UNIVERSITY OF CALGARY

The Effects of Particles Momentum Transfer in Shock Wave/ Boundary Layer Interaction

by

E Jieh Teh

A THESIS

SUBMITTED TO THE FACULTY OF GRADUATE STUDIES

IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE

DEGREE OF MASTER OF SCIENCE

GRADUATE PROGRAM IN MECHANICAL AND MANUFACTURING ENGINEERING

CALGARY, ALBERTA

JANUARY, 2016

© E Jieh Teh 2016

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Abstract

Numerical simulations of solid particles seeded into a supersonic flow containing an oblique

shockwave reflection were performed in order to observe and understand the momentum transfer

effects between solid and gas phases in shock-wave / boundary layer interaction. Particle size and

mass loading were varied to study the problem. It was found that solid particles were capable of

significant modulation of the flow field as the separation bubble was suppressed primarily by the

additional momentum introduced by the particles into the flow field. The particle size controlled

the rate of momentum transfer while the particle mass loading controlled the magnitude of

momentum transfer. As particles are seeded into the flow, a flow transition to unsteady, three-

dimensional flow is observed in the simulations. The seeding of micro and nano-particle upstream

of a supersonic air-breathing propulsion system as a flow control concept is proposed.

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Preface

Preliminary results of the particle-shock interaction were presented at the 22nd Annual Conference

of the CFD Society of Canada in 2014. The final results are being prepared for submission to the

AIAA Journal of Propulsion and Power. The proposed flow-control concept has been filed by

Space Engine Systems, Inc. as a provisional patent. The author is listed as one of the Inventors on

that patent.

iv

Acknowledgements

I would like to thank my advisor, Dr. Craig T. Johansen for trusting and giving me the opportunity

to work on this very interesting project as well as generously providing the crucial financial support

that allows me to complete this work. I would also like to thank my work colleagues (Bill, Dan,

Garrett, & Steve) for all the interesting discussions and banter. My graduate life has been made

easier with the good friends that I have met during my time in Canada. You know who you are –

Hsu Chew, Rif (Rif-meister), Zixiang (Mr. Zee), and of course, Neil (the Doc). I am also grateful

to have make acquaintance with other people that I have come across in the university.

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Dedication

This thesis is dedicated to my parents, Alan and Esther. Without the emotional support that they

have been providing since I left my home, this would have been a fruitless and futile endeavor.

Familia supra omnia.

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Table of Contents

Abstract ............................................................................................................................... ii Preface................................................................................................................................ iii

Acknowledgements ............................................................................................................ iv Dedication ............................................................................................................................v Table of Contents ............................................................................................................... vi List of Tables ................................................................................................................... viii List of Figures and Illustrations ......................................................................................... ix

List of Symbols, Abbreviations and Nomenclature ......................................................... xiii Epigraph ........................................................................................................................... xiv

CHAPTER ONE: INTRODUCTION ..............................................................................1

1.1 Research Motivation ..................................................................................................1 1.2 Thesis Outline ............................................................................................................3

CHAPTER TWO: BACKGROUND & LITERATURE REVIEW ..............................4

2.1 Oblique Shock Wave/Boundary Layer Interaction ....................................................4 2.2 Supersonic Flow, Application & Control of SWBLI ..............................................10

2.3 Effects of Particles Addition in Gas Flow ...............................................................13 2.3.1 Compressible Flow Modulation by Solid/Liquid Particles .............................13 2.3.2 Flow Instabilities & Turbulence Modulation due to Solid/Liquid Particles ...15

CHAPTER THREE: GOVERNING EQUATIONS ....................................................23 3.1 Gas Phase .................................................................................................................23

3.2 Particle Phase ...........................................................................................................25 3.3 Finite Volume Method .............................................................................................29

3.3.1 Domain Discretization .....................................................................................31 3.3.2 Spatial Discretization .......................................................................................32

3.3.2.1 Convective Terms ..................................................................................33 3.3.2.2 Diffusion/Laplacian Term ......................................................................35 3.3.2.3 Source Terms .........................................................................................36

3.3.3 Temporal Discretization ..................................................................................36 3.4 High Resolution FV Method in rhoCentralFoam ...................................................38 3.5 Particle Modeling & Lagrangian Particle/Parcel Tracking ......................................45

CHAPTER FOUR: OBJECTIVE, ASSUMPTION AND LIMITATION..................48

CHAPTER FIVE: PROBLEM DESCRIPTION & NUMERICAL METHODS ......50

CHAPTER SIX: RESULTS AND DISCUSSION .........................................................52

6.1 Verification & Validation ........................................................................................52 6.2 Effects of Particles on the Free Stream & Shock Attenuation .................................62 6.3 Effects of Particles on the Separation Bubble .........................................................63 6.4 Effects of Particles on the Flow Instabilities ...........................................................69 6.5 Effects of Particles on the Gas Phase Temperature and Heat Transfer ...................84

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CHAPTER SEVEN: CONCLUSIONS ..........................................................................91

REFERENCES .................................................................................................................93

viii

List of Tables

Table 1 Grid Spacing Values of Several Levels of Mesh Refinement ......................................... 54

Table 2 Immediate Post Shock Flow Deflection due to Particles ................................................. 63

Table 3 Particle Relaxation times for different particle sizes. ...................................................... 71

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List of Figures and Illustrations

Fig. 1 Schematics of a typical shock wave / laminar boundary layer [1] ....................................... 4

Fig. 2 Typical Wall Pressure Profile for a Complete SWBLI [2] ................................................... 7

Fig. 3 The Reflection of the Incident Oblique Shock Wave [] ....................................................... 7

Fig. 4 Map of Flow Regimes in Particle-Laden Flows [47] ......................................................... 18

Fig. 5 Control Volume for Finite Volume Discretization [] ......................................................... 31

Fig. 6 Face Interpolation (adapted from [75]) .............................................................................. 34

Fig. 7 Vectors d and 𝑺𝒇 on a non-orthogonal mesh (adapted from [75]) ..................................... 35

Fig. 8 FV Method of Updating Cell Average, Q through Fluxes across the Cell Faces

(adapted from [74]) ............................................................................................................... 39

Fig. 9 Schematics of Computational Domain ............................................................................... 50

Fig. 10 Grid Sensitivity Test for Pure Gas SWBLI ...................................................................... 53

Fig. 11 Grid Sensitivity Test for Particle Laden SWBLI.............................................................. 53

Fig. 12 Validation Test with 2D, Steady Laminar SWBLI Experiment [102] ............................. 55

Fig. 13 Velocity magnitude contours of 2D (top) and 3D (bottom) case of SWBLI ................... 55

Fig. 14 Comparison between 2D and 3D case of M=2.15, 𝜷 = 33.18° ........................................ 56

Fig. 15 Static Pressure contours of 2D (top) and 3D (bottom) case of SWBLI ........................... 57

Fig. 16 Streamlines of a Pure Gas SWBLI (M=2.15, β=33.18°) taken at 𝒁𝑳𝒛 = 𝟎. 𝟓 ................. 58

Fig. 17 Near Wall Velocity Streamlines. (left column) M=2.15, 𝜷 = 33.18°, 𝑳𝒛=0.5, (right

column) (taken from [103]) M=2.15, 𝜷 = 32°, 𝑳𝒛= 0.8; (top) t= 10ms, (middle) t =

25ms, (bottom) t = 35ms ....................................................................................................... 58

Fig. 18 Verification of the Drag Law [98] Implementation in OpenFOAM ................................ 60

Fig. 19 Static Pressure of (left) wall near SWBLI region and (right) Post Oblique Shock .......... 61

Fig. 20 Effects of Particles' Sizes on Free Stream Post Shock Condition (left) Static Pressure

(right) Mach number ............................................................................................................. 63

Fig. 21 Effects of Particles on the Separation Bubble .................................................................. 63

x

Fig. 22 Velocity Magnitude Contours (in descending order) (a) φ=0.1,𝒅𝒑=16μm, (b)

φ=0.2,𝒅𝒑=1.6μm, (c) φ=0.1,𝒅𝒑=1.6μm, (d) φ=0.2,𝒅𝒑=1.6μm, (e) ) φ=0.1,𝒅𝒑=880nm .... 65

Fig. 23 Effects of Particles’ Sizes on Time Averaged Wall Static Pressure ................................. 66

Fig. 24 Effects of Particles’ Sizes on Near Wall |U| Streamlines (a) Pure Gas flow, (b)

𝒅𝒑=16μm, φ=0.1, (c) 𝒅𝒑=1.6μm, φ=0.1 , (d) 𝒅𝒑=0.88μm, φ=0.1 (arrow denotes

separation bubble) ................................................................................................................. 72

Fig. 25 Effects of Particles 𝒅𝒑=16μm, φ=0.1 on Separation Bubble (Streamlines) at various

time, t = 9.25ms, 9.5ms, 9.75ms & 10ms. ............................................................................ 73

Fig. 26 Effects of Particles 𝒅𝒑=1.6μm, φ=0.1 on Separation Bubble (Streamlines) at various

time, t = 9.25ms, 9.5ms, 9.75ms & 10ms. ............................................................................ 74

Fig. 27 Streamlines of Separation Bubble of particles of 𝒅𝒑=16μm, φ=0.1, at t = 10ms ............ 75

Fig. 28 Effects of Particles of 𝒅𝒑=0.88μm, φ=0.1 on Separation Bubble (Streamlines) at

various time, t = 9.25ms, 9.5ms, 9.75ms & 10ms. ................................................................ 76

Fig. 29 Effects of Particles’ Sizes on upstream Boundary Layer Profiles close to Leading

Edge of the Flat Plate ............................................................................................................ 77

Fig. 30 Evolution of the Separated Boundary Layer. (black) Pure gas (red) 𝐝𝐩=16μm,

(blue) 𝐝𝐩=1.6μm (green) 𝐝𝐩=0.88μm .................................................................................. 78

Fig. 31 Particle Trajectories for (a) 𝒅𝒑=16μm, φ=0.1, (b) 𝒅𝒑=1.6μm, φ=0.1, (c) 𝒅𝒑=0.88μm,

φ=0.1 ..................................................................................................................................... 79

Fig. 32 Pressure spectra density for several points in the flow (x,y,z) a) (0.625,0,0.5), b)

(0.875,0,0.5), c) (1,0,0.5) d) (1,0.0125,0.5), and e) (1.125,0,0.5) ......................................... 81

Fig. 33 Vorticity magnitude contours of separation bubble for case 𝒅𝒑 = 𝟏. 𝟔𝝁𝒎 at a)∆𝒕𝟏=

0s, b) )∆𝒕𝟏= 25𝝁𝒔, c) ∆𝒕𝟐= 50𝝁𝒔, d) ∆𝒕𝟑= 75𝝁𝒔, 𝒆) ∆𝒕𝟒= 100𝝁𝒔 ...................................... 82

Fig. 34 Streamlines of separation bubble for case 𝒅𝒑 = 𝟏. 𝟔𝝁𝒎 at a)∆𝒕𝟏= 0s, b) )∆𝒕𝟏= 1𝒎𝒔,

c) ∆𝒕𝟐= 2m𝒔, d) ∆𝒕𝟑= 3m𝒔, 𝒆) ∆𝒕𝟒= 4m𝒔 ............................................................................ 82

Fig. 35 Conjecture for topolofical changes of an incompressible separation bubble’s structure

associated with the onset of vortex shedding [111]. ............................................................. 84

Fig. 36 Effects of Particles on the Wall Static Temperature ......................................................... 89

Fig. 37 Effects of Particles on Post Shock Gas Temperature (left) without and (right) with

Heat Transfer......................................................................................................................... 89

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xii

xiii

List of Symbols, Abbreviations and Nomenclature

Symbol Definition

ρ gas density, kg/m3

U

E

T

𝑇𝑂

gas velocity, m/s

gas total energy, j/kg

gas temperature, K

gas total temperature, K

𝑃𝑂 μ

γ

gas total pressure, Pa

gas molecular viscosity, kg/m.s

specific heat ratio

E

M

Kn

S

Re

β

θ

𝑥𝑝

gas constant, j/kg K

gas Mach number

gas Knudsen number

molecular speed ratio

Reynolds number

Shock wave angle

Flow deflection angle

Particle position, m

𝑚𝑝 Particle mass, kg

𝑈𝑝 Particle velocity, m/s

𝑑𝑝 Particle diameter, m

𝜌𝑝 Particle density, kg/m3

𝑀𝑝 Particle Mach number

𝑅𝑒𝑝

g

σ

Particle Reynolds number

Gravitational body force

Stress tensor

ϕ Generic scalar quantity

Γ Diffusivity coefficient

d Vector between P and N

𝑑𝑓𝑁 Vector between cell center and cell face

f Point in the center of the face (cell)

𝑓+

𝑓− β

𝑓𝑥 S

𝑆𝑓

V

𝑉𝑝

Downwind face (pointing outward)

Upwind face (pointing inward)

TVD limiter

Interpolation factor

Outward-pointing face area vector

Face area vector

Volume

Cell volume

xiv

Epigraph

“Everything we hear is an opinion, not a fact.

Everything we see is a perspective, not the truth”.

~ Marcus Aurelius

1

Chapter One: INTRODUCTION

1.1 Research Motivation

Typically in compressible flow, there are shock structures which can prove to be a

challenge to study especially when they interact with other fluid structures. The physical

generation of waves in supersonic flow is basically due to the propagation of information through

molecular collisions in supersonic flow where the hyperbolic nature of the partial differential

equations(PDE) equations that govern the flow dictates that information can only propagate via

certain regions and directions. From being a challenge that had to be overcome in faster-than-

sound-flight (breaking the sound barrier), the formation of shock waves has now become

necessary for the operation of supersonic/hypersonic flight systems such as the ramjet and

scramjet. Shock wave interaction is an active field of study and relatively recently, the scientific

community has already begun to look into shock wave interaction with liquid droplets / solid

particles. This has opened up a new and exciting realm of new discoveries as well as potential

applications.

The use of computational fluid dynamics (CFD) has become ubiquitous in most of the

engineering fields due to improvements in high resolution/accuracy, parallelized CFD codes and

the increased availability of affordable computing resources. From a practical point-of-view, CFD

offers a much more flexible and cost effective means of investigation than experimental methods

for some problems. Instead of sampling data at a few points, CFD is capable of constructing the

whole domain of interest allowing instantaneous fields and any local change of the flow property

2

can be monitored/sampled. CFD is selected as the tool in this study to explore particle-fluid

interaction in the compressible flow regime.

There is a caveat that must be mentioned before going forward. For most engineering

applications, the flows are usually turbulent. However, numerical simulations of turbulent flows

typically consist of one major drawback: inaccuracies that are inherent in the turbulence models

themselves (unless DNS – Direct Numerical Simulation is used where the Navier-Stokes equations

are solved numerically without the use of any turbulence model and the numerical grid system

must be small enough to resolve all the turbulence spatial and temporal scales). When modeling

turbulent flows, experimental data must accompany the CFD results in order to assess the accuracy

or the deviation of the two results. Although DNS is theoretically capable of rivaling experiments

as an alternative to study fluid dynamics, it is also prohibitively computationally expensive at

higher Reynolds numbers. With the addition of particles, a whole new range of complexities are

introduced which will be described in the following sections. Thus, in this study, we must carefully

select the base flow to be well within the laminar flow regime and employ a grid system that is

fine enough to capture the base and additional flow physics that is introduced by the particles

themselves. This is, of course, not the ideal situation but we believe this would suffice as the first

step to explore such interactions. To the author’s knowledge, there is a dearth of available literature

in examining the effects of solid particles when introduced into an oblique shock wave / boundary

layer interaction (SWBLI) system. This work will examine the impact that small scale solid

particles have on the laminar separation bubble, which is induced by a shock wave. The results

extracted from this study could possess potentially useful information in terms of high speed flight

3

system application. Thus, the flow condition selected is typical of most engine inlets of military

aircraft operating at Mach number > 2 and at high altitude.

