Predictive Tools for Turbulent Reacting Flows

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MASTER’S THESIS

MASTER OF SCIENCE PROGRAMME Department of Mechanical Engineering

Division of Fluid Mechanics

Predictive Tools for Turbulent Reacting Flows

A comparison between FLUENT and GENUS

MAGNUS PERSSON

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Abstract

Two different CFD codes, FLUENT 5.5 and GENUS, are compared with independent experimental data. FLUENT is a widely used commercial software package while GENUS is a newly developed modular CFD prediction tool of Continuum Computational Mechanics.

The GENUS turbulence model library features non-linear closures at the second moment level for both velocity and scalar fields whereas FLUENT allows closures at this level only for the velocity field. The GENUS code will be used for turbulent reacting flow calculations in combustor-related geometries at ALSTOM Power. FLUENT and GENUS are here compared on the basis of two test cases relevant to flow in gas turbine combustors.

The first test case features an axisymmetric isothermal swirling flow with central air injection in a dump combustor. Experimental data are provided by the International Flame Research Foundation (IFRF) and include mean and RMS velocities in the axial and tangential (swirl) directions at various cross-stream planes. This case is computed using a 2D axisymmetric and a 3D sector model with FLUENT and only the 3D sector model with GENUS.

Both the Standard k-ε Model and second moment based closures are used to represent turbulent transport. A grid dependency test has been performed in the context of the 2D calculations and no substantial differences were observed. A comparison between 2D and 3D calculations featuring a Reynolds Stress Model and FLUENT revealed significant solution-convergence and prediction improvements in favour of the latter — the k-ε Model solution remained almost unaffected. A comparison between results obtained by FLUENT and GENUS using the k-ε Model reveals significant differences in favour of the latter package. When FLUENT and GENUS are compared using the Reynolds Stress Model the results are similar but FLUENT is found to predict an unphysically large internal recirculation zone.

The second test case features a range of planar, stoichiometric methane/air, premixed flames propagating in a frozen uniform turbulence field.

The latter is in each case obtained by assigning to the turbulence intensity a constant value for the whole domain. The integral length scale is assigned the value of 10 mm. The calculation is transient and integration proceeds until a steady state in flame-space is obtained. Calculated turbulent burning velocities as a function of the turbulence intensity are compared against experimental data provided by Abdel-Gayed et al. This case is in principle simple to calculate and well suited for the evaluation of turbulent combustion models. Using FLUENT with the Eddy-Dissipation concept based closure for the mean reaction rate, extreme difficulties were observed with re- spect to flame front stabilisation. In consequence a combination of Ar- rhenius expressions and the Eddy-Dissipation concept were used as a means to alleviate the above difficulties. In the latter case FLUENT was found to yield substantially under-predicted values for the turbulent burning velocity and over-predicted burnt gas temperature. Surprisingly, calculations were found to require an inordinate amount of CPU time of the order of days.

Using GENUS with an algebraic fractal flame surface based model for the flamelet regime of combustion predictions were found to be in excellent agreement with experimental data while all calculations were performed in a few hours.

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Preface

This Master of Science Thesis documents the final project of the course in mechanical engineering at Luleå University of Technology. The project is carried out during the period from the 31^st of July 2001 to the 10^th of De- cember 2001 at ALSTOM Power, Finspång.

I would like to thank the following people: my supervisor Vladimir Milo- savljevic and Jonas Holmborn at Alstom Power for their help and encour- agement, my examiner Staffan Lundström at Luleå University of Technol- ogy for his support and especially Evangelos M. Váos for his invaluable CFD-expertise.

Finspång, 10 December 2001

Magnus Persson

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Appendix 3: 3D sector with FLUENT...XIII Appendix 4: 3D sector with GENUS... XXII Appendix 5: Comparison between FLUENT and GENUS ...XXX Appendix 6: Swirl number... XXXVIII Appendix 7: Calculation of mass fractions ... XLII Appendix 8: Propagating flame with FLUENT...XLIV Appendix 9: Propagating flame with GENUS...XLVII

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Nomenclature

Latin Symbols

a mass of reactant per cubic centimetre

A constant = 4 in Eddy-Dissipation Model

B constant = 0.5 in Eddy-Dissipation Model

c reaction progress variable

c* quench value of reaction progress variable

c_d integral length scale constant

cp specific heat

C dimensionless Courant number

C_k molar concentration

C_ijk turbulent transport tensor

D diffusion coefficient

Da dimensionless Damköhler number

e_t truncation error

E activation energy

f normalized autocorrelation function

F autocorrelation function

gi body force in the ith coordinate direction

h enthalpy

J_i diffusive flux in the ith coordinate direction

k turbulence kinetic energy

kb backward rate constant

kf forward rate constant

Ka dimensionless Karlovitz number

l integral length scale

lF flame thickness

L length scale

n iteration number

N number of ensambles

N_g number of grid points

p pressure

Pr dimensionless Prandtl number

PrT dimensionless turbulent Prandtl number

q source term

r radius

R universal gas constant

Re dimensionless Reynolds number

ReT dimensionless turbulence Reynolds number

S swirl number

S_ij mean strain-rate tensor

Sc dimensionless Schmidt number

t time

tF flame time

t_T turbulent time

t_η Kolmogorov time

T temperature

Tb temperature of burned gases

T_i ignition temperature

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Tu temperature of unburned gases u_i = {u, v, w} velocity vector

