stability of wall-bounded flow modified to the of distributed · 2004. 11. 29. · stability of...

Stability of wall-bounded flow modified due to the presence of distributed surface roughness

by

Antonio Cabal

Graduate Program in Applied Mathematics

Submitted in partial fulfillment

of the requirements for the degree of

Doctor of Philosophy

Faculty of Graduate Studies

The University of Western Ontario

London, Ontario

October 1998

@ Antonio Cabal 1998

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Abstract

Linear stability of wall-bounded flow modified due to the presence of d i s tributed surface roughness has been considered. The form of the flow in the presence of roughness is determined by two different methods (i) Do- main Perturbation Method (DPM) and (ii) Domain Transformation Method (DTM). Effectiveness of these methods for simulation of flows over cormgated/rough boundaries has been analyzed. DPM, which approximates the shape of the boundary, is applicable to corrugation amplitudes up to and provides accuracy no better than 0 (10-I) for such conditions. Its absolute accuracy significantly increases for lower amplitudes especially when higher-order versions of the method are used. DTM enforces flow boundary conditions exactly and provides spectral accuracy.

A new (by-pass) route to transition has been found by carrying out stability analysis of distributed surface roughness described by one Fourier harmonic with amplitude 2s. A criteria for stability based on the critical roughness Reynolds number (RG) has been formulated for the maximum distributed roughness amplitude that the flow can accommodate without inducing a new type of unstable disturbances (streamwise vortices).

Dedication

To Andrea for the strength of her constant presence and

to Tatiana for the mildness of her smile.

Acknowledgments

I would like to q r e s s my thanks to all the people who had a part in the completion of this thesis, among whom are:

My senior supervisor Dr J. M. Floryan for introducing me to the topic that sprouted this work and for all his ideas, guidance, research abilities, and financial support. I am also deeply gratefiil to Dr J. Szumbarski for our discussions, his valuable questions, comments and for his friendship.

Dr R. D. Russell and Dr M. R. Thmmer for their invaluable friendship. Dr. P. J. Mann for his readiness to help me any time I needed and for

his remarkable programming abilities and in depth computer knowledge. Roger and Jocelyn Coroas, Rolando Linares and Jesus Siu for the good

times I had in my last visit to Vancouver. Dr. D. S. Ross whose personal effort made possible the shaping of my

near future. My family and friends whose consistent moral support helped me tremen-

dously. D. Gayle McKenzie, Pat A. Malone and Audrey J. Kager for their as-

sistance and the Department of Applied Mathematics of The University of Western Ontario for financial support.

Contents

Certificate of Examination ii

Abstract iii

Dedication iv

Acknowledgements v

Table of Contents

List of Figures ix

List of Tables xi

Nomenclature xii

1 Introduction 1 . . . . . . . . . . . . . . . . . . . . . . . 1.1 Representativework 3

. . . . . . . . . . . . . . . 1.1.1 Geometric Characterization 3 . . . . . . . . . . . . . . . . 1.1.2 EarlyExperimentd Work 4

. . . . . . . . . . . . . . . . . 1.1.3 Early Theoretical Work 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Recent Work 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Present Work 8

2 Flows over Corrugated Walls 11 2.1 Model Problem . . . . . . . . . . . . . . . . . . . . . . . . . . I1

2.1.1 Reference Flow . . . . . . . . . . . . . . . . . . . . . . 12 2.1.2 Flow in a Cormgated Channel . . . . . . . . . . . . . . 12

2.2 Domain Perturbation Method ( DPM ) . . . . . . . . . . . . 14 2.2.1 Boundary Conditions . . . . . . . . . . . . . . . . . . . 15

2.3 Numerical .Method . . . . . . . . . . . . . . . . . . . . . . . . 18 . . . . . . . . . . . . . . . . . . . . 2.3.1 Truncated Problem 19

2.4 Discussion of Results . . . . . . . . . . . . . . . . . . . . . . . 24

3 Stability analysis in the case of simulated (blowing/suction)

distributed surface roughness 30 3.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . 31 3.2 Small Amplitude Simulated Distributed Roughness . . . . . . 32

. . . . . . . . . . . . . . . . . 3.2.1 Fourier Mode Separation 34 3.3 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . 39 3.4 Computational Method . . . . . . . . . . . . . . . . . . . . . 40

3.4.1 Generalized Eigenvdue Solvers . . . . . . . . . . . . . 42 3.5 Code Verification . . . . . . . . . . . . . . . . . . . . . . . . 44

4 Domain Transformation Method ( DTM ) 46 4.1 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . 47 4.2 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . 49 4.3 Numerical Solution . . . . . . . . . . . . . . . . . . . . . . . . 51

. . . . . . . . . . . . . . . . . . . 4.3.1 Truncated Problem 52 4.4 Positioning of Roughness . . . . . . . . . . . . . . . . . . . . 61

4.4.1 Pureshift . . . . . . . . . . . . . . . . . . . . . . . . 61 4.4.2 Shift Plus Cormgation . . . . . . . . . . . . . . . . . 63

4.5 Linear Approximation . . . . . . . . . . . . . . . . . . . . . . 64 4.6 Discussion of Results . . . . . . . . . . . . . . . . . . . . . . 67

5 Stability Analysis using DTM 80 5.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . 80

vii

. . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Test Problem 83 . . . . . . . . 5 2.1 Periodic Three-dimensional Disturbances 85

. . . . . . . . . . . . . . . . . . 5.2.2 Boundary Conditions 87 . . . . . . . . . . . . . . . . . . . . . . . 5.3 Numerical Method 87

. . . . . . . . . . . . . . . . . . . . . . . . 3.4 Analysis of Results 91

6 Sllmmary 101 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Conclusions 102

Bibliography

A Appendix 110 . . . . . . . . . . . . . . . . . . . A . 1 Expressions for Un and @. 110

. . . . . . . . A.2 General Problem . Linear Differential Operators 112 . . . . . . . . . . . . . . . . . . . . . . . A.2.1 Coefficients 113

A.3 Test Problem . Linear Differential Operators . . . . . . . . . . 141

B Appendix 143 B.l DTM Stability System . Linear Differential Operators . . . . I43

. . . . . . . . . . . . . . . . . . . . . . . B.1.1 Coefficients 147

Vita 210

List of Figures

. . . . . . . . . . . . . . . . . . . . . Sketch of the flow domain 27 . . . . Variations of lluL(x) 1 1 as a function of Fourier modes 28

. . . . . . DPM2 error distributions and their Fourier spectra 28 Variations of lluL(x) l l m as a function of corrugation amplitude 29

Amplification rate . I m a g ( 0 ) for the suction-induced-disturb- with respect to p . . . . . . . . . . . . . . . . . . . . . . . . . 45

Mapping from physical to computational domain . . . . . . . 69 . . . . . . . . . . . . .. Changes of profiles with respect to a: 70 . . . . . . . . . . . . .. Changes of profiles with respect to a 71

Changes of u. profiles with respect to Re . . . . . . . . . . . 72 v. . . . . . . . . . . . Changes of profiles with respect to Re 73

Stream line patterns of flow modXcations (a = 1) . . . . . . . 74 Stream line patterns of flow modifications (a = 10) . . . . . . 75 Variations of the Chebyshev norm llD@& as a function of Fourier mode number n . . . . . . . . . . . . . . . . . . . . . . 76 Variations of spectral convergence rate p as a function of wave number a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

4.10 Variations of spectral convergence rate p as a function of the corrugation amplitude 2s . . . . . . . . . . . . . . . . . . . . . 77

4.11 Variations of energy mode El ( linear and nonlinear ) with respect to roughness amplitude for different values of the shift parameter y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

4.12 Variations of energy modes Eo, El, E2 with respect to roughness amplitude for different values of the shift parameter y . . 79

Amplification rate -Imag(o) as a function of shift parameter y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 Amplification rate -Imag(a) as a function of the spanwise wavenumber p for different values of the shift parameter y. . 95 Amplification rate -Imag(o) as a function of the spanwise wavenumber p ( Re = 3000 ). . . . . . . . . . . . . . . . . . . 96 Amplification rate -Imag(o) as a function of the spanwise wavenumber p ( Re = 5000 ). . . . . . . . . . . . . . . . . . . 97 Amplification rate -Imag(o) as a function of the Reynolds number Re. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 AmpIification rate -Imag(o) as a function of the distributed roughness amplitude 2s. . . . . . . . . . . . . . . . . . . . . . 99 Amplification rate -[mag@) as a function of the roughness Reynolds number Rg. . . . . . . . . . . . . . . . . . . . . . . 100

List of Tables

5 . Amplification -1mag (o) of roughnessinduced disturbances ( Re=5000, y = 0 , s =0.007, cu=2.0, p =2.0) . . . . . . . . 90

Nomenclature

We Iist only symb 01s that appear frequently-

Symbol Meaning

upper rough-channel wall

lower rough-channel wall

Reynolds number

roughness Reynolds number

velocity vector

vorticity vector

Floquet exponent

wall-shift parameter

streamwise wavenumb er

spanwise wavenumber

half roughness amplitude

disturbance complex frequency

L,-norm

Chebyshev norm

Chapter 1

Introduction

Hydrodynamic stability has been an important area of fluid dynamics research since the nineteenth century when renowned scientists like Helmholtz, Kelvin, Rayleigh and Reynolds first recognized and formulated its essential problems. The phenomenon of transition from laminar to turbulent flow was first investigated by Osborne Reynolds. He described, in the classic series of experiments published in 1883 [37], how the laminar flow breaks down when what is now commonly referred to as the Reynolds number exceeds a certain critical value. The key equation of linear stability theory was arrived at inde- pendently by Orr in 1907 and by Somrnerfeld in 1908. This Orr-Sommerfeld equation remained unsolved for twenty-two years, until Tollmien calculated the first neutral eigenvalues and obtained a critical Reynolds number. About 1930, Prandtl and his collaborators succeeded in attaining the value of the critical Reynolds number for two-dimensional boundary layer in a satisfactory way. Ten years later the theory was experimentally verified by L. Dryden. Since then, efforts to clarify and to explain theoretically the remarkable process of transition have led to different hypothesis and theories with variable degree of success.

