Modeling Turbulent Boundary Layers with Elliptic Relaxation

98 Pages·2011·1.49 MB·English

Modeling Turbulent Boundary Layers
with
Elliptic Relaxation
by
Jørgen Myre
THESIS
for the degree of
MASTER OF SCIENCE
(Master i Anvendt matematikk og mekanikk)
Faculty of Mathematics and Natural Sciences
University of Oslo
May 2011
Det matematisk- naturvitenskapelige fakultet
Universitetet i Oslo

Modeling Turbulent Boundary Layers
with
Elliptic Relaxation
by
Jørgen Myre
THESIS
for the degree of
MASTER OF SCIENCE
(Master i Anvendt matematikk og mekanikk)
Faculty of Mathematics and Natural Sciences
University of Oslo
May 2011
Det matematisk- naturvitenskapelige fakultet
Universitetet i Oslo
Abstract
ThetheorybehindtheRANSequationsandcommonDRSMmodelingterms
ispresented. MostcurrentDRSMturbulencemodelshavemodelsforthere-
distribution tensor which are developed for homogeneous flow. The Elliptic
Relaxation model is presented, which is an attempt to make homogeneous
redistribution models behave better close to walls. This model is then im-
plemented in the CBC.RANS framework, which uses the Finite Element
Method. Results are analyzed with a focus on stability. Further investiga-
tions and improvements are proposed based on successes and failures.

Acknowledgments
Firstly, I would thank my supervisor Bjørn Anders Reif, who provided the
framework upon which this work stands.
MikaelMortensendeservesmorepraisethanthereisspaceforhere. Most
of all for his neverending patience in the face of unreasonable requests and
incomprehensible code, but also for his willingness to provide good advice
and tips whenever I would interrupt his work day. Without his help, this
thesis would have ended at chapter 3.
With this thesis I complete an education that has been my driving pur-
pose for the past 8 years, and it would have been impossible without the
support of my family. They have been helpful in countless ways, and I hope
the work I’ve done lives up to their expectations.
But my most heartfelt thanks goes to my closest friends and our role-
playing games. Our travels through imaginary worlds have kept me sane
through the trials of this all-too-real one. A special mention goes to Nils
Ødeg˚arden,forhissubsidyofmyotherwisemeagerfoodbudgetandhislevel
perspective on life during these past years.
For all these things, and more, I am deeply grateful.