1.2 Thesis Outline

This thesis focuses on a canonical supersonic flow phenomenon i.e. oblique Shock-Wave

/ Boundary-Layer Interaction (SWBLI) and its interaction with solid particles. Chapter 1 is a brief

introduction to frame the tone of this work. Chapter 2 is the background and literature review

portion of this thesis where certain key studies will be mentioned and their relevant conclusions

listed. Chapter 3 will list out the governing equations which are solved numerically in this work.

Chapter 4 will highlight on certain key limitations and assumptions made in this work. Chapter 5

describes the problem studied in this work and the numerical methods employed to solve it. These

will include the discretization practices and other numerical concerns involved in multiphase flow

modeling. Chapter 6 is the focal point of this thesis where the key results are presented and

discussed. This chapter is divided into several subsections. The verification and validation of the

results will first be presented and this includes mesh sensitivity tests and assessing the

implementation of the drag law. Then the impact of particles on the free stream, near wall and the

laminar separation bubble will follow. Lastly, Section 7 will serve as the conclusion and future

potential works will be briefly discussed.

4

Chapter Two: BACKGROUND & LITERATURE REVIEW

A short literature review of each field of research interest will be presented in this chapter.

This survey focuses more on key significant works instead of cataloguing all the previous works

available.

2.1 Oblique Shock Wave/Boundary Layer Interaction

Fig. 1 Schematics of a typical shock wave / laminar boundary layer [1]

The interest in supersonic and hypersonic flow has not been damped since the original

impetus of such study circa 1940s-1950s [1,2,3] where the first systematic investigations of the

SWBLI were carried out by Ackeret et al. [4] and Liepmann (1946)[5] . It was Ferry (1939) [6]

who first observed that the potential flow changes due to boundary layer separation at supersonic

speed. Then, Oswatitsch et al. [3] showed that a local interaction can occur at supersonic speeds

while noting that a local pressure increase and the thickening of the boundary layer can reinforce

5

each other as can be demonstrated through an integral-based calculation. Liepmann [5] discovered

that for an incident oblique shock wave and compression corner, there are fundamental differences

in the interactions between laminar and turbulent cases. Some major conclusions from all these

studies are:

1. A weak shock wave can penetrate the boundary layer but must terminate at the supersonic

part of the flow as shock cannot exist in subsonic region.

2. The thickening of the subsonic part of the boundary layer extends its effects upstream and

downstream, causing the outward displacement of the flow streamlines and local changes

in the external flow.

3. For laminar flows, a small pressure rise/gradient is sufficient to induce a flow separation

and the separation point moves upstream as the strength of the disturbance/incident shock

is increased.

4. The pressure rise to constant plateau pressure (refer to Fig. 2) observed are independent of

the manner in which the disturbance is introduced and this interaction is known as the “free

interaction”.

The physics of SWBLI is described in detail in [7] and by [8]. The SWBLI can be described

through the upstream influence phenomenon. As the incident shock wave propagates through the

supersonic portion of the boundary layer, it imparts such a strong deceleration to the flow that the

shear forces become temporarily negligible compared to much more dominant pressure and inertia

forces. Although the shock penetration event can be viewed as a perfect fluid/inviscid phenomenon

(where boundary layer does not exist) but a more physically realistic description would require the

6

multi-deck (triple deck) models [9,10]. The multi-deck model outlined by [11] divides the flow

field into 3 distinct regions;

1. Upper Deck: Outer irrotational, inviscid layer which obeys the Euler equations.

2. Main/Middle Deck: Delery et al. (2009) [11] describes the middle deck as an inviscid but

rotational flow where viscosity plays a role in creating entropy and hence, vorticity. This

agrees with Crocco’s equation in a steady non-viscous flow. This layer is also a region of

stagnation pressure and stagnation temperature which are constant along each streamline

because entropy is a transported quantity.

3. Inner/Lower Deck: A thin viscous layer which is in contact with the wall. It enforces the

no-slip wall boundary condition and allows non-viscous flow to decelerate.

As the shock wave continues to propagate through the boundary layer towards the wall, it

progressively curves as the local Mach number decreases steadily, weakening itself in the process

and becomes vanishingly weak as it reaches the sonic line of the boundary layer. Furthermore, a

shock wave cannot exist in a subsonic flow regime. The pressure rise associated with the shock

wave is transmitted upstream of the point where the shock would have impinged on the surface for

an inviscid case (where there is no boundary layer) through the subsonic region of the boundary

layer where information propagates upstream at the speed of sound due to nature of parabolic PDE

equations (which essentially governs the subsonic region of the boundary layer). This pressure

increase causes the boundary layer to thicken which in turn redirects the flow in the adjacent

supersonic layer towards the freestream and this generates compression waves that coalesce into a

separation shock and weaken the incident shock wave. Due to the presence of boundary layer,

instead of an instantaneous, pressure step rise on the surface as it would have been in a purely

7

inviscid case, the process is replaced with a continuous evolution of the pre-shock wave pressure

to the post-shock wave pressure that corresponds with the shock jump relations in the outer non-

rotational flow.

Fig. 2 Typical Wall Pressure Profile for a Complete SWBLI [2]

Fig. 3 The Reflection of the Incident Oblique Shock Wave [12]

The incident shock wave that penetrates the supersonic boundary layer will reflect from the

sonic line as an expansion wave because of the continuous pressure change in the bubble and due

to the fact that the outer supersonic flow is turning away from the free stream (a condition for

8

Prandtl-Meyer expansion fan). The fluid above the separation streamline flows downstream while

the small amount of fluid entrained in the shear layer from below is turned back by the pressure

rise immediately ahead of the reattachment point. At the downstream point of the separation bubble

boundary, the upstream flow from the separation point reattaches itself and this process generates

another series of compression waves which will coalesce into a reattachment shock in the outer

flow. The reattachment point marks another pressure rise and by then, the entire flow field differs

from the purely inviscid case of shock wave impingement/reflection and the shock reflection is

now known to be a strong viscous – inviscid interaction.

Adamson et al. [13] quantify several key parameters in an oblique SWBLI. The equation of

the free interaction theory is capable of predicting the pressure at the separation region as it

assumes that a weak incident oblique shock impinging on a plane wall will cause separation if the

deflection angle of the flow, δ = (const.) (Me2 − 1) 0.5Re−0.5. However, if the shock wave is of

greater strength, that is to say if |δ|>>Re−0.5, then several flow regions will develop and become

distinguishable from one another. The pressure will initially rise in a free interaction region, which

contains the separation point and then reaches a constant “plateau” value of 𝒪(Re−0.5) and then

undergoes another rise to its final value through a reattachment point which is downstream of the

shock impingement point. The separation shear layer upstream of the shock impingement point

has the thickness of 𝒪(Re−0.5ℒ) where ℒ is the separation length and it is inclined to the wall at a

constant angle of order, again, Re−0.5. Downstream of the shock impingement point, the shear

layer is then turned towards the wall at an approximate angle of 2δ, thus leading the compression

at the reattachment occurs through an angle of 2δ.

9

One of the most comprehensive studies of the separated region in supersonic and subsonic

flows is compiled by Chapman et al. [14]. This paragraph is to summarize the results for the

oblique shock wave boundary layer interaction. The study’s main objectives were to determine

via experiments whether transition from laminar to turbulent flow regime occurs downstream of

reattachment, between separation and reattachment or upstream of separation. This is to

understand the fundamental mechanism near reattachment and explain the reason behind the

transition location and finally free interaction type of flows. It is found that when transition occurs,

it is marked by an abrupt increase of pressure. Pure laminar separation case usually involves a

small pressure rise accompanied by gradual increase of pressure gradients while transitional flow

separation case involves severe pressure gradients near the transition region and are usually

unsteady. For the former case, after the incident shock wave interacts with the boundary layer near

the impingement point, the shock waves form near the separation and reattachment and they do

not originate within the viscous layer. In the latter case where transition occurs upstream of a

reattachment point, there is a sharp pressure rise and the shock waves originate partially within the

boundary layer flow near reattachment. In estimating the pressure of the recirculation region, the

essential mechanism that needs to be considered is the balance between the mass flow taken from

that separated region by the mixing layer and the mass flow reversed back into the separated region

by the pressure rise through the reattachment zone. This pressure rise to separation is independent

of the mode of inducing separation for either laminar or turbulent supersonic flow. The pressure

rise to pressure plateau in the laminar separation is also independent but the peak pressure rise is

dependent on the model geometry such as shock angle for turbulent separation.

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2.2 Supersonic Flow, Application & Control of SWBLI

The flow around a supersonic/hypersonic vehicle generally contains a myriad of shock wave

structures. Key locations include supersonic intake systems, leading–edge of wings and tails, and

also at the rear portion of the vehicle where the supersonic nozzle jets exit into the ambient

surrounding. These shock waves are a main source of aerodynamic heating and shock interactions

with boundary layers can produce undesirable interferences. Motivation for continued research on

SWBLIs stems from the fact that these phenomena are still not well understood, especially for

turbulent flow regimes [15,16].

Supersonic and hypersonic propulsion systems, such as the ramjet and scramjet, use shock

waves (typically multiple oblique shock waves) to achieve the inlet compression process at

supersonic speeds. SWBLIs can induce a flow separation on the boundary layer which will lead to

the decrease of efficiency of the supersonic air-flow intake due to the thickening of the downstream

boundary layer, unsteadiness in the flow towards the engine or the effectiveness of a control

surface. The unsteadiness that is induced by the SWBLs can lead to engine unstart which can cause

the whole flight system propulsion to fail. Furthermore, in high enthalpy flows, the subsequent

reattachment of the separated boundary layer is responsible for the increase of local heat transfer

rates which can be far higher than rates associated with a typical attached boundary layer [17].

Since SWBLI is unavoidable in the presence of these shock systems, its control is an opportunity

to improve performance.

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In addition to SWBLI, separation in supersonic flows also occurs near deflected flaps of a

wing, in over-expanded rocket nozzles, behind re-entry capsules and other blunt bodies, and on

the leeward side of a flight body at high angle of attack to name a few. There have been many

studies devoted to flow control devices that mitigate the size of induced separation bubble. Flow

separation control devices are typically categorized as active or passive control. Some examples

of these flow control devices and a brief summary of each method are presented below.

1. air jets (blowing) [18,19]

Ability to switch and off.

Require to be at appropriate angle with main flow/stream [20].

Generate streamwise vortices that is weakly dependent on the jet momentum

2. Hybrid cavity/wall ventilation [21]

Combination of suction and blowing (of air) – passive and active portion of control.

Passive control increases drag and losses.

Active control leaves thinner boundary layer.

3. Bleed holes/slots with controlled plenum pressures [22,23]

Removes lower momentum fluid from boundary layer, allowing only higher

momentum fluid to withstand adverse pressure gradient better.

Requires proper alignment or arrangement of the bleed holes because

misalignments can weaken the bleed holes’ ability [24].

Inclination of the bleed holes i.e. inclined or normal bleed holes and bleed rate are

major factors in determining its success in control.

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Can eliminate flow separation but at the cost of flow distortions.

4. Morphing surfaces e.g. mesoflap [25]

A matrix of small flaps which are fixed at one end and cover a cavity.

Designed to achieve proper mass bleed/injection when subjected to gas dynamics

effects of SWBLI.

5. Arc filament plasma actuators [26,27]

Generate perturbations upstream of the flow.

Primary mechanism of flow control is the heating at the near wall region and

degradation of the boundary layer.

6. Micro vortex generator (VG) ramps [28]

Trigger boundary layer mixing and reduce the boundary layer shape factor.

Reduce drag.

Typical size is of or smaller than the order of magnitude of the boundary layer.

Mechanically simple and fail-proof (not detachable from the surface).

CFD results show it can rival the performance of boundary layer bleed systems

[29].

The concept behind most of these devices is simple: increase the momentum of the boundary

layer so that it can better resist the adverse pressure gradient that would otherwise cause separation.

By doing so, the size of the separation bubble is reduced. All the techniques have their own

disadvantages or costs. For an example, bleeding is one of the most widely used methods as it can

13

operate at a wider range of operating conditions [30] but it reduces the mass flow rates of the air

intake into the engine, increases aerodynamic drag and requires additional and complex component

design for the bleeding process.

2.3 Effects of Particles Addition in Gas Flow

2.3.1 Compressible Flow Modulation by Solid/Liquid Particles

When it comes to the subject of flow modulation by particles in compressible flow, Carrier

(1958) [31] was among the first to analyze the flow behind a normal shock wave a dust-gas

mixture through a reduced form equations to estimate the thickness of the transition region. The

transition region in this context is defined as the region where the gas and particle phases are not

in dynamic or thermal equilibrium. Then, Nettleton (1977) [32] compiled a rather comprehensive

review of works from the 1950s to the 1970s of the research involving shock wave interaction with

dusty gases, particle acceleration, droplet break-up by shocks, shock-heating of particles, etc.

Marconi et al. [33] formulated a characteristic-based finite difference scheme to study the

mathematical nature of dusty gas equations and the dependence of shock-tube flowfields on

variables such as particle size, initial particle loading and the frozen shock strength. It was found

that the magnitude of the equilibrium properties can be described by pseudo gas analysis if the

volume occupied by the particles is sufficiently large relative to the size of the relaxation zone.

Olim et al. [34] proposed a general law to describe the instantaneous shock wave Mach number as

it decreases while propagating through a dusty gas mixture. Based on the derivation provided by

14

Olim et al., Aizik et al. [35,36] proposed analytical equations to predict the instantaneous velocity

of a normal and spherical shock wave, respectively, as they propagate through a gas mixture of

high loadings of solid particles.

Sommerfeld (1985) [37] conducted a series of experiments to study the shock wave

deceleration and the pressure jump decay when a shock wave traverses into a dusty gas mixture.

The shock wave decelerates quicker when the particle loading is increased. For the condition of

very strong shock waves, boundary layer effects play an important role. Not surprisingly, it was

found that the numerical predictions of such flowfields are sensitive to the drag and heat transfer

law models applied to the particles. Boiko et al. [38] performed experiments to study the dispersion

of particles when subjected to a travelling normal shock wave. Depending on the particle material

and concentration, the clouds of particles have unique dispersion patterns following the

transmission of a shock wave. Reflected shocks are also found to form upstream of the clouds

when the volume concentration of particles are about 1-3%. A follow-up study conducted by

Kiselev et al. [39] using numerical modeling studied the interaction of a shock wave with a cloud

of finite size particles (where each particle is modeled instead of computational parcel – each is a

collection of particles of zero dimension). When a shock wave passes through a cloud of particles,

the incident shock wave undergoes a double Mach reflection and one or two more shock waves

are formed aft of the cloud due to the gas streams collision in the vortices. These two studies focus

more on the qualitative aspects of shock wave particle interactions especially in terms of the

dispersion of the cloud of particles and the fluid/shock structures that arise from such interactions.

As a clarification, when numerical simulations involved finite size particles, it is meant that the

15

cloud of particles are modeled as a collection of spherical solid particles and not computational

particles where they only serve as markers and do not possess any dimension.

A much more quantitative study of shock wave / cloud of particles interaction was

performed by Park et al. [40]. In their numerical study, a moving shock wave impinges a cloud of

solid particles that is situated on a wedge. Combustion effects were also considered. They showed

that as the particle material density decreases, the gas and particles follow the shock front more

closely and thus, the particle concentration increases behind the shock wave. The thermal and

momentum exchange between the two phases become more intense and this results in the

temperature and velocity difference between the two phases to be smaller. The particle specific

heat only plays an indirect role on the concentration and velocity. In the case of particle

combustion, the gas density decreases due to the heat release.