Rough walls exist in all flow systems, where they may lead either to deterioration or improvement of the desired functionality. Nature provides

for the survival of noted by [3], shark

numerous instances where rough surfaces are essential many species of air and marine animals. For instance, as scales have humplets, so spatial variations in their placement should lead to at least as good a drag reduction to that found recently for streamwise aligned riblets [Ill.

Wall roughness can be increased to promote mixing of the fluid, or reduced to eliminate flow disturbances. The related problem of the laminar-turbulent transition over a rough wall is one of the classical problems in fluid mechanics that has so far defied all analytical efforts. Qualitative understanding of the mechanisms through which surface roughness may affect transition is of considerable practical importance, it touches almost every field of fluid dynamics from biology to aeronautics.

Some experiments [14, 28, 36, 35, 121 have shown roughness enhances transition in the sense that under otherwise identical conditions transition occurs at a lower Reynolds number on a rough surface than on a smooth surface. The existence of roughness elements gives rise to additional disturbances in the laminar stream which have to be added to those coming from the environment. If disturbances created by the roughness elements are bigger than those coming from the environment, one should expect that a lower degree of amplification will be sufficient to effect the transition. On the other hand, if the roughness height is sufficiently small, it has no effect on the transition process; the corresponding walls are considered to be hydraulically smooth. Ability of roughness elements to significantly alter transition depends, beside their size, on their geometrical form and their distribution. The real challenge is to identify mechanisms through which surface roughness may affect transition and to provide conservative transition prediction criteria that can be used in the case of roughness sensitive engineering designs.

Many of the difficulties associated with this problem stem horn the fact that various physical mechanisms may initiate the transition process, depending on the distribution, amplitude and geometry of the roughness, and depending on the flow conditions. Since there are an infinite number of possible forms of roughness elements a methodology for characterization of the roughness is required so that the forms of roughness can be classified, with each class subject to separate investigations. The ( unviable ) alternative is to study each rough surface separately. The related questions are: (i) what

parameters should be used to characterize a particular geometry, and (ii) how to define equivalent geometries.

1.1 Representative Work

The routes to transition are many; some are more understood than others. The existing knowledge about the effects of surface roughness in the laminar- turbulent transition process is mostly of empirical origin. Review of these works can be found in [30, 33, 38, 151. There have been recent theoretical developments shedding some light on the mechanisms through which surface roughness accelerates instability processes. Both of the above aspects of the roughness issue are discussed in this chapter.

1.1.1 Geometric Characterization

The scientific literature regarding this topic deals roughly with three classes of shapes: (i) single isolated two-dimensional roughness, i.e., trip wire, (ii) single isolated three-dimensional roughness, i.e., grain of sand, and (iii) distributed roughness.

Flow Topology

The key feature of single isolated two-dimensional roughness is the presence of separated wall-wakes and their special characteristics: the vortex sheets which become thinner with increasing Reynolds number (Re), the lengths of the recirculation zones, internal secondary flows, etc (see [33, 151). Ef- fects of roughness of height k should depend either on its relative (blocking) height, k /P (6' - boundary layer displacement thickness) or on its dimensionless disturbance Reynolds number to the flow near the wall, Rek, or both. A frequently used criterion is that the roughness Reynolds number Rek = Uk k/u < 25 ([33]), where Uk is the undisturbed velocity at the height k and v the kinematic viscosity.

The topology of the separated regions around a single three-dimensional roughness element of a similar hzight, span and length is qualitatively different from the two-dimensional one. The front separation generates a horseshoe vortex which wraps around the roughness forming streamwise vortices on the downstream end, with a chimney flow in the form of a pair of spiral vortexes escaping vertically horn the downstream separation pocket and forming trail- ing vortices. This flow topology is almost universal for obstacles of various shapes, but symmetric with respect to the oncoming flow [33]. These obstacles include spheres, hemispheres, cones, short cylinders (upright and on the side) and parallelepipeds. The Reynolds number changes with the shape but the dominant topology appears invariant.

There is no single feature that can be associated with distributed surface roughness.

1.1.2 Early Experimental Work

The earlier papers ( see [39] for an account of them ) which addressed the influence of roughness on transition assumed that the point of transition was located at the position of the roughness element, when they were large, or that their presence had no infiuence at dl when they were small. The dependence of the transition point on the roughness height was empirically described by Dryden [Xi]. After analyzing a large amount of experimental work he formulated an empirical law for the determination of the position of the point of transition in terms of the height of the roughness element and its position. He discovered that in incompressible flow all experiment a1 points for the case when transition does not occur at the roughness element itself ar- range themselves on a single curve ( see [13]) in a plot of the Reynolds number Ret, = Ud;,/v formed with the displacement thickness 6: of the boundary layer at the point of transition against the ratio k& (6; - displacement thickness at the position of the roughness element ). As the roughness height k is increased the position of the point of transition moves gradually upstream and the experimental points begin to deviate from this curve as soon as the point of transition has reached the roughness element. They then lie along the family of straight lines defined by Rec, = 3 xk/& x k - roughness element

position. Tani [45] provided arguments for the universality of the correla- tions between height of the roughness, the flow conditions and the critical Reynolds number found in [I31 for certain classes of geometrical forms of the roughness.

Feindt [14] used sand grains of different sizes in his experiments to sim- ulate distributed roughness. Measurements were performed in a convergent and a divergent channel of circular crosssection with a cylinder covered with sand placed axially in them. The walls of the channel were smooth and their slope controlled the pressure gradient. He concluded that for Rek < 120 the roughness has no influence and the transition occurs at the same location as on the smooth surface.

Observations of the flow pattern behind a wire were carried out by Hama et al. [24]. Their experiments revealed the appearance of spanwise, two- dimensional waves which evolve into three-dimensional vortex loops having a well defined wavelength. These systems led to the breakdown into turbulence in the same manner as it would occur on a smooth surface.

Klebanoff and Tidstrom [29] studied in great detail the mechanism by which a two-dimensional roughness induces transition. A cylindrical rod was attached to a surface of a flat plate, with its axis perpendicular to the mean flow direction. Emphasis was placed on measurements within the recovery zone, i.e. the distorted region in the downstream vicinity of the roughness. The boundary layer separated immediately behind the rod and reattached at a certain distance further downstream, while the mean velocity profile returned to Blasius distribution. At the same time the transition point moved forward fkom where it would have been in a similar problem without the rod. They concluded that, according to linear stability theory, the inflexional velocity profiles encountered in the recovery zone caused a rapid amplification of the instability waves. The disturbances reaching the reattachment location are larger than they would be without roughness and lead to a premature transition. The rod does not. introduce new disturbances into the boundary layer, but strongly amplifies the existing ones.

Reshotko and Leventhal [36] found that distributed roughness in the form of sand paper thickens the boundary layer and moves the essentially unde- formed Blasius profile outward. The important growth of disturbances occurs in frequencies lower than those for which Tollmien-Schlichting waves are un-

stable. The ampliiication seems to be driven by the local wake profile at the crest of distributed roughness elements [35].

Tadjfar et al. [43] created distributed roughness using arrays of spheres attached to an otherwise smooth surface and measured the velocity distribution between individual spheres. They found no evidence of Tollmien-Schlichting waves. The results of their measurements suggests that the mechanism of transition is similar to the case of an isolated three-dimensional roughness element, i.e. it is driven by the horseshoe and hairpin vortices generated by the roughness element. The contribution of the downstream element should thus increase the strength of the upstream-generated vortices towards eventual transition.

Corke et al. [12] carried out experiments using sand paper to represent distributed roughness. They reported (i) that it is the low-inertia fluid in the valleys between the grains that responds to free-stream disturbances, (ii) once Tollmien-Schlichting waves appear, they grow faster (although the reason is unknown) than the comparable smooth-wall case, and (iii) there is evidence of roughness-induced three-dimensionalization of the wave fronts leading to earlier secondary inst ability.

1.1.3 Early Theoretical Work The laminar-turbulent transition process in the case of smooth walls begins, typically, with instability in the form of linearly growing two-dimensional Tollmien-Schlichting waves followed by secondary instabilities and the appearance of three-dimensional effects.

Smith [40] used stability theory to confirm an earlier empirical discovery by Michel [32] ( it was empirically confirmed at about the same time by Van Ingen [47] ) which stated a remarkably simple relationship between the rate of amplification and the distance between the theoretical position of the point of instability and the experimentally determined position of the point of transition. Results of Smith's calculations led to the conclusion that the amplification rate of unstable disturbances, integrated along the path from the point of neutral stability to the point of transition, has an amplification factor of e9.

JaEe et al. [27] took the confirmation of Michel's heuristic relation further from temporal to spatial stability ( the method is known as the eN method ). It also extended the st ability theory to produce approximate predictions for act u d transit ion on axisymme t ric or two-dimensional bodies having arbitrary pressure distributions in incompressible flows. After extensive comparisons with experiments for flows over a variety of bodies an average spatial amplification (not the timewise treated by earlier papers) rate of small disturbances of about elo was found. Later research has shown a universal value of elo is not always retained. Horstmann et al. [26] instead took the approach of finding the value of N for which it was possible to obtain agreement with the experiment for a given set of conditions (N = 13.5 correlated both).

Although the precision achievable is not exceptionally good, the eN method provides a satisfactory transition prediction criteria for hydraulically smoot h walls. Mechanisms promoting transition in the presence of distributed surface roughness represent bypasses to the smooth surface scenario when the roughness amplitudes exceed certain critical value and lead to the failure of the eN method. These mechanisms which strongly depend on the geometry of the roughness, are not known in the case of distributed roughness, even though Floryan recently suggested 1161 they may lead to a similar flow response to that found by him in the case of simulated (blowing/suction) distributed roughness.

1.2 Recent Work

Recently, the effects of surface roughness became of interest from the point of view of passive/active flow control strategies, where one is interested in determining the " smallest' possible surface modification that may induce the "largest' possible changes in the flow field. A subject that is hotly pursued at present involves the techniques to reduce the turbulent skin-friction drag. These include large eddy breakup devices, riblets, compliant surfaces, wavy walls, and other surface modifications [6]. With few exceptions, there is little theoretical basis for what is mostly an experimentally based knowledge. Monograph [Zl], published earlier this year, provides a comprehensive review

of these and other flow control strategies as well as the frontiers of the field of flow control.