uL laminar burning velocity

uRMS RMS (root mean square) turbulent velocity

u_T turbulent burning velocity

U axial velocity

vK Kolmogorov velocity

W weighting function

W_α molecular weight of species α

x_i = {x, y, z} Cartesian coordinate vector

Y mass fraction

Z pre-exponential factor in Arrhenius expression Greek Symbols

α ^species

β temperature exponent

δ reaction zone thickness

δij Kronecker delta

ε turbulence dissipation rate

εij dissipation tensor

φ arbitrary variable

γ under-relaxation factor

Γe coefficient of numerical diffusion

η Kolmogorov microscale

η^p rate exponent for products

η^r rate exponent for reactants

λ Taylor microscale

λc thermal conductivity

µ ^viscosity

µT eddy viscosity

ν kinematic viscosity

πij pressure-strain correlation tensor

θi turbulent scalar flux

ρ ^density

ρb density of burned gases

ρu density of unburned gases

σij Reynolds stress tensor

τ heat release parameter

τij viscous stress tensor

υ stoichiometric coefficient

Ψ net effect of third bodies on reaction rate Others

- Reynolds averaged mean value

~ Favre averaged mean value

’ difference from Reynolds averaged mean value

’’ difference from Favre averaged mean value

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1 Introduction

Recirculation zones are commonly used to stabilise flames in gas turbine combustors. The recirculation zone is often achieved by introducing a swirling motion into the combustor.

The turbulent flow field in a combustor includes many complicated features, such as swirl, recirculation and combustion, and to develop a combustor usually both numerical simulations and experiments are required.

However, in numerical simulations, standard models fail in the prediction of swirling flows and it is well known that different turbulence models capture the swirling motion with different levels of accuracy.

In Direct Numerical Simulation (DNS) all the length and time scales of turbulence are resolved. The advantage of DNS is that no turbulence model is needed but the disadvantage is that it requires a lot of computational power and can only handle simple geomtries with low Reynolds numbers.

Recent advances in computational power has meant that the application of CFD is now quite common. Many such simulations uses the k-ε turbulence model to represent turbulent transport. In this approach the Reynolds stress is linearly related to the mean rate of strain via an eddy viscosity, and while it is often capable of providing good results, it does have serious limitations. It is incapable of reproducing for example swirling motions and the effects of strong streamline curvature. Therefore, a more detailed description of turbulent transport is required.

Currently the only tractable approach to the modelling of turbulent com- busting flows is that based on closures through the use of Favre or density- weighted averaging of the conservation equations. However, the averaging process results in that there are more unknowns than equations. In com- busting flows these unknowns are of two types, the density-weighted Rey- nolds stress and turbulent scalar flux – the second moments. For turbulent transport, second moment closures represent an optimum choice, they com- bine a reasonably detailed turbulence representation at an acceptable level of complexity. With such approaches all the second moments are obtained from the solution of modelled partial differential equations. The modelled equations are obtained through approximations to relate higher order unknown correlations to lower order known quantities.

The aim of this Master Thesis is evaluation and validation of the newly developed moment based unstructured 3D code, GENUS (General Non- conformal Unstructured Solver), which contains full closures at the second moment level for velocity and scalar fields. GENUS is a newly developed modular CFD prediction tool, by Dr. Evangelos M. Váos of Continuum Computational Mechanics. The newly developed code will be compared with a commercial software package, FLUENT 5.5, and experimental data from test cases that contains swirl, recirculation and combustion.

2 The Conservation Equations

The governing equations for turbulent reacting flows express conservation of mass, momentum, energy and species. For a reacting flow field, the continuity equation [1] is

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=0

∂ +∂

∂

i i

x u t

ρ ρ

(2-1)

where ρ is the density, u_i is the velocity vector, t is time and x_i is the cartesian coordinate vector. Conservation of momentum yields

i j ij i j

j i i

x g x

p x

u u t

u +

∂ +∂

∂

− ∂

∂ = +∂

∂

∂ρ ρ τ

(2-2)

where τij is the shear stress in the ith coordinate direction on a surface whose outward normal is in the jth coordinate direction and where gi formally rep- resents a body force in the ith coordinate direction and p is the pressure.

Generally it is assumed that the fluid in turbulent reacting flows can be considered Newtonian and the stress tensor becomes













∂

− ∂

∂ +∂

∂

= ∂ ij

k k i

j j i

ij x

u x

u δ

µ

τ 3

2 (2-3)

where bulk viscosity is neglected. µ is the viscosity and δij is the Kronecker delta.