2.3.2 Flow Instabilities & Turbulence Modulation due to Solid/Liquid Particles

Turbulence modulation is a phenomenon where the particles interact with the turbulent

eddies of the flow and thus modulate the turbulent properties of the flow. Although this work will

not involve any turbulence modeling and assume to the flow to be laminar, this section is included

to briefly discuss the role of particles play in turbulent flows as it cannot be simply ignored.

In turbulent flows with particles, the fluid and particle phases are characterized by several

parameters. For the fluid, they are the fluid density, viscosity, the integral length scale, the

Kolmogorov length and time scale, the turbulent kinetic energy and the dissipation rate. The

16

particle phase on the other hand, is characterized by the particle diameter, the particle material

density, and the volume fraction or loading. At times, depending on the study, the particles could

be monodisperse (single size) or polydisperse (multiple sizes).

The earliest studies that examine this interaction are done by Lumley [41], Baw et al. [42]

and Tsuji et al. [43]. However, the experimental results obtained by [43] and Kulick et al. [44]

stand out as the most sought-after data to validate new models or CFD predictions. In [43], it was

found that as the particle size gets smaller, the mean air velocity distribution/profile in the pipe

becomes flatter. The large particles on the other hand increases the turbulence of the air through

the pipe section while the small particles decrease it. However, the medium sized particles enhance

and suppress the turbulence concurrently at the pipe center and near the wall, respectively.

Elghobashi [45] provides an overview of DNS and closure models in particle-laden turbulent

flows. In this paper, he presents a map of flow regimes in particle laden flows where it shows that

through a rule-of-thumb, in a two-way coupling flow where particles and fluid interact with each

other, particles can enhance or attenuate turbulence based on the ratio of the particle response time

to Kolmogorov time scale or the ratio of particle response time to turnover time of large turbulent

eddy.

Eaton and Balachandar et al. [46,47] discuss that for very small Stokes number particles

and in the absence of external forces, particles behave essentially like passive traces and convect

with the carrier flow. With increasing Stokes number or with external forces, particles no longer

respond instantaneously with the carrier flow and the relative motion between the fluid and particle

phases leads to carrier flow’s turbulence modulation - the two-way coupling flow regime. Boivin

17

et al. [48] conducted a DNS of heavy particles suspended in homogeneous isotropic turbulence

and demonstrated that in the absence of external forces, particles with Stokes number in the range

of 1 to 10 can reduce turbulent kinetic energy tremendously (> 50%) at a very small volume

loadings. This result highlights the fact that two-way coupling effects can be significant. Poelma

et al. [49] experimentally investigated relatively light particles settling in grid generated

turbulence. The study showed that particles delay the turbulent decay onset upstream and this

makes the flow becomes anistropic downstream. This is a results of the particles redistributing the

energy of the flow. Daniel et al. [50] formulated a modelling framework that accounts for

preferential particle concentration (described below) observed in experiments. The framework also

reproduces the isotropic and anisotropic turbulence attenuation effects of particles but is restricted

to very low particle volume fraction cases. The framework is outlined for both RANS and joint

probability density function methods.

In turbulent flows, there is a phenomenon known as preferential concentration of the

particles. At moderate or unity Stokes numbers, and with heavy particles, there is a tendency for

particles to accumulate away from flow regions with high vorticity [47]. Several DNS studies have

shown preferential concentration in homogenous turbulence [51,52]. Eaton et al. [53] conducted a

review of such effects in various flows. Bagchi et al. [54] showed that if the particle response time

is greater that the Kolmogorov time scale and the Stokes number is greater than unity, the carrier

flow has no influence on the particle motion. However, when the particles are of sufficient size

and the particle Reynolds number exceeds the critical value of 210, vortex shedding will occur in

the particle wake and this leads to turbulence enhancement. To be even more specific, [46,47]

conclude that particles of diameters larger than 10% of the integral length scale of the flow will

18

augment turbulence whereas smaller particles will attenuate turbulence. The general theory of how

small particles can attenuate turbulence is that small particles which act as passive tracers interact

directly with the turbulent eddies and this leads to energy distribution from these eddies to the

particles and thus, in effect, decrease the turbulent kinetic energy of the flow [55].

Fig. 4 Map of Flow Regimes in Particle-Laden Flows [47]

The main issue when it comes to computationally modeling multiphase flows is the closure

models. Because of cost considerations, most works use a modified Reynolds Averaged Navier-

Stokes (RANS) turbulence model such as the k-ε turbulence model. Modifications to these

formulae usually include an additional source/sink term in the turbulent kinetic energy and

dissipation transport equations. There are numerous derivations and modifications to that

19

particular turbulence model but usually authors disagree on which performs better or best captures

the physics of the flow as it can be seen and discussed in the next paragraph.

In general, these models can be divided into three categories. The main group, i.e. the

standard approach [56], derives the source term due to the particles using the standard approach of

Reynolds averaging method [57,58] and this results in a term that is only capable of predicting

turbulence attenuation. Then there is the “consistent” approach, where the basic theory behind the

derivation of the term is that the instantaneous carrier phase velocity at the surface of the particle

equals the particle velocity and this will result in a term that can only predict turbulence

enhancement. The derivation is basically subtracting the product of the mean velocity and the

momentum equation from the mechanical energy equation of the carrier phase to get an expression

for the turbulent kinetic energy [59]. Finally, the last type is known as semi-empirical or semi-

heuristic models, which are touted to be able to predict both turbulence enhancement and

attenuation, but are criticized for not providing a theoretical basis that has yet to withstand the

fundamental physical principles [60]. Mando et al. [61] introduced a derivation of the source term

that represents a “hybrid” between the standard and consistent approaches which is capable of

predicting turbulence augmentation and dissipation for both small and large particles.

So far, the turbulence modulation studies above are concerned with the effects of particles

in a fully turbulent flow. However, a few studies that investigate the effects of particles on the

instabilities of an initially laminar flow will be briefly explored. All of these studies use a mixing

layer as the base flow and solve a non-linear, averaged, first-moment particle model that is reduced

to either a Rayleigh equation or the Orr-Sommerfeld equation for inviscid and viscous flow,

20

respectively. Saffman (1962) [62] is the first to present an analytical stability analysis of a viscous,

incompressible mixing layer with uniform loading. He found that particles with low Stokes number

destabilize a viscous flow, even in the absence of gravity, because the particles increase the bulk

average mixture density and this results in an increase of the effective Reynolds number by a factor

of (1+φ), where φ is the mass loading of the particles. In contrast, particles with large Stokes

number stabilize the flow due to the added dissipation introduced by them. These analytical

predictions were confirmed through numerical simulations conducted by Tong et al. [63]. They

have also shown that particles of Stokes number at unity give the most stability by adding

maximum dissipation. The direction of interphase energy transfer is different for particles of small,

medium and large inertia.

Yang et al. [64] performed a study on the spatial instability of a developing, particle-laden

mixing layer through a modified Rayleigh equation. Since the mixing layer was treated as inviscid,

the particles were found to increase the flow stability (both spatial and temporal) and decrease the

amplification rate of perturbations in the flow. This stability enhancement increases with particle

loading and decreases with free stream velocity ratio.

Temporal stability analysis of inviscid mixing layers with uniformly laden of heavy

particles was performed by several researchers. Dimas et al. [65] conclude that the presence of a

dynamic particle phase within the shear layer dampens the spatial growth rate of the primary

Kelvin-Helmholtz instability. The magnitude of dampening depends primarily on particle loading

and Stokes number, which is a measure of the particles responsiveness. Particles with low Stokes

number do not affect the growth rate of the shear layer regardless of the loading. Particles with

21

large Stokes number, on the other hand, reduce the growth rate in proportion with the loading

amount. At a critical Stokes number and particle loading, large inertia particles will introduce a

second, low frequency mode, which always remain unstable as the mass loading increases.

Furthermore, the net effect of particle interaction is to decrease and increase vorticity in certain

regions.

Wen et al. [66] investigated the effects of non-uniform particle loading in the shear layer

and discovered that the non-uniformity introduces a long wave mode, which corresponds to the

standard Kelvin-Helmholtz instability, and a short mode, which is similar to Holmboe instability

of a density-stratified mixing layer flow. When the mean particle loading is increased and the

velocity boundary layer drifts into the region of low particle loading, the most unstable mode

changes from a long-wave (broadband) instability to a short-wave (narrow band) instability. The

first unstable mode exists in both homogeneous and particle laden mixing layer flows while the

second mode only exists in differential particle loading mixing layer flow.

It is known that compressibility plays a significant role in the instability of compressible

mixing layers. It has a significant effect on the shape of the mixing layer structure and the three

dimensional perturbations have larger growth rates in compressible than incompressible mixing

layers [67,68]. Thevand et al. [69] analyzed the effects of particles on the temporal development

of compressible, inviscid mixing layers. The study reveals that with particles influence, the

obliquity angle of the most amplified perturbation becomes three dimensional at a lower

convective Mach numbers. The maximum stability is always achieved by particles of Stokes

number of order of unity regardless of the convective Mach number. The growth rate attenuation

22

increases with compressibility effects (increasing convective Mach number) regardless of the

particle mass loading. Particles of small, unity and large Stokes numbers increase the stability of

the flow with increasing mass loading and convective Mach number. However, when the particle

mass loading is increased at high compressibility, particles with large Stokes number attenuate the

maximum perturbation growth much slower than particles of low Stokes number.

23

Chapter Three: GOVERNING EQUATIONS

3.1 Gas Phase

The fluid will be treated as a continuum. The concept of material derivative is used in [75]

to describe the rate of change of an intensive physical property, ϕ in time as the following:

d

dt∫ ρϕ(𝐱, t)dV =

∂

∂t∫ ρϕdV + ∮ d𝐒 ∙ (ρϕ𝐔)

∂V(t)

V(t)

V(t)

3.1

Where V is volume, U is the velocity vector and dS is the outward pointing unit normal on 𝜕𝑉,

surfaces that bound a cell.

The rate of change of 𝜙 in V is equal to its volume source, 𝑄𝑉 and surface source, 𝑄𝑆 is:

∂

∂t∫ ρϕdV + ∮ d𝐒 ∙ (ρϕ𝐔)

∂V(t)

V(t)

= ∫ QV(ϕ)dV + ∮ d𝐒 ∙ 𝐐𝐒(ϕ)

∂V(t)

V(t)

3.2

Which in its differential form is:

∂ρϕ

∂t+ ∇ ∙ (ρϕ𝐔) = QV(ϕ) + ∇ ∙ 𝐐𝐒(ϕ) 3.3

The governing equations which are solved are as follows:

Continuity Equation

∂ρ

∂t+ ∇ ∙ (ρ𝐔) = 0

3.4

Navier – Stokes Equation

∂ρ𝐔

∂t+ ∇ ∙ (ρ𝐔𝐔) = − ∇P + ρ𝐟 + ∇ ∙ 𝛔 + 𝐒𝐏 3.5

Total Energy Equation

∂ρE

∂t+ ∇ ∙ [𝐔(ρE + P)] = ρ𝐟 ∙ 𝐔 − ∇ ∙ 𝐪 + ∇ ∙ (𝛔 ∙ 𝐔) 3.6

24

Where

E is total energy and it is the sum of internal energy and kinetic energy.

E = e +|𝐔|2

2 3.7

The thermodynamic state equations for thermally and calorically perfect gas to close the

NS equations:

P = ρRT, γ =CP

CV= constant 3.8

The Fourier’s law of heat conduction is defined as:

𝐪 = −λ∇T 3.9

The generalized form of the Newton’s law of viscosity for Newtonian fluid is defined as

𝛔 = − (2

3 μ) ∇ ∙ 𝐔 + μ[ ∇𝐔 + (∇𝐔) T] 3.10

f is body force.

The second viscosity coefficient is 2/3 comes from Stokes’s assumption [70]. [72] states that

Stokes’ hypothesis is commonly used in high-speed compressible flows but it has yet to be

confirmed as valid [71]. Although there is insufficient experimental data to model the coefficients

accurately, it is a common practice in CFD to apply Stokes’ assumption.

There is a reason behind why the governing equations are cast in the strong, conservation

form as it comes primarily from experience [72]. When they are combined with shock capturing

method, the numerical prediction of the flow fields are usually smooth and stable. Unlike when

the non-conservative form of the governing equations are use, where unphysical oscillations near

the shock wave region as well as prediction of incorrect shock location are found. The shock

25

capturing method is by far a much more wide-spread in commercial CFD softwares compared to

shock-fitting method. This is mostly due to the fact that the shock wave can be introduced directly

into the numerical solution as an explicit discontinuity without needing to track the shock and the

jump conditions can be described by the Rankine-Hugoniot relations. This is a much more realistic

approach as it allows the computations of complex flow fields with numerous shock structures

without requiring to know the locations of all the shockwaves at firsthand. However, shock

capturing method comes with a price tag as the shocks are usually smeared over a finite number

of grid points.

3.2 Particle Phase

The following is a summary of the significant assumptions that are used in the simulations:

1. The particle motion is essentially governed by the viscous drag force.

2. Although the volume fraction of the particles is negligible but the flow is clearly in the

two-way coupling, dilute flow regime.

3. The thermal and Brownian motion of the particles is neglected.

4. Particle-particle interactions (four way coupling) is assumed not to occur in this flow

regime.

5. Phase change does not occur.

6. The particles are solid spheres with a uniform diameter and constant material density.

26

7. The particles are adiabatic (heat transfer does not occur) and have uniform temperature

distribution.

The Newton's equation of motion (2nd Law) of a single particle in the Lagrangian frame is given

below [93]:

mP

∂𝐔𝐏

∂t = 𝐅𝐩 = −

𝜋ρ DP2

8 𝐶𝐷|𝐔𝐏 − 𝐔|(𝐔𝐏 − 𝐔)+mP𝐠 3.11

∂𝐱𝐩

∂t= 𝐔𝐏

3.12

∂𝐔𝐏

∂t= −

𝐔𝐏−𝐔

τP+ 𝐠

where particle relaxation time, τp = 4

3

ρPDP

ρ𝐶𝐷|𝐔𝐏−𝐔| 3.13

The term F, contains other forcers such as the added mass, Basset, Saffman, Magnus (rotating

particles), pressure and buoyancy force and all together, Equation 3.17 is known as BBO equation,

named after Basset (1888), Boussinesq (1903) and Oseen (1927). However, if the density ratio

between the solid and phases is at least of order ~102, most of these forces can be neglected thus

leaving only the forces acting on the particle to be only drag and gravitational forces [93] which is

shown in the most right hand side of Equation 3.17.

The particle Reynolds number and particle Mach number are defined by the particle slip velocity:

Rep =

ρPDP |𝐔𝐏

− 𝐔|

μ 3.14

Mp =

|𝐔𝐏 − 𝐔|

√γRT 3.15

Where is the specific heat ratio of the gas (air) and μ is the viscosity of the gas.

The particle drag coefficient expression given by [98] is as follows:

27

CD =

24

ReP k [1 + 0.157(kReP)0.687]ζ(Kn)C 3.16

The coefficient k is obtained by solving:

g(k) = a1 k1.687 + a1 k − 1 = 0 3.17

Where

a1 = 0.3375

L

DP

Kn

ε′(

2DP

L

S√Π

Kn)0.687 3.18

a2 = 1 + 2.25

L

DP

Kn

ε′ 3.19

ε′ = 3

8

√Π

S′( 1 + S 2

′)erf(S′) + e−S′2/4 3.20

The molecular speed ratio, S, is defined as:

S = √

γ

2MP 3.21

S′ = (1 − k)S 3.22

As the flow regime in the simulation falls in slip flow regime, and as 𝑅𝑒𝑃 1; L = 2𝐷𝑃. Slip flow,

in this context, is a regime in which, due to rarefaction effects, the no slip condition on the particle

surface (where 𝑈𝑤𝑎𝑙𝑙 = 0 m/s ) is no longer valid. The slip velocity between the particle surface

and the molecules of the fluid surrounding the particle has to be taken into account. A systematic

categorization of flow regimes is described by [98] and [73].

