On the flow characteristics behind a backward-facing step and the ...

On the flow characteristics behind a backward-facing step and thedesign of a new axisymmetric model for their study

by

Jagannath Rajasekaran

A thesis submitted in conformity with the requirementsfor the degree of Master of Applied Science

Graduate Department of Aerospace EngineeringUniversity of Toronto

Copyright © 2011 by Jagannath Rajasekaran

Abstract

On the flow characteristics behind a backward-facing step and the design of a new

axisymmetric model for their study

Jagannath Rajasekaran

Master of Applied Science

Graduate Department of Aerospace Engineering

University of Toronto

2011

An extensive review was made to study the wake characteristics of a backward-facing step.

Experimental and numerical studies of the backward-facing step suggest that the wake of a

separated shear layer to be dependent on parameters such as: expansion ratio, aspect ratio, free

stream turbulence intensity, boundary layer state and thickness at separation. The individual

and combined effects of these parameters on the reattachment length are investigated and

discussed in detail in this thesis. A new scaling parameter, sum of step height and boundary

layer thickness at separation is proposed, which yields significant collapse of the available data.

Based on the literature review, an axisymmetric model is designed for further investigating the

dynamics of the flow independent of aforementioned parameters. Additionally, porous suction

strips are incorporated to study the step wake characteristics independent of Reynolds number.

This model has been built and will be tested extensively at UTIAS.

ii

Dedication

To my parents and sister, for their love and support.

iii

Acknowledgements

I have had help from many people during the course of my masters degree. First, I would like

to start by thanking my supervisor Dr. Philippe Lavoie, for his calm guidance, tremendous pa-

tience and constant encouragement in finishing my thesis. I owe him a debt of gratitude for the

financial support that I have had throughout the program and for being willing to countenance

a delay of nearly a year beyond my original deadline. I would also like to thank the Professors

from my research assessment committee, Dr. Zingg, Dr. Martins and Dr. Ekmekci, for their

invaluable feedback and suggestions for improvement.

I would also like to thank Ronald Hanson for his timely help on countless number of occa-

sions and for being a very good friend. If not for him, I might have had a hard time coping

with the education system in Canada. Special thanks to Arash Lahouti for his time to discuss

regarding the project and for proof reading my thesis. Also, many thanks to Heather Clark

for tolerating and helping with my questions regarding the project. I would also like to ex-

tend thanks to my colleagues Denis Palmeiro, Jason Hearst, Zhou Yuan, Mike Kociolek, Luke

Osmokrovic and Leticia Gimeno for taking interest in my topic and also for sharing knowledge

about theirs during the group meetings.

The research lab at University of Toronto has provided me with several opportunities to

involve both physically and mentally. This was possible because of the regular funding from

the Canadian government, the University and the private firms to our lab. I am very thankful

for that and would like to encourage them to keep funding our lab for many more years to come.

I also am thankful to all the University office staff for their help throughout the course of my

degree. Also special thanks to Mrs. Joan Dacosta and Mrs. Nora Burnett for their assistance

and sincerity.

Last but not the least I would like to thank my parents and my sister for their moral support

and inspiration throughout my degree.

iv

Contents

Abstract iii

Dedication iii

Acknowledgements iv

Contents vi

List of Tables vii

List of Figures ix

1 Introduction 1

1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2 Literature Review 5

2.1 Common Features of the Backward-Facing Step Flow . . . . . . . . . . . . . . . 5

2.2 Shear Layer Region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2.1 Vortex Rolling and Pairing Mechanism . . . . . . . . . . . . . . . . . . . . 7

2.2.2 Coherent Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2.3 Vortex Shedding Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.3 Reattachment Zone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.4 Recirculation Zone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.5 Important Flow Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.5.1 Effect of the Expansion Ratio . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.5.2 Effect of Aspect Ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.5.3 Effect of Free Stream Turbulence Intensity . . . . . . . . . . . . . . . . . . 20

2.5.4 Effect of Boundary Layer State at Separation . . . . . . . . . . . . . . . . 20

2.5.5 Effect of Boundary Layer Thickness at Separation . . . . . . . . . . . . . 21

2.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

v

3 Novel Scaling Analysis 25

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.2 New Normalization Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

4 Model Design 30

4.1 Model Design Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

4.2 Wind Tunnel Facility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

4.3 Model Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

4.4 Porous Suction Strip Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

4.4.1 Evolution of the Turbulent Boundary Layer Thickness . . . . . . . . . . . 33

4.4.2 Effects of Suction on Turbulent Boundary Layers . . . . . . . . . . . . . . 39

4.5 Suction System Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

4.6 Model Support . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

4.7 Nose and Trailing Cone Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

4.7.1 Axisymmetric Nose Design . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

4.7.2 Trailing Edge Cone Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

5 Conclusions and Recommendations 53

5.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

5.2 Future Experimentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

vi

List of Tables

3.1 Important parametric data acquired from different experiments on backward-

facing step flows. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

vii

List of Figures

1.1 Backward-facing step flow features. . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2.1 Flow characteristics behind a BFS. . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2 Initial instability of the shear layer, and the roll-up into discrete vortices. . . . . 7

2.3 Streamlines over a bluff plate with splitter plate configuration. . . . . . . . . . . 8

2.4 Acoustic forcing in backward-facing step flows. . . . . . . . . . . . . . . . . . . . . 11

2.5 Convective and absolute instabilities in the shear layer. . . . . . . . . . . . . . . . 13

2.6 Mean flow structure of one half of the secondary vortex. . . . . . . . . . . . . . . 16

2.7 Planar backward-facing step geometry. . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.8 Sketches of oil film pattern on the floor for the region behind the step. . . . . . 19

2.9 Axisymmetric backward-facing step geometry. . . . . . . . . . . . . . . . . . . . . 20

2.10 Effect of boundary layer state at separation. . . . . . . . . . . . . . . . . . . . . . 21

2.11 Effect of boundary layer state and thickness at separation at ReH = 26,000. . . 22

2.12 Effect of Reynolds number and boundary layer thickness at separation. . . . . . 23

3.1 Combined effect of parameters on the reattachment length. . . . . . . . . . . . . 26

3.2 Boundary layer thickness at separation vs. rescaled reattachment length. . . . . 27

4.1 Closed return tunnel schematic. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

4.2 Schematic of control volume used for momentum integral analysis. . . . . . . . . 34

4.3 Evolution of displacement thickness over a flat plate, with and without suction

at the wall.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4.4 Axisymmetric backward-facing step model. . . . . . . . . . . . . . . . . . . . . . . 42

4.5 Evolution of boundary layer thickness for different suction velocity. . . . . . . . 43

4.6 Plenum chamber design within the suction strip system. . . . . . . . . . . . . . . 44

4.7 Cross sectional view of a plenum chamber. . . . . . . . . . . . . . . . . . . . . . . 44

4.8 Experimental set up to measure pressure drop across a porous plate. . . . . . . . 45

4.9 Velocity profile of a fully developed turbulent pipe flow. . . . . . . . . . . . . . . 46

4.10 Schematic of the axisymmetric model with dimensions. . . . . . . . . . . . . . . . 48

viii

4.11 Pressure distribution over a cylinder and a sphere. . . . . . . . . . . . . . . . . . . 49

4.12 Coefficient of pressure versus the normalized streamwise distance for different

nose models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

4.13 Pressure gradient versus the normalized streamwise distance for different nose

models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

ix

Chapter 1

Introduction

1.1 Background

The discovery of boundary layer theory by Ludwig Prandtl in the early twentieth century was

the beginning to the extensive research on separated flows. Separated flows are common in sev-

eral engineering applications such as aircraft wings, turbine and compressor blades, diffusers,

buildings, suddenly expanding pipes, combustors, etc. The characteristics of a separated flow

has been studied for decades by experimentalists to understand the physics of the separated

shear layers and their instability mechanisms. The instabilities in the free shear layers are the

source to distinctly visible large coherent structures. The existence of coherent structures in

almost every turbulent flows has been well documented [1] and this makes it even more inter-

esting to study such separated shear flows.

Besides the academic interests, knowledge of separated flows can also be applied to many

practical applications. Two of their main applications include the automobile and aircraft in-

dustries, who are developing fuel efficient designs to reduce consumption of the rapidly-depleting

non-renewable resource and minimize green house gas emission. In an aerodynamic perspec-

tive, drag is considered as one of the major reason for inefficient fuel consumption. There are

several types of drag, but in this thesis, the focus will be on the pressure drag created by the

separated flows and the ways to minimize it. Studies by Roos and Keegelman [2] demonstrated

that by actively controlling the flow at separation, characteristics of the coherent structures

can be modified and consequently alter the drag characteristics. These aspects of the flow

make it important to understand the instabilities and characteristics of coherent structures for

controlling flow dynamics to achieve significant drag reduction or lift enhancement. Apart from

drag reduction, understanding the fluid-structure interactions of these separated shear layer

instabilities can be very useful in controlling the noise and vibration characteristics of such

1

Chapter 1. Introduction 2

flows [3].

The physics of separated flows, due to their instabilities, are very complex. In an attempt

to simplify these flow characteristics, researchers conducted experiments on various geometries,

which include rib, fence, bluff body with a splitter plate, suddenly expanding pipes, forward and

backward-facing steps, cavities, and bluff bodies with blunt leading edges. These geometries

simplify the flow characteristics to a certain extent by fixing the separation or the reattachment

point or both, which are otherwise unsteady. The backward-facing step is considered by most

as the ideal canonical separated flow geometry because of its single fixed separation point and

the wake dynamics unperturbed by the downstream disturbances. An illustration of the wake

characteristics behind a backward-facing step is shown in Figure 1.1.

Recirculation bubble!

Shear layer! vortices!

Coherent Structure!

Figure 1.1: Backward-facing step flow features.

In this thesis, an extensive review on the characteristics of the flow features behind the

backward-facing step are discussed. The wake of a backward-facing step has unique features

mainly in two regions: the free shear layer and the low velocity re-circulating bubble. Due to

instabilities, the vortices in the shear layer roll up and pair with the adjacent vortices to form

larger coherent structure [4]. These vortices entrain fluid from the region below and trigger

the recirculation. Due to the adverse pressure gradient in the wake of the step the free shear

layer reattaches at the bottom wall. A more detailed description of the separated shear layer

behind the backward-facing step and the instability mechanisms will be discussed in Section 2.2.

Besides the omnipresent nature of coherent structures in turbulent flows, recent studies

[2, 3, 5, 6] suggest that coherent structures play an important role in altering the flow dynam-

ics to achieve desirable characteristics such as reduced drag, subdued noise, enhanced mixing

and suppressed vibration. Several different passive and active control techniques have been


implemented to date mainly to alter the wake characteristics of the step. The active control

experiment on a backward-facing step by Roos and Kegelman [2], was one of the first experi-

ments to indicate that the coherent structures can be modified to achieve desired results. Since

then, in most of the recent studies the emphasis has been to determine the characteristics of

the coherent structures and explore the external means of modifying their dynamics.

There have been several experiments conducted over a backward-facing step geometry and

yet the results obtained from different experiments are not comparable due to their depen-

dency on flow and geometric parameters. Some of these significant parameters were noted in

the review paper by Eaton and Johnston [7]. These parameters include, the expansion ra-

tio, aspect ratio, free stream turbulence intensity, Reynolds number and the boundary layer

state and thickness at separation. The independent effects of each of these parameters on the

reattachment length were extensively studied by different experimentalists [7–11], which are

discussed in detail in Chapter 2. However, a distinct relationship for a combined effect of all

these parameters at various conditions has not been established yet. It is shown in this thesis

that a unique relationship for the combined parametric effect on reattachment length can be

achieved by rescaling the normalizing parameter for the reattachment length.

Owing to the parametric differences between each experiments the reattachment length at

step wake differs accordingly. The long term goals for our lab are to study and actively control

the wake dynamics behind the step. This requires a state of the art model that is independent

of aforementioned parametric effects. Hence, for the current study a unique model is designed

that can either minimize or control the effects of those parameters. The two key features of

this model are the axisymmetry and the porous suction strip design. Experiments on the ax-

isymmetric backward-facing step (the one where the flow is external) has been conducted by

very few researchers [12–14]. This provides an opportunity to investigate more about the effects

of axisymmetry on the backward-facing step flow. Moreover, the axisymmetric step is useful

in achieving a better case of two-dimensional wake behind the step, since the planar models

require large aspect ratios to achieve two-dimensional characteristics at the center. Porous

suction strips are incorporated in the current model to control the boundary layer thickness

at separation independent of the Reynolds number and consequently study their effects on the

step wake characteristics.

