HYPERSONIC BOUNDARY LAYER STABILITY
ANALYSIS USING MOMENTUM POTENTIAL THEORY
A THESIS
SUBMITTED TO THE FACULTY OF THE
UNIVERSITY OF MINNESOTA
BY
Mary L. Houston
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE OF
MASTER OF SCIENCE
Joseph W. Nichols, Advisor
September 2020
© 2020
Mary L. Houston
All rights reserved
iACKNOWLEDGEMENTS
I would first and foremost like to thank my research adviser, Dr. Joseph W. Nichols, who
guided me from an undergraduate researcher to the completion of this masters degree.
I would like to thank my roommate Morgan Foote whom supported and tolerated me for
the past four years, as well as the meaningful community of friends I’ve made during my
time here.
Special thanks to John Thome for generating the geometry and baseflow used in the present
analysis and to David Cook for the input-output results.
This work was supported by the Office of Naval Research (ONR) grant no. N00014-17-1-
2496.
ii
ABSTRACT
Linear Stability Theory (LST) and the Parabolized Stability Equations (PSE) have provided
valuable tools for analysis and prediction of laminar to turbulent transition for plates,
sharp cones, and geometries for which parallel-flow or a slowly-varying boundary layer
can be assumed. However, these techniques struggle to capture the complex flow-physics
present near the tip of blunt-cones. Input-output analysis has been used in conjunction
with direct numerical simulation to capture the effects of nose bluntness on downstream
stability. Using the results of the input-output analysis we apply momentum potential theory
(MPT) to preform fluid-thermodynamic (FT) decomposition, separating disturbances into
their vortical, thermal and acoustic components. A reference case of Mach 6 flow over a
flat-plate is computed and output responses are compared to the results for Mach 6 flow
over a blunt-cone of 7표 half angle. Perturbation eigenfunctions and structures are examined
in the areas of second-mode amplification. For both the flat-plate and blunt-cone the
vortical components are the largest, followed by the thermal then acoustic components.
Fluid-thermodynamic structures in the second-mode amplification region of blunt-cone
show wall-normal stretching above the critical layer. Fluid-thermodynamic decomposition
of full-domain input and output results for the blunt-cone geometry are considered. It is
found that input sensitivity is highest at the top of the entropy layer and along the boundary
layer edge for the fore-half of the cone. Output response in the streamwise direction is
highest in the regions between the generalized inflection point (GIP) and the boundary layer
edge and dissipates near the surface, whereas wall-normal response extends to the surface
and shows a local minimum between the GIP and boundary layer edge. To compliment
iii
existing studies on hypersonic boundary layer response to surface roughness/ vibration we
look at input sensitivity and output response at the surface. It is found that there is greater
sensitivity to wall-normal forcing than streamwise forcing at the surface and among the
three FT components in this direction the vortical had the highest relative output amplitude.
Finally, total fluctuating enthalpy (TFE) is considered for both the flat-plate and blunt-cone,
in both cases the thermal terms provides the strongest source of TFE.
iv
TABLE OF CONTENTS
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii
Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi
Chapter I: Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Past research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
Chapter II: Theory and Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.1 Input-Output Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2 Momentum potential theory . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.3 Numerical Solution to Differential Equations . . . . . . . . . . . . . . . . . 12
2.4 Disturbance Phase Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.5 Significant boundary layer locations . . . . . . . . . . . . . . . . . . . . . . 14
2.6 Non-dimensionalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
Chapter III: Flow Configuration and Validation . . . . . . . . . . . . . . . . . . . . 17
3.1 Baseflow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.2 Verification and Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
Chapter IV: Fluctuations in Flow Variables . . . . . . . . . . . . . . . . . . . . . . 28
4.1 Baseflow Profiles and Eigenfunctions . . . . . . . . . . . . . . . . . . . . . 28
4.2 Second-Mode I/O Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
Chapter V: Fluid-Thermodynamic Features . . . . . . . . . . . . . . . . . . . . . . 38
5.1 Eigenfunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
5.2 Perturbation Structures in Second-Mode Growth . . . . . . . . . . . . . . . 38
Chapter VI: Total fluctuating enthalpy . . . . . . . . . . . . . . . . . . . . . . . . . 51
6.1 Total Fluctuating Enthalpy in Flat-Plate Boundary Layer . . . . . . . . . . . 51
6.2 Total Fluctuating Enthalpy in blunt-cone Boundary Layer . . . . . . . . . . 53
Chapter VII: Summary and Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . 58
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
Appendix A: Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
A.1 The viscous stress tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
vA.2 FT components of TFE . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
Appendix B: Tensor derivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
Appendix C: Mode 1 features . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
C.1 Mode 1 - Eigenfunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
C.2 Mode 1 - flow var I/O . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
C.3 Mode 1 - FT I/O . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
C.4 Surface profiles - mode 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
Appendix D: Mode 3 features . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
D.1 Mode 3 - Eigenfunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
D.2 Mode 3 - flow var I/O . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
D.3 Mode 3 - FT I/O . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
D.4 Surface profiles - mode 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
vi
LIST OF FIGURES
Number Page
3.1 baseflow temperature field over blunt-cone. . . . . . . . . . . . . . . . . . . 17
3.2 blunt-cone grid. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.3 Validation of Laplace operator. . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.4 Validation of gradient operator. . . . . . . . . . . . . . . . . . . . . . . . . 20
3.5 Solutions to the Poisson equation. . . . . . . . . . . . . . . . . . . . . . . . 21
3.6 Pressure perturbations on plate surface . . . . . . . . . . . . . . . . . . . . 22
3.7 baseflow profile flat-plate . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.8 Second-mode eigenfunctions of flow quantities over flat-plate . . . . . . . . 23
3.9 Second-mode eigenfunctions of FT quantities over flat-plate . . . . . . . . . 24
3.10 FT perturbation structures in region of second-mode amplification over flat-
plate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
4.1 Baseflow profiles at point of maximum first-mode instability over cone. . . . 30
4.2 Baseflow profiles at point of maximum second-mode instability over cone. . 31
4.3 Baseflow profiles at point of maximum third-mode instability over cone. . . . 33
4.4 second-mode eigenfunctions of flow quantities over blunt-cone. . . . . . . . 34
4.5 Second-mode pressure I/O results. . . . . . . . . . . . . . . . . . . . . . . . 36
4.6 Second-mode streamwise momentum I/O results. . . . . . . . . . . . . . . . 36
4.7 Second-mode wall-normal momentum I/O results. . . . . . . . . . . . . . . 37
5.1 Second-mode Eigenfunctions of FT quantities at over blunt-cone. . . . . . . 39
vii
5.2 Second-mode perturbation structures in region of amplification over blunt-cone. 41
5.3 Second-mode 퐵′휉 I/O results. . . . . . . . . . . . . . . . . . . . . . . . . . . 43
5.4 Second-mode 퐵′휂 I/O results. . . . . . . . . . . . . . . . . . . . . . . . . . . 44
5.5 Second-mode ∇휉휓′퐴 I/O results. . . . . . . . . . . . . . . . . . . . . . . . . 44
5.6 Second-mode ∇휂휓′퐴 I/O results. . . . . . . . . . . . . . . . . . . . . . . . . 45
5.7 Second-mode ∇휉휓′푇 I/O results. . . . . . . . . . . . . . . . . . . . . . . . . 46
5.8 Second-mode ∇휂휓′푇 I/O results. . . . . . . . . . . . . . . . . . . . . . . . . 46
5.9 baseflow entropy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
5.10 Second-mode 퐵′휉 I/O results on cone surface. . . . . . . . . . . . . . . . . . 47
5.11 Second-mode 퐵′휂 I/O results on cone surface. . . . . . . . . . . . . . . . . . 48
5.12 Second-mode ∇휉휓′퐴 I/O results on cone surface. . . . . . . . . . . . . . . . 48
5.13 Second-mode ∇휂휓′퐴 I/O results on cone surface. . . . . . . . . . . . . . . . 49
5.14 Second-mode ∇휉휓′푇 I/O results on cone surface. . . . . . . . . . . . . . . . 49
5.15 Second-mode ∇휂휓′푇 I/O results on cone surface. . . . . . . . . . . . . . . . 50
6.1 Total fluctuating enthalpy over flat-plate. Sonic line (——), critical layer (–
– –), generalized inflection point (——) and boundary layer edge (– – –). . . 52
6.2 Second-mode fluctuating enthalpy over plate - component-wise. . . . . . . . 54
6.3 Total fluctuating enthalpy over cone. . . . . . . . . . . . . . . . . . . . . . . 54
6.4 Second-mode fluctuating enthalpy over cone - component-wise. . . . . . . . 57
C.1 First-mode eigenfunctions of flow quantities over blunt cone. . . . . . . . . . 69
C.2 First-mode pressure I/O results. . . . . . . . . . . . . . . . . . . . . . . . . 70
C.3 First-mode streamwise momentum I/O results. . . . . . . . . . . . . . . . . 70
C.4 First-mode wall-normal momentum I/O results. . . . . . . . . . . . . . . . . 71
C.5 First-mode FT structures in region of amplification over blunt cone. . . . . . 72
C.6 First-mode 퐵휉 I/O results. . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
C.7 First-mode 퐵휂 I/O results. . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
C.8 First-mode ∇휉휓′퐴 I/O results. . . . . . . . . . . . . . . . . . . . . . . . . . . 74
viii
C.9 First-mode ∇휂휓′퐴 I/O results. . . . . . . . . . . . . . . . . . . . . . . . . . . 74
C.10 First-mode ∇휉휓′푇 I/O results. . . . . . . . . . . . . . . . . . . . . . . . . . . 75
C.11 First-mode ∇휂휓′푇 I/O results. . . . . . . . . . . . . . . . . . . . . . . . . . . 75
C.13 First-mode 퐵′휉 I/O results on cone surface. . . . . . . . . . . . . . . . . . . . 76
C.14 First-mode ∇휉휓′퐴 I/O results on cone surface. . . . . . . . . . . . . . . . . . 77
C.15 First-mode ∇휂휓′퐴 I/O results on cone surface. . . . . . . . . . . . . . . . . . 77
C.16 First-mode ∇휉휓′푇 I/O results on cone surface. . . . . . . . . . . . . . . . . . 78
C.17 First-mode ∇휂휓′푇 I/O results on cone surface. . . . . . . . . . . . . . . . . . 78
D.1 Third-mode eigenfunctions of flow quantities over blunt cone. . . . . . . . . 80
D.2 Third-mode pressure I/O results. . . . . . . . . . . . . . . . . . . . . . . . . 81
D.3 Third-mode streamwise momentum I/O results. . . . . . . . . . . . . . . . . 81
D.4 Third-mode wall-normal momentum I/O results. . . . . . . . . . . . . . . . 82
D.5 Third-mode FT structures in region of amplification over blunt cone. . . . . . 83
D.6 Third-mode 퐵′휉 I/O results. . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
D.7 Third-mode 퐵′휂 I/O results. . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
D.8 Third-mode ∇휉휓′퐴 I/O results. . . . . . . . . . . . . . . . . . . . . . . . . . 85
D.9 Third-mode ∇휂휓′퐴 I/O results. . . . . . . . . . . . . . . . . . . . . . . . . . 85
D.10 Third-mode ∇휉휓′푇 I/O results. . . . . . . . . . . . . . . . . . . . . . . . . . 86
D.11 Third-mode ∇휂휓′푇 I/O results. . . . . . . . . . . . . . . . . . . . . . . . . . 86
D.12 Third-mode 퐵′휉 I/O results on cone surface. . . . . . . . . . . . . . . . . . . 87
D.13 Third-mode 퐵′휂 I/O results on cone surface. . . . . . . . . . . . . . . . . . . 87
D.14 Third-mode ∇휉휓′퐴 I/O results on cone surface. . . . . . . . . . . . . . . . . 88
D.15 Third-mode ∇휂휓′퐴 I/O results on cone surface. . . . . . . . . . . . . . . . . 88
D.16 Third-mode ∇휉휓′푇 I/O results on cone surface. . . . . . . . . . . . . . . . . 89
D.17 Third-mode ∇휂휓′푇 I/O results on cone surface. . . . . . . . . . . . . . . . . 89
ix
LIST OF TABLES
Number Page
2.1 Boundary layer edge values . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.2 free-stream values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
x
xi
NOMENCLATURE
Constants
훾 ratio specific heats 1.4
i imaginary unit
√−1
R푎푖푟 ideal gas constant 287.06 퐽/푘푔퐾
Sub and Superscripts
∗ complex conjugate.
