Development of a generic three dimensional model for stratified flows

The model solves the original Navier-Stokes equations NSE with a variable fluid density without the employment of Boussinesq approximation.. Keywords: 3-D numerical model, stratified flo

DEVELOPMENT OF A GENERIC

THREE-DIMENSIONAL MODEL FOR STRATIFIED FLOWS

WANG DONGCHAO

(M Eng., TIANJIN)

A THESIS SUBMITTED FOR THE DEGREE OF MASTER OF ENGINEERING

DEPARTMENT OF CIVIL ENGINEERING NATIONAL UNIVERSITY OF SINGAPORE

2003

ACKNOWLEDGEMENTS

I would like to give my sincere thankfulness to my supervisors Assistant Professor Pengzhi Lin and Professor N Jothi Shankar They are always open to suggestions and very flexible in dealing with my specific needs This greatly helps me keep the motivation high and guarantee the best possible outcome of my project

I acknowledge all people in Environmental & Hydraulic Division and Civil Engineering Department at the National University of Singapore for their cooperation and efforts throughout my education and research work

I would like also to acknowledge the National University of Singapore for its benevolence in providing a full scholarship grant for my study

This thesis is dedicated to my family and my friends for their much-needed support and caring during these past months I would not have accomplished this without them

TABLE OF CONTENTS

ACKNOWLEDGEMENTS i

TABLE OF CONTENTS ii

SUMMARY iv

NOMENCLATURE vi

LIST OF FIGURES viii

LIST OF TABLES x

Chapter 1 Introduction and Literature Review…… ……….1

1.1 General Background and Purpose……….1

1.2 Stratified Flows……… 2

1.3 Sediment Dumping……… ……3

1.4 Review of Stratified Flow Models……… … 6

1.5 Objectives and Scope of the Present Study ……… 8

1.6 Outline of the Present Work……….9

Chapter 2 Mathematical Review of Hydrodynamic Model……… 11

2.1 The Navier-Stokes Equations……….11

2.2 Turbulence Modelling……….12

2.2.1 Reynolds Averaged Navier-Stokes Equations…… ………12

2.2.2 The Eddy Viscosity Concept……… ……… 15

2.2.3 The Mixing Length Model……….16

2.2.4 The Standard k-ε Model……… ……… 17

2.2.5 Large Eddy Simulation Model (LES)……… ……….19

2.2.6 Direct Numerical Simulation (DNS)……… ……… 20

2.3 Concluding Remarks………21

Chapter 3 Development and Formulations of the 3D Hydrodynamic Model……….…22

3.1 Governing Equations in Cartesian Coordinate……… 22

3.2 Governing Equations in σ-Coordinate………25

3.3 Numerical Approximations………30

3.3.1 Advection Step……… 32

3.3.2 Diffusion Step………34

3.3.3 Pressure-updating Step……….………35

3.3.4 Velocity-correction Step……… ……… 39

3.3.5 Density Tracking Step………39

3.3.6 Free Surface Tracking Step………42

3.4 Boundary Conditions in σ-Coordinate ……… 43

3.5 Stability Criterion……… ……… 46

Chapter 4 Results and Discussions……… ………47

4.1 Additional Convective Effect in a Stratified Flow……….………….… 47

4.2 Density-driven Flow……….……….…….50

4.3 Computation of the Sediment Dumping into Water… ……….53

4.3.1 Drift Velocity Assumption……….…54

4.3.2 Turbulence Model…… ……… ………54

4.3.3 Computational Conditions in 2-D Cases………55

4.3.4 Calculated Results in 2-D Cases……… 57

4.3.5 3-D Demonstration with Free Surface……….65

4.4 Future Works on Buoyant Jets and Plumes.…… ……….72

4.4.1 A Horizontal Buoyant Jet ……….72

4.4.2 Preliminary Results for Jet Centerline Trajectory…… ……… 74

4.4.3 Mixing of a Buoyant Plume with and without Waves…….……… 78

Chapter 5 Conclusions and Recommendations……… ……82

5.1 Conclusions……….82

5.2 Recommendations………84

References………85

Appendix ………90

SUMMARY

A three-dimensional numerical model has been developed to simulate stratified flows with free surface The model solves the original Navier-Stokes equations (NSE) with a variable fluid density without the employment of Boussinesq approximation The modified NSE are solved in a transformed σ-coordinate system with the use of operator-splitting method (Lin & Li, 2002), which splits the solution procedure into advection, diffusion and pressure-correction steps Assuming the free surface is the single function

of the horizontal plane, a slightly modified σ-coordinate introduced by Blumberg and Mellor (1983) would map the irregular computation domain to a regular computational domain By introducing density variation, a transport equation for fluid density is added, which is solved by the Cubic-Interpolated propagation (CIP) method as developed by

