Seminar 1b - 1. letnik, II. stopnja

Phenomenology and modelling of turbulent thermal

plumes

Author: Rok KrpanMentor: doc. dr. Ivo Kljenak

May 18, 2016

Abstract

Turbulent thermal plumes are encountered in many industrial applications. The use of numericalmodelling allows the investigation of the flow field, turbulence and gas mixing in the plume with theenvironment. Two-equation Reynolds-Averaged Navier-Stokes turbulence models are most widely usedmodels to simulate fluid dynamics problems. Turbulent plume is mainly driven by thermal buoyancyand its main characteristics are unsteadiness, energy nonequlibrium, counter-gradient diffusion andlack of universal scaling. All these phenomena are believed to be associated with distinct large coherenteddy structures, which are hardly captured by steady Reynolds-Averaged-Navier-Stokes equations. InUnsteady-Reynolds-Averaged-Navier-Stokes approach, the large scale structures are fully resolved,whereas the stochastic small scales are modelled. The buoyancy-corrected standard k− ε model is themost commonly used two equations model for modelling the effects of turbulence. Since buoyancy playsa significant role in the production and destruction of turbulence in a plume, additional turbulenceproduction terms due to buoyancy effects must be introduced. From comparison of the computationalresults with experimental data from several articles it is observed that the buoyancy corrected k − εmodel is a good choice to model a turbulent thermal plume. This model is able to capture the Gaussiannature of the plume and empirical plume spread coefficients.

Contents

1 Introduction 2

2 Computational fluid dynamics - CFD 2

3 Phenomenology of turbulent thermal plume 3

4 Modelling of turbulent thermal plume 54.1 Mathematical model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

4.1.1 Governing equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64.1.2 Buoyancy-corrected standard k − ε model . . . . . . . . . . . . . . . . . . . . . 10

5 Conclusions 11

1 Introduction

A plume in hydrodynamics is a column of one fluid moving through another. Turbulent plume differsfrom a jet in terms of its driving forces. A jet is momentum driven while a plume is buoyancy driven.The buoyancy force in such flows can either be caused by a heat source or by the introduction of onefluid into another fluid of different density or temperature. A forced plume is a type of flow betweena plume and jet and therefore driven by momentum and buoyancy. A forced plume is generated bydischarging a hot fluid at low initial momentum through a nozzle into a quiescent environment. [1]

Thermal buoyancy-dominated turbulent flows are frequently encountered in various industrial ap-plications and consequently have been the subject of research due to their technological and environ-mental importance in many physical processes. The prediction of fluid flow and heat transfer in abuoyancy driven flow is important in space heating and cooling, smoke and fire spreading, nuclearreactor containment, pollutant dispersion etc. [1, 2]

Experiments over a wide range of Rayleigh and Prandtl numbers, to simulate different physicalphenomena in industrial and geophysical applications, are very difficult to perform. In laboratoryexperiments, problems arise when a large buoyancy force is required to maintain a given Rayleighnumber as the Prandtl number increases. In numerical experiments, with increased Rayleigh andPrandtl number smaller grid sizes and time stepping is needed, which has effect only on calculationtimes [3].

A computational investigation of thermal plume is important because such flows are very difficultto achieve in experiments. Computational Fluid Dynamics (CFD) is used to generate flow simulationswith the help of computers. CFD involves the solution of the governing laws of fluid dynamicsnumerically. The complex set of partial differential equations are solved in geometrical domain dividedinto small volumes, commonly known as a mesh (or grid). CFD has given us a tool to understand theworld in new ways. We are able to simulate and understand fluid flows without the help of instrumentsfor measuring various flow variables at desired locations [4].

A plume and its environment in general can be liquid or gas. This seminar is limited only togas plumes and environments with two different species of gases, for example helium plume rising inquiescent air environment.

In Section 2 a short explanation of Computational Fluid Dynamics is presented. Plume phe-nomenology is described in Section 3. Models used in CFD are described in Section 4.

2 Computational fluid dynamics - CFD

Computational fluid dynamics (CFD) is used for solving fluid dynamic problems with numerical meth-ods. Computers are used to perform the calculations required to simulate the interaction of fluids withsurfaces defined by boundary conditions [5].