′ perturbation quantity.
∞ freestream value.
푒 edge value.
푤 surface value.
 base flow quantity.
Variables
휔 circular frequency.
휓 total irrotational field.
휓퐴 acoustic component of momentum density.
휓푇 thermal (entropic) component of momentum density.
휌 mass density.
t time.
휉,휂 streamwise and wall-normal coordinates blunt-cone.
B vortical (solenoidal) component vector.
u velocity vector.
xii
a local speed of sound.
c disturbance phase speed.
I generalized inflection point (GIP).
x,y streamwise and wall-normal coordinates flat-plate.
A note on terminology: As anyone familiar with the topic can attest, every flow phenomena
and their uncle has at one point or another been referred to as a mode. Therefore, to
avoid future confusion, the author will delineate the various labels we will be applying to
the various phenomena for the remainder of this dissertation. Fluid-thermal components;
vortical, thermal and acoustic (퐵푖, ∇푖휓푇 and ∇푖휓퐴). Stabilitymodes such that the first mode
is Tollmien-Schlichting and the second and third modes are Mack mode. Fluctuation will
be used to designate variation from mean flow in flow quantities (푢푖, 휌, p, T ect.), whereas
perturbation refers to disturbances of the fluid-thermodynamic components.
1Chapter 1
INTRODUCTION
1.1 Motivation
Hypersonic boundary layer (HBL) transition has been an area of considerable interest for the
past several decades. Increased surface heat flux generated by turbulent flow, compared to
laminar flow, necessitates increased thermal protective measures for any high-speed aerial
vehicle. As well as increased heating, turbulent flow increases skin friction and drag on the
vehicle and reduces the effectiveness of control surfaces. Therefore delaying or preventing
the transition to turbulence would reduce the cost and complexity of high-speed aerial
vehicle construction. Understanding the physical mechanisms which trigger this transition
is a crucial step in designing these vehicles.
There has been compelling research to suggest that transition can be delayed through
modification of the geometry and surface properties of these vehicles [1]. Specifically,
blunting of the leading edge or tip of the vehicle has been shown to delay the onset of
transition [1–3], this will be discussed further in section 1.2. A noise absorbing surface,
be it through numerical filtering or use of an ultrasonically absorptive coating (UAC), was
also found to delay transition[1, 3, 4].
Past numerical research has relied on linear stability theory (LST) and parabolized stability
equations (PSE) to predict the onset of transition. These techniques have provided significant
2advances in understanding the physical mechanisms which lead to transition, and are still
used today to guide both computational and experimental research [3, 5]. Continued
gains in processing power have allowed compuationalists to solve the complete Naviar-
Stokes (N-S) equations and build high-fidelity models of fluid-dynamical systems. Direct
numerical simulation (DNS) is one such powerful tool we can use to explore themechanisms
responsible for transition. It allows us to capture the entire flow field and directly measure
fluid and thermodynamic quantities with high degree of resolution. A further advantage of
DNS is that it does not require the assumptions parallel flow or a similarity profile [5].
The goals of this research are to investigate energy transfer between fluid-thermodynamic
(FT) components and correlate the stream-wise location at the onset of transition with
the relative amplitudes of the FT components or their interactions. This understanding
could inform the design of future high-speed aerial vehicles and guide research efforts into
laminar flow control. Choosing total fluctuating enthalpy as the energy variable, we can
map enthalpy fluxes and their constituent sources and sinks [5–7].
1.2 Past research
Linear stability theory (LST) was first used to investigate the interaction of acoustic waves
with a hypersonic boundary layer by Mack in 1975 [8]. Since then numerous numerical
investigations into boundary layer stability of high-speed flows using eigenvalue methods
and linear stability analysis have been carried out byMalik (1990)[9], Fedorov andKhokhlov
(1991, 2002)[10, 11] and Fedorov (2003)[12]. An excellent summary of research in HBL
transition is given by [1].
The mechanisms underpinning transition of hypersonic flows are fundamentally differ-
ent than those of subsonic and moderate-supersonic flows [11]. Instabilities in subsonic
and moderate-supersonic boundary layer flows are generated by excitement of first-mode
Tollmien-Schlichting (T-S) waves [1]. Other modes of the discrete spectrum are stable and
so coupling between T-S waves and other modes is insignificant [1]. Solving the boundary
3layer receptivity problem in these flows becomes identifying the scale-convergence mecha-
nism which tunes external disturbances to match the frequency and wavelength of the T-S
waves [11].
For hypersonic flows the second-mode instability is dominant [10, 11], although the first
and higher modes coexist in the boundary layer [1, 13]. At high enough speeds there is a
region of the flow which is supersonic with respect to the disturbance phase velocity. The
wall normal location where the disturbance phase speed is exactly sonic with respect to the
mean flow is the sonic line, that is: 푦 = 푦푎 such that 푈 (푦푎) = 푐 − 푎(푦푎), where U(y) is the
mean flow, c is the disturbance phase velocity and 푎(푦푎) is the local speed of sound. This
causes the boundary layer to act as a wave guide, wherein acoustic rays are trapped between
the wall and the sonic line [1]. The free-stream Mach number for which second-mode
growth occurs is a function of the wall thermal condition, such that a cooled wall facilitates
second-mode growth at a lower Mach number than an insulated wall [1, 8].
For small nose radii, blunting the tip of the vehicle has been shown to delay the onset
of transition on cones [1–3] as well as reduce stagnation heat flux [2, 14] 1. Bluntness
may delay transition by reducing local edge Mach number and unit Reynolds number
in the entropy layer, which itself is a result of bluntness. This moves the neutral point
downstream, stabilizing the second-mode [15]. According to this model, with sufficient
bluntness transition should be completely avoidable, unfortunately this is not the case.
Beyond a critical value, a further increase in tip bluntness results in the reversal of this
trend. Laurence et. al.[3] conducted experiments on blunt-cones with 7표 half angle and
observed the trend of delayed transition with increasing nose tip radius.
Though observed experimentally, this reversal cannot be predicted by LST or PSE by
considering the second-mode instability alone. Using input-output (I/O) analysis, Cook et.
al.[16] found qualitative changes in the dominant instabilitymodeswith increased bluntness.
While the instability of the second-mode diminishes, a broadband unstable mode develops
1As compared to sharp cones at the same flow conditions.
4in the entropy layer. With increasing bluntness the growth of this entropy layer instability
becomes high enough to trigger transition, giving rise to the transition reversal phenomenon
[16]. Using a similar approach to Cook et. al.[16], Goparaju et. al.[17] recently conducted
DNS on a blunt-cone geometry to computationally capture the reversal phenomenon. They
also found that for a given local Reynolds number, perturbation growth of the most amplified
frequency first decreases then increases with increasing nose bluntness, indicating transition
reversal.
Recently, Unnikrishnan and Gaitonde (2019) [5] applied the momentum potential theory
proposed by Doak (1989) [6] to separate energy flux into turbulent, acoustic and thermal
components, which will be referred to as fluid-thermodynamic (FT) components. The flux
splitting was carried out using DNS results for a flat plat with zero angle of attack and free-
stream of Mach 6. Both adiabatic and cold wall boundary conditions were used. Stability
modes are named according to their phase speed near the leading edge such that what is
called mode S has the same phase speed of the slow acoustic wave, 푐 = 1− 1/푀 , and mode
F matches the speed of the fast acoustic wave, 푐 = 1 + 1/푀 . Mack’s second-mode was
found to be dominant in the flow regime presented and is most receptive to mode S.
After splitting the flux into the three FT components it was found that the vortical component
was the largest, followed by the thermal component, with the acoustic component being
the smallest. The vortical component exhibits peak amplitude at the generalized inflection
point (GIP) where the mean flow gradients are large and was significant for the length of the
plate. Vortical perturbations are also generated by free-stream acoustic waves interacting
with strong gradients in the shock being partly converted into vortical perturbations, creating
a region of high vorticity just past the shock. The DNS results show vortical phase speed
reduces after the F-S synchronization point, this is thought to be due to non-parallel effects
which are not captured by linear stability theory. This reduction of phase velocity is a
potential precursor to transition and identification of second-mode instability growth. It
was also found that the thermal and acoustic FT components show distinct pre- and post-
5synchronization behaviour.
Themagnitude of the acoustic component in the boundary-layer is minimal until the location
of second-mode growth, at which point the acoustic perturbations are amplified in the
vicinity of the critical layer. Although the acoustic component is present both above and
below the sonic line, there is no acoustic flux across the sonic line due to the waveguide
nature between it and the wall. The non-trapped acoustic component radiates away from
the boundary layer but with rapid attenuation. The thermal component also exhibits trapped
behaviour beneath a wavy line centered on the GIP. The internal reflection of the thermal
component may be linked to the acoustic component, however the sinusoidal nature of
the demarcation is the result of vortex tilt, influencing the thermal component by way of
Crocco’s theorem.
Change in energy associated with each FT component is a consequence of interactions
between different disturbance components and the mean flow. We quantify these changes
with the total fluctuating enthalpy (TFE) budget. Variations in TFE due to each FT
component can be correlated with corresponding source terms in the energy balance.
Sources (positive changes) contribute to production of TFE and sinks (negative changes)
dissipate TFE. ForMack’s second-mode disturbances the vortical component acts as a source
prior to second-mode growth but as a sink subsequently, whereas the thermal and acoustic
components are sources. The thermal source term in particular sustains the production
of TFE in the trapped acoustic region of the second-mode [5]. Wall cooling can reduce
the changes in TFE and delay transition, but the thermal term is the least sensitive to this
condition [5].
In this work we broadly follow the DNS analysis presented in [5] for the flat plat boundary
layer and extend it to flow around a blunt-cone. We also discuss the results of input-output
analysis for three modes of instability on the blunt-cone.
6Chapter 2
THEORY AND METHODOLOGY
Prior to the analysis which will be the focus of this paper, perturbation and mean flow fields
were generated by applying input-output analysis to a converged DNS baseflow solution.
2.1 Input-Output Analysis
Input-output analysis is advantageous in the analysis of blunt-body flows because, like DNS,
it does not require the assumption of parallel flow or slow boundary layer growth [14, 18].
Input-output analysis captures modal growth as well as non-modal stability effects, such
as transient growth and disturbance amplification [18, 19]. For these reasons input-output
analysis has been better able to predict transition on blunt-cones, which can occur upstream
of the location predicted by LST or PSE, as well capture the transition reversal phenomenon
[14, 16].
In I/O analysis we consider the dynamics of small fluctuations about a steady baseflow
governed by the linearized compressible Navier-Stokes equations. After discretization, the
linearized governing equations are a represented as a matrix referred to as the Jacobian.
The Jacobian is defined as
7A =
휕푅푖
휕푈 푗
(2.1)
where R is the residual vector produced by the CFD code at each time step and U is the
conserved vector. Instead of simply finding the eigenfunctions of ¤푥 = A푥 , in I/O analysis
we consider the response to forcing f in the system below,
¤푥 = A푥 + B 푓 (2.2a)
푦 = C푥, (2.2b)
where B controls the effects of the forcing and C specifies how output y is measured. After
a temporal Fourier transform, the outputs can be related to the inputs by
푦ˆ = C(−푖휔I − A)−1B 푓ˆ = H 푓ˆ , (2.3)
where H is the transfer function. The maximum gain is determined though a singular value
decomposition of the transfer function,
H =푈Σ푉 ∗, (2.4)
where U and V are matrices of left and right singular vectors and Σ is a diagonal matrix of
singular values. The principal left singular vector in U gives the optimal response, and the
principal right singular vector in V gives the optimal forcing. The principal singular vectors
are connected through 휎1, the principal singular value of Σ. Details relating to methods
used in preforming the the singular value decomposition can be found in Dwivedi et. al.