Theoretical analysis of the effects of distributed surface roughness faces the same problem as experiments, i.e., how to realistically model all possible geometrical forms of the roughness. Floryan [15] suggested using spectral and fractal models. In [16] the former was used to study the stability characteristics of wall-bounded shear layers in the presence of simulated (by means of blowing/suction) distributed roughness. In all studied cases (Poiseuilte flow, plane Couette flow and Blasius boundary layer) suction was able to induce a new type of instability characterized by the appearance of streamwise vortices as the dominant mode. The author concluded that the linear model of suction induced flow modifications is sufEcient for analysis of the flow stability, i.e., it is s ac i en t to carry out stability analysis on the mode- by-mode basis once and for ever rather than studying each particular suction distribution on a case-by-case basis.

1.3 Present Work

Analysis of flows over rough/comgated boundaries is conditional upon avail- ability of algorithms capable of determining spectral composition of flow with spectral accuracy. The available approaches can be divided into two cate- gories. In the first one, the irregular flow domain is mapped onto a regular computational domain. Imposition of the flow boundary conditions becomes simple, however, one has to work with a much more complex form of the %ow field equations. In the second approach, one works directly in the physical domain. The field equations have a very simple form, however, one has to develop special procedures for the imposition of boundary conditions.

The determination of a mapping between the flow and the computational domain is equivalent to the construction of a coordinate system (orthogonal or nonorthogonal) where one of the coordinate Lines overlaps with the cormgated boundary [46]. Lack of spectrally accurate coordinate generation techniques limits the development of spectrally accurate algorithms. Alter- natively, one may use analytical mappings or specialized numerical mappings

that provide the desired absolute accuracy. Sobey [41] analyzed furrowed p e riodic channeIs and used an analytical mapping resulting in a non-orthogonal reference system. Caponi et . al. [9] employed an orthogonal transformation expressed in terms of aa infinite series in their analysis of boundary layers over wavy surfaces. Benjamin [4] considered a coordinate system based on streamlines of an inviscid flow over a wavy wall in his analysis of shear flows over wavy walls. Balasubranian and Orszag [2] employed numerically generated conformal mapping in their simulations of flows over wavy walls.

The direct approach, where the computations are carried out in the physical space, poses a challenge in terms of the imposition of boundary conditions. The best available met hod that does not introduce additional simplification is due to Szumbarski and Floryan [42]. The irregular flow domain is submerged into a regular computational domain and the resulting internal (rather than boundary) value problem is solved with spectral accuracy by taking advantage of the implicit function formulation. An alternative method, which we shall refer to as the domain perturbation method, involves transfer of boundary conditions to a certain mean location of the boundary, resulting in a regular computational domain. The accuracy of the domain perturbation depends on the amplitude of the cormgation and the type of boundary conditions transfer procedure. A first-order procedure is well described in [17]. This method is of interest due to its simplicity and due to the fact that in many applications the interest is in small roughness/cormgations where the domain perturbation method may be able to provide the required absolute accuracy.

There are three gods for the present investigation. The first one involves determination of the limits of applicability of the domain perturbation method and evaluation of the errors associated with various boundary condition transfer procedures. This will be accomplished by comparing the domain perturbation solution with the complete solution. The second goal is to propose a new general method based on analytical mapping of the physical domain into a rectangular computational domain. The capabilities of the domain transformation method are assessed by measuring its modal efficiency and rate of convergence. The third goal, which is the main motivation for this research, is to determine whether similar conclusions to those found in [16] regarding the stability characteristics of the flow in the case of simulated

distributed roughness can be made in the case of real distributed roughness. The thesis is organized as follows. Chapter 2 gives a description of the

model problem used for testing the various methods and discusses the domain perturbation method (DPM). Chapter 3 describes the procedures proposed by Floryan in 1161 to study the stability characteristics of the flow modified by simulated (blowing/suction) distributed roughness. Chapter 4 discusses a method based on domain transformation (DTM). Chapter 5 analyzes the linear stability characteristics of the modified flow due to wall roughness using DTM.

Chapter 2

Flows over Corrugated Walls

Analysis of flows over rough/corrugated boundaries is conditional upon avail- ability of algorithms capable of determining spectral composition of flow with spectral accuracy. A simple first-order method which involves transfer of the boundary conditions to a certain mean location of the boundairy, resulting in a regular computational domain is well described in 1171. Similar higher- order methods, which we shall refer to as domain perturbation methods, will be used in this chapter to deal with the irregular boundaries. The accuracy of the domain perturbation depends on the amplitude of the corrugation and the type of boundary conditions transfer procedure. This method is of interest due to its simplicity and due to the fact that in many applications the interest is in small roughness/cormgations where the domain perturbation method may be able to provide the required absolute accuracy.

2.1 Model Problem

Discussion of different methods that can be used to determine flow over an irregular problem.

boundary will be carried out in the context of a convenient model This selected problem consists of a viscous flow driven by a pressure

gradient through a channel with rough walls.

2.1.1 Reference Flow

Consider plane Poiseuille flow confined between flat rigid walls y = &l and extending to infinity in the x-direction (Fig. 2.1 a). The fluid motion is described by the following velocity and pressure fields

where the fluid is directed towards the positive x-axis and the Reynolds number Re is based on the half-channel height and the maximum x-velocity. This flow is driven by a constant negative pressure gradient.

2.1.2 Flow in a Corrugated Channel

Consider the upper and lower walls to have arbitrary shapes described by hU(x) and hL (x) (Fig. 2.1 b), respectively, and characterized by certain periodicity with wavelength A, = 2 n/a. The shape of the walls can be expressed in terms of Fourier series in the form

where (A,).y = (An),); and (An)L = (An); in order for hU(x) and hL(x) to be real, and star denotes the complex conjugate. The subscript L refers to the bottom wall, while the subscript U refers to the upper wall of the channel. In addition, it is assumed that

~ r l a ~ ~ hu(x) 5 1, rnin hL (x) 2 -1 , 05x52 .n/a 05x52 ?r/a

i.e. the flow domain b) . The fluid motion

is bounded by -w 5 x 5 00, -1 5 y 5 1 (see Fig. 2.1 is described by the following velocity and pressure fields

where vo, Po denote the velocity and pressure fields associated with the flow over the smooth walls, vl,pl stand for the velocity and pressure field modifications associated with the presence of the wall roughness, and UT, v~ denote the total x, y velocity component, respectively.

Equations of motion and continuity corresponding to the above representation of the flow have the form

with boundary conditions

Introduction of the stream function

and elimination of pressure from equations (2.7), (2.8) result in the following equation for +

where A denotes Laplace operator. The problem formulation has to be completed by specifying two additional

conditions for II, at the walls. Problem (2.10), (2.12) is closed under the assumption that the volume flux in the corrugated channel is the same a s the volume flux in the absence of roughness, i.e.

Q ( h ~ ( x ) ) + + ( x , h ~ ( x ) ) = B (B-arbitraryconst.) , (2.13)

Q ( h ~ ( x ) ) + $(z, h ~ ( x ) ) = B + Qo ( Qo - volume flux), (2.14)

where Q - stream function corresponding to plane Poiseuille flow (Qo = 413, ( - 1 ) = 0 In all test calculation presented in this Chapter, constant B was taken to be zero.

2.2 Domain Perturbation Method ( DPM ) Because of periodic character of channel corrugation, the unknown stream function (2.11) can be expressed in terms of Fourier expansion as

Substitution of (2.16) into (2.12) and separation of Fourier modes yields an h h i t e system of nonlinear ordinary differential equations for Q,, n 2 0, in the form

where

2.2.1 Boundary Conditions

The treatment of the boundary conditions shall be explained by focusing on the lower wall whose location is expressed, for the purposes of this discussion (see Eq. (2.3))) as

where E << 1, f (x) = O(1). Here, E denotes a measure of the magnitude of the lower as well as the upper wall corrugation. Since the flow modifications are of the order of the corrugation amplitude, one can write

where - 0(1). It is convenient to center our Taylor expansion around the point y = -1 and this requires the functions uq and u r to allow ma- lytic continuation on a small neighborhood of y = -1. The reader should remember that these expansions are carried out for purely numerical reasons. Substituting (2.18) into boundary conditions (2.10) and expressing them in terms Taylor expansions ce~tered a t y = -1, one obtains

+e2 (El f (x) + m) y=- 1 2 dy2 y=-1

Retention of terms 0 (E) , 0 (c2) 0 (c3) , etc., results in first-, second-, third-, etc., order boundary condition transfer procedure. One could introduce a similar (asymptotic) approximation into the field equations and solve a se- quence of resulting perturbation problems. Since our interest is in the maximum possible absolute accuracy, we shall always work with the complete

equations subject to various approximation of boundary conditions. Assum- ing that the field equations can be solved exactly, the distance between our "replacement" problem and the original problem is determined by the various levels of Taylor series truncations in (2.19), (2.20). This " distance", whose precise definition will be given later in the text, is the measure of the error of the domain perturbation method.

Substitution of (2.3) and (2.15) into (2.19) - (2.20), return to the original notation (i-e., elimination of E ) and separation of Fourier modes give boundary conditions in the form

where

+a dQn-m

Pz,, = C (n - m) dy

(Am) L r m=-oo

Expressions (2.2 I), (2.22) result from the no-slip and no-penetration conditions, respectively. Retention of L1,, and f i In only gives the first-order

boundary condition transfer second-order procedure, and

procedure, adding L2, and P2,, produces the incorporation of L3,n and P3,n results in a the

third-order procedure. One could construct higher-order procedures in a similar fashion. Boundary conditions for the upper wall can be constructed in the same manner, i.e.,

Expressions (2.22) and (2.32) do not provide boundary conditions for mode n = 0. These conditions have to be determined from (2.13), (2.14). Substitution of (2.3) and (2.15) into (2.13)-(2.14), introduction of Taylor expansions and separation of mode zero gives the boundary conditions in the form

F L + F 2 + F 3 = O , at y = - 1 , (2.41)

G L + G 2 + G 3 = 0 , at y = I , (2 -42)

where

In the above, retention of Fl and GI only gives the first-order procedure, addition of F2 and Gz produces the second-order procedure and inclusion of F3 and G3 results in the third-order procedure.

2.3 Numerical Method Numerical tests have been carried out for the upper and lower walls in the form

- (A0)u = (Ao) L = 7 , ( A L ) U = ( A ~ ) L = S > (An)(i = (An)L = 0 for n 2 2, (2.5 1 )

i.e., they have the same shape described by one Fourier harmonic with amplitude 2s and with the average locations moved into the interior of the channel by a distance y.