Yabe et al (1991) The CIP method has the advantage to capture the moving sharp

interface front Without Boussinesq assumption, no restriction is made for the rate of density variation and the model can be used to simulate both continuously stratified flows and layered flows The numerical model is validated against the one-dimensional convection-diffusion problem, which displayed fairly the influence of strong density stratification to the diffusion term Another comparison of the model prediction to an analytical solution was conducted for a horizontal density-gradient flow in a closed basin Excellent agreements are obtained between numerical results and analytical solutions The model is then used to study transport phenomena of dumped sediments into a quiescent water body, which is modeled in this study as a strongly stratified flow A mixing length model is applied to represent the induced turbulence with a well-defined

length scale For the two-dimensional problem, the numerical results are compared well with experimental data in terms of the mean particle falling velocity and the spreading rate of the sediment cloud for both fine and coarse sediments The present model is further extended to study the dumping of sediments in a 3D environment with the presence of free surface A reasonable explanation is given by the current model on the behaviours of particle dumping The model simulation results reveal an inverse relationship between the rate of spreading of the cloud and the settling velocity, and show that the frontal velocity approaches the settling velocity in the ultimate stage It is found that an annulus-like cloud will be formed for fine sediments whereas a plate-like cloud for coarse sediments The model is proven to be a good tool to simulate strongly stratified free surface flow, in which the conventional Boussinesq assumption may become invalid

Keywords: 3-D numerical model, stratified flows, σ-coordinate, free surface, sediment dumping, non-Boussinesq

NOMENCLATURE

b nominal width of the cloud

C mixing length coefficient

C empirical constant in SGS model

d diameter of the particle

Fr densimetric Froude number

FX momentum flux in x-direction

FY momentum flux in y-direction

R strain rate of the mean flow

u velocity component in x-direction

i

u velocity component in i-th direction

0

U initial mean jet velocity

v velocity component in y-direction

* * * *

x y z t spatial and temporal coordinates in physical domain

x y σ t spatial and temporal coordinates in computational domain

η free surface displacement

λ horizontal density gradient

µ dynamic molecular viscosity

τ viscous stress tensor

ϕ′ time-varying fluctuating component of a flow property

ϕ time-averaged component of a flow property

Φ steady mean component of a flow property

χ convergence number in diffusion step

Figure 3.2 Schematic plot of mesh definition σ−coordinate 31

Figure 4.1 Comparisons of numerical velocity profile at t= 1,2,5,10,20

Figure 4.2 Schematic diagram of density-driven flow in a tank 51

Figure 4.3 Comparisons of numerical velocity profile at x=10 km, y=2.5

km with analytical solution for constant horizontal density gradient

53

Figure 4.4 Illustration of sediment dumping 55

Figure 4.5 Variation of falling velocity of cloud normalized by

characteristic velocity scale against dumping distance with d50

at 5.0 mm

58

Figure 4.6 Variation of falling velocity of cloud normalized by

characteristic velocity scale against dumping distance with d50 at 0.8 mm

59

Figure 4.7 Dispersion width of the dumping clouds normalized by square

Figure 4.8 Velocity fields and density contours at intervals of one tenth

of density difference between thermal maximum value and ambient water at a cross-section in the middle of y-

coordinates for the case d50=5.0 mm with q0 = 5 cm2 at time

t= 0.3, 0.7, 1.0, 1.5 (s)

61

Figure 4.9 Velocity fields and density contours at intervals of one tenth

of density difference between thermal maximum value and ambient water at a cross-section in the middle of y-

coordinates for the case d50=5.0 mm with q0 = 10 cm2 at time

62

t= 0.3, 0.7, 1.0, 1.5 (s)

Figure 4.10 Velocity fields and density contours at intervals of one tenth

of density difference between thermal maximum value and ambient water at a cross-section in the middle of y-

coordinates for the case d50=0.8 mm with q0 = 5 cm2 at time

t= 0.3, 0.7, 1.0, 1.5 (s)