The Knudsen number is defined as:

Kn = √

γπ

2

MP

ReP 3.23

The function 𝜁(𝐾𝑛) is to account for free molecule flow.

28

ζ(Kn) = 1.177 + 0.177

0.85Kn1.16 − 1

0.85Kn1.16 + 1 3.24

Finally, the coefficient C is defined to account the drag force associated with compressibility (𝑀𝑃

> 0.3):

C = 1 +

ReP2

ReP + 100 e

−0.225

MP2

3.25

Thus, the drag force which is the source term, 𝑆𝑃 for the Navier-Stokes equation is defined as:

𝐒𝐏 =

1

𝑉𝑐𝑒𝑙𝑙

3

4 ∑ [CDReP

mPμ

ρPdp2

(𝐔𝐏 − 𝐔)]

𝑛𝑝

3.26

29

3.3 Finite Volume Method

There are several numerical methods that are currently widely employed in Computational

Fluid Dynamics. They are Finite Difference (FD), Finite Volume (FV), Finite Element (FE) and

Spectral Methods and more recently, the Lattice-Boltzmann method (LBM). The methods are

distinguished by the discretization technique. Finite Difference is the discretization of the partial

differential equations while Finite Volume is the discretization of the integral form of the

equations. Discretization [72] on the other hand is just the process where “a closed-form

mathematical expression, may it be a differential or integral equation, is transformed into

corresponding systems of algebraic expressions and the numerical solutions are prescribed at a

finite number of points or volumes in the computational domain”.

In most compressible gas dynamics problems, there are usually discontinuities in the form

of shock waves. This leads to computational difficulties and there is a need to accurately

approximate such solutions. As a short summary [72], Finite Difference method requires special

treatment near the discontinuities as the differential equations do not hold. This is where Finite

Volume can overcome this problem with much ease due to the fact that instead of pointwise/node-

wise approximations, the computational domain is discretized into cells and the total integral or

cell average of a variable is approximated over each grid cell which are updated in each time step

through flux approximation at each cell boundaries. This makes the primary objective of any FV

method is to determine and derive good numerical flux functions that approximate the correct

fluxes accurately based on the cell averages. All the FV methods require high resolution techniques

in order to predict discontinuous solutions.

30

The Riemann problem is the basis of finite volume methods and it is a hyperbolic equation

with a single jump discontinuity as an initial condition [74]. The Riemann problem, for the Euler

equations, that is centered on 𝑥 = 𝑥𝑜 at 𝑡 = 𝑡𝑜 with a single jump discontinuity can be described

as:

𝑢(𝑥, 𝑡𝑜) = {𝑢𝐿 , 𝑥 < 𝑥𝑜

𝑢𝑅 , 𝑥 > 𝑥𝑜

3.27

The exact solutions of the Riemann problem for both the Euler equation and non-linear problems

are computationally expensive because it is nonlinear and implicit. Thus, usually approximate

Riemann solvers are used in implementing numerical methods such as Roe’s or Osher’s

approximate Riemann solvers.

The following sections regarding the discretization techniques used in OpenFOAM as well

as rhoCentralFoam are taken from Hrjove Jasak’s [75] and Eugene de Villiers’s [76] Ph.D

dissertations. The readers of this thesis of course can refer to both dissertations to have a complete

understanding of OpenFoam framework but the following subsections are written just for

completeness and as an attempt of the author to reconcile a few discrepancies and elaborate further

the formulations found in the theses with the formulations stated in [77].

31

3.3.1 Domain Discretization

Fig. 5 Control Volume for Finite Volume Discretization [77]

From [75]: the computational domain discretization can be subdivided into spatial and

temporal discretization. Spatial discretization defines the computational domain as a system of

control volumes or computational cells as shown in Fig. 5. Each of these cells or control volumes

(CV) encompasses a point, P at its centroid. This typical CV is bounded by a set of faces which

can be of arbitrary shape and each face is shared with only one neighboring CV. d is the vector

that connects adjacent cell centers of P and N while 𝑑𝑓𝑛 connects the center of the cell boundary

to the cell centroid. 𝑆𝑓 is the outward pointing face normal area vector of the owner cell at the

boundary between the two cells. In this collocated system framework, all dependent variables and

material properties are stored at the cell centroid while the numerical fluxes are evaluated at the

cell faces.

32

3.3.2 Spatial Discretization

What is unique about the OpenFOAM framework is that instead of the entire discretization

procedure for each governing equation, the discretization for each generic transport equation is

carried out on term by term. This basically allows the users of OpenFOAM to discretize each term

using different numerical schemes of different order of accuracy if the user so chooses. From

equation (3.1), now cast in this current form:

3.28

The 1st term: Temporal derivative.

The 2nd term: Convection.

The 3rd term: Diffusion and

The 4th term: Sources.

Again, ϕ is the transported quantity while 𝜞𝝓 is the diffusivity coefficient. When equation (3.28)

is in its differential form, it becomes a second order equation as it carries the diffusion term which

contains a second order derivative. Thus to represent this term with sufficient accuracy, the

discretization order also has to be equal or higher than of the temporal discretization. The

generalized form of Gauss’s theorem [75] is invoked throughout the discretization procedure of

the spatial terms with these identities:

∫ 𝛻 ∙ 𝒂 𝑑𝑉

𝑉

= ∮ 𝑑𝑺 ∙ 𝒂

𝑑𝑉

3.29

∫ 𝛻𝒂 𝑑𝑉

𝑉

= ∮ 𝑑𝑺 𝒂

𝑑𝑉

3.30

∫ [𝜕

𝜕𝑡∫ 𝜌𝜙𝑑𝑉 + ∫ 𝛻 ∙ (𝜌𝑼𝜙)𝑑𝑉 − ∫ 𝛻 ∙ (𝜌𝛤𝜙𝛻𝜙)𝑑𝑉

𝑉𝑝

𝑉𝑝

𝑉𝑝]

𝑡+∆𝑡

𝑡𝑑𝑡 =

∫ (∫ 𝑆𝜙𝑑𝑉)

𝑉𝑝

𝑡+∆𝑡

𝑡𝑑𝑡

33

∫ 𝛻𝜙𝑑𝑉

𝑉

= ∮ 𝑑𝑺 𝜙

𝑑𝑉

3.31

Where 𝑑𝑉 is the infinitesimal volume and 𝑑𝑆 denotes the infinitesimal surface element that points

outward-normal to 𝑆 and a is some vector variable. The gradient term can be formulated through

Gauss Theorem or the Least Squares Fit (LSF).

The volume integrals and surface intergrals can be defined respectively [75] as:

∫ 𝜙(𝒙)𝑑𝑉

𝑉𝑝

= ∫ [𝜙𝑃 + (𝒙 − 𝒙𝑷) ∙ (𝛻𝜙𝑃)]𝑑𝑉

𝑉𝑝

= 𝜙𝑃 ∫ 𝑑𝑉

𝑉𝑝

+ ∫ [(𝒙 − 𝒙𝑷)]𝑑𝑉 ∙ (𝛻𝜙𝑃)

𝑉𝑝

= 𝜙𝑃𝑉𝑃 3.32

∫ 𝛻 ∙ 𝒂𝑑𝑉

𝑉𝑝

= ∮ 𝑑𝑺 ∙ 𝒂

𝑆

= ∑ ∫ 𝑑𝑺 ∙ 𝒂

𝑓

𝑓

= ∑ (∫ 𝑑𝑺

𝑓

∙ 𝒂𝒇 + ∫ [(𝒙 − 𝒙𝑷)]𝑑𝑺: (𝛻𝒂)𝑓)

𝑉𝑝

)

𝑓

= ∑ 𝑺 ∙ 𝒂𝒇

𝑓

= (∇ ∙ 𝒂)𝑉𝑃 3.33

The subscript f is to indicate the value of the variable that is in the middle of the face and S is the

normal outward pointing area vector. The sum over the faces is divided into “owner” and

“neighboring” faces.

∑ 𝑺 ∙ 𝒂𝒇

𝒇

= (𝛁 ∙ 𝒂)𝑉𝑃 = ∑ 𝑺𝒇 ∙ 𝒂𝒇

𝒐𝒘𝒏𝒆𝒓

− ∑ 𝑺𝒇 ∙ 𝒂𝒇

𝒏𝒆𝒊𝒈𝒉𝒃𝒐𝒖𝒓

3.34

3.3.2.1 Convective Terms

The convective terms in the governing equations are 𝛻 ∙ (𝜌𝑼),𝛻 ∙ [𝑼(𝜌𝑼)],𝛻 ∙ [𝑼(𝜌𝐸)], and

34

𝛻 ∙ (𝑼𝑃). The discretization procedure for this set of terms is:

∫ 𝛻 ∙ (𝜌𝑼𝜙)𝑑𝑉

𝑉𝑝

= ∑ 𝑺𝒇

𝑓

∙ (𝜌𝑼𝜙)𝑓 = ∑(𝑺𝒇

𝑓

∙ 𝜌𝑼𝑓)𝜙𝑓 = ∑ 𝐹

𝑓

𝜙𝑓 3.35

Where F = 𝑺𝒇 ∙ (𝜌𝑼)𝑓 is the mass flux through the face and, 𝑆𝑓 is the outward pointing surface

face centered vector. The flux is evaluated through the interpolation values of ρ and 𝑈𝑓. This can

be solved in the same way as 𝜙𝑓 but it must satisfy the continuity equation [76]. Linear

interpolation of neighboring cell center values is employed to evaluate the value on the cell face

as shown in the figure below.

Fig. 6 Face Interpolation (adapted from [75])

Using the linear interpolation of 𝜙 between P and N,

𝜙𝑓 = 𝑓𝑥𝜙𝑃 + (1 − 𝑓𝑥)𝜙𝑁 3.36

Where the interpolation factor, 𝑓𝑥 is defined as:

𝑓𝑥 =𝑓𝑁

𝑃𝑁

3.37

Using such linear interpolation to determine the face value of 𝜙 is known as Central Differencing

(CD). However CD is known to introduce unphysical oscillations in convection-dominated

problems [78] and furthermore, in near region of a shock wave, CD breaks down completely if

35

one of the point/stencils is taken across the shock. Thus, it will be shown that in a way,

rhoCentralFoam will use a form of flux splitting method to switch from central to upwind schemes

near areas of large gradient such as near shock regions. In the context of the polyhedral mesh as

shown in Fig. 5, then it becomes a weighting function [77] which is defined as:

𝑓𝑥 =𝑺𝒇 ∙ 𝒅𝒇𝑵

𝑺𝒇 ∙ 𝒅 3.38

3.3.2.2 Diffusion/Laplacian Term

The diffusion term is discretized using the same linear interpolation of ϕ:

∫ 𝛻 ∙ (𝜌𝛤𝜙𝛻𝜙)𝑑𝑉

𝑉𝑝

= ∑ 𝑺𝒇

𝑓

∙ (𝜌𝛤𝜙𝛻𝜙)𝑓

= ∑(𝜌𝛤𝜙)𝑺𝒇

𝑓

∙ (𝛻𝜙𝑓) 3.39

Equation 3.43 is 2nd order accurate and this discretized form will preserve the boundedness

properties of FV if the mesh is orthogonal. When the mesh is orthogonal, then vectors d and 𝑆𝑓 are

parallel and this allows

𝑺𝒇 ∙ (𝛻𝜙𝑓) = |𝑺|𝜙𝑃 − 𝜙𝑁

|𝐝| 3.40

Fig. 7 Vectors d and 𝑺𝒇 on a non-orthogonal mesh (adapted from [75])

36

In the case of a non-orthogonal mesh, then

𝑺𝒇 ∙ (𝛻𝜙𝑓) = ∆̅ ∙ (𝛻𝜙)𝑓 + 𝐤 ∙ (𝛻𝜙)𝑓 3.41

Where the 1st term on the right is the orthogonal contribution and the 2nd term is the non-orthogonal

correction. ∆̅ = 𝒅|𝑺|𝟐/|𝒅 ∙ 𝑺| and k = 𝑺 − ∆̅

3.3.2.3 Source Terms

Any terms in the governing equations that cannot be written in the form of convection,

diffusion or temporal terms will be treated as sources. This is the discretization procedure of the

source term outlined in [79]:

𝑆𝜙(𝜙) = 𝑆𝑢 + 𝑆𝑝𝜙 3.42

∫ 𝑆𝜙(𝜙)𝑑𝑉

𝑉𝑝

= 𝑆𝑢𝑉𝑃 + 𝑆𝑝𝑉𝑃𝜙𝑃 3.43

3.3.3 Temporal Discretization

As rhoCentralFoam is a transient solver and not a steady state solver, there is a need to

discuss how the temporal derivative is solved. Thus, drawing from [75] and [76] where it is

outlined in detailed but here, only discretization through backward differencing will be discussed.

Starting with the integral form of the transport equation:

With the assumption that the volume of the cell/control volume is constant and cast in the “semi-

discretized” [78] form, the equation becomes:

∫ [𝜕

𝜕𝑡∫ 𝜌𝜙𝑑𝑉 + ∫ ∇ ∙ (𝜌𝑼𝜙)𝑑𝑉 − ∫ ∇ ∙ (𝜌𝛤𝜙∇𝜙)𝑑𝑉

𝑉

𝑉

𝑉] 𝑑𝑡

𝑡+∆𝑡

𝑡 = ∫ ∫ 𝑆𝜙𝑑𝑉

𝑉𝑑𝑡

𝑡+∆𝑡

𝑡

37

∫ [(𝜕𝜌𝜙

𝜕𝑡)

𝑃𝑉𝑃 + ∑ 𝐹𝜙𝑓

𝑓

− ∑(𝜌𝛤𝜙)𝑓𝑺 ∙ (∇𝜙)𝑓

𝑓

] 𝑑𝑡𝑡+∆𝑡

𝑡

= ∫ (𝑆𝑢𝑉𝑃 + 𝑆𝑝𝑉𝑃𝜙𝑃)𝑡+∆𝑡

𝑡

𝑑𝑡

3.44

There are several methods in discretizing time derivatives due to the practice of neglecting the

variation of the face values 𝜙 and 𝛻𝜙 in time [80]. This leads to discretized transport equations

where the previous and current time-level convection, diffusion and source terms are present

together. However this comes with at a price as this equation is only 1st order accurate in time:

𝜌𝑃𝑛𝜙𝑃

𝑛 − 𝜌𝑃𝑛−1𝜙𝑃

𝑛−1

∆𝑡𝑉𝑃 + ∑ 𝐹𝜙𝑓

𝑓

− ∑(𝜌𝛤𝜙)𝑓𝑺 ∙ (∇𝜙)𝑓 = 𝑆𝑢𝑉𝑃 + 𝑆𝑝𝑉𝑃𝜙𝑃

𝑓

3.45

Backward differencing in time is a temporal numerical scheme which is 2nd order accurate

in time and also neglects the temporal variation of the face values. [75] states that backward

differencing has cheaper computational costs and easier in code implementation but possess a

truncation error that is four times larger than Crank-Nicolson which results in additional diffusion.

To avoid such issue to a minimum, the cell-face CFL (Courant-Friedrichs-Lewy) number has to

be kept below 1. The CFL number is a stability criterion which states that the numerical

information propagation speed must be greater than or equal to the physical information

propagation speed [81].