[20, 21] and Jovanovic and Baheim [18].
In the present work, we are forcing at one constant frequency at a time across the entire
domain. Sweeping through a range of forcing frequencies allows us to determine which
8frequency is the most most amplified overall. Solving the transfer function in (2.3) for 푓ˆ
gives us the spatial distribution for this optimal forcing. In this way the input-output analysis
informs us where to force to get the largest response.
2.2 Momentum potential theory
Flux splitting
Here we follow the methodology developed by Doak (1989) [6] and applied to hypersonic
transition by Unnikrishnan and Gaitonde (2019) [5]. Momentum potential theory allows
for an exact decomposition because the continuity equation is already linear in 휌u:
휕휌
휕푡
+ ∇ · (휌u) = 0. (2.5)
We can assume fluctuations follow the harmonic ansatz and so can be Fourier transformed
into the form
휙′ = 휙ˆ(푦)푒푖(훼푥−휔푡) (2.6)
where 휙′ represents any primitive perturbation quantity.
Helmholtz decomposition is used to separate the momentum field into unique irrotational
and solenoidal components [6]. The irrotational component is represented as scalar potential
휓′, which is a linear combination of acoustic potential, 휓′퐴, and thermal potential, 휓
′
푇 ,
whereas the rotational component is designated by the solenoidal vector quantity B. The
Helmholtz decomposition yields
휌u = B + B′ − ∇휓′, (2.7a)
where we have assumed ∇휓 = 0. Subtracting the mean flow and re-arranging gives
B′ = (휌u)′ + ∇휓′. (2.7b)
9Taking the divergence of (2.7a) to remove B and combining with (2.5) yields a Poisson
equation for 휓′:
∇2휓′ = 휕휌
′
휕푡
(2.8)
The scalar potential 휓′ is then further reduced to a linear sum of the acoustic (isentropic)
and thermal (isobaric) contributions:
휓′ = 휓′퐴 + 휓′푇 , (2.9)
where 휓′퐴 and 휓
′
푇 are defined by
∇2휓′퐴 =
1
푎2
휕푝′
휕푡
(2.10a)
∇2휓′푇 =
휕휌
휕푆
휕푆′
휕푡
(2.10b)
and 푎 is the local speed of sound calculated from baseflow values. Assuming the same
harmonic ansatz, equation (2.6), for density and pressure fluctuations, 휌′ and 푝′, respectively,
their derivatives can be taken to be −푖휔휌 and −푖휔푝, respectively, where 휔 is the circular
frequency of the perturbation. Solving (2.8) and (2.10a) for 휓′ and 휓′퐴 respectively and
thus (2.9) for 휓′푇 , gives the acoustic and thermal potentials. Taking the gradients of 휓
′
퐴 and
휓′푇 in the streamwise and wall-normal directions giving the streamwise and wall-normal
components of the corresponding momenta. For the flat-plate case, used for verification,
these directions will be x and y. On the blunt-cone the streamwise coordinate will be
휉 and the wall-normal coordinate 휂. Substituting the gradients into (2.7b) gives vortical
component B.
Energy transfer between FT components
Momentum potential theory also allows us to derive an energy budget as perturbations
grow downstream. The flow is assumed to be time-stationary, which is defined by Doak as
10
meaning that both the time derivative and the time integral of any fluctuating quantity has
zero mean value over a given time interval (푡1 ≤ 푡 ≤ 푡2) [7].
The variable used to represent the energy budget is the total fluctuating enthalpy (TFE), H’
[6, 7]. We are able to express the generic equation for conservation of TFE in terms of flux
gradients, sources and sinks:
∇ · [ 푓 푙푢푥] = 푠표푢푟푐푒푠 + 푠푖푛푘푠. (2.11)
For a time-stationary flow, total enthalpy, H, can be split into mean and fluctuating compo-
nents as below
퐻 = ℎ + 1
2
u · u = ℎ + 1
2
(u · u) + ℎ′ + 1
2
(u · u)′. (2.12)
Where we use Reynolds’ (equation (2.13a)) decomposition and assume the product of two
fluctuating quantities is negligible (equation (2.13b)) such that the fluctuating component
of the product of two quantities is as defined by equation (2.13c).
u · u = (u − u′) · (u − u′) (2.13a)
u′ · u′ ≈ 0 (2.13b)
(u · u)′ = 2 ∗ uu′. (2.13c)
Now the left hand side of equation (2.11) becomes
∇ ·
[
퐻′B′ + 퐻′(−∇휓′퐴) + 퐻′(−∇휓′푇 )
]
, (2.14)
11
and the right hand side
−
{
B′ · 훼′ + [(−∇휓′퐴) · 훼′] + [(−∇휓′푇 ) · 훼′ − (휌푇)′휕푆′휕푡 ]
}
, (2.15)
where 휕푆′휕푡 is the time rate of change in entropy fluctuation. Here 훼
′ is the ’acceleration’
vector, which can be split into the following components,
훼′ = 훼1′ + 훼2′, (2.16)
훼1
′ = (휔 × u)′, (2.17)
the fluctuating Lamb vector which accounts for the effects of vorticity fluctuations, and
훼2
′ = −
(
푇∇푆 + 1
휌∞푅푒∞
∇ · 휏
)′
, (2.18)
which has not been given a name in the literature but accounts for the effects of the entropy
gradient and the viscous stress tensor. We will eventually find that in the regimes of the
present work the entropic terms, ∇푆 in equation (2.18), eclipse the viscous terms, ∇ · 휏 in
equation (2.18), and so 훼2′ will be referred to as the entropic acceleration vector. In the
above 휔 = ∇ × u is the vorticity vector, ∇푆 is the entropy gradient and 휏 is the viscous
stress tensor, the entries of which can be found in Appendix A. The denominator of the
second term in (2.18) is the result of non-dimensionalization. We make the assumption that
entropy, S, is a time-stationary quantity and can be decomposed as 푆 = 푆 + 푆′ [6]. Total and
fluctuating entropy are defined
푆 = 푆∞ + 푐푝 ln
(
푇
푇∞
)
− 푅 ln
(
푝
푝∞
)
(2.19a)
푆′ = 푐푝 ln
(
1 + 푇
′
푇∞
)
− 푅 ln
(
1 + 푝
′
푝∞
)
≈ 푐푝 푇
′
푇∞
− 푅 푝
′
푝∞
(2.19b)
12
where 푐푝 is the specific heat at constant pressure and R is the ideal gas constant. Entropy
only appears as a gradient or time derivative in the governing equations, therefore without
loss of generality we can let 푆∞ = 0.
The terms within the square brackets of equation (2.14) are the total mean flux of energy
from the turbulent, thermal and acoustic components of the momentum fluctuations. For
a free jet it is hypothesized that equation (2.14) is zero [6], however a jet is not subject to
the high gradients generated by the wall as is the case in boundary layer flows, so it is not
clear that the hypothesis can be generalized to other flows. In fact, we would assume that
the TFE would need to increase to cause transition.
The terms in equation (2.15) give the sources and sinks of TFE broken down by FT compo-
nent. The first term, 퐵′ · 훼′, provides the vortical FT contribution to the TFE. The second
term, (−∇휓′퐴) · 훼′, provides the acoustic contribution and the third term, (−∇휓′푇 ) · 훼′, pro-
vides the thermal FT contribution to the TFE. The final term in equation (2.15), (휌푇)′ 휕푆′휕푡 ,
concerns entropy fluctuations but does not directly represent the effects of a FT component.
2.3 Numerical Solution to Differential Equations
To solve the Poisson equation, such as equation (2.8) and equation (2.10a), a discrete
Laplacian operator is constructed and inverted. Discrete divergence and gradient operators
are also needed to find all desired quantities, although there is no reason these operators
cannot be inverted, their inversion is never required in the calculations. 1
Operator Construction
The computationalmesh surrounding the blunt-cone body is non-Cartesian, non-orthogonal,
and with a high degree of stretching. Therefore, the Laplacian, divergence and gradient
operators used for solving the governing equations must be built using a generalized co-
1For the sake of completeness we do test both forward and inverse Laplace and gradient operators in
chapter 3
13
ordinate system. This can be achieved most elegantly using tensor calculus, details of the
derivations can be found in Appendix B.
Ultimately, we are able to craft gradient and divergence operators;
∇휙 = 푔푖 푗 휕휙
휕휈푖
e 푗 (2.20a)
and
∇ · v = 1√
푔
휕
휕휈푖
(√
푔v 푗
)
(2.20b)
for scalar 휙 and vector v with contravariant components v 푗 . As well as a generalized
Laplacian operator;
∇2휙 = 1√
푔
휕
휕휈 푗
(√
푔푔 푗 푘
휕휙
휕휈푘
)
, (2.21)
where g is the determinant of the metric tensor, 푔푖 푗 , which provides the mesh scaling factors.
On a side note, because the components the basis set are in general neither orthogonal nor
of unit length, the dot product also requires the use of the metric tensor,
a · b = 푔푖 푗푎푖∗푏 푗 . (2.22)
The ∗ above the first component in the right hand side designates the complex conjugate, as
required for the complex dot product.
Boundary Conditions
All FT components go to zero at the wall since there is zero momentum flux at the wall.
This does not imply that vorticity and pressure fluctuations are zero at the wall. The wall
vorticity is accounted for in the curl of the solenoidal component, as calculated at the wall.
The pressure fluctuations on the wall are dependent on interior values, through the boundary
condition. The boundary conditions used in solving ∇2휓′ and ∇2휓′퐴 were Neumann at the
14
wall and Dirichelet elsewhere, as the fluctuating potential, 휓′, can assumed to be zero in
the free-stream.
2.4 Disturbance Phase Velocity
The speed at which disturbances move through the medium with respect to the solid body
is the phase velocity. We are able to extract the phase velocity from the DNS flow data by
taking the Fourier transform along the GIP to find the wave number then using the relation
푐 =
휔
훼
(2.23)
we are able to obtain the phase velocity, c. Here휔 is the non-dimensional circular frequency
and 훼 is the wave number. For these calculations 휔 is taken as a constant input. To create
a vector of values for c a 201 element sub-set of values lying on the GIP were used for
the transform, with the index of the middle value corresponding to the location for which
the phase speed is sought. To blur the effects of anomalous behaviour or numerical error
that may occur, the process was repeated directly above and directly below the GIP and the
results averaged.
2.5 Significant boundary layer locations
Sonic Line
The sonic line is the wall normal location in the boundary layer at which the mean flow, u
is sonic relative to the disturbance phase velocity [1], as defined below;
푢 = 푐 − 푎 (2.24)
where a is the local baseflow speed of sound. As mentioned in §1.2 this layer in the
flow traps acoustic rays, acting as a wave guide. The baseflow values u and a are easily
determined from the converged baseflow solution, and the disturbance phase speed is found
using a spatial Fourier transformation on the density perturbation field, as detailed in §2.4.
15
Critical Layer
The critical layer is the wall normal location in the boundary layer at which the mean flow
matches the phase velocity of the disturbance;
푢 = 푐. (2.25)
Due to the synchronisation of velocities, it is at this location that the disturbances can most
effectively capture energy from the baseflow. As will be demonstrated later, the second-
mode eigenfunctions in the second-mode dominated region tend to display either peak
magnitude or zero crossing at the critical layer.
