Fluid Mechanics and Its Applications
László Könözsy
A New Hypothesis
on the Anisotropic
Reynolds Stress Tensor
for Turbulent Flows
Volume II: Practical Implementation
and Applications of an Anisotropic
Hybrid k-omega Shear-Stress Transport/
Stochastic Turbulence Model
Fluid Mechanics and Its Applications
Volume 125
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László Könözsy
A New Hypothesis
on the Anisotropic Reynolds
Stress Tensor for Turbulent
Flows
Volume II: Practical Implementation
and Applications of an Anisotropic Hybrid
k-omega Shear-Stress Transport/Stochastic
Turbulence Model
123
László Könözsy
Centre for Computational
Engineering Sciences
Cranfield University
Cranfield, Bedfordshire, UK
ISSN 0926-5112 ISSN 2215-0056 (electronic)
Fluid Mechanics and Its Applications
ISBN 978-3-030-60602-2 ISBN 978-3-030-60603-9 (eBook)
https://doi.org/10.1007/978-3-030-60603-9
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I dedicate this book to the memory of my
father—who was a mechanical engineer and
a mathematician—who passed away during
that period of time when I was working on
this manuscript. I will never forget our
fruitful discussions on scientific questions and
on the meaning of life.
Preface
The subject of this self-contained book is interdisciplinary which encompasses
mathematics, physics, computer programming, analytical solutions and numerical
modelling, industrial Computational Fluid Dynamics (CFD), academic benchm ark
problems and engineering applications in conjunction with the research field of
anisotropic turbulence. In other words, this book focuses on theoretical approac hes,
computational examples and numerical simulations, including computer program-
ming techniques, to demonstrate the strength of a new hypothesis and anisotropic
turbulence modelling approach for academic benchmark problems and industrially
relevant engineering applications. The reader can learn quickly how to use a new
turbulence model in engineering practice to obtain accurate and reliable numerical
results for elementary and advanced turbu lent flow problems where the physics of
anisotropic turbulence is indispensable. This book contains MATLAB codes and C
programming language-based User-Defined Function (UDF) codes which can be
compiled in the ANSYS-FLUENT environment. The computer codes help to
understand and use efficiently a new concept which can also be implemented in any
other soft ware packages. The simulation results are compared to classical analytical
solutions and experimental data taken from the literature. A particular attention is
paid to how to obtain accurate results within a reasonable computational time for
wide range of benchmark problems. The provided examples and programming
techniques help to graduate and postgraduate students, engineers and researchers to
further develop their technical skills and knowledge that have posed a challenge to
many people in the past and present.
The content of Chap. 1 returns to the original roots of turbulence modelling,
which, thus, intentionally goes back in time. For graduate and postgraduate stu-
dents, the understanding of classical analytical approaches to turbulence is indis-
pensable, because the current state-of- the-art turbulence models would not exist
without them. Osb orne Reynolds (1883) [17] observed the transition from laminar
to turbulent flow in a pipe. Thus, turbulent flows in long horizontal circular smooth
pipes are one of the most well-known and experimentally studied problems in fluid
mechanics. Therefore, in addition to external flows, axisymmetric turbu lent shear
flows in horizontal circular pipes and three-dimensional turbulent flows in a
vii
cylindrical pipe are investigated in Chaps. 4 and 5, respectively. To compare
simulation results of the anisotropic hybri d k-x SST/STM closure model [10] in
Chaps. 4 and 5 with classical analytical solutions for internal flows, the first chapter
focuses on the analytical solutions of the simplified Reynolds momentum equation
for wall-bounded turbulent flows. The introductory chapter is devoted to demystify
the strength and weaknesses of classical analytical solutions including even ele-
mentary intermediate mathematical derivation steps in the investigations which
cannot be found in most textbooks. It means that the full step-by-step mathematical
derivations of classical analytical solutions for the simplified Reynolds equation are
discussed in depth in which cases the hydrodynamic and hydraulic aspects of the
fluid flow are considered together. The deriv ed analytical solutions are relying on
the momentum transfer theory of Prandtl (1925) [15] by employing a first-, second-
and third-order turbulent length-scale function proposed by Prandtl (1933) [16], von
Kármán (1930) [9] and Czibere (2001) [5], respectively. Using the Newton–
Raphson iterative method, the numerical solution of the implicit formula for the
resistance coefficient for smooth circular pipes is explained in detail. For educa-
tional purposes, the analytically derived fully develo ped dimensionless turbulent
mean velocity profiles have been implemented in a MATLAB code (see Appendix
B). The simplified analytical solutions derived in the first chapte r can be used for
code verification and turbulence model validation purposes in an initial stage of a
model development and computer code implementation (see Chaps. 4 and 5). It is
important to note that these classical analytical approaches can predict the physi-
cally correct turbulent shear-stress distribution; however, these zeroth-order (alge-
braic) turbulence models are not capable of predicting the normal Reynolds
stresses. Therefore, the reader must keep in mind that real turbulent flows are
always three dimensional, unsteady and anisotropic.