The 2nd order of the discretized form of the temporal derivate can be found using Taylor

series expansion of ϕ in time around 𝜙𝑛+1 = 𝜙(𝑡 + ∆𝑡):

38

𝜙(𝑡) = 𝜙𝑛−1 = 𝜙𝑛 −

𝜕𝜙

𝜕𝑡∆𝑡 + 0.5

𝜕2𝜙

𝜕𝑡2∆𝑡2 + 𝒪(∆𝑡3) 3.46

Thus now, the temporal derivative can be defined as:

𝜕𝜙

𝜕𝑡=

𝜙𝑛 − 𝜙𝑛−1

∆𝑡+ 0.5

𝜕2𝜙

𝜕𝑡2∆𝑡 + 𝒪(∆𝑡2)

3.47

To increase the order of accuracy, three time levels are used:

𝜙(𝑡 − ∆𝑡) = 𝜙𝑛−2 = 𝜙𝑛 − 2 (

𝜕𝜙

𝜕𝑡)

𝑛

∆𝑡 + 2 (𝜕2𝜙

𝜕2𝜙)

𝑛

∆𝑡2 + 𝒪(∆𝑡3) 3.48

By combining the equations 3.51 and 3.52, the 2nd order approximation of the temporal

derivative becomes:

(𝜕𝜙

𝜕𝑡)

𝑛

=

32 𝜙𝑛 − 2𝜙𝑛−1 +

12 𝜙𝑛−2

∆𝑡

3.49

So now, the final form of the transport equation, which is fully implicit, to solve for 𝜙𝑃𝑛 is:

32

𝜌𝑃𝑛𝜙𝑃

𝑛 − 2𝜌𝑃𝑛−1𝜙𝑃

𝑛−1 +12

𝜌𝑃𝑛−2𝜙𝑃

𝑛−2

∆𝑡𝑉𝑃 + ∑ 𝐹𝜙𝑓

𝑛

𝑓

− ∑(𝜌𝛤𝜙)𝑓

𝑺 ∙ (∇𝜙)𝑓𝑛

𝑓

= 𝑆𝑢𝑉𝑃 + 𝑆𝑝𝑉𝑃𝜙𝑃𝑛

3.50

3.4 High Resolution FV Method in rhoCentralFoam

In this chapter and the previous sections, the discretization methods of the terms have been

discussed and shown. Now, the time is nigh to discuss how the systems of equations of the FV

method are solved, which is the basis of rhoCentralFoam. So again, this section will draw

materials and derivations that are found in [75, 76, 77].

39

The way FV method works is through the approximation of the integral of the cell averages

of interest, Q over each of these cells where they are tracked and updated in each time step through

the approximation of the fluxes across the faces of these cells. This is easily illustrated through the

figure below:

Fig. 8 FV Method of Updating Cell Average, Q through Fluxes across the Cell Faces

(adapted from [74])

Starting with the integral form of the conservation law of a 1D region bounded by [x,b] [74]:

∫ [𝜙(𝑥, 𝑡2

𝑥𝑖+1/2

𝑥𝑖−1/2

) − 𝜙(𝑥, 𝑡1)]𝑑𝑥 = − ∫ [𝑓 (𝜙(𝑥𝑖−1/2, 𝑡)) − 𝑓 (𝜙(𝑥𝑖+1/2, 𝑡))] 𝑑𝑡𝑡2

𝑡1

3.51

or its conservation form:

𝜕𝜙

𝜕𝑡 = −

𝜕𝑓(𝜙)

𝜕𝑥 3.52

With the cell averages,𝑄𝑖𝑛 at time 𝑡𝑛, 𝑄𝑖

𝑛+1 can be approximated by integrating in time and obtain

∫ 𝜙

𝐶𝑉

(𝑥, 𝑡𝑛+1)𝑑𝑥 − ∫ 𝜙

𝐶𝑉

(𝑥, 𝑡𝑛)𝑑𝑥

= ∫ 𝑓 (𝜙 (𝑥𝑖−

12

, 𝑡)) 𝑑𝑡𝑡𝑛+1

𝑡𝑛

− ∫ 𝑓 (𝜙 (𝑥𝑖+

12

, 𝑡)) 𝑑𝑡𝑡𝑛+1

𝑡𝑛

3.53

And rearranging it and dividing by ∆𝑥:

40

1

∆𝑥∫ 𝜙

𝐶𝑉

(𝑥, 𝑡𝑛+1)𝑑𝑥

=1

∆𝑥 ∫ 𝜙

𝐶𝑉

(𝑥, 𝑡𝑛)𝑑𝑥

− 1

∆𝑥[∫ 𝑓 (𝜙 (𝑥

𝑖+12

, 𝑡)) 𝑑𝑡𝑡𝑛+1

𝑡𝑛

− ∫ 𝑓 (𝜙 (𝑥𝑖−

12

, 𝑡)) 𝑑𝑡𝑡𝑛+1

𝑡𝑛

]

3.54

The numerical conservation form is defined as (which high resolution methods are often cast in

such form):

𝑄𝑖𝑛+1 = 𝑄𝑖

𝑛 - ∆𝑡

∆𝑥 (𝐹𝑖+1/2

𝑛 - 𝐹𝑖−1/2𝑛 ) 3.55

Where 𝐹𝑖−1/2𝑛 and 𝐹𝑖+1/2

𝑛 are flux average approximation along 𝑥𝑖−1/2 and 𝑥𝑖+1/2 respectively.

𝐹𝑖−1/2𝑛 can be obtained through only the values of 𝑄𝑖−𝑖

𝑛 and 𝑄𝑖𝑛:

𝐹𝑖−1/2𝑛 = ℱ (𝑄𝑖−1

𝑛 , 𝑄𝑖𝑛) 3.56

Finally, the numerical method becomes:

𝑄𝑖𝑛+1 = 𝑄𝑖

𝑛 - ∆𝑡

∆𝑥 [ ℱ (𝑄𝑖

𝑛, 𝑄𝑖+1𝑛 ) - ℱ (𝑄𝑖−1

𝑛 , 𝑄𝑖𝑛)] 3.61

For this case, 𝓕 is based on the explicit method with a three-point stencil.

There are two numerical methods that can be selected in rhoCentralFoam; Kurganov and

Tadmor (KT) [82] method and Kurganov, Noella, and Petrova (KNP) [96].

KNP method is based on the Godunov-type central schemes. Godunov type schemes,

which are described in [83], are based in the integral framework that bridge the cell averages

41

evolution and the flux evaluations at the spatial cell boundaries. Through cell

averages, [𝑄(𝑥𝑗 , 𝑛)]𝑗,𝑤ℎ𝑒𝑟𝑒 𝑥𝑗 = ((j-1/2) ∆𝑥 and j = 1,2,3,…2

𝜋

∆𝑥 a global reconstruction, 𝑤(𝑥, 𝑡𝑛)

is formed to approximate 𝜙(𝑥, 𝑡𝑛) (instantaneous cell values) which is then evolved in time

through Equation 3.60. There are 2 different types of framework that can be used for this evolution

step; upwind and central schemes. In upwind scheme, Riemann solvers are required to evaluate

the flux intergrals in Equation 3.60. However, in Godunov central scheme framework, a staggered

cell average is used instead and its evolution is defined as:

�̅� 𝑗+1/2𝑛+1 = �̅� 𝑗+1/2

𝑛 − 1

∆𝑥[∫ 𝑓( 𝑤𝑗+1

𝑛 (𝑥(𝑡))𝑑𝑡𝑡𝑛+1

𝑡𝑛

− ∫ 𝑓( 𝑤𝑗𝑛(𝑥(𝑡))𝑑𝑡

𝑡𝑛+1

𝑡𝑛

] 3.57

where

�̅� 𝑗+1/2𝑛 = ∫ 𝑃𝑗(𝑥, 𝑡𝑛)𝑑𝑥 +

𝑥𝑗+∆𝑥/2

𝑥𝑗

∫ 𝑃𝑗(𝑥, 𝑡𝑛)𝑑𝑥 𝑥𝑗+1

𝑥𝑗+∆𝑥/2

3.58

And 𝑷𝒋(𝒙, 𝒕𝒏) is the fixed order polynomial. It is apparent from Equation 3.63 that the flux

integrals are computed at the smooth midpoints of the reconstruction and this totally avoids the

need to use Riemann solvers. Thus, KNP is called central schemes because it uses exact evolution

and averaging over Riemann fans and Riemann solver free. One important thing to note is that it

possesses an upwind character because one-sided information is used to estimate the width of the

Riemann fans and thus this method is called central upwind. The reconstruction is based on the

use of the CFL number related to the local speeds of propagation.

KT method is purely a (2nd order) central schemes method that is based on the Nessyahu-

Tadmor [84] scheme which is itself based on the 1st order central Lax-Friedrichs scheme. The 1st

order accurate in time and space Lax-Friedrichs (LxF) flux approach method preserves

42

monotonicity, is TVD (Total Variation Diminishing) and essentially non-oscillatory (ENO) [85].

However, LxF is also known to contain the most amount of artificial viscosity or numerical

diffusion. This (LxF) method has the form of:

𝜙𝑖

𝑛+1 = 1

2(𝜙𝑖−1

𝑛 +𝜙𝑖+1𝑛 ) -

∆𝑡

2∆𝑥 (𝐹(𝜙𝑖+1

𝑛 ) - 𝐹(𝜙𝑖−1𝑛 )) 3.59

The idea behind the KT scheme is to replace the 1st order piecewise constant solution with van

Leer’s MUSCL type piecewise linear 2nd order approximation which is then combined with the

LxF solver. Local wave speed propagation is used to average the non-smooth parts of the predicted

solution over small cells in the KT method.

In compressible gas dynamics, the solution has to take the propagation of waves into

account. So in rhoCentralFoam, the flux interpolation discretization procedure is split into two

direction depending on the direction of the flow relative to the face owner cell (inward (-)/upwind

& outward(+)/downwind):

∑ �⃗�𝑓 𝜙𝑓 = ∑ [𝛼𝜙𝑓+𝐹+ 𝑓 + (1 − 𝛼)𝜙𝑓−𝐹− + 𝜃𝑓(𝜙− − 𝜙+)] 3.60

Where

�⃗⃗�+ = 𝑚𝑎𝑥(𝑐𝑓+|𝑆𝑓| + 𝐹𝑓+, 𝑐𝑓−|𝑆𝑓| + 𝐹𝑓−, 0)

�⃗⃗�− = 𝑚𝑎𝑥(𝑐𝑓+|𝑆𝑓| − 𝐹𝑓+, 𝑐𝑓−|𝑆𝑓| − 𝐹𝑓−, 0) 3.61

𝒄𝒇 is nothing more that the speed of sound of the gas at the faces depending on the direction.

In the KT method, the fluxes are weighted equally and thus 𝛼 =0.5 and thus it is satisfies its

description as a central scheme. In the KNP method, 𝛼 is calculated based on the upwind-biased

43

local speeds of propagation; 𝛼 = 𝜙𝑓+/ (𝜙𝑓+ + 𝜙𝑓−) and this satisfies its description as central

upwind.

The diffusive flux is calculated as:

𝜃𝑓 = 𝛼 𝑚𝑎𝑥(𝜙𝑓+, 𝜙𝑓−) − KT

𝜃𝑓 = 𝛼 (1 − 𝛼)(𝜙𝑓+ + 𝜙𝑓−) − KNP 3.62

Based on [75], the concept of Convection Boundedness Criterion [86] and Normalized Variable

Approach [87] are introduced to provide the most general way to ensure local boundedness and

unphysical oscillations in the solutions do not occur by making the cell value of ϕ of a is locally

bounded by the values between the upwind and downwind cells. [72] has made modifications of

the NVA in terms of gradient of the variable of interest to enable NVA to be used on arbitrarily

unstructured, polygonal meshes. Thus, 𝜙�̆� the normalized variable is defined as:

𝜙�̆�=

𝜙𝑖−𝜙𝑓−

𝜙𝑓+−𝜙𝑓

− =1- 𝜙𝑓

+−𝜙𝑖

𝜙𝑓+−𝜙𝑓

− 3.63

Where 𝜙𝑓− refers to the upwind/inward cell face value and 𝜙𝑓

+ refers to the downwind/outward cell

face value. To reformulate the equation above in terms of gradients across cell face and upwind

cell, the interpolation factor, 𝑓𝑥 (Equation 3.41) , the vector, d and the relationship where

[(𝜙𝑖+1 − 𝜙𝑖)/(𝑥𝑖+1 − 𝑥𝑖)] = (∇𝜙)𝑓 ∙ 𝒅) are used and the interpolation procedure used in

rhoCentralFoam to interpolate face f + and f – of scalar variables of interest, 𝜙 is cast in the form

of :

𝜙�̆�=

2(∇𝜙)𝑓

∙𝑑

(∇𝜙)𝑓

∙𝑑 - 1 3.64

44

The face interpolations of the transported variable also uses a flux limiter, β in order to

switch between low and high order schemes which is a function of 𝜙�̆�. The limiters chosen to be

implemented in rhoCentralFoam are Total Variation Diminishing (TVD). As a brief discourse,

the TVD condition, first suggested by Harten [88], is a non-linear stability condition that preserves

monotonicity and addresses the stability of both monotone and nonmonotone solutions [85]. A

new pair of local maximum-minimum and if the existing maximum increases or the local minimum

decreases can increase the total variation. The numerical approximation which is proposed that

inherits the TVD property from Sweby [89]:

TV(𝑢𝑛+1) < TV(𝑢𝑛) = ∑ |𝑢𝑖+1𝑛 − 𝑢𝑖

𝑛 |∞𝑖=−∞

This implies that for each time step, the total variation of a numerical approximation on an infinite

domain is a sum of extrema, maxima counted positively and minima counted negatively.

Using Equation 3.34, the gradient term is discretized as follows [77]:

∫ 𝛻𝜙𝑑𝑉

𝑉𝑝

= ∮ 𝑑𝑺 𝜙

𝜕𝑉

≈ ∑ 𝑺𝒇𝜙𝑓

𝑓

3.65

As the KT and KNP schemes take into account the direction of the face interpolation, the gradient

term is calculated as follows:

∑ 𝑺𝒇𝜙𝑓

𝑓

= ∑ 𝛼𝑺𝒇𝜙𝑓+ + (1 − 𝛼)

𝑓

𝑺𝒇𝜙𝑓− 3.66

Finally, in rhoCentralFoam the f+ interpolation of the scalar variable is calculated as:

𝜙𝑓+ = (1 − 𝜁𝑓+) 𝜙𝑃 + 𝜁𝑓+ 𝜙𝑁 3.67

Where 𝜁𝑓+ = 𝛽(1 − 𝑓𝑥). So when 𝛽 = 0, the equation above uses upwind interpolation and when

𝛽 = 1, then central interpolation is used. This is nothing more than a form of flux-splitting

45

approach. All the other terms are evaluated according to the discretization found in Subsection

3.3.2.

Now there is a huge difference between how the conservation equations are solved in Jasak’s

dissertation and in rhoCentralFoam. The problem studied and solved in Jasak’s dissertation is

incompressible gas flow. Hence, the PISO algorithm is employed to solve for transient problem

and SIMPLE algorithm for steady state problem. Thus, it can be safely stated that the solver he

used can be classified as an incompressible pressure based solver because the variable pressure, p

and flux, F are the first ones to be solved first. However, rhoCentralFoam is a compressible

density-based solver because first, the inviscid portion of the solver is used to obtain density, ρ and

the momentum density, 𝒖⏞ = 𝜌𝑼 and total energy density, 𝑬⏞ = 𝜌𝐸. Then the full Navier-Stokes

equation are solved along with other equations. The full description of the solver algorithm of

course can be found in detail in [77]. Thus, the author would not go any further and the description

regarding the Eulerian solver will end here.