Generalized Inflection Point
The generalized inflection point (GIP) is the wall normal location in the boundary layer
with the highest mean flow gradients. The GIP is defined as the location where
퐼 =
휕
휕푦
(
1
푇
휕푢
휕푦
)
(2.26)
is zero. In this region we anticipate significant interaction between FT components. In
particular, the acoustic component in the second-mode dominated region can receive inputs
from both thermal and vortical disturbances [5].
Boundary Layer Edge
The boundary layer edge is found by calculating as the velocity displacement thickness
corresponding to 99% of edge (post-shock) velocity, 훿99 [5, 10], such that the boundary
layer thickness is
훿 =
∫ 훿99
0
(
1 − 푢
푢푒
)
푑푦. (2.27)
Fluid properties above the boundary layer are assumes to be homogeneous (apart from
shock, of course).
16
Entropy Layer
Blunting the tip or leading edge of a hypersonic vehicle results in the creation of an
entropy layer, 훿푠. The entropy layer typically begins at the detached shock, and continues
downstream until it merges with the boundary layer at a streamwise location know as the
swallowing length. We define the entropy layer edge to be the location at which the local
increment with respect to the free-stream value is 25% of the entropy increment between
the surface and the free-stream [15], such that
푐푝 ln
(
푇 |훿푠
푇∞
)
+ 푅푎푖푟 ln
(
푝 |훿푠
푝∞
)
= 0.25
[
푐푝 ln
(
푇푤
푇∞
)
+ 푅푎푖푟 ln
(
푝푤
푝∞
)]
(2.28)
2.6 Non-dimensionalization
Flow variables are non-dimensionalized by boundary layer edge values. Pressure was
calculated as 휌푇/(훾푀2) using non-dimensional values of 휌 and 푇 . Edge values are in SI
units, spatial dimensions in figures are in centimeters.
density 휌푒 = 0.0477 (kg/m3)
velocity 푢푒 = 856.59 (m/s)
temperature 푇푒 = 62.63 (K)
edge Mach number 푀푒 = 5.196
unit Reynolds number 푅푒푒 = 6.2 × 106 (/m)
Table 2.1: Boundary layer edge values
As the entropy layer is partially defined using free-stream values, those quantities are
presented below.
density 휌∞ = 0.03 (kg/m3)
velocity 푢∞ = 870.1 (m/s)
temperature 푇∞ = 56 (K)
edge Mach number 푀∞ = 6.0
Table 2.2: free-stream values
17
Chapter 3
FLOW CONFIGURATION AND
VALIDATION
3.1 Baseflow
The present analysis uses a converged baseflowwith edgeMach number 5.19, corresponding
to pre-shock free-stream Mach 6, generated using finite-volume flow solver US3D by Cook
et. al. [14]. Flow variables are non-dimensionalized using their edge values. The surface
boundary condition is isothermal with the wall temperature held at 300K, which is close to
the adiabatic value away from the nose, which comes out to ∼ 303K. The temperature field
of this baseflow is presented in figure 3.1. The isothermal surface condition as well as the
presence of a detached oblique shock are readily observed.
(a) (b)
Figure 3.1: Temperature field of baseflow 푇/푇푒.
18
The geometry is an axisymmetric 90cm cone with half-angle of 7◦ and hemispherical tip of
0.38cm radius. The computational domain is discretized by a body fitting mesh with 3251
and 280 points in the streamwise and wall-normal directions, respectively. Grid points are
clustered along the surface and blunt tip of the cone. Figure 3.2 shows a cross section of
the computational grid, significantly coarsened for the sake of illustration.
(a) (b)
Figure 3.2: Computational grid, coarsened for clarity.
Sub-shock region
To avoid the complication due to the shock wave, stability analysis was carried out on
the sub-shock region. The flow was forced at 135 kHz, the frequency determined by I/O
analysis to trigger maximum second-mode growth in this flow.
3.2 Verification and Validation
Verification is achieved in three ways. First, basic computational techniques employed in
this work are tested with the method of manufactured solutions. Second, we compare to the
DNS results attained by [5] using the same geometry and flow conditions. For validation
we compare results from the blunt-cone simulation to experimental results.
Method of Manufactured Solutions
Discrete Poisson and gradient/divergence operators are used in solving the governing equa-
tions of this work. We are required to solve the Poisson equation and so need a discrete
19
Poisson operator which can be numerically inverted. The gradient/divergence operators
are also tested. These operators are applied to data sets representing 2D analytic functions
such that the computational and exact analytical solutions may be compared. Using the
mesh generated for the 2D blunt-cone geometry, figure 3.2, we apply the forward and inverse
Poisson operator to a cosine function. As seen in 3.3 the original function, 푓 = 푐표푠(푁휋/퐿),
is recovered from the analytical expression ∇2 푓 = −(푁/퐿)2푐표푠(푁휋/퐿).
(a) (b)
(c) (d)
Figure 3.3: Validation of Laplace operator on computational mesh with sinusoidal test
function. (a) Analytic sinusoid and (b) analytic second derivative of the sinusoid. (c)
Computational recovery of original sinusoid and (d) computational second derivative of
sinusoid.
The same technique is used to test the discrete gradient operator, with the results seen in
figure 3.4.
Furthermore, to assess the uniformity of the operator behaviour across the domain, we
sweep a Gaussian pulse across the surface, recording the normalized difference from the
analytical solution at each location, as can be seen in figure 3.5d. The average difference is
0.16%. 1
1There was an anomalous max at one point which was around 100% probably from interpolation between
points, therefor giving the max error would be misleading.
20
(a) (b)
(c)
(d)
Figure 3.4: Validation of gradient (here in x) operator on computational mesh with sinu-
soidal test function. (a) Analytic sinusoid and (b) analytic first derivative of the sinusoid.
(c) Computational recovery of original sinusoid and (d) computational first derivative of
sinusoid.
Matching Previous Results
Verification of computationalmethodswas achieved bymatching the results ofUnnikrishnan
and Gaitonde on Mach 6 flow over a flat-plate with 0표 angle of attack forced with non-
dimensional circular frequency of 260 [5]. Eigenfunctions of flow fluctuations and FT
quantities at specific streamwise locations are considered, as well as instantaneous snapshots
showing FT components in the two dimensional flow field. Experimental results for Mach
5.92 flow over a flat-plate at 0표 angle of attack are used to validate results [22].
Baseflow profiles in streamwise velocity, temperature and the generalized inflection point
are taken at the location where second-mode pressure fluctuations on the plates surface are
highest. The location and magnitude of the maximum in figure 3.6 agree with the results
in referenced work [5]. The baseflow profile shapes in 3.7a-3.7c likewise match those in
the referenced work. At ∼ 87cm from the leading edge, the maximum fluctuation pressure
21
(a) (b)
(c) (d)
Figure 3.5: Solutions to the Poisson equation. (a) Analytical Laplacian of Gaussian pulse.
(b) Computational recovery of Gaussian pulse. (c) analytical recovery of original pulse. (d)
Normalized difference between computational and analytical recoveries of Gaussian pulse.
occurs. At this streamwise location the GIP is 1.1cm above the plate.
Eigenfunctions of second-mode flow quantities are given in figure 3.8. The locations
of maxima and zero-crossings match those in the reference work. Being instantaneous
eigenvalues of fluctuating quantities, the specific maximums are unlikely to match exactly
since the data sets are neither identical nor can we assume we are selecting precisely the
same x-location. Nonetheless, the amplitudes are of the correct magnitude. We notice that
x-momentum (figure 3.8a), pressure (figure 3.8c) and temperature (figure 3.8f) fluctuations
show similar patterns, peaking at or slightly above the critical layer with the real component
crossing zero at the sonic line. The real components of the y-momentum (figure 3.8b),
22
Figure 3.6: Pressure perturbation relating to second-mode instability taken at the plates
surface.
(a) (b) (c)
Figure 3.7: baseflow similarity solution taken at 푥 = 87cm. (a) Velocity profile, (b)
temperature profile and (c) generalized inflection point.
x-velocity (figure 3.8d) and y-velocity (figure 3.8e) cross zero at the critical layer and both
real and imaginary components show maxima between the sonic line and the wall. The
imaginary component of y-momentum fluctuation peaks above the critical layer, another
feature matching the reference work.
Figure 3.9 shows the the second-mode eigenfunctions for the FT components. The vortical
23
(a) (b) (c)
(d) (e) (f)
Figure 3.8: Second-mode eigenfunctions of select fluctuation quantities 푥 = 0.87 on flat-
plate. Real and imaginary parts of (a) streamwise momentum, (b) wall-normal momentum,
(c) pressure, (d) streamwise velocity, (e) wall-normal velocity and (c) temperature fluctua-
tions. Sonic line (——) and critical layer (– – –).
perturbations (figures 3.9a and 3.9d) and the y-acoustic (figure 3.9e) and y-thermal (figure
3.9f) perturbations all have zeros around the critical layer, as in the reference work. The
x-acoustic (figure 3.9b) and y-thermal (figure 3.9c) do not possess zeros crossings in the
reference work, however there is what appears to be a zero crossing of the real part of the
24
x-thermal component near the critical layer.
(a) (b) (c)
(d) (e) (f)
Figure 3.9: Second-mode eigenfunctions of FT quantities at 푥 = 87cm on flat-plate. Real
and imaginary parts of vortical (a,d), acoustic (b,e) and thermal (c,f) perturbations. Sonic
line (——) and critical layer (– – –),
Focusing on the region of maximum pressure fluctuation, figure 3.10 shows the second-
mode FT dynamics in detail. As was seen in the eigenfunction results in figure 3.9, the
x-components are larger than the y-component, though the distinction is the least marked
25
for the thermal components (figures 3.10e and 3.10f). The dominant component is again
vortical with the maxima straddling the GIP (figures 3.10a, 3.10b). The "rope-shaped"
structures observed in figures 3.10a and 3.10b have been observed in experimental data of
Mack-mode growth [3, 23], indicating that the primary feature of second-mode amplification
in this region may be connected to distortions of the vortical content. The acoustic (figures
3.10c, 3.10d) and thermal (figures 3.10e, 3.10f) components have similar overall appearance,
although the acoustic component is smaller, as was observed in figure 3.9. The acoustic
(figure 3.10c) and thermal (figure 3.10e) x-components of both do not exhibit the distortion
across the GIP as was present in the vortical x-component (figure 3.10a), indicating a
connection in the x-components of the acoustic and thermal perturbations. On the other
hand, the y-components of the acoustic and thermal perturbations (figures 3.10d, 3.10f) are
distorted, causing a phase shift of 휋 across the GIP. As a result of this distortion, the upper
parts of the x-components are out of phase from the upper part of the y-components by 휋/2
and the lower parts are out of phase by −휋/2. These qualitative observations are consistent
with the results in the referenced work [5].
Matching Experiment
In their 2019 paper, Unnikrishnan and Gaitonde cited the experimental work of Stetson and
Kimmel [24]; Laurence, Wagner and Hannemann [3]; Kennedy et al. [23], to validate their
findings on the FT dynamics in the region of second-mode growth. Streamwise elongated
and flattened regions of alternating positive and negative peaks, referred to as "rope-like"
structures, are found in the split vortical field, and constitute the highest amplitude feature
in the second-mode amplification zone in the boundary layer.
Rufer and Schneider took hot-wire measurements of instability waves for Mach 6 flow
around sharp and blunt-cones [2]. Two cones of 7표 half-angle were tested, one with sharp
tip and one with blunt tip of radius 0.20 inches (5.08 mm). Both of the cones were 22
26
(a) (b)
(c) (d)
(e) (f)
Figure 3.10: Perturbation structures in region of second-mode amplification over flat-plate.