This document including the introduction is divided into five chapters. An extensive lit-

erature review on the flow characteristics behind the backward-facing step and the effect of

parameters on their characteristics are discussed in the Chapter 2. The combined effect of


parameters on the reattachment length with the inclusion of new scaling length are discussed

in Chapter 3. The Chapter 4 focuses on the design of the axisymmetric backward-facing step

model to be used in future studies at the Flow Control and Experimental Turbulence lab at

UTIAS. A brief conclusion, followed by a future scope is given in Chapter 5. The major findings

of this thesis has been published in a conference [15].

Chapter 2

Literature Review

2.1 Common Features of the Backward-Facing Step Flow

The flow behind the backward-facing step (BFS) is complex and involves various instability

mechanisms. Some of the most common features behind the step recognized in the literature

are illustrated in the Figure 2.1.

!

H

U"

Secondary vortices Separation bubble

Shear layer

Time averaged

dividing streamline Vortex merging

Coherent

structure

Vortex shedding Reattachment

zone

Figure 2.1: Flow characteristics behind a BFS (adapted from Driver et al. [16]).

Based on the important flow features studied by previous researchers in a planar BFS geom-

etry, the flow wake can be distinguished into three main regions namely, the shear layer region,

separation bubble or recirculation zone and the reattachment zone. The general characteris-

tics of a BFS flow begins with an upstream boundary layer separating at the step edge due

to the adverse pressure gradient that develops into a thin shear layer. As the flow progresses

5

Chapter 2. Literature Review 6

downstream, the shear layer grows in size with the amalgamation of the turbulent structures

contained within. This region where the shear layer develops and grows is referred to as the

shear layer region and is shown in Figure 2.1. The turbulent structures in the shear layer

entrain irrotational fluid from the non-turbulent region outside the shear layer. This flow en-

trainment causes the formation of a low velocity recirculation in the region, which is located

between the shear layer and the adjacent wall. The recirculation zone is mainly comprised of

a primary vortex in the center and a secondary vortex adjacent to the corner of the step as

shown in Figure 2.1. Due to the favorable pressure gradient created by the fluid entrained,

the shear layer eventually curves down towards the wall and impinges at a location known

as the reattachment point. The horizontal distance between the step and the reattachment

point is defined as the reattachment length. The reattachment length is unsteady due to the

inherent oscillatory motion of the shear layer known as the flapping. As a result, the reattach-

ment point spreads within a certain span along the streamwise distance, which is referred to

as the reattachment zone. These three regions in a whole comprise the important features of

a BFS flow that can be altered or controlled to achieve desirable outcome such as, enhanced

mixing characteristics and reduced drag, noise and vibrations. Hence it is essential to under-

stand these flow features to control the flow dynamics. In most earlier studies [7, 9–11, 17],

the reattachment length has usually been the primary parameter of interest to study the wake

characteristics of a BFS flow. With the advancement in technology and the recent discovery

of the coherent structures in the shear layers, more researchers have shifted their interests to

studying these turbulent structures. The fact that these coherent structures can be altered

to get desired characteristics highlights the importance of understanding their dynamics. In

the following sections, a detail description of the flow characteristics in the wake of a step with

more emphasis on the evolution and the significance of the coherent structures will be discussed.

2.2 Shear Layer Region

The layer of fluid with a velocity gradient subjected to viscous shearing is known as the shear

layer. The shear layers are of two types, wall-bounded and free shear flows; the latter is the

one of interest in BFS flows. The free shear layer in the BFS originates at the separation point

and eventually curves down towards the wall to impinge at the reattachment point. This layer

is created due to the fast moving fluid (free stream velocity) on the top and the low momentum

fluid in the wake of the step. The separated free shear layer behind the BFS involves numerous

instabilities. In the following section, a description of the free shear layer features behind the

BFS and related instabilities are discussed.


2.2.1 Vortex Rolling and Pairing Mechanism

The rolling and amalgamation of vortices are characteristics of every free shear layer flows such

as, the wakes, mixing layers and the jets. Browand [18] was one of the earlier researchers to

have noted the existence of vortex pairing mechanism in the shear layer of a BFS flow. How-

ever, a better description of the vortex pairing mechanism was available only when Winant and

Browand [19] observed them in a mixing layer using flow visualization. In their experiments,

they observed a periodic train of vortical structures, also known as vortices, in the turbulent

mixing layer. The vortices in the shear layer are formed by the rolling and pairing of the ad-

jacent turbulent structures. This pairing process was seen to continue until the mixing layer

thickness increased to the channel height. In the case of a BFS, the experiments suggested that

these vortices grew at most to the height of the step [20].

A model to describe the physics behind the roll up process of the vortices in a mixing

layer was developed by Winant and Browand [19]. Since the initial growth of the separated

shear layer behind the step resembles the mixing layer flows, this model can also be used to

describe the roll up of vortices in BFS flows. The model is described through an illustration

shown in Figure 2.2. To describe this phenomenon they considered the inviscid instability of

a constant-vorticity layer between two parallel streams. According to their theory, the initial

distortion in the boundary of the region containing the vorticity requires an initiation from a

small amplitude wave, as shown in Figure 2.2(a). It is known that the vorticity region induces

vertical velocity thus resulting in the growth of these perturbations. Due to these perturba-

tions the region containing the vorticity becomes periodically flatter and thinner as shown in

Figure 2.2(b) and 2.2(c). As the instability grows the vortical areas between the two streams

get to the verge of pinching off. Finally they break into discrete vorticity lumps as shown in

Figure 2.2(d). This process of forming vortical lumps is known as the roll-up process.

(a) (b)

(c) (d)

Figure 2.2: Initial instability in the shear layer, and the roll-up into discrete vortices [19].


The next and the most dominant stage in this interaction is the merging mechanism of adja-

cent vortices commonly known as the vortex pairing. Winant and Browand [19] suggested that

the pairing process in a turbulent free shear layer is a result of a mutual interaction between

adjoining vortices due to slight imperfections in the vortex spacing and strength. In the pairing

process, neighboring pairs of vortices roll around each other and due to viscous diffusion their

identities are smeared out leaving a single large vortex. It has been noted in several BFS flows

[2, 13, 21], like in the mixing layer flows, most vortices double, triple and quadruple, as shown in

Figure 2.3, before the reattachment of the shear layer. Nevertheless, it is not always necessary

for two adjacent vortices to merge in a shear layer. There are some odd ones referred to as

the ‘drop outs’ that do not merge initially but do further downstream with a larger adjacent

vortical structure. This instability mechanism involving vortex rolling and pairing observed in

the mixing layer is related to the Kelvin-Helmholtz Instability.

Figure 2.3: Streamlines over a bluff plate with splitter plate configuration [4].

The amalgamation of vortices plays an important role in the growth of separated free shear

layers. The growth of a shear layer is mainly caused by the increase in size of vortices due to

the vortex pairing mechanism. However, there are other mechanisms, like the fluid entrainment

process that contribute to the growth of a shear layer. As the flow progresses downstream the

shear layer grows larger in size, due to the incessant pairing of adjacent vortices that advect

downstream with the shear layer. The vortex pairing continues as long as it is not inhibited

by a wall or other boundaries. The resulting large scale turbulent structure, which is phase

correlated over its spatial extent, is commonly termed as a coherent structure. The features

and significance of coherent structures are discussed in section 2.2.2.

The process of vortex roll up and the pairing mechanism in the separated shear layer of a

BFS flow was investigated by Troutt et al. [20]. During their study, they detected the presence of


large scale structures in the separated shear layer close to reattachment. These structures were

formed by vortex pairing interactions in the separated shear layer similar to the mechanism

observed in mixing layer flows. Unlike mixing layer flows, the vortex pairing in BFS flows

cease to occur close to reattachment of the shear layer. They believed that the pairing process

stopped due to the proximity of the bottom wall in a BFS. Thus suggesting that the length

scale of coherent structures formed by vortex pairing close to reattachment is at most one step

height. The vortex pairing mechanism in BFS shear layers was also noted by Lee and Sung [22],

when they measured the wall pressure fluctuations at various streamwise and spanwise location

downstream to the step. Their results suggested that the flow behind the step were organized

and coherent along the spanwise direction. The non-dimensional frequency (St) for the vortex

pairing process in a BFS flow was noted by Liu et al. [23] to be approximately equal to 0.13. It

was later confirmed that similar non-dimensional frequency were also observed in several other

BFS flow experiments. The Strouhal number for these experiments is given by St = fH/U∞,

where f is the vortex pairing frequency, H is the step height and U∞ is the free stream velocity.

2.2.2 Coherent Structures

A coherent structure is a connected turbulent fluid mass with instantaneously phase-correlated

vorticity over its spatial extent, and the vorticity is termed as the coherent vorticity [1]. Coher-

ent structures are spatially non-overlapping and have their own boundaries. It has been noted

in many BFS flows that the vortices in the unstable shear layer roll-up and pair to form a single

large coherent structure. Hence a turbulent shear layer can be decomposed into coherent and

incoherent turbulence. Although the turbulence at Kolmogorov scale are the most coherent

[1], the large-scale structure are the ones that are often referred to as the coherent structures

because of their dynamical significance. The existence of a coherent vorticity is the primary

indicator of a coherent structure and hence is important to be able to identify them in a flow

amidst the incoherent turbulence. Winant and Browand [19] described that the property of a

coherent structure can be determined by obtaining the ensemble average of sufficiently large

number of structures. The ensemble averaging is critical to identify a coherent structure be-

cause it filters the effects of incoherent turbulence. This process of measuring the properties of

a coherent structure over its spatial extent is commonly known as eduction.

Significance of Coherent Structures

Coherent structures have been a hot topic for research in the past few decades. These structures

are almost omnipresent in flows involving turbulence such as, boundary layer flows, mixing lay-

ers, jets and wakes. Hussain [1] describes coherent structure as a tool to find order in apparent


disorder in turbulent flows. Several studies have been conducted to understand the proper-

ties of a coherent structure because of its significant influence on the flow. It is known from

literature that the coherent structures play an important role in heat, mass and momentum

transfer, and hence can be useful in controlling drag. Some of the other benefits of these struc-

tures as described by Hussain [1] are understanding of entrainment phenomena, explanations

for excitation-induced enhanced mixing, turbulence and noise suppression. There are instances

where knowledge of coherent structures have been used for controlling BFS flows.

There have been several active control studies conducted on BFS flows in which the char-

acteristics of the coherent structures are altered to achieve desired results. In active control

studies the forcing on the separated shear layer is usually applied through two methods: flaps

or by acoustic forcing [24], the latter being the most common. The flap was implemented by

Roos and Kegelman [2] in a backward-facing step flow at the step edge. They investigated the

effects of active flow control on coherent structures and the wake of the BFS by oscillating the

flap periodically at a certain frequency. They observed remarkable change in the reattachment

length when the flap was activated at a non dimensional frequency (St) 0.29. Their spectral

map results of shear layer velocity fluctuations showed that the strong concentration of energy

at the excitation frequency evolved into a peak at half the excited frequency farther down-

stream. This suggests enhanced amalgamation process within the shear layer due to forcing.

They attributed the reduction in reattachment length to the enhanced vortex pairing mecha-

nism and formation of coherent structures slightly upstream in comparison to the unforced case.

Similar observations were made when Bhattacharjee et al. [5] tried to modify the vortex

interaction mechanism in a reattaching separated shear layer behind the BFS. They succeeded

in altering the flow characteristics behind the wake of a step by applying forcing through a

single acoustic speaker located on the top of the step edge as shown in Figure 2.4(a). When the

acoustic speaker was forced at St = 0.2 they observed high turbulent activity within the shear

layer including enhanced merging of vortices and organization of turbulent structures along the

spanwise width. The reattachment length reduced by at most 15% from the unforced case.

The wake characteristics, however, remained nearly unchanged when the speaker was forced at

St = 1.2.