3.5 Particle Modeling & Lagrangian Particle/Parcel Tracking

There are several approaches that can be undertaken to pursue this goal. As described above,

the Lagrange-Euler approach is chosen in this study where the gas flow is solved in the Eulerian

framework while the particles are solved in the Lagrangian framework which necessitates the need

of a Lagrangian Particle Tracking. The other approach is known as Euler-Euler approach where

the particles themselves are treated as volume/mass fractions in the flowfield. Of course there are

debates in determining which approach is superior in terms of accuracy or computational costs

46

[90] and as always, there are advantages and disadvantages inherent in either approach. In

numerical prediction multiphase flow in turbulent flow regime, Direct Numerical Simulation

(DNS) approach is generally taken [91,92] where flow over only a few “particles” are studied.

Nordin (1996) [93] is the first, as far as the author’s knowledge, to describe multiphase flow

modeling in the OpenFOAM framework, which is in the Euler/Lagrange framework. In it, parcels

or computational groups of particles are used to account for the solid/liquid phase and are solved

in the Lagrangian framework. Each parcel is a representation of a large number of real particles

and is initially characterized by a material type, diameter, velocity, and temperature. These

points/parcels basically have zero dimension only serve as markers in which computational cells

where there are interactions taking place between the two phases in order to allow the interaction

terms to be correctly distributed. These parcels are tracked through the flow field and their local

mean properties and values will be the source terms in the gas phase equations. The motion of the

parcels are governed by Newton’s second law, where viscous and pressure forces are primarily

approximated through an overall drag coefficient. The particle tracking algorithm is described and

formulated in Macpherson et al. [94]. The tracking algorithm is formulated for 3D simulations and

is well suited for parallel computations with complex, unstructured meshes. The tracking algorithm

is defined as "generic" as it can be suitably applied in CFD, granular flow simulations, ray tracing

and molecular modeling. In short, there are 2 branches of LPT technique i.e. Lose-Find and Face

to Face algorithm [93] where the latter is the one implemented in OpenFOAM. The Face-to-Face

algorithm can be succinctly described through 4 steps [95]:

1. The parcel is allowed to travel up till the boundary of the cell it is located in or for

the entire time step should it still remain in that particular cell.

47

2. The time for the parcel to move out of the cell will be calculated and its properties

will be updated.

3. Then, the momentum change of the cell previously occupied by the parcel will be

updated.

4. Repeat step 1 for the new cell.

A very detailed discussion regarding the Lagrangian particle tracking (LPT) algorithm, as it

was mentioned above, can be also found in [93] where it addresses all the possible issues regarding

Eulerian/Lagrangian approach in OpenFoam framework. Thus, further discussion and

mathematical expressions of this particle LPT are neglected from this thesis and the author refers

the readers to the aforementioned PhD. Dissertation.

48

Chapter Four: OBJECTIVE, ASSUMPTION AND LIMITATION

This study is primarily interested in the exploring the momentum transfer effects of particles

on the laminar separation bubble and shedding some light on the mechanism of the particles in

suppressing the separation bubble when the flow is still steady and 2D. It is well noted that when

the shock angle is large, the separation bubble induced by the shock is probably unstable and would

undergo transition to turbulent phenomenon. In fact in the Boin et al. study [103] which partly

serves as the basis of this study, highlights the fact that a low frequency oscillation exists from the

intrinsic dynamics of the shock and separation bubble interaction will “trip” the flow and this is

likely the case due to the selection of the shock angle and lateral (spanwise) dimension, Lz of the

computational domain in this study. Thus to avoid further complexities which will arise from the

laminar-turbulent transition, this study is limited to the duration when the particle free separation

bubble remains 2D and steady [102]. However the flow will become three-dimensional and

unsteady once smaller particles are introduced into the flow as it will be shown in the following

results section.

The goals for this work is twofold. First, it is to investigate the impact of dilute suspension of

solid particles on the post-shock properties of the oblique shock wave (OSW). The second

objective would be to observe the effects of the particles on the modulation of the size of the

separation bubble induced by SWBLI. We are particularly interested in observing solely the

momentum transfer between the solid and gas phase to investigate its effects on the suppression

of the separation bubble. This is because similarly in the flow control devices mentioned

49

previously, only the introduction of additional momentum into the upstream boundary layer is the

main mechanism behind the control.

Furthermore, in order to streamline discussion, the assumption of the energy transfer between

the solid and gas phase is neglected is invoked and is discussed in the subsection below while flow

conditions are selected such that the momentum-transfer mechanism between the particles and gas

flow is dominant (especially near the OSW). Furthermore, if the heat transfer between the two

phases is ignored, the only source term left in the energy equation is the work done by particles’

drag. This term, which can be shown through order-of-magnitude analysis, can be ignored for this

analysis as it is four orders magnitude lower than the temporal and convective terms of the energy

transport equation.

The most important key point regarding this numerical study is that it is not a Direct Numerical

Simulation due to the fact that global instabilities convection is not taken into account. Due to the

huge computational resources that involves particle tracking, the number of parcels being tracked

is only limited to ~200,000. This is still significantly much larger than the number of particles

being tracked in most literature however this is to remind that parcels/ groups of computational

particles are being used to represent the particles. There will be inevitable errors associated with

such approach.

50

Chapter Five: PROBLEM DESCRIPTION & NUMERICAL METHODS

In order to generate the oblique shock wave / laminar boundary layer interaction, the

incident shock wave is generated by an adiabatic, inviscid shock generator. This shock wave will

impinge on the developing laminar boundary layer and a separation bubble will form due to the

adverse pressure gradient imposed on the boundary layer. The freestream inflow Mach number, M

stagnation pressure, 𝑃𝑂 and stagnation temperature, 𝑇𝑂 are, 2.15, 10916 Pa and 300°K,

respectively. The Reynolds number, Rex,sh based on the distance Xsh between the leading edge

and the inviscid shock impingement location is 1x105. The shock wave angle and shock generator

half angle imposed are 33.18° and 6.5° with respect to the horizontal, respectively.

Fig. 9 Schematics of Computational Domain

The particles will be seeded along with the incoming flow at sufficient mass loadings where

the interactions between the gas and solid phases need to be accounted for. This flow regime is

known as dilute flow with two-way coupling and the Eulerian - Lagrangian approach is selected

to model these interactions. The gas phase will be treated as a continuum while the dispersed

(solid) phase is treated as a set of computational particle groups, known as parcels. The Eulerian

51

(gas phase) is solved through OpenFOAM v.2.3.0 high speed flow solver known as

rhoCentralFoam and its implementation in OpenFOAM framework is discussed in detail in [77].

The algorithm uses a density-based finite volume method where the inviscid flux vectors in the

conservation equations were calculated with the 2nd order central-upwind scheme of Kurganov,

Noelle and Petrova [96]. The temporal discretization scheme is backward differencing. The

Eulerian part of the problem described is modeled by the three-dimensional conservation equations

of mass (continuity), momentum (“fully compressible” Navier Stokes), and total energy. These

equations are closed by the thermally and calorically perfect equation of state for the gases (ideal

gas law). Sutherland’s viscosity model as well as JANAF based polynomial fit for specific heat

constant are used for the air [97]. The flow is assumed to be laminar. Based on the assumption that

the particles/parcels are adiabatic (where the justifications behind it shall be discussed in the

chapter’s last subsections), the heat transfer between the gas and solid phase is neglected. The drag

model, which serves as the momentum source term for the gas phase, chosen in the numerical

simulation is based on Tedeschi el al. [98] formulation., which has shown to be able to accurately

tracking particles in flow fields with shock structures.The Lagrangian Particle Tracking algorithm

which is used to model and track the particles is described in the previous sections. There are

multiple multiphase solvers in OpenFOAM framework but none of them are designed specifically

to handle particles in supersonic flow. As OpenFOAM is nothing more than a collection of C++

libraries and design to be modular as in portions of code can be easily adapted together as the user

see fit, thus, a new solver was created by combining rhoCentralFoam with the LPT library and it

is named as rhoLPTFoam.

52

Chapter Six: RESULTS AND DISCUSSION

6.1 Verification & Validation

In the field of Computational Fluid Dynamics, it is proper to conduct some verification and

validation tests. Verification, whether it is the verification of a calculation (error estimation) or a

code (error evaluation), typically involves grid independency/sensitivity tests [99]. The grid cells

will be increased successively until the numerical predictions offer no discernable changes or

changes of a very small magnitude. Validation, on the other hand, is to determine if the correct

partial differential equations are solved. Typically, this involves the numerical solutions to be

compared with available experimental results and any difference should be analyzed to decide if

the errors are due to missing equations which are necessary to describe the entire physics of the

problem or uncertainties associated to models (e.g. turbulence models).

Performing grid sensitivity tests on multiphase flow is a slightly more complicated issue

than it is in a standard single phase flow simulations. This is because the computational truncation

errors associated with Lagrangian-Eulerian based simulations do not follow the common CFD rule

of grid independence. They do not decrease when the number of computational grid points is

increased [100]. The discretization/truncation error is minimized at a specific grid size, due to the

coupling of the Eulerian cells and Lagrangian parcels. Unfortunately this minimum is determined

on a case-to-case basis [101]. Hence, in an ideal case scenario, the numerical predictions would

have to rely solely on available experimental results as part of the validation process. However, a

grid sensitivity test has to be carried out to determine that the grid provides a grid independent

result when the flow is unladen or when the particles are not present. The results of that study are

53

shown in Fig. 10 and Fig. 11 Grid Sensitivity Test for Particle Laden SWBLIwhere the static

pressure on the wall are plotted against one another for the unladen and particle-laden case,

respectively. It can be seen that when the grid is of “medium” level, any more grid refinement

doesn’t yield anymore changes in the numerical prediction. Thus, based solely on this results, it is

concluded that the grid is or close to grid independent. The details of the grid is summarized in the

table below.

Fig. 10 Grid Sensitivity Test for Pure Gas SWBLI

Fig. 11 Grid Sensitivity Test for Particle Laden SWBLI

54

Grid ∆𝒙+ ∆𝒚+ (Free-Stream) ∆𝒚+ (Near Wall) ∆𝒛+

Coarse 38 76 4.8 40

Medium 13 56 2.4 25

Fine - X 9.5 38 2.4 25

Fine - Y 13 38 1.2 25

Table 1 Grid Spacing Values of Several Levels of Mesh Refinement

It was found that beyond the medium mesh grid specification, further grid refinement in

the x-direction does not yield any change to the numerical prediction. Thus, the grid spacing for

x-axis is kept at 0.5mm. However, when the grid resolution is increased by nearly twofold in the

y-direction, the numerical prediction alters only slightly (refer to Fig. 10 and Fig. 11) and the grid

resolution is similar to the one used in [103].Further discussion about the solutions obtained both

for the pure gas and particle laden cases will be discussed in depth in the results section. It should

be noted that the study does not attempt to conduct a full DNS as the convection of global

instabilities in the flow is not taken into account.

To evaluate the capability of OpenFOAM’s rhoCentralFoam solver, a numerical case

study is conducted where the numerical prediction is compared with available experimental data

of a fully laminar SWBLI [102] and Fig. 12 shows the results of the validation test. Fig. 12 also

shows that rhoCentralFoam predicted the same numerical solution as ANSYS Fluent and both

codes were capable of capturing the physics of a laminar SWBLI whose results fall well within

the uncertainty of the study which is 2%. For the simulation conducted through ANSYS Fluent

v.14.0, Roe flux differencing scheme was used to evaluate the inviscid numerical fluxes and the

55

2nd order upwind where employed to calculate the convective terms and least squared method was

used for cell-to-cell interpolation.

Fig. 12 Validation Test with 2D, Steady Laminar SWBLI Experiment [102]

Fig. 13 Velocity magnitude contours of 2D (top) and 3D (bottom) case of SWBLI

[98]

56

The base flow for this study is partly based on the numerical investigations conducted by

[103]. In this study, we wish to induce a separation bubble with a significant global size in order

to examine the effects of particles on it. The study shows that at M = 2.15 and 𝑅𝑒𝑥 = 1 x105, the

separation bubble will exhibit unsteady characteristics when the shock generator deflection angle,

θ is further increased, which increases the shock angle, β. To summarize, with the aforementioned

flow inlet condition, the separation bubble induced will become unsteady when the shock angle is

increased ≥ 32°. However, we have examined and found that when the recirculation bubble is

steady and two-dimensional, i.e. β ≤ 32°, it is very small and it would be difficult to examine the

particles effects on it. Thus, we chose the deflection angle of θ = 6.5° and β = 33.18° but in the

case of particle-free simulation, the flow remains 2D and steady for the duration of analysis [103].

Fig. 14 Comparison between 2D and 3D case of M=2.15, 𝜷 = 33.18°

As for the 3D, unsteady characteristics which will appear as the SWBLI is allowed to

progress and described in the previous section, [103] only shows the numerical results of the case

when the lateral/spanwise dimension of the plate is 0.8 and the shock angle is 32°. With the “fine”

grid level system, it was found that it is capable of capturing such instabilities and 3D

57

characteristics although the previous study employs a higher order (3rd) numerical scheme as well

as a slightly finer grid.

Fig. 15 Static Pressure contours of 2D (top) and 3D (bottom) case of SWBLI

58

Fig. 16 Streamlines of a Pure Gas SWBLI (M=2.15, β=33.18°) taken at 𝒁

𝑳𝒛= 𝟎. 𝟓

Fig. 17 Near Wall Velocity Streamlines. (left column) M=2.15, 𝜷 = 33.18°, 𝑳𝒛=0.5, (right

column) (taken from [103]) M=2.15, 𝜷 = 32°, 𝑳𝒛= 0.8; (top) t= 10ms, (middle) t = 25ms,

(bottom) t = 35ms

Now, Boin et al. (1996) also shows that when the shock angle is significant and the bubble

induced is unsteady, the numerical prediction will result in unphysical results in terms of creation

of false secondary and even tertiary vortices in the flow separation region. However, Boin et al.

(1996) did not present the static wall pressure when such situations occur. Fig. 13 shows

59

qualitatively that there is only no observable difference between the two-dimensional and three-

dimensional case in terms of the velocity magnitude contours. However, when the static wall

pressure and the static pressure contours were plotted as shown in Fig. 14 and Fig. 15, it is clear

that the two-dimensional simulation predicted a lower static wall pressure. With such difference,

it is interesting to note that both 2D and 3D simulations predicted nearly similar separation and

reattachment points or rather a similar separation length only when the separation bubble remains

two-dimensional and steady.

In Fig. 17, the evolution of the flow through the near wall streamlines are shown. The most

interesting feature by comparing the two sets of flow evolution is that despite the different

spanwise length, and shock angle thus different jump conditions (𝑀2 = 1.91 vs. 𝑀2 = 1.957),

the streamlines in the separation bubble evolved in a similar manner in the beginning (in the first

two instances). It is important to first state that in this work, the spanwise dimension is 62.5% of

the spanwise dimension of Boin et. al. (2006) study and symmetry boundary condition is imposed

on both of the lateral boundaries. Thus, when the first bifurcation occurred as shown in Fig. 17

(middle) at t = 25ms, the flow pattern of the larger spanwise flat plate is just a reflection at the axis

of flow pattern of the smaller spanwise flat plate. When the flow is allowed to evolve further, there

are some discrepancies between the left and right sets of case but the general features such as the

deflection of the streamlines are still quite similar.

This is curious because as it was mentioned before, despite the different post shock flow

conditions, the flow field in the separation bubble behaves and evolves in quite a similar manner.