63

Figure 4.11 Velocity fields and density contours at intervals of one tenth

of density difference between thermal maximum value and ambient water at a cross-section in the middle of y-

coordinates for the case d50=0.8 mm with q0 = 10 cm2 at time

t= 0.3, 0.7, 1.0, 1.5 (s)

64

Figure 4.12 Spatial profiles of surface elevation and density isosurface of

sediment cloud in 50% density difference between thermal maximum value and ambient water with d50=0.15 mm at time

t = 0.1, 0.3, 0.6, 1.0, 1.5, 2.0, 2.5, 3.0 (s)

68

Figure 4.13 Spatial profiles of surface elevation and density isosurface of

sediment cloud in 50% density difference between thermal maximum value and ambient water with d50 =1.3 mm at time t

= 0.1, 0.3, 0.6, 1.0, 1.5, 2.0, 2.5, 3.0 (s)

69

Figure 4.14 Velocity fields and density contours at intervals of one tenth

of density difference between thermal maximum value and ambient water at a cross-section through the cloud center for the case with d50=0.15 mm at time t = 0.6, 1.0, 2.0, 3.0(s)

70

Figure 4.15 Velocity fields and density contours at intervals of one tenth

of density difference between thermal maximum value and ambient water at a cross-section through the cloud center for the case with d50=1.3 mm at time t = 0.6, 1.0, 2.0, 3.0(s)

71

Figure 4.16 Sketch of a horizontally discharged buoyant round jet in a still

Figure 4.17 Velocity fields and density contours and comparison of the jet

trajectory determined experimentally with the prediction of this study and mathematical model by Davidson for various Froude numbers (a)Fr=10, (b)Fr=15, (c)Fr=20

75, 76

Figure 4.18 Surface elevation profiles and density isosurface of the jet

evolution at ρ =1004 kg/m3 with Fr=15 78

channel

Figure 4.20 Density contour of Case B (affected by wave) in a vertical

plotting window (x-z plane at y=0.1 m) in the middle of the

CHAPTER ONE

INTRODUCTION AND LITERATURE REVIEW

1.1 General Background and Purpose

Complex stratified flows are encountered in natural processes and engineering practices,

in which non-uniform distribution of temperature, solute (e.g salinity) or suspended particles (e.g sediment) exists The non-uniformity of these physical variables reflects themselves through the change of effective fluid density The studies of stratified flow are especially important in estuarial and coastal regions, where density stratification can be induced by many natural or man-made processes For instance, ocean disposal operations

of dredged material usually occur in the dredging of navigation channels and harbors, or

in the construction of breakwaters are well as offshore man-made islands Since contaminants are typically bound to the sediments, an understanding and predictive capability of the movement of the sediment as suspended solids can lead to insight into the fate of containments In addition, the basic estuary problems of salinity intrusion, pollution by discharged wastes, shoaling, and sediment transports are associated with diffusion processes in a stratified flow From a water quality point of view, there is commonly great interest in modeling the density field of water bodies Therefore, there are of great importance to environmental engineers to model stratified flows

1.2 Stratified Flows

The term “stratified flow” encompasses fluid motions in a gravitational field which are initiated or influenced by variations in density within the fluid Due to the buoyancy effect, the velocity and density profiles in density different fluid are significantly different from that in a still unstratified fluid Variations in vertical density with respect

to water depth can occur in natural aquatic environments Examples of stratified flows include sediment-laden flows above seabed, wave-river flow interaction in estuaries, internal waves on the continental shelf, buoyant plumes & jets around marine outfalls and sediment dumping by barges

In the presence of a gravitational field, density differences result in variable buoyant forces throughout a fluid and a strong feedback to the dynamical equations This really means that light fluid will tend to move above heavy fluid In other words, in the presence of gravity, a density stratified fluid has a preferred organization The preference

to have heavy fluid below light fluid allows us to describe the characteristic stability states of hydrostatic stratified fluids Shown in Figure 1.1 the density structure can lead to three possible stability states Stably-stratified fluids tend to maintain their structure, and the stabilizing forces enforce this preferred organization For neutrally stable fluids there

is no preferred structure and fluid particles experience similar resistances to motion in all directions Unstable stratification occurs when heavy fluid overlays light fluid Although such a density structure is theoretically possible, it is very sensitive to small scale disturbances The density structure is said to be unstable and will tend toward the stable state

Figure 1.1 Stability definitions showing related hydrostatic density distributions

