LARGE EDDY SIMULATIONS OF LAMINAR SEPARATION BUBBLE

FLOWS

by

Francois Cadieux

A Dissertation Presented to the

FACULTY OF THE USC GRADUATE SCHOOL

UNIVERSITY OF SOUTHERN CALIFORNIA

In Partial Fulllment of the

Requirements for the Degree

DOCTOR OF PHILOSOPHY

(AEROSPACE ENGINEERING)

May 2015

Copyright 2015 Francois Cadieux

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Acknowledgments

A heartfelt thanks to my supervisor Andrzej for his guidance, support and sense

of humour. To Giacomo, thank you for countless in-depth and useful discussions.

I am also grateful to those who shared their code with me without which this

endeavour might have taken yet another year: Tawan, Brian, Tak, and Peter. I

am deeply indebted to Vina, who was always by my side when I needed her most.

The support of my parents Johanne and Yves and my sister Genevieve and their

eagerness for me to join them in their travels kept me sane and focused. Finally, a

special thanks to Dr Philippe Spalart for sharing his DNS data and answering my

many questions. This research was supported by the National Science Foundation

through grant CBET-1233160.

ii

Contents

Acknowledgments ii

List of Tables v

List of Figures vi

Abstract viii

1 Introduction 1

1.1 Laminar Separation Bubble Flows . . . . . . . . . . . . . . . . . . . 11.2 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2 Background 7

2.1 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2 Direct Numerical Simulation . . . . . . . . . . . . . . . . . . . . . . 82.3 Reynolds-averaged Navier-Stokes . . . . . . . . . . . . . . . . . . . 9

2.3.1 RANS for Laminar Separation Bubble Flows . . . . . . . . . 112.4 Large Eddy Simulation . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.4.1 Filtered Navier-Stokes Equations . . . . . . . . . . . . . . . 142.4.2 Subgrid-scale Models . . . . . . . . . . . . . . . . . . . . . . 152.4.3 LES Results for Laminar Separation Bubble Flows . . . . . 21

2.5 Wall-modeled LES and Hybrid RANS-LES . . . . . . . . . . . . . . 23

3 Methodology 25

3.1 Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.2 Flow Specication . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.3 Numerical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.3.1 Center for Turbulence Research Code . . . . . . . . . . . . . 303.3.2 Spectral code in vorticity form . . . . . . . . . . . . . . . . . 323.3.3 Spectral code in skew-symmetric form . . . . . . . . . . . . 36

3.4 Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

iii

4 Center for Turbulence Research Results 46

4.1 Numerical Dissipation . . . . . . . . . . . . . . . . . . . . . . . . . 494.1.1 Estimating Numerical Dissipation due to Filtering . . . . . . 524.1.2 Quantifying Numerical Dissipation . . . . . . . . . . . . . . 54

5 Spectral Results I 59

5.1 LES at 3% of DNS Resolution . . . . . . . . . . . . . . . . . . . . . 615.2 LES at 1% of DNS Resolution . . . . . . . . . . . . . . . . . . . . . 635.3 Spanwise Structures . . . . . . . . . . . . . . . . . . . . . . . . . . 72

6 Spectral Results II 83

6.1 LES-V at 3% of DNS Resolution . . . . . . . . . . . . . . . . . . . 836.2 LES-V at 1% of DNS Resolution . . . . . . . . . . . . . . . . . . . 85

7 Conclusions 96

7.1 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . 967.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

Reference List 101

iv

List of Tables

3.1 Validation using linear stability theory: vorticity spectral code . . . 40

3.2 Validation using linear stability theory: skew-symmetric spectral code 45

4.1 CTR simulation parameters . . . . . . . . . . . . . . . . . . . . . . 46

4.2 CTR numerical dissipation with ltering . . . . . . . . . . . . . . . 57

4.3 CTR numerical dissipation without ltering . . . . . . . . . . . . . 58

5.1 Spectral simulation parameters . . . . . . . . . . . . . . . . . . . . 60

5.2 Spectral 1% LES performance . . . . . . . . . . . . . . . . . . . . . 69

6.1 Vorticity spectral simulation parameters . . . . . . . . . . . . . . . 84

6.2 Spectral 1% LES-V performance . . . . . . . . . . . . . . . . . . . . 90

v

List of Figures

1.1 Laminar separation bubble ow sketch . . . . . . . . . . . . . . . . 3

1.2 Laminar separation bubble on an airfoil . . . . . . . . . . . . . . . . 4

3.1 Cp of laminar separation bubble on airfoil and at plate . . . . . . . 41

3.2 Anatomy of a at plate laminar separation bubble ow . . . . . . . 42

3.3 Visualization of a at plate laminar separation bubble ow . . . . . 43

3.4 Flat plate laminar separation bubble ow computational setup . . . 43

3.5 Numerical dissipation example . . . . . . . . . . . . . . . . . . . . . 44

3.6 Spectral LES boundary conditions . . . . . . . . . . . . . . . . . . . 44

4.1 CTR boundary conditions . . . . . . . . . . . . . . . . . . . . . . . 48

4.2 Visualization of CTR DNS . . . . . . . . . . . . . . . . . . . . . . . 49

4.3 CTR mean velocity contour plot . . . . . . . . . . . . . . . . . . . . 50

4.4 CTR time-averaged Cp and Cf . . . . . . . . . . . . . . . . . . . . . 51

4.5 CTR energy rate-of-change in turbulent region . . . . . . . . . . . . 53

5.1 Spectral 3% LES ow visualization . . . . . . . . . . . . . . . . . . 61

5.2 Spectral 3% LES Cp and Cf . . . . . . . . . . . . . . . . . . . . . . 64

5.3 Spectral 3% LES boundary layer thicknesses . . . . . . . . . . . . . 65

5.4 Spectral 3% LES mean and RMS velocity proles . . . . . . . . . . 66

vi

5.5 Spectral 1% LES Cp and Cf . . . . . . . . . . . . . . . . . . . . . . 70

5.6 Spectral 1% LES subgrid-scale dissipation . . . . . . . . . . . . . . 75

5.7 Spectral 1% LES boundary layer thicknesses . . . . . . . . . . . . . 76

5.8 Spectral 1% LES mean velocity proles in wall units . . . . . . . . 77

5.9 Spectral 1% LES mean and RMS velocity proles . . . . . . . . . . 78

5.10 Spectral 1% LES maximum RMS velocity . . . . . . . . . . . . . . 79

5.11 Spectral 1% LES velocity auto-correlation functions . . . . . . . . . 80

5.12 Spectral 1% LES velocity auto-correlations for wider domain . . . . 81

5.13 Spectral 3% LES velocity auto-correlation functions . . . . . . . . . 82

6.1 Spectral 3% LES-V Cp and Cf . . . . . . . . . . . . . . . . . . . . . 86

6.2 Spectral 3% LES-V boundary layer thicknesses . . . . . . . . . . . . 87

6.3 Spectral 3% LES-V mean and RMS velocity proles . . . . . . . . . 88

6.4 Spectral 1% LES-V Cp and Cf . . . . . . . . . . . . . . . . . . . . . 91

6.5 Spectral 1% LES-V boundary layer thicknesses . . . . . . . . . . . . 92

6.6 Spectral 1% LES-V mean velocity proles in wall units . . . . . . . 93

6.7 Spectral 1% LES-V mean and RMS velocity proles . . . . . . . . . 94

6.8 Spectral 1% LES-V maximum RMS velocity . . . . . . . . . . . . . 95

vii

Abstract

The ow over blades and airfoils at moderate angles of attack and Reynolds

numbers ranging from 104 to 105 undergoes separation due to the adverse pressure

gradient generated by surface curvature. In many cases, the separated shear layer

then transitions to turbulence and reattaches, closing o a recirculation region -

the laminar separation bubble. To avoid body-tted mesh generation problems

and numerical issues, an equivalent problem for ow over a at plate is formu-

lated by imposing boundary conditions that lead to a pressure distribution and

Reynolds number that are similar to those on airfoils. Spalart & Strelets (2000)

tested a number of Reynolds-averaged Navier-Stokes (RANS) turbulence models

for a laminar separation bubble ow over a at plate. Although results with

the Spalart-Allmaras turbulence model were encouraging, none of the turbulence

models tested reliably recovered time-averaged direct numerical simulation (DNS)

results. The purpose of this work is to assess whether large eddy simulation (LES)

can more accurately and reliably recover DNS results using drastically reduced

resolution on the order of 1% of DNS resolution which is commonly achiev-

able for LES of turbulent channel ows. LES of a laminar separation bubble ow

over a at plate are performed using a compressible sixth-order nite-dierence

code and two incompressible pseudo-spectral Navier-Stokes solvers at resolutions

corresponding to approximately 3% and 1% of the chosen DNS benchmark by

viii

Spalart & Strelets (2000). The nite-dierence solver is found to be dissipative

due to the use of a stability-enhancing lter. Its numerical dissipation is quan-

tied and found to be comparable to the average eddy viscosity of the dynamic

Smagorinsky model, making it dicult to separate the eects of ltering versus

those of explicit subgrid-scale modeling. The negligible numerical dissipation of the

pseudo-spectral solvers allows an unambiguous assessment of the performance of

subgrid-scale models. Three explicit subgrid-scale models dynamic Smagorinsky,

, and truncated Navier-Stokes (TNS) are compared to a no-model simulation

(under-resolved DNS) and evaluated against the benchmark DNS data focusing on

two quantities of critical importance to airfoil and blade designers: time-averaged

pressure (Cp) and skin friction (Cf) predictions used in lift and drag calculations.