There is a huge possibility that such behavior involves Hopf bifurcation and it would be an exciting

60

and potentially insightful study to pursue to understand how a laminar, steady and 2D separation

bubble will undergo some bifurcation that eventually leads to breakdown and turbulent separation

bubble. Unfortunately, this is well beyond the scope of this study.

Away from the wall, in the freestream, the particles are also expected to have an effect on

the oblique shock wave. To ensure that the drag law was properly implemented into OpenFOAM,

a simple verification and validation test was performed (Fig. 18). The trajectory of a single particle

through the oblique shock wave in the freestream was compared from the numerical simulations,

from a 1D analytical solution implemented in MATLAB, and from Tedeschi’s experimental

results. Good agreement indicates that the drag model was implemented correctly and is accurately

predicting particle lag at these conditions.

Fig. 18 Verification of the Drag Law [98] Implementation in OpenFOAM

61

Fig. 19 Static Pressure of (left) wall near SWBLI region and (right) Post Oblique Shock

OpenFOAM possesses several non-linear flux limiters which are made available to the

solver rhoCentralFoam. Flux limiters are in essence, numerical techniques employed to obtain high

resolution, second order, oscillation free or more readily known as Total Variation Diminishing

(TVD) solutions through the addition of a limited anti-diffusive flux to a first order scheme [104].

An exploratory study is conducted in order to evaluate the effects of flux limiters on the

rhoCentralFoam solutions. Of all of the available flux limiter schemes, four were only selected to

be assessed; Gamma [75], Van Leer [105], Superbee [106], & Van Albada [107].

Van Leer scheme is the default selection in OpenFOAM. However, as it is shown in figures

below, Van Leer & Superbee introduce spurious/artificial oscillations across the shock wave but

predict a smooth solution for the wall static pressure. This defeats the very purpose of employing

a TVD scheme in high speed flow simulations. Gamma differencing, in another hand, provides

different solutions when its parameter is tweaked. Gamma differencing scheme allows input to be

set from value of 0-1, based on the need of convergency or accuracy. In other numerical tests, it

was found that Gamma differencing at 1, introduces too much dissipation while Gamma

differencing at 0.5, again, generates oscillatory solutions, in this instance, on the wall. Hence, it is

62

Van Albada that predicts smooth and physical solutions on both wall and free stream. The Van

Albada limiter was used for all of the simulations.

6.2 Effects of Particles on the Free Stream & Shock Attenuation

The ability of particles to modulate the flow with shock wave is well-known as there have

been many previous studies where a normal shock traversed into a dust/particles - gas mixture. In

these studies, the travelling shock wave decays as the energy of the gas is expended to accelerate

the particle cloud. The higher the mass loading, the faster it decays and the lower its shock velocity

at equilibrium. However, when the particles traverse across an oblique shock wave (OSW), as in

our numerical simulations, an opposite phenomenon occurs.

When the particles are of 16μm diameter, regardless of the mass loading, there are negligible

effects on the post shock conditions of the gas. However, when the incoming flow was seeded with

1.6μm diameter particles, the shock jump conditions and distribution of properties through the

shock (y/H=0.5) alter (see Fig. 20). The Mach number and static pressure ratios decrease as the

mass loading increases or when the size of particles decreases. For a description of the cases, see

Table 2. It can be said that particles attenuate the oblique shock wave, which is confirmed by the

decrease in shock wave angle. This is due to the fact that, unlike the situation when the gas loses

energy as a shock wave travels into a dusty gas mixture, the particles undergo momentum exchange

and provides additional momentum to the flow behind the OSW. This causes the OSW angle to

change as well to satisfy the new post shock wave condition.

63

Fig. 20 Effects of Particles' Sizes on Free Stream Post Shock Condition (left) Static

Pressure (right) Mach number

Case Particle

Diameter

Mass Loading (φ) Flow Deflection (𝜽°) Effective 𝑴𝟐

1-1 16μm 0.1 6.46 1.91

1-2 16μm 0.2 6.41 1.912

2-1 1.6μm 0.1 5.55 1.944

2-2 1.6μm 0.2 5.02 1.963

3-1 880nm 0.1 5.38 1.95

Table 2 Immediate Post Shock Flow Deflection due to Particles

6.3 Effects of Particles on the Separation Bubble

Fig. 21 Effects of Particles on the Separation Bubble

64

As it will be shown later that the flow near the wall is 3D and unsteady when particles are

introduced, it is necessary to show that the sampling duration and length taken was sufficient to

present the average values of the flow. Fig. 21 shows that when the sampling were taken beyond t

= 7ms at the sampling time step of 25μs, they provide the same average values, regardless of the

duration of the sampling was taken. The sampling duration from t = 7ms to 10ms corresponds to

8 flow residence time and the sampling duration from t=10ms to 20ms corresponds to 27 flow

residence time.

Given that the particle diameter and mass loading are important parameters governing the OSW

interaction in the free-stream, a similar analysis was performed on the SWBLI. Fig. 22 shows how

these parameters affect the global separation bubble at the wall at the SWBLI. Velocity magnitude

contours (plane at z/Lz = 0.5) are shown for various particle diameter and mass loading conditions.

The separation bubble is almost completely suppressed when the mass loading is moderate and

when the particle diameter is small (φ≥0.1; 𝑑𝑝≤0.16μm). Similarly, separation is almost

completely suppressed when the mass loading is higher and when the particle size is moderate

(φ≥0.2; 𝑑𝑝≤1.6μm). For reference, the maximum height of the separated region in the gas-only

simulation is approximately 10 mm. Fig. 22 illustrates the suppression of the bubble more clearly

through the gas pressure profile on the wall. Due to the fact that in these inlet flow conditions, i.e.

Mach number and shock angle, the incident shock wave induced boundary layer separation and

such flow is known as strongly interacting flows and the shock reflection is considered to be strong

viscous-inviscid interaction [7]. In these flows, just upstream of the separation bubble, the wall

pressure first undergoes a steep pressure rise and then plateaus which characterizes the separated

65

region. Then a second steep pressure follows and this marks the reattachment of the separated

boundary layer. This is clearly shown in the pure gas wall pressure plot in Fig. 22 below.

Fig. 22 Velocity Magnitude Contours (in descending order) (a) φ=0.1,𝒅𝒑=16μm, (b)

φ=0.2,𝒅𝒑=1.6μm, (c) φ=0.1,𝒅𝒑=1.6μm, (d) φ=0.2,𝒅𝒑=1.6μm, (e) ) φ=0.1,𝒅𝒑=880nm

66

Fig. 23 Effects of Particles’ Sizes on Time Averaged Wall Static Pressure

It can be observed that when either the particle size is decreased (increase in rate of momentum

transfer) or the mass loading is increased (increase in magnitude of momentum transfer), the length

of plateau (separated region) decreases, which indicates that the length of the separation bubble is

reduced. Table 2 lists the post shock Mach numbers, 𝑀2 which increase due to the presence of the

particles. It is clear that in the region of relatively very low velocity, the high velocity particles

impart their momentum to the recirculation flow. Now the mechanism of flow separation

suppression through the addition of momentum is unclear at best in this region. This is because in

67

the upstream boundary layer, again due to velocity and hence, momentum disparity, the particles

transfer their momentum into the boundary layer. This increases the momentum of the boundary

layer and thus allow it to resist the flow separation more effectively.

In the extreme case (1.6μm, φ=0.2), the plateau nearly vanishes and the upstream wall pressure

steadily rises to the level of the downstream pressure, indicating that the shock reflection has

become one without separation (weak interaction). The boundary layer and the subsonic region

near the point of shock impingement thicken but no flow separation occurs. This opens up the

potential possibility of using particle injection as a control mechanism to alter or prevent flow

separation at locations of shock reflection. In supersonic/hypersonic flight system, this would be a

further enhancement to its performance as the particles can be used not only as a mean of flow

control but the particles prior to entering the combustor section can be ignited provided the

temperature is sufficiently high for autoignition. The application of solid particles to enhance the

supersonic combustion has been explored in SCHRAMJET (SHock-induced Combustion

RAMJET) [108].

Interestingly, the addition of particles does not seem to significantly affect the averaged

gas phase properties upstream and downstream of the separation bubble. With particles at the

highest mass loading (20%), the post shock gas static pressure only deviates from the gas-only

predictions by ~ 10%, while the post shock gas Mach number only changes by ~ 5%. The presence

of particles only affect the separation bubble significantly and the momentum transfer between the

two phases is only important at the interaction location.

68

The interaction between the shock wave and each particle can be characterized by the study of

Sun et al. [109]. When heat transfer and viscous effects are neglected, shock-particle interaction

usually involves a combination of shock reflection, diffraction and focusing. A regular shock

reflection is formed when the particle crossed the shock wave. As the particle traverse the across

the shock wave, a Mach reflection will form on the surface of the particle. Then as the foot of the

shock start to reach the symmetric axis, the shock “focuses” and a region of high pressure is

generated which will expand and then the pressure decreases. Although the particles in the dilute

regime are far apart, but it is possible that the reflections of the shock waves from each particle

could still interact and these multiple shock interactions could result in pressure drop seen just

downstream of the OSW with the unsteady wakes generated from each particle reduce the pressure

even further.

69

6.4 Effects of Particles on the Flow Instabilities

The suppression of the flow separation cannot be only attributed to the increase of the post

shock Mach number and change in jump conditions, which in essence imposes a weaker adverse

pressure gradient on the wall and thus induced a smaller separation. In addition to facilitating

momentum transfer, the particles also provide individual sources for perturbations to the flow.

Studies have shown that the presence of particles can increase or decrease the growth rate of

instabilities of the flow [57-69]. Depending on the particle’s Stokes number, St, the particles can

either stabilize or destabilize the flow. Although these studies are limited to incompressible flow,

in the region of the separated bubble in the current work, there is a relatively low velocity, i.e.

Mach No. <0.3 and the flow can be treated as incompressible. With this assumption, several

observations made in the literature can be applied to the current results. The Stokes number of the

particles in the current work, with the fluid time scale, 𝜏𝑓 based on the separation bubble length

and recirculation region mean velocity, 𝑈𝑠𝑒𝑝 is St ~ 𝒪(10−2). Saffman (1962) made important

observations regarding flows with addition of such fine particles. The particles increase the

effective inertia i.e., the density of the mixture and this effectively leads to an increase of the flow

Reynolds number by a factor of (1 + φ). Applied to the current work, the addition of particles

destabilizes the flow, which is clearly depicted in Fig. 21. Most of the previous work [62,63,65]

are limited only to the study of particles with Stokes numbers of St = 0.01, 0.1, 1 and 10, but it can

be inferred that the particles with decreasing Stokes number further destabilize the flow.

In the gas-only flow (Fig. 24 a), the separation bubble is two-dimensional and steady within

the time frame of the simulation. The flow field in the separation bubble exhibits unsteady

70

characteristics when particles are introduced due the reasons discussed above. When larger

particles (𝑑𝑝=16μm) are suspended in the flow, the separation bubble remains quasi-steady and

streamlines near the wall closely align with the outer free stream except near the reattachment

location.

As the particles decrease in diameter, the entire separation bubble breaks down into a three-

dimensional flow. For the case of 𝑑𝑝=1.6μm, the separation bubble decomposes into several

recirculation regions or vortices as it can be seen in Fig. 26, the appearance of which is unsteady.

Fig. 27 further illustrates how these vortices are three-dimensional at this condition and do not

span though the entire lateral (x-z) plane. For very fine particle suspensions (𝑑𝑝=88nm), the

recirculation region also contains several vortices (Fig. 28). The newly formed recirculation

regions dynamically interact with each other and no longer separate from one other as in the

previous larger-particle cases by fixed locations. In this case, the separated bubble consists of

several vortices that consistently merge with each other and then break up into smaller ones. All

of these vortices also propagate freely within the separated bubble. This is quite dissimilar with

the case of 𝑑𝑝=1.6μm particles. Although there are several vortices inside the separated bubble,

but there is consistently a single vortice that is larger than the other at a somewhat fixed location

just beneath the shock impingement point. This larger vortex changes in size and shape in time but

it remains the largest among other vortices formed.

The simulation results showed that the particles, regardless of size, were able to penetrate

the separation bubble from the boundary layer and from the freestream. The extent of momentum

71

transfer in the separation bubble was dictated by the ratio of the particle relaxation time to the

characteristic time of the separated flow, 𝜏𝑝/𝜏𝑔 (see Table 3). For larger particles (𝜏𝑝/𝜏𝑔 > 1), the

rate of momentum transfer to the gas phase is not sufficient to bring the particles and the gas into

equilibrium inside the separated region. However, as the particle diameter decreases (𝜏𝑝/𝜏𝑔 < 1),

there is sufficient time for the particles to completely transfer their momentum surplus to the gas

flow and bring the particles and gas into equilibrium. As expected, as 𝜏𝑝/𝜏𝑔 decreases, the average

gas velocity in the separated region increases.

Particle Diameter, 𝑑𝑝 𝜏𝑝/𝜏𝑔

16μm 813μs

1.6μm 8.13μs

880nm 2.1μs

Table 3 Particle Relaxation times for different particle sizes.

The simulation results showed that the particles, regardless of size, were able to penetrate

the separation bubble from the boundary layer and from the freestream. The extent of momentum

transfer in the separation bubble was dictated by the ratio of the particle relaxation time to the

characteristic time of the separated flow, 𝜏𝑝/𝜏𝑔 (see Table 3). For larger particles (𝜏𝑝/𝜏𝑔 > 1),

the rate of momentum transfer to the gas phase is not sufficient to bring the particles and the gas

into equilibrium inside the separated region. However, as the particle diameter decreases (𝜏𝑝/𝜏𝑔

< 1), there is sufficient time for the particles to completely transfer their momentum surplus to

the gas flow and bring the particles and gas into equilibrium. As expected, as 𝜏𝑝/𝜏𝑔 decreases,

the average gas velocity in the separated region increases.

72

Fig. 24 Effects of Particles’ Sizes on Near Wall |U| Streamlines (a) Pure Gas flow, (b)

𝒅𝒑=16μm, φ=0.1, (c) 𝒅𝒑=1.6μm, φ=0.1 , (d) 𝒅𝒑=0.88μm, φ=0.1 (arrow denotes separation

bubble)

73

Fig. 25 Effects of Particles 𝒅𝒑=16μm, φ=0.1 on Separation Bubble (Streamlines) at various

time, t = 9.25ms, 9.5ms, 9.75ms & 10ms.

74

Fig. 26 Effects of Particles 𝒅𝒑=1.6μm, φ=0.1 on Separation Bubble (Streamlines) at various

time, t = 9.25ms, 9.5ms, 9.75ms & 10ms.

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Fig. 27 Streamlines of Separation Bubble of particles of 𝒅𝒑=16μm, φ=0.1, at t = 10ms

In the study conducted by Rist et al. [110], the authors have demonstrated that instability

waves which are introduced periodically upstream of the separation region can be used as means

to control incompressible, laminar separation bubbles. The amplitude and the frequency of the

upstream disturbances has its own distinct effects on the separation bubble. The basic principle

behind the control process is to incite a faster laminar to turbulence transition (LTT) event which

will effectively causes the laminar separation bubble to undergo a quicker turbulent reattachment.

Analogous to this current study, the particles, depending on its size and mass loading, can be

viewed as upstream disturbances instead of just carriers of additional momentum for the gas phase

to absorb. Larger particles can be viewed as disturbances of large amplitudes while small particles

as high frequency disturbances. However, compared to these perturbation waves, particles have

the capability of re-distributing the momentum of the entire system. This is more advantageous

because any beneficial effects that it would introduce into the flow system would not be just

confined to the near wall region. Furthermore, once these instability waves trigger LTT, the

downstream boundary layer would be turbulent and the waves themselves might even amplify the

turbulence while small particles are known to attenuate turbulence. However, this aspect of the

flow is beyond the scope of this study and shall not be discussed any further.