In some other BFS flow control studies, the acoustic speaker was installed close to the step

edge functioning like a synthetic jet [3, 23]. Chun and Sung [3] investigated the most effective

forcing frequency for influencing the reattachment length by varying the forcing frequency from

0 to 5. Their results suggested that the reattachment length reached its minimum at St ≈ 0.27


and a maximum at St ≈ 1.5. They believed that the reduction in reattachment length is

caused due to the increased entrainment in the recirculation region. Furthermore, it can be

seen that in most of these flow control experiments the ideal forcing frequency for reducing the

reattachment length is determined to be St ≈ 0.27 and this frequency is the first harmonics

of the amalgamation process in the separated shear layer [23]. It is evident from the above

studies that the knowledge on coherent structures is key to controlling BFS flows and hence is

important to study the characteristics of coherent structures further.

(a) Acoustic forcing from the top wall [5].

(b) Acoustic forcing from the step edge [3].

Figure 2.4: Acoustic forcing in backward-facing step flows.

2.2.3 Vortex Shedding Mechanism

The shear layer characteristics on reattachment and after were studied extensively by Brad-

shaw and Wong [25]. Their results suggested that the shear layer after reattachment continues

to spread downstream into a new shear layer with a sub-boundary layer underneath. The

key behavior they observed in the relaxing boundary layer was the region of rapid distortion

which depended on the amount of mass flow deflected upstream upon shear layer reattach-

ment. In another experimental study, Farabee and Casarella [26] measured the wall pressure

fluctuations using static pressure taps downstream of the step at various streamwise locations

to understand the wake characteristics. Their results indicated the presence of coherent, highly

energized, velocity fluctuations within the shear layer. They also noted that these coherent

structures after attaining the maximum length scale, which in their case was the step height,

shed and convected downstream due to the momentum of the free stream velocity. These co-


herent structures were still identifiable as far as 72 step heights downstream of the step location.

In a comparative study, Sigurdson [6] noted that the shedding instability were a common

phenomenon in free shear layer flows. He suggested that the shedding instability resembled

the von Karman vortex shedding that is commonly seen in the wake of bluff bodies. This

vortex shedding characteristics of separated shear layers were noted in most of the recently

conducted BFS experiments. In an experiment by Roos and Kegelman [2] they analysed their

hot-wire data acquired from within the shear layer at different streamwise locations. Their re-

sults suggested that the size of the vortex increased due to amalgamation as the flow progressed

downstream. These results were also observed by Liu and Sung [23], however they also noted

those structures to shed after attaining maximum size. The non-dimensional vortex shedding

frequency in the wake of a BFS was measured by Lee and Sung [22] to be approximately equal

to 0.07. Similar non-dimensional frequency for vortex shedding phenomenon in BFS flows were

also observed by many other researchers [13, 23, 27].

Based on the above-mentioned evidence, it was commonly believed that small scale vortices

in the shear layer grew in size due to the Kelvin-Helmholtz instability as it advected down-

stream. In other words, it was believed that the growth of the shear layer structures occurred

spatially and then shed downstream after attaining their maximum possible length scale, the

step height. Contrary to this view, Hudy et al. [13] observed the temporal growth of stationary

coherent structures to occur intermittently along with the spatially growing coherent structures.

They noted similar shedding mechanism for temporally growing coherent structures once they

attained the size of the order of step height. They drew similarities between the temporally

growing coherent structure with the development of the vortex structure in the wake of bluff

bodies. To distinguish between these two shedding modes, they referred to the temporally grow-

ing coherent structure as the wake mode, while the other as the shear-layer mode. Sketches

illustrating both these modes are shown in Figure 2.5.

The important behavior to note is that regardless of which mode is prevalent, the flow

structures eventually grow to a length scale of the order of the step height and then shed down-

stream with a certain advection velocity. Hence, Hudy et al. [13] believed that the distinction

between the two modes is likely to affect flow characteristics within the separation bubble.

Similar observations were also noted in a numerical study conducted by Wee et al. [27]. In

their study, they noted that the large-scale structures formed periodically in the middle of the

recirculation zone before shedding downstream. They attributed the temporal growth of the

structures to the existence of an absolute instability in the flow. Wee et al. [27] stated that


xr

H

x

y

Free stream

velocity (U!)

Vortical structures in

shear layer Edge of

shear layer

(a) Shear mode.

xr

H

x

y

Free stream

velocity (U!)

Vortical structures

developing in place Coherent structures

accelerating downstream

(b) Wake mode.

Figure 2.5: Convective and absolute instabilities in the shear layer (referred from Hudy [28]).

absolute modes are most likely to originate in the region with strongest back flow. It was also

suggested by Huerre and Monkewitz [29] that absolute modes were likely to appear when the

velocity ratio, R is greater than 1.315, where R is the ratio of the difference and the sum of

velocities of the two co-flowing streams. Based on those results, Hudy et al. [13] suggested that

the absolute modes most likely existed around two to three step heights downstream of the step.

Although they were able to identify the possible cause for the intermittently appearing

temporally growing structures, the question still remains as to why this shift between modes

occurred intermittently. The switching behavior between the shear-layer mode and the wake

mode was also seen in numerical computations in cavities by Rowley et al. [30].

2.3 Reattachment Zone

The reattachment point of an uncontrolled separating and reattaching shear layer in a BFS is sel-

dom fixed at a single point. This unsteadiness in the point of reattachment is associated with the

low-frequency oscillations detected in the shear layer [31]. This unsteadiness in the shear layer

trajectory is referred to as the flapping of the shear layer. Several researchers [2, 5, 23, 26, 31],


who have experimented on BFS flows, have observed highly energized low-frequency peaks in

the power spectrum obtained near the separation point. The non-dimensional frequency (St)

of the flapping phenomenon were noted to be of the order of 0.02 in many experiments [21–

23, 32]. Eaton and Johnston [31] were among the first to associate low frequency oscillations to

the flapping of the shear layer. Although many researchers have proposed several hypothesis

(based on their observations), the source of the flapping phenomenon is not clearly understood

yet. In this section, some of these theories are discussed briefly.

Eaton and Johnston [31] studied the source of the low frequency disturbances in a flow

behind a BFS using hot-wires and thermal tuft probes. Their results suggested that these dis-

turbances could be caused due to the non-periodic, vertical fluctuation of the reattaching free

shear layer. In other words this can be realized as the flapping of the shear layer resulting in

the fluctuation of reattachment point. They suspected that this low-frequency motion could be

caused due to the imbalance in the flow injection and entrainment rate within the recirculation

region. This theory is the most widely accepted for the likely cause of the flapping phenomenon

in the separated shear layer. The occurrence of flapping in the shear layer was also confirmed

by Durst and Tropea [33], while measuring the probability density function of the velocity at

various streamwise locations behind the step.

A different theory was proposed by Troutt et al. [20] based on the coherence of data ob-

tained from hot-wires placed at several streamwise locations. Their results suggested that the

coherence at lower frequencies were the unsteadiness involved within the recirculation region.

They also believed that the unsteadiness could be caused due to the passage of large-scale vor-

tices through the reattachment region, which was later supported by Simpson [34]. According

to Driver et al. [16], flapping is caused by a particularly high-momentum structure moving far

downstream before reattachment. They noticed the abnormal contraction and elongation of

the separation bubble as a result of the shortening and lengthening of the reattachment length.

They suggested that the moving structures in the shear layer intermittently creates greater

pressure gradient within the recirculation region resulting in the higher recirculation rate and

shorter reattachment length. More recently, Spazzini et al. [35] studied the characteristics of a

BFS flow using Particle Image Velocimetry (PIV). Their analysis on the low frequency motion

using a wavelet transform suggested that a frequency comparable to the flapping phenomenon

was also observed within the secondary vortices. Similar to their results Hall et al. [36] also

observed activities in the secondary vortices that were closely related to the low-frequency shear

layer flapping.


Although many theories have been proposed based on the experimental observations, they

are difficult to verify. One of the reason being the lack of more detailed information regarding

the flow-field behind the step especially in the recirculation region. It is known from various

experiments that the separation bubble intermittently changes shape and size with the flapping

of the shear layer. Hence measuring data from the recirculation region to determine the entrain-

ment rate and the injection rate is complex and difficult. With the advent of technology, new

instruments, such as time-resolved PIV, are capable of acquiring the velocity information from

the wake of the BFS flow for different instances of time separated by a small duration. This

will be helpful in tracking the transfer of fluid between the shear layer and the recirculation

zone through entrainment and injection and thus serve to advance our understanding of these

flow.

2.4 Recirculation Zone

The region behind the step, which is bounded by the separating and reattaching shear layer on

the top and by the wall on the bottom is the recirculation zone. Due to the presence of vortices

in the separated shear layer fluid from the recirculation zone is entrained and thus creates a

low pressure triggering recirculation. This region also referred to as the separation bubble is

dominated primarily by a large two-dimensional vortex (or primary vortex) that possesses a

low circulation velocity. The other significant turbulent structure commonly seen in the recir-

culation region is the secondary vortex located at the corner of the step. A sketch illustrating

the recirculation region in a BFS wake can be seen in Figure 2.1.

The literature describing the recirculation region and its prominent features were very few

in the past [17, 25, 37] mainly because of difficulties in measuring recirculating flows. Hot-wires

and pitot tubes were amongst the common instruments used to measure flow properties in early

experiments. These instruments are insensitive to the direction of the flow and hence inaccurate

in highly turbulent regions. Later, with the introduction of new instrumentation techniques

such as the Laser Doppler Anemometer and the PIV, more measurements were acquired from

the recirculation region. Some of the recent studies on the characteristics of the recirculation

region are discussed below.

The characteristics of the recirculation region behind the BFS were studied by Scarano

and Reithmuller [32] using a Digital Particle Image Velocimetry. Their flow streamline results

behind the step suggested that the primary vortex extends from the step edge to the reattach-

ment point while the counter-rotating secondary vortex remains in the corner of the step wall.


Figure 2.6: Mean flow structure of one half of the secondary vortex [36].

Similar results were also noted by other researchers [13, 21, 35, 36]. According to Hudy et al.

[13], the primary vortex is driven by the mass and momentum of the fluid moving upstream

from the shear layer on reattachment, while the secondary vortex is driven by the fluid from the

primary vortex. Due to the frequent oscillation of the reattachment point the size and shape

of the separation bubble is subjected to change. In a flow visualization experiment conducted

by Spazzini et al. [35], they observed the secondary vortex in the recirculation region of a BFS

wake undergoing changes in shape and size in a quasi-periodic cycle. During this cycle, the

secondary vortex was seen to increase in size as large as the step height and then break down to

its smaller size. They believed that the cycle of change in the size and shape of the secondary

vortices and the flapping motion of the shear layer to be two different aspects of a same motion.

Another important property of these vortices were noted by Hall et al. [36], while studying

the characteristics of a BFS wake using PIV. Based on their velocity streamline results, they

suggested that in both primary and secondary vortices the mass flow spiraled towards the center.

This fluid inflow by conservation of mass produces a spanwise directional motion leading to the

transport of the fluid from the core to the step edge as shown in the Figure 2.6. They also

believed that a significant spiral inflow for the primary vortex can have pronounced effect, like

the flapping, on the shear layer and the reattachment length.

2.5 Important Flow Parameters

The most commonly observed characteristics of the flow behind a BFS were discussed in the

previous sections. Apart from the complexity involved due to the instabilities in the flow, the


flow characteristics behind the step were also found to be dependent on certain flow and geomet-

ric parameters. The parameters that significantly affect this flow are the aspect ratio (w/H),

expansion ratio (Y1/Y0), free stream turbulence intensity (√u′2 + v′2 +w′2/U∞), Reynolds num-

ber (U∞H/ν), and the boundary layer state and thickness (δ) at separation [7]. It is essential to

understand the effects of these parameters mainly to be able to compare the results from various

BFS experiments conducted under different conditions. Moreover, in the current research the

knowledge of these effects are used in designing a unique BFS model with wake characteristics

less dependent on such governing parameters. An illustration of the planar BFS geometry and

the aforementioned parameters are shown in Figure 2.7

U!

H

w

Y0 Y1

"

y

z

x

Top wall

Bottom wall

xr

Separation

streamline

Figure 2.7: Planar backward-facing step geometry.

