Effects of Wall Roughness on Adverse Pressure Gradient Boundary Layers

Pouya Mottaghian

A thesis submitted to the

Department of Mechanical and Materials Engineering

in conformity with the requirements for

the degree of Master of Applied Science

Queen's University

Kingston, Ontario, Canada

December, 2015

Large-eddy Simulations were carried out on a at-plate boundary layer over smooth and rough surfaces in the presence of an adverse pressure gradient, strong enough to induce separation. The inlet Reynolds number (based on freestream velocity and momentum thickness at the reference plane) is 2300. A sand-grain roughness model was implemented and spatial-resolution requirements were determined.

Two roughness heights were used and a fully-rough ow condition is achieved at the reference plane with roughness Reynolds numbers 60 and 120. As the friction velocity decreases due to the adverse pressure gradient the roughness Reynolds number varies from fully-rough to transitionally rough and smooth regime before the separation. The double-averaging approach illustrates how the roughness contribution decreases before the separation as the dispersive stresses decrease markedly compared to the upstream region.

Before the ow detachment, roughness intensies the Reynolds stresses. After the separation, the normal stresses, production and dissipation substantially increase through the adverse pressure gradient region. In the recovery region, the ow is highly three dimensional, as turbulent structures impinge on the wall at the reattachment region.

Roughness initially increases the skin friction, then causes it to decrease faster than on a smooth wall, generating a considerably larger recirculation bubble for rough cases with earlier separation and later reattachment; increasing the wall roughness also leads to larger separation bubble. In addition, roughness causes early ow reversal upstream of the real

i separation (which occurs when the zero-velocity line moves away from the wall) because the small-scale separation regions downstream of the roughness elements become larger and merge together as a result of the APG. However, this ow reversal remains below the roughness crest.

The reasons for the earlier separation are larger momentum decit in rough-wall ows and the shutting down of the production of −hu0v0i both before and after the separation, mainly due to the decrease in the velocity gradient in the outer layer. After the separation, roughness eects can be felt throughout the boundary layer because of the advection of near-wall uid around the recirculation region.

ii Acknowledgment

I would like to express my sincere gratitude to my advisor Professor Ugo Piomelli for the continuous support of my master's study and related research, for his patience, motivation, and immense knowledge. His guidance helped me in all the time of research and writing of this thesis. I could not have imagined having a better advisor and mentor for my master's study.

I am thankful to Junlin Yuan for her support and encouragement whenever I was in need and completion of this project was impossible without her assistance. I would like to thank Amirreza Rouhi, who as a good friend, was always willing to help and give his best suggestions. It would have been a lonely lab without him. I thank my fellow colleagues at Turbulence Simulation and Modelling laboratory Rayhaneh, Rabijit, Wen, Mojtaba and

Divya for their help and support in this project.

Also I would like to thank High Performance Computing Virtual Laboratory, Queens

University site, for the computational support throughout my research.

At the end I would like to thank my family; my parents, Nahid and Mohammad Ali, and to my brother, Nima, and my aunt, Jaleh, for supporting me spiritually throughout writing this thesis and my life in general.

iii Nomenclature

Acronyms

APG Adverse pressure gradient

CFD Computational uid dynamics

DA Double-averaging

DNS Direct numerical simulation

FPG Favourable pressure gradient

IBM Immersed boundary method

LES Large-eddy simulation

MPI Message passing interface

RHS Right-hand-side

RMS Root-mean-square

RANS Reynolds-averaged Navier-Stokes

SFS Sub-lter stress

TKE Turbulent kinetic energy

ZPG Zero pressure gradient

Roman symbols

B Mean-velocity prole intercept in the logarithmic region

Ce Model parameter

iv Cf Friction coecient d Zero-plane displacement

Fi IBM body force in i direction G Filter function h Channel half-height

H Shape factor

Lij Germano identity k Certain quantication of roughness height in an average sense ks Equivalent sand-grain height k+ Roughness Reynolds number kc Roughness crest kR Top of roughness sublayer K Acceleration parameter

K Turbulent kinetic energy

Lx, Ly, Lz Domain size

Nx, Nz Horizontal resolution of a single roughness element P Pressure

P Shear production

Q Second invariant of the velocity tensor

Re Reynolds number

Reθ Reynolds number based on θ and U∞

Reτ Reynolds number based on channel half height (h) and uτ

Sij Strain rate tensor t Time

T Total simulation time

Tij Resolved turbulent stresses

v uτ Friction velocity

Ui Mean velocity components

Ui,∞ Freestream velocity

Uc Convection velocity

Ucrest Streamwise velocity at roughness crest

Uδ Streamwise velocity at boundary layer thickness xi Direction xs Separation location xr Reattachment location

Greek symbols

∆x, ∆y, ∆z Grid spacing

∆U + Roughness function

∆ Grid lter width

∆b Test lter width δ Boundary layer thickness

δ∗ Displacement thickness

δν Viscous length scale Viscous dissipation

θ Momentum thickness

κ Von Kármán constant

ν Kinematic viscosity

νt Turbulent eddy-viscosity ρ Density

µ Dynamic viscosity

τ Total shear stress

vi τij Subgrid lter stress a Anisotropic residual-stress tensor τij

τw Wall shear stress φ Fraction of a grid cell occupied by uid

ω Turbulent vorticity

Ω Rotation rate tensor

ψ Streamline

Others symbols

(·) Filtering at grid level

(c·) Filtering at test level h(·)i Averaging in time and spanwise

(f·) Spatial variation of time-averaged quantity (·)0 Turbulent uctuations

(·)+ Non-dimensional quantity normalized by inner scaling

(·)o Quantity at reference plane of simulation domain

vii Table of Contents

Abstract i

Acknowledgements iii

Nomenclature iv

Table of Contents viii

List of Tables x

List of Figures xi

Chapter 1:

Introduction ...... 1

1.1 Motivation ...... 1

1.2 Literature review ...... 5

1.3 Objectives ...... 15

Chapter 2:

Problem Formulation ...... 16

2.1 Introduction ...... 16

2.2 Governing Equation ...... 16

2.3 Time-advancement and discretization ...... 21

2.4 Boundary conditions ...... 24 viii 2.5 Immersed-boundary method (IBM) ...... 24

2.6 Calculation of wall shear-stress ...... 26

2.7 Time averaging and double averaging (DA) ...... 27

Chapter 3:

Model Validation ...... 29

3.1 Introduction ...... 29

3.2 Rough-wall channel ow ...... 29

3.3 Adverse pressure gradient boundary layer ...... 33

Chapter 4:

Results ...... 38

4.1 Introduction ...... 38

4.2 Case setup ...... 38

4.3 Smooth wall APG ...... 41

4.4 Rough wall APG ...... 49

4.5 Separation physics ...... 58

Chapter 5:

Conclusions ...... 65

Bibliography ...... 68

ix List of Tables

3.1 Open channel grid size, grid resolution and number of grids per each roughness

element...... 31

3.2 Summary of current LES and DNS by Na & Moin (1998a)...... 34

4.1 Summary of simulation parameters...... 39

4.2 Number of grid for majority of roughness ellipsoids...... 41

4.3 Drag force per spanwise length...... 50

4.4 Separation bubble size. In the table, xs denotes the position of separation

(Cf = 0), xr is the position of reattachment, ls is the length of the separated

region, and Hsep is the shape factor at the position of separation...... 54

x List of Figures

1.1 Flow over airfoil, accelerating (favourable pressure gradient) and decelerating

(adverse pressure gradient)...... 2

1.2 Massive blade erosion...... 2

2.1 Schematic of computational domain...... 17

2.2 The staggered-grid arrangement. The u, v and pressure cells are indicated with blue, red and green colors respectively...... 23

3.1 Grid renement study, roughness function. DNS of Scotti (2006), ∗ experiment of Colebrook & White (1937), LES of + , LES . ks = 20, 96 × 96 × 96 / of + , LES of + ...... 31 ks = 20, 128 × 140 × 192 O ks = 20, 192 × 208 × 256 3.2 Mean streamwise velocity prole in wall coordinate at + . DNS of ks = 20 Scotti, smooth; , LES + , , LES + ks = 20, 192 × 208 × 256 ks = 20, 192 × 208 × 256...... 32 3.3 Proles of (a) streamwise (b) wall-normal (c) spanwise uctuations; ,

Scotti's DNS, smooth; Scotti DNS, + ; , LES + ks = 20 ks = 20, 192 × 208 × 256...... 33 3.4 Computational setup...... 34

3.5 Streamwise and wall-normal velocity prole along the freestream...... 35

xi 3.6 (a) Development of friction coecient. (b) Proles of the streamwise veloc-

ity component before separation (c) after separation. Each prole is shifted

upwards by 20 units for clarity; • DNS by Na & Moin (1998a); , LES; , logarithmic law of the wall ...... 35

3.7 (a) Development of friction coecient. (b) Proles of root-mean square pres-

sure uctuations shifted upwards by 0.01 units for clarity; • DNS by Na & Moin (1998b); , LES ...... 36

4.1 Location of the rescaling and the gradual imposing of the roughness...... 40

4.2 Sand-grain roughness obtained by Scotti's model at iso-surface of φ = 0.9 for

case 3...... 40

4.3 Mean velocity proles at the reference plane; , Smooth; , k/θo =

+ + 0.47 (k = 60); , k/θo = 0.95 (k = 120) ...... 41

4.4 , freestream velocity U∞,o and , acceleration parameter K for all the cases...... 42

4.5 Distribution of mean (a) streamwise and (b) wall-normal velocity for case 1

at Reθ = 2300; , mean streamline; , zero velocity line ...... 42 4.6 Streamlines through the domain...... 43

4.7 (a) ,U∞; ,Uδ; (b) distribution of Cf for case 1...... 44

4.8 History of the location of zero Cf of the spanwise-averaged (a) separation

(XS) and (b) reattachment (XR) point • for case 1...... 44 4.9 Distribution of mean and turbulence statistics for case 1; (a),(b) before the

separation at x/θo = 120; (c),(d) inside the separation bubble at x/θo = 250;

(e),(f) in the recovery region at x/θo = 400; , mean streamwise velocity; , hu0u0i; , hv0v0i; , −hu0v0i...... 45

xii 4.10 Distribution of turbulence statistics for case 1; , mean streamline; ,

zero velocity line: (a) turbulent kinetic energy; (b) production of turbulent

kinetic energy; (c) dissipation of turbulent kinetic energy...... 46

4.11 Isosurface of instantaneous Q > 0.05 for case 1. The visualization domain

size shown here is 330θo × 50θo × 60θo...... 48 4.12 Isosurface of instantaneous u0 for case 1; light pink is u0 = −0.15 and purple

0 is u = 0.15. The visualization domain size shown here is 330θo × 50θo × 60θo. 48

0 + 4.13 Instantaneous contour of u normalized by uτ,o for case 1 at y = 12, the

visualization domain size shown here is 330θo × 50θo × 60θo...... 49

4.14 (a) Distribution of Cf ; (b) streamwise velocity at roughness crest; ,

Smooth; , k/θo = 0.47; , k/θo = 0.95 ...... 50

4.15 Total stress above the crest; , Smooth; , k/θo = 0.47; ,

k/θo = 0.95...... 51 4.16 Streamwise velocity contours; , mean streamline; , zero velocity

line: (a) case 1; (b) case 2; (c) case 3 at table 4.1...... 51

4.17 Time averaged streamwise velocity at y/θo = 0.4 and y/θo = 0.8; , zero

velocity line for k/θo = 0.47 (case 2)...... 53

4.18 The evolution of the shape factor for , Smooth; , k/θo = 0.47;

, k/θo = 0.95...... 53 4.19 Proles of mean streamwise velocity; (a) before the ow detachment (shifted

upward by 10 units); (b) after the ow reattachment (shifted upward by

20 units); , smooth; , k/θo = 0.47; , k/θo = 0.95; , logarithmic law of the wall...... 55

4.20 Mean streamwise velocity proles normalized by local freestream velocity

(U∞); , smooth; , k/θo = 0.47; , k/θo = 0.95; shifted rightward by 1 unit...... 56

xiii 4.21 Contour of streamwise Reynolds stress hu0u0i normalized by mixed scaling at

reference plane (uτ,oU∞,o); (a), smooth; (b), k/θo = 0.47; (c), k/θo = 0.95;