In Chap. 2, a brief summary of the improved mathematical formulation of the
anisotropic hybrid k-x Sh ear-Stress Transport (SST)/ Stochastic Turbulence Model
(STM) related to a new hypothesis on the Reynolds stress tensor (2.41) proposed in
the first volume of this book [10] is presented. The aim of the second chapter is to
bring closer a novel anisotropic turbulence modelling approac h to the wider
audience. The mathematical formulation of the aniso tropic hybrid k-x SST/STM
closure model [10]—in conjunction with the anisotropic Reynolds stress tensor
(2.41)—is relying on the unification of the generalised Boussinesq hypothesis
(deformation theory) [1, 8] and the three-dimensional similarity theory of turbulent
velocity fluctuations [6, 7]. In other words, the anisotropic hybrid k-x SST/STM
closure model assumes that the anisotropic Reynolds stress tensor
s
RA
is related to
a) the product of twice the dynamic eddy viscosity coefficient and the mean
rate-of-strain (defor mation) tensor 2l
t
S; b) the anisotropic distrib ution of the
dominant turbu lent Reynolds shear stress l
H
HG and c) the turbulent kinetic energy
k term 2=3ðÞqk
I
"#
. To support the reader to understand quickly the computer
code implementation of the new hybrid model in Chap. 3, all differences and
viii Preface
similarities between the anisotropic hybrid k-x SST/STM approac h [10] and the
k-x SST model of Menter [12, 13] are highlighted in Chap. 2. Those derivations
and equations are summarised and discussed briefly which are relevant to com-
putational purposes.
The content of Chap. 3 focuses on the computer code implementation of the
anisotropic hybrid k-x Shear-Stress Transport/Stochastic Turbulence Model
(SST/STM) including computer programming aspects. Since graduate and post-
graduate students do not necessarily have extensive experience in anisotropic tur-
bulence modelli ng, including how to implement a new anisotropic engineering
turbulence model in an in-house, open-source or comm ercial software package, the
third chapter has a unique feature from this point of view. In the first part of Chap. 3,
the numerical computation of the elements of the symmetrical anisotropic similarity
tensor
H (2.4) has been discussed through a modified version of the Stochastic
Turbulence Model (STM) of Czibere [6, 7]. The importance of the anisotropic
similarity tensor
H (2.4) and its modified deviatoric part H
H
(2.30) is to provide
physically correct model constants to describe the mechanically similar local velocity
fluctuations (2.1) related to the new anisotropic Reynolds stress tensor (2.41)
(see Chap. 2). As a practical approach, the computation of the scalar elements of the
symmetrical anisotropic similarity tensor
H (2.4) is explained b y the implementation
of an example MATLAB code. The difference between the implemented STM and
the original STM of Czibere [6, 7] is that the Bradshaw constant a
1
[2, 3, 12] has been
considered in the modified STM for the convergence criterion instead of the von
Kármán constant j. In the second part of Chap. 3, the implementation of the aniso-
tropic hybrid k-x SST/STM closure model [10] has been described through a
C programming language-based User-Defined Functions (UDF) code which can be
compiled in the ANSYS-FLUENT environment. This software package is widely
used for teaching engineering turbulence modelling, because already existing clas-
sical and advanced turbulence modelling approaches are available in this software
package. Furthermore, it is possible to implement new anisotropic turbulence models
through UDF C codes. However, a step-by-step guidance, tutorial and UDF C
computer code examples are still missing from the literature to support the reader to
understand quickly how to implement a new anisotropic turbulence model through
UDF C functions in the ANSYS-FLUENT environment. Therefore, the second part
of Chap. 3 is intended to explain the UDF C code of the new anisotropic hybrid k-x
SST/STM closure model [10] including the discussion on computer programming
techniques to teach how to implement a new anisotropic turbulence modelling
approach in the ANSYS-FLUENT environment. The additional source terms of the
anisotropic hybrid k-x SST/STM closure model [10] are added to k-x SST model of
Menter [12, 13, 14] in conjunction with the scalar momentum equations and the
additional production terms of the turbulent kinetic energy k and specific dissipation
rate x transport equations, respectively. The reader can gain experience how to use
existing turbulence models and learn how to implement a new anisotropic turbulence
model which can be implemented in any other software packages. Since there is no
practical guide currently available in the literature about how to implement
Preface ix
technically an anisotropic turbulence model throu gh UDF C functions in the
ANSYS-FLUENT environment, the objective of Chap. 3 is to fi ll the knowledge gap
in this subject area.