Vortical components (a) 퐵′푥 and (b) 퐵′푦, acoustic components (c) ∇푥휓′퐴 and (d) ∇푦휓′퐴 (e)∇푥휓′푇 (f) ∇푦휓′푇 . Sonic line (——), critical layer (– – –), generalized inflection point (——)
and boundary layer edge (– – –).
inches in length (55.9 cm). 2 For the blunt-cone, transition was found to occur between 20
and 22 inches (50.8 to 55.9 cm) and at 130 kHz. In comparison the sharp cone transitioned
between 14 and 17 inches (35.6 to 43.2 cm) from the tip and for a second-mode frequency
of ∼240 kHz. In the present work, maximum second-mode instability occurs at 45cm from
2Length of and positions upon the blunt-cone are measured from the extrapolated sharp tip location.
27
tip. Transition likely occurred further upstream then in the experiment because the nose
tip in the computational work is slightly smaller but both are in the "small radius" regime
where increasing bluntness delays transition [2, 14, 17].
28
Chapter 4
FLUCTUATIONS IN FLOW
VARIABLES
4.1 Baseflow Profiles and Eigenfunctions
It was found that optimal growth of the second-mode instability occurred at at frequency
of 135 kHz or 휔 = 900 in non-dimensional circular units. First and third-mode surface
pressure response will be briefly discussed as well. The location of maximum perturbation
growth varies depending on the perturbation mode, consequently the state of the boundary
layer will be different at each maxima. As in figure 3.7, baseflow profiles are taken at the
steamwise location at which the pressure perturbation at the surface in at its maximum.
First-Mode
For the first-mode, the profiles are taken at the streamwise location 휉 = 86cm. While this is
the location where the pressure profile at the wall reaches its peak in figure 4.1d, in actuality
the peak would likely be further downstream. This is a result of the numerical damping used
at the outflow boundary. The amplitude of these fluctuations is also very high as compared
to the second and third-modes, as will be discussed below. This energetic yet delayed
response is characteristic of first-mode instability [8]. The velocity and temperature profiles
29
at 휉 = 86cm appear as expected, the temperature profile in 4.1b shows a small inflection at
the wall, reflecting the isothermal surface condition. The generalized inflection point seen
in 4.1c stands at 휂 ∼ 2.5 × 10−2cm from the surface of the cone. Both the velocity and
temperature profiles begin to deviate from their edge-values above the GIP.
Second-Mode
The second-mode surface pressure fluctuations, figure 4.2d, show a distinct maximum at
streamwise coordinate 푝′ ∼ 2.8 × 10−4 휉 = 45cm. This is significantly further upstream
compared to the second-mode amplification region on the flat-plate. A possible reason
for the early transition is that the half angle of the cone means there is an effective angle
of attack for the incoming air, causing a change in direction. The GIP as seen in 4.2c at
this location is at 휂 ∼ 2.0 × 10−2cm from the surface. Between the first and second-mode
locations the boundary layer has grown by 20%.
The amplitude of the pressure fluctuation increases steadily from 휉 ∼ 15cm until its max-
imum, at which point it drops sharply to a local minimum 휉 ∼ 55cm. From this local
minimum, the pressure fluctuates form wave packets of decreasing amplitude and length
until the end of the domain. The peak value surface pressure fluctuation for the second-
mode is about three orders of magnitude less than the corresponding first-mode value. The
surface sensitivity, seen in the input, figure 4.2e, aligns with the output region in figure
4.2d, with little to no sensitivity upstream of 30cm.
For the sake of consistency, the baseflow velocity, temperature and GIP at the third-mode
peak are included in figures 4.3a-4.3c, but since the third-mode maximum is just 6cm
downstream from the second-mode maximum the difference in these and their second-
mode counterparts is insubstantial.
30
(a) (b) (c)
(d)
(e)
Figure 4.1: Baseflow profiles at point of maximum first-mode instability. (a) Streamwise
velocity, (b) temperature and (c) generalized inflection point, taken at 87cm. Perturbation
pressure at surface, (d) output, (e) input.
Third-Mode
The pressure profile for the third-mode, given in figure 4.3d, has similar amplitude and
location at its maximum, 푝′ ∼ 2.7 × 10−4 and 휉 = 51cm, respectively, as the second-mode
31
(a) (b) (c)
(d)
(e)
Figure 4.2: Baseflow profiles at point of second-mode instability. (a) Streamwise velocity,
(b) temperature and (c) generalized inflection point, taken at 45cm. Perturbation pressure
at surface (d) output, (e) input.
response in 4.2d. Modes two and three are both inviscid, acoustic instabilities which might
explain the similarity in the pressure response. The third-mode response does differ from
the second-mode response in that the pressure peak less distinct. There is an initial peak
32
휉 ∼ 24cm from the tip with amplitude half that of the main peak. The first half of the
mode 3 domain, in fact, is almost the same as the mode 2 domain but with a bite taken out
around 휉 = 33cm . In further similarity to figure 4.2d a local minimum at 휉 = 55cm and
amplitude oscillations follow the main peak. These oscillations in 4.3d do not attenuate
downstream as was seen in 4.2d and remain high overall until 휉 ∼ 82cm. In the last 15cm
of the domain the pressure amplitude shows an increase similar to the first-mode response
seen in 4.1d. The input sensitivity shape in figure 4.3e, like the second-mode, shows
peak sensitivity marginally upstream of the peak output response, and with little surface
sensitivity upstream of 30cm.
We notice that the zero crossings of the eigenfunctions in figure 4.4 are, in general, higher
relative to the critical layer, closer to the GIP, than they were in figure 3.8. This may be
in part due to the boundary layer displacement effect, which shifts vortical and thermal
disturbances to the boundary layer edge [1]. Also, the boundary layer over a cone is thinner
than that over a plate for the same Mach number [25]. Another difference is that, in general,
the blunt-cone eigenfunctions exhibit more zero-crossings, indicating that the fluctuations
are more distorted.
The amplitudes of the fluctuations in figure 4.4 are, in general, an order of magnitude lower
than they were for the flat-plate, however the relative amplitudes within each set are similar.
The exception of this the wall-normal momentum fluctuation (figure 4.4b) which is nearly
the same as its flat-plate counterpart (figure 3.8b). The relatively high amplitude of the wall
normal momentum fluctuation could be caused by the mean flow changing direction around
the cone. The disparity in overall amplitude between the flat-plate and blunt-cone may be
explained by considering the energy held in the waveform. The energy of a wave depends
on both its frequency and amplitude. The second-mode instability is excited at 135 kHz
for the blunt-cone but only ∼ 40 kHz for the flat-plate, for a similar amount of energy to be
held, the higher frequency wave will need to compensate by a reduction in amplitude [26].
33
(a) (b) (c)
(d)
(e)
Figure 4.3: Baseflow profiles at point of third-mode instability. (a) Streamwise velocity, (b)
temperature and (c) generalized inflection point, at 55cm. Perturbation pressure at surface
(d) output, (e) input.
34
(a) (b) (c)
(d) (e) (f)
Figure 4.4: second-mode eigenfunctions of select fluctuation quantities 푥 = 48cm on
blunt-cone. Real and imaginary parts of (a) streamwise momentum, (b) wall-normal mo-
mentum, (c) pressure, (d) streamwise velocity, (e) wall-normal velocity and (c) temperature
fluctuations. Sonic line (——), critical layer (– – –) and GIP (——).
35
4.2 Second-Mode I/O Results
The viscous first-mode instability has the most energetic response, as seen in figure 4.1d.
However, growth does not occur until well downstream, past 휉 = 80cm, thus the first-mode
is not responsible for transition in hypersonic flows. This delayed first-mode response is
consistent with the predictions of Mack [8] for high Mach number flow. Instead it is the
second, inviscid/acoustic mode, peaking around 45cm from the tip, which is of interest
when considering the stability HBL flows. For reference, first-mode and third-mode I/O
results are included in appendices C and D.1, respectively.
Pressure fluctuations as well as stream-wise and wall-normal momentum fluctuations are
seen in figures 4.5, 4.6 and 4.7, respectively. In the referenced figures, as will be the
convention, the output field is the upper of the figures and the input field the lower, i.e.
figure 4.5a shows the second-mode output shape of the pressure fluctuations, where as
figure 4.5b is the second-mode input field.
An interesting feature of the second-mode pressure result is the splitting of the output into
two regions. Beginning around 휉 = 20cm the stronger region of pressure fluctuations follow
the surface until around 55cm. This region of disturbance is bounded at the top by the sonic
line, demonstrating the wave-guide phenomenon present in acoustic-type instabilities to
which pressure fluctuations are tied [1, 5]. The secondary region of pronounced disturbance
follows the GIP. Like the fluctuations at the surface, this region begins around 20cm from
the tip, but unlike the fluctuations at the surface, it extends the length of the cone. Between
휉 = 20cm and 휉 = 50cm there is visible pressure disturbance up to the boundary layer edge.
The hypersonic boundary layer is very sensitive to vertical velocity disturbances [11]. This
can be seen in figure 4.7 which depicts the input and output fields for the second-mode
instability for wall-normal momentum. The amplitude of the fluctuation in the sensitivity
region in figure 4.7b is much lower than the amplitude of the output response in figure 4.7a.
Comparing to the stream-wise momentum as seen in figure 4.6, we can see the gain in the
36
(a)
(b)
Figure 4.5: Fluctuation in pressure, mode 2. (a) output and (b) input. Sonic line (——),
critical layer (– – –), generalized inflection point (——) and boundary layer edge (– – –).
Entropy layer edge traced in white.
(a)
(b)
Figure 4.6: Fluctuation in 휌푢, mode 2. (a) output and (b) input. Sonic line (——), critical
layer (– – –), generalized inflection point (——) and boundary layer edge (– – –). Entropy
layer edge traced in white.
37
wall-normal component is relatively large.
Aside, regarding figures 4.5, 4.6 and 4.7 the sonic line and critical layer toward the end of the
domain are extrapolation. The sonic line and critical layer both involve phase speed which
is derived from the spatial Fourier transform. Outflow boundary conditions and numerical
damping at the end of the domain make the transform unreliable in that region and so true
calculations of sonic line and critical layer cannot be preformed. The GIP is extracted from
the baseflow alone and is not dependent on the spatial Fourier transform of the fluctuations.
(a)
(b)
Figure 4.7: Fluctuation in 휌푣, mode 2. (a) output and (b) input. Sonic line (——), critical
layer (– – –), generalized inflection point (——) and boundary layer edge (– – –). Entropy
layer edge traced in white.
38
Chapter 5
FLUID-THERMODYNAMIC
FEATURES
5.1 Eigenfunctions
Figure 5.1 shows the the second-mode eigenfunctions for the fluid-thermodynamic pertur-
bations over the blunt-cone. Similar to what we saw in figure 4.4, the values peak around
the GIP rather than at the critical layer as was seen in the flat-plate case. We also see that,
again, the eigenfunctions have in general more zero crossings. The real and imaginary parts
of the 휉-acoustic (5.1b) component now possesses zero crossings between the critical layer
and the sonic line. The amplitudes of the functions are again reduced as compared to the
flat-plate reference case. The higher degree of reduction in the 휉-components as compared
to the 휂-component can be attributed to the geometry of the cone.
5.2 Perturbation Structures in Second-Mode Growth
Referring to figure 5.2 we notice immediately that the disturbance wavelength on the cone
is much smaller than that on the flat-plate. The wavelength of the second-mode disturbance
on the plate is 2.2cm whereas on the cone the wavelength is 0.6 cm. This is because
the boundary layer is thinner in the region of second-mode amplification and the optimal
39
(a) (b) (c)
(d) (e) (f)
Figure 5.1: Second-mode Eigenfunctions of FT perturbation quantities at 휉 = 0.48m on
blunt-cone. Real and imaginary parts of vortical (a,d), acoustic (b,e) and thermal (c,f)
perturbations. Sonic line (——), critical layer (– – –) and generalized inflection point
(——).
frequency of trapped acoustic waves is higher at 135 kHz (900 non-dim) vs 42 kHz (260
non-dim) [1, 2]. In fact the ratio of the wavelengths is about half of the ratio of the distance
between the plate and the critical layer. We also see, as was suggested by the eigenvalue
analysis, that the perturbations extend higher from the the surface, relative to the critical
40
layer and GIP than was seen in the reference case. There is significant stretching above the
GIP, perhaps a result of the asymmetric base flow around the cone.