2

Buoyancy may play a significant role in the physics of turbulent flows. It affects the productionand dissipation of turbulent kinetic energy of the flow. Therefore the accuracy of computationalsimulation of thermal plume fundamentally depends on how well the effect of buoyancy on turbulenceis modelled.[2] Direct numerical simulation (DNS) and large eddy simulation (LES) in which largeeddies are fully resolved can provide the flow field more accurately than other methods. However theapplication of DNS to flows with very high Reynolds number and Rayleigh number is still not feasibledue to the requirements of high computational resources. LES also requires higher computationalresources compared to those for Reynolds-averaged Navier-Stokes (RANS) computations. On theother hand, the RANS approach in which inherently there is no structural and spectral informationis not suitable for large eddy structures. In the unsteady Reynolds-averaged Navier-Stokes (URANS)approach the large eddy deterministic structures are fully resolved, whereas, the rest of the scales ofturbulence are modelled by conventional closure models. Therefore, it can be ascertained that theuse of URANS equations can be a good and promising tool for predicting flow physics in a buoyancydriven turbulent plume [1].

With CFD it is possible to model gas release and mixing, for example hydrogen release in nuclearreactor containment during the loss of coolant accident (LOCA). Hydrogen release and dispersioncould occur at the early stage of a severe nuclear accident and its potential explosion could seriouslythreaten the integrity of the containment or auxiliary buildings as it has happened in the Fukushimaaccident in Japan in 2011. The three-dimensional nature of the physical processes involved (turbulenttransport, mixing, buoyancy and heat transfer) in hydrogen dispersion make an excellent case for CFDapplication in nuclear safety [6].

Figure 1: Simulation of helium molar concentration in reactor containment calculated using Open-FOAM CFD code [6].

3 Phenomenology of turbulent thermal plume

The buoyancy force in such (thermal) flows can either be caused by a heat source or by the introductionof one type of gas into another type of gas of different density or temperature. Once the gas is set inmotion, its velocity field affects the thermal field, and vice versa. The initial state of laminar motionquickly changes into turbulence and the flow starts to spread radially by entraining ambient gas intothe main flow. [7]

A single laminar plume develops as the gas near a point or line heat source begins to rise. In thebeginning of the spatial evolution of a buoyant plume the flow remains laminar. As the plume rises itwidens due to the diffusion of both heat and momentum in the lateral direction. A horizontal profileof velocity and temperature across the plume boundary layer is Gaussian (Fig. 2). [3]

3

Figure 2: Temperature (T ), vertical velocity (vz) and Nusselt number in an ideal laminar 2D plume.δT denotes the thermal boundary layer and δvz the velocity boundary layer.[3].

A plume tends to become turbulent as it rises and diffusive processes becomes less important. Theplume becomes turbulent within a distance 2D from a source, where D is the source diameter. Asthe flow evolves, after a distance of 10D from the source, the flow tends to be in the self-similar statewhere the shape of the mean flow quantities can be represented by a single profile when normalizedby scales of velocity and length. [7]

Since the mass flow across the plume increases with height, surrounding gas must be entrained intothe plume. A line source plume forms a wedge, with the angle controlled by the rate of entrainment[3]. The edge of the wedge forms a sharp boundary that separates the turbulent and buoyant gasinside the plume from the surrounding ambient gas. The plume appears to behave as a line sinkin potential flow theory with respect to the surrounding gas [3]. However, entrainment in plumes ismore complicated than in jets. In jets, entrainment is accomplished completely by horizontal flow. Inplumes the decreased pressure in the buoyant region also causes upward entrainment from below [3].