2.3.1 Truncated Problem

Approximate solutions of (2.16) can be found by cutting the sum (2.16) at a finite number N of terms and solving a coupled system of N + 1 ordinary differential equations. To illustrate the type of systems that have to be solved, lets take N = 5. From (2.16) one gets a system of six, fourth order ordinary differential equations, five of which are complex and only the equation corresponding to mode zero is real

with boundary conditions at y = I given by

d 3 a O d3 \zr

dy3 dy2 d3Q; s2 d3Q2 s2 d3Q;

+ - - = 2 s 2 - ~ ( 2 - y ) , ( 2 . 6 0 ) -' (ydyJ-g)+Tdy3 2 dy3

d BQl d3Q1 + s t - -- d3 %P2

- - d~ ( ) dy3 S(ydy3-s) d3Q0 & P o s2 d3Q3 s2 d3Q; +--+-- = + 2 s ( l - y ) , (2.61) ( ) 2 dy3 2 dy3

-- d3Q2 Y-

= s2 , (2.62)

dQ, dQ, -- + s t - 7 -- 8% d3@n+1 7dy1 ( 2 2 ) dy3 s(' dy3

- dy

and at y = - 1 given by

System (2.52)-(2.69) is solved using a variable-stepsize finite difference discretization with deferred corrections [34, 101. The complex equations are written as a first order nonlinear system of eight real ordinary differential equations the solution of which is computed with the two-point boundary value problem solver D02RAE form the NAG Library. The solution strategy used in order to guarantee convergence of the iterative process consists of obtaining at first a solution to the problem when N = 1, then these results u e used as an initial approximation of the solution for N = 2. Solution for N = 2 is used as an initial guess of the solution for N = 3, and so on. Due to the nature of the solution around the boundaries, a continuation procedure had to be employed for N 2 2. A continuation method solves a set of intermediary problems by Newton's iteration, using as a starting approximation for the k - th problem the final iterate of the problem k - 1 extrapolated by Euler's method. All calculations have been carried out with the machine-level accuracy.

2.4 Discussion of Results Results presented in this section have been obtained for the wavy upper and lower walls described by (2.3) and (Zjl), with the amplitude of the waviness equal to 2 s (distance from the top to the bottom of the corrugation is 4s) and with the average location of the wall moved into the interior of the channel by distance y = 2s, unless otherwise noted.

We begin this discussion by considering the domain perturbation method (DPM) . First (DPM1)-, second (DPM2)-, and third (DPM3) - order boundary condition transfer procedures have been implemented. The replacement problems (i.e. problems resulting fiom the domain perturbation procedures) have been solved with machine accuracy. Our main concern here is the d e termination of the " distance" between the original problem (i-e. flow over a corrugated wall) and the replacement problems. We shall refer to this " distance" as the error of DPM. This error will be measured by looking at the error in fulfilling the flow boundary conditions at the actual location of the

wall. The L,-norm defined as

provides a convenient measure of this error. Similar norms can be defined for the y-velocity component and for the upper wall.

Figure 2.2 displays variations of 1 luL (x) 1 loo as a function of the number of Fourier modes N retained in the expansion (2.15). One may note that each variant of DPM leads to a certain minimum error, which is of the order of truncation of Taylor expansions (2.19)-(2.20). This error can be reduced to its minimum value only by retaining a sufficient number of Fourier modes in (2.15). Use of additional modes does not improve modeling of flow boundary conditions. The error decreases exponentially with N and very few modes are sufEcient to reduce it to its minimum. For DPMl N = 1 is sufficient, while N = 3,4 is required for DPM2, DPM3, respectively (see Fig. 2.2). This conclusion can also be reached by looking at the order of magnitude analysis of (2.21)-(2.22) and noting that QI = O(s) and the magnitude of the other modes decrease according to the magnitude of the inhomogeneities in the boundary conditions and the character of the nonlinearities. The reader should keep in mind that Qo = O(s) because of the reduction of the average height of the channel.

Error distribution along the wall is illustrated by distributions of uT(x, hL (I)) and vr(x, hL (x)) displayed in Fig. 2.3 for DPM2 with N = 10. The maximum error is located approximately around the bottom of the corrugation. Fourier spectra of the error shown in the same figure demonstrate that, indeed, there is no contribution from higher modes.

Variations of I[uL(x) as a function of the amplitude 2s of wall corrugation are shown in Fig. 2.4 . For 2s small enough, the error increases with s as dictated by truncation of Taylor expansions (2.19), (2.20), i-e., it is 0(s2) for DPM1, 0(s3) for DPM2, 0(s4) for DPM3. Use of higher-order methods significantly improves accuracy, i.e., DPM3 provides absolute error 0 (lod7) for 2 s = while DPMl reaches error 0(10-~) only. Use of insufficient number of Fourier modes increases the error of the high-order methods in absolute terms and reduces their order, e.g., compare curves for DPM3 with N = 2,3,4 in Fig. 2.4. All versions of the method provide a similar absolute

error O(10-l) for high corrugation amplitudes with 2s FS 0.1, i.e., increase of the order of the method only marginally improves its accuracy for such amplitudes. Error variation as a function of s departs from regularity dictated by truncation of Taylor expansions for high enough s and this can be used as a criterion for determination of the limits of applicability of DPM. Results displayed in Fig. 2.4 suggest that DPMl can be used for corrugation amplitudes 2s 5 0.2, DPM2 for 2s 5 0.1 and DPM3 2s 5 0.06. These results suggest that increase of the order of the method decreases rather than ex- pands its range of applicability. It should be clear from the above discussion that DPM can be used up to 2s = 0.1 with the resulting error being O(lO-L) for such high corrugation amplitudes regardless of the version of the method being used. If higher accuracy is required, either the domain transformation method (DTM) or the direct method described in [42] must be used. DTM will be discussed in Chapter 4.

Figure 2.1: Sketch of the Bow domain. (a) - straight reference^ channel. jb)

- channel with corrugated walls.

0 1 2 3 4 5 6 7 8 9 1 0 Number of Fourier modes N

boundary condition transfer procedures-

Fourier mode number n 1 2 3 4 5 6 7

M DPMl - N=l -3 DPMl - N=2 M D P M l - X=5 +--+DPM2 - N=1 (1-4 DPM2 - N=2 6-4 DPM2 - N-3 * - + D P M 2 - N 4 t-- DPM2 - N=5 *--* DPM3 - N=l m-- - DPM3 - N=2 * - -4 DPM3 - N=3 *--a DPM3 - N 4 - - -q DPM3 - N=5

10-~ lo-' 2s - corrugation amplitude

Figure 2.4: Gtriations of lluL(x) 11, as a function of corrugation amplitude 2s

for = 1, Re = 100 for different implementations of the domain perturbation

method with various number of Fourier modes retained in the calculations.

Note: tan = 2, tan ,& = 3. tan & = 4.

Chapter 3

Stability analysis in the case of

simulated (blowing/suct ion)

distributed surface roughness

In Chapter 2 an upper bound for the applicability of DPM was given with the resulting absolute error being O(l0-I). As will be shown in Chapter 5 the instability region when Re = 2000 begins at 2s x 0.026 which in the case of DPM2 and DPM3 would yield absolute error no better than 0(10-~) and O(IO-~), respectively. If one considers that the absolute error will be compounded by errors from DPM-like expansions of disturbance boundary conditions, it is clear that DPM is not the best choice for stability calculations of flow modified due to distributed surface roughness.

This Chapter describes the procedures followed to study the linear stability characteristics of the Poiseuille flow modified by simulated distributed roughness (blowing/suction) along the walls. Results given in [16] were reproduced in order to test and refine the numerical methods needed to perform stability calculations for a much more complex system described in Chap-

ter 5.

3.1 Problem Formulation Equations of motion and continuity in the form of vorticity transport have represent at ions

dw, d u du du h, dw, dw, 1 -W,-- w,--w,-+u- +w- +u- = -AU,, (3.1)

dt ax dz 8~ d x dz dy Re

8% -- dzu dw dw h z h z dw, 1 at

w, - -az-- W y - + u p +w- + V - = -Au,, (3.2) ax dx a~ dx dz d y Re

Assume unsteady, three-dimensional disturbances are superimposed on the mean ( modified ) part of the flow in the form

where subscripts 2 and 3 refer to the modified mean flow and the disturbance field, respectively. Substitution of (3.6), (3.7) into (3.1) - (3.4) results after linearization in:

where

Notice that, since the mean part satisfy equation

it has been subtracted from equation (3.9).

3.2 Small Amplitude Simulated Distributed

Roughness

As it was discussed in [16], in the case of small amplitudes of blowing/suction ( 2s 5 ) nonlinear effects are completely negligible and the mean flow can be represented by

where U represents plane Poiseuille flow profile introduced in Chapter 2 by (2-I), q, a, are solutions to the h e a r problem described in [16] and C.C.

stands for complex conjugate. The suction distribution studied by Floryan was given by

The disturbance equations (3.8) - (3.11) have coefficients that are functions of x and y only. This permits separation of variables and representation of the t and z dependence of the solution in the form

The exponent p is real and accounts for the spanwise periodicity of the disturbance field. The exponent 0 is assume to be complex and its imaginary part describes the rate of growth of the disturbances while its real part describes the kequency of the disturbances.

Since the coefficients in (3.8) - (3.10) are periodic in x with periodicity 2 a / a , g3 is written, following the Floquet theory, as

where f3 is periodic in x with the same periodicity 2nla and 6 is referred to as the Floquet exponent. Our interest is in the temporal stability theory and thus b is assumed to be real. One should note that g3 is a product of two functions periodic in I, one with a period 27r/a and one with a period 2 s / 6 . This product is periodic only if b/a! is rational.