The overall shape of the structures in the 휉-thermal component shown in figure 5.2e are
comparable to those seen in figure 3.10e, albeit at the a lower amplitude. The structures
seen in both the 휉- and 휂-components of vorticity (figures 5.2a and 5.2b), as well as the y-
component of the thermal perturbations (5.2f) are analogous to their flap-plate counterparts
(figures 3.10a, 3.10b, 3.10f). They are, however, less skewed at the critical layer. Instead
of the almost horizontal shift of y-components by 휋 seen in figures 3.10b, 3.10d and 3.10f,
the analogous 휉-components stretch in the 휂-directions while still accomplishing a phase
shift of 휆/2. The streamwise vortical components show the same behaviour, but the overall
phase shift across the boundary layer is 휆 (figures 3.10a in the flat-plate case and 5.2a for
the blunt-cone). This is interesting because as Laurence et. al. [3] observed in wind-tunnel
testing on blunt-cones, the rope-like structures in the boundary layer fold over and become
braided in appearance as they become more unstable. The lower rate of skewing at the GIP
could be because the most unstable location on the cone is more stable than that on the
plate, or the development of the boundary layer on the cone is such that the structures do
not fold over to the same degree for the same degree of instability.
The acoustics components show the most intriguing structures. What appears to be a
waveguide form is visible in figure 5.2c, which corresponds to the streamwise acoustic
component. Alternating mono-poles lie along the surface, almost entirely separate from the
cells above and out of phase by 휆/2. However, when we examine the full field below, we
will see the waveguide behavior appears to be localized to the amplification region. The
휂-acoustic components likewise exhibits a line of alternating monopoles along the wall, out
of phase by 휆/2 from the cells above (figure 5.2d). These separations can be seen as the
zero crossings in figures 5.1b and 5.1e.
The vortical component dominates the perturbation field for the second-mode, followed by
41
(a) (b)
(c) (d)
(e) (f)
Figure 5.2: Second-mode perturbation structures in region of second-mode amplification
over blunt-cone. Vortical components (a) 퐵′휉 and (B) 퐵
′
휂, acoustic components (c) ∇휉휓′퐴
and (d) ∇휂휓′퐴 (e) ∇휉휓′푇 (f) ∇휂휓′푇 . Sonic line (——), critical layer (– – –), generalized
inflection point (——) and boundary layer edge (– – –).
the thermal then acoustic components. The output response is most pronounced at and
above the GIP and is insignificant below the critical layer. As was hypothesized for the
flat-plate reference case, vorticity is generated by shear stress which is greatest about the
GIP.
42
Input-Output Results
The flow is most unstable along the GIP where the shear stress is highest, as determined
by output intensity in the region. In the 휉-directions output responses (figures 5.3a, 5.5a
and 5.7a) are maximized between the GIP and boundary layer edge, with growth starting
∼ 20cm from the tip. In contrast, the 휂-direction responses (figures 5.4a, 5.6a and 5.8a)
have the feature that the perturbations bifurcate, following the GIP and boundary layer edge,
leaving a local minimum between them in the upstream portion. Moving downstream the
two sections converge at around 50cm, which was approximately the location of maximum
second-mode growth. The 휂-acoustic component (5.6a) has a third region of high output
amplitude directly at the wall, again with a local minimum between the wall and the
GIP. Generally, it is only the 휂-direction components of the output momentum that show
noticeable response along the surface. Within each FT component the overall length of the
output is consistent, tapering toward the end of the domain while remaining centered the
boundary layer edge or GIP.
Turning our attention to the inputs, figures 5.3b-5.8b, we notice that a region of sensitivity
begins above the boundary layer just after the shock. This corresponds to the region just
above the entropy layer. For reference, baseflow entropy with the same axial scaling as the
inputs is shown in figure 5.9.
For the vortical (figures 5.3b and 5.4b) and thermal components (figures 5.7b and 5.8b),
the upstream region of receptivity corresponds closely to the top of the entropy layer. The
pattern of acoustic receptivity is somewhat different in that it seems more localized to the
forward half of the cone and begins at a greater distance from the surface. For the 휂-acoustic
component there are three distinct fingers of receptivity at the front of the domain emanating
from the entropy layer, as can be seen in figure 5.6b.
Over the entire domain, the wall-normal thermal component is the most amplified, followed
by the the streamwise thermal component. Thewall-normal and streamwise vortical compo-
43
nents come next, followed by the wall-normal and finally streamwise acoustic components.
These results are consistent with the findings of Unnikrishnan and Gaitonde regarding the
total fluctuating enthalpy associated with each FT component [5].
(a)
(b)
Figure 5.3: Second-mode (a) output response and (b) input receptivity of the vortical
component of momentum density in the 휉-direction, 퐵′휉 . Sonic line (——), critical layer (–
– –), generalized inflection point (——) and boundary layer edge (– – –). Entropy layer
edge traced in white.
Surface Response
Although our results indicate that the flow is most sensitive to input forcing near the
entropy layer and boundary layer edge, surface roughness/vibrations are one way forcing
can physically enter the system. Therefore, sensitivity to surface disturbances will remain
a critical area of research.
Numerous efforts have been made to investigate the effects surface roughness and/or forcing
akin to engine vibrations on the stability of the hypersonic boundary layer [11, 16, 27, 28].
If we wish to inspect the input receptivity and output response on the plates surface we can
take the magnitude of the perturbations at that location. We can see in figures 5.10 to 5.15
44
(a)
(b)
Figure 5.4: Second-mode (a) output response and (b) input receptivity of the vortical
component of momentum density in the 휂-direction, 퐵′휂. Generalized inflection point
(——) and boundary layer edge (– – –).
(a)
(b)
Figure 5.5: Second-mode (a) output response and (b) input receptivity of the acoustic
component of momentum density in the 휉-direction, ∇휉휓′퐴. Sonic line (——), critical layer
(– – –), generalized inflection point (——) and boundary layer edge (– – –). Entropy layer
edge traced in white.
45
(a)
(b)
Figure 5.6: Second-mode (a) output response and (b) input receptivity of the acoustic
component of momentum density in the 휂-direction, ∇휂휓′퐴. Sonic line (——), critical layer
(– – –), generalized inflection point (——) and boundary layer edge (– – –). Entropy layer
edge traced in white.
that the regions of input receptivity on the plate surface are roughly in the same location
as the output growth regions. The amplitude increase in all cases is O(104). However,
because the I/O analysis was not performed with forcing restricted to only the surface, we
cannot assume that the amplification of the surface output is a direct response to the surface
input. We do notice that the receptivity and response in 휂-direction FT components (figures
5.11, 5.13 and 5.15) is greater than the 휉-direction (figures 5.10, 5.12 and 5.14) by O(102).
Therefore wall-normal momentum forcing is important to boundary layer stability, likely
tied to the sensitivity to wall-normal velocity forcing [11].
46
(a)
(b)
Figure 5.7: Second-mode (a) output response and (b) input receptivity of the thermal
component of momentum density in the 휉-direction, ∇휉휓′푇 . Sonic line (——), critical layer
(– – –), generalized inflection point (——) and boundary layer edge (– – –). Entropy layer
edge traced in white.
(a)
(b)
Figure 5.8: Second-mode (a) output response and (b) input receptivity of the thermal
component of momentum density in the 휂-direction, ∇휂휓′푇 . Sonic line (——), critical layer
(– – –), generalized inflection point (——) and boundary layer edge (– – –). Entropy layer
edge traced in white.
47
Figure 5.9: baseflow entropy. Generalized inflection point (——) and boundary layer edge
(– – –).
(a)
(b)
Figure 5.10: Second-mode (a) output response and (b) input receptivity of the vortical
component of momentum density in the 휉-direction, | | 퐵′휉 | | on cone surface.
48
(a)
(b)
Figure 5.11: Second-mode (a) output response and (b) input receptivity of the vortical
component of momentum density in the 휂-direction, | | 퐵′휂 | | on cone surface.
(a)
(b)
Figure 5.12: Second-mode (a) output response and (b) input receptivity of the acoustic
component of momentum density in the 휉-direction,| | ∇휉휓′퐴 | | on cone surface.
49
(a)
(b)
Figure 5.13: Second-mode (a) output response and (b) input receptivity of the acoustic
component of momentum density in the 휂-direction, | | ∇휂휓′퐴 | | on cone surface.
(a)
(b)
Figure 5.14: Second-mode (a) output response and (b) input receptivity of the thermal
component of momentum density in the 휉-direction, | | ∇휉휓′푇 | | on cone surface.
50
(a)
(b)
Figure 5.15: Second-mode (a) output response and (b) input receptivity of the thermal
component of momentum density in the 휂-direction, | | ∇휂휓′푇 | | on cone surface.
51
Chapter 6
TOTAL FLUCTUATING ENTHALPY
Momentum potential theory provides the conservation laws which govern the behavior of
instabilities as described by their FT content. The change in energy contained in each FT
component is a consequence of interactions between different components as well as with
the baseflow. We quantify these changes using the total fluctuating enthalpy (TFE) budget
as derived in §2.2. Changes in the TFE due to each FT component can be correlated with
the corresponding source terms in the energy balance, as represented by the TFE equation
2.15. A positive source indicates that the TFE is gaining energy from the baseflow or other
components, while a negative source means the opposite and is an energy sink.
6.1 Total Fluctuating Enthalpy in Flat-Plate Boundary Layer
We first look at the flat-plate enthalpy results. These results were left out of the validation
section because a different scaling was used in the referenced work, however the overall
trends and mode shapes are acceptably comparable. The sum of the three sources, the
total TFE, can be seen in figure 6.1. We notice that sources and sinks are almost equally
represented with slight bias toward sources, indicating that the TFE is gaining some energy
from the mean flow. This of course is required for instability to increase. The total TFE
appears as a wave pattern tilted in the streamwise direction but without the skewing across
52
the critical layer that was seen in the FT components from §3.2. The amplitude of the
enthalpy wave follows the streamwise pattern of the fluctuating pressure profile in figure
3.6, with growth starting around 65cm and the peak around 90cm. The disturbance is also
localized in the wall-normal direction as it is contained from above by the GIP and just
overlaps the critical layer below. This is the same wall-normal location that the skewing of
the FT components occurred in figure 3.10.
Figure 6.1: Total fluctuating enthalpy over flat-plate. Sonic line (——), critical layer (– –
–), generalized inflection point (——) and boundary layer edge (– – –).
In figure 6.2 we present the individual TF components dotted with the two terms of the
acceleration vector, 훼′ = 훼′1 + 훼′2, such that the column to the left depicts the vortical,
acoustic and thermal sources of TFE attributed to 훼′1, and the column to the right depicted
the vortical, acoustic and thermal sources of TFE attributed to 훼′2. Recall this breakdown
was applied to determine the relative contributions from 훼′1 which accounts for effects of
vorticity fluctuations and is referred to as the Lamb vector, and 훼′2 which represents the
effects of the entropy and viscous stress, and is referred to as the entropic acceleration vector.
It is plainly seen that the contributions from the latter are ∼ 1−2 orders of magnitude above
the contributions of the former, and so entropic and viscous effects are more important than
the effects of vorticity fluctuations in this flow.
The contributions from 훼1, figures 6.2a, 6.2c and 6.2e are almost entirely TFE sinks. But
53
they are in general two orders of magnitude less that the 훼′2 and so unlikely to hold much
physical importance for this case.
