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An Alternate eddy viscosity formulation based on anisotropy

using fluent User Defined Function

B.Tech Project Report submitted in partial fulfillment for the award of degree of

Bachelor of Technology

in

Ocean Engineering and Naval Architecture

Submitted By:

MAYANK RUNTHALA

10NA10017

Under the guidance of:

Prof. Hari V Warrior Department of Ocean Engineering and Naval Architecture

Indian Institute of Technology, Kharagpur, India

July’2013 –May’2014.

Department of Ocean Engineering and Naval Architecture Indian Institute of Technology, Kharagpur, India

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ACKNOWLEGMENT

I am heartily thankful to Prof. Hari V Warrior who amidst his busy schedule spared his

valuable time to guide me through the project. Without his support, motivation and

useful suggestions, this project would not have taken its present shape. I feel proud

and honored to be a student of such personality.

I also want to express my sincere gratitude and sincere thanks to all the M.

Tech scholars who helped with their valuable inputs towards completing the project.

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Abstract

An Alternate eddy viscosity formulation based on anisotropy using fluent User Defined Function

Turbulence is a flow regime characterized by chaotic property changes. Lots of

research has been done to try and understand it and come up with a mathematical

form that can replicate it to some level of accuracy.

This project has been my attempt to formulate a code for an alternate eddy viscosity

model based on anisotropy using a two equation model, as the existing two equation

models does not incorporate turbulence anisotropy in their codes. To script the code I

have used the user defined function of an established coding tool for fluid dynamics,

FLUENT ®.A user defined function, or UDF is a function that you program, that can

be dynamically loaded with the FLUENT solver to enhance the standard features of the

code. UDFS are written in C programming language.

Turbulence has been called the last unsolved problem in classical mechanics and in it,

a complete closure for eddy viscosity (and diffusivity) is the most daunting feature.

Many types of turbulence models have been formulated starting from mixing length

hypothesis going up to the most complicated Direct Numerical Simulations. Hence a

lot of research has been carried out in this topic and is the motivation behind the present

thesis work. In this thesis, we provide a solution for eddy viscosity along the lines of

Reynolds stress anisotropy. A transport equation for second invariant of anisotropy (II)

is developed which takes into account a new and improved model for the slow pressure

strain rate. The new formulation for the slow pressure strain rate uses the anisotropy of

the dissipation tensor which is not negligible compared to the anisotropy of the

turbulent kinetic energy. The improved slow pressure strain rate model performs well

when compared with the established Reynolds stress model. The formulation so

implemented is advantageous in that it is simpler and more complete than the existing

eddy viscosity models and retains the accuracy of the existing Mellor Yamada two

equation model.

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CONTENTS

1. INTRODUCTION ……………………………………………………..……………………………...….. 5

2. LITERATURE REVIEW………………………………………………..…………………………..… 11

2.1 Overview of Turbulence Modeling……………………………………………………... 11

3. Eddy Viscosity Models………………………………………………...…………….………….…. 17

3.1 Introduction………………………………………………………………………………….….…17

3.2 Anisotropy in Turbulence………………………………………….……………….…….….18

3.3 Eddy Viscosity Formulation……………………………………………………….….…….19

4. Fluent ………………….……………………………………………………………………………….…24

4.1 Governing Equations………………………………………………………………………..…24

4.2 The Universal Law of the Wall……………………………………………………….........26

4.3 Reynolds Stress Tensor....................................................................................................26

4.4 FLUENT UDF.........................................................................................................................27

5. Result & Conclusion…. …………………………………………………….………….…………. 33

5.1 k-ε model results...............................................................................................................33

5.2 Anisotropic UDF model Results..................................................................................34

6. REFERENCES ………………………………………………………………………………………….. 35

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Introduction

1.1 Background of Turbulence There are many opportunities to observe turbulent flows in our everyday surroundings,

whether it be smoke from a chimney, water in a river or waterfall, or the buffeting of a

strong wind. In observing a waterfall, we immediately think that the flow is unsteady,

irregular, seemingly random and chaotic, and surely the motion of every droplet or eddy

is unpredictable. In the plume seen by a solid rocket motor turbulent motions of many

scales can be observed, from eddies and bulges comparable in size to the width of the

plume, to the smallest scales that the camera can resolve. The features mentioned in

these two examples are common to all turbulent flows.

Study of this natural phenomenon is one of the most complicated and exciting fields of

research that raises many issues and this is a key feature in a large number of application

fields, ranging from engineering to geophysics to astrophysics. It is still a dominant

research topic in fluid dynamics, and several conceptual tools developed in the

framework of turbulence analysis have been applied in other fields dealing with

nonlinear chaotic phenomena (e.g. non-linear optics, non-linear acoustics etc.).