, Cf = 0; , boundary layer thickness...... 56 4.22 Contour of wall-normal Reynolds stress hv0v0i normalized by mixed scaling at

reference plane (uτ,oU∞,o); (a), smooth; (b), k/θo = 0.47; (c), k/θo = 0.95;

, Cf = 0; , boundary layer thickness...... 57 4.23 Contour of spanwise Reynolds stress hw0w0i normalized by mixed scaling at

reference plane (uτ,oU∞,o); (a), smooth; (b), k/θo = 0.47; (c), k/θo = 0.95;

, Cf = 0; , boundary layer thickness...... 58 4.24 Mean streamwise velocity prole before separation; , Smooth; ,

k/θo = 0.47; , k/θo = 0.95...... 59 4.25 (a) Streamwise velocity contour and streamline near the roughness crest for

case 2; (b) production of turbulent kinetic energy along the streamline; (c)

−hu0v0i along the streamline; (d) ∂hUi/∂y along the streamline; , case

1; , case 2; , case 3 ...... 60

4.26 (a) Streamwise velocity contour and streamline near the roughness crest for

case 2; (b) production of −hu0v0i along the streamline; (c) hv0v0i along the

streamline; (d) ∂hUi/∂y along the streamline; , case 1; , case 2;

, case 3 ...... 61

4.27 Double average components of streamwise velocity for k/θo = 0.47 (case 2) at

y/θo = 0.4; (a) total velocity; (b) time-averaged velocity; (c) wake velocity; (d) velocity uctuations...... 62 4.28 Streamwise dispersive stress 2 + inside the roughness sublayer ( ); hue i 2ks (a)

k/θo = 0.47 (case 2); (b) k/θo = 0.95 (case 3); , cf = 0...... 62

4.29 Proles of the dispersive stresses at x/θo = 40 normalized by reference friction

velocity (uτ,o); k/θo = 0.47 (empty symbols) and k/θo = 0.95 (lled symbols). 2 +; 2 +; 2 +; + ...... 63 (a) hue i (b) hve i hwe i huevei xiv 4.30 Proles of the dispersive stresses at x/θo = 40 ( ) and x/θo = 110 ( )

for k/θo = 0.47 (empty symbols, a,b) and k/θo = 0.95 (lled symbols, c,d) roughness heights. 2 +; 2 +; 2 +; +...... 64 hue i hve i hwe i huevei

xv Chapter 1

Introduction

1.1 Motivation

Separation of the turbulent boundary layer is among the most critical phenomena that play role in the eciency of ow devices, ranging from airplane wings to turbine and compressor blades, to curved ducts; their maximum eciency is often at an operational point close to the onset of separation. The accurate prediction of incipient separation and reattachment in these congurations is critical. This unwanted interaction causes a reduction in the performance of the ow device of interest (e.g., a loss of lift on an airfoil, a loss of pressure rise in a diuser and increase the heat load in gas turbine).

The canonical boundary layer is the zero pressure gradient (ZPG) boundary layer over a smooth wall; however, in realistic cases (e.g. ow around turbine blades, airfoils) other effects are present, including pressure gradient, rotation, surface curvature and surface roughness (or a combination of these) that can induce separation; a boundary layer subject to these perturbations often shows non-equilibrium characteristics. Surface curvature induces a pressure gradient (gure 1.1) that inuences the separation and subsequent evolution of the boundary layer. In favourable pressure gradient (FPG) the ow is accelerating, decelerating in adverse pressure gradient (APG); if the deceleration is suciently large, the ow

1 Figure 1.1: Flow over airfoil, accelerating (favourable pressure gradient) and decelerating (adverse pressure gradient).

Figure 1.2: Massive blade erosion. separates.

In engineering applications, surface roughness is also important. For instance, the presence of ice on an aircraft surface can lead to increased drag and decreased maximum lift.

In the atmosphere the Earth's surface is rough, and the variation of surface altitude creates localized ow acceleration and deceleration. In turbine blade, surface roughness increases signicantly due to erosion, corrosion, and deposition during the operation under high pressure and temperature conditions (Tarada & Suzuki, 1993) and even exists on new-made blades. Hence, the eciency of turbines is aected by surface roughness (gure 1.2) and the expected life can be reduced as well.

2 The majority of previous roughness studies have been performed over a at plate boundary layer with ZPG, in a channel or a pipe (Krogstad et al., 1992; Krogstad & Antonia, 1999;

Schultz & Flack, 2005; Flack et al., 2005, 2007; Schultz & Flack, 2007; Yuan & Piomelli,

2014). The main eect of surface roughness here is to increase the wall friction, due to the contribution of pressure drag. In many cases, wall roughness delays laminar boundary layer separation by making the boundary layer turbulent and increasing the mixing (shear stress).

Boundary layers subjected to strong APG on a rough wall have been studied experimentally and numerically in ows over a ramp and hill (Song et al., 2000; Durbin et al.,

2001; Aubertine et al., 2004; Cao & Tamura, 2006). The roughness was shown to cause earlier separation and a larger separation bubble compared to the smooth-wall ow. This was attributed to a larger mean momentum decit on the rough wall. However, roughness is also expected to generate higher turbulence intensity, which could contribute to delay the separation. The cause of the earlier separation is not yet fully understood and the knowledge of the interaction between the APG leading to the separation (which is an outer layer eect) and roughness (an inner layer eect) is limited.

In this study, combined eects of adverse pressure gradient and surface roughness are studied. The eects of the surface curvature are eliminated at this work by considering a

at-plate boundary and imposing suction and blowing at the freestream (Perry & Fairlie,

1975; Spalart & Watmu, 1993; Spalart & Coleman, 1997; Na & Moin, 1998a,b; Skote &

Henningson, 2002).

Numerical simulation can contribute to the understanding of these phenomena; however, up to now, work on APG boundary layer has concentrated on smooth walls (Spalart &

Watmu, 1993; Alving & Fernholz, 1996; Simpson et al., 1977), although the coupling of

FPG and roughness has been considered (Cal et al., 2008; Tachie & Shah, 2008; Piomelli &

Yuan, 2013).

Various types of simulation can provide the required detail. Direct numerical simulation

(DNS), the numerical solution of uid ow governing equations without introducing any kind

3 of modelling, provides valuable information, that may be as accurate as the experiments, and makes the calculation of some quantities possible that cannot be normally measured through an experiment. The DNS approach, however, is often limited to the computation of simple geometries at relatively low Reynolds numbers because of the expensive computational cost which scales as the cube of the Reynolds number. The Reynolds-averaged Navier-Stokes

(RANS) simulation also has shortcomings: rst ignoring the ow details and, second due to the inaccuracies inherent in models that simulate the behavior of all eddies globally.

Therefore, RANS simulations are not appropriate for this study due to their diculty in predicting separated ows at high Reynolds numbers, and roughness eects.

In Large-eddy Simulation (LES) only the large, energy-carrying eddies are computed, while the small scales are modelled; thus the computational cost of representing the small- scale motions is avoided. The basic idea in LES is to separate the large scales of the uid motion from the small scales and is based on the assumption that small-scale turbulent eddies are more isotropic than the large ones. In LES there is a reduced error in modelling the small eddies; eliminating that from the solution makes LES a computationally feasible tool.

Accordingly, LES has been chosen for this study. This thesis provides a benchmark data set for advanced turbulence model development for separated boundary layers with roughness at high Reynolds number; this type of non-equilibrium boundary layer is challenging the current computational uid dynamics (CFD) capabilities.

4 1.2 Literature review

In this section, the literature on APG boundary layers with strong and mild separation is reported. Then, a summary of existing work on roughness eects on ZPG and APG boundary layers is given and Townsend's wall similarity hypothesis for two-dimensional

(2D) and three-dimensional (3D) roughness types is reviewed.

1.2.1 Turbulent boundary layers with adverse pressure gradient over smooth wall

Although there are comprehensive studies for zero pressure gradient boundary layer, the

APG boundary layer is more challenging because measurements are dicult and ow is inhomogeneous in two directions. In boundary layers with an APG or surface curvature, separation occurs when the ow near the surface can no longer resist the downstream pressure rise. Most studies in separation refer to the work by Prandtl (1904), who showed that streamlines in a steady ow past a two-dimensional streamlined body separate where the skin friction or wall shear stress vanishes and the velocity gradient at the wall becomes negative. There are dierent denitions of the separation, in addition to the one based on zero streamline or contour of the zero velocity line (Prandtl, 1904). 50% back ow fraction also can imply ow separation (Dengel & Fernholz, 1990). The value of the shape factor can also be used as an indication of how close the boundary layer is to separation: the ow separates when the shape factor reaches values between 1.8 and 2.5 (Kline et al., 1968; Elsberry et al., 2000).

Alving & Fernholz (1996) conducted hot wire measurements on an axisymmetric body with turbulent boundary layer separation in a short region. The Reynolds stresses in the inner region are reduced; turbulent kinetic energy peak occurs outside of the recirculation zone and after reattachment the inner layer responds were more gradually than the outer

5 layer. According to these authors, two dierent types of separation can be dened: mild-

APG-induced separation and strong-APG-induced separation; geometry induced separation is in the latter category. In mild APG, the wall shear stress is close to zero with a small separation bubble whose height is much smaller than the boundary layer thickness before the ow detachment (Stratford, 1959; Dengel & Fernholz, 1990). Mild APG is a stable turbulent boundary layer with near zero skin friction coecient and has applications in ow and drag control for stabilizing the turbulent boundary layer.

Dengel & Fernholz (1990) investigated the sensitivity of ow separation to the pressure- gradient in an axisymmetric boundary layer. They studied three cases by controlling pressure distribution. The rst case has a slightly positive skin friction, the second a slightly negative one and the third a zero skin friction coecient. They concluded that this kind of ow is sensitive to small changes in upstream condition. Also, the skin friction is zero when the fraction of time the ow travels in the upstream direction is 50%. In a strong-APG-induced separation, the height of the separation bubble and the boundary layer thickness before separation are of the same order of magnitude and the separation is accompanied by large streamline curvature and strong ow reversal. Pioneering work was conducted by Simpson and his co-workers for separation by strong APG (Simpson et al.,

1977, 1981; Simpson, 1989).

Geometry induced separation can be divided into two categories. In the rst one, separation takes place at a sharp corner (e.g., backward-facing step), while in the second category, separation is not xed and uctuating, as is the reattachment point (ow over hill and ramp); in both categories strong separation exists.

The normal stresses change signicantly in strong APG, as reported in literature (Simp- son et al., 1977; Na & Moin, 1998a; Skote & Henningson, 2002; Dianat & Castro, 1991), contributing to increase the production of turbulent kinetic energy in boundary layer. A separated turbulent boundary layer subjected to an increasingly adverse pressure gradient was examined by Simpson et al. (1977) with an airfoil-type pressure distribution using an

6 adjustable top wall suction. They reported that streamwise and wall-normal stresses change signicantly before the separation. Abe et al. (2012) and Elsberry et al. (2000) mentioned that turbulence intensity peaks are aligned with the inection point of streamwise velocity, which can be useful for the scaling.

Song & Eaton (2004a) investigated experimentally the Reynolds number eects on a separated turbulent boundary layer for the ow over a ramp; the Reynolds number varies from 1100 to 20100 (based on the momentum thickness, θ). Reynolds number eects on the mean ow are small, separation and reattachment locations are found to be only a very weak function of the Reynolds number (except for the lowest Reynolds number) while Reynolds number eects on turbulence quantities are considerable and none of the scales proposed works for the separated region due to highly non-equilibrium nature of the ow in this region.

Song & Eaton (2004b) examined the PIV measurements of the ow over the contoured ramp at Reθ = 3400 (a similar geometry to the one used by Song & Eaton (2004a)). They found evidence of the existence of roller vortices, generated by the KelvinHelmholtz instability in the separated shear layer; the boundary layer recovery is aected by the instability of these vortices produced in the separated region. In addition, two-point correlation proles indicate that the size and shape of turbulence eddies are strongly aected by the curvature of the ramp.