In incompressible thermal convection the velocity is generated solely due to buoyancy forces cre-ated by thermal density variations. The buoyancy force allows the plume to rise, and both viscousand thermal diffusivity oppose the plume’s rise. Viscous diffusion creates viscous drag, and thermaldiffusion can cause heat to be lost faster than the gas can rise. Both of these effects must be overcomefor convection to occur. Rayleigh number (Ra) is dimensionless number that expresses the ratio ofbuoyancy and viscosity forces multiplied by the ratio of momentum and thermal diffusivity: [8]

Ra =buoyancy forces

viscosity forces· momentum

thermal diffusivity=gβ

να(Ts − T∞)x3 (1)

where g is gravitational acceleration, β thermal expansion coefficient, ν kinematic viscosity, α thermaldiffusivity, Ts surface temperature, T∞ gas temperature far from the surface of the object and x isthe characteristic length. The magnitude of the Rayleigh number (Ra) dictates whether the flow islaminar or turbulent [3].

The Prandtl number (Pr), which is also dimensionless number and expresses the ratio of kinematicviscosity and the thermal diffusivity, is also needed to fully describe the convection: [3]

Pr =kinematic viscosity

thermal diffusivity=µcpk

=ν

α(2)

where cp is specific heat, k thermal conductivity and µ is dynamic viscosity.Prandtl number can be related to the thickness of the thermal and velocity boundary layers. It is

actually the ratio of velocity boundary layer to thermal boundary layer. When Pr = 1, the boundary

4

layers coincide. If Pr is small, it means that heat diffuses very quickly compared to the velocity(momentum). [5]

The Prandtl number represents only physical properties of the fluid (gas), in contrast to theRayleigh number, which is related to both the physical properties of the fluid and the externallyimposed conditions [3].

Plume growth is related to both advective and diffusive effects. The relative balance of these twoeffects affects on varying Prandtl and Rayleigh numbers. Most of the transport properties of theplumes vary continuously as the relative effects of viscosity and thermal diffusivity vary during theplume development [3].

4 Modelling of turbulent thermal plume

Turbulent plume is primarily driven by buoyancy forces, therefore it is necessary to incorporate theeffect of buoyancy on the production and dissipation of turbulent kinetic energy [2]. This is done byadding a source term in the transport equations for turbulent kinetic energy (k) and turbulent kineticenergy dissipation rate (ε). The effect should be in such way that it produces turbulence in stablystratified flows whereas it suppresses turbulence in unstably stratified flows. The buoyancy productionterm in the transport equations for turbulent kinetic energy and dissipation rate is modelled usinggeneralized gradient diffusion hypothesis (GGDH). [1]

A comparison with the experimental measurements reported in the literature shows that the GGDHalong with turbulence models correctly predicts the mean flow field, temperature field and spread rates[2].

There are two complexities involved in modelling buoyancy driven flows. The first is the co-existence of stagnant gas, transitional regime and fully turbulent regime in one flow. Transitionalflow is challenging and difficult to compute because of the complex physics and small scale motionsinvolved. The second difficulty is due to low momentum in buoyancy driven flows and coupling ofenergy equation and therefore it becomes difficult to reach a steady state. [1]

At the outer edge of the plume, turbulence serves to entrain and mix the surrounding environmentwith the plume. The “entrainment velocity” is the horizontal velocity of the surrounding gas, whichis being entrained into the plume, measured at the plume edge. The entrainment assumption is thatthe entrainment velocity at a given height is proportional to the mean vertical velocity at that sameheight. This approximation is based upon the observation that the entrainment depends upon theintensity of the turbulence at the plume edge which scales with the mean vertical velocity as it is ameasure of the strength of the shear between the rising plume and the quiescent surroundings.[9]

Figure 3: Axisymmetric coordinate system. [1]

5

4.1 Mathematical model

4.1.1 Governing equations

To mathematically describe turbulent flow we need to know what happens in such flow. Turbulenceis state of fluid motion which is characterized by random and chaotic three-dimensional vorticity.When turbulence is present, it usually dominates all other flow phenomena and results in increasedenergy dissipation, mixing, heat transfer, and drag. The essential property of turbulence is its abilityto generate new vorticity from old vorticity. Turbulent flows are not only time-dependent but spacedependent as well. Because of this chaotic and apparently random behavior of turbulence, we needstatistical techniques for study of turbulence. [8]