In the following section, the effect of each individual parameter on the reattachment length

are studied extensively. The choice of a reattachment length as an indicator of the step wake

characteristics was preferred in the earlier literature [25, 37–39] mainly because it is easy to

measure and also useful in predicting the base drag. Technological advancements in the in-

strumentation field and analytical schemes have make it possible to measure readily more

information such as the coefficient of pressure, the turbulent structure characteristics in the

shear layer, power spectral data, to name only a few. Nonetheless, the reattachment length is

used in the following discussion as the primary parameter of comparison since it is the most

commonly reported parameter in previous studies. This section serves as an extension to the

study conducted by Eaton and Johnston [7] and includes several more recent results.


2.5.1 Effect of the Expansion Ratio

The expansion ratio is defined as the ratio of downstream to upstream height of the channel

at the step location, i.e. Y1/Y0 as shown in Figure 2.7. The effect of the expansion ratio on

separating and reattaching shear flows was reviewed by Eaton and Johnston [7]. In their re-

view, they compared the results from various experiments on backward-facing step operating

at nearly similar conditions. Their comparative study included expansion ratios ranging from

1.1 to 1.67, although other parameters such as the relative boundary layer thickness at sepa-

ration (δ/H), free stream turbulence intensity and the Reynolds number (ReH) varied slightly

between each experiments. The boundary layers for all the experiments used in this comparison

were fully developed turbulent at separation. Based on those results, Eaton and Johnston [7]

observed a trend that the reattachment length increased with the increase in expansion ratio.

The effect of expansion ratio on the reattachment length was also investigated by Kuehn [40]

by increasing the angle of upper wall divergence at the step location and by changing the step

height. Their results were similar to Eaton and Johnston’s findings, i.e. reattachment length

increased with the increase in expansion ratio. Kuehn [40] attributed the effect on reattachment

length by expansion ratio to the change in adverse pressure gradient. He further confirmed the

role of adverse pressure gradient by observing similar results that were obtained by changing

expansion ratio, while rotating the channel wall.

Later, Otugen [11] studied the effects of expansion ratio on the reattachment length by

changing the step heights between each experiment. The expansion ratios for his experiments

were 1.5, 2 and 3.13, which is higher than most of the previously obtained results. While the

expansion ratio was altered, other flow parameters such as the boundary layer thickness (δ)

and state at separation, Reynolds number (ReH), and the free stream turbulence intensity were

all maintained constant. Their results suggested that the reattachment length decreased with

increasing expansion ratio. They attributed this reduction in the reattachment length for higher

expansion ratios to the increase in turbulence levels in the separated shear layer. Although

Otugen’s [11] results appear to contradict the results observed by Eaton and Johnton [7], the

higher expansion ratios and the different relative boundary layer thickness at separation(δ/H)

in their experiments makes a direct comparison impossible.

2.5.2 Effect of Aspect Ratio

In the case of a planar BFS geometry, like the one shown in Figure 2.7, the wake adjacent to

the step is two-dimensional. This however is not the case for a BFS with finite span. The

two-dimensionality of the flow behind the step with finite span is usually affected by vortices


developing from the side corners of the step or the boundary layers developing from the side

walls. Extensive study was conducted by Brederode and Bradshaw [8] to understand the effect

of aspect ratio (the ratio of width to height of the BFS) on the two-dimensionality of the step

wake. They traced the oil-film patterns on the floor of the turbulent region downstream of

the step for different aspect ratios. Here, the oil-film patterns obtained for aspect ratios 15

and 7.5 are shown in Figure 2.8 for reference. Their results suggested that the flow remained

two-dimensional but not uniform along the width, when the aspect ratio was greater than or

equal to 10:1. They, however, observed improvements in spanwise uniformity with increasing

aspect ratio.

(a) Aspect ratio = 15. (b) Aspect ratio = 7.5.

Figure 2.8: Sketches of oil film pattern on the floor for the region behind the step [8].

Alternative to the above approach, there has also been several studies on the separating

and reattaching shear layer characterisitics behind an axisymmetric BFS geometry. Most have

been on suddenly expanding pipes [41–44] (Figure 2.9(a)) while only a few recent experiments

were conducted on concentric cylinders that are aligned axially to the flow [12–14], as shown

in Figure 2.9(b). The mean flow characteristics surrounding the axisymmetric geometry is

the same as in any finite width planar BFS geometry except the mean flow characteristics are

similar around all azimuthal plane for the axisymmetric geometry [28]. Recently, Li and Naguib

[12] studied the flow characteristics of a separating and reattaching flow over an axially aligned

concentric cylinder. In their experiments, they observed smaller reattachment lengths compared

to the flow over a finite width planar BFS geometry. They hypothesized that the reduction in

reattachment length were caused due to the change in circumferential lengths upstream and

downstream at the step location of the axisymmetric geometry, which was referred to as the

curvature effect [12]. A similar effect was also observed by Hudy and Naguib [13]. In order to


U!

r R

(a) Expanding pipe.

U!

R r

(b) Concentric cylinder.

Figure 2.9: Axisymmetric backward-facing step geometries.

avoid confusion of this effect with the “transverse curvature effect” mentioned in Willmarth et

al. [45] (which will be discussed later in section 4.4.1), the curvature effect suggested by Li et

al. [12] will be referred to as the “circumferential effect” for the remainder of this thesis.

2.5.3 Effect of Free Stream Turbulence Intensity

The free stream turbulence intensity, which is the ratio of the turbulent fluctuation to the

free stream velocity (√u′2 + v′2 +w′2/U∞), has a significant effect on the reattachment of the

shear layer behind the BFS. In a review by Eaton and Johnston [7], while comparing different

experimental results they suspected that the increase in free stream turbulence intensity caused

reduction in the reattachment length. Later, a more systematic and comprehensive study was

conducted by Isomoto and Honami [10] to investigate the effects of free stream turbulence

intensity on the reattachment length. In their work, they changed the free stream turbulence

by using different turbulent grids upstream of the step while maintaining other significant

parameters nearly constant. They observed results similar to Eaton and Johnston’s [7], where

the reattachment length decreased monotonically with the increasing free stream turbulence

intensity. They attributed the reduction in reattachment length to the increase in turbulence

levels within the separated shear layer immediately downstream of the step.

2.5.4 Effect of Boundary Layer State at Separation

The boundary layer state at separation plays an important role in characterizing the flow behind

the step. The effect of boundary layer state on BFS flows is highly nonlinear and difficult to


isolate due to its dependency on Reynolds number and boundary layer thickness at separation.

Hence the combined effect of boundary layer state and thickness on the reattachment length

was studied by Eaton and Johnston [7]. Their results are shown in Figure 2.10. It can be seen

from the figure that the reattachment length increases sharply to a peak value with the change

in boundary layer state from laminar to transitional and then decreases gradually to a steady

value as the boundary layer becomes turbulent.

!"

#"

$"

%" &%%%" '%%%" (%%%"

)*+" ,-*./0,01.*)" ,2-32)4.,"-567689:5;<"=5;><9"?@ ABCD"

+E:5;<F:"<9G8H;5II"-5J;E=KI";F:L5A""?-5MD"

Figure 2.10: Effect of boundary layer state at separation [7].

Similar results (shown in Figure 2.11) were also obtained by Adams and Johnston [46]

when they studied the effect of boundary layer state and thickness on the reattachment length

for different Reynolds number (ReH). In Figure 2.11, it can be seen that the reattachment

length increases to a peak as the boundary layer state changes from laminar to turbulent and

then remains nearly constant. Based on the above results, it can be seen that the effect of

turbulent boundary layer thickness on the reattachment length is negligible. This suggests that

the effects of boundary layer state on reattachment length can be minimized by maintaining a

fully developed turbulent boundary layer at separation.

2.5.5 Effect of Boundary Layer Thickness at Separation

The effect of boundary layer thickness at separation on the reattachment length is not evident

because of its dependence on the boundary layer state and Reynolds number (ReH). Adams

and Johnston [46] attempted to study the effect of boundary layer thickness independent of

the Reynolds number on the reattachment length by applying suction through porous plates

installed upstream of the step. Their BFS model consisted of 24 porous plates which were


0 0.5 1 1.5 24.5

5

5.5

6

6.5

7

Boundary layer thickness at separation (δ/H)

Rea

ttach

men

t len

gth

(Xr/H

)

laminarturbulent

Figure 2.11: Effect of boundary layer state and thickness at separation; ReH = 26,000 [46].

installed approximately 5.98H to 0.01H upstream of the step. When all the plates were in use

they were able to achieve a boundary layer thickness less than 0.005H at the step edge. With

the application of suction Adams and Johnston [46] were able to study the effect of boundary

layer thickness at separation on the reattachment length for different Reynolds number (ReH).

Their results for different Reynolds number and turbulent boundary layer at separation are

shown in Figure 2.12. It can be seen from the figure that the Reynolds number, has significant

impact on the reattachment length. The other key feature to note is that the boundary layer

thickness appears to have a limited effect on the reattachment length due to its turbulent state.

However, the application of suction very close to the step edge is not advisable because

of the change in boundary layer properties immediately after suction thus altering the flow

characteristics behind the step [47]. Experiments conducted by Antonia et al. [47] suggested

that the application of suction to remove the boundary layer resulted in a significant alteration

of the boundary layer properties downstream of the suction. The parabolic variation observed

in the reattachment length with increasing boundary layer thickness could be due to change in

boundary layer turbulence intensity caused by suction. More details on the effects of suction

on the boundary layer will be discussed later in this thesis. Based on the above discussion, it

is clear that further experimentation is required to verify the effect of boundary layer thickness


0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.85.2

5.4

5.6

5.8

6

6.2

6.4

6.6

6.8

7

Boundary layer thickness at separation (δ/H)

Rea

ttach

men

t len

gth

(Xr/H

)

ReH = 8,000

ReH = 13,000

ReH = 20,000

ReH = 26,000

Figure 2.12: Effect of Reynolds number and boundary layer thickness at separation [46].

independent of Reynolds number on the reattachment length.

2.6 Summary

In this chapter, the flow characteristics in the wake of a BFS were reviewed and discussed in

detail. From earlier studies, it is understood that the free shear layer behind the step involves

several instability mechanisms such as, vortex rolling and pairing, shedding and flapping. The

rolling and pairing of vortices in the shear layer results in the formation of large coherent struc-

tures. Coherent structures are significant features observed in flow over a step. It has been

demonstrated in several studies that by altering the coherent structures desired characteristics

such as enhanced mixing, reduced drag and subdued noise can be achieved. Hence it is nec-

essary to understand the characteristics of the coherent structures in order to control the flow

actively. The wake characteristics behind a step differs between experiments due to different

operating conditions. Consequently, the individual effects of each significant parameters on the


reattachment length were reviewed. Studies suggests that a number of parameters such as free

stream turbulence intensity, aspect ratio and boundary layer state at separation have a promi-

nent effect on the reattachment length. In Chapter 3, the combined effect of these parameters

with a novel scaling length for reattachment length are discussed.

Chapter 3

Novel Scaling Analysis

3.1 Introduction

In the previous sections, the effect of each significant BFS flow-affecting parameter was re-

viewed. For some parameters such as the free stream turbulence intensity, boundary layer state

at separation and the Reynolds number, their independent effect on the reattachment length

were observed to have a unique relationship. This, however, does not exist while studying

the combined effects of the aforementioned parameters on the reattachment length. In other

words, it has been difficult to predict the reattachment length for a combination of wide range

of BFS flow-affecting parameters. As an example, the effect of the boundary layer thickness at

separation on the reattachment length is studied here with sets of data acquired from several

experiments conducted under different parametric conditions. The results thus obtained are

shown in Figure 3.1 and the data sets used for the current study are included in Table 3.1.

The data set considered for the current analysis is limited to the experimental results with

fully developed turbulent boundary layers at separation and aspect ratios greater than 10. In

Figure 3.1, the experimental uncertainty associated with each measurement is illustrated us-

ing error bars for the few cases where this information is available. The larger error bars are

associated with the changes in the reattachment length due to the flapping of the shear layer,

while the smaller ones represent the uncertainty on the time averaged reattachment length.