76

Fig. 28 Effects of Particles of 𝒅𝒑=0.88μm, φ=0.1 on Separation Bubble (Streamlines) at

various time, t = 9.25ms, 9.5ms, 9.75ms & 10ms.

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Fig. 29 Effects of Particles’ Sizes on upstream Boundary Layer Profiles close to Leading

Edge of the Flat Plate

It seems that the upstream boundary layer did not show any sign of being perturbed. In

fact, by taking the velocity profile of the boundary layer upstream of the separation bubble, close

to the leading edge of the flat plate shown in Fig. 29, there was no oscillation or perturbation in

the boundary layer profile but there were slight increase of the momentum thickness of the

boundary layer. This can also be explained by the observations made by Thevand et al. [69]. The

compressible boundary layer experienced a very strong compressibility effects unlike the flow in

the separated region. The compressibility effect actually has a stabilizing effect on the growth of

disturbance of the particle laden flow regardless of the Stokes number of the particles and for this

case, the upstream boundary layer.

Fig. 31 shows the trajectories of the particles at various particle diameters at an instantaneous

time of t = 10ms. It was found that larger particles enter and exit the separation bubble without

transferring much momentum. This can be seen from the relatively high velocity of the 16 μm

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particles compared to the other two cases of smaller particles. The particles of smaller size

appeared to “conform” more to the general shape of the separation bubble (lower Stokes number)

and they imparted most of their momentum to the lower velocity gas phase. In these two smaller

particle cases, there seems to be a region (the top/roof of the separation bubble) where the particles

had a significantly higher velocity than the rest of the separation bubble. This is likely due to

expansion fans near that region where the gas was expanding and accelerating, and momentum

transfer back to the particles occurred (i.e. particles accelerated). It was found that the finest

particles (𝑑𝑝=88nm) behaved nearly as true tracer particles and because of this, there is a small

region void of particles at the center of the separation bubble.

Fig. 30 Evolution of the Separated Boundary Layer. (black) Pure gas (red) 𝐝𝐩=16μm,

(blue) 𝐝𝐩=1.6μm (green) 𝐝𝐩=0.88μm

79

Fig. 31 Particle Trajectories for (a) 𝒅𝒑=16μm, φ=0.1, (b) 𝒅𝒑=1.6μm, φ=0.1, (c) 𝒅𝒑=0.88μm,

φ=0.1

80

In the efforts of attempting to understand more of the instabilities that are observed in the

recirculation region, a fast Fourier transform (FFT) was performed on the temporal pressure

evolution to ascertain the dominant frequency(s) at certain locations of the near wall region. In the

upstream boundary layer (Fig.32 a), the base frequency is ~ 100 Hz and the secondary frequencies

are the harmonics of this fundamental frequency. Just downstream of the point of separation

(Fig.32 b), the fundamental frequency is increased to ~150 Hz and the secondary frequencies at

this point correspond to the harmonics of the base frequency.

81

Fig. 32 Pressure spectra density for several points in the flow (x,y,z) a) (0.625,0,0.5), b)

(0.875,0,0.5), c) (1,0,0.5) d) (1,0.0125,0.5), and e) (1.125,0,0.5)

82

Fig. 33 Vorticity magnitude contours of separation bubble for case 𝒅𝒑 = 𝟏. 𝟔𝝁𝒎 at a)∆𝒕𝟏=

0s, b) )∆𝒕𝟏= 25𝝁𝒔, c) ∆𝒕𝟐= 50𝝁𝒔, d) ∆𝒕𝟑= 75𝝁𝒔, 𝒆) ∆𝒕𝟒= 100𝝁𝒔

Fig. 34 Streamlines of separation bubble for case 𝒅𝒑 = 𝟏. 𝟔𝝁𝒎 at a)∆𝒕𝟏= 0s, b) )∆𝒕𝟏= 1𝒎𝒔,

c) ∆𝒕𝟐= 2m𝒔, d) ∆𝒕𝟑= 3m𝒔, 𝒆) ∆𝒕𝟒= 4m𝒔

83

In Fig. 32 b, c & d, there is a distinct frequency of 30 kHz. This frequency is only observed

in locations at and downstream of the shock impingement location. In order to discover the source

of this frequency, snapshots of the vorticity contours are taken at 25μs (which corresponds to 40

kHz). It can be seen from Fig. 33 that the source of this frequency is the vortex shedding that is

occurring at the tip of the “roof” of the boundary layer encasing the separation bubble. From these

snapshots, it can be seen that the vortices being shed do no break down into smaller vortices but

instead a pair of them will merge and form an elongated vortex that travels downstream. Aside

from the 30 kHz frequency, there are other dominant frequencies that is below 10 kHz range.

Again, in order to determine the flow structure that is associated with this frequency, snapshots of

the streamlines at intervals of 25μs are performed as shown in Fig. 34. It can be seen that in each

time interval of such size, the 10 kHz is associated with the behavior of the largest vortex in the

separation bubble. At each 1ms, the largest vortex deforms, elongates and shifts downstream and

then reshapes itself as it moves back towards its relative original location. It can be deduced from

Fig. 34 b that after each time the large vortex elongates, it sheds off a smaller vortex before

reshaping itself.

In summary, the behavior of such three-dimensional and unsteady recirculation region

agrees with the conjecture formulated by Theofilis et al. [111] which proposes that before the flow

and separation bubble becomes unsteady and three-dimensional and the occurrence of vortex-

shedding, “multiple recirculation regions will occur inside the primary bubble which then will

lead to a global change of the flow structure with multiple structurally unstable saddle-to-saddle

connections”. Although this conjecture only has intentions to be applied to and to explain the

origin of unsteadiness of incompressible laminar separation bubble, this conjecture can be

84

extended to this study especially considering that in the recirculation zone, the velocity of the gas

is very low i.e. M < 0.1 and thus can be treated as incompressible.

Fig. 35 Conjecture for topolofical changes of an incompressible separation bubble’s

structure associated with the onset of vortex shedding [111].

6.5 Effects of Particles on the Gas Phase Temperature and Heat Transfer

The role that small sized particles play not only in turbulence modulation but in nanofluid

heat transfer enhancement has the scientific community delve vigorously into understanding the

mechanism behind it. While there seems to be very little disagreement that the thermal

conductivity of nanofluids is higher than just pure fluid alone, a schism is formed when it comes

to the subject of mechanisms that enhance it. One group believe that it is the clustering or

aggregation of nanoparticles and such, thermal conductivity enhancement agrees with the theories

of effective medium based on conduction for well-dispersed mixtures of spherical nanoparticles.

They postulate that through aggregation, these nanoparticles form high aspect ratio particles or

chains of highly conductive particles. The opposing group identifies the Brownian motion of

nanoparticles as the origin of the observed anomalous thermal conductivity enhancement. They

85

postulate that it is the micro-convection of the interfacial interaction energy that contribute to this

enhancement.

It all started with the experiments conducted by Lee et al. [112] where they show that the

thermal conductivity behavior of dilute nanofluids is increased with just a small amount of

nanoparticles and the thermal conductivity ratios increase almost linearly with volume fraction but

at a different rate for different mixtures. Then recently, a study has been done by Seok et al. [113]

to investigate the role of Brownian motion in enhancing thermal conductivity through a theoretical

model and found that it is a key nanoscale mechanism that governs the thermal behavior and

fluctuations of measured macroscale and molecular/nanoscale phenomena. These findings are

further supported with the results obtained by Gupta et al. [114] where through Brownian

dynamics simulations, it is shown that Brownian motion can increase the thermal conductivity of

the nanofluid by 6% primarily through “random walk” motion and not only through diffusion.

However, almost as soon as these previous results are presented, there are numerous rebuttals

to the findings which argue that the Brownian motion’s contributions to thermal conductivity

enhancement is either insignificant or downright provide no contribution at all. Evans et al. [115]

use a kinetic theory based analysis and molecular dynamics simulations of the heat transfer of fluid

suspended with solid nanoparticles and concludes that Brownian motion has a minor effect on the

thermal conductivity and their results agree with the experimental results obtained by Putnam et

al. [116]. They instead suggest that particle clustering would be a much more significant

contributor to such enhancement. Shukla et al. [117] derive a general expression for the effective

thermal conductivity of a colloidal suspension through ensemble averaging and show that the

86

microscopic model predicts that thermal conductivity enhancement is dependent on particle size

and temperature as well as highlights the significance of long range repulsive potentials. Finally,

Babaei et al. [118] use equilibrium molecular dynamics simulations to study the role that micro-

convection plays in increasing the thermal conductivity. They demonstrate that the individual

terms in the heat current autocorrelation function associated with nanoparticle diffusion cancel out

each other if average enthalpy expression are correctly defined and subtracted and thus, negating

the enhancements attributed to Brownian motion-induced micro-convection.

In this study, heat transfer effects are neglected and should be reminded again that it is not

the focus of this work to investigate heat transfer enhancement. In traditional CFD of multiphase

flows, the dispersed phase is extremely dependent on the models that are prescribed to them

because in essence, they are nothing more than computational parcels and not “real” particles.

Their behaviors, i.e. velocity, trajectory and temperature are governed by the drag, tracking

algorithm and heat transfer model respectively. When it comes to the heat transfer, more

specifically the Nusselt number correlation, most of previous studies favor the use of the

correlation formulated by Ranz et al. [119] where they investigate the factors that influence the

rate of evaporation of pure liquid drops and liquid drops that contain dissolved and suspended

solids. In hindsight, this is technically the earliest research that actually look into the effects of

suspended particles in fluids on heat transfer rates. In the author’s opinion, the correlation is

somewhat antiquated and there are other and more recent correlations that have been already

devised. Unfortunately, in OpenFOAM, only Ranz formulation is available. And this is due mostly

to the fact that the portion of the code adapted into performing these simulations were taken from

87

liquid jet modeling code, i.e., sprayFoam. Nonetheless, additional simulations analyses were

performed in order to assess the contribution of heat transfer of particles to the gas phase.

By conducting a scale or order of magnitude analysis, it is possible to approximate the

magnitude of individual terms in the equations. The two-dimensional (for simplicity) total energy

equation in conservative form is given below:

𝜕𝜌𝐸

𝜕𝑡+

𝜕𝑈(𝜌𝐸 + 𝑝)

𝜕𝑥+

𝜕𝑉(𝜌𝐸 + 𝑝)

𝜕𝑦= −𝑢𝑃𝐹𝑃𝑥 − 𝑣𝑃𝐹𝑃𝑦 − 𝑄𝑃

Where 𝑄𝑃 = 𝜇𝐶𝑝

𝑃𝑟𝑁𝑢𝑑𝑝(𝑇 − 𝑇𝑝)

For now, let’s assume the flow is steady and in the free stream, when the particles crosses

the oblique shock, it is assumed that the y-component terms are negligible and thus reduce the

equation to

𝜕𝑈(𝜌𝐸 + 𝑝)

𝜕𝑥= −𝑢𝑃𝐹𝑃𝑥 − 𝑄𝑃

Now we attempt to estimate the scale of each term by introducing the values of the relevant

physical properties: ρ = 0.0328 kg/𝑚3, p = 1600 Pa, U = 504 m/s, E = 256652 𝑚2/𝑠2 and let X be

the transition region of between the incident and reflected shock ~ 0.1m

𝜕𝑈(𝜌𝐸 + 𝑝)

𝜕𝑥=

𝑈(𝜌𝐸 + 𝑝)

𝑋=

𝒪(102)[𝒪(10−2)𝒪(105) + 𝒪(103)]

𝒪(10−1)

= 𝒪(106)

88

The work done by particle to fluid term for 𝑑𝑝 = 16𝜇𝑚, 1.6𝜇𝑚 𝑎𝑛𝑑 0.8𝜇𝑚 particles and

their particle numbers, 𝑁𝑝 are ~ 1x106, 2x109 and 2x1010 respectively:

∑ 𝑁𝑝1𝑢𝑃𝐹𝑃𝑥 = 𝒪(106)𝒪(102)𝒪(10−8) = 𝒪(10 )

∑ 𝑁𝑝2𝑢𝑃𝐹𝑃𝑥 = 𝒪(109)𝒪(102)𝒪(10−9) = 𝒪(102)

∑ 𝑁𝑝3𝑢𝑃𝐹𝑃𝑥 = 𝒪(1010)𝒪(102)𝒪(10−10) = 𝒪(102)

The simple order-of-magnitude analysis above shows that the contribution of the work done

by particles to the gas phase can be considered to be insignificant, i.e.; 𝒪(106) 𝑣𝑠 𝒪(102) as it is

four order of magnitude smaller than that total energy convective term in the free stream.

However, the situation is very different when it comes to the near wall and in the boundary

layer. When the particles encounter the leading edge shock and the initially developing boundary

layer, the particles play a significant role in altering its growth and shape as shown in Fig. 29. The

effects of particle-fluid heat transfer on the near wall fluid are shown in Fig. 36. The particles have

a more impact in terms of temperature and heat transfer in the boundary layer compared to the free

stream because when the boundary layer starts to grow, its growing thickness is very relative to

the particles’ size.

It is interesting to note that despite the particle mass loading or size, the gas near wall

temperature is raised at the same magnitude at the leading edge and continues to increase until

X/Xsh ~ 0.08. Beyond the point, the near wall gas temperature decreases. This is likely due to the

fact that at X/Xsh ~ 0.08, it is the approximate region where the separation bubble vortices are

located where the smaller particles act like tracer particles and follow the recirculating motion of

89

the separated region and gas now expends its energy in order to keep the particles entrained within

the recirculation region.

Fig. 36 Effects of Particles on the Wall Static Temperature

Fig. 37 Effects of Particles on Post Shock Gas Temperature (left) without and (right) with

Heat Transfer.

90

Fig. 37 shows the post shock gas temperature with and without heat transfer. The largest

particles have very negligible effect on the gas temperature compared to the smaller particles.

When the heat transfer model is switched on, there are changes to the gas temperature albeit very

small. For the particles of dp = 1.6μm, φ=0.1, when the heat transfer effects are accounted for, it

only yields a 1.36% change of gas temperature which is about ~ 2K°. When the mass loading is

increased to 0.2, the change is only ~ 0.9%. It is the nano sized particles that yield the most relative

change of temperature which is about 2.5%. In summary with the heat transfer model activated,

the difference of the temperature does not exceed 5K°.

91

Chapter Seven: CONCLUSIONS

Numerical simulations of nano-particle suspensions interacting with an oblique shock wave and

shock wave/ boundary layer interaction were performed. The momentum transfer mechanism

between the solid and gas phases was isolated for study by inhibiting heat transfer between the

phases. The rate of momentum transfer was controlled through the choice of particle diameter and

the overall magnitude of momentum transfer was controlled by the choice of particle mass loading.

In general, it was found that only combinations of large mass loadings (> 0.1) and small particle

sizes (< 16 μm) could have a significant effect on the flow. In the freestream, particles were shown

to decrease both the strength and angle of the oblique shock wave. At the wall, particles were

shown to decrease the size of, and in some cases completely suppress the formation of a separation

bubble induced by a shock wave boundary layer interaction. It was found that particles decrease

the shape factor of the boundary layer upstream of the interaction, introduce unsteadiness into the

separated flow, and rapidly transfer momentum to the gas at the interaction location. For the first

time, the concept of using small-scale particles nano-particle injection as a means of flow control

to suppress separation in an isolator of a hypersonic air-breathing propulsion system has been

proposed. The particle mass loading requirements can be alleviated by judiciously seeding

particles only into the boundary layer itself, upstream of where the flow separation occurs.

92

93

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