1.3 Sediment Dumping

In land reclamations or dredging projects large amount of sediments and mud are often disposed of in designated areas of coastal waters As a prerequisite to the environmental impact due to these projects, the dynamics and dispersion of the sediments in water should be determined One important aspect is determining where and how the disposal material is initially dispersed and deposited after disposal Therefore, accurate prediction

of the settling processes of disposed material in open waters, such as the descending and transport or dispersion of the material, is of practical importance for both engineering management and environmental conservation

The description of a disposal typically divides the behaviour of dumping material into three distinct transport phases or stages, according to the physical forces or processes that dominate during each period (Truitt, 1988) The most common terminology in use today for these stages is convective descent, dynamic collapse, and long-term or passive diffusion Figure 1.1 is a schematic sketch of these stages

Figure 1.2 General overview of the disposal of dredged material (Pequegnat et al 1981)

The behaviour of sediment dumping has been studied both theoretically and experimentally for many years The motion induced by a cloud of particles released in a heavier or lighter fluid is referred to as a particle thermal Early laboratory experiments

on thermals were conducted by Scorer (1957), Richards (1965) and Sullivian (1972), who modeled these flows by releasing salt water into a body of fresh water of uniform density The width of the cloud of buoyant fluid and the frontal position of the cloud were measured and empirical equations on thermal motions were obtained Turner (1973) considered thermals as special cases of buoyant vortex rings and showed that these differences in spreading rates can arise due to subtle differences which are generated in

the release process Krishnappan (1975) developed an empirical model based on a dimensional analysis Koh and Chang (1973) developed a mathematical model for predicting the short-term fate of ocean disposal material from barges

non-Sediment dumping is similar to the release of a particle thermal that has been researched

by many investigators recently A number of studies have conducted on the differences between thermals, which are due to temperature or salinity differences, and those that derive their buoyancy from heavy particles suspended in them Nakatsuji et al (1990)

found the dynamic behaviour of a cloud of particles is close to thermal motion if the initial volume of the cloud is relatively large and the sizes of the particles are relatively small The particles move independently and motion is dominated by the balance between the buoyant force and the drag force on each particle if the volume of the cloud

is relatively small and the settling velocity of the particles is relatively large Buhler& & Papantoniou (1991) demonstrated that suspension thermals eventually evolve into particle swarms, which sink at a constant velocity, and grow slightly in size along their path In an unstratified ambient fluid the transition from the thermal to the swarm stage was found to occur when the frontal velocity in line with the similar thermal theory has decayed to the settling velocity of individual particles in calm fluid More systematic measurements of the velocity and growth rates were carried out by Tamai et al (1991) as

well as Noh & Fernando (1993) Tamai et al showed the initial stage of diffusion of

turbidity is dominated by flow fields which falling motions of soil particles induce Noh

& Fernando proposed a slightly different concept for the transition length, which they determined as the position of the cap at the time when particles start to fall out from it

Along with recent developments in numerical computations, there has been considerable progress in wide applicability of thermal Oda & Shigematsu (1994) introduced a numerical method, which combined the Marker and Cell method and the Discrete Element Method (DEM), to propose the dispersion width of the settling particles and deposition width on the water bottom become wider with an increase of the volume of dumped particles However the computational results by DEM are of doubtful accuracy because it is difficult to determine accurately the stiffness and damping coefficients Li (1997) performed numerical simulations for particle clouds to demonstrate that velocity

of the thermal approaches the thermal velocity of the individual particles and the degree

of lateral spreading of the cloud varied inversely with magnitude of the settling velocity Chen & Lee (2002) suggested that the advected line thermal is characterized longitudinally by a flat trajectory with scalar dilution taking place near the jet exit, and transversely by a vortex-pair flow and a kidney-shaped concentration structure with double peak maxima corresponding to stronger buoyant effect These studies indicate that the problem of sediment dumping is not fully understood

1.4 Review of Stratified Flow Models

There are a few numerical approaches that solved the Navier-Stokes equations with varying density, most of which, however, were conducted with two-dimensional models For example, Gu (1998), using a two-dimensional (2D) unsteady model, examined a submerged warm water discharged into a stratified lake or reservoir with an ice cover; Liu et al (2002) simulated the salinity field and tidal current by two-dimensional models with different types of closure models; Shen et al (2003) used a turbulent two-phase flow model to simulate turbulent stratified flows Although two-dimensional models are

computationally efficient and their results are generally reasonable, most practical applications generally require a full three-dimensional analysis due to complex geometrics and topographies