Curvature eects dierentiate concave and convex surfaces, which cause dierent ow physics (Muck et al., 1985). Concave curvature enhances the growth of the boundary layer

(i.e, is destabilizing); it changes the turbulence structures notably and increases mixing by generation of Taylor-Gortler vortices. Convex curvature suppresses the boundary layer growth (i.e, is stabilizing). The main dierence between APG and longitudinal curvature eects are introduced by ∂V/∂x, which in curved surfaces can be comparable to ∂U/∂y. Eects of longitudinal curvature and adverse pressure gradient are present in most of the studies including both mild and strong separation. By considering a at plate surface with an APG imposed through suction at the top wall, APG eects are conned and curvature

7 eects are removed (Perry & Fairlie, 1975; Spalart & Coleman, 1997; Na & Moin, 1998a,b;

Skote & Henningson, 2002).

Perry & Fairlie (1975) applied an inviscid model to the steady ow formation of a separation-reattachment bubble within an adverse pressure gradient turbulent boundary layer introduced by a exible roof; the separation of the boundary layer is dominated by the interaction between the vertical region and the irrotational eld outside of the boundary layer.

Spalart & Coleman (1997) carried out the DNS simulation for rapid separation and reattachment of the boundary layer over a at plate with heat transfer. The separation bubble is induced by suction and blowing along an inviscid boundary. Reynolds stresses and turbulent kinetic energy increased markedly over the separation bubble as the uid structures lift up from the wall and ride over the bubble.

Na & Moin (1998a,b) performed DNS simulation of a separated boundary layer over a

at plate at Reθ = 300; separation is again induced by suction at the free-stream similar to Perry & Fairlie (1975). After the suction (decelerating) part, there is a blowing (accelerating) region to obtain rapid reattachment to keep the size of the domain. Both separation and reattachment are uctuating (Na & Moin, 1998a,b); the separation point shows a large saw- tooth variation both in time and spanwise direction. The reason for the dierent behaviour of detachment and reattachment is the existence of the streaks upstream of the separation: the low speed streaks cause the earlier separation, while high speed streaks delay the separation.

Moreover, at detachment, large structures lift into the shear layer, ride over the separation bubble, impinge at the reattachment region and break apart, generating a highly three- dimensional ow (Na & Moin, 1998a,b).

Abe et al. (2012) carried out DNS simulation similar to Na & Moin (1998a,b) and studied

eects of the Reynolds number ranging from Reθ = 300, 600 and 985. The recovery region after the reattachment has a strong Reynolds number dependence and the streak spacing is

smaller at higher Reynolds number. The magnitude of the turbulence quantities decreases

8 at top of the bubble, an eect of streamline convex curvature.

Skote & Henningson (2002) performed DNS of a separated boundary layer induced by pressure gradient. Production of shear and normal Reynolds stresses are equal in strength and are negative at upstream of reattachment which results in elimination of turbulence energy. The near-wall streaks are weakened by the adverse pressure gradient before the separation; the spacing of the streaks is increased, also reported by Simpson et al. (1977).

Inside the separated region the streaks disappear. However, after the reattachment they reappear further downstream.

1.2.2 Eects of roughness

The main eect of the roughness is to increase the drag by creating a wake region downstream of a roughness element, leading to increase of the skin friction coecient, Cf . The eect of roughness depends on the shape, size, and distribution of roughness elements. Sand-grain roughness with roughness height is considered the reference for dierent roughness types and is the most widely studied type of roughness

Nikuradse (1933) rst conducted a series of experiments with pipes roughened by sand- grain. The Moody Chart correlates extensive experimental data obtained by Nikuradse

(1933) and Colebrook (1939), and gives the Darcy friction factor as a function of Reynolds number and relative roughness for the description of friction losses in pipe ow as well as open channel ow. An extensive review of the theoretical and experimental knowledge of rough-wall turbulent boundary layers were conducted by Raupach et al. (1991) and Jiménez

(2004).

The important scale in rough-wall ows is the roughness Reynolds number; that is the ratio of the roughness height to the viscous length scale and is the most important parameter to characterize the roughness eect:

k u k+ = s τ (1.1) s ν 9 where ks is the equivalent sand-grain height, dened as the mean height of the roughness that produces the same drag as the sand-grain roughness (Nikuradse, 1933). As the Reynolds number increases, the eects of roughness become more signicant because the viscous length scale decreases (ν/uτ ) while the roughness Reynolds number increases. Hence, extremely smooth surfaces become hydro-dynamically rough at very high Reynolds number. Following the experiments by Schlichting (1968) and Nikuradse (1933), three regimes in rough wall

ows were identied and the roughness Reynolds number can be used as the indicator of the

ow regimes in rough ows. Hydraulically smooth wall for + : shear is entirely (1) 0 < ks ≤ 5 viscous and the roughness eect on mean velocity proles is negligible. (2) Transitionally rough regime for + : both viscous and form drag are important. Fully rough 5 < ks < 60 (3) regime for +: drag is dominated by the form drag; viscous drag becomes irrelevant as 60 ≤ ks the buer layer is destroyed by the roughness; the ow obeys Reynolds number similarity and skin friction is independent of Reynolds number.

Clauser (1956) mentioned that the inner layer of rough wall ows must have a logarithmic region with the same slope as for a smooth surface, but with dierent log-law intercept. The shift of the velocity prole in wall coordinates compared to smooth walls is called "roughness function", ∆U +, and is a function of roughness Reynolds number. In addition to roughness function, roughness causes a displacement of the velocity prole, d, in wall-normal direction.

1 U + = log(y − d)+ + C − ∆U +(k+) (1.2) κ

where κ = 0.41 is the Von-Kármán constant, C ' 5 is the smooth-wall log-law intercept, the subscript "+" denotes the normalization using wall units (friction velocity, uτ , and

+ + ν/uτ ). ∆U , the roughness function, depends on k . The most common roughness types typically used in experiments and simulations in-

clude: Sand-grain (Nikuradse, 1933), sand paper (Song & Eaton, 2002; Schultz & Flack,

2005), wire mesh (Krogstad et al., 1992) and arrays of rods (Krogstad & Antonia, 1999).

10 Perry et al. (1969) dened two groups for 2D roughness, in the rst, k−type, roughness function depends on the roughness height. In the second, d−type, the cavities between the roughness elements are narrow, the ow depends on the outer scales and the roughness function depends on the outer scale not to roughness height (Jiménez, 2004). For both roughness types, it is necessary to nd the eective roughness height equal to the sand-grain roughness.

Townsend's wall similarity hypothesis states that at suciently high Reynolds number, turbulent motions outside the roughness sublayer (the region where roughness causes spatial variations of time-averaged turbulent statistics) are independent of the viscosity, ν, and the wall roughness, k (Townsend, 1976). Extensive experimental and numerical studies have attempted to support or contradict the wall-similarity hypothesis. Raupach (1981) studied arrays of rough surfaces with dierent densities and observed similarity for second and higher-order moments of turbulent uctuations. The research by Song & Eaton (2002),

Flack et al. (2005), Schultz & Flack (2005), Ashraan et al. (2004), Flack et al. (2007),

Schultz & Flack (2007) and Lee et al. (2011) supports the wall similarity, as the eect of surface roughness is conned to the roughness sublayer. Song & Eaton (2002) performed an experiment of a at-plate boundary layer with separation over a ramp. They concluded that

Reynolds stress components are weakly aected by wall roughness normalized by the friction velocity squared. Flack et al. (2005) studied a boundary layer over three-dimensional rough surfaces covered with sand-grain and woven mesh roughness (fully rough regime and y/δ >

40, δ is boundary layer thickness). Their results showed that the Reynolds stresses of rough cases collapse well with smooth wall above y/ks > 3 when normalized by friction velocity squared; the dierence of the higher moment turbulence statistics and quadrant analysis

are conned to y < 5ks. Schultz & Flack (2005) examined fully rough ow over closely packed spheres with an additional roughness length scale. Reynolds stresses show excellent agreement above y > 5ks and the additional secondary roughness length scale had no eect

11 on the Reynolds stresses. Ashraan et al. (2004) performed DNS over rod-roughened (k- type) channel for transitionally rough regime; they found that outside the roughness sublayer

(beyond 5k), wall-similarity is observed for mean velocity and second order statistics. DNS of Scotti (2006) for the transitionally rough regime (sand-grain roughness) in an open channel also illustrated that the eect of surface roughness on Reynolds stresses is limited to the roughness sublayer.

Other researchers have observed roughness eects alter the outer layer (Krogstad et al.,

1992; Krogstad & Antonia, 1999; Antonia & Krogstad, 2001; Tachie et al., 2003). Krogstad et al. (1992) examined zero pressure gradient turbulent boundary layer over the mesh-screen

(k-type) roughness. There is a signicant increase in the wall-normal turbulence intensity hvvi and a mild increase in the Reynolds shear stress huvi normalized by inner scale over the rough wall compared to the smooth wall. Krogstad & Antonia (1999) investigated two dierent roughness geometries (rod and mesh) that give an identical roughness function.

They observed dierences in turbulent energy production and turbulent diusion (advection and dissipation terms are similar) for dierent roughness types. Lee & Sung (2007) carried out the DNS of the turbulent boundary layer over a 2D rod-roughened wall with k/δ = 8−22. They found that the rod roughness inuences turbulent Reynolds stresses and velocity triple products both in inner and outer layer.

The disagreement may be due to the dierences in characteristics of the roughness elements. Jiménez (2004) suggested that the blockage ratio (δ/k) is also important for rough- wall boundary layer in addition to k+ or ∆U +; rough surfaces with δ/k < 40 may exhibit roughness eects into the outer layer. Flack et al. (2005) mentioned that δ/ks is more appropriate parameter than δ/k and they applied similar roughness type (woven wire mesh) to Krogstad et al. (1992) with contradictory results.

Flack et al. (2007) studied two 3D roughness types (sand-grain and woven mesh) with blockage ratio varies 16 < k/δ < 110 (9 < ks/δ < 91) and roughness Reynolds number + . The eect of increasing roughness height on the outer layer turbulence 36 < ks < 1150 12 statistics was investigated; they observed that the roughness eect is conned to the region y < 5k (or 3ks) from the wall and wall similarity was reached in the turbulence quantities for smooth and rough boundary layers beyond the roughness sublayer. They concluded that there is no critical roughness height for reaching self-similar behavior.

According to the literature, there is a fundamental dierence in the response of boundary layers to 2D and 3D roughness; 2D roughness is controversial in literature. For example, Lee et al. (2011), Krogstad & Antonia (1999) and Schultz et al. (2009) stated that roughness

gives rise to signicant changes in the turbulence structure well into the outer layer, while,

Ashraan et al. (2004) found that roughness eects in outer layer were negligible. These

contradictory observations demonstrate that the outer-layer similarity may depend on the

ow types, such as whether they are internal or external ows. Another reason could be the

diculty in estimating friction velocity for the rough cases. In 3D roughness the outer-layer

similarity is reached in the regions of y > 5k or y > 3ks at high Reynolds numbers. In this study sand-grain (3D type) roughness is used.

1.2.3 Decelerating boundary layers over rough walls

Most of the previous rough and smooth wall studies were conducted in fully developed chan-

nel ows and zero pressure gradient boundary layers, however; pressure gradient present for

majority of the realistic boundary layers. One of the rst studies addressing the simultane-

ous eects of the roughness and APG was carried out by Perry et al. (1969). They studied

the turbulent boundary-layer development over rough walls in both zero and adverse pres-

sure gradients and noted that, due to the additional roughness parameters, the Clauser plot

technique for determining the friction velocity can be inaccurate. The existing rough wall

literature also indicates that accurate measurement of skin friction still poses a challenge to

experimentalists.

To study the combined eects of pressure gradient and roughness experiment on ow

separation, geometries with uctuating separation (similar to the current study) such as 13 the ow over the ramp and hill are appropriate. Experimental studies of the eects of both adverse pressure gradients and roughness were carried out by Song & Eaton (2002);

Aubertine et al. (2004); Cao & Tamura (2006); Tay et al. (2009) and numerical studies by

Durbin et al. (2001); Tamura et al. (2007); Cao et al. (2012).