The scientific study of turbulence did not begin until late in the nineteenth century. The

first substantial step was the publication in 1883 of the paper by Osborne Reynolds. He

described how a smooth flow of water through long circular tubes with diameters

ranging from about 0.6 to 2.5 cm is disrupted and cannot be sustained when the mean

speed of flow, U, exceeds a value that is related to the tube diameter, d and to the

viscosity of water. In his laboratory experiments Reynolds introduced a thin line of dye

into the water entering through one end of the horizontal tube from a large tank of

stationary water. He described his observations as follows;

When the velocities were sufficiently low, the streak of color extended in a beautiful

straight line through the tube. But as the velocity was increased by small stages, at

some point in the tube, always at a considerable distance from the intake, the color

band would all at once mix up with the surrounding water; and fill the tube with a mass

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Of colored water. On viewing the tube by light of an electric spark, the mass of color

itself resolved itself into a mass of more or less distinct curls, showing eddies.

Reynolds’ remarkable experiments show that the laminar flow, the smooth flow

through the tube at low flow speeds breaks down into a random eddying turbulent

motion at higher speeds when a non-dimensional number now known as Reynolds

number

Re= ��

�

exceeds a value of 1.3 × 10�.

Despite more than a century of research and a number of important insights, a complete

understanding of turbulence remains elusive, as witnessed by a lack of fully satisfactory

theories of such basic aspects such as transition and the Kolmogorov spectrum. Most

of the problems raised in turbulence are often explained based on theories and models

that introduce closure issues and are supported by more and more numerical

experiments.

Turbulence is arguably the most challenging area in fluid dynamics and the most

limiting factor in accurate computer simulation of engineering flows. It constitutes a

classical multi-scale problem, which is far beyond human intuitive understanding and

beyond resolution capabilities of even the most powerful modern parallel computers.

Turbulence has been described by Nobel-prize winning physicist Richard Feynman as

the “most important unresolved problem in classical physics.” An even more

pronounced quote is associated to Werner Heisenberg: “When I meet God, I am going

to ask him two questions: Why relativity? And why turbulence? I really believe he will

have an answer for the first.”

From a more pragmatic standpoint, however, one could argue that a complete

understanding of turbulence is not required (and there is actually no indication that

humans can comprehend complex nonlinear problems), but a sufficiently accurate

solution of the underlying equations (better, a general method for achieving those)

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would suffice. Such numerical methods exist and allow a direct numerical simulation

(DNS) of Navier–Stokes equations for all turbulence scales in space and time.

However, due to the inherent scaling laws of turbulence, DNS can be applied only to

very low Reynolds (Re) numbers and very simple and limited geometries. The

numerical effort for DNS scales with Re3, and with technical Re numbers in the range of

104 to 109, practically no numerical solution for flows of interest to engineers can be

obtained. Turbulence modeling is the attempt to develop approximate formulations that,

despite our incomplete understanding and limited computational resources, allow

engineers to obtain approximate solutions for their pressing technological applications.

When dealing with turbulence models, keep in mind that they often need to bridge a

gap of many orders of magnitude in computing power relative to DNS. The order of

CPU reduction of RANS methods relative to DNS for technical applications is

astronomic (easily reaching 1010 and more) — such models, therefore, are not simply

“models,” but they alter and redefine the equations solved. Industrial users are often

disturbed by RANS-related inaccuracies in their CFD solution relative to data.

Considering the above challenge, such differences cannot always be avoided. Finally,

not all differences between numerical results and experimental data are automatically a

result of turbulence modeling; there are many other sources of error that should be

considered before casting a judgment on a model.

Industrial CFD codes have to cover a wide range of applications from aerodynamics to

internal flows, flows with heat and mass transfer to inherently unsteady applications.

There are numerous areas of turbulence interaction with other physical effects, like

combustion or acoustics. Obviously, no single model (or modeling approach) can cover

all such applications, and numerous modeling concepts need to be developed. On the

other hand, it would not be appropriate to simply program an indiscriminate number of

models into industrial CFD codes, as it would have severe negative effects on the user

community. The first is that most models published are not “industrial-strength,”

meaning they have severe weaknesses that prevent efficient use for complex problems.

The most prominent limitation is numerical robustness, as many new models are

applied only to generic test cases with relatively simple geometries and high-quality

grids. When faced with less-optimal conditions, they often pose severe challenges to

the solver and, in many cases, lead to numerical instabilities. Most models based on the

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low-Re ε-equations fall into this category. Furthermore, models often feature many

complex nonlinear terms, which are designed to resolve a very specific problem in

generic tests, but the side effects of such terms in complex flows are not sufficiently

considered. From a coding standpoint, it is desirable to limit the number of models, as

otherwise implementation quality will suffer and the user cannot rely on the correct and

optimal programming of equations. Support teams need to understand turbulence

models to ensure proper response to customer problems in a short timeframe. An

oversupply of models severely limits the engineering team’s ability to properly help

customers and supply best-practice advice. Finally, the different models and sub-

models have to work properly in combination (for example, turbulence and transition,

etc.). For these reasons, it is necessary to provide a limited number of preselected

models that are:

• Well understood and tested

• Correctly implemented and documented

• Accurate for certain classes of flows

• Robust even for non-optimal grids

• Interoperable with other models and sub-models

• Supported by test cases

In both engineering and academia the most frequent employed turbulence models are

the Eddy-Viscosity-Models (EVMs). Although the rapidly increasing computer power

in the last decades, the simplistic EVMs still dominate the CFD community.