Study of the turbulence requires statistics and modelling of stochastic processes. Because ofthe chaotic nature of turbulent flow the instantaneous motions are complicated to understand andmathematically describe. This does not mean that the governing equations (Navier-Stokes equations)are stochastic, but in turbulent cases even simple non-linear equations have deterministic solutionsthat look random. To say in a different way, even though the solutions for a given set of initial andboundary conditions can be perfectly repeatable and predictable at a given time and point in space, itmay be impossible to guess from the information at one point or time how it will behave at another (atleast without solving the equations). Moreover, a slight change in the initial or boundary conditionsmay cause large changes in the solution at a given time and location. [8]

Most of the statistical analyses of turbulent flows are based on averaging of the quantities neededto describe such flows. Let ϕ be any time dependent variable. Reynolds averaging is the averaging ofa variable or an equation in time: [5]

ϕ =1

T

∫Tϕ(t)dt (3)

where T is time interval long enough to average out the fluctuations of ϕ.In turbulent flow it is convenient to analyse the flow variables in two parts: a mean (or average)

component (ϕ) and a fluctuating component (ϕ′). The instantaneous quantity can be written as: [8]

ϕ = ϕ+ ϕ′ (4)

In the velocity case the Eq. 4 is written as:

u = u+ u′ (5)

Figure 4: Time dependence of velocity in a turbulent flow. [10].

Favre averaging is another tool used to simplify the calculations. Favre averaging is defined as: [5]

ϕ =

∫T ρ(t)ϕ(t)dt∫

t ρ(t)dt=ρϕ

ρ(6)

6

where the overline represents Reynolds averaging.We want to compute the evolution of a round turbulent plume of one type of gas in a quiescent

environment of another gas. At first we need to make certain assumptions to simplify the full Navier-Stokes equations, continuity equation and passive scalar equations. In addition, since the Machnumber, the ratio between flow velocity to speed of sound, of the flow is sufficiently low, a low Machnumber (LMN) version of Navier-Stokes equations can be used. [1]

Low Mach number formulation describes weakly compressible fluid and includes density as anexplicit variable in the computations. This formulation filters the acoustic compressibility effect andretains other compressibility effects and allows inertial effect of a variable density. The Favre averagedcontinuity equation using the Einstein summation convention can be written as: [1]

∂ρ

∂t+∂ρuj∂xj

= 0 (7)

where ρ and uj are Reynolds-averaged density and Favre-averaged velocity component, respectively.The low-Mach-number version of Favre-averaged Navier-Stokes equation can be expressed as: [1]

∂ρui∂t︸ ︷︷ ︸1

+∂

∂xj(ρuiuj)︸ ︷︷ ︸2

= − ∂pd∂xi︸ ︷︷ ︸3

+∂

∂xj

(τ ij − ρu′iu′j

)︸ ︷︷ ︸

4

− gkxk∂ρ

∂xi︸ ︷︷ ︸5

(8)

where pd and τij are time-averaged component of total pressure and viscous shear stress tensor com-ponent, respectively.

Terms in Eq. 8 represent:

1. The change in mean momentum of fluid element.

2. Convective transport.

3. Pressure forces.

4. Viscous forces, which are the result of molecular viscosity and turbulent stresses.

5. Gravitational forces.

Viscous shear stress tensor defines the flow of momentum due to molecular viscosity, which isperceived as stress. It can be written as:

τ =

τxx τxy τxzτyx τyy τyzτzx τzy τzz

(9)

and its components are defined as:

τxx = −2µ∂vx∂x

+2

3µ(∇ · ~v), τyy = −2µ

∂vy∂y

+2

3µ(∇ · ~v), τzz = −2µ

∂vz∂z

+2

3µ(∇ · ~v),

τxy = τyx = −µ(∂vx∂y

+∂vy∂x

), τxz = τzx = −µ(∂vx∂z

+∂vz∂x

), τyz = τzy = −µ(∂vz∂y

+∂vy∂z

)

(10)

Term ρu′iu′j is stress caused by fluctuating velocity field, generally referred to as the Reynolds stress.