The final form of the disturbance velocity vector is written as

3.2.1 Fourier Mode Separation

Substitution of (3.15) and (3.19) into the disturbance equations (3.8) - (3.11) results in:

+oo

~ i c t ( h e i a z - ,: , - i a z ) ] c i ~ g u (m) ,i(tmx+fiz+ut) =

+- -ia ( h e a x - e - i a x ) ] c iPgv (m) ,i(tm z+pz+at) =

where

Since ody two of equations (3.20) - (3.22) are independent, a combi- nation of the first two equations is obtained by differentiating (3.20) with respect to z and subtracting it from the I-derivative of (3.21). Separation of Fourier modes in the continuity equation (3.23) together with the operations mentioned above, results in

+oo x i a [ - ( ~ 2 - k ; ) ~ , ( m ) ] e i ( t m z + P z + f l t ) f [U + a,, e'"" m=-oo

+oo

x C it, [ D ~ , ( ~ ) - i t m g v (4 ] e i ( t , x+pz+at) + i a (% e i a ~

- i t, D ~ ~ ( ~ ) ] e i(tmx+pz+at)

-ia ( ~ e i a x - ,: , - i o z ) ]

Collection of terms with the same exponential factors in equations (3.24) and (3.22) gives after some rearrangement

Separation of the Fourier components in (3.25), (3.26) and (3.23) results in an infinity system of complex ordinary differential equations governing the disturbance amplitudes gu(m), gw(m), gv(m), rn 2 0, in the form

where

3.3 Boundary Conditions Boundary conditions describing the disappearance of disturbances at the walls have the form

3 ( x , 2, 1 = 0 , w 3 ( x , z , H ) = 0 , u ~ ( x , Z , 4 3 ) = 0 .

Substitution of the disturbance velocity components expansion (3 .19) into (3.30) - (3.32) , yields

Exponential collection and mode separation results for m 2 0 in

The extra conditions needed for gv(m) are obtained combining the above conditions with the continuity equation. They have the form

3.4 Computational Method

Approximate solutions of the infinite set of coupled complex linear homogeneous differentid equations (3.27) - (3.29) with homogeneous boundary conditions (3.36) - (3.39) can be found by truncating the sum in (3.19) after a finite number M of terms and solving 6 M + 3 differential equations of type (3.27) - (3.29).

The eigenvalue problem to be solved after truncation is discretized using a pseudospectral method (the reader is referred to [23], [5], [8], [19], [20], [31], [44] for comprehensive reviews of the method) based on Chebyshev polynomials. The truncated Chebyshev expansions can be written as

Using collocation points {yi)L2 defined by the standard formula for Chebyshev- Gaussian quadrature, i.e. T N - P ( ~ i ) = 0 on ( -1 , l ) for

yi = - COS ((2;:: 1;) the matrix discretization of (3.27) - (3.29) is of the form

where C is a block band matrix with block order 6 M + 3

and the column vector g is given by

Notice that the same notation has been used to dehed the block-matrix elements of C as earlier to d e h e d their corresponding linear differential operators in (3.27) - (3.29). The continuity equation block-matrices have been assigned the notation

To illustrate how the bIock-matrix elements of C are form, lets show the structure of block-elements B,('), -M 5 1 5 M. Since the operator B$) is defmed by

B,(') = -p ( ( LI2 - k:) - i ~e ( t i ~ ( y ) + 0)) then the coeEcients of the block-element B,(') are given by

( u ' ) i , j = - ~ ~ y . ~ ( g i ) + p [i Re ( t l U ( y i ) + o ) +k:] Tj-l(yi)

2 = l , 2 , --• , N - 2 j = l , 2 , --• , N

The bottom two rows of the block-element matrix B,(') impose the bounc conditions corresponding to gu('), i.e.

B~ ('1 ( ) M - l , j = Tj-l(- l ) , j = l , Z , --• , N

( B ) = Tj-1(l), j = 1 , 2 , - , N N, j

The remaining boundary conditions are imposed as follows:

The bottom four rows of the block-element matrices A,('), K,(') and L,(') impose boundary conditions (3.36) and (3.39), i-e., boundary conditions corresponding to gv(i).

The bottom two rows of the block-element matrices corresponding to the continuity equation take care of the boundary conditions for gw(').

3.4.1 Generalized Eigenvalue Solvers

From the set of eigenvalues Re, a, 7, 6, p, s and 0 only two real quantities can be determined from the complex algebraic equation (3.40). Thus, the strategy for the computation of the non-trivial solutions of (3.40) is to select specific values for the parameters Re, a, y, 6, p and s, and compute the spectrum of equation

Since (3.41) has only linear dependence on a, i.e. C can be expanded as

the generalized eigenvalue solver CGEGV from LAPACK is used to compute the fidl spectrum T (C) (see [I], [22] for details). Once T (C) is found, one can select the most unstable eigenvalue and determined the amplification rate for disturbances with respect to other parameters, for instance p, Re and s, using inverse iterations ( [22], [48] ). Here is the algorithm that was used to implement it:

Set ItCount = 1 ( iteration counter ) Set MaxIt = 40 ( max. # of iterations ) Set LExp = TRUE ( logical expression ) Set TOL = 10-l4 ( tolerance value ) X E T(C) - given b - eigenvector corresponding to X Find LU decomposition of C d + b/llb112 REPEAT

b + B d Solve Cv = b b + vlllvllz if ( I (d*b ) - 11 5 TOL )

Set LEkp = FALSE h - b H ~ b / b H ~ b

else d + b ItCount t ItCount + 1 i f ( ItCount = MaxIt )

Set LExp = FALSE Print 'Max. # of i terations reached

without convergence! ' UNTIL ( LExp = FALSE )

Since the LU decomposition ( n3/3 - flops ) has been performed outside the loop, each iteration consists of a matrix-vector multiplication, forward substi-

tution, back substitution, and normalization. Therefore the most expensive operation has to be done only once.

3.5 Code Verification The remaining of this Chapter will be used to make some comments about the test performed to tune up the computational method using the blowing/suction results given in [16] as reference.

The eigenvalue problem to be solved is equivalent to the system described in [16]. The only difference is that some operators in (3.27) - (3.29) are written in a simpler form compared to their counterpart in [16] by taking advantage of the continuity equation. Rather than reducing the number of unknowns as it was done in [16], the system was solved in terms of all three disturbance amplitudes gu(m), gw(m) and gv(m). This is due to the fact that in the method to be described in Chapter 5, three consecutive modes of g,(m) we coupled through the continuity equation, making it so diilicult and laborious to reduce the number of system unknowns by one that its benefits become questionable. Therefore the system is solved the long way using primitive vaxiables for the benefit of a more difficult problem to be solved later.

Results described in [16] for the plane Poiseuille flow case were reproduced using all three disturbance amplitudes as unknowns. A larger number of Chebyshev polynomials, 50 rather than 41, was needed to achieve the same level of accuracy. Figure 3.1 shows the amplification rate -Imag(o) of the most dangerous mode in the suction distribution, in the sense that it induces the most amplXed disturbances, as a function of the spanwise wavenumber p for Reynolds number 5000 and suction amplitude 2s = 0.006.

Figure 3.1: -1mplification rate -Imag(o) as a function of the spanwise

wavenumber p for the suction-induced-disturbances in the Poiseuille flow

with Re = 5000 and 2s = 0.006.

4,343

26-03

Oe+00 0

&1.8 ' - - - - - - -----------,; &---% - - - - - - - - - - - - - - - - - - - - -

/ / \ -<:- \ -7

/ \ / \

1 \ - - - - - - - - - - / - - - - - - - - - - - - . - - - - - - - \\ - - - - - - - - - - - - - - - - - - - - -

/ \ I

/ \ 1 \

/ \ ' \

I - - - - - - - - / - - - - - - - - - - - - - - - - - - - - - - - - - - - - * - \ - - - - - - - - - - - - - - * I

/ 8 I \ I

I ' I I \ I ; t \ , L

I . \ ' - - - - - - / - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - I - - - - - - - - \ '

I t 1 \I---""- I

I \

I Re=5000,2~=0.006 : I \, I I \ - - - - r - - - ' - - - - - - - - - - - - - - - - - - - - - - - - - - - . - - - - - - - - - - - - \ - - - - - '

I \

I 1 1 \

I \ \ .

2 4 6 p - spanwise wavenumber

Chapter 4

Domain Transformat ion

Method ( DTM )

In this approach difficulties due to presence of irregular boundaries are r e solved by mapping the physical domain into a rectangular computational domain as shown in Fig. 4.1. Here, we use an analytical mapping in order to avoid additional errors associated with numerical mappings. A homeomor- phism

carries out the mapping. The restrictions imposed upon hU(x) and h L ( x ) are due to their differentiability requirements implied by equation (2 .12) , namely hrr(x) ,hL(") E C4-

4.1 Governing Equations

Application of the coordinate transformation

(x) Y) + (L 7))

to the system of equations (2.7) - (2.9) results in

where

a Differentiating (4.4) by + all a and subtracting it from & & of (4.3) ?q

the pressure is eliminated, reduclng the system to

du du 2 dv - + a ~ - + - - 0 . (4.1 1)

87 hu-hL & Introduce the stream function $ such that

Notice that the stream function definition (4.12) has not changed, i.e., it is exactly (2.11) written in the ( 5 , ~ ) coordinates.

a Using the commutative properties of differential operators A, +al* & and -- * a equations (4.10), (4.11) become h 7 - h ~ %

4.2 Boundary Conditions

Stream function definition (4.12) allows for the adherence condition (2.10) at the boundaxy to be written in the form

Since the shape of the walls is given by (2.3), it is convenient to represent Poiseuille flow and the corresponding stream function 9 in terms of Fourier expansions in the form

where expressions for U, , a, are given in Appendix A.1. The unknown stream function can be represented as

The reader may note that (2.15) and (4.17) are not equivalent due to different domains of validity. Substitution of (4.15) and (4.17) into boundary conditions (4.14) and separation of Fourier modes results in boundary conditions for each mode in the form

where B, = (An)u - (A,) L. The impermeability condition at the walls forces the volume flux to be determined by mode n = 0 only. Thus the fixed volume flux condition leads to

Selection of @(I(-1) = 0 (one of the conditions must be imposed arbitrarily) yields

Substitution of (4.17) into field equation (4.13) and separation of Fourier components leads to an infinite systems of complex ordinary differential equa- t ions

where the linear differential operators are described in Appendix A.2. Com- parison of (2.16) and (4.23) demonstrate the increased complexity of equations resulting f?om domain transformation.

4.3 Numerical Solution Numerical tests have been carried out for the upper and lower walls in the form (2.51). For this particular case the Fourier representation (4.15) of the Poiseuille flow has the form

The differential equations for an reduce to the form

where the linear differential operators are described in Appendix A.3. The corresponding boundary conditions have the form for q = f 1

and are supplemented by the fixed volume flux conditions in the form

Conditions in (4.30) result from evaluation of

3 2 = ( b y ) [ ( 1 - 2 ~ ~ ) ~ - ( 1 - y ) ~ % ] + 3 (4.31)

in formula (4.22) at 17 = f 1.