The landmark model is the k-ε model of Jones and Launder which appeared in 1972.

This model has been followed by numerous EVMs, most of them based on the equation

and an additional transport equation, such as the k-ω, models. With the emerging Direct

Numerical Simulations (DNSs), it has now been possible to improve the EVMs,

especially their near-wall accuracy, to a level not achievable using only experimental

data. The first accurate DNS was made by Kim et al. albeit at a low Reynolds number

Re = 180 and for a simple fully developed Channel flow test-case. Today however,

DNS's are made at both interesting high Reynolds numbers, and of more complex flows,

enabling accurate and advanced EVMs to appear.

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This paper will try to explain these newly developed EVMs which are based on DNS-

data. A number of turbulence models are compared with both DNS-data and

experimental data for different flows. Particularly interesting is how these newer

models compare to the older, non-DNS tuned EVMs. The majorities of the different

ideas when modifying/tuning turbulence models, such as damping functions, boundary

conditions, etc. are included.

Although there is neither any hope nor intention to including all two-equation EVMs,

quite a number of them are tabulated and referenced.

ES Time 3D Aniso Trans

DNS Y Y Y Y Y

LES Y Y Y Y Y

RANS-RSM N N N Y Y

RANS-EVM N N N N Y

RANS-Algebraic N N N N N

Table 1: Turbulence models and physics. ES: The ability to predict the Energy-Spectrum, Time: whether

or not the computation is time accurate, 3D: if a 3D solution is required, Aniso: if the model predicts

anisotropic Reynolds stresses, Trans: if turbulence is a transported or local quantity.

1.2 Motive behind the Project The state-of-art numerical ocean models like solves for turbulent quantities by solving

for the eddy viscosity (defined as the rate of mixing) through two equation models of

Mellor and Yamada (1982) or Canuto et al (2001). Full-fledged details of the terms

used in this section are given later. They use stability function based method to

determine the turbulent viscosity after solving for the turbulent quantities. This involves

computation of complex forms of stability functions. In the present thesis the author

deviates from this usual method and implement a formulation for eddy viscosity which

is based on Reynolds stress anisotropy. The II equation or the second invariant of

anisotropy, takes into account an improved model for the slow pressure strain rate.

The new model developed for the slow pressure strain rate uses the anisotropy of the

dissipation tensor which is not negligible as compared to the anisotropy of the turbulent

Reynolds stress tensor. The improved slow pressure strain rate model performs well as

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compared to the established Reynolds stress model of Speziale, Sarkar and Gatski

(abbreviated as SSG model) and in some of the cases it better reflects the experiments

than the SSG model. The slow pressure strain rate is then utilized to modify the II

equation model developed by Maity et al (2011). The II-equation model so developed

has been implemented in POM by replacing the Mellor-Yamada scheme with the new

formulation. Though the results do not show any significant progress than the existing

results the formulation so implemented is advantageous in that it is simpler and more

complete than the MY model where the return to isotropy is assumed to be

instantaneous.

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Literature Review

2.1. Overview of Turbulence Modeling The primary emphasis in this dissertation is upon the time-averaged Navier-Stokes

equation. The origin of this approach dates back to the end of the nineteenth century

when Reynolds (1895) published results of his research on turbulence. His pioneering

work proved to have such profound importance for all future developments based on

the concept of Reynolds averaging.

The earliest attempts at developing a mathematical description of turbulent stress

sought to mimic the molecular gradient diffusion process. In the spirit, Boussinesq

(1877) introduced the concept of a so-called eddy viscosity. As with Reynolds,

Boussinesq has been immortalized in turbulence literature. The Boussinesq eddy-

viscosity approximation is so widely known that few authors find a need to reference

his original paper.

Reynolds or Boussinesq did not attempt a solution of the Reynolds-averaged Navier-

Stokes equation in any systematic manner. Much of the physics of viscous flows was a

mystery in the nineteenth century until Prandtl’s discovery of the boundary layer in the

year 1904. Focusing upon turbulent flows, Prandtl (1925) introduced the mixing length

(an analog of the mean free path of a gas) and a straightforward prescription for

computing the eddy viscosity in terms of the mixing length. The mixing-length

hypothesis, closely related to the eddy-viscosity concept, formed the basis of virtually

all turbulence modeling research for the next twenty years. Important early

contributions were made by several researches, most notably by von Kármán (1930).

In modern terminology, we refer to a model based on the mixing-length hypothesis as

an algebraic model or a zero-equation model of turbulence. By definition, an n-

equation model signifies that a model requires solving n additional differential

transport equations in addition to the transport equations that express conservation mass

conservation, momentum and energy conservation for the mean flow.

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With the concept involved in mixing length hypothesis several modifications were

made to the model in successive years of research in turbulence. Prandtl (1945)

postulated a model in which the eddy viscosity depends upon the kinetic energy of the

turbulent fluctuations, k. This improved the ability to predict properties of turbulent

flows and it was a step forward to develop a more realistic mathematical description of

the turbulent stresses. Prandtl proposed a modeled partial-differential equation

approximating the exact equation for k. This improvement, on a conceptual level, takes

account of the fact that the turbulent stresses, and thus the turbulent viscosity or the

eddy viscosity, are affected by where the flow has been, i.e., upon flow history. This

led to the evolution of one-equation models in the area of turbulent research.