The data presented in the figure exhibit a significant amount of scatter due to variations in the

Reynolds number, free stream turbulence intensity, boundary layer thickness at separation and

expansion ratio between each data set. This was to be expected given the discussion of the

previous sections.

25

Chapter 3. Novel Scaling Analysis 26

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

5

5.5

6

6.5

7

7.5

8

Boundary layer thickness (δ/H)

Rea

ttach

men

t len

gth

(xr/H

)

Figure 3.1: Combined effect of parameters on the reattachment length. The legends of thefigure are, + Driver et al. [16]; ● Adams and Johnston [46]; ○ Kim [17]; ◻ Tani et al. [48]; ×Otugen [11]; ◇ Jovic and Driver [49]; △ Lee and Sung [22] ; ▽ Scarano and Reithmuller [32];▷ Scarano and Reithmuller [50]; ◁ Schram et al. [51]; ⊙ Le et al. [52]; ⊡ Heenan and Morrison[53]; � Jovic and Driver[54]; � Roos and Kegelman [2]; ⧫ Chun and Sung [3]; ∎ Troutt et al.[20]; ◂ Adams and Eaton [55]; ▸ Kang and Choi [56]; ⟐ Hudy [28].

3.2 New Normalization Scheme

For the present analysis, a novel normalization scheme is proposed, which includes the bound-

ary layer thickness as part of the characteristics length scale of the flow. The new length scale

parameter is stated as the sum of step height, a geometric parameter, and the boundary layer

thickness at separation, a flow parameter. This contrasts from the common approach, where the

step height is the only length scale considered. The results obtained with the new normalizing

factor are shown in Figure 3.2. It can be seen from the figure that the collapse of the data on

this normalization is greatly improved.

The collapsed data shown in Figure 3.2 is fitted with second-order polynomial, included in

the figure for reference. The curve is given by,

xrH + δ

=

xr

H

1 + δH

≈ −5.73⎛

⎝

δH

1 + δH

⎞

⎠

2

− 2.59⎛

⎝

δH

1 + δH

⎞

⎠+ 6.04 (3.1)


which has a root mean square error of approximately 0.89% with the plotted data and a maxi-

mum error percentage of ≈ 17%. This suggests that there is an excellent collapse of data. This

curve fit serves as a good visual guide and may be a potential model for predicting the reattach-

ment length with only knowledge of the boundary layer thickness and step height at separation.

0.1 0.2 0.3 0.4 0.5 0.6 0.71.5

2

2.5

3

3.5

4

4.5

5

5.5

6

6.5

Boundary layer thickness (δ/H)/(1 + δ/H)

Rea

ttach

men

t len

gth

(xr/H

)/(1

+δ/H

)

Figure 3.2: Boundary layer thickness at separation vs. rescaled reattachment length. Thelegends of the figure are, + Driver et al. [16]; ● Adams and Johnston [46]; ○ Kim [17]; ◻ Taniet al. [48]; × Otugen [11]; ◇ Jovic and Driver [49]; △ Lee and Sung [22] ; ▽ Scarano andReithmuller [32]; ▷ Scarano and Reithmuller [50]; ◁ Schram et al. [51]; ⊙ Le et al. [52]; ⊡Heenan and Morrison [53]; � Jovic and Driverl [54]; � Roos and Kegelman [2]; ⧫ Chun andSung [3]; ∎ Troutt et al. [20]; ◂ Adams and Eaton [55]; ▸ Kang and Choi [56]; ⟐ Hudy [28]; - -- −5.73((δ/H)/(1 + δ/H))2 − 2.59((δ/H)/(1 + δ/H)) + 6.04.

The addition of step height and the boundary layer thickness as a scaling parameter phys-

ically represents the dimension of a geometric and viscous boundary. It is possible that the

momentum or displacement thickness would be more appropriate length scales to add to the

step height since they are the length scales governing the coherent structures in the separated

shear layer. In the current analysis, the boundary layer thickness is used because it is more

readily found in the cited literature. Besides, knowing that the ratios of displacement and

boundary layer thickness (δ∗/δ), and momentum and boundary layer thickness (θ/δ) are nearly

constant, the changes to the existing curve will be small when either one is used. The bound-

ary layer thickness is known to be affected by the Reynolds number (Rex) based on streamwise


Table 3.1: Important parametric data acquired from different experiments on backward-facingstep flowsAuthors Symbols ReH × 10−4 Y1/Y0 δs/H u′/U∞(%) xr/H

Tani et al. [48] ◻ 6 1.07 0.28 na 6.7 ± 0.2Kim [17] ○ 3.00 1.33 0.45 0.60 7 ± 1

4.50 1.5 0.30 0.60 7 ± 1Troutt et al. [20] ∎ 4.5 1.1 0.18 0.6 7Roos and Kegelman [2] � 3.9 1.31 0.38 < 0.05 6.8Driver et al. [16] + 3.7 1.125 1.5 na 6.11 ± 1Adams and Johnston [46] ● 0.8 1.25 0.7 0.3 - 0.4 6.15 ± 0.1

0.8 1.25 1.3 0.3 - 0.4 6 ± 0.10.8 1.25 1.62 0.3 - 0.4 5.85 ± 0.11.3 1.25 0.72 0.3 - 0.4 6.45 ± 0.11.3 1.25 1.16 0.3 - 0.4 6.55 ± 0.11.3 1.25 1.78 0.3 - 0.4 6.25 ± 0.12 1.25 0.14 0.3 - 0.4 6.2 ± 0.12 1.25 0.18 0.3 - 0.4 6.35 ± 0.12 1.25 0.7 0.3 - 0.4 6.6 ± 0.12 1.25 1.1 0.3 - 0.4 6.8 ± 0.12 1.25 1.62 0.3 - 0.4 6.45 ± 0.1

2.6 1.25 0.14 0.3 - 0.4 6.4 ± 0.12.6 1.25 0.2 0.3 - 0.4 6.55 ± 0.12.6 1.25 0.54 0.3 - 0.4 6.8 ± 0.12.6 1.25 1.1 0.3 - 0.4 6.7 ± 0.12.6 1.25 1.62 0.3 - 0.4 6.45 ± 0.1

Adams and Eaton [55] ◂ 3.6 1.25 1 0.2 - 0.4 6.6Otugen et al. [11] × 1.66 1.50 0.72 0.70 6.90

1.66 2.50 0.36 0.70 6.651.66 3.13 0.17 0.70 6.35

Jovic and Driver [54] � 0.5 1.2 1.17 1 6 ± 0.15Jovic and Driver [49] ◇ 0.68 1.09 2 1 5.35

1.04 1.14 1.27 1 6.352.55 1.14 1.2 1 6.92.55 1.2 0.82 1 6.73.72 1.2 0.82 1 6.84

Chun and Sung [3] ⧫ 2.3 1.5 0.38 < 0.6 7.2Le et al. [52] ⊙ 0.51 1.2 1.2 na 6.28Heenan and Morrison [53] ⊡ 19 1.1 0.21 < 0.05 5.88 ± 0.3Scarano and Reithmuller [32] ▽ 0.5 1.2 1.2 0.8 5.9Scarano and Reithmuller [50] ▷ 0.5 1.2 1.5 0.3 6Lee and Sung [22] △ 3.3 1.5 0.41 < 0.8 7.4Kang and Choi [56] ▸ 0.51 na 1.34 na 6.2Schram et al. [51] ◁ 0.51 1.25 1.3 2 5.24Hudy et al. [28] ⟐ 0.6 1.07 ≈1.49 < 1 4.29

0.8 1.07 ≈1.53 < 1 4.481.5 1.07 ≈1.22 < 1 4.791.9 1.07 ≈1.07 < 1 4.94


distance and free stream turbulence, and hence its inclusion in the scaling appears to account

for the effect of these two parameters on the reattachment length. This suggest that these two

parameters are more important in as much as they affect the state of the flow at separation,

rather than play an important role in the separated shear layer evolution. Nonetheless, more

targeted experiments are necessary to evaluate the validity of this scaling unambiguously due

to the limited information available from the studies listed in the Table 3.1.

Chapter 4

Model Design

4.1 Model Design Objective

The wake flow characteristics behind the backward-facing step are dependent on several geomet-

ric and flow parameters such as, the expansion ratio (Y1/Y0), aspect ratio (w/H), free stream

turbulence intensity (u′/U∞), and the boundary layer state and thickness at separation (δ/H).

The effects of each aforementioned parameters on the reattachment length were discussed in

section 2.5. With this knowledge it will be possible to design a BFS model such that the effect

of these parameters are either controllable or eliminated.

4.2 Wind Tunnel Facility

The experiment will be conducted in a subsonic wind tunnel at Flow Control and Experimental

Turbulence (FCET) Lab, University of Toronto Institute for Aerospace Studies. The exper-

imental model was designed taking into account the wind tunnel specifications. The FCET

facility is a closed return wind tunnel with a contraction ratio of 9 ∶ 1 and a test section 5

m long, 1.2 m wide and 0.8 m high. The test section has adjustable corner plates installed

to minimize the disturbance due to the corner vortices. The tunnel is also designed with a

heat exchanger unit to maintain the temperature constant within the test section. A schematic

drawing of the wind tunnel is shown in Figure 4.1. The fan is powered by a 60 hp motor and

can generate wind speeds up to 30 m/s in the test section. The free stream turbulence intensity

within the test section is measured to be approximately 0.05% of the free stream velocity.

30

Chapter 4. Model Design 31

Primary diffuser

Secondary diffuser

Test section

Motor housing

Contraction

Fine mesh

Honeycomb chamber

Heat exchanger

Figure 4.1: Closed return tunnel schematic.

4.3 Model Design

In this section, the key features of the model and rational for their selection are briefly stated

before discussing the details of the model design. Choosing the right geometry is the crucial

decision while designing a backward-facing step model. One of the primary objective while

designing the geometry is to ensure minimal disturbance in the wake due to the boundary

layers from the test section walls and the vortices from the side edges of the step. In other

words, the wake of the step must remain uniform and two-dimensional throughout the span-

wise width. It was discussed earlier in section 2.5.2 that the above criterion can be satisfied if

the geometry were to be axisymmetric. Based on that requirement, two concentric cylinders

forming an axisymmetric backward-facing step is the geometry selected for the current research.

Experiments on an axisymmetric backward-facing step geometry were previously conducted

by Hudy et al. [13]. Their results suggested that the reattachment length was shorter for the


axisymmetric geometry than for those measured for the planar case. They believed that the

reduction in reattachment length was caused by the circumference effect. This effect due to

the cylinder curvature can be minimized by choosing big radius for both the cylinders while

maintaining a small step height. However, the maximum size of the model is also limited by

the test section dimensions to maintain an appropriate blockage ratio and an expansion ratio

close to unity.

The bigger radii and step height for the current model were chosen as 125mm (twice as

large as the radii of Hudy’s [28] model) and 12.5mm (equal to Hudy’s [28] model) respectively,

to minimise possible circumference effects. For the above mentioned dimensions, the blockage

ratio is approximately equal to 5.57 %, while the blockage ratio was 3.6 % for Hudy’s [28]

model. Nevertheless, the blockage ratio for the current model is less than 6 %, which according

to West and Apelt [57] is the minimal limit to determine an approximate blockage correction.

The expansion ratios for this model are 1.039 along the Y-axis and 1.024 along the Z-axis.

Although the expansion ratios are not symmetric circumferentially, the values are expected to

be small enough not to have a significant impact on the flow characteristics.

The state and thickness of the boundary layer at separation are the other parameters known

to have significant effect on the characteristics of the step wake. Based on the discussion in

section 2.5.4 it is understood that a fully developed turbulent boundary layer profile at separa-

tion has little impact on the reattachment length. Hence the effects of boundary layer state can

be eliminated by either choosing a longer upstream model length or by tripping the boundary

layer near the leading edge to obain a self similar turbulent boundary layer profile at the sepa-

ration point. For a fixed upstream model length, the boundary layer thickness still varies with

the change in Reynolds number (Rex) and hence will continue to affect the step wake unless

controlled.