Results obtained with these explicit subgrid-scale models conrm that accurate

LES of laminar separation bubble ows are attainable with as low as 1% of DNS

resolution, and the poor performance of the no-model simulation underscores the

necessity of subgrid-scale modeling in coarse LES with low numerical dissipation.

ix

Chapter 1

Introduction

Laminar ow separation, transition to turbulence, and reattachment coined

laminar separation bubble because of the recirculation region the phenomena create

together directly aect the performance of increasingly important applications

ranging from small unmanned aerial vehicles to low pressure turbines found in

jet engines and gas generators. The presence of a laminar separation bubble on

a small unmanned aerial vehicles wing can increase the lift over drag ratio L/D

and thus the eciency of the craft. Airfoil shapes that maximize this eect are

sought at the design stage. On the other hand, laminar separation bubbles cause

detrimental unsteadiness and transient structural loads on low pressure turbine

blades, so blade shapes and ow control schemes to mitigate this phenomenon

are sought at the design stage. Faster, more accurate computational tools are

necessary to enable the design optimization of technologies to better control the

onset of laminar separation bubbles on blades and airfoils.

1.1 Laminar Separation Bubble Flows

The Reynolds number (Re) is a dimensionless ratio of inertial to viscous forces

characterizing the ow regime. Reynolds numbers for small unmanned aerial vehi-

cles and micro-air vehicles are low to moderate. Based on wing chord length, they

are typically less than 2106 and are in some cases only on the order of 104 to 105.

By comparison, civilian airplanes are characterized by Reynolds numbers ranging

1

from a few million to 80 106 for the Boeing 747 at cruising velocity. Recent

experimental investigations of low Reynolds number aerodynamics by Hu et al.

(2007); Hain et al. (2009); Spedding & McArthur (2010) reveal that low to moder-

ate Reynolds number ows over airfoils are often dominated by the eects of ow

separation and reattachment. The appearance of unsteady recirculation regions

due to separation and reattachment greatly inuence the aerodynamic forces the

wing is subjected to. They change the lift and drag characteristics and thus the

ight stability of small unmanned aircraft and micro aerial vehicles. Separation-

induced transition is also seen in higher Reynolds number ows. For example,

ow separation and turbulent reattachment is sometimes observed on the blades

of wind turbines as well as on low pressure gas turbine blades due to the inuence

of the interior wall. This phenomenon causes unsteadiness in the ow, which is a

determining factor in high cycle fatigue of turbomachinery components. Although

most wind and gas turbines operating Re are higher than 105, the physics gov-

erning separation-induced transition and the stability of the recirculation region it

creates are akin to those seen in laminar separation bubbles that occur on lower

Re applications like micro aerial vehicles.

A number of experiments have been performed to elucidate laminar sep-

aration bubble ows. The seminal works by Gaster (1963) and Horton

(1968) established the foundation for the physical understanding of the lam-

inar separation bubble and advanced some semi-empirical predictions for the

location of separation and reattachment. Since then, a number of experi-

ments have been conducted to study in more detail the structure, stability,

and dynamics of laminar separation bubble ows Alving & Fernholz (1996);

Hggmark (2000); Marxen et al. (2003); Burgmann et al. (2006); Yarusevych et al.

(2006); Burgmann et al. (2007); Burgmann & Schrder (2008); Hu et al. (2007);

2

Figure 1.1: Features of the ow eld in the vicinity of a transitional separationbubble Horton (1968); Lin & L.Pauley (1996); Castiglioni et al. (2014).

Hain et al. (2009); Spedding & McArthur (2010). As a result, the physical origin

of laminar and transitional ow separation is now qualitatively well understood.

As gure 1.1 illustrates, the attached laminar boundary layer developing on a wing

or blade is subjected to an adverse pressure gradient due to the airfoils curvature,

which causes it to separate. Immediately behind the separation point there is an

eectively stagnant ow region, the so-called dead air region. A reverse ow vor-

tex develops downstream of the dead air region, leading to an inectional mean

velocity prole in the boundary layer. This triggers the growth of convective and

secondary instabilities which quickly break down to turbulence. Kelvin-Helmholtz

rolls are indeed visible in a snapshot of the ow eld of a laminar separation bubble

ow over a NACA0012 airfoil shown in gure 1.2. As the separated shear layer

transitions to turbulence, its interaction with the reverse ow vortex causes it to

3

Figure 1.2: Instantaneous iso-surfaces of spanwise vorticity on a NACA0012 airfoilat 5 degrees angle of attack and Rec = 5 104 taken from a STAR-CCM+ largeeddy simulation Castiglioni (2015).

reattach, thereby closing o the recirculation region. Aft of the laminar separa-

tion bubble, unlike in the two dimensional case, the ow is fully turbulent and

three-dimensional. Clear spanwise vortices are not shed, but the size and shape

of the bubble changes in time due to the complex balance between the eects of

the pressure gradient, convective instabilities, and viscous dissipation. This pic-

ture emerges from these experimental investigations as well as from direct numer-

ical simulations (DNS) results Lin & L.Pauley (1996); Spalart & Strelets (2000);

Alam & Sandham (2000); Marxen & Rist (2010); Jones et al. (2008, 2010).

4

Many DNS were carried out to match and augment experimental data. These

DNS shed light on the mechanisms of energy transfer at work in laminar separation

bubble ows Skote et al. (1998); Spalart & Strelets (2000); Skote & Henningson

(2002), the process of transition Na & Moin (1998); Wu & Moin (2010);

Alam & Sandham (2000); Marxen & Rist (2004), the stability characteristics of

such ows Marxen et al. (2003); Marxen & Rist (2010); Jones et al. (2010), and

the eect of disturbances and forcing on its dynamics Herbst & Henningson (2006);

Marxen & Henningson (2011); Jones et al. (2008). Agreement with experiments

was found to be very favorable in most cases. Key to getting agreement with

experimental results was resolving the reverse ow region near the wall, and the

shear layer which transitions to turbulence above the separation bubble.

Although three-dimensional laminar separation bubble ows have been shown

to dier substantially from their two-dimensional analog, there is some disagree-

ment on the degree of importance of spanwise structures, their dominant wave-

length, and the role they play in laminar separation bubble ows. Recent simula-

tion results show that airfoil sections may require one chord length in the spanwise

direction to ensure the full three-dimensional nature of the laminar separation bub-

ble can be captured Eisenbach & Friedrich (2008). Experimental studies conrm

that the dynamics and turbulent statistics of laminar separation bubble ows are

sensitive to incoming levels of turbulence, boundary conditions, and even acoustic

vibrations. This impedes the direct comparison of dierent experiments, and the

search for universal empirical parameters and predictors for its impact on lift and

drag as well as the level of turbulence it generates in the ow. Researchers devising

technologies to control or mitigate the onset of separation bubbles face the same

issues. A predictive tool fast enough to reliably explore the sensitivity to these

parameters and ow control schemes is needed.

5

1.2 Motivation

In order to produce more ecient airfoil or blade designs, to create control

schemes to reduce separation eects, and to better predict high cycle fatigue,

numerical prediction tools for laminar separation bubble ows are needed. Airfoil

and blade designers are primarily interested in obtaining accurate time-averaged

quantities relating to the aerodynamic forces on the airfoil or blade, namely the

coecient of pressure to measure the lift and drag, and the skin friction, another

important component of drag. To enable optimization in a realistic industrial

setting, these quantities must be calculated in a matter of a few hours or less. Such

computationally aordable, accurate and reliable numerical predictions for laminar

separation bubble ows had not been obtained until this study. As a proof of

concept and to avoid numerical issues associated with body-tted mesh generation,

an equivalent problem for ow over a at plate is formulated by imposing boundary

conditions that lead to a pressure distribution and Reynolds number based on

bubble length similar to those observed on the airfoils of small unmanned aircraft

and low-pressure turbine blades. The objective of this research is thus to test the

accuracy of large eddy simulation at resolutions drastically reduced compared to a

benchmark DNS for a laminar separation bubble ow over a at plate. Corollary

objectives are to identify key factors in obtaining accurate predictions, and to

investigate the performance of dierent subgrid-scale models in this highly under-

resolved environment.

6

Chapter 2

Background

2.1 Governing Equations

The relevant governing equations are the incompressible Navier-Stokes equa-

tions because the uid density and temperature may be considered constant for

laminar separation bubble ows at moderate Reynolds numbers (Re = 104 to

Re = 2 105).

uit

+

xj(uiuj) =

p

xi+

xj

(uixj

+ujxi

)+ Fi, i = 1, 2, 3 (2.1)

uixi

= 0, (2.2)

The velocity eld u = (u1, u2, u3) expressed in reference to a Cartesian coordinate

system x = (x1, x2, x3) is a solution of the momentum and continuity equations,

(2.1) and (2.2). p = P/ is the static pressure and is the kinematic viscosity

which is assumed to be constant and uniform in space.

No accepted analytical closed form solution to the Navier-Stokes equations for

laminar separation bubble ows is known. A numerical solution to the Navier-

Stokes equations is thus sought on a discrete set of grid points xijk or discrete cell

volumes. A number of approaches exist to discretize the equations such as the

nite dierence, nite volume, and nite element methods to name only a few.

Despite fundamental dierences in their approach, each discretization methods

conservation properties are directly dependent on the accuracy of the schemes

7

used to approximate each of the terms in the equations solved. The accuracy of

the solution depends both on the exact formulation of the numerical scheme, the

quality (level of non-orthogonality) of the grid, and the level of resolution on which

the discretized equations are solved.