Though an eddy viscosity model that depends upon the flow history provides more

physically realistic model, there is still the need to specify a turbulence length scale.

Now, on dimensional grounds, viscosity has the dimensions of velocity times length. It

is thus essential to have an idea of the length scale of flow since it more or less

represents characteristic eddy size in a flow. Since such scales are different for each

flow, turbulence models that do not provide a length scale are referred to as incomplete

models. In other words, turbulence models should provide some advance information

about the flow, other than initial and boundary conditions. It has been found out that

though the incomplete models do not render much information, still they are not without

merit. In fact in many engineering applications they have proven to be of worth.

Keeping in view the above discussions, it is desirable to develop a turbulence model

that can be applied to a given turbulent flow by prescribing at most the appropriate

boundary and/or initial conditions. That is, no advance knowledge of any property of

the turbulence should be required for determining the flow features. Such a model is

referred to as a complete model. These models give an idea of the turbulent length

scale and hence the characteristic size of the eddy.

It was in 1942 that Kolmogorov introduced the first complete model of turbulence. He

introduced a second parameter ω, which he referred to as rate of dissipation of energy

in unit volume in unit time, other than having a modeled equation for k, the turbulent

kinetic energy. The inverse of ω served as the time scale in turbulence, ��

�/ω served as

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the analog of mixing length and kω served as analog of dissipation rate, ε. In this model,

ω satisfied a differential equation which is very much similar to that of the transport

equation for the turbulent kinetic energy k. The model, known as the k- ω model, is thus

referred to as two-equation model of turbulence as it solves for two transport

equations other than the transport equations for conservation of mass, momentum.

Though this model offered great anticipation, because of the insufficient computational

facilities to compute the differential equations these models went with practically no

applications in turbulence research.

The two equation models developed determined eddy viscosity with the aid of

Boussinesq assumption. It was only by Chou (1945) and Rotta (1951) that turbulence

models were developed which avoided the use of Boussinesq approximation which was

a major assumption in simplifying the turbulent dynamics. Rotta developed a

reasonable model for the differential equation of the turbulent stresses or the Reynolds-

stress. These models are popularly referred to as stress-transport models or second

order closure models or second-moment closure models. These are also referred to

as seven equation models as these solve for six additional transport equations for the

turbulent stresses other than the transport equation for the turbulent length scale. The

introduction of the turbulent stress transport models straightway gives an advantage of

considering the history effects and non-local effects. Though these models involve

computational complexities they automatically accommodate complicating effects in

the flow like streamline curvatures or rigid body rotations. In eddy viscosity models,

these complications are taken care of through introduction of additional empirical

terms. As with Kolmogorov’s, k- ω model, stress transport models also awaited

adequate computational requirements.

Thus all the above can be categorized into four main categories of turbulence models,

viz.

A. Algebraic (Zero-Equation) Models

B. One-Equation Models

C. Two-Equation Models

D. Reynolds Stress-Transport Models

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The following discussions briefly introduce to the different models that are popularly

referred in turbulence modeling related research.

Algebraic Models. In these models, an algebraic equation is used to compute the

turbulent viscosity or the eddy viscosity. The Reynolds stresses are then calculated

using the Boussinesq approximation. This correlates the turbulent stresses with the

mean velocity gradient of the flow. Van Driest (1965) introduced a viscous damping

correction to the mixing-length model or the zero equation model or the algebraic

model, which has been included in virtually all algebraic models that are in use today.

Cebeci and Smith (1974) also refined the eddy-viscosity or mixing-length model to a

point that it can be used with great confidence for most attached boundary layers.

Though this model possesses algebraic simplicity in determining the turbulent length

scale, Baldwin and Lomax (1978) proposed an alternative algebraic model that enjoyed

widespread use in turbulence research for many years.

One-Equation Models. As discussed earlier when defining n-equation model, one

equation model refers to solving one additional transport equation for one turbulent

parameter, viz. the turbulent kinetic energy. The turbulent length scale is obtained from

an algebraic expression. Here also the Boussinesq assumption is used to calculate the

eddy viscosity. Of the four types of turbulence models described above, these one

equation model has enjoyed the least popularity and success. Perhaps the most

successful early model of this type was formulated by Bradshaw, Ferriss and Atwell

(1967). The reason being this model reproduced flow properties in best agreement with

the experimental data available then in the 1968 Stanford Conference on Computation

of Turbulent Boundary Layers [Coles and Hirst (1969)]. In the early nineties of the

twentieth century there had been renewed interests in one-equation models based on a

postulated equation for eddy viscosity [c.f. Baldwin and Barth (1990), Goldberg (1991)

and Spalart and Almaras (1992)]. This was due to the ease with which the one-equation

models gave a rough estimate of the eddy viscosity in comparison with the two equation

models or the Reynolds stress transport models. Of the one-equation models, literature

survey reveals that Spalart and Almaras model was perhaps the most accurate for

practical turbulent-flow applications.