There have been several control techniques implemented in the past to maintain the bound-

ary layer thickness independent of the Reynolds number. Mass transfer is described as the

most prominently used boundary layer control technique by Schlichting and Gersten [58]. For

the current design, the boundary layer will be controlled through suction. The application of

suction to control the boundary layer thickness independent of the Reynolds number has been

implemented before in a BFS geometry by Adams and Johnston [9]. In their experiments, they

managed to achieve a boundary layer thickness less than 0.005H at the separation point by

extending the porous strips up to the step edge. Similar suction control experiments on the

boundary layer were also conducted by Antonia et al. [47] on a flat plate. According to Antonia


et al. [47], the boundary layer profile changes due to suction and requires some distance to

recover to its unperturbed state. More details on the effects of suction on the boundary layer

profile will be discussed in section 4.4.2. Based on the above information, it is understood that

the presence of suction strip close to the separation point could introduce different effects in

the step wake. Hence for the current model, the suction strip must be located carefully to allow

sufficient relaxing distance downstream of the strip for the boundary layer to recover. Therefore

to choose the upstream length of the model the porous strips are to be designed first and their

details are discussed in following section.

4.4 Porous Suction Strip Design

The objective for the porous suction strip design is to be able to control the boundary layer

thickness independent of the Reynolds number, while avoiding the change in boundary layer

profile due to suction affecting the separated shear layer. The first step to suction strip design

is to derive an analytical expression to estimate the evolution of boundary layer thickness along

the streamwise direction with suction. The estimate is essential for locating the porous strips to

achieve a wide range of boundary layer thickness at separation for a constant Reynolds number

(Rex).

4.4.1 Evolution of the Turbulent Boundary Layer Thickness

An approximate relationship between the boundary layer thickness (δ) and the streamwise

distance (x) with suction is derived using the momentum integral method. In the case of

an axisymmetric geometry, for e.g. a cylinder, the boundary layer development is not usu-

ally the same as the one over a flat plate due to the transverse surface curvature [45, 59, 60].

However, this curvature effect can be neglected on two conditions; if the boundary layer thick-

ness is smaller than or equal to the radius of the cylinder, i.e., δ/r ≤ 1 [61], and if the non-

dimensionalized radius of the cylinder (r+ = r/(ν/uτ)) is greater than or equal to 250 [62, 63].

In the subsonic wind tunnel operating at 5 m/s, assuming the boundary layer to be turbulent

from the leading edge, the model will require more than 5 m to grow a boundary layer as

thick as the radius of the cylinder (125 mm). Also, for the same velocity and an estimated

upstream model length of 1 m the Reynolds number based on the streamwise distance (Rex) is

calculated to be of the order of 3.2 × 105. The local coefficient of friction (cf ) for a turbulent

boundary layer (1/7th power law profile) over a flat plate is given as 0.0592Re−1/5x [64]. From

the above information cf can be calculated to be approximately equal to 0.0047. Assuming the

radius of curvature of the axisymmetric model has negligible effect on cf , the friction velocity


(uτ =√τw/ρ), where τw is the wall shear stress, can be calculated to be approximately equal

to 0.2421. The non-dimensionalized radius (r+) based on the above information is calculated

to be 1940. This suggests that the transverse curvature effect to be negligible for the current

model as long as the upstream length is less than 5 m.

The evolution of turbulent boundary layer on the axial cylinder can thus be derived assuming

the flat plate conditions. The growth of turbulent boundary layer for an incompressible flow

along a flat plate with suction is derived using momentum integral method [65]. The control

volume under study is illustrated in Figure 4.2.

mda

mbc

mab

x

y

b

a

c

d

mcd

V0

δ + dδ

δ

dx

Figure 4.2: Schematic of control volume used for momentum integral analysis.

Conservation of Mass

The mass flow rate into the system through ‘ab’ surface is given by,

mab = ρ

δ

∫

0

u(y)dy, (4.1)

where ρ is the density of the incompressible fluid and u(y) is the velocity profile within the

boundary layer. The mass flow rate out of the system through ‘cd ’ surface can be derived using

the Taylor series expansion (neglecting the higher order terms) as,

mcd = ρ

δ

∫

0

u(y)dy +d

dx

⎛⎜⎝ρ

δ

∫

0

u(y)dy⎞⎟⎠

dx. (4.2)

The mass flow rate out of the system through ‘da’ surface due to suction can be given as,

mda = ρ (V0dx) , (4.3)


where V0 is assumed to be uniform suction velocity along the porous strip. Finally, the mass

flow rate into the system through ‘bc’ surface can be determined using the continuity equation

to be,

mbc =d

dx

⎛⎜⎝ρ

δ

∫

0

u(y)dy⎞⎟⎠

dx + ρ (V0dx) . (4.4)

Conservation of Momentum

The conservation of momentum is established along the x-direction. The momentum along the

x-direction is due to the mass flow through the surfaces ab, bc and cd. The momentum due to

the mass flow into the control volume can be given as,

Fab + Fbc = ρ

δ

∫

0

u2(y)dy +

ddx

⎛⎜⎝ρ

δ

∫

0

u(y)U∞dy⎞⎟⎠

dx + ρ (V0dx)U∞, (4.5)

where U∞ is the free stream velocity along the streamwise direction. The momentum along

x-direction due to the mass flowing out of the control volume can be given as,

Fcd = ρ

δ

∫

0

u2(y)dy +

ddx

⎛⎜⎝ρ

δ

∫

0

u2(y)dy

⎞⎟⎠

dx. (4.6)

Fcd − Fab − Fbc gives the total momentum acting along streamwise direction.

F =d

dx

⎛⎜⎝ρ

δ

∫

0

u2(y)dy

⎞⎟⎠

dx −d

dx

⎛⎜⎝ρ

δ

∫

0

u(y)U∞dy⎞⎟⎠

dx − ρ (V0dx)U∞. (4.7)

Since the pressure gradient is assumed to be zero, the only force acting on the flat plat (i.e. the

axisymmetric model) will be the shear stress acting along the suction strip. Therefore, the F

on the flat plate can be written as

F = −τwdx, (4.8)

−τw = ρd

dx

δ

∫

0

u2(y)dy − ρ

ddx

δ

∫

0

u(y)U∞dy − ρ (V0U∞) . (4.9)

The boundary layer thickness (δ) is defined as the distance normal from the wall to the

point where u(y) = 0.99U∞. The expressions for displacement thickness (δ∗), and momentum

thickness (θ) are given as


δ∗ =δ

∫

0

(1 −u(y)

U∞)dy, (4.10)

θ =

δ

∫

0

u(y)

U∞(1 −

u(y)

U∞)dy. (4.11)

Using the above definitions, the following equations can be derived

δ

∫

0

u (y)dy = U∞ (δ − δ∗) , (4.12)

δ

∫

0

u2(y)dy = U∞

δ

∫

0

u (y)dy −U2∞θ = U

2∞ (δ − δ∗ − θ) . (4.13)

With the above two relationships (4.12),(4.13), (4.9) can be simplified to,

τwρ

= U2∞

dθdx

+U∞V0. (4.14)

This can be represented in terms of δ by replacing θ with the relation in (4.13) as

τwρ

= U2∞

ddx

⎡⎢⎢⎢⎢⎣

δ

∫

0

u (y)

U∞(1 −

u (y)

U∞)dy

⎤⎥⎥⎥⎥⎦

+U∞V0. (4.15)

The turbulent boundary layer profile is assumed to remain self-similar along the length

of the suction strip and is approximated by u(y)/U∞ = (y/δ)1/7. Assuming the power-law

approximation, an empirical expression for the coefficient of friction of a turbulent boundary

layer over a flat plate without suction can be given as cf ≈ 0.02Re−1/6δ [64]. Based on the above

expression the wall shear stress can be written as,

τw = 0.01ρU2∞ (

ν

U∞δ)

16

. (4.16)

Substituting (4.16) in (4.15) and rewriting in terms of η, where η = y/δ, we get

0.01ρU2∞ (

ν

U∞δ)

16

= ρU2∞

dδdx

1

∫

0

η17 (1 − η

17 )dη + ρU∞V0, (4.17)

0.01(U5∞νδ

)

16

=772U∞

dδdx

+ V0. (4.18)


Let A = 0.01 (U5∞ν)16 , B = 7

72U∞ and h = δ16 , such that (4.18) can be expressed as,

dx = B6h6dhA − V0h

. (4.19)

The evolution of the boundary layer thickness can be calculated from (4.19) by numerically

integrating using a fourth-order Runge-Kutta method. However, to get more insight on the

expression, the above equation is further simplified by assuming s = A − V0h. The modified

equation is given as,

dx = −6BV 7

0

[A6dss

− 6A5ds + 15A4sds − 20A3s2ds + 15A2s3ds − 6As4ds + s5ds] . (4.20)

The above assumption is only valid for V0 /= 0. Choosing V0 = 0 makes s = A (a constant),

and thus ‘dh’ cannot be replaced by ‘ds’. Hence for the case where V0 = 0, the following

derivation is not applicable. For further simplification, (4.20) is then integrated on both sides

to get,

x = −6BV 7

0

[A6 ln ∣s∣ − 6A5s +152A4s2 −

203A3s3 +

154A2s4 −

65As5 +

s6

6] +C, (4.21)

where C is the integral constant. Substituting and expanding s and h in (4.21) gives,

x = −6BV 7

0

[A6 ln ∣A − V0δ16 ∣ −

14760

A6+A5V0δ

16 +

12A4V 2

0 δ13 +

13A3V 3

0 δ12 ]

−6BV 7

0

[14A2V 4

0 δ23 +

15AV 5

0 δ56 +

16V 6

0 δ] +C.

(4.22)

The constant ‘C’ in (4.22) can be determined by substituting the appropriate boundary

conditions. For instance, at x = 0 let the initial boundary layer thickness before the suction

strip δ = δ1, therefore the constant can be calculated as,

C =6BV 7

0

[A6 ln ∣A − V0δ161 ∣ −

14760

A6+A5V0δ

161 +

12A4V 2

0 δ131 +

13A3V 3

0 δ121 ]

+6BV 7

0

[14A2V 4

0 δ231 +

15AV 5

0 δ561 +

16V 6

0 δ1]

(4.23)

On substituting (4.23) in (4.22) the equation is simplified to,


x = −6BV 7

0

⎡⎢⎢⎢⎢⎢⎣

A6 lnRRRRRRRRRRRRR

A − V0δ16

A − V0δ161

RRRRRRRRRRRRR

+A5V0(δ16 − δ

161 ) +

12A4V 2

0 (δ13 − δ

131 ) +

13A3V 3

0 (δ12 − δ

121 )

⎤⎥⎥⎥⎥⎥⎦

−6BV 7

0

[+14A2V 4

0 (δ23 − δ

231 ) +

15AV 5

0 (δ56 − δ

561 ) +

16V 6

0 (δ − δ1)] .

(4.24)

It is interesting to note that in (4.24) when V0 → 0.01U∞Re−1/6δ , the streamwise distance

x →∞. This implies that, δ will asymptote to a value equal to (0.01 (U5∞ν)16 /V0)

6

as x →∞.

The asymptotic state of the boundary layer on suction is physically reasonable because it is

the condition when a balance exists between the free stream mass flow entering the boundary

layer due to entrainment and the mass flow leaving due to suction. This also suggests that

theoretically complete suction of the boundary layer is not possible, since it implies infinite

shear stress at the wall.

In the following, the validity of (4.24) is tested by comparing analytical predictions to ex-

perimental measurements. The experimental measurements on the evolution of boundary layer

with streamwise distance can be obtained from the suction control study conducted by Yoshioka

et al. [66] on a flat plate. Due to the complexity involved in measuring the turbulent boundary

layer thickness with suction, they measured the velocity profile at several streamwise distance

to calculate the displacement thickness.

The analytical expression given in equation (4.24) is based on a major assumption that the

turbulent boundary layer with suction follows a power law profile, i.e., u(y)/U∞ = (y/δ)1/7.

Hence the analytical and the experimental results can be comparable only if both have similar

velocity profiles. This can be verified by comparing the boundary layer evolution obtained

from analytical and experimental methods for the no suction case. The results are shown in

Figure 4.3.

For the cases when suction was applied the results suggest that the theory slightly over

predicts the effect of suction on displacement thickness in comparison to the experiments. The

percentage of error between the experimental and analytical results are maximum when the

suction velocity is large. The highest error recorded was approximately 27%. The displacement

thickness were measured at various streamwise locations (x/δ∗0 ) along the suction strip, where δ∗0is the displacement thickness at the leading edge of the strip. The analytical results are mostly

within 10% of the experimental value and therefore suggesting acceptable match between them.


