realizations of the optimal forcing in the free stream.
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Figure 20
Schematic showing the dependence of fluctuating velocity on incidence angle for different types of vortical waves. Velocities $u_{n}$ and $u_{t}$ comprise the first type of vortical wave, while $u_{p}$ comprises the second type of vortical wave.
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Figure 21
Isosurfaces of (a), (c) pressure and (b), (d) $w$ velocity generated by incident $D_{1}$ vortical waves in the near-tip region of the (a), (b) sharp cone and (c), (d) 3.6mm blunt cone at 10kHz.
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Figure 22
Absolute value of fluctuating wall pressure as a function of the azimuth for several streamwise positions as a result of $D_{1}$ forcing of the (a)sharp cone and (b)3.6mm blunt cone at 10kHz. The disturbances enter the boundary layer upstream and amplify along the $θ = 60^{\circ}$ azimuth. Profiles at $x = 1.0 m$ and $θ = 60^{\circ}$ of the absolute value of fluctuating velocity for (c) the sharp cone and (d) the $R_{N} = 3.6 mm$ cone. Profiles are shown along with the Mack first mode at the same streamwise position at $m = 18$ .
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Figure 23
Spatial amplification of (a), (b) pressure and (c), (d) Chu energy amplitude at the wall at several azimuthal positions. Profiles are shown for (a), (c) the sharp cone and (b), (d) the 3.6mm blunt cone along with the best fit with the LST N-factor at appropriate wavenumbers. Significant amplification occurs upstream of the first-mode neutral point.
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Figure 24
H-IO results at 40kHz for (a)–(c) the 3.6mm blunt cone and (d)–(f) the 3.6mm blunt cone. Shown are (a), (d) gains vs I/O direction, (b), (e) $D_{1}$ input directions, and (c), (f) physical realizations of the optimal forcing in the free stream.
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Figure 25
Global response of (a)the 3.6mm blunt cone and (b)the 7.2mm blunt cone to $D_{1}$ forcing directions at $40 kHz$ . Contours on the outermost surface are in the free stream, while temperature isosurfaces show the downstream response.
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Figure 26
Streamwise growth of the Chu energy amplitude in the entropy layer for blunt cones with (a) $R_{N} = 3.6 mm$ and (b) $R_{N} = 7.2 mm$ . Also shown are streamline heights above the wall ( $δ_{str}$ ) and the boundary layer edge height ( $δ_{b l}$ ) as a visualization of where the entropy layer is interacting with the boundary layer.
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Figure 27
Contours of spatially amplifying (a), (b) velocity, (c), (d) temperature, and (e), (f) pressure for the (a), (c), (e) $R_{N} = 3.6 mm$ blunt cone and (b), (d), (f) $R_{N} = 7.2 mm$ blunt cone at 40kHz. The solid streamlines are extracted at the boundaries of the injected velocity packet, and the dashed lines show the boundary layer edge.
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Figure 28
$D_{1}$ and $D_{2}$ forcing directions at frequencies from 10–90kHz from H-IO analysis of the (a)sharp, (b)3.6mm blunt, and (c)7.2mm blunt cones. Dot-dashed lines are the $D_{2}$ forcing directions and solid lines are the $D_{1}$ forcing direction. Arrows denote increasing frequency.
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Figure 29
Receptivity $N$ -factors computed from the H-IO responses to the $D_{1}$ input directions for each cone at (a)10kHz, (b)40kHz, and (c)70kHz. The initial amplitude is determined by the peak of the free-stream forcing wave packet. Arrows indicate increasing nose-tip bluntness.
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