Tay et al. (2009) examined rough surfaces (sand-grain and gravel roughness) in asym-

metric diuser and reported APG and roughness increases the boundary layer thicknesses (δ,

δ∗ and θ) more than each eect considered individually; it also promotes Reynolds stresses and production of turbulent kinetic energy compared to the smooth wall. Additionally, the combined eect of APG and the roughness leads to the higher momentum decit in rough surfaces.

The eects of sand paper roughness (fully rough in the upstream boundary layer) were studied experimentally by Song & Eaton (2002) for a at plate boundary layer on a ramp with separation. They found that the separation region is considerably larger for the rough case, producing smaller Reynolds stress peaks when normalized by friction velocity upstream of the at plate. The increase in boundary layer thickness and momentum decit in rough cases compensate the increase of the shear stress. Upstream of the separation, the Reynolds stresses normalized by the friction velocity are slightly dierent in smooth and rough case, however; the dierence is much smaller than observed by Krogstad & Antonia (1999), which is attributed to the uncertainty in the determination of the friction velocity.

Aubertine et al. (2004) considered two dierent rough-wall heights (sand-paper) over the ramp with separation and reattachment. As the roughness height increases, the skin friction increases and the separation bubble becomes larger. Peaks of Reynolds stresses collapse at reattachment if normalized by height of the inection point, similar to Song & Eaton (2002).

Cao & Tamura (2006); Tamura et al. (2007) carried out LES simulation over 2D and 3D hill; rough surfaces were modeled by placing small cubes on the hill surface. The recirculation zone for the rough hill becomes larger with earlier separation compared to smooth case.

14 1.3 Objectives

The overall purpose of the research is to improve the understanding of the eects of surface roughness on both the mean velocity and turbulence elds in an adverse pressure gradient boundary layer. Based on the literature review, several questions are still open and this work will attempt to bring more insight into how roughness (inner layer eect) and deceleration

(outer layer eect) interact. We will consider two issues:

• Reasons for the earlier separation in rough wall ow as a result of two counteraction phenomena: Roughness enhances turbulent mixing, presumably increasing the resistance to separation; on the other hand, roughness thickens the boundary layer and makes the ow more likely to separate and reasons of the earlier separation still is not well understood.

• Does separation communicate the roughness eects away from the wall?

15 Chapter 2

Problem Formulation

2.1 Introduction

In this chapter, the governing equations of the ow and the Large-Eddy Simulation (LES) technique are presented, then the numerical method and the boundary conditions are discussed. The details of the the immersed boundary method (IBM), used to simulate the roughness, are described. Moreover, the double averaging (DA) approach is introduced.

Finally, the technique used to calculate the wall shear-stress on rough surfaces (including both viscous and form drag) is illustrated.

2.2 Governing Equation

The governing equations are the non-dimensional continuity and Navier-Stokes equations for incompressible ow.

∂u i = 0 (2.1) ∂xi ∂u ∂u u ∂p 1 ∂2u i + j i = − + i (2.2) ∂t ∂xj ∂xi Re ∂xj∂xj

16 Figure 2.1: Schematic of computational domain.

x1, x2 and x3 (or x, y andz) are, respectively, the streamwise, wall-normal and spanwise directions, and ui (or u, v and w) are the velocity components in those directions. They are normalized by the freestream velocity at the inlet U∞,o ; p is the pressure and is normalized with dynamic pressure, 2 where is the density of the uid which is a constant value. ρU∞,o ρ

The Reynolds number is dened as Reθ = ρU∞,oθo/µ where µ is the dynamic viscosity of

the uid and θo (gure 2.1) is the momentum thickness at the inlet .

Z ∞ U U θ = (1 − )dy (2.3) 0 U∞ U∞

The ltering operation (Leonard, 1974) is introduced to separate the large (or resolved) scales, denoted by overbar, is dened as:

Z f(x) = f(x0)G(x, x0, ∆)dx0, (2.4) D

where G is the lter function and D is the entire domain. Applying the ltering to

17 equations (2.1) and (2.2) gives:

∂u i = 0 (2.5) ∂xi ∂u ∂u u ∂p ∂τ 1 ∂2u i + j i = − − ij + i (2.6) ∂t ∂xj ∂xi ∂xj Re ∂xj∂xj

The eect of the small scales appears through a residual stress term, which is usually called the sub-lter stress (SFS),

τij = uiuj − uiuj, (2.7)

the anisotropic residual-stress tensor is

δ τ a = τ − ij τ (2.8) ij ij 3 kk

The linear eddy-viscosity is used to relate the anisotropic residual-stress tensor (2.8) to the ltered rate of the strain (2.9).

a 2 (2.9) τij = −2νT Sij, νT = (Ce∆) S

1/3 Where ∆ is the grid lter width ∆ = (∆x∆y∆z) , νt is eddy viscosity and S is the ltered rate of the strain,

1/2 1 ∂ui ∂uj S = (2Sij Sij) , Sij = ( + ) (2.10) 2 ∂xj ∂xi

The model parameter Ce is calculated using the dynamic procedure (Germano et al., 1991) with Lagrangian averaging (Meneveau et al., 1996). In the dynamic model two lters are used, the test lter width ∆b and the grid lter width ∆. They are related in this work

18 by √ ∆b = α ∆, α = 6 (2.11)

The stress associated with the test lter ∆b is

Tij = udiuj − ubiubj (2.12)

Resolved turbulent stresses, Tij is dened as Germano identity (Germano et al., 1991),

(2.13) Lij = Tij − τbij

2 2 2 2 (2.14) τij = 2Ce ∆ |S||Sij|,Tij = 2Ce (α∆) |Sb||Scij|

The Germano identity can be used to relate the resolved turbulent stresses Lij, which can be computed, to Tij and τij by assuming a model for Tij of the form of equation (2.12).

The coecient Ce can be calculated in terms of known variables and the system is over determined.

The error associated with use of the Smagorinsky model in the Germano identity is dened as,

2 (2.15) eij = Lij − Ce Mij

This error should be minimized in equation (2.15) and various formulation of the dynamic model can be interpreted to minimize the error (eij) in dierent ways (Ghosal et al. (1995), Lilly (1992))

∂heijeiji 2 hLijMiji (2.16) 2 = 0 =⇒ Ce = ∂Ce hMmnMmni 2 2 Mij = 2∆ |\S|Sij − α |cS|Scij (2.17)

Lij = udiuj − ubiubj (2.18)

19 where hi denotes an appropriate average. In this work, Lagrangian averaging (Mene- veau et al., 1996) is taken: hMijLiji and hMmnMmni are averaged along particle paths by integrating over the particle pathline based on the following equation.

Z t 0 0 0 0 0 0 E = eij(z(t ), t )(eij(z(t ), t )W (t − t )dt (2.19) −∞

The weighting function W (t−t0) is introduced here in order to control the relative impor-

tance of events near time t with those of earlier times (Meneveau et al., 1996). Minimizing the error in equation (2.15) leads to,

2 JLM (2.20) Ce = JMM

where JLM and JMM are

Z t 0 0 0 0 JLM = LijMij(z(t ), t )W (t − t )dt (2.21) −∞

Z t 0 0 0 0 JMM = MijMij(z(t ), t )W (t − t )dt . (2.22) −∞

With the aid of a weighting function with the form of W (t−t0) = T −1e−(t−t0)/T , equations (2.21), (2.22) are convertible to the two auxiliary dierential equations,

DJ ∂J 1 LM ≡ LM + u · 5J = (L M − J ) (2.23) Dt ∂t LM T ij ij LM

DJ ∂J 1 MM ≡ MM + u · 5J = (M M − J ) (2.24) Dt ∂t MM T ij ij MM

Numerical integration of equations (2.23) and (2.24) is equivalent to taking the inte-

grals in equations (2.21) and (2.22). The time scale T controls the memory length of the

20 Lagrangian averaging in time (Meneveau et al., 1996):

−1/8 T ∼ ∆[JLM JMM ] (2.25)

2.3 Time-advancement and discretization

The second order Crank-Nicolson time advancement scheme is applied on the wall-normal viscous and SFS terms. The explicit Adams-Bashforth time-advancement is applied to all other terms. The fractional step method (Chorin, 1968; Kim & Moin, 1985) is used to solve the governing equations (2.5), (2.6). First the velocity eld is predicted (denoted by ∗) from ui the momentum equation without pressure term (Helmholtz equation) by using information from the current time step (n) and the previous time step (n − 1). For the x-momentum equation,

∆t ∂ ∂ 3 1 ∆t ∂ ∂un 1 − (ν + νn) ) u∗ = un + ∆t F n − F n−1 + (ν + νn) 2 ∂y T ∂y 2 u 2 u 2 ∂y T ∂y (2.26)

Where

∂unun ∂unvn ∂unwn ∂ ∂un F n = − − − + (ν + νn) (2.27) u ∂x ∂y ∂z ∂x T ∂x ∂ ∂un ∂ ∂un ∂ ∂vn ∂ ∂wn 1 ∂P n + (ν + νn) + νn + νn + νn − ∂z T ∂z ∂x T ∂x ∂y T ∂x ∂z T ∂x ρ ∂y

For the y-momentum,

∂ ∂ 3 1 ∂ ∂vn 1 − ∆t (ν + νn) ) v∗ = vn+∆t F n − F n−1 +∆t (ν + νn) (2.28) ∂y T ∂y 2 v 2 v ∂y T ∂y

21 Where

∂vnvn ∂vnun ∂vnwn ∂ ∂vn F n = − − − + (ν + νn) (2.29) v ∂x ∂y ∂z ∂x T ∂x ∂ ∂vn ∂ ∂un ∂ ∂wn 1 ∂P n + (ν + νn) + νn ) + νn − ∂z T ∂z ∂x T ∂y ∂z T ∂y ρ ∂y

For the z-momentum,

∆t ∂ ∂ 3 1 ∆t ∂ ∂wn 1 − (ν + νn) ) w∗ = wn + ∆t F n − F n−1 + (ν + νn) 2 ∂y T ∂y 2 w 2 w 2 ∂y T ∂y (2.30)

Where

∂wnun ∂wnvn ∂wnwn ∂ ∂wn F n = − − − + (ν + νn) (2.31) w ∂x ∂y ∂z ∂x T ∂x ∂ ∂wn ∂ ∂un ∂ ∂vn ∂ ∂wn 1 ∂P n + (ν + νn) + νn + νn + νn − ∂z T ∂z ∂x T ∂z ∂y T ∂z ∂z T ∂z ρ ∂z

After the prediction step the Poisson equation is solved,

1 ∂u∗ − ∇2Φn+1 = i (2.32) ∆t ∂xi

where Φ is dened by

∂Φ ∂ ∂2Φ ∂P + ν∆t = . (2.33) ∂xi ∂xj ∂xj∂xi ∂xi

The velocity is then corrected to obtain a divergence-free velocity eld (2.34),

n+1 ∗ ∂Φ (2.34) ui = ui + ∆t ∂xi The solution of the Poisson equation is obtained by a Fourier-transform of the equation in the spanwise direction, followed by a cosine transform of the resulting equation in x, and

22 Figure 2.2: The staggered-grid arrangement. The u, v and pressure cells are indicated with blue, red and green colors respectively. by direct solution of the resulting tri-diagonal matrix, at each wavenumber. Note that when transforming the equations in Fourier space, the modied wavenumber corresponding to the consistent central scheme is used in place of the actual wavenumber. The code is parallelized using the Message Passing Interface (MPI) protocol.

A second-order central dierencing scheme on a staggered grid is used for all the spatial gradient terms in the equations above. The staggered grid arrangement is shown in Figure

2.2. where xi−1/2 is the node in the middle of xi and xi−1, and yj−1/2 between yj and yj−1. Therefore,

x + x x = i i−1 (2.35) i−1/2 2 y + y y = j j−1 (2.36) j−1/2 2

For the convective terms, a volume-weighting conservative method by Ham et al. (2002) is employed. Based on this method

23 n n x0 (xi+1/2 − xi)φi+1/2,j,k + (xi − xi−1/2)φi−1/2,j,k φ |i,j,k,n = (2.37) xi+1/2 − xi−1/2 n n x φ + φ φ | = i+1/2,j,k i−1/2,j,k (2.38) i,j,k,n 2

x where φ is a staggered variable, and φ , denotes interpolation in the x direction. Therefore as an example the convection term in the x−momentum equation is calculated as,

x x0 ∂ δui uj (uiuj) ' (2.39) ∂xj δxj where δ denotes discrete dierencing and summation is applied for repeating subscript.