Choosing to focus on the more significant contributions from 훼′2, we see that the vortical
component in figure 6.2b shows the highest amplitude, but is overall a sink of TFE. On
the other hand, the acoustic and thermal contributions, figures 6.2d and 6.2f, both, on
average, contribute positively to the TFE. However, each component exhibits both source
and sink behaviour within the domain, with non-uniform distribution of this behavior. Take
figure 6.2b for instance, where the interaction between vortical FT component and entropic
acceleration vector is shown. Above the critical layer the net change in TFE is around zero
as the strong shear at the GIP strengthens vorticity, but below the critical layer the vortical
component is almost entirely a sink. In contrast the interaction of the acoustic and thermal
components with the entropic acceleration vector, figures 6.2d and 6.2f, show bias towards
TFE loss above the critical layer, but strong gain below. Furthermore, between the sonic
line and the surface, there exists a regions of weakly varying TFE at the streamwise location
of the second-mode instability.
The interaction between the thermal component and 훼′2 provides the strongest source term.
therefore we hypothesis that thermal instabilities are main factor in the resulting second-
mode instability growth, as was likewise posited by Unnikrishnan and Gaitonde (2019)[5].
6.2 Total Fluctuating Enthalpy in blunt-cone Boundary Layer
Now considering the blunt-cone entropy sources, the sum of the three sources of TFE can
be seen in figure 6.3. In contrast to what was observed for the plate, significant TFE is
being gained from the mean flow. The maximum amplitude of the TFE correlates with
the maximum amplitudes of the FT components, between 20 and 60 cm in the streamwise
direction, and peaking slightly above the GIP. As was likewise seen over the plate, the wall-
normal maximum of TFE aligns with the wall-normal location at which the corresponding
FT components are skewed.
54
(a) (b)
(c) (d)
(e) (f)
Figure 6.2: Second-mode fluctuating enthalpy sources for the flat-plate, divided into specific
FT components and types of acceleration. (a) 퐵′ ·훼′1, (b) 퐵′ ·훼′2, (c) ∇휓′퐴 ·훼′1, (d) ∇휓′퐴 ·훼′2,
(e) ∇휓′푇 ·훼′1, (f) ∇휓′푇 ·훼′2. Sonic line (——), critical layer (– – –) and generalized inflection
point (——).
Figure 6.3: Total fluctuating enthalpy over cone.
55
In figure 6.4 we again present the individual TF components dotted with the two terms of
the acceleration vector. As before, the TFE sources associates with 훼′2, containing entropic
and viscous terms, are more significant than the sources associated with 훼1. However,
the difference is less than in the flat-case, and so the contributions from the Lamb vector
cannot be as readily ignored. Indeed, we surmise that the Coriolis force 1 plays a greater
roll in blunt-cone instability. This appears to be in contrast with the reduced dominance
of the vortical FT component; the ratio of peak amplitudes of the vortical and thermal
eigenfunctions in figure 5.1 is less than they are in figure 3.9, which are from the flat-plate
results, giving the impression that the relative influence of vorticity is both greater and
lesser for the cone compared to the plate.
Both parts of the vortical source term are negative. In the case of the source term which
is correlated with the Lamb vector (figures 6.4a), the values are nearly uniformly negative,
indicating that the interaction between the vortical component and vortical acceleration
work to dissipate TFE. The vortical source term correlated with entropic acceleration
vector (6.4b) displays both source and sink behaviour in the domain. The sink region lies
along the GIP and the source region above them sink. The regions of positive and negative
TFE change appear separate, in contrast with figure 6.2b where the source and sink regions
are partially woven together.
Both parts of the acoustic source term (figures 6.4c and 6.4d) are positive, albeit weaker
than the other source terms. Both of the 훼′1 and 훼
′
2 correlated terms are strongest upstream
of the second-mode growth region, then dissipate moving downstream. This suggests that
instability is may be triggered but not sustained by acoustic sources of TFE. Indeed, along
the downstream GIP in figure 6.4d we see the development of a TFE sink.
The thermal source term corresponding to the fluctuating Lamb vector (figure 6.4e) is net
negative and the thermal source term corresponding to the entropic acceleration vector
1The Coriolis acceleration here is caused by the interaction of the velocity field with its own rotation [29],
not the rotation of the Earth or a football-esq spiral toss of the cone.
56
(figure 6.4f) is net positive. Both have regions of increasing and decreasing TFE. For the
source term related to the Coriolis force, a broad area of decreasing TFE can be observed
to sit above the GIP, peaking in magnitude around 35 cm. A region of weak growth exist
between 20 and 40 cm just below the boundary layer edge, as well as a spot between the sonic
line and critical layer at 48 cm. The strongest source comes from the thermal component
correlated with the entropic acceleration vector, along the GIP. A region of TFE loss as
well can be observed directly above the source, but is weak in comparison. In contrast with
the behaviour of the acoustic source terms, the thermal-entropic source peaks closer to the
Mack mode region, with strong growth extending to 45cm, where second-mode growth was
strongest. Thus it is the thermal TFE source which maintains the instability. However the
correlation of the thermal FT component and the Coriolis acceleration provided a net sink
of TFE before the second-mode growth region, potentially stabilizing the flow.
57
(a) (b)
(c) (d)
(e) (f)
Figure 6.4: Second-mode fluctuating enthalpy sources for the blunt-cone, divided into
specific FT components and types of acceleration. (a) 퐵′ · 훼′1, (b) 퐵′ · 훼′2, (c) ∇휓′퐴 · 훼′1, (d)∇휓′퐴 ·훼′2, (e) ∇휓′푇 ·훼′1, (f) ∇휓′푇 ·훼′2. Sonic line (——), critical layer (– – –) and generalized
inflection point (——).
58
Chapter 7
SUMMARY AND CONCLUSION
The assumptions required for LST or PSE do not hold for flow around complex geometries.
Therefore, new techniques are necessary to capture the complexflow-physics present in high-
speed flow around blunt-cones. Applying input-output analysis to a non-linear baseflow
solution generated by the DNS solver, US3D, has allowed us to capture the effects of a
0.38cm radius blunt tip on the boundary layer stability.
Fluid-thermodynamic decomposition separated the fluctuations into a linear sum of vortical,
thermal and acoustic components. The blunt-cone results were compared to those for a
flat-plate boundary layer with the same free-stream Mach number. It was found that in
the area of second-mode amplification the boundary layer on the blunt-cone was thinner
than that of the plate, and consequently the optimal forcing frequency was higher and the
wavelength shorter. Possibly because of this, the amplitude of the FT perturbations in the
amplification region are lower than was seen for the flat-plate. For both the flat-plate and
blunt-cone, the vortical components were found to be largest, followed by the thermal then
finally the acoustic components. The vortical dominance is consistent with experimental
results which show boundary layer structures resembling that of the vortical component.
Above the critical layer, the structure of all FT components on the cone appear stretched in
the wall-normal direction.
59
Considering the FT decomposition of the entire domain for both input and output fields, we
find that the flow is most sensitive to input forcing at the top of the entropy layer. This region
of sensitivity extends downstream following the boundary layer edge, dissipating at around
휂 = 50cm. Output response begins between 휂 = 15cm to 휂 = 20cm. The output response in
the streamwise direction is highest in the regions between the generalized inflection point
(GIP) and the boundary layer edge, but dissipates towards the surface. On the other hand,
the wall-normal response extends to the surface and shows a local minimum between the
GIP and boundary layer edge. Over the entire domain, the wall-normal thermal component
is the most amplified, followed by the streamwise thermal component. The wall-normal
and streamwise vortical components come next, followed by the wall-normal and finally
streamwise acoustic components.
To compliment studies on hypersonic boundary layer response to surface conditions, input
sensitivity and output response at the surface was considered. As was seen with the full
domain, the flow was most sensitive to wall-normal forcing. However, the surface of the
cone was found not to be very receptive to input forcing. This does not mean that surface
conditions should be neglected, just that all things being equal, the flow is much more
sensitive near the entropy layer and boundary layer edge than the surface.
Total fluctuating enthalpy was considered for both the flat-plate and blunt-cone. In both
cases the thermal terms provided the strongest source of TFE. The acoustic term were also
net sources and the vortical terms net sinks of TFE. In aggregate the TFE added up to a
weakly positive source over the flat-plate and a strongly positive source over the blunt-cone.
Separating the acceleration vector into its vorticity component and its entropic component
it was found that the latter contributed more to the TFE. This distinction was greater for the
plate than the cone, indicating that Coriolis accelerationmay play a greater roll in stability of
flow over blunt bodies. However, for the cone, the correlation of the thermal FT component
and the Coriolis acceleration provided a net sink of TFE in the second-mode growth region,
potentially stabilizing the flow.
60
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63
Appendix A
EQUATIONS
A.1 The viscous stress tensor
휏 =
©­­«
휏11 휏12
휏21 휏22
ª®®¬ (A.1)
휏11 = 2휇
휕푢
휕푥
− 2
3
(
휕푢
휕푥
+ 휕푣
휕푦
)
(A.2a)
휏12 = 휇
(
휕푢
휕푦
+ 휕푣
휕푥
)
(A.2b)
휏21 = 휏12 (A.2c)
휏22 = 2휇
휕푣
휕푦
− 2
3
(
휕푢
휕푥
+ 휕푣
휕푦
)
(A.2d)
A.2 FT components of TFE
For an ideal gas following 푝 = (푐푝 − 푐푣)푇 and 훾 = 푐푝/푐푣, the total fluctuating enthalpy can
also be broken down into the three FT components, 퐻 = 퐻 + 퐻′퐵 + 퐻′퐴 + 퐻′푇 such that,
퐻′퐵 =
(
푎
휌
)
푀 · 퐵′ (A.3a)
퐻′퐴 =
(
푝′
휌
) (
1 −푀 ·푀
)
−
(
푎
휌
) (
푀 · ∇휓′퐴
)
(A.3b)
64
퐻′푇 =
[
푎2
(훾 − 1)
] (
푆′
푐푝
) [
1 + (훾 − 1)푀 ·푀
]
(A.3c)
where the subscripts B, A and T designate vortical, acoustic and thermal components,
respectively. Overbar quantity푀 is the base flowMach number vector and 휌 and 푎 are base
flow values of density and speed of sound, respectively [5, 6].
65
Appendix B
TENSOR DERIVATIONS
The following is a brief overview of tensor operations in general coordinates following [30].
For a three-dimensional coordinate system any point P in space can be expressed in terms
of three curvilinear coordinates 휈푖 where 푖 = 1, 2, 3. If we express the position vector to
point P as r(휈푖), 푖 = 1, 2, 3 then there exist two sets of basis vectors
e푖 =
휕r
휕휈푖
푎푛푑 e푖 = ∇휈푖 (B.1)
where e푖 is referred to as the covariant basis set and e푖 the contravariant basis set. At given
point P in the curvilinear coordinate system the covariant and contravariant basis sets are
reciprocal, such that
e푖 · e 푗 = 훿 푗푖 (B.2)
where 훿 푗푖 , or 훿푖 푗 , is the Kronecker delta, the tensor equivalent of the identity matrix and
therefor having the property that
훿
푗
푖 =

1 if 푖 = 푗
0 if 푖 ≠ 푗 .
(B.3)
66
It should be noted that in general there is no requirement for the basis vectors to have unit
length or form an orthogonal set. This property is of course necessary for calculations on
non-orthogonal systems such as the one in the present work.
For a curvilinear coordinate system, each point in space is also characterized by nine
quantities which form the metric tensor, g, which has (covariant) components
푔푖 푗 = e푖 · e 푗 . (B.4)
The metric tensor is symmetric, 푔푖 푗 = 푔 푗푖 and second order. In general, 푔푖 푗 ≠ 0 for 푖 ≠ 푗 ,
unless the coordinate system is orthogonal, in which case 푔푖 푗 = ℎ2푖 for 푖 = 푗 and 0 otherwise,
where ℎ푖 are the coordinate system scale factors. For Cartesian coordinates 푔푖 푗 = 훿푖 푗 . Using
the metric tensor, the square infinitesimal arc length is found to be (푑푠)2 = 푔푖 푗푑휈푖푑휈 푗 and
the infinitesimal volume element 푑푉 = √푔푑휈1푑휈2푑휈3 where g is the determinant of the
matrix [푔푖 푗 ], which has as its elements the covariant components of g.