4.3.1 Truncated Problem Approximate solutions of (4.24) can be found by cutting the sum (4.17) at a finite number N of terms and solving a coupled system of N + 1 ordinary differential equations. From (4.24) when N = 5 one gets a system of six, fourth order complex ordinary differential equations

System (4.32) - (4.37) is solved in terms of the fourth order derivatives of the six given modes. The resulting set of six fourth order ordinary differential equations is written as a first order system of forty four real equations, the solution of which is computed with the two-point boundary value problem solver D02RAE form the NAG Library. The solution strategy used in order to guarantee convergence of the iterative process is the same as described in section 2.3.1. AU cdculations have been carried out with the machine-level accuracy.

4.4 Positioning of Roughness

Understanding how positioning of roughness with respect to a reference channel affects the stability characteristics of the flow is important for practical reasons. From the applications point of view one has to understand whether, for instance, mounting sand on the channel walls or digging down identical shapes from the walls will make any difference as far as the stability prop erties are concerned. Theoretically this translates into the question of how the stability characteristics of the flow behave with respect to the roughness shift parameter 7 defined earlier-

4.4.1 Pure Shift The linking factor in the family of channels to be studied here will be the volume flux Q f. In all cases it is assumed that Q is the same and constant, thus the subscript f stands for fixed. Suppose we have a series of infinite flat channels with half-channel height given by L ( 1 - y) and flow driven by a constant volume flux Qf . Here L, Qf are dimensional parameters and y is a dimensionless scalar. Let U,, be the maximum x-veloci ty corresponding to the reference channel 7 = 0.

The characteristic length, velocity and time are chosen as follows: L, U . , T = L/U-,, hence Re = LUmm/u, where the kinematic viscosity v remains unchanged in al l cases. The reader may notice that restrictions (2.4) considered earlier to evaluate the range of applicability of DPM are no longer in place. Thus, if

the resulting set of dimensionless Poiseuille profiles is given by

Notice that if one chooses a dehition of the Reynolds number based on half-channel height and maximum x-velocity in each particular case, then

obviously all the characteristic parameters would be y-dependent , i-e.,

Nevertheless, Re remains unchanged, namely

Renew

Similarly,

thus n

A relationship between the standard and shrunk/expanded dimensionless remaining parameters ( wave numbers and amplification rate ) is given by

where * denotes dimensional quantities. Parameter p is real and accounts for the spanwise periodicity of the disturbance field. Parameter 0 is assume to be complex and accounts for the rate of growth of the disturbances. Both parameters will be defined and analyzed in detail when we study the stability properties of the modified flow due to distributed roughness in Chapter 5.

4.4.2 Shift Plus Corrugation

In the case of shift with distributed surface roughness the components of the Poiseuille flow in the transformed domain for the y-family of profiles (4.50) have the form

Boundary conditions corresponding to (4.25) - (4.30) become for 7 = 3~1

and are supplemented by the fixed volume flux conditions in the form

The contributions of different modes can be assessed by calculating their energy. Modes of the velocity components are given by

Thus, energy of mode n will be defined as follow

From (4.12) and (4.17) one gets

therefore, the energy modes in the (E , 7)-domain are given by

and

4.5 Linear Approximation

When amplitude 2s of the distributed roughness at the walls is small enough all flow quantities can be represented as asymptotic expansions in terms of

S, namely

Substitution of these expansions into the original Navier-Stokes equations and retention of O(s) terms gives linear description of the flow modifications due to roughness. The leading order problem ( terms O(1) ) yields

The 0 (s) problem is given by


Change of variable

results in


Introduction of the stream function ?,bl such that

reduces system (4.67) - (4.69) to


In equation (4.72) the Laplacian operator is defined as follow

Consider our test problem now, i.e.,

We seek a solution in the form

67

where c.c. stands for complex conjugate. Substitution of (4.74) into (4.72), (4.73) results in


Linear problem (4.75), (4.76) can also be obtained by keeping only one term corresponding to n = 1 in (4.17) and retaining the O ( s ) terms in the resulting equation and boundary conditions. Problem (4.75), (4.76) is solved following the same numerical procedures described earlier for the full nonlinear system.

4.6 Discussion of Results Figures 4.2 and 4.3 illustrate velocity field changes as a function of the wavenumber a. These graphs depict a rapid evolution of the flow field with respect to a. Figures 4.4 and 4.5 show variations of the flow field as a function of the Reynolds number Re. Results demonstrate that the character of the flow field changes rather weakly with respect to Re. Stream lines of the velocity modification vl in the physical and transformed domain for two different values of the wave number a! are shown in figures 4.6 and 4.7.

The domain transformation method enforces flow boundary conditions exactly. Fig. 4.8 shows that DTM provides spectral accuracy. The convergence of DTM is assessed using Chebyshev norm applied to the derivative of the modal function, i.e.,

The modal by introducing

efficiency of the method can be measured in absolute terms asymptotic rates of spectral convergence p defined as

*- lim log llD%+~ llu -- n,,

11DQnllw '

Limit p describes slopes of lines in Fig. 4.8 demonstrating exponential convergence of DTM. Variations of p as a function of a are illustrated in Fig. 4.9, and as a function of s are shown in Fig. 4.10. These results have been obtained for a channel with fiat upper wall and lower wall in the same form as in the previous tests. It can be seen that the modal eficiency of the method decreases with increase of both s and a. In [7] the modal efficiency and convergence rate of DTM is compared with what is called Direct Method (DM) introduced by Szumbarski and Floryan in [42].

A comparison between the energy mode obtained from the linear approximation and the same energy mode (El ) resulting from the fuIl nonlinear model is depicted in fig. 4.11 for three different values of parameter y. For amplitudes 2s = 1.5 lo-* a deviation from the linear approximation starts to occur. This effect increases gradually with the increase of s. Variations of energy modes EO, El , El defined by (4.60), (4.61) with respect to roughness amplitude 2s are depicted in fig. 4.12 for three different values of the shift parameter y. All calculations were performed using six modes, i-e., stream function expansion (4.17) was truncated at N = 5 . It can be seen that for Re = 5000 mode interaction is rather weak for roughness amplitudes 2s 5 The reader may also note that nonlinear effects are more visible with the increase of s. The expanded channel (y = -0.1) has slower velocity profile and thus the values of energy modes are lower as shown in fig. 4.12. The opposite occurs with the shrinkage of the channel width. Since the volume flux is kept constant, the velocity profile must be accelerated and therefore the energy levels are higher. In Chapter 5 more information regarding the role of mode interaction will be described when the stability analysis is performed.

4.02 1 I -i 0 -0.5 0.5 1 .o

q - chaGei width

Figure 4.2: Modifications of Poiseudle flow in the transformed domain due to

the presence of distributed wall roughness as a function of the wave number,

cr ( s = 0.007, Re = 5000 ). ul = ul eiat + ui e-la' .

Figure 4.3: Modifications of Poiseuille flow in the transformed domain due to

the presence of distributed wall roughness as a function of the wave number.

a ( s = 0 . 0 0 7 , Re=jOOO). u, = ~ , e ~ a f + v ; e - ' ~ E .

-1.0 4.5 1 .o q - ~ h a G e i width O5

Figure 4.4: Modifications of PoiseuiUe flow in the transformed domain due

to the presence of distributed wall roughness as a function of the Reynolds

number, Re ( s = 0.007, a = 2.0 ). ul = ul e iuc + ui e-la' .

-5.5 0 -5 q - chankl width

Figure 4.5: Modifications of Poiseuiile flow in the transformed domain due

to the presence of distributed wall roughness as a function of the Reynolds

number. Re(s=0.007, a = 2 . 0 ) .

Figure 4.6: Stream line patterns of flow modifications in the physical (a) and

the transformed ( b ) domains calculated for Re = 100, a = 1, 2s = 7 = 0.1.

Figure 4.7: Stream line patterns of flow modifications in the physical (a) and

the transformed (6) domains calculated for Re = 100, a = 10, 2s = r =

0,025.

. ,

A.. I I ID@J la (DTM)

2 4 6 8 10 Fourier mode number n

Figure 4.8: Variations of the Chebyshev norm IID@n/lw as a function of

Fourier mode number n for Re = LOO, a = I, 2s = 0.1.

Figure 4.9: Variations of spectral convergence rate p of DTM as a function

of wave number cr i Re = 100) for channel with upper mall flat and lower wall

in the form of a simple harmonic with (.4& = -/ = 2s. ( - & ) L = -5-

0.01 0.02 ts-cormgation amplitude

Figure 4.10: Variations of spectral convergence rate p of DTM as a function

of the corrugation amplitude 2s. Other conditions as in Fig- 4.9.

lo- ' 2s - rog&;;ess amplitude

Figure 4-11: Energy associated with the modifications of the Poiseuille flow

due to the presence of wall roughness as a Function of the roughness amplitude

2s for differem values of the shift parameter y. Calculations have been carried

out with W = 5 ( Re = 5000, a = 3.0 ). Linear cases are described by

different types of lines and nonlinear cases by different types of symbols.

Chapter 5

Stability Analysis using DTM

This Chapter describes the linear stability analysis of the Poiseuille flow modified by the distributed surface roughness along the walls. The analysis is performed using the domain transformation method and refers to the Floquet theory owing to the periodic character of the flow modifications as shown in Chapter 4.

5.1 Problem Formulation

Equations of motion and continuity in transformed ( (, 6, 7, t ) domain have the form

where

Unsteady, three-dimensional disturbances are superimposed on the mean part in the form

where subscripts 2 and 3 refer to the mean flow and the disturbance field, respectively. D e h h g vorticity in the new variables a s

relations (5.11) and (5.12) yield

where

The assumed form (5.11) of the %ow field is substituted into the vorticity transport form of the governing equations (5.1) - (5.4), the mean part is subtracted and the equations are linearized. The resulting linear disturbance equations have the form

with adherence condition at the boundaries given by

One should notice that only two of the vorticity equations (5.15) - (5.17) are independent.