2.4 Boundary conditions

A no-slip boundary condition is applied at the bottom wall; the turbulent inow at the inlet is generated from a rescaling/recycling region (Lund et al., 1998). Periodic boundary conditions are applied in the spanwise direction, and a convective condition (Orlanski, 1976) is used at the outlet. At the freestream (gure 2.1), a physically realistic boundary condition is to prescribe V∞(x) and to let the streamwise velocity adjust itself by imposing the zero- vorticity (irrotational) condition (Na & Moin, 1998a).

dU dV dW ∞ = ∞ , ∞ = 0 (2.40) dy dx dy

2.5 Immersed-boundary method (IBM)

The roughness is presented by an Immersed Boundary Method (IBM) based on the volume-

of-uid (VOF) approach. The VOF method was introduced by Hirt & Nicols (1981) to

study the interface between dierent types of uid. In this method, the volume fractions

24 φ in surface cells are calculated (for incompressible uids) from the following conservation equation,

∂φ + · (φ~v) = 0 (2.41) ∂t O

In this study, to represent the random roughness elements, the virtual sandpaper model

proposed by Scotti (2006) and the IBM based on VOF approach was used. Both the time-

derivative term and the convection term in equation (2.41) equal zero for the interface

between surface roughness and a uid to ensure conservation of mass for each type of uid.

The volume fraction of each cell occupied by uid is φ=1 for a cell fully in the uid and

φ = 0 for a cell inside the roughness.

∂u ∂u u ∂p ∂τ 1 ∂2u i + j i = − − ij + i + F n (2.42) ∂t ∂xj ∂xi ∂xj Re ∂xj∂xj

A force is imposed on the right-hand side of the momentum equation (F n) to reduce the velocity proportionally to the solid volume in each cell. This method is rst order accurate; it is, however, adequate for boundary layers over rough surfaces (Scotti, 2006; Yuan, 2011), since the description of the rough surface is only an approximation to real sandpaper.

The immersed boundary method is imposed by calculating the forcing term,

n ∗ ∗ (2.43) F = (φui − ui )/∆t

Afterwards, the prediction step is carried out for a second time (u∗∗) using the forcing term as the source term, and the modied intermediate ltered velocity is obtained. For the x-momentum equation as an example,

25 ∆t ∂ ∂ ∆t ∂ ∂ 1 − (ν + φνn) ) u∗∗ = 1 + (ν + φνn) ) un 2 ∂y T ∂y 2 ∂y T ∂y 3 1 +∆t (RHSn − F n) − (RHSn−1) − F n−1 (2.44) 2 x 2 x

Where

∂unun ∂unvn ∂unwn ∂ ∂un RHSn = − − − + (ν + φνn) (2.45) ∂x ∂y ∂z ∂x T ∂x ∂ ∂un ∂ ∂un ∂ ∂vn ∂ ∂wn 1 ∂P n + (ν + φνn) + φ νn + νn + νn − ∂z T ∂z ∂x T ∂x ∂y T ∂x ∂z T ∂x ρ ∂y

After the prediction step, the Poisson equation (2.33) is solved; the eddy viscosity is also multiplied by the volume of the uid at each cell.

2.6 Calculation of wall shear-stress

The wall shear stress, τwall, over smooth surfaces is entirely caused by viscous drag, and is calculated directly based on wall-normal velocity gradient at the wall. In the rough surfaces however, the wall shear stress comprises both viscous and form drag (Raupach et al., 1991) due to the small ow separation after the each roughness elements; thus, the wall shear stress can not be calculated from wall-normal velocity gradient only, but is determined by integration of the streamwise momentum equation (2.42), assuming the eddy viscosity equation (2.9), from y = 0 to the top of the domain y = h.

26 Z h ∂ ∂u Z h τwall(x, z, t) ≡ (ν + νT ) dy + Fxdy = 0 ∂y ∂y 0 Z h ∂u Z h ∂u¯u¯ ∂u¯v¯ ∂u¯w¯ 1 Z h ∂P Z h ∂2u ∂2u dy + + + dy + dy − ν 2 + 2 dy 0 ∂t 0 ∂x ∂y ∂z ρ 0 ∂x 0 ∂x ∂z Z h ∂ ∂u ∂ ∂v ∂ ∂u ∂ ∂w − 2 νT + νT + νT + νT dy (2.46) 0 ∂x ∂x ∂y ∂x ∂z ∂z ∂z ∂x

Where Fx represents the form drag force in the streamwise direction. The instantaneous

τwall can be averaged in both the spanwise and time to get hτwalli, then the friction coecient

Cf is computed from

2hτwalli (2.47) Cf = 2 ρU∞

2.7 Time averaging and double averaging (DA)

For the smooth case, a ow quantity can be decomposed into two componenets by the time-averaging.

θ(x, y, z, t) = hθi(x, y) + θ0(x, y, z, t) (2.48)

Where hθi and θ0 are the spatial, the temporal average and the turbulent uctuations. Due to the small-scale ow eddies and their irregular arrangement on the bottom wall,

the time-averaged ow characteristics close to the bed are highly three-dimensional. The

double averaging (DA) approach can be applied from the ow/bed interface up to the free

surface, since it potentially oers new insights into eects of macro-roughness element on

mean hydrodynamics. The DA approach decomposes a ow quantity, θ, into the space-time

average (Mignot et al., 2008), h(·)i, the spatial disturbance of the temporal average and

represents wakes of the macro-roughness elements (f·), and the turbulent uctuations, (·)0.

27 θ(x, y, z, t) = hθi(x, y) + θe(x, y, z) + θ0(x, y, z, t) (2.49)

28 Chapter 3

Model Validation

3.1 Introduction

In this chapter, the model is validated by applying it to two problems. Large eddy simulations are carried out of an open-channel ow with dierent roughness heights to validate the implementation of the IBM and to determine the spatial-resolution requirements; the results are compared to DNS results by Scotti (2006). In the second part, LES simulation of ow over a smooth adverse pressure gradient boundary layer with ow separation is performed to validate the simulation using the DNS results by Na & Moin (1998a).

3.2 Rough-wall channel ow

In this study, to represent the random roughness elements, the virtual sandpaper roughness model proposed by Scotti (2006) is applied. The bottom surface is divided into Nx ×Nz tiles with 2ks × 2ks sides. There is an ellipsoid in each tile with semiaxes ks, 2ks (wall-normal direction) and 1.4ks, randomly oriented; its center is located at the center of the tile. This geometry is inspired by sand grain roughness, which is characterized by a single parameter, ks.

29 Large eddy simulations of open-channel have been carried out with various resolutions to validate the roughness model with the DNS results (Scotti, 2006) and to nd out the adequate grid resolution for roughness elements. The ow conguration is similar to that used by Scotti (2006); the channel dimensions are 6h × h × 2h. A symmetry boundary condition is imposed at the top boundary; periodic boundary conditions are applied in streamwise and spanwise directions and no-slip boundary condition is applied at the bottom wall, where the roughness elements are modeled using immersed boundary method (IBM).

The Reynolds number is Reτ ≡ uτ h/ν = 1, 000 (uτ and h denote the friction velocity and

channel half height, respectively, and ν is the kinematic viscosity). The roughness heights, ks are and , corresponding to + and + , in the transitionally 0.02h 0.04h ks ≡ ksuτ /ν = 20 ks = 40 rough regime. For the streamwise and spanwise directions, a uniform mesh is used; for the

+ wall-normal direction ∆y < 1 within the region y < 1.5ks (below the roughness crest) for all the cases. Grid size, grid resolution and number of the grid points per each roughness

element (N) are compared with the DNS of Scotti (2006) for each case at table 3.1. The velocity prole for the rough wall boundary layer in the overlap region is

1 U +(y+) = log(y+ − d) + B − ∆U +(k+), (3.1) κ s

where κ is the Von-Kármán constant, B is a mean-velocity prole intercept in the logarithmic region.

Roughness eects can be measured by the roughness function ∆U +, which is plotted against + in gure 3.1 for each resolution. Case (coarse grid) dose not capture roughness ks 1 eects near the wall. Case 3 (ne grid), however, shows good agreement with the DNS of

Scotti (2006) and experiment of Colebrook & White (1937). Hence, resolution of the case 3 is considered adequate for resolving the mean ow.

The location of the virtual wall, d is also important: d is the equivalent arm of the moment exerted by the ellipsoids on the ow (Jackson, 1981):

30 10

6 + U

∆ 4

0 100 101 102 + ks

Figure 3.1: Grid renement study, roughness function. DNS of Scotti (2006), ∗ experiment of Colebrook & White (1937), LES of + , LES of + . ks = 20, 96 × 96 × 96 / ks = 20, 128 × , LES of + . 140 × 192 O ks = 20, 192 × 208 × 256

Table 3.1: Open channel grid size, grid resolution and number of grids per each roughness element. Cases 1 2 3 Scotti (2006) Grid size 96 × 96 × 96 128 × 140 × 192 192 × 208 × 256 386 × 256 × 386 N(k = 0.02h) 0.64 × 31 × 1.92 0.85 × 38 × 3.84 1.28 × 72 × 5.12 2.57 × 7.72 N(k = 0.04h) 1.28 × 39 × 3.84 1.70 × 64 × 7.68 2.56 × 83 × 10.14 5.14 × 15.44 ∆x+ 62.5 46.9 31.2 15.5 ∆z+ 20.8 10.4 7.8 5.2

31 30

+ 20 U ∆ +

+ 10 U

0 10−1 100 101 102 103 (y − d)+

Figure 3.2: Mean streamwise velocity prole in wall coordinate at + . DNS of Scotti, ks = 20 smooth; , LES + , , LES + . ks = 20, 192 × 208 × 256 ks = 20, 192 × 208 × 256

R kc yF (y)dy d = 0 , (3.2) R kc 0 F (y)dy

where F (y)dy is the horizontally averaged drag force and kc is the maximum height of the roughness elements (1.5ks). Due to the self-similar nature of the sand-grain roughness,

+ d is close to 0.8ks (Scotti, 2006). By considering roughness function ∆U = 2.85 from gure 3.1 at + and equals to , there is a reasonable collapse between case , and ks = 20 d 0.8ks 2 3 the DNS by Scotti (2006) at gure 3.2 for the smooth wall at k+ = 20.

The three components of turbulent uctuations for case 3 are compared to the smooth DNS results in gure 3.3. Wall-normal and spanwise uctuations agree very well with the

reference data. Nevertheless, for the streamwise direction the LES results under-predicts

the DNS results since the grid is much coarser than the DNS one. The vertical velocity

uctuations decay rapidly to zero, due to the v = 0 condition at the freestream. The results satisfy the wall similarity hypothesis proposed by Townsend (1976). In conclusion, for this

roughness height (k = 0.02h) and Reynolds number (Reτ = 1000), at least two and ve grid points (case 3) per roughness element in the streamwise and spanwise directions, respectively, are required to resolve the roughness eects for the rst and second order statistics.

32 4 (a)

+ rms 2 u 0 100 101 102 103 2 (b)

+ 1 rms v 0 100 101 102 103 2 (c) + rms 1 w 0 100 101 102 103 (y − d)+

Figure 3.3: Proles of (a) streamwise (b) wall-normal (c) spanwise uctuations; , Scotti's DNS, smooth; Scotti DNS, + ; , LES + . ks = 20 ks = 20, 192 × 208 × 256

3.3 Adverse pressure gradient boundary layer

Large-eddy Simulations are carried out on the at-plate boundary layer in the presence of an adverse pressure gradient, strong enough to induce separation. An APG is achieved by a suction and blowing velocity prole imposed at the freestream (gure 3.4). This case was studied numerically with DNS by Na & Moin (1998a), Na & Moin (1998b), Abe et al.