For a general coordinate system the basis vectors e푖, 푖 = 1, 2, 3 are a function of the
coordinates and therefor must be considered in the differentiation of general tensors. For a
point P the non-zero derivative of a basis vector is itself a vector and so much be able to be
represented as a linear combination of the basis vectors. The 27 coefficients necessary to
uniquely specify all linear combinations of the basis vector at a point are contained in what
is know as a Christoffel symbol, Γ푘푖 푗 , such that in the expression
휕e푖
휕휈 푗
= Γ푘푖 푗e푘 , 푖, 푗 , 푘 = 1, 2, 3, (B.5)
Γ푘푖 푗 is the k component of the covariant derivative vector 휕e푖/휕u 푗 . Through the reciprocal
relation (B.2) the 27 coefficients can be isolated and given by
Γ푘푖 푗 = e푘 ·
휕r푖
휕휈 푗
, 푖, 푗 , 푘 = 1, 2, 3. (B.6)
67
However, a more tractable expression for the Christoffel symbol given in terms of the metric
tensor and its derivatives,
Γ푚푖 푗 =
1
2
푔푚푘
(
휕푔 푗 푘
휕휈푖
+ 휕푔푘푖
휕휈 푗
− 휕푔푖 푗
휕휈푘
)
, 푖, 푗 , 푘, 푚 = 1, 2, 3, (B.7)
was used in calculation.
Ultimately we are able to craft gradient and divergence operators;
∇휙 = 푔푖 푗 휕휙
휕휈푖
e 푗 (B.8a)
and
∇ · v = 1√
푔
휕
휕휈푖
(√
푔v 푗
)
(B.8b)
for scalar 휙 and vector v with contravariant components v 푗 . Taking the gradient of a scalar
results in a vector, we require this vector to have contravariant components on the covariant
basis set, hence why the metric tensor is used to raise the index of the vector components
in (B.8a). Replacing ∇휙 with v 푗 in (B.8b) we obtain an expression for Laplacian in general
coordinates,
∇2휙 = 1√
푔
휕
휕휈 푗
(√
푔푔 푗 푘
휕휙
휕휈푘
)
, (B.9)
where we are reminded that g is the determinant of the metric tensor with covariant
components as its elements.
On a side note, because the components the basis set and, in general, neither orthogonal nor
of unit length, the dot product also requires the use of the metric tensor,
a · b = 푔푖 푗푎푖∗푏 푗 . (B.10)
where the ∗ above the first component in the right hand side designates the complex
conjugate, as required for the complex dot product.
68
Appendix C
MODE 1 FEATURES
C.1 Mode 1 - Eigenfunctions
69
(a) (b) (c)
(d) (e) (f)
Figure C.1: First-mode eigenfunctions of select fluctuation quantities 푥 = 86cm on blunt
cone. Real and imaginary parts of (a) streamwise momentum, (b) wall-normal momen-
tum, (c) pressure, (d) streamwise velocity, (e) wall-normal velocity and (c) temperature
fluctuations. Sonic line (——), critical layer (– – –) and GIP (——).
70
C.2 Mode 1 - flow var I/O
(a)
(b)
Figure C.2: Fluctuation in p, mode 1. (a) output and (b) input. Sonic line (——), critical
layer (– – –), generalized inflection point (——) and boundary layer edge (– – –).
(a)
(b)
Figure C.3: Fluctuation in 휌푢, mode 1. (a) output and (b) input. Sonic line (——), critical
layer (– – –), generalized inflection point (——) and boundary layer edge (– – –).
71
(a)
(b)
Figure C.4: Fluctuation in 휌푣, mode 1. (a) output and (b) input. Sonic line (——), critical
layer (– – –), generalized inflection point (——) and boundary layer edge (– – –).
72
C.3 Mode 1 - FT I/O
(a) (b)
(c) (d)
(e) (f)
Figure C.5: First-mode FT perturbation structures in region of second-mode amplification
over blunt cone. Vortical components (a) 퐵′휉 and (B) 퐵
′
휂, acoustic components (c) ∇휉휓′퐴
and (d) ∇휂휓′퐴 (e) ∇휉휓′푇 (f) ∇휂휓′푇 . Sonic line (——), critical layer (– – –), generalized
inflection point (——) and boundary layer edge (– – –).
73
(a)
(b)
Figure C.6: First-mode (a) output response and (b) input receptivity of the vortical compo-
nent of momentum density in the 휉-direction, 퐵′휉 . Sonic line (——), critical layer (– – –),
generalized inflection point (——) and boundary layer edge (– – –).
(a)
(b)
Figure C.7: First-mode (a) output response and (b) input receptivity of the vortical compo-
nent of momentum density in the 휂-direction, 퐵′휂. Sonic line (——), critical layer (– – –),
generalized inflection point (——) and boundary layer edge (– – –)..
74
(a)
(b)
Figure C.8: First-mode (a) output response and (b) input receptivity of the acoustic com-
ponent of momentum density in the 휉-direction, ∇휉휓′퐴. Sonic line (——), critical layer (–
– –), generalized inflection point (——) and boundary layer edge (– – –).
(a)
(b)
Figure C.9: First-mode (a) output response and (b) input receptivity of the acoustic com-
ponent of momentum density in the 휂-direction, ∇휂휓′퐴. Sonic line (——), critical layer (–
– –), generalized inflection point (——) and boundary layer edge (– – –).
75
(a)
(b)
Figure C.10: First-mode (a) output response and (b) input receptivity of the thermal com-
ponent of momentum density in the 휉-direction, ∇휉휓′푇 . Sonic line (——), critical layer (–
– –), generalized inflection point (——) and boundary layer edge (– – –).
(a)
(b)
Figure C.11: First-mode (a) output response and (b) input receptivity of the thermal com-
ponent of momentum density in the 휂-direction, ∇휂휓′푇 . Sonic line (——), critical layer (–
– –), generalized inflection point (——) and boundary layer edge (– – –).
76
C.4 Surface profiles - mode 1
(a)
(b)
[First-mode 퐵′휉 I/O results on cone surface.]nFirst-mode (a) output response and (b) input
receptivity of the vortical component of momentum density in the 휉-direction, | | 퐵′휉 | | on
cone surface.
(a)
(b)
Figure C.13: First-mode (a) output response and (b) input receptivity of the vortical com-
ponent of momentum density in the 휂-direction, | | 퐵′휂 | | on cone surface.
77
(a)
(b)
Figure C.14: First-mode (a) output response and (b) input receptivity of the acoustic
component of momentum density in the 휉-direction,| | ∇휉휓′퐴 | | on cone surface.
(a)
(b)
Figure C.15: First-mode (a) output response and (b) input receptivity of the acoustic
component of momentum density in the 휂-direction, | | ∇휂휓′퐴 | | on cone surface.
78
(a)
(b)
Figure C.16: First-mode (a) output response and (b) input receptivity of the thermal com-
ponent of momentum density in the 휉-direction, | | ∇휉휓′푇 | | on cone surface.
(a)
(b)
Figure C.17: First-mode (a) output response and (b) input receptivity of the thermal com-
ponent of momentum density in the 휂-direction, | | ∇휂휓′푇 | | on cone surface.
79
Appendix D
MODE 3 FEATURES
D.1 Mode 3 - Eigenfunctions
80
(a) (b) (c)
(d) (e) (f)
Figure D.1: Third-mode eigenfunctions of select fluctuation quantities 푥 = 87cm on blunt
cone. Real and imaginary parts of (a) streamwise momentum, (b) wall-normal momen-
tum, (c) pressure, (d) streamwise velocity, (e) wall-normal velocity and (c) temperature
fluctuations. Sonic line (——), critical layer (– – –) and GIP (——).
81
D.2 Mode 3 - flow var I/O
(a)
(b)
Figure D.2: Fluctuation in p, mode 3. (a) output and (b) input. Sonic line (——), critical
layer (– – –), generalized inflection point (——) and boundary layer edge (– – –).
(a)
(b)
Figure D.3: Fluctuation in 휌푢, mode 3. (a) output and (b) input. Sonic line (——), critical
layer (– – –), generalized inflection point (——) and boundary layer edge (– – –).
82
(a)
(b)
Figure D.4: Fluctuation in 휌푣, mode 3. (a) output and (b) input. Sonic line (——), critical
layer (– – –), generalized inflection point (——) and boundary layer edge (– – –).
83
D.3 Mode 3 - FT I/O
(a) (b)
(c) (d)
(e) (f)
Figure D.5: Perturbation structures in region of third-mode amplification over blunt cone.
Vortical components (a) 퐵′휉 and (B) 퐵
′
휂, acoustic components (c) ∇휉휓′퐴 and (d) ∇휂휓′퐴 (e)∇휉휓′푇 (f) ∇휂휓′푇 . Sonic line (——), critical layer (– – –), generalized inflection point (——)
and boundary layer edge (– – –).
84
(a)
(b)
Figure D.6: Third-mode (a) output response and (b) input receptivity of the vortical com-
ponent of momentum density in the 휉-direction, 퐵′휉 . Sonic line (——), critical layer (– –
–), generalized inflection point (——) and boundary layer edge (– – –).
(a)
(b)
Figure D.7: Third-mode (a) output response and (b) input receptivity of the vortical com-
ponent of momentum density in the 휂-direction, 퐵′휂. Sonic line (——), critical layer (– –
–), generalized inflection point (——) and boundary layer edge (– – –).
85
(a)
(b)
Figure D.8: Third-mode (a) output response and (b) input receptivity of the acoustic
component of momentum density in the 휉-direction, ∇휉휓′퐴. Sonic line (——), critical layer
(– – –), generalized inflection point (——) and boundary layer edge (– – –).
(a)
(b)
Figure D.9: Third-mode (a) output response and (b) input receptivity of the acoustic
component of momentum density in the 휂-direction, ∇휂휓′퐴. Sonic line (——), critical layer
(– – –), generalized inflection point (——) and boundary layer edge (– – –).
86
(a)
(b)
Figure D.10: Third-mode (a) output response and (b) input receptivity of the thermal
component of momentum density in the 휉-direction, ∇휉휓′푇 . Sonic line (——), critical layer
(– – –), generalized inflection point (——) and boundary layer edge (– – –).
(a)
(b)
Figure D.11: Third-mode (a) output response and (b) input receptivity of the thermal
component of momentum density in the 휂-direction, ∇휂휓′푇 . Sonic line (——), critical layer
(– – –), generalized inflection point (——) and boundary layer edge (– – –).
87
D.4 Surface profiles - mode 3
(a)
(b)
Figure D.12: Third-mode (a) output response and (b) input receptivity of the vortical
component of momentum density in the 휉-direction, | | 퐵′휉 | | on cone surface.
(a)
(b)
Figure D.13: Third-mode (a) output response and (b) input receptivity of the vortical
component of momentum density in the 휂-direction, | | 퐵′휂 | | on cone surface.
88
(a)
(b)
Figure D.14: Third-mode (a) output response and (b) input receptivity of the acoustic
component of momentum density in the 휉-direction,| | ∇휉휓′퐴 | | on cone surface.
(a)
(b)
Figure D.15: Third-mode (a) output response and (b) input receptivity of the acoustic
component of momentum density in the 휂-direction, | | ∇휂휓′퐴 | | on cone surface.
89
(a)
(b)
Figure D.16: Third-mode (a) output response and (b) input receptivity of the thermal
component of momentum density in the 휉-direction, | | ∇휉휓′푇 | | on cone surface.
(a)
(b)
Figure D.17: Third-mode (a) output response and (b) input receptivity of the thermal
component of momentum density in the 휂-direction, | | ∇휂휓′푇 | | on cone surface.