5.2 Test Problem Consider the case when h&) and h&) are given by relations (2.51) horn Chapter 2. One can define h; = hc ( h; = h; , ' stands for derivative with respect to E ) and under this assumption equations (5.15) - (5.18) simplify to ( a o = 1 / ( 1 - y ) , a 1 2 = - h F / ( 1 - ~ ) , a 2 2 = ( l + h ; ) / ( l - y ) 2 , a 2 = - h t X / ( l - ~ ) ):

where

Notice that in the test case considered here the mean flow satisfies equations

The mean flow is assumed to have the form

i.e. it is described by the first three modes ( N = 0,1,2 ) from (4.17). Three is the minimum number of modes needed to describe the Poiseuille flow in the transformed domain. Description (5.27) provides sufficiently accurate representation of the flow in the case of small amplitudes of distributed roughness being of interest here, as discussed in Chapter 4. In the above C.C. stands for complex conjugate. It is important to clarify that the difference between the various nonlinear contributions in the modSed flow is determined by the composition of the mode amplitudes U, V, f,, f,, F, and F, given in (5.27).

In the N = 1 case mode amplitudes are formed by the three components of the PoiseuilIe flow expansion in the transformed domain plus a modification to mode one coming from the solution to the full nonlinear system of equations with six modes. If higher nonlinear cases are considered extra contributions to the zero mode ( 0, 1 - case ) or to the zero and second mode ( 0, 1, 2 - case ) have to be added.

5.2.1 Periodic Three-dimensional Disturbances

The disturbance equations (5.20) - (5.23) have coefficients that are functions of E and 7 only. This permits separation of variables and representation of the t and C dependence of the solution in the form

The exponent p is real and accounts for the spanwise periodicity of the disturbance field. The exponent a is assume to be complex and its imaginary part describes the rate of growth of the disturbances while its real part describes the frequency of the disturbances.

Since the coefficients in (5.20) - (5.23) are periodic in 6 with periodicity 2*/(ry u3 is written, following the Floquet theory, as

where w3 is periodic in 5 with the same periodicity 2 r / a and 6 is referred to as the Floquet exponent. Our interest is in the temporal stability theory and thus 6 is assumed to be real. One should note that us is a product of two functions periodic in E, one with a period 2s/a and one with a period 2 ~ / 6 . This product is periodic only if &/a is rational.

The final form of the disturbance velocity vector is written as

where gu(m) = gu(-m) * 9 gw (m) - - gw (-m) *, gv(m) = gV(-m)* in order for v3 to

be real ( * - denotes complex conjugate ). Substitution of (5.27) and (5.30) into the disturbance equations (5.20) -

(5.23) and separation of the Fourier components in the same fashion as it was carried out in Chapter 3 results, after rather lengthy algebra, in a system of linear ordinary different i d equations governing gu(m) , gw(m) , gv(m), rn 2 0 , in the form

with linear differential operators defined in Appendix B.

From (5.19), (5.30) and (5.33) one gets

Notice that in the absence of wall roughness (s = 0 ) all modes from the Fourier series (5.30) decouple and equations (5.31) - (5.33) with boundary conditions (5.34) - (5.35) describe the classical three-dimensional instability of the plane PoiseuilIe flow. The coupling due to roughness in the field equations involves eleven consecutive modes of the Fourier expansion (5.30). The coupling due to the presence of roughness in the boundary conditions involves three consecutive modes.

5.3 Numerical Method

The eigenvalue problem to be solved is described by an inh i te set of coupled linear homogeneous ordinary differential equations (5.31) - (5.33) with homogeneous boundary conditions (5.34) - (5.35). Approximate solutions can be found by truncating the sum in (5.30) after a finite number of terms and solving 6 M + 3 differential equations of type (5.31) - (5.33).

The finite system obtained after truncation is discretized by employing a pseudospectral method based on Chebyshev polynomials mentioned in Chap- ter 3. The truncated Chebyshev expansions are given by

with collocation points {%)Ly2 defined by

q+ = - COS (;;;:;;) The matrix discretization of (5.31) - (3.33) is of the form

where C is a block band matrix with block order 6 M + 3

and the column vector g is given by

I . .

Notice that the same notation has been used to defined the block-matrix elements of C as earlier to defined their corresponding linear differential operators in (5.31) - (5.33). The continuity equation block-matrices have been assigned the notation

The boundary conditions are imposed as follows:

The bottom four rows of the block-element matrices A , ( ~ ) , K , ( ~ ) and L , ( ~ ) correspond to the boundary conditions

The bottom two rows of the block-element matrices B , ( ~ ) impose the boundary conditions corresponding to gu(k), i.e., gu(k) = 0 , q = &1.

The bottom two rows of the block-element matrices k(k) take care of the boundary conditions gw(k) = 0 , 71 = f 1 .

Number of Chebyshev

Table 5.1: AmpMcation -Imag(o) of roughness-induced disturbances (

Number of disturbance modes

J I 1 .L1

Correction due to roughness described by the first two modes N=O,l

Various tests were carried out to determine the number of Chebyshev polynomials required to obtained the desired accuracy. Table 5.1 shows the amplification rate -Imag(o) of roughnessinduced disturbances for the nonlinear models with different number of Chebyshev Polynomials. The tests showed that for the values of roughness amplitudes of interest in this study, the eigenvdues can be determined with accuracy no worse than 0.1%.

i

60 70 80

Polynomials

90 0.1856 0.1842 - 0.1856 Correction due to roughness described by the first three modes

N = 0,1,2

0.2155 0.1843 - 0.1842

0.1992 - 0.1872 0.1845

M = 0 , 1 , 2 ( M = 0 , 1 , 2 , 3 1 M = 0 , 1 , 2 , 3 , 4

0.2069 0.1851 0.1848

Correction due to roughness described by mode one N = l

0.2494 lom3 0.2454

0.2575 0.2439*10-~

4

60 70

0.2400 0.2481

0.2379 lo-" 0.2384 80 0.2380

5.4 Analysis of Results

We consider temporal stability theory, i-e. the exponent b in (5.30) is assumed to be real. Exponent p is red and accounts for the spanwise periodicity of the disturbance field. Exponent a is complex and its imaginary part describes the rate of growth of the disturbances. A flow field that is modified by distributed surface roughness in the form (2.3), (2.51) is considered, i-e. the roughness is periodic in the streamwise direction with period 2 a/a and it is the same at the upper and lower walls.

Results of the present analysis demonstrate Floryan's [16] suggestion that the flow response in the case of distributed surface roughness may be similar to the form of disturbances he found in the case of simulated (blowing/suction) distributed surface roughness, namely that the presence of wall distributed roughness lead to the appearance of growing disturbances at Reynolds number Re < ReL = 5772.22 ( critical Reynolds number at which the Poiseuille %ow with hydraulically smooth walls becomes linearly unstable ). The disturbances have the form of streamwise vortices, i.e., the dominant mode corresponds to m = 0 in (5.30). No subharmonics have been found, i.e., 6 = 0 in (5.30), which is also in agreement with what was found in [16] and [18] in the case of simulated wall roughness. The disturbances are fixed with respect to the wall and do not propagate, i.e. Real(0) = 0 in (5.30).

Figure 5.1 illustrates amplification curves for nonlinear modifications to mode 1 (solid line) and modes 0,l (dash line) in (5.27) as a function of roughness shift parameter y. A comparison between the nonlinear models N = 1 and N = 0 , l will be carried out in most of the computations throughout this discussion. In all cases studied here mode 1 model overpredicts the unstable regions computed using the 0, l model. Percentages of the overprediction change with variations of the parameters Re, p, a, 2s and y. The rectangle in fig. 5.1 marks the shift sizes that correspond to the range of distributed roughness amplitudes of interest here.

Amplification rates - Imag (a) as functions of the spanwise wavenumber p are compared for three different values of y ( 7 = 0.1, y = 0, y = -0.1 ) in fig. 5.2. All computations were performed in terms of the same fixed dimensional roughness. Thus in the cases when 7 # 0 plots in fig. 5.2 dis- play dimensionless amplification rates -Imag(a,,) with respect to k, as

defined earlier in section 4.4.1. The wall roughness period a was chosen to be equal to 3 for reasons that will be described shortly. Disturbances with p - 2.15 appear to have the largest amplification rates.

Plots in figures 5.3 and 5.4 show that a whole band of spanwise wavenumbers p is amplified and the width of this band increases with an increase of both the roughness amplitude 2s as well as the Reynolds number Re ( see figures 5.5 and 5.6 ). The distributed roughness with the wavenumber a = 3 appears to be the most dangerous in the sense that it induces disturbances with the highest ampliftcation rates. The most amplified spanwise wavenumber corresponds to p = 2.15.

The distributed-roughnessinduced instability described above represents the initial stage of a new bypass route to transition found in 1161, [18] for simulated wall roughness. The form of the disturbances is such that it leads to a rapid three-dimensionalization of the flow field. Flow evolution is driven by a different mechanism to the one that give rise to the classical Tollmien- Schlichting traveling waves ( TS-waves ). It can arise at Re < RelvL -- 2700 ( critical Reynolds number for the nonlinear growth of the disturbances found by Herbert in 1977 [25] ) and its occurrence is not related to the subharmonic character of the TS-instability.

Amplification rates of two-dimensional TS-waves ( cw = 1, p = 0 ) as functions of the Reynolds number Re are compared in fig. 6.5 with disturbances induced by the distributed roughness for the most amplified wavenumbers ( cr = 3, p = 2.15 ) . Results show that the TS-waves are slightly stabilized by the roughness. Similar comparison, this time with respect to the distributed roughness amplitude 29, can be made from the results presented in fig. 5.6. TS-waves growth rate decreases slightly but overall the changes are rather small.

Results depicted in fig. 5.5 show that disturbances due to distributed roughness can become unstable for Reynolds numbers smaller than the critical Re of the TS-waves. In the case of distributed roughness the critical Reynolds number can be lowered by increasing the roughness amplitude 2s, as shown in fig. 5.5. Comparison between nonlinear model plots for N = 1 and N = 0 , l given in fig. 5.5 confirms the statement made earlier about the overprediction of model with N = 1.

Effects of roughness amplitude on the disturbance amplification rates

are illustrated in fig. 5.6 for the nonlinear model N = 1. Curves for the N = 0 , l model ( not shown ) were found to be below the N = 1 model in all the cases studied here. Figure 5.6 shows that an increase of roughness amplitude 2s leads to an almost linear increase of the amplification rates of the distributed-roughnessinduced disturbances while the growth rates for the TS-waves practically remain unchanged.