(2012) and Raiesi et al. (2011); the same conguration is used here. The computational domain size is ∗ ∗ ∗ in streamwise ( ), wall-normal ( ) and spanwise ( ) 460δo × 65δo × 50δo x y z direction respectively. 640 × 192 × 128 grid points are used, resulting in a resolution in wall units + , + and + . All dimensions are normalized by ∗, ∆x < 20 ∆yo,min < 0.6 ∆z < 11 δo the displacement thickness (equation (3.3)) of the boundary layer at the reference location,

∗ ; the Reynolds number at the reference plane is ∗ corresponding x = 0 Reδo ≡ u∞δo /ν = 540

33 Table 3.2: Summary of current LES and DNS by Na & Moin (1998a).

Case Grid Domain size + + + Reθ,o ∆xmax ∆yo,min ∆zmax LES 300 640 192 128 ∗ ∗ ∗ 20 0.6 11 × × 460δo × 64δo × 50δo DNS (Na & Moin) 300 512 192 128 ∗ ∗ ∗ 18 0.1 10 × × 350δo × 64δo × 50δo

to Reθ,o ≡ u∞θo/ν = 300. Z ∞ U δ∗ = 1 − dy (3.3) o U∞ Summary of the current LES and Na & Moin (1998a) DNS are presented at table 3.2.

Figure 3.4: Computational setup.

A schematic diagram of the three-dimensional computational domain is shown in gure

3.4. The inlet is located ∗; both reference plane and inlet are in ZPG region. x = −80δo APG region is not signicant until ∗; well downstream of the recycling plane, the x > 40δo deceleration is imposed by the suction prole at the freestream (gure 3.5) which causes the ow to separate and then to reattach downstream due to the FPG caused by blowing

( ) that occurs for ∗ ; this prole is uniform in the spanwise direction. V∞ < 0 x/δo > 220 V∞

U∞ is obtained based on zero vorticity boundary condition discussed in section 2.4. After the suction prole, V∞, there is a blowing prole in order to obtain rapid reattachment to keep the size of the domain.

)

y 1 ( U∞ ∞ 0.5 ,V ) x ( 0 V∞ ∞ U

−0.5 0 50 100 150 200 250 300 350 δ∗ x/ o

Figure 3.5: Streamwise and wall-normal velocity prole along the freestream.

(a)

f 5 ,c 3 10 0 0 50 100 150 200 250 300 350 δ* x/ 0 120 80 (b) (c)

100 δ*=330 x / 0 60 80

* δ*=145 x /δ =300 x / 0 0 + 60 40 u

δ*=130 x / 0 40 δ*=285 x / 0 20 x /δ*=115 20 0

* x δ*=270 x /δ =100 / 0 0 0 0 10-1 100 101 102 103 10-1 100 101 102 103 y+ y+

Figure 3.6: (a) Development of friction coecient. (b) Proles of the streamwise velocity component before separation (c) after separation. Each prole is shifted upwards by 20 units for clarity; • DNS by Na & Moin (1998a); , LES; , logarithmic law of the wall

35 (a)

f 5 ,c 3 10 0 0 50 100 150 200 250 300 350 x/ * 0.06 o (b) x/ =320 0.05 x/ =270 0.04

2 o x/ =220

/ u 0.03 r.m.s

p x/ =160 0.02

x/ =120 0.01

x/ =80 0 0 10 20 30 40 50 * y/ o

Figure 3.7: (a) Development of friction coecient. (b) Proles of root-mean square pressure uctuations shifted upwards by 0.01 units for clarity; • DNS by Na & Moin (1998b); , LES .

The friction coecient, 2 is shown in gure 3.6 (a) and the agreement Cf = 2τw/ρU∞,o between the current LES and DNS is reasonable. There is a disagreement in the Cf predicted at the beginning of the domain because of the dierent inlet boundary conditions for these two cases: while Na & Moin (1998a) assign a mean velocity with superposed synthetic turbulence at the inlet, a recycling condition is applied for LES, while at ∗ has a x/δo = 0 ∗ fully developed boundary layer that matches the values of δ and Cf used by Na & Moin

(1998a). While in their case a short adjustment zone, with increased Cf , is present. Our is a fully developed boundary layer from the beginning.

First the mean ow decelerates and separates at ∗; after ow separation, the mean 155δo ow accelerates and reattaches at ∗. Note that the velocity proles in the downstream 251δo region deviate signicantly from the standard logarithmic law and the slope of the velocity

36 proles increases due to the APG eects (gure 3.6(b)). After the reattachment, this slope decreases again because of FPG eects (gure 3.6(c)). LES predicts the main ow well.

However, the velocity proles deviate near the detachment ( ∗ ) and the reattach- x/δo = 145 ment ( ∗ ) (gure 3.6); this is probably due to the errors in the prediction of the x/δo = 270 friction factor.

Comparison of root mean square pressure (RMS) uctuations as a function of wall-normal distance for dierent streamwise locations are presented in gure 3.7; the wall pressure uctuations decrease inside the separation bubble and increase above the bubble as a result of the motion of structures above the separation region. In the reattachment region, these uctuations reach the highest value compare to upstream and downstream, where the structures impinge in the reattachment region.

37 Chapter 4

Results

4.1 Introduction

In this chapter, three large-eddy simulations of APG boundary layer over smooth and rough surfaces are studied in detail with the main goal of investigating the earlier separation that occurs in rough walls. First a summary of all cases is presented, with an introduction and problem setup followed by presentation of the results, including mean ow characteristics, turbulent statistics and structures, with focus on the combined eects of roughness and

APG.

4.2 Case setup

To study the eects of roughness and APG on ow separation, simulations of smooth and rough wall ow (for two dierent roughness heights) were carried out at ( is Reθo = 2300 θo the momentum thickness at the reference plane). The domain size is 760θo × 90θo × 70θo in x, y and z directions, respectively; the summary of the simulations is presented in table 4.1. The domain size and grid resolution are chosen based on LES of Raiesi et al. (2011). The wall-normal freestream velocity prole has similar prole to Na & Moin (1998a) and Raiesi

38 Table 4.1: Summary of simulation parameters.

Case Grid Domain size + + + Reθo k/θo ∆xmax ∆ymin ∆zmax

1.Smooth 2300 0 2560×384×384 760θo × 90θo × 70θo 30 1 19

2.Rough 2300 0.47 2048×384×384 760θo × 90θo × 70θo 50 1.2 26

3.Rough 2300 0.95 2560×432×384 760θo × 90θo × 70θo 44 0.8 28 et al. (2011).

Two roughness heights were used here, k/θo = 0.47 and 0.95, corresponding to roughness

+ Reynolds numbers in the ZPG region, k ≡ kuτ,o/ν = 60 (case 2) and 120 (case 3); k here is the averaged sand-grain roughness height and both cases are in the fully rough regime in

the ZPG region. Between 302 and 425 million grid points are used, resulting in ∆x+ < 30, + , + for case , + , + , + for case , and ∆ymin < 1 ∆z < 19 1 ∆x < 50 ∆ymin < 1.2 ∆z < 26 2 + , + , + for case 3 at the reference plane. The total simulation ∆x < 44 ∆ymin < 0.8 ∆z < 28 time is more than ∗ for all the cases, sucient to achieve sample convergence for 120δo /uτ second order turbulent statistics.

Figure 4.1 shows the location of the rescaling region and the gradual implementation

of the roughness at the bottom wall. The recycling region starts at x = −140θo and ends

at x = −80θo; it is long enough not to aect the reference plane. The roughness model described in section 3.2 is used and commences after the recycling region with gradual

increase from x = −60θo until x = −45θo where the roughness reaches its maximum height. The roughness height is the only parameter describing the sand-grain surface; the vi-

sualization of how these ellipsoids are implemented is presented in gure 4.2 as isosurface

of φ = 0.9 (φ is the volume of uid) (section 2.5) for case 3. Roughness ellipsoids have

semi-axes equal to k, 2k and 1.4k; the number of grid points used for resolving most of the roughness elements is provided in table 4.2, showing that the roughness elements are

resolved fairly well with 1621 and 16706 grid volumes per roughness element for cases 2 and

3; more than 15000 roughness tiles with size 2k × 2k are used for each case.

39 Figure 4.1: Location of the rescaling and the gradual imposing of the roughness.

Figure 4.2: Sand-grain roughness obtained by Scotti's model at iso-surface of φ = 0.9 for case 3.

At the reference plane (x/θo = 0) uid properties are well developed; the plane displacement is d = 0.8k and the roughness function is ∆U + = 6.9 and 8.5, respectively for case 2 and case 3 from gure 3.1; mean velocity proles for all the cases are in good agreement in the log-law region (gure 4.3) indicating that the boundary layer is well-developed at the reference plane.

40 Table 4.2: Number of grid for majority of roughness ellipsoids.

Case k/θo Streamwise Vertical Spanwise Grid Per Ellipsoid 2 0.47 3.9 54 7.7 1621 3 0.95 9.6 113 15.4 16706

30 ) + k

( 20 + U ∆

+ 10 + U 0 101 102 103 (y − d)+

Figure 4.3: Mean velocity proles at the reference plane; , Smooth; , k/θo = 0.47 + + (k = 60); , k/θo = 0.95 (k = 120).

4.3 Smooth wall APG

4.3.1 Mean ow characteristics

Statistics are presented at Reθ = 2300; the sux 1, 2, 3 denotes the streamwise, wall-normal and spanwise components, respectively; ; and 0 are instantaneous and uctuating U, V, W Ui ui quantities; angle brackets denote averaging in time and spanwise direction hUii.

Figure 4.4 shows the streamwise variation of the free-stream velocity, U∞, and acceleration parameter, K.

ν ∂U∞ (4.1) K = 2 U∞ ∂x

U∞ decreases at the center of the domain and then increases, resulting in deceleration (negative K) and acceleration (positive K) of the ow. Contours of mean streamwise and

41 5 1.2 4 1.1 3 1 2 0.9 o ,

K 1 0.8 ∞ u × 0 0.7 / 6 −1 0.6 ∞ 10 −2 0.5 u −3 0.4 −4 0.3 −5 0.2 0 50 100 150 200 250 300 350 400 450 500

x/θo

Figure 4.4: , freestream velocity U∞,o and , acceleration parameter K for all the cases. wall-normal velocities are shown in gure 4.5; the mean streamwise velocity signicantly decreases in the freestream after x/θo = 100 (gure 4.4). The APG results in the ow separation; the wall-normal velocity is positive up to the middle of the separation bubble.

However, after x/θo = 280 the freestream accelerates (gure 4.4) due to the negative wall-

normal velocity, causing the reattachment of the ow. Near x/θo = 280 the streamlines are parallel to the wall and their wall-normal velocity is negligible. The zero velocity line

highlights the size of the recirculation bubble.

Figure 4.5: Distribution of mean (a) streamwise and (b) wall-normal velocity for case 1 at Reθ = 2300; , mean streamline; , zero velocity line .

Figure 4.6 is the visualization of the streamlines to highlight the eects of suction and

80 80

60 60 o θ 40 40 y/

20 20

0 0 0 100 200 300 400 500 x/θo

Figure 4.6: Streamlines through the domain.

blowing. Streamlines are directed upward in the APG region until x/θo = 280. On the other hand, streamlines are coming inside the domain from the freestream due to the blowing part (FPG region). In addition, gure 4.6 shows how the streamlines behave around the separation bubble.

The mean skin friction coecient (Cf ), and the streamwise velocity at freestream and boundary layer edge are shown in gure 4.7. Cf decreases signicantly after x/θo = 100 as the ow decelerates and eventually separation occurs at x/θo = 178; the separation bubble ends, due to the accelerating streamwise velocity, at x/θo = 344 (separation here is dened based on the point where τw = 0). Figure 4.8 shows the time history of the location of zero Cf of the spanwise averaged ow eld. The reattachment point uctuates, similar to backward-facing step; the separation point also uctuates as low speed streaks cause an earlier separation, while high speed streaks delay it. Detachment and reattachment points are uctuating respectively around x/θo = 178 and x/θo = 344 for the smooth case.