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Two-Equation Models. The two equation models solve for two additional transport

equations other than the continuity and momentum transport equations. While

Kolmogorov’s k- ω model was the first of this type, it remained unutilized till the

development of computers. The most useful two equation model is that of Launder and

Spalding (1972). Launder’s k-ε model is very well known and is the mostly used two

equation model, � being the turbulent dissipation. Even the model’s demonstrable

inadequacy for flow with adverse pressure gradient [Rodi and Scheuerer (1986),

Wilcox (1988a, 1993b) and Henkes (1998a)] could hardly do anything to discourage its

widespread use. Several modifications were adopted in this model by the continuing

succession of his students and colleagues. Without any prior knowledge of

Kolmogorov’s work, Saffman (1970) formulated another k- ω model that enjoyed

advantages over the k- ε model. This is mainly due to the integration carried out through

the viscous sublayer and due to taking into account the effects of adverse pressure

gradient. Many further developments in the k-ω model have been carried out by Wilcox

and Alber (1972), Saffman and Wilcox (1974), Wilcox and Traci (1976), Wilcox and

Rubesin (1980), and Wilcox (1988a). In 1986, it was pointed out by Lakshminarayana

that the k-ω models are the second most widely used type of two-equation turbulence

model.

Reynolds Stress-Transport Models. By early seventies in the twentieth century,

computational resources slowly became more and more available to compute the

complex computations in solving transport equations for the Reynolds stresses. Among

the noted works, those of Donaldson [Donaldson and Rosenbaum (1968)], Daly and

Harlow (1970) and Launder, Reece and Rodi (1975) are praise worthy. The latter has

become the baseline stress-transport model: more recent contributions by Lumley

(1978), Speziale (1985, 1987a, 1991) and Reynolds (1987) have also provided

mathematical rigor to the turbulence closure process. Although significant

contributions have been made till then, still the complexity involved limits the use of

stress transport models to simple engineering and industrial applications when

compared to algebraic models or the two equation models.

Turbulence models have been created that fall beyond the bounds of the four categories

cited above. This is true because model developers have tried unconventional

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approaches in an attempt to remove deficiencies of existing models of the four basic

classes. In this dissertation focus is on one type of Reynold’s stress model depending

on anisotropy. The primary emphasis is upon examining the underlying physical

foundation and upon developing the mathematical tools for analyzing and testing the

model.

2.2. Summary In the current chapter, we have given a literature survey about the various developments

in turbulence starting with the work of Prandtl. Though a lot of research has gone into

the study of mixed length models, one equation models and two equation models, little

work has been done in the topic of anisotropic models. In this thesis, we have taken the

anisotropy of turbulence as the basic variable in the turbulence modeling and gone

ahead with devising the Reynolds stress model.

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Eddy Viscosity Models

3.1 Introduction

The two-equation EVMs, which all use the turbulent kinetic energy as one of the solved

turbulent quantities. Apart from the transport equation for k, the models add another

transport equation for a second turbulent quantity. The main difference between these

EVMs is the choice of this quantity.

The commonly accepted idea is that the eddy-viscosity may be expressed as the product

of a velocity scale and length scale. Thus the obvious choice would be, to combine√�,

with l. Although its logical construction this combination has not been used with any

two-equation turbulence model.

Instead of velocity-length scale model the overwhelmingly majority of the used two-

equation EVMs are based on the k-ε concept, which use the dissipation, ε, in the k-

equation to construct the eddy-viscosity. The subsequent relation is based on

dimensional reasoning, and as such is no improvement compared to a k-l model.

However using the k-ε concept one avoids the additional complication of how to model

the dissipation rate in the k-equation.

The k-ω models originally developed by Kolmogorov, however more recently

promoted by Wilcox, uses the reciprocal to the time scale or vorticity. This secondary

quantity is however more commonly referred to as the specific dissipation rate of

turbulent kinetic energy.

The major differences and also benefits of using either of the above mentioned types of

two-equation EVMs are:

1. The used secondary turbulent quantity, and its boundary condition.

2. The way the turbulent is modelled.

3. Modelling of the exact terms in the ε-equation.

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3.2 Anisotropy in Turbulence

The value of anisotropy is a determining factor in the development of two equation

models. Anisotropy varies throughout the energy cascade spectrum. Smaller eddies at

high Reynolds numbers are highly anisotropic than the large eddies. These ideas lead

to the concept of a complicated variable known as eddy viscosity. Eddy viscosity is

defined as a proportionality constant between the Reynolds stresses and the mean strain

rate. This is what is famously referred to as Boussinesq eddy viscosity assumption

which postulates that Reynolds stress tensor,���, is proportional to the mean strain rate

tensor, Sij and is represented as :

��� = 2�����

−�

� ����� (3.1)

where ��� = −���������� , ��������� represents the Reynolds stress components,

��� =�

��

���

���+

���

���� is the strain rate of the mean flow, �� represents a scalar property

called the eddy viscosity and is normally computed from the two transported variables,

k which is the turbulent kinetic energy and ���which is the Kronecker delta.