This nonlinear term requires additional modelling to close the Navier-Stokes equation for solving.The density weighted Favre-averaged transport equations of passive scalars (mass fraction of gas

and enthalpy) can be written as: [1, 11]

∂

∂t

(ρYk

)+

∂

∂xi

(ρuiYk

)=

∂

∂xi

(µ

Sc

∂Yk∂xi− ρu′iY ′

k

)(11)

7

∂ρh

∂t+

∂

∂xj

(ρuj h

)=

∂

∂xj

(µ

Pr

∂h

∂xj− ρu′jh′

)(12)

where µ, p, Yk, Sc, h and Pr are molecular viscosity, total pressure, mass fraction of k-th gas, Schmidtnumber, enthalpy and Prandtl number, respectively [11]. These are just a part of the basic transportequations of fluid mechanics, written in a specific form.

Schmidt number is dimensionless number and expresses the ratio of the viscous diffusion rate tothe mass diffusion rate: [12]

Sc =viscous diffusion rate

mass diffusion rate=

µ

ρD(13)

where D is mass diffusivity.In Eq. 8 gravitational acceleration (gk) points along the negative vertical direction and xi is the

distance from a reference point in the direction of the gravitational acceleration. It is clear that thebuoyancy term is zero in the horizontal direction where the gravitational acceleration is zero. Thedensity can be calculated from the equation of state of an ideal gas p = ρRT , where R and T are idealgas constant and Favre-averaged temperature, respectively. [1]

Turbulent kinetic energy (k) is the mean kinetic energy per unit mass associated with eddiesin turbulent flow. Physically, the turbulent kinetic energy is characterised by measured root-mean-square (RMS) velocity fluctuations. Generally k can be quantified by the mean of the turbulencenormal stresses (velocity fluctuations): [8]

k =1

2u′iu

′i (14)

Turbulent kinetic energy is produced by fluid shear, friction or buoyancy and then is transferred downthe turbulence energy cascade to smaller scales, when at some point it is dissipated by viscous forces.[8]

Rate of dissipation of turbulent kinetic energy per unit mass due to viscous stresses is modelledas: [8]

ε = 2ν ˜SijSij (15)

Sij is the strain rate tensor defined as: [8]

Sij =1

2

(∂ui∂xj

+∂uj∂xi

)− 1

3

(∂ui∂xi

)δij (16)

where δij is the Kronecker delta.ν is the kinematic viscosity defined as: [8]

ν =µ

ρ(17)

Dissipation of the turbulent kinetic energy is always positive. It reduces the kinetic energy of the flow,therefore it causes a negative rate of change of kinetic energy; hence the name dissipation. Physically,energy is dissipated because of the fluctuating viscous stresses in resisting deformation of the fluidmaterial by the fluctuating strain rates. [8]

The problem with the non-linearity of the instantaneous equations is that they introduce theReynolds stress into the averaged equations. There are six individual stress components (u′21 , u′22 ,

u′23 , u′1u′2, u

′1u

′3 and u′2u

′3 ) which must be related to the mean motion before the mean momentum

equation and equations for passive scalars can be solved, since the number of unknowns and numberof equations must be equal. The absence of these additional equations is often referred to as theTurbulence Closure Problem. The objective of the turbulence models for the RANS equations is tocompute the Reynolds stresses. [8]

8

The Reynolds stress tensor (ρu′iu′j) appearing in the mean momentum equation (Eq. 8) is unknown

and therefore has to be modelled. With the k − ε type of turbulence model the Reynolds stress interms of the mean flow properties is modelled using the Boussinesq hypothesis. [1, 2]

Boussinesq approximation is used to make assumptions in buoyancy driven flows and consequentlyto simplify equations which describe such flows. This approximation ignores density differences exceptwhere they appear in terms multiplied by gravitational acceleration (g). The essence of the Boussinesqapproximation is that the difference in inertia is negligible but gravity is sufficiently strong to make thespecific weight appreciably different between the two fluids [13]. The momentum transfer caused byturbulent eddies can be modelled with an eddy viscosity. This is in analogy with how the momentumtransfer caused by the molecular motion in a gas can be described by a molecular viscosity. TheBoussinesq approximation states that the Reynolds stress tensor is proportional to the mean strainrate tensor (Sij) as: [8]

−ρu′iu′j = 2µtSij −2

3ρkδij (18)

where Sij is the time-averaged strain rate tensor and µt is the turbulent viscosity coefficient, which isderived with turbulent kinetic energy and turbulent dissipation as:[1]

µt = ρCµk2

ε(19)

In Eq.19 Cµ is a model constant, which is obtained from comparison of the model results with ex-perimental data. The turbulent kinetic energy and turbulent dissipation rate depend on the type ofturbulence model used.