4.3.2 Turbulent characteristics

The development of mean (normalized by U∞,o) and turbulence quantities (normalized by 2 ) is presented in gure 4.9. Before the separation, the peaks of 0 0 0 0 0 0 oc- uτ,o hu u i, hv v i, −hu v i cur near the wall (gure 4.9 (b)). In the recirculation region, the back ow in the separation

43 1.5 (a) δ 1 ,U

∞ 0.5 U 0 100 200 300 400 500 (b) 4 f C

× 2 3

10 0 100 200 300 400 500 x/θo

Figure 4.7: (a) ,U∞; ,Uδ; (b) distribution of Cf for case 1. 190 (a) 180 S X 170

160 3200 3400 3600 3800 4000 350 (b) 340 R X 330

320 3200 3400 3600 3800 4000 Time

Figure 4.8: History of the location of zero Cf of the spanwise-averaged (a) separation (XS) and (b) reattachment (XR) point • for case 1.

bubble is clearly visible in gure 4.9 (c); once in the adverse pressure gradient, the boundary

layer thickness grows rapidly and the streamwise velocity prole develops an inection point

(gure 4.9 (c)); hu0u0i, hv0v0i peaks increase in magnitude above the separation bubble due to the passage of large scale structures (Na & Moin, 1998a), their peaks are aligned with

44 50 50 (a) (b) o θ y/ 0 0 0 0.5 1 0 5 10 50 50 −3 (c) x 10(d) o θ y/ 0 0 0 0.5 1 0 5 10 50 50 −3 (e) x 10(f) o θ y/ 0 0 0 0.5 1 0 5 10 × 3 −3 U Rii 10 x 10

Figure 4.9: Distribution of mean and turbulence statistics for case 1; (a),(b) before the separation at x/θo = 120; (c),(d) inside the separation bubble at x/θo = 250; (e),(f) in the 0 0 0 0 recovery region at x/θo = 400; , mean streamwise velocity; , hu u i; , hv v i; , −hu0v0i.

the inection point (y/θo ≈ 25). Three stress components are relatively small within the separation bubble. In the recovery region the peak levels of hv0v0i, −hu0v0i increase signicantly over the upstream levels before separation. There are double peaks in the streamwise

Reynolds stress (gure 4.9 (e)), one near the wall and the other in the outer region, showing

the development of an inner layer after the reattachment.

Figure 4.10 presents turbulent kinetic energy (TKE), K, its production, Pk and its dissipation rate normalized by inner scales at the reference plane ( 2 and 4 ). uτ,o uτ,o/ν

hu0 u0 i K = i i (4.2) 2 0 0 ∂hUii (4.3) Pk = −huiuji ∂xj 0 0 (4.4) = 2νhsijsiji

45 A xed scaling based on the reference value uτ,o is used for two reasons. First, the xed scaling shows absolute variations of the stress level through the ow eld. In addition, the local value of uτ is not an appropriate scale in the separated ow region where the skin friction vanishes. Large magnitudes of K, Pk and occur in the separated shear layer and reattached region (gure 4.10 (a), (b), (c)). Their magnitude, however, is attenuated at the

top of the bubble due to the decrease of the mean shear (∂hUi/∂y) in this region. Further-

more, ∂hUi/∂x becomes positive after x/θo = 280 due to acceleration and −huui∂hUi/∂x turns negative, contributing to decreasing the production of TKE (gure 4.10 (b)) and

consequently TKE (gure 4.10 (a)) (Skote & Henningson, 2002). Additionally, turbulence decays in the separation region and transitions towards a quasi laminar regime.

Figure 4.10: Distribution of turbulence statistics for case 1; , mean streamline; , zero velocity line: (a) turbulent kinetic energy; (b) production of turbulent kinetic energy; (c) dissipation of turbulent kinetic energy.

46 To view the ow structures, the second invariant of the velocity gradient tensor is calculated,

−1 ∂Ui ∂Uj −1 Q = = (SijSij − ΩijΩij) (4.5) 2 ∂xj ∂xi 2

S and Ω are respectively, the resolved strain and rotation-rate tensors. Q > 0 identifying regions of high vorticity in which the rotational motion is stronger than the shear (Dubief

& Delcayre, 2000).

After the separation the shear-layer instability leads to the formation of turbulent eddies; they are lifted above the separation for x/θo > 180, grow in size and travel around the separation bubble. In the middle of the separation bubble (x/θo = 280) the decaying of the vortices is reduced (gure 4.11) and no turbulence is generated, coinciding with the beginning of acceleration phase (gure 4.4) and decrease of the production of TKE shown in gure 4.10. Eventually, the structures impinge on the wall in the reattachment region; they then are advected downstream (x/θo = 340). Structures are distorted and not elongated in the streamwise direction after the reattachment.

Low speed streaks appear before the separation in gure 4.12; they are the footprints of hairpin structures. Larger scales form in the detachment region (x/θo = 180) and advect downstream; there are large-scale positive and negative isosurface of streamwise uctuations

(u0) appearing alternatively in the spanwise direction.

Figure 4.13 shows contours of instantaneous streamwise uctuations at y+ = 12. Streamwise- elongated streaky structures of alternating high and low streamwise velocity exist before x/θo = 150; these structures grow in size and merge before the detachment region. There are also footprints of the large-scale structures in the ow reversal region when structures impinge on the wall at the reattachment (x/θo ' 344), and the attached boundary layer develops a larger streak spacing compared to the upstream region.

47 Figure 4.11: Isosurface of instantaneous Q > 0.05 for case 1. The visualization domain size shown here is 330θo × 50θo × 60θo.

Figure 4.12: Isosurface of instantaneous u0 for case 1; light pink is u0 = −0.15 and purple is 0 u = 0.15. The visualization domain size shown here is 330θo × 50θo × 60θo.

48 0 + Figure 4.13: Instantaneous contour of u normalized by uτ,o for case 1 at y = 12, the visualization domain size shown here is 330θo × 50θo × 60θo.

4.4 Rough wall APG

Figure 4.14 (a) shows the skin-friction coecient for the three cases; the wall stress in the rough cases includes both the contributions of viscous and form drag (due to the existence of the small separation regions behind each roughness element); In this study Cf is obtained by integration of the streamwise momentum equation as discussed at section 2.6.

If one considers the standard denition of separation (τwall = 0 and dτwall/dx < 0), separation happens notably earlier in rough wall ows and the separation bubble for the rough cases extend further downstream (gure 4.14 (a)), resulting in a larger reattachment length than in the smooth case, consistent with Song & Eaton (2002) and Cao & Tamura

(2006). However, the ow streamlines, shown in gure 4.16, do not reect the earlier separation; the zero-velocity line does not lift up until x/θo ≈ 150, much downstream of the point were Cf crosses zero (gure 4.14 (a)). This is also evident in gure 4.14 (b), in which the mean streamwise velocity above the roughness crest is shown. The point where Ucrest = 0 is much closer to the point where the ow lifts from the wall. Note that in experiments the occurrence of separation is usually measured using Ucrest (Song et al., 2000) and (Song & Eaton, 2002), since the wall stress is not generally known.

49 15 (a)

f 10 C

× 5 3

10 0 −5 100 200 300 400 500

(b) 0.4 0.2 crest U 0 −0.2 100 200 300 400 500 x/θo

Figure 4.14: (a) Distribution of Cf ; (b) streamwise velocity at roughness crest; , Smooth; , k/θo = 0.47; , k/θo = 0.95 .

Table 4.3: Drag force per spanwise length.

Case k/θo Total Entrance Separation bubble Recovery Smooth 0 0.59 0.21 -0.01 0.39 Rough 0.47 1.01 0.26 -0.05 0.80 Rough 0.95 1.29 0.27 -0.09 1.11

Drag force for all the cases are arranged in table 4.3 and is calculated based on integration of the skin friction coecient (gure 4.14 (a)).

Z x 2 ρV∞ F = Cf,x dx (4.6) 0 2 Total denotes integration through whole domain, entrance is drag force from reference plane until separation, separation bubble denotes integration of the recirculation bubble and recovery is for after the reattachment. Roughness augments the total drag force and the increase of the drag force is more signicant after the reattachment comparing to before

50 10 total

τ 5 × 3 10 0 100 200 300 400 500 x/θo

Figure 4.15: Total stress above the crest; , Smooth; , k/θo = 0.47; , k/θo = 0.95.

Figure 4.16: Streamwise velocity contours; , mean streamline; , zero velocity line: (a) case 1; (b) case 2; (c) case 3 at table 4.1.

51 separation.

Figure 4.15 shows the skin friction for case 1 and the total stress (equation 4.7) above the roughness crest for case 2 and 3:

1 ∂hUi ∂hV i τ = −hu0v0i + + (4.7) Re ∂y ∂x

Experimental techniques measure and turbulence models predict the total stress above

the roughness crest and they are not able to capture the ow reversal inside the roughness

elements. The negative shear happens further downstream (gure 4.15) for rough cases

comparing to the negative skin friction (gure 4.14 (a)). Roughness results in a wider and taller recirculation zone (gure 4.16), and a thicker

boundary layer. The early occurrence of Cf = 0, unaccompanied by detachment of the ow, represents a ow reversal inside the roughness sublayer without separation, and is due to the

recirculating-ow regions that occur behind each roughness element. Their size increases in

the APG region (gure 4.17); the ow outside of the sublayer, however, is mostly unaected

and streamlines remain roughly parallel to the wall, until the recirculating regions occupies

most of the roughness sublayer (the region where roughness causes spatial variations of

time-averaged turbulent statistics). At this point, the outer ow separates. Figure 4.17

also indicates that recirculation remains inside the roughness layer until x/θo ≈ 165 and

the velocity above the crest changes sign far downstream of the point where Cf = 0 (at

x/θo = 130). The value of the shape factor H = δ∗/θ (where δ∗ is the displacement thickness and θ is the momentum thickness) can be used as an indication of how close the boundary layer is

to separation. Figure 4.18 shows the evolution of the shape factor for all the cases. Shape

factors for smooth and rough cases during the ZPG region are respectively 1.4 and 2.1, consistent with the values for turbulent boundary layers. The stronger adverse pressure

gradient leads to the higher value of H; the shape factor increases very rapidly near the

52 Figure 4.17: Time averaged streamwise velocity at y/θo = 0.4 and y/θo = 0.8; , zero velocity line for k/θo = 0.47 (case 2).

15 130 170 x/θo = 117

10 H 5

H = 2.1 = 1 42 0 H . 100 200 300 400 500 x/θo

Figure 4.18: The evolution of the shape factor for , Smooth; , k/θo = 0.47; , k/θo = 0.95.

53 Table 4.4: Separation bubble size. In the table, xs denotes the position of separation (Cf = 0), xr is the position of reattachment, ls is the length of the separated region, and Hsep is the shape factor at the position of separation.

Case xs xr ls Hsep I 170 345 175 2.3 II 130 358 228 2.5 III 117 360 243 2.3

separation and reaches its maximum at x/θo = 280. Then it decreases as the FPG region starts. Separation of the ow occurs when H reaches a value between 1.8 and 2.5 (Kline et al., 1968); H in all the cases is in this range.

The mean velocity proles in wall units (normalized by local uτ ) are shown in gure 4.19 before the detachment (deceleration) region and after the reattachment (acceleration) region

for all the cases. As the ow decelerates and slows down, the slope of the logarithmic region

increases. Acceleration, on the other hand, leads to decrease of the slope of the logarithmic

layer. The change in the slope before and after the separation bubble is more signicant for

case 3. Figure 4.20 compares the proles of streamwise mean velocity at several locations nor-

malized by the local freestream velocity, U∞. Before the separation, rough walls streamwise velocity proles deviate from the smooth case; momentum decit can be observed due to

the thicker boundary layer and higher Reynolds stress in rough cases and the ow reversal

is clear after the separation for all the cases. After the reattachment, the mean velocity

prole does not recover its original state as in the upstream ow even at x/θo = 500 (Kim & Chung, 1995), (Song & Eaton, 2002) and (Essel & Tachie, 2015).

Figures 4.21, 4.22 and 4.23 show Reynolds-stress contours along the streamlines normalized using mixed scaling (uτ,oU∞,o, DeGraa & Eaton (2000)) at the reference plane.