From its very inception the Boussinesq eddy viscosity assumption helped in the

simplification of the complicated phenomenon of turbulence. It is a powerful

assumption in that it is a huge simplification allowing one to think of the effect of mean

flow on turbulence in the same way as molecular viscosity affects a laminar flow

(referred to as the Newtonian law). However, the main principle in two equation models

is that the eddy viscosity depends on the complex structure parameter(��). These two-

equation models (including the k-kl model known as Mellor-Yamada model (MYM))

approximate the eddy viscosity as,

�� = ����

� (3.2)

In the above expression �� is the turbulent eddy viscosity �= ��

�� �� is the structure

parameter, k is the turbulent kinetic energy and � is the rate of turbulent kinetic energy

dissipation. The above relation is reflected in the works of Kolmogorov et.al (1942) and

Prandtl (1945). Thus a lot of information on second moments is now contained in the

rather-complicated, non-dimensional structure parameter �� . Therefore it is very

important that we model this structure parameter or stability function accurately.

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Pioneering work on geophysical turbulence started with Mellor and Yamada (1982)

proposing their k-kl model, k being the turbulent kinetic energy and l being the integral

length scale. The two unknowns, k and kl are solved using transport equations just like

in the k-ε model. In addition, they proposed eddy viscosities as functions of stability

functions (structure parameters). It is to be mentioned here that Launder in his work on

two equation (k-ε) models set the structure parameters as constants. In MYM (1982),

they are treated explicitly as functions of shear and buoyancy, the two quantities on

which turbulence production depends. This parameterization was subsequently

improved upon over the years with notable contribution coming from Kantha and

Clayson (1994) followed by Canuto et al 2001, whereby the form of the structure

parameter was modified to include the effects of vorticity and anisotropy. These

parameterizations led to improved prediction for viscosities because the value of critical

Richardson's number (Ricr) for buoyancy is increased to unity as it should in the

presence of non-linear instabilities. This value of unity is both a necessary and sufficient

condition for turbulence to die out (Abarbanel, 1984).

We have come up with an eddy viscosity formulation that obviates the need for

calculating the structure parameter. Jovanovic et al. (2000) has already come up with

formulation by a totally different method. Our derivation here is very simple and easy

to use. In the current chapter the focus is on that formulation. This is done by making

the eddy viscosity a function of the anisotropy invariant (defined in the following

pages).

However the transport equation for the second invariant of anisotropy, II , is simplified

using Craft et al (1997) for our study of geophysical domain. It should be stressed here

that this formulation is more exhaustive than the used structure parameter approach in

that it also incorporates the degree of anisotropy in addition to the turbulent stresses

which makes it superior to any two equation model. With it we have the additional

advantage of not having to model the complex structure parameters.

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3.3 Eddy Viscosity Formulation

The Reynolds stresses appearing as unknowns in the momentum equations are

determined in the turbulence model either by turbulent eddy viscosity hypothesis or

directly from modeled Reynolds stress transport equation. As stated before, definition

of eddy viscosity follows Boussinesq (1877) approximation, which stands as the

building block of two-equation model,

ij

i

j

j

itji k

x

U

x

Uuu

3

2

(3.3)

This Boussinesq assumption provides closure to RANS equations.

General time-averaged Navier Stokes equations can be written in tensor form as

� ����

��� = �� −

��

���+

�

������

with ��� = ����

���+ � ��� �

���

���+

���

���� −

�

������ (3.4)

Now we rewrite the equation 2 in terms of anisotropy aij, defined as;

1

2 3

i j

ij ij

u ua

k (3.5)

This gives,

22

ij

ij t

Sa

k (3.6)

There is a slight problem with this equation. Equation (3.6) implies that the Reynolds

stress is aligned with the mean strain rate, which is found to be untrue by many scientists

(Tao et al., 2000). In fact, it has been found that the stress tensor has a preferred

orientation of 34 - 45º with respect to the mean strain rate. It is seen that the non-

alignment of the vectors mainly arises due to the ik kj ik kj ik kj jk ki(a S + a V + S V + S V ) term

which will exist in (3.6). Here Vij is vorticity. Therefore it goes beyond doubt that the

anisotropy should depend on vorticity as well, and we are working on improving this

model by inclusion of extra terms in equation (3.6). It has already been shown that once

21 | P a g e

these extra vorticity terms are removed from the model, the stress rate aligns parallel to

the mean strain rate, as it does in the current model.

From equation (3.6) it follows directly that

2

22

ij

ij ji t

Sa a II

k

(3.7)

where II is defined as the second invariant of anisotropy tensor, Lumley (1978). From

this, we get the expression for eddy viscosity as,

1/ 2

t

kII

S (3.8)

where 1 2

ij jiS S S , ijS is the mean rate of strain tensor which is analogous to viscous

stress in the Newtonian fluid.

A similar expression was derived by Jovanovic et al (2000) but it was after a lot of

unnecessary assumptions and mathematics.