To close the mean energy equation (Eq. 12) and the equation for turbulent scalar flux of massfraction (Eq. 11), specifications of the turbulent enthalpy flux, ∂h/∂t, and the scalar flux of massfraction are needed. The magnitudes of the fluctuations in this simple models are related to thegradient of the average. The turbulence enthalpy and turbulent scalar flux of mass fraction can beclosed using gradient diffusion hypothesis: [11]

ρu′iY′k = − µt

Sct

∂Yk∂xi

(20)

ρu′jh′ = − µt

Prt

∂h

∂xj(21)

For computational simplification a constant value for turbulent Prandtl number, Prt = 0.85, isusually taken. Turbulent Prandtl number is a non-dimensional number defined as the ratio betweenthe momentum eddy diffusivity and the heat transfer eddy difussivity. The effect of stability on thediffusivity of the plume can be incorporated by modelling the turbulent Prandtl number as a functionof Richardson number Rf : [1]

Prt =0.7(1−Rf )2

(1−Rf/0.15)(22)

Richardson number is the dimensionless number which expresses the ratio of the buoyancy term tothe flow gradient term: [8]

R =buoyancy term

flow gradient term=g

ρ

∇ρ(∇u)2

(23)

9

4.1.2 Buoyancy-corrected standard k − ε model

The Reynolds averaged Navier-Stokes (RANS) approach has been widely used in modelling turbulentplumes in CFD to simulate mean flow characteristics for turbulent flow conditions. The standardk−ε model is the simplest two equation model used for modelling a variety of engineering flows and iscommonly used to close the RANS equations. This model has been popular due to its computationalrobustness, simplicity, and well documented validation cases. [1]

The evolution equatins for k (Eq. 24) and ε (Eq. 25) are obtained from Navier-Stokes equation(Eq. 8). Navier-Stokes equation is multiplied by the appropriate quantity (velocity) and then isaveraged over time.

In this model, the turbulent kinetic energy k and its dissipation rate ε are considered as transportproperties. Therefore, in order to describe turbulence, the two evolution equations are needed tocompute two variables. [2]

However, the k−ε model requires modifications to model the effects of buoyancy on the productionand dissipation of turbulent kinetic energy in a thermal plume. The effect of buoyancy is usuallyincorporated by adding a source term in the transport equations for turbulent kinetic energy anddissipation rate. With this source terms we get the buoyancy-corrected k − ε model. The evolutionequations for stratified flows can be written as: [2]

∂(ρk)

∂t︸ ︷︷ ︸1

+∂(ρkuj)

∂xj︸ ︷︷ ︸2

=∂

∂xj

[(µ+

µtσk

)∂k

∂xj

]︸ ︷︷ ︸

3

+P +B − ρε︸︷︷︸4

(24)

Terms in Eq.24 represent:

1. Local change of turbulent kinetic energy.

2. Convective transport of turbulent kinetic energy.

3. Diffusion of the Reynolds stresses.

4. Dissipation of turbulent kinetic energy.

∂(ρε)

∂t+∂(ρεuj)

∂xj=

∂

∂xj

[(µ+

µtσk

)∂ε

∂xj

]+ C1ε

ε

kP + C1ε(1− C3ε)

ε

kB − C2ερ

ε2

k(25)

B is a source term, which denotes the production of turbulent kinetic energy due to buoyancy. P isthe production of the turbulent kinetic energy due to the mean shear and is specified on the basis ofthe Boussinesq approximation, as seen in Eq. 18, as: [2]

P = −ρu′iu′j∂ui∂xj

= µt

[(∂ui∂xj

+∂uj∂xi

)− 2

3

(∂u1∂x1

)δij

]− 2

3ρkδij (26)

The source term B is usually modelled using generalized gradient diffusion hypothesis (GGDH).