Nomenclature
Notation and physical variables
v vector, vector notation
e standard basis vector, i’th direction
i
(1 = x, 2 = y, 3 = z), vector notation
A second order tensor, vector notation
v vector, index notation
i
A second order tensor, index notation
ij
M fourth order tensor, index notation
ijkl
a (Reynolds stress) Anisotropy tensor, index notation (dimensionless)
ij
A (Reynolds stress) Anisotropy tensor, vector notation (dimensionless)
f Model tensor for redistribution, index notation (s−1)
ij
F Model tensor for redistribution, vector notation (s−1)
k Turbulent kinetic energy (m2/s2)
L Model length scale (m)
P˜ Total pressure field (kg/[ms2])
P Mean pressure field (kg/[ms2])
p Fluctuating pressure field (kg/[ms2])
P Production of Reynolds stresses, index notation (m2/s3)
ij
P Production of Reynolds stresses, vector notation (m2/s3)
R Reynolds stresses, vector notation (m2/s2)
S Rate of strain tensor, index notation (s−1)
ij
S Rate of strain tensor, vector notation (s−1)
T Model time scale (s)
T Turbulent transport of Reynolds stresses, index notation (m2/s3)
ij
U˜ Total velocity field, index notation (m/s)
i
U Mean velocity field, index notation (m/s)
i
U, V, W Mean velocity field, individual components (m/s)
u Fluctuating velocity field, index notation (m/s)
i
u, v, w Fluctuating velocity field, individual components (m/s)
u Mean velocity field, vector notation (m/s)
u u Reynolds stresses, index notation (m2/s2)
i j
u2, v2, uv Reynolds stresses, individual components (m2/s2)
W Rate of rotation tensor, index notation (s−1)
ij
W Rate of rotation tensor, vector notation (s−1)
(cid:15) Diffusion of turbulent kinetic energy (m2/s3)
(cid:15) Diffusion of Reynolds stresses (m2/s3)
ij
φ Redistribution tensor, index notation (m2/s3)
ij
φh Homogeneous model redistribution tensor, index notation (m2/s3)
ij
φh Homogeneous model redistribution tensor, vector notation (m2/s3)
Model Constants and numerical quantities
C SSG, 3.4
1
C∗ SSG, 1.8
1
C SSG, 4.2
2
C SSG, 0.8
3
C∗ SSG, 1.3
3
C SSG, 1.25
4
C SSG, 0.4
5
C Length scale, 0.25
L
C LRR-IP production term, 0.6
P
C LRR-IP, anisotropy term, 1.8
R
C∗ (cid:15)-equation production, 1.44
(cid:15)1
C∗ (cid:15)-equation dissipation, 1.9 or function
(cid:15)2
C Nonconstant C∗ -equation, 0.045
(cid:15)d (cid:15)1
C Length scale, 80
η
C Turbulent transport, Daly-Harrow model, 0.22
µ
C Stabilizing eddy viscosity ν -term, 0.09
ν T
e (cid:15)-linearization in k-equation, = 0.5
d
σ (cid:15)-equation, scaling of Laplacian, 1.3
(cid:15)
θ Relaxation parameter for Picard Iteration
E Matrix in nonlinear system of equations
E∗ Linearized matrix for nonlinear system
f Vector in nonlinear system of equations
f∗ Linearized vector for nonlinear system
r Residual of nonlinear system of equations
x Vector solution of nonlinear system of equations
xn Vector holding numerical solution, n’th iteration
Contents
1 Introduction 9
1.1 What is turbulence? . . . . . . . . . . . . . . . . . . . . . . . 9
1.2 The Energy Cascade and modeling . . . . . . . . . . . . . . . 10
1.3 Boundary layers and their importance . . . . . . . . . . . . . 12
1.4 The Elliptic nature of pressure and near-wall effects . . . . . 14
1.5 The Elliptic Relaxation model . . . . . . . . . . . . . . . . . . 14
1.6 Computational Approach . . . . . . . . . . . . . . . . . . . . 14
1.7 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.7.1 Vector Notation . . . . . . . . . . . . . . . . . . . . . 15
1.7.2 Outer products . . . . . . . . . . . . . . . . . . . . . . 15
1.7.3 Index Notation . . . . . . . . . . . . . . . . . . . . . . 15
1.7.4 Derivatives . . . . . . . . . . . . . . . . . . . . . . . . 16
1.7.5 An Apology . . . . . . . . . . . . . . . . . . . . . . . . 16
1.8 The author’s ’we’ . . . . . . . . . . . . . . . . . . . . . . . . . 17
2 Navier-Stokes and RANS/DRSM 19
2.1 The Navier-Stokes Equations . . . . . . . . . . . . . . . . . . 19
2.2 Averages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.3 Reynold-Averaged Navier-Stokes . . . . . . . . . . . . . . . . 20
2.4 The Reynolds stresses . . . . . . . . . . . . . . . . . . . . . . 21
2.5 Modeling unknown terms . . . . . . . . . . . . . . . . . . . . 23
2.6 The near-wall production of kinetic energy . . . . . . . . . . . 24
2.7 Modeling (cid:15) and turbulent transport . . . . . . . . . . . . . . . 25
2.8 The tensors a , S , W and scales . . . . . . . . . . . . . . . 27
ij ij ij
2.9 The magical mystical φ . . . . . . . . . . . . . . . . . . . . . 28
ij
3 Elliptic Relaxation 31
3.1 The non-local nature of pressure effects . . . . . . . . . . . . 31
3.2 Elliptic pressure and near-wall effects . . . . . . . . . . . . . . 31
3.3 Derivation of the Elliptic Relaxation Model . . . . . . . . . . 32
3.4 Further modifications . . . . . . . . . . . . . . . . . . . . . . 33
3.5 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . 34
3.6 The final model . . . . . . . . . . . . . . . . . . . . . . . . . . 36
5