We show below in Fig 3.1 the results of equation (3.6) with DNS data of Rogallo

(1981). The performance of this new equation (3.7) is as good as possible with the

Boussinesq assumption. The results are this accurate because the only assumption in

deriving the expression for eddy viscosity is the Boussinesq approx. whereas in the

stability function method, many more approximations and corrections are used.

A very promising tool for representing the effect of anisotropy is the anisotropy

invariant map (or it can also be represented as the Lumley triangle, Lumley, 1978)

which is a invariant space between II and III (the second and third invariants). Figure

2 represents this triangle. The physically realizable turbulence possibilities occur inside

the triangle. The different sides represent axisymmetric expansion or contraction and

2-component turbulence. The three vertices of the triangle represent the case for

isotropic two-dimensional turbulence, isotropic three-dimensional turbulence and one-

dimensional turbulence. The limiting cases of eddy viscosity for these three vertices

will be (The values of II have been derived from Pope, 2000, page 395).

22 | P a g e

Fig 3.1: A plot of anisotropy ija against 1 2. ijII S S for axisymmetric turbulence from Direct

Numerical Simulations of Rogallo (1981).

For isotropic 2D turbulence:

1 6II Implies 1/ 2

1.

6t

k

S

(3.9)

For isotropic 3D turbulence;

0II Implies 0t (3.10)

For 1D turbulence;

2 3II Implies 1/ 2

2.

3t

k

S

(3.11)

23 | P a g e

Fig 3.2: Lumley’s triangle

3.4 Summary

For anisotropic turbulence, the Reynolds stress tensor, i ju u , is usually anisotropic.

The second and third invariances of the Reynolds-stress anisotropic tensor aij are

nontrivial, where 1

2 3

i j

ij ij

u ua

k and k is the turbulent kinetic energy.

It is natural to suppose that the anisotropy of the Reynolds-stress tensor results from the

difference in various Reynolds stress components which arises from turbulent

production, dissipation, transport, pressure-stain-rate, and the viscous diffusive tensors.

In the current chapter, we introduced the concept of anisotropy in the calculation of

eddy viscosity. The applications of this equation will be carried out in the coming

chapters.

24 | P a g e

Fluent

4.1 Governing Equations

Mean and Instantaneous Velocities

If we recorded the velocity at a particular point in the real (turbulent) fluid flow, the

instantaneous velocity (U) would look like this:

At any point in time: � = �� + ��

The time average of the fluctuating velocity �� must be zero: �� = 0

But , the RMS of �� is not necessarily zero : ���

�����≠ 0

K this is the sum of the 3 fluctuating velocity components : k =

0.5*�������� + ������� + ���������

Reynolds-Averaged Navier Stokes (RANS) in Fluent

This is the main tool used by engineers.

Equations are solved for time-averaged flow behavior and the

magnitude of turbulent fluctuations

25 | P a g e

RANS Equations and the Closure Problem

The time-averaging is defined as

�̅ = lim�⟶�

1

�� �(��

�

�

, �)��

The instantaneous field is defined as the sumo of the mean and the fluctuating

component , such as � = �̅ + �� �� = ��� + ���

By averaging the Navier-Stokes equations, we obtain the Reynolds averaged

Navier-Stokes (RANS) equations:

��

��+

�(���� )

���= 0

�(���� )

��+

������ ��� �

���= −

��̅

���+

�

����� �

����

���+

����

���−

2

3���

�������

���� +

�

����−���

������������

Reynolds stress tensor, ��� = −������

�������

Eddy Viscosity Models (EVM)

These assume the ‘stress’ is proportional to the ‘strain’ (strain being the

gradients of velocity). The only new (unknown) quantity needed by EVMs is an

effective viscosity ��

−������

������� = �� �����

���+

����

����� −

2

3��� ��� + ��

�������

����

Eddy viscosity is similar to molecular viscosity in its effect of diffusing

momentum.

Eddy viscosity is NOT a fluid property; it is a turbulent flow characteristic.

Unlike an isothermal laminar flow in which viscosity is a constant which

varies with position throughout the flow field

EVMs are the most widely used turbulence models for CFD.

26 | P a g e

4.2 The Universal Law of the Wall

The size of your grid cell nearest to the wall (value of y+) is very important.

The value you need depends on the modelling approach chosen.

In the near-wall region, the solution gradients are very high, but accurate

calculations in the near-wall region are paramount to the success of the

simulation.

4.3 Reynolds Stress Tensor

����������� ���������� ����������

���������� ���������� ����������

���������� ���������� ������������

27 | P a g e

Rij is a symmetric, second-order tensor; it comes from averaging the

convective acceleration term in the momentum equation

Reynolds stress thus provides the averaged effect of turbulent (randomly

fluctuating) convection, which is highly diffusive

Reynolds stress tensor in the RANS equations represents a combination of

mixing due to turbulent fluctuation and smoothing by averaging.

SST k-ω Model Background

Many people, including Menter (1994), have noted that:

The k–ω model has many good attributes and performs much better than k–ε

models for boundary layer flows

Wilcox’ original k–ω model is overly sensitive to the free stream value of ω,

while the k–ε model is not prone to such problem

Most two-equation models, including k–ε models, over-predict turbulent

stresses in the wake (velocity-defect) regions, which leads to poor

performance in predicting boundary layers under adverse pressure gradient

and separated flows

The basic idea of SST k–ω is to combine SKW in the near-wall region with

SKE in the outer region

4.4 FLUENT UDF

What is a User Defined Function?