Several recent advances in numerical techniques are incorporated into the models which solved the three-dimensional Navier–Stokes equations for a stratified flow subject to the Boussinesq approximation (Huang 1995; Fung 1996; Winters 2000; Galmiche 2001; Paisley & Bhatti 1998; Torres et al., 2002) Under the Boussinesq approximation, only small density fluctuations and stratification are considered In the case of fluids with strong variation and fluctuation of density, the assumption needs to be reconsidered

Conventional stratified flows are solved with Boussinesq assumption (e.g Rubin & Atkinson, 2001) Boussinesq assumption dramatically simplified the problem but its validity relies on the physical condition of weak stratification However, it has been observed (Ricou & Spalding, 1961) experimentally that in the non-Boussinesq case, where the initial density difference between the plume and ambient flow is large, the entrainment velocity partly depends on the ratio of the plume density to the ambient fluid density Woods (1997) suggested that non-Boussinesq effects may have a great impact on the shape and density evolution of a plume over a certain distance above the source Also

in 1997, Mlaouah et al proposed that the solutions with Boussinesq approximation for thermally-driven turbulent flows deviated gradually from exact solutions The recent work of Larrazábal et al (2003) indicated that the earlier Marker-and-Cell (MAC) method is only applicable to solve fluid flow problems for a limited range of Reynolds Numbers However, if the flow is strongly stratified, it is necessary to use a more robust

equations (NSE) without Boussinesq assumption should in principle provide more accurate result

1.5 Objectives and Scope of the Present Study

The main objective of the present study is to numerically investigate the dynamic behaviours of particle clouds released into waters and other natural phenomena relevant

to density variation with the developed 3D model The objectives of the present study are

as bellows,

1) To develop a three dimensional hydrodynamic model with variable density that solves the original Navier-Stokes equations in a transformed σ-coordinate for simulating stratified flows in natural water environment without the employment of Boussinesq approximation;

2) To solve the density transport equation by using selected high-accuracy hyperbolic scheme, namely, Cubic-Interpolated propagation (CIP) method which has the advantage to capture the moving sharp interface front;

3) To validate the resulting numerical model by comparing its solutions with available experimental data, empirical formulas and theoretical solutions;

4) To specify the evolution of 3-D sediment dumping process with free surface variation

1.6 Outline of the Present Work

A general introduction to this study is given in Chapter One The definition of stratified flows, their application in coastal engineering and literature on sediment dumping process are reviewed Mathematical overview of hydrodynamic model is presented in Chapter Two The universal governing equations of natural fluid flow and practical turbulence closure models are outlined Some remarks of typical turbulence models are summarized Chapter Three gives the detailed formulations of 3D hydrodynamic model developed in

the present study Firstly, the original NSE in Cartesian-coordinate are transferred in a coordinate system, then the NSE are solved with the use of operator-splitting method (Lin

σ-& Li, 2002), which splits the solution procedure into advection, diffusion, and correction steps Secondly, a transport equation for fluid density is updated in line with density variation Finally, the free surface displacement is modified by tracing momentum flux

pressure-Chapter Four concentrates on the model verification and its application on sediment dumping study A one-dimensional convection-diffusion problem is employed to illustrate the potential influence of strong density stratification to the diffusion step for solving NSE Subsequent comparison of the present model results for sediment dumping study with experimental data testifies the accuracy of the resulting model

Chapter Five presents the summary of conclusions and recommendations for further works The preliminary results for buoyant jet rising a still fluid and wave effect on jet dilution are discussed

CHAPTER TWO

MATHEMATICAL REVIEW OF HYDRODYNAMIC

MODEL

2.1 The Navier-Stokes Equations

The Numerical solution of any flow problem requires the solution of the general

equations of fluid motion, that is, the Navier-Stokes and the continuity equations Fluid

flow problems are described mathematically by these equations which are a set of

coupled non-linear partial differential equations with appropriate boundary conditions