54 80 180 (a) (b) 70 160 60

x/θo = 470 50 140

+ 40

U θ x/θo = 100 x/ o = 440 30 120 x/θo = 70 x/θ = 410 20 o θ x/ o = 40 100 10 θ x/ o = 10 x/θo = 380 0 80 100 102 104 100 102 104 (y − d)+ (y − d)+

Figure 4.19: Proles of mean streamwise velocity; (a) before the ow detachment (shifted upward by 10 units); (b) after the ow reattachment (shifted upward by 20 units); , smooth; , k/θo = 0.47; , k/θo = 0.95; , logarithmic law of the wall.

The vertical ordinate is the stream function,

Z y ψ(x, y) = U(x, y0)dy0 (4.8) 0

This transformation, introduced by Blackwelder & Kovasznay (1972), highlights the forces acting on a uid particle as it moves along its path.

The zero skin friction lines divides the Reynolds stresses contours into three regions; rst, before separation, roughness enhances all the Reynolds stresses. In the second region, the recirculation region, all Reynolds stresses increase quickly in the adverse pressure gradient region; streamwise and spanwise Reynolds stresses are more signicant. This increase is due to the shear-layer instability and the formation of turbulent eddies that are lifted after separation and travel around the separation bubble. Rough wall cases have stronger shear stress after the separation, due to the roughness eects in the outer layer. Nevertheless, the

Reynolds stresses are damped after x/θo = 280 where the streamline is at its most convex,

55 10 100 150 200 250 300 350 400 450 500 x/θo = 50 8

6 o θ

y/ 4

0 0 1 2 3 4 5 6 7 8 9 10 U/U∞

Figure 4.20: Mean streamwise velocity proles normalized by local freestream velocity

(U∞); , smooth; , k/θo = 0.47; , k/θo = 0.95; shifted rightward by 1 unit.

Figure 4.21: Contour of streamwise Reynolds stress hu0u0i normalized by mixed scaling at reference plane (uτ,oU∞,o); (a), smooth; (b), k/θo = 0.47; (c), k/θo = 0.95; , Cf = 0; , boundary layer thickness.

56 Figure 4.22: Contour of wall-normal Reynolds stress hv0v0i normalized by mixed scaling at reference plane (uτ,oU∞,o); (a), smooth; (b), k/θo = 0.47; (c), k/θo = 0.95; , Cf = 0; , boundary layer thickness.

57 Figure 4.23: Contour of spanwise Reynolds stress hw0w0i normalized by mixed scaling at reference plane (uτ,oU∞,o); (a), smooth; (b), k/θo = 0.47; (c), k/θo = 0.95; , Cf = 0; , boundary layer thickness. coincident with the beginning of the acceleration phase. All the Reynolds stresses reach their maximum in region three, after the reattachment, as the structures impinge on the wall and advect downstream; signicant Reynolds stress for all the cases exist in the outer layer and advect downstream as the boundary layer redevelops.

4.5 Separation physics

The mean-streamwise-velocity proles at two streamwise locations, the reference plane and a point shortly before separation (τw = 0) are shown in Figure 4.24. Roughness leads to lower mean momentum in the boundary layer due to the higher Reynolds shear stress and thicker boundary layer thickness. The momentum decit is the main reason for the early separation in the rough cases.

58 20

15 o

θ θ = 0 θ = 110 10 x/ o x/ o y / 5

0 0 0.5 1 1.5 2 U/U ∞, o

Figure 4.24: Mean streamwise velocity prole before separation; , Smooth; , k/θo = 0.47; , k/θo = 0.95.

Figure 4.25 shows proles of the production of q2 = 2K (twice the turbulent kinetic energy), normalized by inner layer scales (uτ,o and ν/uτ,o),

Pq2 = 2Pk (4.9) and its main components, the Reynolds shear stress hu0v0i and ∂hUi/∂y, along the streamline passing through a point slightly above the roughness crest. Twice of production of turbulent kinetic energy (equation 4.9) presents dierent behaviors for smooth and rough walls in gure

4.25 (b). The production increases in smooth wall (case 1) along the streamline before the separation. In the rough walls (case 2, 3), however, the production decreases before the separation. In other words, smooth and rough walls have dierent responses to the adverse pressure gradient. The dominant term in the production is hu0v0i∂hUi/∂y, the rst element

hu0v0i (gure 4.25 (c)) is similar for all the cases; however, decrease of the production is mainly due to the velocity gradient (gure 4.25 (d)) and reduction of the velocity gradient

is more signicant than the Reynolds stress, indicating that advection of the near wall uid

commute the eects of roughness to the outer layer.

The production of −hu0v0i, ∂hUi P ' −hv0v0i (4.10) uv ∂y

59 Figure 4.25: (a) Streamwise velocity contour and streamline near the roughness crest for case 2; (b) production of turbulent kinetic energy along the streamline; (c) −hu0v0i along the streamline; (d) ∂hUi/∂y along the streamline; , case 1; , case 2; , case 3 . is also aected by the increase of momentum decit in rough cases and shuts down before the detachment (gure 4.26 (b)), while the normal stresses −hv0v0i, are not strongly

aected until the reattachment region. The fact that both −hu0v0i and −hv0v0i do not decrease substantially over their path while their production does, indicates that in rough

wall boundary layers, frozen shear stresses are convected over the recirculation region, while

active turbulence production mechanisms are suppressed.

In the double averaging (DA) eld in section 2.7, the wake eld (f·) is the dierence between time and time-spanwise averaged eld and is highly correlated with the roughness

geometry. Figure 4.27 shows the total mean and uctuating velocities in a ow inside

the roughness layer; it supports the conjecture that roughness eects are reduced before

60 Figure 4.26: (a) Streamwise velocity contour and streamline near the roughness crest for case 2; (b) production of −hu0v0i along the streamline; (c) hv0v0i along the streamline; (d) ∂hUi/∂y along the streamline; , case 1; , case 2; , case 3 .

61 Figure 4.27: Double average components of streamwise velocity for k/θo = 0.47 (case 2) at y/θo = 0.4; (a) total velocity; (b) time-averaged velocity; (c) wake velocity; (d) velocity uctuations.

Figure 4.28: Streamwise dispersive stress 2 + inside the roughness sublayer ( ); hue i 2ks (a) k/θo = 0.47 (case 2); (b) k/θo = 0.95 (case 3); , cf = 0.

(a) 0.6 (b) 3 kR kR 0.4 2 0.2 1 0

0 −0.2 0 1 2 3 0 1 2 3 y/ks y/ks

Figure 4.29: Proles of the dispersive stresses at x/θo = 40 normalized by reference friction velocity ; (empty symbols) and (lled symbols). 2 +; (uτ,o) k/θo = 0.47 k/θo = 0.95 (a) hue i 2 +; 2 +; + . (b) hve i hwe i huevei the separation region (τw = 0 at x/θo = 130). There is a low-speed region downstream of most of the roughness elements in averaged velocity at gure 4.27(b). Upstream of the separation, wake and turbulence uctuations (gure 4.27 (c,d)) are of the same order of magnitude; however, wake uctuations drop near separation. Figure 4.28 shows the streamwise dispersive stress ( 2 ) the stress arising due to the spatial heterogeneity of hue i the rough surface near the wall normalized by reference friction velocity ( 2 ); the vertical uτ,o

axis is normalized by the roughness height (ks). In the ZPG region, signicant spatial variations of time-averaged ow quantities are generated due to the three-dimensionality of the roughness geometry. As the APG becomes signicant, the wake velocity decreases approximately by a factor of four before the separation (compared to the ZPG region gure

4.28 (a,b)). Disappearing of the streamwise dispersive stress results in attenuation of the contribution of the roughness and dropping down roughness Reynolds number (k+) (as a consequence of decreasing uτ (gure 4.14)). In general, for case 2 and 3 roughness regime starts from fully rough regime at the inlet then transitionally rough regime and nally smooth before the separation.

Dispersive stresses resulting from the wake component, 2 and , are shown for huei i huevei both rough cases at gure 4.29 (x/θo = 40). Their peaks appear near the roughness height

(ks) and increase as the roughness height increases. In contrast to the Reynolds stresses, the dispersive stresses are negligible above the roughness sublayer.

63 3 (a) 0.6 (b)

2 0.4 0.2 1 0

0 −0.2 0 1 2 3 0 1 2 3 0.6 3 (c) (d) 0.4 2 0.2

1 0

0 −0.2 0 1 2 3 0 1 2 3 y/ks y/ks

Figure 4.30: Proles of the dispersive stresses at x/θo = 40 ( ) and x/θo = 110 ( ) for k/θo = 0.47 (empty symbols, a,b) and k/θo = 0.95 (lled symbols, c,d) roughness heights. 2 +; 2 +; 2 +; +. hue i hve i hwe i huevei

Figure 4.30 shows the dispersive stresses normalized by 2 at ZPG ( ) and uτ,o x/θo = 40

APG (x/θo = 110) regions for two dierent roughness heights. Streamwise and spanwise dispersive stresses decrease markedly (more than 75%) in the APG region comparing to

ZPG region because of decreasing of the friction velocity (uτ ) in decelerating region. The decrease of the wake uctuations in APG regions indicates that roughness eects are less

signicant in the APG.

64 Chapter 5

Conclusions

Large-eddy simulations of APG boundary layer over smooth and rough surfaces were performed at Reθ = 2300 to study combined eects of roughness and APG on turbulent boundary layer with separation. A sand-grain roughness model (Scotti, 2006) was implemented with two roughness heights, k/θo = 0.47 and 0.95 in fully-rough regime; the roughness Reynolds numbers are + and in the zero pressure gradient region, respectively. ks = 60 120 The no-slip boundary condition was imposed via an immersed boundary method based on the volume of uid for the roughness.

The ow separates due to the large adverse pressure gradient induced by suctionblowing velocity distribution along the upper boundary, ow separation is oscillating, and so is the reattachment. The domain can be divided into three regions based on the recirculation bubble:

(i) Before the separation, peaks of Reynolds stresses are located near the wall and roughness intensies all the Reynolds stresses.

(ii) After the separation, normal stresses increase in the adverse pressure gradient region as a result of the passage of large scale structures; the increase is more signicant for streamwise and spanwise directions; peaks of normal stresses are located above the separation bubble, aligned with the height of the streamwise velocity inection point; these 65 stresses are comparatively small within the separation bubble. The shear layer is stronger after the separation for the rough cases, indicating the presence of roughness eects in the outer layer. Moreover, production and dissipation reach their maximum after the separation. However, above the recirculation bubble, when streamlines are parallel to the wall, production of turbulent kinetic energy decreases as ∂U/∂y attenuates and ∂U/∂x goes up (start of the acceleration region).

(iii) The recovery region, hv02i and −hu0v0i increase signicantly and are comparable to

hu02i because of the impingement of the large structures in the reattachment region and all

the Reynold stresses reach to their maximum, producing highly 3D ow. Earlier separation takes place for the rough cases; roughness initially increases the skin

friction, but later makes it decrease faster than in the smooth wall. As the roughness height

increases, the skin friction increases and the separation bubble becomes larger. There is

a discrepancy between the separation point dened based on Cf = 0 with the one corre-

sponding to change the sign of Ucrest (which is closer to smooth wall separation), the earlier

occurrence of Cf = 0 illustrates ow reversal inside the roughness without the ow separation, due to the recirculation regions behind roughness elements. Thus, a more universal

denition of separation point needs to be investigated as a potential topic to pursue.

Rough cases have the larger boundary layer thickness and higher Reynolds shear stress,

causing the larger momentum decit that is the primary reason for inducing the earlier

separation in rough cases. A second reason of the earlier separation is the dierent responses

of smooth and rough walls to the adverse pressure gradient. −hu0v0i and −hv02i along the streamwise close to the roughness crest do not deviate from smooth wall before the

separation. However, production of TKE diers from the smooth case and production of

−hu0v0i shuts down for the rough cases both before and after the separation, mainly due to the decrease in the velocity gradient in the outer layer. Adverse pressure gradient propagates

the roughness eects outside of the roughness sublayer and roughness eects can be felt

throughout the boundary layer after the separation because of the advection of near-wall

66 uid around the recirculation region.

Results conrm that APG changes the roughness Reynolds number as friction velocity decreases before the separation and roughness regime starts from fully rough regime at the inlet, then transitionally rough and nally smooth near the separation. This is proven by the attenuation of the dispersive stresses before the separation comparing to upstream.

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