0 0.5 1 1.5 2 2.54

5

6

7

8

9

10

11

12

Suction Strip Width (m)

Dis

plac

emen

t Thi

ckne

ss (

mm

)

V0 = 0.024 (m/s)

V0 = 0.016 (m/s)

V0 = 0.008 (m/s)

V0 = 0 (m/s)

V0 = 0.024 (m/s)

V0 = 0.016 (m/s)

V0 = 0.008 (m/s)

V0 = 0 (m/s)

Figure 4.3: Evolution of displacement thickness over a flat plate, with and without suction atthe wall. Symbols denote the experimental results and the lines denote the analytical resultscomputed from 4.19.

The slight mismatch between these results can be attributed to the several assumptions made to

arrive at 4.24. One of the major assumptions in the analytical method is that the velocity profile

of the boundary layer would remain similar with or without suction. The changes in boundary

layer properties and the velocity profile, however, has been well documented by Antonia et al.

[47]. Despite these assumptions, the analytical model predicts the evolution of boundary layer

within certain level of accuracy and hence is adequate for designing the porous strip for the

model.

4.4.2 Effects of Suction on Turbulent Boundary Layers

The method of controlling the boundary layer thickness through mass transfer is a commonly

used technique. The application of suction to control the boundary layer has its pros and cons

and hence it is important to understand the limitations of suction control for the current study.

The effect of suction can be understood by observing the characteristics of the boundary layer

evolving downstream of the suction strip. There have been several studies both experimental


[67], [47] and numerical [68] on the evolution of the boundary layer after suction. It can be

realized from these studies that the effect of suction causes suppression in turbulence level by

removing the near-wall eddies from the turbulent boundary layer, which causes a significant

change in the boundary layer characteristics. For higher suction rates, the turbulent boundary

layer has been observed to undergo relaminarization close to the wall. Nevertheless, the changes

that occur in the boundary layer due to suction fades away downstream as the boundary layer

recovers to its original velocity profile, however, with a smaller boundary layer thickness than

it would have had without suction. If those changes in the turbulence intensity within the

boundary layer were to happen near the separation point of the model it will alter the shear

layer and consequently the flow characteristics behind the step [10]. Hence it is necessary to

locate the suction strip such that the boundary layer recovers to a full-developed state before

separation.

The evolution of boundary layer properties at various streamwise locations downstream of

suction strip were extensively studied by Antonia et al. [47]. Their experiments were limited to

the application of suction to a flow with self preserved turbulent boundary layer over a flat plate.

Their measurements included variables such as skin friction coefficient (Cf ) and the Reynolds

stresses u′2, v′2 and −u′v′ at different streamwise locations. They conducted the experiments

for various levels of non-dimensionalized suction rate (σ). The suction rate, σ, is given by,

σ =V0w

U∞θ0, (4.25)

where V0 is the suction velocity, w the suction strip width and θ0 the momentum thickness

of the turbulent boundary layer at the leading edge of the suction strip when no suction is

applied. According to Antonia et al. , the suction rate (σ) required for completely removing the

boundary layer can be calculated to be approximately 10. In their experiments, they applied

different suction rates namely, 2.6, 5.2 and 6.5, and observed the changes in the boundary layer

properties immediately downstream of the strip. Based on the Reynolds stress measurements

acquired from different downstream locations, they were able to predict the approximate recov-

ery distance required for the boundary layer to develop back to its original state. For a suction

rate, σ = 2.6, the recovery length for the boundary layer was determined to be approximately

40δ0, where δ0 is the boundary layer thickness at the leading edge of the suction strip for the

no suction case. As for higher suction rates, the boundary layer was observed to be still in the

process of recovery at the farthest measurement location, which was 63δ0 downstream from the

strip.


The application of suction to control boundary layer thickness in a BFS flow provides two

benefits. Firstly, it is essential in studying the characteristics of the step wake independent of

the boundary layer thickness effects. Secondly, it provides an opportunity to extend the study

conducted by Adams and Johnston [9] in understanding the effects of boundary layer thickness

independent of Reynolds number on the reattachment length. To be able to perform both, it

must be possible to achieve a wide range of boundary layer thicknesses at the separation point

independent of the Reynolds number. Based on the above discussion, the turbulent boundary

layer thickness achievable after recovery with one suction strip is dependent on δ0 and the

suction velocity. Therefore, the range of boundary layer thickness that can be attained with

one suction strip is limited by those parameters. This range can be increased by incorporating

multiple porous strips. Experiments with double suction strip to delay the transition in bound-

ary layer were performed by Oyewola et al. [69]. They demonstrated that the transition of the

relaminarized boundary layer from the first suction strip can be delayed by introducing a second

suction strip before the transition starts. However, for the current design the objective is to

hasten the recovery process while smaller boundary layer thickness can be achieved. Therefore,

with multiple suction strip in place it must be ensured that strips downstream of the first one

are placed near the location where the boundary layer has nearly fully recovered from the effect

of suction. However, choosing fixed locations for the porous strips is difficult as the boundary

layer recovery process depends on the operating Reynolds number of the flow.

For the current model design, the length of the model from the stagnation point at the nose

to the step was selected to be 1450 mm. The model is designed long so that the boundary layer

will be fully-developed turbulent at separation and also contain multiple porous strips if neces-

sary. To have a large range of boundary layer thickness at the separation point, three porous

strips 40 mm wide each are incorporated in the upstream length. The porous strip width was

chosen based on the model by Antonia et al. [47], which was adequate for complete removal of

the boundary layer. The location for the three porous strips were predicted based on equation

(4.19), but due to the variable Reynolds number range to be used in the experiments, some

flexibility was required. The porous strips were located at 750 mm, 1000 mm and 1200 mm

from the trailing edge of the nose cone. The first suction strip is placed near the center of

the bigger cylinder to ensure the boundary layer is fully developed and turbulent. The porous

strips are designed to be interchangeable so that their widths can be adjusted by placing two or

three strips adjacent to each other. Moreover, the bigger cylinder is made of five interchange-

able parts as shown in Figure 4.4 so that the location of the porous strips can be adjusted if

necessary. Furthermore, the suction velocity through each strip is controlled independently via

individual control valves to provide further flexibility to the suction system.


!"#"$%&%'#()%&

*'+)&

&,"-+.&

*$))"#'&

Figure 4.4: Axisymmetric backward-facing step model.

The evolution of boundary layer thickness upstream of the step is calculated using [4.19] for

certain free-stream and suction velocities. The boundary layer thickness at different conditions

are shown in Figure 4.5. The porous strip location were also altered to achieve different bound-

ary layer thickness at separation. The first four cases shown in the legend, are for strip location

placed at 750 mm, 1000 mm and 1200 mm from the trailing edge of the nose cone respectively.

The last two cases shown in the legend are when the strips are located at 450 mm, 750 mm

and 1000 mm from the trailing edge of the nose cone respectively. The option of changing the

porous strip location also provides more distance for the boundary layer to recover to its origi-

nal state. The range of boundary layer thickness that can be achieved by altering the suction

velocity and free stream velocity is between 2.88 δ/H and 0.33 δ/H.


Figure 4.5: Evolution of boundary layer thickness for different suction velocity.

4.5 Suction System Design

Apart from choosing the porous strip width and their locations in the axisymmetric model, it

is also essential to ensure that the application of suction is uniform along the surface of the

model. Similar to the experimental set up conducted by Antonia et al. [47], for the current

model plenum chambers are installed between the porous strip and a vacuum pump. An illus-

tration of the suction system design used for the current model is shown in Figure 4.6. Each

Plenum chamber is built with two holes, outer diameter 5/32”, for air tubings to achieve uni-

form suction along its circumference. The metal link between each plenum chamber, as seen

in figure, is required for the structural integrity of the model. Nevertheless, the metal link is

designed concave (curved inwards) on the top surface, as shown in Figure 4.7, to allow air flow

between chambers and thus create a 360○ suction surface without discontinuity.

Furthermore, in order to estimate the approximate power required to activate the suction

strip, a simple experiment was conducted to determine the pressure drop across a porous strip.

A sketch of the experimental set up is shown in Figure 4.8. A square, 4” × 4”, porous sample


Figure 4.6: Plenum chamber design within the suction strip system.

Figure 4.7: Cross sectional view of a plenum chamber.

with pore density 40 µm was used for this experiment. The porous strip was installed in front

of a 2 inch inner diameter (D) and 6 feet long acrylic pipe. The dynamic pressure was measured

using a pitot static tube, which was connected to a 223-B MKS Baratron differential pressure

transducer with full scale range up to 1 mbar and accuracy of 5% of full scale reading. The

pitot static tube was installed at about 20D downstream from the porous strip to allow the flow

to achieve developed turbulent state. The static pressure at the same location was separately

monitored using a Betz manometer. During the experiment, each measurement was sampled

thirty times for a time period of one second to minimize the precision uncertainty. The precision

uncertainty for 95% confidence is given by 2σ/N , where σ is the standard deviation and N is

the total number of samples (in this case 30) per data. The total uncertainty for the velocity

measured in this experiment was calculated to be within ±6%


Vacuum pump!

Ball valve!

Pressure transducer!

Porous plate!

Pitot static tube!

Figure 4.8: Experimental set up to measure pressure drop across a porous plate.

The velocity profile thus obtained from the measured dynamic pressure is shown in Fig-

ure 4.9. The experimental data is in good agreement with a power law fit of a fully developed

turbulent pipe flow profile. The power law fit is given by U/Umax = (r/R)(1/7), where U is the

instantaneous velocity, Umax is the maximum velocity, r is an instantaneous position in the

radial direction of the pipe and R is the radius of the pipe. From the velocity profile, the mean

velocity is calculated to be 3.73 m/s. Using Bernoulli’s equation the pressure drop (Pd) across

the porous plate can be given as,

Pd = P1 − P2 −12ρV 2

2 − Pl, (4.26)

where P1 is the atmospheric pressure, P2 is the static pressure near the pitot static tube, ρ is the

density of air, V2 is the average flow velocity inside the pipe and Pl is the pressure loss due to

the sudden entrance. The difference between the static pressures, P1−P2 was measured using a

Betz manometer to be approximately 107.91 Pa. The pressure loss due to the sudden entrance

can be calculated using the formula, Pl = KV 22 /2, where K is 0.78 for reentrant type pipe

entrance [70]. Based on the above results, the ratio of pressure drop to dynamic pressure at the

location of measurement per square inch of the porous plate is determined to be approximately

4.13. Therefore the non-dimensionalized pressure drop across three porous strips in the model

for the same suction velocity can be calculated to be approximately 525.5.

4.6 Model Support

The support is essential for centering the model inside the test section. It is also necessary for

holding the model firmly to minimize any deflection due to its own weight and avoid the unde-


−0.2 0 0.2 0.4 0.6 0.8 1 1.20

0.2

0.4

0.6

0.8

1

1.2

1.4

r/R

U/U

max

experimental datapower law fit

Figure 4.9: Velocity profile of a fully developed turbulent pipe flow.

sirable structural vibration. In the current research, the field of study is at the wake of the step

and hence placing the model support upstream of the step could affect the wake characteristics.

Alternatively, the model support can be located downstream from the step.

In an experimental study conducted on a BFS, Farabee and Casarella [26] observed the

large scale turbulent structures shedding and convecting downstream from the separated shear

layer. These turbulent structures were detected to be steadily decaying and diffusing away as

they progressed downstream and yet were identifiable as far as 72H from the separation point.

Based on these observations, it will be preferable to locate the support at least 72H to minimize

noise due to the support from propagating upstream. Simultaneously, locating the support too

far downstream could lead to large deflections closer to the leading edge of the model due its

own weight. Such deflections can be a major issue for the current study because of their effects

on the flow symmetry over the model.