2.2 Direct Numerical Simulation

Once discretized, the Navier-Stokes equations are integrated in time. Doing

so without further approximations is called direct numerical simulation (DNS).

DNS is the most accurate and reliable uid dynamics simulation tool available.

However, to obtain an accurate solution using DNS requires that the mesh or

cell size captures all relevant scales of motion in the problem. If the ow to be

simulated is turbulent, then the mesh or cell size must be within one order of

magnitude from the Kolmogorov length scale, the smallest length scale at which

turbulence exists. Dominated by viscosity, these small scales are largely responsible

for the dissipation of turbulent motion into heat. Capturing Kolmogorov scales

imply solving the Navier-Stokes equations on a very large number of cells or a

very ne mesh. The computational work required may take a prohibitively large

amount of time even on the best available supercomputer. Indeed, DNS requires

substantial computational resources, long wall-clock runs, and long analysis times;

e.g. a relatively simple 3-D airfoil conguration at a Reynolds number of 5 104

required over 170 million grid points Jones et al. (2008). For laminar separation

bubble ow over a at plate at a Reynolds number of 105, DNS results required

over 16000 processor-hours Cadieux et al. (2012). A number of 3-D congurations

and angles of attack need to be quickly investigated to allow for the optimization

8

of airfoil and turbine blade designs. For this case, a DNS approach is generally

impractical and other simulation approaches must be considered.

2.3 Reynolds-averaged Navier-Stokes

A widely-used computational uid dynamics simulation approach for moderate

to high Reynolds number ows is to solve for the mean ow quantities directly

instead of calculating the primary quantities at each instant in time as is done in

DNS. The primary quantities are split into uctuating and mean ow components

ui = Ui + ui (2.3)

where Ui = ui is the ensemble average of ui satisfying Reynolds averaging con-

ditions. Substituting eq.(2.3) into the Navier-Stokes equations and noting that

ui = 0 gives the Reynolds-averaged Navier-Stokes (RANS) equations for incom-

pressible ow:

Uit

+

xj(UiUj) =

1

P

xi+

xj

(

(Ujxi

+Uixj

) ransij

), (2.4)

Uixi

= 0 (2.5)

where ransij = uiu

j. (2.6)

All information about the instantaneous velocity elds is lost through this averag-

ing in favor of obtaining an estimate of the mean ow. Unfortunately, averaging

procedures used in practice, namely time and spatial averaging, do not strictly

satisfy Reynolds last condition that f(x)g(x) = f(x)g(x) for any ow that

is not fully developed with a large separation of scales, limiting its applicability to

9

such ows Wilcox (2006). In other words, if the mean and uctuating components

are correlated, then the time or spatial average of their product does not vanish

and the RANS equations are no longer valid Wilcox (2006). As seen in eq. (2.4),

the RANS equations have an additional unknown term, the Reynolds stress tensor

ransij , which must be modeled using prior knowledge about the ow being simu-

lated and/or additional equations. The majority of RANS models for ransij rely on

the Boussinesq eddy viscosity approximation. The Boussinesq hypothesis states

that by analogy to momentum transfer in the molecular motion of a gas which

can be described by a molecular viscosity , the Reynolds stress tensor should be

proportional to the mean strain rate tensor using a turbulent eddy viscosity rans:

ransij rans

(Ujxi

+Uixj

)(2.7)

This reduces the number of extra unknowns from six to one, the eddy viscosity.

Most RANS models use dimensional arguments and analogy to other physical

processes to set the value for rans. For example, Prandtls mixing length model

uses the Boussinesq analogy to molecular momentum transport and assumes that

there exists a turbulent mixing length analogous to the mean free path Wilcox

(2006). Using dimensional analysis, the mixing length model for a boundary layer

simplies to

rans = ransU

y, (2.8)

rans = 2mix|U

y|, (2.9)

mix = =

0

(1

U(y)U

)dy 1.72x/

Rex, (2.10)

10

where the mixing length is assumed to be proportional to the displacement thick-

ness of the boundary layer. Such a simple turbulence model provides results in

agreement with experiments for boundary layers Wilcox (2006). However, it is

incomplete because the appropriate mixing length for the ow being simulated

must be known a priori. Moreover, the mixing length model is only valid for sim-

ple ows with slow-varying properties (so-called equilibrium turbulent ows) due

to the assumptions used in deriving it. Modern models are far more complex and

perform better in a wider range of ows, but still struggle with accurately pre-

dicting transition from laminar to turbulent ow and the lengths of recirculation

zones.

2.3.1 RANS for Laminar Separation Bubble Flows

Laminar separation bubble ows present a challenge to RANS models because

boundary layer separation and reattachment involves subtle interactions between

viscous, advective, and pressure eects and is inherently a non-equilibrium process,

especially when driven by an adverse pressure gradient instead of geometry (e.g.

backward facing step). Although the rate of strain changes rapidly as the ow

separates, the turbulence adjusts to changes in the ow on an unrelated, longer

time scale. A perturbed turbulent boundary layer was experimentally shown not

to return to equilibrium for at least 10 boundary-layer thicknesses downstream of

the perturbation Wilcox (2006). Since the Reynolds stresses modeled in typical

two-equation RANS models are adjusted based solely on the mean rate of strain,

they preclude any such transient eects from ow history. Attempts were made to

adjust for this eect by relaxing the eddy viscosity in the region behind separation.

However, this method requires prior knowledge of the separation point - and thus

11

precludes strong predictive capabilities even for unsteady RANS Howard et al.

(2000).

Spalart & Strelets (2000) tested a number of typical RANS turbulence model,

namely Spalart-Allmaras, Menters shear stress transport, modied shear stress

transport, and Secundovs 92t , for a simple laminar separation bubble ow on

a at plate driven by suction from the top. All RANS turbulence models except

Spalart-Allmaras predict earlier transition and reattachment than observed in their

spectral DNS results. Obvious disagreement in the location and magnitude of the

peak negative skin friction is observed between the results of dierent models.

All RANS results were found to under-predict the level of skin friction down-

stream of reattachment. Without modication, Menters shear stress transport

model transitions even before the expected separation point, predicting attached

ow throughout the domain. Although Spalart-Allmaras results were encouraging,

none of the RANS turbulence models Spalart & Strelets (2000) tested recovered

all the important features of the time-averaged skin friction DNS curve: location

of the separation point, location and magnitude of the peak negative skin friction,

reattachment point, and turbulent skin friction levels immediately downstream of

reattachment. The large variations observed in skin friction predictions depending

on the RANS turbulence model chosen may be symptomatic of a lack of robust-

ness of the typical RANS turbulence modeling approach when applied to inherently

unsteady phenomena.

To better predict transition, Howard et al. (2000) sensitized two-equation

RANS model coecients of the Launder and Sharma and kg models to the local

turbulent Reynolds number as proposed by Wilcox (2006), removing the need for

a priori knowledge of the separation point. With the local turbulent Reynolds

12

number sensitivity modication, two-equation models were shown to predict tran-

sition more reliably in an unsteady RANS solver and showed improved agreement

for separation bubble length. However, Launder and Sharma and k g model

results were still inconsistent amongst each other and amongst dierent sensitiza-

tion approaches. None matched the DNS results for peak negative skin friction

and its level immediately downstream of reattachment Howard et al. (2000).

RANS turbulence models using second moment closures show improvement

over single and two-equation models without the need for articially triggering

transition, but require tuning of their closure coecients for lower Reynolds num-

ber ows with separation and transition Hadi & Hanjali (2000). While results

show better agreement for the reattachment point and turbulent region at coarser

resolution than the Spalart-Allmaras model, second-moment closure RANS results

still fail to recover the peak negative skin friction. Since the authors mention but

do not present results for a ner grid, it can be surmised that results did not

improve signicantly Hadi & Hanjali (2000). That the method does not con-

verge to DNS results with increased resolution is further evidence that it is not

well-suited for the problem.

Similarly, results for a laminar separation bubble on a at plate with a semi-

circular leading edge using a two-layer model matched experimental results reason-

ably well, but required a number of empirical correlation along with special modi-

cations to the model to capture transition Papanicolaou & Rodi (1999). Although

RANS can predict separation and transition reliably with models optimized for

such ows (second-moment closure, two-layer model), it still struggles to recover

wall skin friction accurately for this proof-of-concept laminar separation bubble

ow over a at plate.

13

2.4 Large Eddy Simulation

Another option is to employ large eddy simulation (LES) techniques. LES tech-

niques were developed based on the observations made by Kolmogorov that the

smallest scales of turbulence are dominated by viscosity, behave mostly isotropi-

cally and account for the majority of turbulent dissipation. The understanding of

the energy cascade from large to small scales led to the idea of modeling the more

universal small scales of turbulent motion while resolving the energy containing

eddies directly aected by the ow boundary conditions. This greatly relaxes the

DNS requirement that the mesh or cell size be on the same order as the Kolmogorov

length scale. LES hinges on the use of subgrid-scale models to predict the correct

small scale dissipation rate based solely on information from the larger resolved

scales.

2.4.1 Filtered Navier-Stokes Equations

A low-pass lter operation is used to separate small scales from larger ones

and derive the ltered Navier-Stokes equations. This ltering operation can be

described by a convolution integral with the lter function or kernel G,

ui(x, t) = G ui = +

G(x x; )ui(x, t) d3x (2.11)

where the resolved scale or ltered velocity is ui, and the subgrid-scale velocity

is dened as ui = ui ui. is the lter width which is generally taken to

14

be proportional to the grid or cell size = (xyz)1

3 Sagaut (2006). The

incompressible ltered Navier-Stokes equations can then be written as follows:

uit

+

xj(uiuj + sgsij ) =

p

xi+

xj

(uixj

+ujxi

)+ Fi, (2.12)

uixi

= 0, (2.13)

where

sgsij = uiuj uiuj (2.14)

is the subgrid-scale stress tensor. It contains the term uiuj which is a new unknown.