Plots depicted in fig. 5.6 permit making the conclusion that given a particular value of the Reynolds number, it is always possible ( in the range of parameters studied ) to find a distributed roughness amplitude 2s that gives rise to the vortex-like disturbances. Presence of these streamwise vortices results in uplifting of the low-momentum fluid away from the walls and cre- ation of a highly distorted streamwise velocity profile that is a function of both the streamwise and the spanwise coordinates. Such profiles are subject to very strong secondary instabilities ( see [49] ) which may lead to a rapid transition to turbulence. The above scenario was described by Floryan in [16] and it is the most likely ( bypass ) route to turbulence when Poiseuille flow is modified by distributed surface roughness, in view of results given in figures 5.1 - 5.6

Information regarding the maximum roughness amplitude the flow can accommodate for a given Re without inducing streamwise vortices has already been given in figures 5.1 - 5.6. However the results can be better interpreted using the roughness Reynolds number Re, mentioned in section 1.1.1 ( i.e., Re, = Utr -2s/v, where Utr - undisturbed velocity at the top of the roughness, 2s' - roughness height, v - kinematic viscosity ).

Figure 5.7 displays amplification curves for different values of Re, as a function of the Reynolds number Re for the most amplified wall-roughness wavenumber cr = 3 and the most amplified spanwise wavenumber p = 2.15. One can observe that the disturbances are not amplified if Re, < 1 ( in the range of parameters studied ). When Re, < 3 there is a monotonic increase of the amplification rates approaching a constant asymptotic value whose magnitude depends on Re,. For Re, > 4 this trend is reversed in the sense that there is a gradual monotonic decrease of the amplification rates towards their asymptotic limit.

Figure 5.1: -1mplification rare -hag@) as a function of shift parameter 7

for the roughness-induced disturbances in the modified Poiseuiile flow with

Re = 5000.2s = 0.014.

0 1 2 3 4 p - spanwise wavenumber

Figure 3.2: .linplification rate -rmag(o) as a iunction of the spanwise

wavenumber p for different values of the shift parameter n: ( Re = 5000,

2s = 0.014 ). Solid lines correspond to iV = 1 while dash lines describe

iv = 0 , l .

1 2 3 4

p - spanwise wavenumber

F i e r e 5.3: -1mplification rate -Imag(o) as a function of the spanwise

mvenumber C( for the roughness-induced disturbances in the modified

Poiseuille flow with Re = 3000.2s = 0.019. Solid lines correspond to -V = 1

while dash lines describe N = 0 , l .

1 2 3 p - spanwise wavenumber

Figure 5.4: Amplification rate -Imag(o) as a func~ion of the spanwise

wavenumber p for the roughness-induced disturbances in the n-~~dified

Poiseuille flow with Re = 5000,2s = 0.014. Solid lines correspond to iV = 1

while dash lines describe N = 0,1.

Figure 5.5: Amplification rate - Imag(oj as a function of the Reynolds num-

ber Re for the roughness-induced disturbances in the modified Poiseuiiie flow

( p = 2.15: a = 3.0 ) and two-dimensional TS-waves ( p = 0 , ct = 1.0 ). Solid

lines correspond to N = 1 while dash lines describe :\i = 0,1.

Figure 5.6: -Amplification rate -Trnag(o) as a funcrion of the distributed

roughness amplitude 2s for the roughness-induced disturbances in the mod-

ified Poiseuille flow ( p = 2.15, a = 3.0 ) and two-dimensional TS-waves (

p = O , a = l . O )

2000 4000 6000 8000 Re - Reynolds number

Figure 5.7: -Amplification rate -Imag(a) as a funccion of the roughness

Reynolds number Re, with p = 2 .15 , CY = 3.0.

Chapter 6

Summary

We have analyzed two types of methods that can be used for simulation of flows over rough/corrugated boundaries. Domain Perturbation Method (DPM) approximates shape of the boundary through various boundary condition transfer procedures. It is applicable to corrugation amplitudes up to 2s = 10-I and delivers accuracy no better than U(10-I) under such conditions even when higher-order boundary condition transfer procedures are used. Accuracy of this method significantly increases for lower cormgation amplitudes Zs, with the error of the first-, second-, and third-order versions of the method decreasing proportionally to s2, s3 and s4, respectively.

Very accurate (i.e., with machine accuracy) simulations of flows with wall corrugations with amplitudes 0(10-=) can be carried out using Domain Transformation Method (DTM). DTM models surface geometry and enforces flow boundary conditions exactly. DTM holds very good modal efficiency and provides spectral accuracy. The method is laborious in implementations due to a very complex form of the transformed field equations.

Stability of wall-bo-mded Poiseuille flow modified by distributed surface roughness has been considered. The analysis focused on a test problem with wall-roughness in the form of a single two-dimensional Fourier mode with the same shape for the upper and lower walls. The two main parts of this work

were (i) determination of the new roughness-modified flow, followed by (ii) a linear stability analysis of this flow.

It has been found in this study that the flow correction due to roughness changes weakly with respect to the Reynolds number Re but evolves rapidly with respect to the wall-roughness wavenumber a.

Stability diagrams for the Fourier modes that give rise to the instability are presented in this work. It has been shown that an increase of the roughness amplitude 2 s results in reduction of the critical Reynolds number. The strength of the instability, as measured by disturbance amplification rates, increases almost linearly with 2s. The dominant mode has the form of streamwise stationary vortices. A whole band of spanwise wavenumbers is amplified and the width of this band increases with 2s and Re. The most amplified spanwise wavenumber corresponds to p = 2.15. The most dangerous mode in the wall-roughness distribution, in the sense that it induces the most amplified disturbances, corresponds to the wavenumber a = 3. The classical TS-waves were found to be little affected by the roughness amplitude levels considered.

Appearance of streamwise vortices results in a sigdicant rearrangement and a rapid three-dirnensionalization of the flow. Uplifting of the low-momentum fluid away from the w d s leads to the formation of highly distorted streamwise and spanwise velocity profiles that are functions of the streamwise and spanwise coordinates and are subject to very strong secondary inst abilities.

Since the presence of streamwise vortices is a strong harbinger of turbulence, one is interested in determining the maximum roughness amplitude that the flow can accommodate without inducing such vortices. It has been shown that the instability does not occur if the roughness Reynolds number Re, < 1.

6.1 Conclusions From the summary of results given above four points can be selected as the most signiscant contributions of this work, which are:

0 The distributed-roughness-induced inst ability in the form of st reamwise

vortices described above represents the initial stage of a new bypass route to transition.

The strength of the inst ability increases almost linearly with the roughness amplitude 2s.

Instability does not occur if the roughness Reynolds number Re, c 1.

0 The most dangerous mode in the roughness distribution corresponds to wavenumber a! = 3 .

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Appendix A

A. 1 Expressions

u,= -(77-q2

- 7 7 - ( 2

for U, and G,

where B, = (An)u - (An)L .

A.2 General Problem - Linear Differential Op-

erators

A.2.1 Coefficients

00

C rBr r=-00

((6-6Sb ( s=-cxl s.) - n u 2 (5 s=-00 S A . ) ) ) )

00

~ ~ ~ = 2 4 a ! ~ ( n - m - ~ ) ~ ~ , p=-00

where Bk = (A& - (&)L and A k ( A k ) w

A.3 Test Problem - Linear Differential Oper-

ators

Appendix B

DTM Stability System - Linear Differ-

ential Operators

B.l.l Coefficients

+ ( - 3 2 ~ ~ s ~ + 3 2 y s a:) Re (z2 -V(v) ) + (

( f ) + (

d2 d3 ) R ( f ) ) + (16isa- 1 6 i y s a ) R e (-p+fu(71))

+ (-16s2 a2 + Eys2 a') Re (Xf,(g)) dv3

2 2 + ( - 1 6 s cr + l 6 y s a!) Re (z3 - U ( v ) ) + (

3 2 4 2 (64ia2+384i?a2 - 2 3 6 i y a 2 - 2 5 6 i y a + 6 4 i y a ) h - 2

+ ( 3 2 i - j ' ~ - 1 2 8 i y 3 a + 1 9 2 i n ~ a - 128iycr+32ia) p2

3 3 + 1 2 8 i y 4 a 3 + 1 2 8 i a 3 - 5 1 2 i y (r +768i -?a3-512iya3) Re

3 2 2 2 F v ( q ) + ( ( 1 4 4 i 7 ~ 2 ~ - 4 8 i ~ 2 ~ + 4 8 i y cy s - 1 4 4 i y cr S )

+ ( - 1 6 i a s + 1 6 i y 3 a s - 4 8 i ~ ( r s + 4 8 i y a s ) ,u2

+ 3 3 6 i ( ~ ~ ~ ~ - 3 3 6 i a ~ ~ s + 1 1 2 i a ~ y ~ s - 112ia3s)Re

d 2 2 d2 ) R ( q ) ) + ( - 3 2 a 2 s 2 + 3 2 ? s a ) Re (;i;jifu(q))

3 2 + l 6 i y a a s - 1 6 i a 2 a s - 4 8 i v d a s + 4 8 i 7 ( r 2 a s ) Re

+ ( 6 4 y 3 a s + 1 9 2 y s a - l 9 2 q s a - 6 4 ~ 3 ) t:-L

+ ( 2 8 8 y a 2 s + 9 6 ~ a 2 s - 2 8 8 q s a 2 -96a2s) tz-, + ( (64y3crs+192ysa- 1 9 2 ~ s ( r - - 6 4 c r s ) ,? - 6 4 a 3 s

+192y(r3s- 192q(r3s+64y30r3s) &-L

+ ( 9 6 y a 2 s - 9 6 ~ s ( r 2 + 3 2 y 3 d s - 3 2 a 2 s ) C 1 2 - 1 6 a 4 ~

+ 1 6 y 3 a 4 s - 4 8 ~ a 4 s + 4 8 y a 4 s

( f ) dq2 + (

( f ) ) + ( ( 9 6 y 2 a - 3 2 7 3 r r - 9 6 y a + 3 2 a ) &+2 -1927%~

) ( f ) +

stability of wall-bounded flow modified to the of distributed · 2004. 11. 29. · stability of...

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