Generalized gradient diffusion hypothesis (GGDH) The generalized gradient-diffusion hy-pothesis (GGDH) approach is consistent with the Boussinesq hypothesis used in the mean momentumequation. In this approach the buoyant turbulence production term is approximated as: [1]

B = gjρ′u′j = −3

2

CµPrt

k

ε

(u′ju

′k

∂ρ

∂xk

)gj (27)

In Eq. 27 Prt denotes the ratio of turbulent and thermal eddy viscosities. From this equation it is seenthat the GGDH formulation includes cross-stream density variation and allows both density gradients,vertical and horizontal, to affect turbulent kinetic energy. The term ∂ρ/∂xk describes the variation ofdensity in the direction perpendicular to the gravity vector. [1, 2]

10

The Richardson number, which was mentioned in the previuos section, is modelled as: [1]

Rf = − B

(P +B)(28)

For the record, several studies on turbulent plume indicated that the value of C3ε in the range of0.3− 1.0 did not affect the computations. The values of the model constants in standard k− ε modelare: [1]

Cµ = 0.09, σk = 1.0, σε = 1.3C1ε = 1.44, C2ε = 1.92, C3ε = 0.8

It should be emphasized, that the models, presented in this seminar, are just proposals by differentauthors, that seem sensible and were shown to provide adequate results. However, these models arenot necessarily the best that can be achieved (at least in principle). Also, the presented models arenot entirely empirical, but are based on some physical intuiton.

5 Conclusions

The use of numerical modelling allows the investigation of the flow field and turbulence in those regionsof the plume of most interest, that is the plume edge and near source regions [9].

Direct numerical simulation (DNS), Large Eddy Simulation (LES) and Reynolds-stress transportmodel are few approaches to turbulence modelling, which can offer a deeper insight of the flow field.These tools are very useful but they require expensive computational resources, therefore two-equationReynolds-Averaged Navier-Stokes (RANS) turbulence models are still most widely used to simulatefluid dynamics cases. [2]

Turbulent plume is mainly driven by thermal buoyancy and its main characteristics are unsteadi-ness, energy nonequlibrium, counter-gradient diffusion and lack of universal scaling. All these phe-nomena are believed to be associated with distinct large coherent eddy structures, which are hardlycaptured by steady Reynolds-Averaged-Navier-Stokes (RANS) equations. In Unsteady Reynolds-Averaged-Navier-Stokes approach, the large scale structures are fully resolved, whereas the stochasticsmall scales are modelled. [1]

The buoyancy-corrected standard k − ε model is one of the most commonly used two equationmodels for modelling the effects of turbulence and buoyancy. These model consists of equations forturbulent kinetic energy and another equation for turbulent kinetic energy dissipation rate. Sincebuoyancy plays a significant role in the production and destruction of turbulence in a plume, addi-tional turbulence production terms due to buoyancy effects are introduced. From comparison of thecomputational results with experimental data from several articles it is observed that the buoyancycorrected k − ε model is a good choice to model a turbulent thermal plume. This models are able tocapture the Gaussian nature of the plume and empirical plume spread coefficients. The mean flowstatistics obtained from unsteady RANS are also in compliance with the experimental measurements.[1]

11

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[4] CFD online, “Introduction to CFD.” http://www.cfd-online.com/Wiki/Introduction_to_CFD. [Ac-cessed: 20.10.2015].

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[9] D. Hargreaves, M. Scase, and I. Evans, “A simplified computational analysis of turbulent plumes and jets,”Environmental Fluid Mechanics, vol. 12, no. 6, pp. 555–578, 2012.

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[11] S. Abe, M. Ishigaki, Y. Sibamoto, and T. Yonomoto, “RANS analyses on erosion behavior of densitystratification consisted of heliumair mixture gas by a low momentum vertical buoyant jet in the PANDAtest facility, the third international benchmark exercise (IBE-3),” Nuclear Engineering and Design, vol. 289,pp. 231 – 239, 2015.

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[13] Wikipedia, “Boussinesq approximation (buoyancy).” https://en.wikipedia.org/wiki/Boussinesq_

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