In Chap. 4, two-dimensional computational examples are presented using the
anisotropic hybrid k-x Shear-Stress Transport/Stochastic Turbulence Model
(SST/STM) [10]. It is important to highlight that the physics of turbulence is always
a three-dimensional phenomenon and strictly speaking two-dimensional turbulence
does not exist in nature at all. However, when a Reynolds-Averaged Navier–Stokes
(RANS) engineering turbulence model is under development, its validation for
two-dimensional well-established classical benchmark problems is still recom-
mended and is in the centre of interest. Therefore, numerical simulations have been
performed for turbulent flows (a) over a flat plate with zero pressure gradient
(Klebanoff problem), (b) over a NACA 0012 airfoil (NASA test problem), (c) in
axisymmetric straight smooth circular pipes at eight different Reynolds numbe rs,
(d) in a rotationally symmetric coaxial curved duct and (e) over a plane
backward-facing step, respectively. The description of these two-dimensional
numerical simulations can be considered as tutorials for solving simple anisotropic
turbulent flow benchmark problems with the use of the ANSYS-FLUENT R19.1
software package. In Chap. 4, all simulation results have bee n compared to
experimental data taken from the literature. For two-dimensional flows, the
dimensionless scalar weight parameter l
H
ð0\ l
H
jj
1Þ and the anisotropic scale
factors k
H
11
and k
H
22
in the modified deviatoric similarity tensor H
H
(3.99) of the
anisotropic hybrid k-x SST/STM closure model [10] are calibrated and validated
related to meas urements. The sim ulation results indicate that the physical behaviour
of the turbulent mean flow is captured correctly with the isotropic k-x SST model
of Menter [12, 13, 14] and the anisotropic hybrid k-x SST/STM closure model [10]
as well. However, further improvements can be achieved wi th the anisotropic
approach in terms of predicting the physically correct Reynolds stress distributions
in the framework of RANS turbulence modelling. The objective of the fourth
chapter is to show the capabilities of the anisotropic approach. Therefore, 156
simulation results are presented and analysed in Chap. 4.
In the short closing Chap. 5, three-dimensional simulation resul ts have been
presented with the anisotropic hybrid k-x Shear-Stress Transport/Stochastic
Turbulence Model (SST/STM) [10] to make an attempt and a first step towards
more complex real-world engineering applications. Therefore, the objective of the
fifth chapter is to shed light on possible research areas where further improvements
can be made as a future work on the new anisotropic turbulence modelling
approach investigated in this book. Numerical simulations have been performed for
three-dimensional turbulent flows (a) in a horizontal smooth cylindrical pipe,
(b) over a NACA 0013 wing and (c) over the Jetstream 31 aircraft, respectively. In
Chap. 5, all simulation results have been compared to the k-x SST model [12, 13,
14] and experimental data taken from the literature. The dimensionless scalar
weight parameter l
H
ð0\ l
H
jj
1Þ and the anisotropic scale factors k
H
11
, k
H
22
and k
H
33
in the modified deviatoric similarity tensor H
H
(3.99) have been calibrated and
x Preface