For the current study, the support is located at 932 mm, i.e., 74.5H, downstream from the

step and the total length of the smaller cylinder is 1.5 m. Based on the finite element analysis

(carried out by the manufacturer), for the aforementioned dimensions the cantilever type model

was ensured minimal deflections, which is approximately 0.36 mm at the leading edge. The

height of the support is chosen as 287.5 mm to center the model inside the test section. The


support cross-section was designed using the NACA 0012 profile. The aerodynamic section

provides minimum flow blockage, while being structurally stiff and allowing enough space to

pass the suction system piping.

4.7 Nose and Trailing Cone Design

The nose and the trailing cone are integral parts of the axisymmetric model. They are necessary

to keep the flow attached and therefore minimize aerodynamic disturbances. In the following

section, the selection of nose and trailing cone using potential flow theory will be discussed

briefly.

4.7.1 Axisymmetric Nose Design

The objective for the nose design is for the flow to remain attached along the surface at any

given velocity within the operating range. Although the nose model is axisymmetric, a two-

dimensional analysis is used for the current study because it is the most stringent with respect

to separation and is simpler to perform with available tools. For better understanding over

the choice of two-dimensional analysis, as an example the variation of coefficient of pressure

Cp = 2(P−P∞)/(ρU2∞) data over the surface of a cylinder and a sphere, shown in Figure 4.11, are

studied here. It can be seen from the figure that the change in pressure is smoother in the case

of a sphere, implying a less chance of separation in comparison to the cylinder. This suggests

that designing a nose profile based on two-dimensional analysis will only ensure better chances

for the flow to remain attached. Therefore for the current study a 2D analysis is sufficient. The

other requirements for the nose design include short length and light weight to minimize the

deflections at the leading edge of the model. The nose design procedure involves two steps and

they are,

• creating co-ordinates for the nose profile using potential flow theory and

• analysing the flow characteristics over the profile using Xfoil.

Some of the shapes considered for the nose cone design are hemispherical nose, elliptical

nose and the leading edge of the symmetrical NACA airfoil profiles.

Hemispherical Nose Profile

A hemispherical profile for the nose can be designed using potential flow theory by assuming

it as the combination of a uniform flow and a source flow. For the current design, with the


! ! !

Fig

ure

4.10

:Sc

hem

atic

ofth

eax

isym

met

ric

mod

elw

ith

dim

ensi

ons.


Figure 4.11: Pressure distribution over a cylinder and a sphere [71].

diameter of the model known the hemispherical nose can be easily designed. The co-ordinates

for the nose are obtained using the equation of a circle, which is given by,

x2+ y2

= R2, (4.27)

where x and y are the co-ordinates of the circle and R is the radius of the circle. The flow over the

resulting profile is then analysed using Xfoil and the coefficient of pressure (Cp = 2(P−P∞)ρU2∞ )

along the surface of the nose are obtained, where P is the local static pressure and P∞ is the free

stream static pressure. Since the flow is more likely to stay attached when the flow is turbulent

the analysis is tested for the lowest operational velocity, 5 m/s. A graphical representation of

the change in Cp with normalized streamwise distance x/xn, where xn is the length of the nose

profile, is shown in Figure 4.12. The length for the hemispherical model is equal to the radius

of the hemisphere, i.e., 125 mm.

Rankine Oval

The combination of uniform, source and sink flow creates an elliptical profile, which is also

commonly referred to as the Rankine oval. The co-ordinates for the Rankine oval can be

generated by knowing the major and minor diameters of the ellipse. The minor radius (Ra) of


the nose profile is equal to the radius of the cylinder attached to the nose, which is 125 mm.

The major radius (Rb) is dependent on two variables and they are the strength of the source or

sink flow (Λ) and the distance between the source and the sink (2d). For the current case, the

strength of the source and the sink are the same and can be determined based on minor radius

and different values of d. The source or sink strength Λ is given by,

Λ =2πU∞Ra

tan−1 ( 2dRa

R2a−d2 )

. (4.28)

Knowing the source strength and the distance between the source and sink, the major radius

can be determined using the following expression,

Rb =

√

d2 +ΛdπU∞

. (4.29)

Rankine oval profiles were considered for five different d values namely, 0.1, 0.2, 0.3, 0.4 and

0.5. The corresponding major radius or the length of the nose profile for all the above cases

are 169 mm, 254 mm, 349 mm, 447 mm and 0.545 mm respectively. Their equivalent major

to minor radius ratios are 1.352, 2.032, 2.792, 3.576 and 4.36 respectively. The front half of

the ellipse profile is then combined with the profile of a long cylinder and is then inputted into

Xfoil for analysis. The change in Cp with x/xn, for the elliptical leading edge are included in

Figure 4.12.

Symmetrical NACA Airfoil

The symmetrical NACA airfoils chosen for the current study are 0012, 0015, 0018, 0021, 0024,

0027 and 0030. The co-ordinates for the above mentioned symmetrical airfoil profiles can be

obtained from the following equation,

y =t

20(0.2969

√x − 0.126x − 0.3516x2

+ 0.2843x3− 0.1015x4) , (4.30)

where t is the thickness to chord ratio of the airfoil. The lengths of the leading edge in the afore-

mentioned NACA airfoil profiles are 625 mm, 500 mm, 417 mm, 357 mm, 313 mm, 277.8 mm and

250 mm respectively. The results of Cp versus x/xn for these airfoils are included in Figure 4.12.

It can be observed from Figure 4.12 that along the surface of each nose profiles the Cptends to decrease initially to a minimum value, Cpmin , and then increase to a constant value

on the straight section of the model. For the hemispherical model, Cpmin reaches the largest

amplitude in comparison to the rest of the nose model designs. This suggests that the wall

static pressure on the hemispherical nose model varies largely along the curvature and hence is


0 0.5 1 1.5 2−2

−1.5

−1

−0.5

0

0.5

1

Normalized streamwise distance (x/xn)

Coe

ffici

ent o

f Pre

ssur

e (C

p)

doubletRankine−0.1Rankine−0.2Rankine−0.3Rankine−0.4Rankine−0.5airfoil−0012airfoil−0015airfoil−0018airfoil−0021airfoil−0024airfoil−0027airfoil−0030

Figure 4.12: Coefficient of pressure versus the normalized streamwise distance for different nosemodels

most likely for the flow to separate. This is also evident from Figure 4.13, which suggests that

the flow experiences adverse pressure gradient close to the junction connecting the nose and

the cylinder, i.e., x/xn = 1.

In the case of the Rankine oval models the Cpmin value increases with the increase in the

length of the nose, i.e., the major axis radius of the ellipse. It can be seen from the Figure 4.13

that the pressure gradient becomes positive for small nose lengths. This suggests that the

probability of the flow separating are higher. For the NACA symmetrical airfoils, with the

decrease in thickness to chord ratio the Cpmin increases gradually. Consequently, the pressure

gradient over the surface changes gradually and reaches positive but small values close to the

nose and cylinder junction for higher t. Although NACA airfoils with smaller thickness performs

better, they have large lengths. Based on the information from the plots, NACA 0027 with a

nose length of 0.2778 m is preferred for the axisymmetric model because of its reactively small


0 0.5 1 1.5 2−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

Normalized streamwise distance (x/xn)

Pre

ssur

e gr

adie

nt (

dP/d

x)/1

000

N/m

3

doubletRankine−0.1Rankine−0.2Rankine−0.3Rankine−0.4Rankine−0.5airfoil−0012airfoil−0015airfoil−0018airfoil−0021airfoil−0024airfoil−0027airfoil−0030

Figure 4.13: Pressure gradient versus the normalized streamwise distance for different nosemodels

length and small positive pressure gradient.

4.7.2 Trailing Edge Cone Design

The tail cone forms the trailing edge of the axisymmetric model. In place of a blunt trailing

edge, a tail cone plays an important role in minimizing the aerodynamic disturbances due to

separation. For the current model, the half cone angle of the tail is chosen as 11 degrees,

based on Hudy’s [28] model. The cone angle was appropriate for Hudy’s study and besides the

resulting length, 580 mm, is appropriate for the space available in the tunnel.

Chapter 5

Conclusions and Recommendations

5.1 Conclusions

The characteristics of a separating and reattaching flow on a BFS geometry was reviewed exten-

sively in this thesis. Despite numerous experimental studies on a BFS, definite understanding of

the flow characteristics has not been accomplished because of the different parametric conditions

in each experiments. In a review by Eaton and Johnston [7], they identified few geometrical

and flow parameters such as, expansion ratio, aspect ratio, free stream turbulence intensity,

boundary layer state and thickness at separation to have significant effect on the reattachment

length. The individual effects of each aforementioned parameters were reviewed in this thesis.

The effect of aspect ratio was extensively investigated by Brederode and Bradshaw [8]. They

suggested that three-dimensional effects in the wake can be minimized close to the centerline

by using a planar model with an aspect ratio greater than 10. Even though more larger aspect

ratio in a planar BFS model can produce nearly two-dimensional wake, a better alternative is

to use an axisymmetric geometry with an expansion ratio close to unity. Hudy [28] studied

the flow over an axisymmetric model which not only had minimal three dimensional effects,

but also uniform wake along the circumference. The effects of free stream turbulence intensity

on the reattachment length was studied by Isomoto and Honami [10] by using different grids

upstream of the step. They found that the reattachment length grew shorter with increasing

free stream turbulence intensity and attributed it to the increase in turbulence within the shear

layer. In the case of a boundary layer state, Eaton and Johnston [7] demonstrated that it had a

nonlinear relationship with the reattachment length. However, an interesting observation was

that the reattachment length had little or no effect on the boundary layer thickness when it

was fully developed turbulent.

In the current study, using the data obtained from different experiments, the combined ef-

53

Chapter 5. Conclusions and Recommendations 54

fect of parameters on the reattachment length was investigated. A new scaling factor, the sum

of boundary layer thickness at separation and step height, was proposed for the reattachment

length. The novel scaling parameter physically represents the dimension of a geometric and

viscous boundary. With the new scaling and by studying the effects of boundary layer thick-

ness on the reattachment length, it was observed that the data collapsed with a second order

polynomial. This curve fit acts as a visual guide and can be a potential model for predicting

the reattachment length. The boundary layer thickness is known to be affected by both the

Reynolds number (Rex) and the free stream turbulence, and hence its inclusion in the length

scale appears to account for their effects on the reattachment length. This also suggests that

these two parameters are important as much as they affect the state of the flow at separation.

Since the information of the data used in this analysis was limited, more targeted experiments

will be required to further verify this collapse.

The extensive review on the effects of parameters on separated shear layer can be useful in

designing a model to either eliminate or control the parametric effects. In the current study, an

axisymmetric BFS model is designed. The model dimensions such as the bigger cylinder radii

and the step height were chosen as 125 mm and 12.5 mm respectively. These aforementioned

dimensions were particularly designed based on Hudy’s [28] set up, to further investigate the

circumference effect for a bigger radii ratio. The upstream cylinder is designed 1.45 m long to

accommodate three porous strips and also ensure turbulent boundary layer at separation. The

porous strips were installed to control the boundary layer thickness at separation independent

of the Reynolds number. The nose cone for the axisymmetric geometry is designed from a

NACA 0027 profile based on two-dimensional flow analysis to avoid flow separation.

5.2 Future Experimentation

The literature review study conducted on the flow characteristics in the wake of a BFS provides

a number of interesting topics to investigate in the future.

• The unusually short reattachment length observed in the previously tested axisymmetric

geometries were believed to be due to the circumference effect. The current model specif-

ically designed twice as big as the Hudy’s model [28], while maintaining the same step

height, can provide some insight on the circumference effect.

• In the previously conducted boundary layer thickness control experiment by Adams and

Johnston [46], they did not account for the change in boundary layer properties due to

suction. Hence with the suction control available in the current model, the effects of

Chapter 5. Conclusions and Recommendations 55

boundary layer thickness independent of Reynolds number on the reattachment length

can be conducted to provide more quantitative data. Further more, the effect of Reynolds

number on the reattachment length can also be examined.

• The effect of turbulence intensity on the reattachment length has been previously stud-

ied by Isomoto and Honami [10]. Nonetheless, conducting a similar investigation using

structured grids upstream of the model will not only provide more data to support their

results, it can also be useful to validate the new scaling length proposed in this work.

• The effect of wall suction on the boundary layer properties over a flat plate were studied

by Antonia et al. [47]. In the current model the effect of wall suction on the boundary

layer over a curved surface can be investigated.

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