A UDF is a function (programmed by the user) written in C which can be dynamically

linked with the Fluent solver.

Standard C functions (Trigonometric, exponential, control blocks, do-loops, file

i/o, etc.)

Pre-Defined Macros (Allows access to field variable, material property, and cell

geometry data and many utilities)

All data exchanged between the UDF and the solver must be in SI units

Why program UDFs?

28 | P a g e

a) Customization of boundary conditions, source terms, reaction rates,

material properties, etc.

b) Customization of physical models

c) User-supplied model equations

d) Adjust functions (once per iteration)

e) Execute on Demand functions

f) Solution Initialization

Interpreted vs. Compiled UDFs

UDFs can either be run compiled or interpreted.

a) The supported compiler for Fluent on Windows platforms is Microsoft Visual

Studio

b) Most Linux systems provide a C compiler as a standard feature.

Interpreted code vs. compiled code

Interpreted

C++ Interpreter bundled with Fluent

Interpreter executes code on a “line by line” basis instantaneously.

Advantage – Does not require a third-party compiler.

Disadvantage – Interpreter is slow, and cannot do some functions.

Compiled

UDF code is translated once into machine language (object modules).

Efficient way to run UDFs.

Creates shared libraries which are linked with the rest of the solver.

Does require a compilation step between creating/editing your UDF and using it.

UDF Step by Step

The basic steps for using UDFs in Fluent are as follows:

1. Identify the problem to be solved

2. Check the usability & limitations of the standard models of Fluent

29 | P a g e

3. Program the UDF (must be written in C) i.e. Prepare the Source Code

4. Compile the UDF in the Fluent session

5. Hook the UDF in Fluent GUI

30 | P a g e

6. Assign the UDF to the appropriate variable and zone in BC panel

7. Solution Initialization

8. Run the calculation

9. Examine the results

31 | P a g e

User Access to Fluent Solver

32 | P a g e

CFD Flow Chart

Pre Processor

Geometry or Computational Domain

Grid Generation

Solver

Approximation of unknown variables

Discretisation

Solution

Fluent Solver using FVM

Post Processor

Vector plots

Line & Shaded contour plots

2D & 3D surface plots

Particle tracking

33 | P a g e

RESULT & CONCLUSION

For flow in a pipe various turbulence models were used to analyze the turbulence

characteristics like velocity profile, variation of Turbulence Kinetic Energy,

Turbulence eddy dissipation and pressure contours. Eddy viscosity variation along the

length of the pipe were then finally obtained and compared for k-ε model & eddy

viscosity formulation based on anisotropy.

5.1 k-ε model results

Velocity contours along a transverse section located midway inlet & outlet

Velocity contour along the length of the pipe

Variation of Turbulence Kinetic Energy and Turbulence Eddy Dissipation

along the length of the pipe

34 | P a g e

Eddy Viscosity variation for k-ε model

5.2 Anisotropic UDF model Results

Velocity contours

along a transverse

section located

midway inlet & outlet

Variation of Turbulence Kinetic Energy and Turbulence Eddy Dissipation

35 | P a g e

Eddy Viscosity variation for Anisotropic UDF model

36 | P a g e

REFRENCES

1. Carrica P, Wilson RV, Stern F (2006) Unsteady RANS simulation of the ship

forward speed diffraction problem. Comput Fluids 35:545–570

2. Stern F, Wilson RV, Coleman HW, et al (2001) Comprehensive approach to

verification and validation of CFD simulations—part 1: methodology and

procedures. J Fluids Eng 123:793

3. Wilson RV, Paterson E, Stern F (2000) Verification and validation for RANS

simulation of a naval combatant. In: Gothenburg 2000: A Workshop on

Numerical Ship Hydrodynamics. Chalmers University of Technology,

Gothenburg, Sweden

4. Wilson R, Carrica P, Stern F (2005) RANS simulation of a container ship using

a single-phase level set method with overset grids. CFD Workshop Tokyo 2005,

9–11 March, Tokyo, Japan, pp 546–551

5. Carrica P, Wilson RV, Stern F (2006) An unsteady singlephase level set method

for viscous free surface flows. Int J Numer Methods Fluids (in press)

6. Rhee S, Stern F (2001) Unsteady RANS method for surface ship boundary

layer and wake and wave field. Int J Numer Methods Fluids 37:445–478

7. Wilson R, Stern F (2002) Unsteady RANS simulation of a surface combatant

with roll motion.24th Symposium on Naval Hydrodynamics, July 8–13,

Fukuoka, Japan (CDROM)

8. Campana EF, Peri D, Tahara Y, et al (2004) Comparison and validation of CFD-

based local optimization methods for surface combatant bow. 25th Symposium

on Naval Hydrodynamics, August 8–13, St. John’s, Newfoundland and

Labrador, Canada (CD-ROM)

9. CFD ONLINE Forum http://www.cfd-online.com/