Just as turbulence models are required to close the RANS equations, the l-

tered Navier-Stokes equations require a subgrid-scale model for sgsij . However,

the derivation of subgrid-scale models generally rely only on the assumption that

the eect of the scales of motion smaller than the lter width on the larger scales

are small and mostly dissipative. This assumption is more robust than those made

in deriving the RANS equations and most RANS turbulence models because it is

satised for a much wider array of ow conditions and relative lter widths.

2.4.2 Subgrid-scale Models

There exists a variety of dierent subgrid-scale models to close equations (2.12)

and (2.13) that generally belong to one of two distinct categories: structural and

functional modeling. Structural modeling directly approximates the subgrid-scale

stress tensor or subgrid-scale velocities based on the resolved velocities or a formal

series expansion. This approach assumes that the structure of the small scales

is universal and the energy contained in the subgrid-scales are a function of the

15

resolved scales Sagaut (2006). Examples include models based on the deconvolu-

tion procedure, stress transport models, and subgrid-scale velocity reconstruction

models. A recent example is the velocity estimation model by Dubois et al. (2002).

Instead of directly approximating sgsij , functional modeling seeks to approxi-

mate the eects of inter-scale energy transfer on the resolved scales. Rather than

assuming that the structure of the small scales is universal, this approach posits

that the eects of the small scales on the larger resolved scales are universal Sagaut

(2006). Knowledge of the turbulent energy cascade and the concepts of forward

and backscatter are used to justify the assumption that the eect of the small

scales on the large is universal and depends only on the energy of the large scales

driving the ow. Recent examples of such subgrid-scale models are the and the

interscale energy transfer models Nicoud et al. (2011); Anderson & Domaradzki

(2012). The increasingly popular implicit LES (ILES) approach also generally falls

into this category, where the numerical scheme is adjusted such that its truncation

errors and associated dissipative and dispersive eects have the desired impact on

the resolved scales. The most famous functional modeling example remains the

Smagorinsky model.

The Smagorinsky Model

The Boussinesq approximation used in many RANS models is invoked again,

but this time to calculate a turbulent eddy viscosity sgs describing the subgrid-

scale dissipation based solely on the resolved scales of motion.

sgsij 13 sgskk ij = 2sgs(Sij

13Skkij), (2.15)

16

where

sgs = (CS)2|S|Sij, (2.16)

|S| = (2SijSij)1

2 (2.17)

Sij =12

(uixj

+ujxi

), (2.18)

= (xyz)1

3 . (2.19)

CS is a closure coecients determined ahead of simulation by matching avail-

able experimental or DNS data. This is a limitation to the models applicability

because foreknowledge of the ow being simulated is required to set the closure

coecient. It also cannot account for the local reduction of eddy viscosity near

physical boundaries without the use of explicit Van Driest damping functions. A

modication of the Smagorinsky model which eliminates these issues and makes it

universal consists of letting the closure coecients be functions of time and space

(e.g. C2S = Cd(x, t)). In this dynamic version of the Smagorinsky model, these

coecients are computed dynamically using the Germano identity.

The Dynamic Procedure

The dynamic procedure used to compute the local instantaneous closure coe-

cient Cd(x, t) for the Smagorinsky model is used here as an example, but it can be

applied to other models formulated in a similar fashion. Its purpose is to provide

this coecient based on local resolved strain rate without prior knowledge of the

ow. It eectively reduces the model contribution in laminar shear ows where

17

the stress tensor is not zero, but no turbulence exists, removing the issues that the

static Smagorinsky model faces with physical boundaries.

sgs = Cd2|S|Sij (2.20)

Cd = min

(0.22, max

[LijMij

MijMij, 0

]), (2.21)

Lij = uiuj ui uj , (2.22)Mij = 2 2|S|Sij 2

2|S|Sij, (2.23)

= 2 = 2(xyz)1

3 . (2.24)

The overbar is used to represent grid-ltered terms (lter width ) and the hat is

used to indicate test-ltered quantities using Simpsons rule (lter width 2):

f(x) 16f(xx) +

23f(x) +

16f(x+x). (2.25)

The weights of the lter are adapted to the non-uniform vertical grid using

quadratic interpolation. The symbol used in (2.21) denotes averaging in any

uniform direction (if one exists) or local averaging in all directions. Although small

negative values of Cd may be physically justied, mimicking backscatter phenom-

ena, such negative values can quickly lead to numerical instability. So in practice,

averaging is used in tandem with clipping to avoid negative as well as rapidly

oscillating values of Cd. The constant computed through this dynamic procedure

is local in space and time. It eectively reduces the model contribution in lami-

nar shear ows where the stress tensor is not zero, but no turbulence exists. The

dynamic procedure is particularly computationally expensive due to the applica-

tion of a test-lter in three-dimensions on at least two tensorial quantities uiuj and2|S|Sij as well as local spatial averaging, each ltering and averaging operation

18

often requiring communication among dierent processes or blocks in parallelized

implementations. Despite this shortcoming, the dynamic Smagorinsky model has

become the benchmark against which other subgrid-scale model are tested due to

its success in academia and its universality.

The Model

The model follows denitions set forth in (2.15) and (2.18) but computes

sgs using the singular values i of the velocity derivative tensor gij. This choice is

motivated by the desire to improve on the dynamic Smagorinsky model by provid-

ing more appropriate near-wall behavior, as well as providing zero contributions

in pure two-dimensional shear or pure rotation cases Nicoud et al. (2011).

sgs = (C)23(1 2)(2 3)

21is the subgrid-scale eddy viscosity,

(2.26)

where

1 2 3 0, are the singular values of gij =uixj

, and (2.27)

C = 1.35 is the closure coecient. (2.28)

C is determined from homogeneous turbulence and validated using channel ow

simulations Nicoud et al. (2011). The singular values i are obtained using the

invariants of Gij = gkigkj and their angles to avoid the overhead of linear algebra

library calls to an eigenvalue solver for each cell in the domain at each time step.

19

The truncated Navier-Stokes Approach

The truncated Navier-Stokes approach follows the method developed by

Domaradzki et al. (2002) in which periodic ltering is used as a substitute for

a subgrid-scale model. Periodic ltering is used to remove energy from the small-

est resolved scales by the use of a low-pass approximate deconvolution method

(ADM) lter Stolz et al. (2001). The ltering operation is implemented using

the product of an approximate deconvolution lter QN G1 described in

Tantikul & Domaradzki (2010) with lter G:

QNG = I (I G)N+1. (2.29)

The order N = 5 is chosen such that the lter only aects scales smaller than lter

width = x when using a simple three point lter in physical space G(x):

G(x) f(x) 18f(xx) +

34f(x) +

18f(x+x). (2.30)

The lter weights are adjusted for the non-uniform vertical grid using quadratic

interpolation. Since the energy cascades from large to small scales and accumulates

there slowly in a high order under-resolved simulation, it is only necessary to lter

after a fraction of a percent of the large eddy turnover time. Hence, ltering is only

applied when the kinetic energy at high wave numbers reaches unphysical levels

in the truncated Navier-Stokes approach. This is fundamentally dierent from

implicit LES, where the inherent approximation errors can be said to act as a low-

pass lter at each time step. Excessive energy accumulating in the small scales is

detected using a criterion based on the ratio of energy removed I(x)/I(2x) by

20

two ADM test-lters with lter widths = x denoted by the tilde and = 2x

denoted by the hat as follows

I(x)I(2x)

=V

E E

E EdV (2.31)

Y

0

3i=1

12(ui ui)(ui ui)3

i=112(ui ui)(ui ui)

(y)dy (2.32)

ui = (Q5G(x)) ui (2.33)

ui = (Q5G(2x)) ui (2.34)

G(2x) f 14f(xx) +

12f(x) +

14f(x+x). (2.35)

The ratio I(x)/I(2x) represents the energy contained in the small scales com-

pared to the larger scales. When it reaches values in excess of those obtained

for a typical dissipation, inertial, or Batchelor energy spectrum determined to

be 0.007 to 0.009 from theory by Tantikul & Domaradzki (2010, 2011) primary

variables are ltered in physical space with lter Q5G(x). Using this criterion,

the lter is applied at varying intervals centered around 200 hundred time steps for

the coarsest resolution simulation presented here, corresponding to approximately

0.5% of one non-dimensional time unit t = t U0Lx.

2.4.3 LES Results for Laminar Separation Bubble Flows

A number of LES of laminar separation bubble ows over at plates and

airfoils have been completed recently by Wilson & Pauley (1998); Yang & Voke

(2001); Roberts & Yaras (2005); Eisenbach & Friedrich (2008); Xu et al. (2010);

Kojima et al. (2013). For instance, LES results Yang & Voke (2001) obtained with

the dynamic Smagorinsky model were reported to be in good agreement with exper-

iments for boundary-layer separation and transition caused by surface curvature

21

at Re = 3, 450. Yet even for this relatively low Reynolds number, the two critical

issues in getting agreement were a numerical resolution (4727264 mesh points)

comparable to DNS of the same ow, and a high order numerical method. Such

strict requirements are dicult to satisfy in simulations of practical ows often

performed with low order nite dierence or nite volume methods (e.g. commer-

cial codes). Similarly, LES of ow separation on an airfoil at a high angle of attack

was performed at Re = 105 using Cartesian grids Eisenbach & Friedrich (2008).