These equations are derived from Newton’s Second Law and describe the conservation of

momentum in the flow

The general form of the three dimensional incompressible instantaneous Navier-Stokes

equations is as follows, in Cartesian tensor form:

and the continuity equation:

i i

Refer to the nomenclature page for details of the individual terms

For low speed laminar flow without heat transfer, the equations detailed above can be

used to describe the flow exactly However, in turbulent flows the velocity components

vary rapidly in both time and space and difficulties arise in the numerical discretization of

the flow field Furthermore, there is a major problem in simply representing or modeling

turbulence This is due to the fact that turbulence is an extremely complex and little

understood phenomenon which is defined by a number of highly complex mechanisms

including irregularity, diffusivity, three dimensional vorticity fluctuations and dissipation

A brief introduction to turbulence theory follows: a detailed description is referred to

Tennekes & Lumley (1972)

2.2 Turbulence Modeling

The crucial difference between visualizations of laminar and turbulent flows is the

appearance of eddying motions of a wide range of length scales in turbulent flows The

computing requirements for the direct solution of the original Navier-Stokes equations at

high Reynolds numbers are too computationally expensive Alternatively, the averaged

Navier-Stokes Equations are solved extensively in engineering field

The time averaged form of the Navier-Stokes equations are called the Reynolds Averaged

Navier-Stokes (RANS) equations These models calculate a mean, steady state velocity

and pressure field and account for the velocity and pressure fluctuations through

additional modeled variables These equations express only the movement of large scale

eddies thus allowing the use of coarse grids and making the models relatively economical

to use A number of models are available under this general heading which range from

closure models based on the eddy viscosity concept to full second moment closure

models which represent the effect of each component of the Reynolds stress tensor on the

mean flow

To investigate the effects of the fluctuations we replace in equation (2.1) and (2.2) the

flow variable u (hence also u , v , w ), i p, ρ by the sum of a mean and fluctuating

Equation (2.4) is generally referred to as the Reynolds equation and differs from the

equation describing a laminar flow only by the presence of the term containing averaged

products of fluctuating velocity The process it represents is the additional transfer of

momentum due to turbulent fluctuations The first term in brackets is the viscous term and the second term −ρu u i′ ′j is the turbulent stress or the Reynolds stress tensor

In addition to the three flow equations provided by equation (2.4), there is also the continuity equation, which for an incompressible fluid takes the form

It is therefore the main aim of the turbulence model to predict the effect of these Reynolds stresses on the mean flow Consequently the next step in the turbulence modeling process is the formulation and application of a suitable model that can accurately represent these stresses over a range of flow fields

2.2.2 The Eddy Viscosity Concept

In terms of the turbulence modeling perhaps the most important research is attributed to

the earliest worker in this field, Boussinesq (1877), who postulated that the Reynolds stresses should be proportional to the mean strain rate This concept is based on the assumption that both the viscous stresses and the Reynolds stresses act on the mean flow

in a similar manner Referring to Equation (2.4), it can be seen that both these stresses

appear on the right hand side of the momentum equation

For an incompressible fluid this relationship can be expressed in mathematical terms as:

j i ij

j i

u u

the mean rate of deformation increase (Versteeg & Malalasekera 1996) Therefore, these

statements led Boussinesq to propose a linear relationship between Reynolds stresses and

the rate of deformation of a fluid linked by a coefficient of proportionality µt as follows:

j i

ij i j t

j i

u u

seems a natural assumption to conclude that an extra viscosity can adequately represent the effects of turbulence

The right hand sides of Equations (2.8) and (2.9) are the same except for the coefficient used linking the two sides of the equations The main difference between these two coefficients is that µ is a function of the fluid properties only while µt is a function of the turbulence

If Equation (2.9) is substituted into Equation (2.1) then the mean flow equation now has

an enhanced additional viscosity µt due to the turbulence of the flow Using this approach the modeling process can be completed if the turbulent viscosity can be found from other variables There are a number of methods available of deriving the value of turbulent viscosity as will be briefly detailed in the following section

In 1925 Prandtl suggested that the eddy viscosity is proportional to a mean fluctuating velocity V and a mixing length l

Combining (2.11) and (2.12) and absorbing the two constants C and c we obtain

Prandtl’s mixing length model as follows:

2

t m

U l

The k- ε model is the most widely used and validated turbulence model It has achieved

notable successes in calculating a wide variety of thin shear layer and recirculating flows The standard k-ε model (Launder & Spalding, 1974) has two model transport equations,

which assume the local turbulence characterized by two parameters: one for the turbulent kinetic energy of the flow, k and one for the dissipation rate of k, ε These values are used

to define the velocity scale and the length scale, at any given point and time in the flow field, representative of large scale turbulence as follows: