Effects of Wall Roughness on Adverse Pressure Gradient Boundary Layers

Effects of Wall Roughness on Adverse Pressure Gradient Boundary Layers by 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 Copyright © Pouya Mottaghian, 2015 Abstract 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 U1 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;1 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 ! Turbulent vorticity Ω Rotation rate tensor 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), ∗ exper- iment 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 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 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 . 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 U1;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) ;U1; ;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.

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