This case also required very high resolutions between 50 and 100 million mesh

points to obtain good agreement. Using LES with such high resolution and higher

order methods implies a time-to-solution on the same order as DNS. Therefore,

the question remains: can LES produce suciently accurate results for laminar

separation bubble ows with drastically reduced resolution, around 1% of DNS

resolution, commonly achievable for fully turbulent ows?

While a handful of other investigators also performed low resolution LES, the

eect of dierent subgrid-scale models on the quality of important results such as

time-averaged skin friction and pressure coecient remains largely unknown for

laminar separation bubble ows. Only the constant Smagorinsky model has been

compared to its dynamic counterpart and a no-model case in Wilson & Pauley

(1998). Other investigators like Eisenbach & Friedrich (2008); Yang & Voke

(2001); Xu et al. (2010) relied entirely on the dynamic Smagorinsky model, or

on the numerical dissipation of their chosen scheme as in implicit LES or ILES

in Kojima et al. (2013) and even without any prior knowledge of the dissipative

schemes eects on the resolved scales in the case of Roberts & Yaras (2005). With-

out benchmark DNS data or a baseline case with no subgrid-scale model active to

compare directly to, the performance of their models or ILES results could not be

evaluated quantitatively.

22

2.5 Wall-modeled LES and Hybrid RANS-LES

For LES to be accurate, it requires a mesh nearly as ne as a DNS near physi-

cal boundary where boundary layers develop. To mitigate this stringent resolution

requirement, wall models are developed to give approximate boundary conditions

to the LES solver away from the surface. A wide variety of models with dier-

ent assumptions have been proposed and reviewed by Piomelli & Balaras (2002);

Piomelli (2008) and Sagaut & Deck (2009). The use of wall models generally

mitigates noise generated by poor approximation of curved surfaces or highly non-

orthogonal body-tted meshes in low resolution settings and permit simulations to

reach much higher Reynolds numbers that are closer to operating conditions for

most turbomachinery blades McMullan & Page (2012). However, wall models have

historically had a poor track record in predicting separation and reattachment, and

introduce another source of error into LES due to further approximations made at

the wall Bose & Moin (2014). But more importantly, when wall models are used

it becomes dicult to distinguish between the performance of the wall model and

that of the subgrid-scale model because obtaining correct amount of turbulent con-

tent near the wall is key to the overall accuracy of LES. Detached eddy simulation

(DES) solves the RANS equations near the wall and smoothly transition to LES

on a single grid using a single hybrid RANS-LES turbulence model developed by

Spalart (2006). This removes the need for wall-layer modeling while still drastically

reducing near-surface resolution requirements, allowing the simulation of higher

Reynols number ow. Despite many successes, dicult issues such as modeled

stress depletion in the log law region, and non-monotonic grid convergence arise

in DES as well as its derivatives delayed DES and zonal DES as acknowledged by

Spalart (2006, 2009); Deck et al. (2011); Deck (2012). Partially averaged Navier-

Stokes (PANS) developed by Girimaji & Abdol-Hamid (2005); Basara et al. (2011)

23

and other variable resolution approaches such as the scale-adaptive simulation

(SAS) proposed by Menter & Egorov (2010); Egorov et al. (2010) and turbulence-

resolving RANS (TRANS) put forth by Shur et al. (2008) all avoid these issues by

tying the eddy viscosity to physical quantities like energy grid density or an integral

length scale. However, their turbulence modeling approaches and by consequence

their results are often closer to unsteady RANS than LES and are thus limited in

their ability to capture transient and unsteady eects accurately as pointed out in

Menter & Egorov (2010). In fact, neither hybrid approaches nor wall models have

been validated to the same extent as LES, even when in pure RANS or pure LES

mode (when the method allows it) Sagaut & Deck (2009). To investigate the eects

of dierent subgrid-scale models without the unknown inuence of wall models or

hybrid RANS-LES approaches, wall-resolved LES is chosen for this work.

24

Chapter 3

Methodology

3.1 Approach

Laminar separation bubble ows occur on blades and airfoils at low to mod-

erate Reynolds numbers ranging from 104 to 105 due the curvature of the airfoils

and blades. Simulating ow over blades and airfoils requires the creation of non-

orthogonal body tted meshes, unstructured grids, or the use of immersed bound-

ary methods to properly represent the airfoil or blades surface. Grid creation not

only presents its own challenges, but also often limits numerical solution methods

to second order accuracy in space and time, with the exception of nite element and

discontinuous Galerkin methods. The choice of meshing technique may also have

inherent approximations that in turn aect stability and accuracy of the numeri-

cal methods at low resolution. For example, approximating a curved surface by a

series of connected straight lines as opposed to bezier curves or splines can give rise

to inaccurate results in second order codes, and catastrophic numerical instability

when using higher order methods.

In order to investigate the capacity of LES to reduce the resolution require-

ment for laminar separation bubble simulations and the performance of dierent

subgrid-scale models free from the numerical issues associated with geometry, ow

over a at plate with an adverse pressure gradient strong enough to cause separa-

tion as described in section 3.2 is studied. This approach has been used sucessfully

to study laminar separation bubble ows both in experiments by Hggmark (2000);

25

Marxen et al. (2003); Sohn et al. (1998) and in simulations by Spalart & Strelets

(2000); Alam & Sandham (2000); Wilson & Pauley (1998); Herbst & Henningson

(2006); Na & Moin (1998); Skote et al. (1998); Wu & Moin (2010). The resulting

pressure distribution is qualitatively comparable to what is seen on blades and

airfoils as is shown in gure 3.1: a smooth increase in pressure is followed by a

plateau over the separation bubble. The plateau ends with a sharp rise in pressure

indicating the ows transition to turbulence and reattachment. Downstream of

the sharp rise, the pressure plateaus again over the developing attached turbulent

boundary layer. The only dierence to note between the airfoil and the at plate

laminar separation bubble pressure distribution is that the sharp peak in pressure

at the stagnation point of the airfoil is not observed on the at plate. Despite

this dierence, the Reynolds number based on bubble length (Re 67, 000) is

similar to those found on airfoils and blades indicating a degree of physical equiva-

lence. In fact, at plate laminar separation bubble ows display the same physical

features as those seen on airfoils as evidenced by the similarity of the mean veloc-

ity contours and mean velocity proles in gure 3.2 to those in gure 1.1. The

mechanisms for separation-induced transition are also the same as in the airfoil

case. Notice that the Kelvin-Helmholtz rolls visible on a at plate with suction

from the top shown in gure 3.3 near x = 3.5 closely resemble those seen on an

airfoil shown in gure 1.2, each gure displaying iso-surfaces of spanwise vorticity.

As such, general conclusions reached from investigating at plate separation and

reattachment should also be applicable to blades and airfoils.

26

3.2 Flow Specification

The computational setup used by Spalart & Strelets (2000) to study separation-

induced transition ow over a at plate is followed. The physical domain is a

rectangular box with height Y , length 7.5Y , and width 0.6Y (see gure 3.4). At

the inow a laminar Blasius boundary layer velocity prole is imposed with the

free stream velocity U0. At the top boundary, a vertical suction velocity is imposed

in a narrow slot oriented perpendicular to the mean ow direction. The suction

produces an adverse pressure gradient that causes ow separation. The ow then

transitions to turbulence and reattaches downstream. The vertical suction velocity

is specied as

V (x) = a exp([(x xs)/(0.24Y )]2), (3.1)

where a is the peak velocity and xs is its streamwise location Spalart & Strelets

(2000). The resulting separation bubble is sensitive only to the upper-wall bound-

ary conditions through the nominal ow deceleration parameter S,

S =1

Y U0

V (x)dx. (3.2)

Using the height Y to non-dimensionalize all relevant lengths the parameters in

the equations above are set such that xs = 3, S = 0.3 and the Reynolds number

at xs is Rexs = 105, giving a 0.7U0 and ReY = Rexs/3, matching those in

Spalart & Strelets (2000). These choices are driven by the requirement that the

ow separates naturally, without additional forcing mechanisms like those used in

Alam & Sandham (2000).

27

3.3 Numerical Methods

The numerical schemes used to approximate the equations in space and inte-

grate them in time have an impact on three important quantities: the rate of con-

vergence, numerical dissipation and dispersion. The choice of numerical scheme

determines the rate of convergence to the true solution. For example, a second

order scheme in space implies that doubling the number of mesh points over a

given solution domain should decrease the error of the simulation by a factor of 4

over a constant time of integration.

The concepts of numerical dissipation and dispersion are linked to the exact

form of the truncation error terms of any approximation made by the scheme. For

example, a spatial derivative approximated by a central dierence has a truncation

error E calculated from its Taylor series expansion as follows.

xf(x) =f(x+ h) f(x h)

2h+ h23xf(x) +O(h

3) (3.3)

f(x+ h) f(x h)

2h(3.4)

E = h23xf(x) +O(h3) (3.5)

The omission of these higher order terms in the simulation have eects that are

unknown a priori and depend on the governing equations, the particular ow simu-

lated and the degree of under-resolution. A scheme is described as dissipative if its

total energy kinetic decreases faster than the exact solution. Its visible eect can be

likened to articially increasing viscosity. The classic example is the reduction in

amplitude of a half sine wave, and its increasing wave length during linear convec-

tion in space using a simple nite-dierence upwind scheme as seen in gure 3.5a

reproduced from Fletcher (1991). Numerical dispersion is linked to the scheme

28

amplifying and attenuating dierent Fourier modes of a derivative approximation

causing oscillations that travel with dierent wave speeds. For example, the same

half-sine wave linear convection problem solved using a Crank-Nicolson scheme

results in spurious oscillations as seen in gure 3.5b reproduced from Fletcher

(1991). Such spurious oscillations are characteristic of a scheme with a dominat-

ing dispersive term. Dissipative schemes have been preferred historically for their

robustness: their ability to articially smooth out any sharp changes or disconti-

nuities in solutions where less dissipative or more dispersive schemes might become

numerically unstable and not provide any results. This is problematic for three

reasons. Solutions for ows with shocks where discontinuities are physical will be

increasingly inaccurate over time. For ows where transition to turbulence occurs

naturally, as in laminar separation bubble ows, excessive dissipation may inhibit

transition to turbulence and reattachement completely. Finally, any highly under-

resolved simulation with a dissipative scheme will likely be particularly inaccurate.

Under-resolution already implies that not all length scales of motion relevant to

the problem will be captured. The amount of numerical dissipation is generally

inversely proportional to the resolution the more under-resolved, the higher the

numerical dissipation. Combined, under-resolution and dissipative schemes may

even preclude the development of high wave number content in primary quantities.

Since most LES models are predicated on the ability to predict the correct subgrid-

scale dissipation rate based solely on coarser resolved scales, energy conservation

in the numerical methods used is paramount as evidenced in Kravchenko & Moin

(1997). Second-order methods often damp and deform high wave number con-

tent of the primary variables. Subgrid-scale models are generally not designed to

account for these eects, and given such awed input are unlikely to compute the

29

correct subgrid-scale dissipation. As such, understanding, controlling, or remov-

ing numerical dissipation at low resolution is of critical importance in predicting

laminar separation bubble ows accurately and quickly. For these reasons, two

Navier-Stokes solvers that employ high-order numerical methods were chosen to

investigate the capability of LES to reduce the resolution requirements for accurate

laminar separation bubble predictions.

3.3.1 Center for Turbulence Research Code

Developed by graduate students at the NASA Center for Turbulence Research

(CTR) at Stanford, this code solves the compressible LES equations for a perfect

gas Nagarajan et al. (2007). Henceforth this solver will be referred to as the CTR

code. Derivatives are computed using a sixth-order nite dierence approximation

similar to a Pad scheme. The free parameters are chosen such that the resulting

derivative approximations resolve higher waves numbers than otherwise possible

Lele (1992). These high wave numbers are generally not well approximated or

even severely damped in standard nite dierence schemes of the same order.

To maintain the spectral-like eciency and high order of convergence of these

derivative approximations, the horizontal directions are treated as periodic and

sponge regions are used to simulate non-periodic ow. A split implicit-explicit

time integration is used. For explicit time advancement in the freestream ow,

a third-order Runge-Kutta scheme (RK3) is employed. A second-order A-stable

implicit scheme is used near the wall to allow for larger stable integration time

steps. Compact ltering as described in Lele (1992) is employed at each time step,

both in the freestream and wall-normal directions. Filtering is necessary to remove

aliasing errors, for overall stability and to ensure a smooth continuous solution at

the interface of the implicit and explicit computational domains Nagarajan (2004).

30

The numerical scheme is constructed on a structured curvilinear grid, and the

variables are staggered in space. The freestream Mach number is chosen to be 0.2.

For compressible ows subgrid-scale models are developed in terms of Favre-ltered

quantities, as addressed for the rst time in detail by Erlebacher et al. (1992),

and in this work the dynamic Smagorinsky model is used in a form described in

Sayadi & Moin (2012).

Due to the periodic boundary conditions in the streamwise direction, numerical

sponges are necessary to simulate spatially evolving ow. This sponge region allows

the ow to be recycled from outlet to inlet by forcing a return to the desired inlet

boundary layer prole. Due to the code being compressible, a numerical sponge

at the top boundary is also necessary to ensure sound and vortical waves are not

reected back into the computational domain Mani (2012). The inlet sponge region

spans from x = 0.03 to x = 0.5, whereas the outlet sponge starts at x = 8 and ends

at x = 9.2. The top sponge extends the domain from y = 1 to y = 1.8. Sponge

regions account for one third of the total number of mesh points. These sponge

layers relax the computed Navier-Stokes solution to the scale-similar compressible

boundary layer case obtained a priori as a reference solution. From y = 1 to

y = 1.4, the reference solutions wall-normal velocity is changed to the suction

prole specied in eq. (3.1). It is then smoothly brought back to its precomputed

scale-similar value from y = 1.4 to y = 1.8. The sponge relaxation parameter

increases from zero at y = 1, the end of the physical domain, and reaches its

maximum close to the end of the sponge region at y = 1.8. The suction velocity is

thus enforced indirectly through the inuence of the forced solution above the top

of the physical domain.

31

3.3.2 Spectral code in vorticity form

A pseudo-spectral incompressible Navier-Stokes solver originally developed by

Domaradzki & Metcalfe (1987) was modied and parallelized to perform LES of

laminar separation bubble ow. The derivatives in the horizontal directions are

approximated using Fourier expansions, whereas the vertical is approximated using

collocated Chebyshev polynomials. The ow variables are integrated in time using

a fractional time step method described in Orszag & Kells (1980). It solves the

ltered Navier-Stokes equations in rotational form:

uit

= [ijkujk]xi

+

xj

(uixj

+ujxi

)+ Fi

sgsijxj

, i = 1, 2, 3 (3.6)

uixi

= 0, (3.7)

where

i = ijkukxj

is the vorticity, (3.8)

= P/+ 1/2uiui is the pressure head, (3.9)

Fi = fi(x, t), is a body force, and (3.10)

sgsij = uiuj uiuj is the subgrid-scale stress tensor. (3.11)

Strong algebraic grid stretching is used in the vertical to redistribute the Chebyshev

grid points such that approximately 2/3 of the points lie inside the boundary layer at

its thickest while still resolving the inow Blasius boundary layer. The streamwise

velocity is then decomposed into a base inow prole and velocity defect as follows:

utotal = u + u, where u = [u(y), 0, 0] is known and computed to be the Blasius

32

solution at the inlet x = x0. Hence, u is the departure away from the inow prole

at each location in space that is solved for at each time step.

The non-linear term in u is computed in rotational form as in eq. (3.6). It is

advanced in time using Adams-Bashforth scheme, whereas the advection due to the

base ow u is computed separately using Crank-Nicolson to obtain better stability

and kinetic-energy-preserving characteristics as seen in eq. (3.12) and (3.13).

ui = uni +

32t

[Ni + Fi

sgsijxj

]n

12t

[Ni + Fi

sgsijxj

]n1(3.12)

ui = ui +

12uit

(uixj

+uixj

)(3.13)

where (3.14)

Ni = ijkujk uiuixj

represent the non-linear term contributions in ui.

(3.15)

The base ow advection, pressure and viscous steps are performed using the Fourier

representation of the ow variables

ui(x, y, z, t) =

|m|

The pressure step consists of a correction to ensure the resulting ow eld is

divergence-free where the pressure is treated implicitly. Used together with the

continuity equation (3.7), this results in a Poisson equation for vertical velocity

with known Dirichlet boundary conditions as shown in eq. (3.18). The scheme

of Orszag & Kells (1980) removes the need to impose explicit pressure boundary

conditions at the expense of a small numerical error near the wall which is O(t).

If a consistent boundary condition for the vertical velocity is used as in eq. (3.22),

O(t3

2 ) may be reached Guermond et al. (2006).

(D2 k2

)v = D[ikx u+ ikz w] k2 v, (3.18)

where (3.19)

D =

y, (3.20)

k2 = k2x + k2z , (3.21)

The above Poisson equation (3.18) is solved with Dirichlet boundary conditions

v|b = v|b t[D (ikx(u un) + ikz(w wn)) + k2(v vn)

]|b (3.22)

such that the pressure head and horizontal velocities can then be obtained alge-

braically as follows

= [Dv + ikx u + ikz w]/(k2t) (3.23)

u = u ikxt (3.24)

w = w ikzt (3.25)

34

The viscous terms are then treated implicitly, which results in one Helmholtz

equation for each direction in spectral space as shown in eq. (3.26).

(D2 k2

2t

)un+1i =

1t

(ui + ui) (3.26)

Numerical sponge regions are implemented in the streamwise direction using the

fringe method formulation of Spalart & Watmu (1993) to damp all turbulence

and return the outow to the desired inow Blasius prole u(y) using the body

force term (3.27):

Fi = (x)(ui ui), (3.27)

where

(x) = f(e((xx0)/l)

2

+ e((xxmax)/r)2), tends to zero outside inow and outow,

(3.28)

f = 10U0Lf

, is the forcing strength, (3.29)

l =110Lf , r =

38Lf , are the widths of sponge regions,

(3.30)

Lf =15100

(xmax x0) is the total size of the sponge regions.

(3.31)

This forcing formulation enables the simulation of a spatially evolving boundary

layer while maintaining periodic boundary conditions and spectral accuracy. The

sponge region at the inow extends from x = 0.25 to x = 0.45, whereas the sponge

at the outow spans x = 8.7 to x = 10. Fringe method forcing terms are active

35

over 15% of the total domain. Their inuence does extend somewhat beyond where

the terms are active, so to be safe the region x = 0.5 to x = 7.5 is considered to

be the physically representative region of our computational domain.

A ceiling suction boundary condition that matches Spalarts and is designed

to cause a 30% free stream ow deceleration are enforced. The resulting top

streamwise velocity is computed during the pressure step by applying the continuity

equation and imposing zero spanwise velocity. In practice, the eective deceleration

is closer to 25% due to viscous eects as reported by Spalart & Strelets (2000).

To ensure mass is conserved in the complete domain, the amount of uid removed

by suction from the top of the domain is injected in a narrow slot outside of

the region of interest (from x = 8.5 to x = 9.5). The streamwise and vertical

velocity boundary conditions are shown in gure 3.6. Blowing is applied at the

wall to avoid any stability problems associated with disturbances near the top of

the domain where the resolution is very coarse. Blowing through the plate severely

limits the CFL number used in our simulations, increasing overall computational

time. The upshot of this strict CFL restriction is that it removes any possibility

of implicit ltering due to large time steps, a phenomenon observed in previous

results using implicit schemes.

3.3.3 Spectral code in skew-symmetric form

The vorticity form of the discretized Navier-Stokes equations was shown

to be prone to aliasing errors and more sensitive to Gibbs phenomena by

Kravchenko & Moin (1997), which can lead to persistent numerical oscillations at

the smallest resolved scales in a spectral code due to the discretization methods

inherently negligible numerical dissipation. Concern that these sources of error

36

might be exacerbated at low resolution and aect the ability of explicit subgrid-

scale models to provide the correct eddy viscosity led to the development of a spec-

tral code with a higher order time integration scheme where the non-linear term is

computed in skew-symmetric form. This form is known to be less prone to alias-

ing errors and has better energy-conservation characteristics Kravchenko & Moin

(1997). The solver is based a on a classic two-step pressure-correction scheme with

a third-order backward dierence formula (BDF3) outlined in Guermond et al.

(2006), with boundary conditions used in the Zang-Hussaini algorithm outlined

in Canuto et al. (2007) that should lead to O(t3) accuracy. The ltered Navier-

Stokes equations are integrated in time using a combined non-linear viscous step

1t

0ui J1q=0

qunqi

xj

(uixj

+ujxi

)=

J1q=0

qNnqi (3.32)

with boundary conditions

u|y=0 =t0

[2p

x

n

p

x

n1]y=0

(3.33)

D2u|y=Y = 0 (3.34)

v|y=0 =t0

D2vn|y=0 (3.35)

v|y=Y = V (x) +t0

D2vn|y=Y (3.36)

w|y=0 =t0

[2p

z

n

p

z

n1]y=0

(3.37)

D2w|y=Y = 0 (3.38)

where 0, q, q are backward dierence formula of order J coecients, and where

Ni =12ujuixj

+

xj(12uiuj + sgsij ) Fi. (3.39)

37

The resulting Helmholtz equations for ui are solved in spectral space. A pressure-

correction step

0t

(un+1i u

i

)+

p

xi

n+1

= 0 (3.40)

uixi

= 0 (3.41)

is then performed to obtain divergence free elds. To avoid applying explicit

Neumann boundary conditions to the pressure, the continuity equation is used to

derive a Poisson equation for vn+1 in spectral space that is equivalent to eq. (3.18),

and subsequently obtaining the pressure and horizontal velocities algebraically

as in equations (3.23), (3.24), and (3.25). The resulting velocity elds are then

divergence free in the entire computational domain including any active sponge

regions and satisfy the boundary conditions to O(t5

2 ) Guermond et al. (2006).

This time integration scheme is stiy-stable for higher CFL numbers that the

previous vorticity-based scheme Guermond et al. (2006). This property was only

fully utilized in the initialization phase of simulations, whereas the time step was

kept to the same order of magnitude as the vorticity-form spectral code during the

time-averaging phase to ensure direct comparisons could be made.

3.4 Validation

The CTR code was validated against analytical predictions for the growth rate

of the inviscid mixing layer instability, the growth rate of Tollmien-Schlichting (T-

S) waves in a non-parallel boundary layer, the level of distortion in representing

a convecting Taylor vortex in time, and the pressure uctuations due to sound

scattered by a circular cylinder by Nagarajan (2004). It has been used in a number

38

of DNS and LES of bypass transition in Nagarajan et al. (2007) which reported

good agreement with experiments.

The pseudo-spectral incompressible code in vorticity form was previously vali-

dated against linear stability theory for parallel boundary layers (ubli = {U(z), 0, 0}

and ubci = 0) where errors of less 2% were reported Domaradzki & Metcalfe (1987)

for the T-S wavelengths tested. This validation test was performed again after

modications were made to parallelize the code and replace obscure fast Fourier

transform and linear algebra subroutines with call to standard open source libraries

like LAPACK and FFTW. Results are shown in table 3.1 were consistent and

obtained numerical growth rates matching growth rates predicted by linear stabil-

ity theory within 5% for a wide range of Reynolds numbers, domain sizes and T-S

wavelengths, discarding 4 outliers (out of 25 tests) where the growth rate predicted

was too close to zero. The same validation test was also performed for the spectral

code in skew-symmetric form, which performed slightly better, with errors less

than 2% with the exception of a few outliers as shown in table 3.2.

39

Table 3.1: Numerical (Et) vs theoretical (i) growth rates of most unstable

Orr-Sommerfeld modes in vorticity-form spectral code.

Reynolds number i Et % error 1805.00 0.500000 -0.115938E-01 -0.115268E-01 0.5776991805.00 0.675000 -0.658913E-02 -0.653586E-02 0.8083851805.00 0.850000 -0.226416E-02 -0.223016E-02 1.501711805.00 1.02500 -0.159557E-03 -0.154442E-03 3.206001805.00 1.20000 -0.108623E-02 -0.111583E-02 2.724472555.00 0.500000 -0.485721E-02 -0.481590E-02 0.8504132555.00 0.675000 -0.970083E-03 -0.935548E-03 3.560022555.00 0.850000 0.185558E-02 0.187327E-02 0.9533772555.00 1.02500 0.267053E-02 0.266628E-02 0.1588982555.00 1.20000 0.920218E-03 0.888659E-03 3.429443305.00 0.500000 -0.211583E-02 -0.208443E-02 1.484323305.00 0.675000 0.921599E-03 0.945822E-03 2.628373305.00 0.850000 0.283207E-02 0.284275E-02 0.3770463305.00 1.02500 0.293234E-02 0.292242E-02 0.3384103305.00 1.20000 0.760838E-03 0.729025E-03 4.181284055.00 0.500000 -0.793901E-03 -0.769478E-03 3.076454055.00 0.675000 0.162942E-02 0.164762E-02 1.116934055.00 0.850000 0.294678E-02 0.295130E-02 0.1533824055.00 1.02500 0.261108E-02 0.259848E-02 0.4824934055.00 1.20000 0.192827E-03 0.164394E-03 14.74554805.00 0.500000 -0.915698E-04 -0.715104E-04 21.90614805.00 0.675000 0.188049E-02 0.189410E-02 0.7239024805.00 0.850000 0.279080E-02 0.279260E-02 0.645979E-014805.00 1.02500 0.216841E-02 0.215651E-02 0.5487164805.00 1.20000 -0.414861E-03 -0.435599E-03 4.99871

40

xCp

0 0.2 0.4 0.6 0.8 1

-1.5

-1

-0.5

0

0.5

1

Jones et al.UDNSDyn. Smag.WALE

(a) NACA0012 airfoil at 5 degree angle of attack at Rec = 0.5 105 Castiglioni et al.

(2014).

1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 60

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

x

C p

(b) Flat plate with suction from the top applied at Rex = 105.

Figure 3.1: Time-average coecients of pressure for airfoils is approximated usingsuction boundary conditions for a at plate.

41

xy

1 2 3 4 5 6 70

0.1

0.2

0.3

0

0.5

1

(a) Contour plot of mean streamwise velocity.

0 0.5 10

0.05

0.1

0.15

0.2

0.25

(x=2)

y

0 0.5 10

0.05

0.1

0.15

0.2

0.25

(x=3)

y

0 0.5 10

0.05

0.1

0.15

0.2

0.25

(x=4)

y

0 0.5 10

0.05

0.1

0.15

0.2

0.25

(x=6)

y

(b) Profiles of mean streamwise velocity before separation (x = 2), after separation(x = 3), at the recirculating vortex (x = 4), and in the turbulent boundary layer (x = 6).

Figure 3.2: The anatomy of a typical laminar separation bubble over a at platetaken from large eddy simulation (LES) results using the truncated Navier-Stokesapproach at 3% of DNS resolution.

42

Figure 3.3: Instantaneous iso-surfaces of spanwise vorticity on a at plate due tosuction from the top at Rex = 105 from spectral LES results with the -model at3% of Spalart & Strelets (2000) DNS resolution.

Figure 3.4: Physical domain, boundary and inlet conditions used to investigatelaminar separation bubble ow over a at plate.

43

(a) Upwind scheme solution (b) Crank-Nicolson scheme solution

Figure 3.5: Solutions for the convection equation tT + uxT = 0 after 40 timesteps with CFL=0.8 reproduce from Fletcher (1991)

1 2 3 4 5 6 7 8 9 100.60.50.40.30.20.1

00.10.20.30.40.50.60.70.80.8

x

velo

city

Figure 3.6: Spectral LES boundary conditions. Line: top vertical velocity v(x, y =Y ); dashed line: top streamwise velocity minus unity u(x, y = Y )1; dash-dottedline: wall vertical velocity v(x, y = 0).

44

Table 3.2: Numerical (Et) vs theoretical (i) growth rates of most unstable

Orr-Sommerfeld modes in skew-symmetric spectral code.

Reynolds number i Et % error 1805.00 0.500000 -0.115938E-01 -0.115823E-01 0.994404E-011805.00 0.675000 -0.658913E-02 -0.658011E-02 0.1368151805.00 0.850000 -0.226416E-02 -0.225150E-02 0.5589281805.00 1.02500 -0.159557E-03 -0.143196E-03 10.25431805.00 1.20000 -0.108623E-02 -0.106467E-02 1.984802555.00 0.500000 -0.485721E-02 -0.485301E-02 0.864110E-012555.00 0.675000 -0.970083E-03 -0.963559E-03 0.6725202555.00 0.850000 0.185558E-02 0.186313E-02 0.4068722555.00 1.02500 0.267053E-02 0.268117E-02 0.3983652555.00 1.20000 0.920218E-03 0.933770E-03 1.472693305.00 0.500000 -0.211583E-02 -0.211210E-02 0.1766243305.00 0.675000 0.921599E-03 0.925912E-03 0.4679923305.00 0.850000 0.283207E-02 0.283767E-02 0.1978923305.00 1.02500 0.293234E-02 0.293782E-02 0.1867243305.00 1.20000 0.760838E-03 0.768801E-03 1.046654055.00 0.500000 -0.793901E-03 -0.791426E-03 0.3117954055.00 0.675000 0.162942E-02 0.163252E-02 0.1901164055.00 0.850000 0.294678E-02 0.294896E-02 0.737562E-014055.00 1.02500 0.261108E-02 0.261351E-02 0.930939E-014055.00 1.20000 0.192827E-03 0.199857E-03 3.645574805.00 0.500000 -0.915698E-04 -0.896177E-04 2.131824805.00 0.675000 0.188049E-02 0.188215E-02 0.883248E-014805.00 0.850000 0.279080E-02 0.279187E-02 0.382952E-014805.00 1.02500 0.216841E-02 0.217091E-02 0.1151364805.00 1.20000 -0.414861E-03 -0.403731E-03 2.68299

45

Chapter 4

Center for Turbulence Research

Results

Results for three cases computed using the CTR code are reported here: a

benchmark DNS case (DNS), a wall-resolved LES with the dynamic Smagorin-

sky model (LES), and an under-resolved DNS. Parameters for these simulations

are summarized in Table 4.1. Both the DNS and LES were set up and run by

a collaborator at CTR, Dr Taraneh Sayadi, whereas I set up and performed the

UDNS cases. The DNS by Spalart & Strelet (2000) was initially intended to be

the benchmark case. However, it was run using an incompressible spectral code

with an imposed vorticity-free boundary condition at the top boundary. These top

boundary conditions could not be matched exactly in simulations with the CTR

Spectral DNS CTR DNS CTR LES CTR UDNSNx 1022 1536 512 240Ny 120 300 140 90Nz 120 128 32 32Ntotal 106 14.7 59.0 2.3 0.7% of spectral DNS 100 401 15.6 4.7% of CTR DNS 25 100 3.9 1.2x+ 20 9.7 26.4 57.0y+ at X = 7Y 1 0.5 1.0 1.6z+ 6.7 7.6 27.5 29.6Seffective 0.25 0.21 0.21 0.20

Table 4.1: Resolution and parameters for all cases run with the CTR code com-pared to the spectral DNS by Spalart & Strelets (2000).

46

code due to the fringe layer formulation used. The eective top boundary condi-

tion is compared with the spectral DNS boundary condition of Spalart & Strelets

(2000) in gure 4.1; the nominal deceleration parameter Seffective = 0.21 is less

than for the spectral DNS case (see Table 4.1). Results for additional simulations

performed with a dierent numerical code are reported in Cadieux et al. (2012).

For those additional cases, however, only LES and under-resolved DNS were per-

formed because direct comparisons to the benchmark DNS by Spalart & Strelets

(2000) were made.

CTR DNS, LES, and UDNS were run until the separation bubble stabilized

and turbulent ow was well established downstream of reattachment as illustrated

in gure 4.2 and 4.3. Results were then averaged over multiple bubble breathing

periods. All time-averaged results relating to pressure and friction coecients

obtained are in good qualitative agreement with the DNS benchmark (see g-

ure4.4a and 4.4b).

The wall pressure coecients Cpw = (Pw P)/(12U20 ) shown in gure 4.4a

for the UDNS and LES cases are both in good quantitative agreement with the

DNS benchmark with the exception of a slight dierence in bubble length. The

downward slope in Cp in gure 4.4a after x = 5 indicates the existence of a slight

favorable pressure gradient which extends to the end of the physical domain.

This favorable pressure gradient is caused by blowing at the top boundary

seen in gure 4.1. This presents a limitation in the applicability of results

to the suction side of airfoils in MAVs and blades in turbo-machinery where

such persistent favorable pressure gradients are seldom encountered downstream

of the separation bubble Jones et al. (2010). Although weak, the favorable

pressure gradient may also articially improve agreement of LES and UDNS

47

1 2 3 4 5 6 7

0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

x

Nor

mal

ized

vel

ocity

Figure 4.1: Normalized wall-normal velocity top boundary condition (V/U0 atY = 1): S & S 2000 Spalart & Strelets (2000) (circles), and UDNS (dashed line).Normalized mean streamwise dierence from freestream velocity ((U U0)/U0 atY = 1): UDNS (line).

results with the DNS benchmark because of its eect on the reattachment location.

At resolutions on the order of 1% of their respective benchmark DNS, and even

without models, all simulations predict the separation point seen in DNS bench-

marks exactly. This can be observed in the rst zero-crossing on the wall skin

friction Cf = Uy |y=0/(12U20 ) plots in gure 4.4b. The UDNS predicts the same

shape and maximum value of the peak negative skin friction as the benchmark

DNS. Wall-resolved LES with dynamic Smagorinsky modeling performs slightly

worse than the UDNS run, but still reaches within 15% of the DNS peak negative

skin friction coecient value. UDNS and LES predict the location of the reattach-

ment point with less than 5% dierence with the DNS. UDNS recovers benchmark

48

Figure 4.2: DNS snapshot of iso-surfaces of vorticity: Kelvin-Helmholtz rollsare visible over the separated shear layer leading to transition to turbulence andsubsequent turbulent ow reattachment, closing of the separation bubble.

DNS results almost exactly for the turbulent Cf in the region downstream of the

bubble whereas LES results underpredict the skin friction in that region.

4.1 Numerical Dissipation

The good quantitative agreement between the no-model highly under-resolved

DNS and benchmark DNS results suggests that the code used may belong to a

category of implicit LES (ILES) where the numerical dissipation plays the role of

subgrid-scale models. As is evident in the results presented in gures 4.4a and 4.4b,

the addition of a subgrid-scale model, even when coupled with higher resolution,

visibly worsens agreement with the DNS benchmark compared to the no-model

case. Such behavior is expected for codes that already provide enough dissipation

49

xy

1 2 3 4 5 6 70

0.2

0.4

0

0.5

1

Figure 4.3: Contour plot of normalized average streamwise velocity U/U0 fromthe UDNS case. Notice the laminar boundary layer growth followed by a clearseparation bubble spanning from x = 2.8 to x 4.6.

through their numerics so that additional explicit subgrid-scale dissipation is not

required.

The code used has two primary sources of numerical dissipation: truncation

error in derivative approximations, and explicit ltering. Since the code uses sixth

order compact nite dierences and is claimed to be conservative Nagarajan et al.

(2003), focus is placed on quantifying the amount of eective numerical viscosity

introduced by the explicit ltering at each time step Nagarajan (2004). The explicit

high wave number ltering is based on the formulation of a sixth order compact

lter Lele (1992). It is used to remove spurious and unstable high frequency

oscillations that may develop at the interface of the implicitly and explicitly treated

regions due to the codes use of high order nite dierences Nagarajan (2004). This

is done to stabilize the code, and to ensure the implicit and explicit grid solutions

match at their interface. It replaces the use of articial viscosity or newer weighted

essentially non-oscillatory (WENO) type schemes, as well as penalty-type methods

used to stabilize physical or numerical interfaces. Numerical dissipation from this

ltering operation is quantied using two dierent methods.

50

1 2 3 4 5 6 70

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

x

C pw

(a) Coefficient of pressure at the wall.

1 2 3 4 5 6 7

2

1

0

1

2

3

4

5

6 x 103

x

C f

(b) Wall coefficient of friction.

Figure 4.4: Time-averaged Cp and Cf . CTR DNS (circles), CTR LES withdynamic Smagorinsky model (line), and CTR UDNS (dashed line).

51

4.1.1 Estimating Numerical Dissipation due to Filtering

The amount of eective viscosity the ltering operation imparts to the sim-

ulation is estimated by comparing the energy decay rates of runs with ltering

and without ltering. The number of time steps in such an analysis is limited

to 10 to ensure that no numerical instabilities develop. First, two runs are per-

formed using the same value of molecular viscosity, one with ltering and the

other without. Second, the run without ltering is then repeated several times

with larger values of the molecular viscosity until its energy decay curve matches

that of the ltered case. The excess of the molecular viscosity in a run for

which the best match is achieved provides an estimate of the eective viscosity

that can be attributed to the ltering operation Diamessis et al. (2008). Since

this approach was developed and validated for wakes and isotropic turbulence by

Domaradzki et al. (2003); Domaradzki & Radhakrishnan (2005); Bogey & Bailly