The equation describing conservation of a scalar quantity, where Y_α is the mass fraction of species α^{, is}

α α α

α ρ

ρ q

x J x

Y u t

Y

i i i

i +

∂

−∂

∂ = +∂

∂

∂ ⁽ ⁾

(2-4)

J_i^(α) is the diffusive flux of the scalar Y_α in the ith coordinate direction and the quantity q_α is a chemical source term. Diffusive transport is always present, and it is usually described by a gradient approximation, Fick’s law for mass diffusion

i

i x

D Y

J ∂

− ∂

= _α ^α

α⁾ ρ

( (2-5)

where D_α is the diffusion coefficient (the binary coefficient between the species α and the fluid). Eq. (2-5) in Eq. (2-4) gives

α α α α

α ρ ρ

ρ q

x D Y x x

Y u t

Y

i i

i

i +





∂

= ∂

∂ +∂

∂

∂ (2-6)

The mixture enthalpy, h, is conserved in the energy equation [2]

R j

i i

x q h Y Sc

x h x

t p

x h u t

h

+













∂

− ∂

∂ +

∂

∂ + ∂

∂

∂ = +∂

∂

∑

α α α

µ µ ρ ρ

) (

Pr) 1 ( 1 Pr

(2-7)

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where h_α denotes the enthalpy of species α^{and q}R is a source term with the origin from thermal radiation. Pr and Sc are the dimensionless Prandtl and Schmidt numbers respectively.

The final major equation is the equation of state for ideal gases [1]

∑

=

α α

ρ α

W RT Y

p (2-8)

where R is the universal gas constant, T is the temperature and W_α is the molecular weight of species α^.

3 Turbulence

There are three possible flow situations, laminar, transitional and turbulent. Laminar flows are smooth and structured, transitional flows occurs when laminar flow becomes turbulent. Turbulent flows seem irregular, cha- otic and unpredictable and are characterised by a rapid rate of diffusion of momentum and heat. Most flows in engineering problems are turbulent.

In 1883 [13] Osborne Reynolds performed experiments on pipe flow.

This was the first systematic work on turbulence, and he found that the flow becomes turbulent when the non-dimensional ratio

ν

=UL

Re (3-1)

exceeds a certain value. The ratio is called Reynolds number, U is the ve- locity, L is the length scale (here, the diameter of the pipe) and ν^{is the} kinematic viscosity. Turbulence occurs at high Reynolds numbers and it consists of random fluctuations in both space and time. It should be noted that turbulence is a property of the flow and not of the fluid.

3.1 Scales of Turbulence

A turbulent flow field consists of a spectrum of length scales and is therefore often described by eddies of various sizes. The eddies can be connected to a local scale of velocity, time and length. The length scales varies from the largest scales that depends on the geometry to the smallest scales that depends on the process of dissipation of energy. Three different length scales are often referred to in turbulent flow fields, the integral scale, l, the Taylor microscale, λ, and the Kolmogorov microscale, η^.

The integral length scale is defined [2] by

∫

∞

=

0

) (X dX f

l (3-2)

where f ( X ) is a normalized autocorrelation function defined by

) ( ) (

) ( ) ) (

( u x u x

X x u x X u

f ′ ′

′ +

= ′ (3-3)

where

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) ( ) ( )

(x u x X F X

u′ ′ + = (3-4)

is the autocorrelation of a single variable u’(x) at two positions. u’ is the ve- locity fluctuation. The correlation can be computed as follows: Obtain a number of records u’(x) and read off the values of u’ at x and x+X. Then multiply the two values of each record and calculate the average value.

The Taylor microscale is related to the mean rate of strain and is defined [6] by

2 1 2











 



 





∂

= x u u_RMS

λ ^(3-5)

where the denominator represents the mean strain rate. uRMS is the RMS turbulent velocity.

The Kolmogorov microscale is the smallest length scale associated with a turbulent flow and is the scale at which molecular effects are significant.

The Kolmogorov microscale is defined [13] by

4 1 3





= ε

η ν ^(3-6)

where ε is the turbulent dissipation rate.

4 Combustion

There are basically two types of flames, premixed flames or non- premixed (diffusion) flames. In a premixed flame, the fuel and the oxidizer are mixed before any significant chemical reaction occurs. In a diffusion flame, the reaction occurs only at the interface between the fuel and the oxidizer, where mixing and reaction both take place. Both types of flames can be either laminar or turbulent. Only turbulent premixed flames will be discussed here.

An important factor in the understanding of turbulent combustion is the different time and length scale ratios. The interaction between chemistry and turbulence can be classified by two criteria: premixed or non-premixed combustion, slow or fast chemistry. Slow chemistry has few practical appli- cations and this implies that fast chemistry occur in nearly all situations. To have stable combustion it must be rapid and all chemical time scales associated with the process must be small. In chapter 3.1, scales of turbulence were discussed and this concept will be expanded here. If Favre-averaged (discussed in chapter 6.1.2) quantities are used (denoted by a tilde), the integral length scale can be expressed [1] as

ε^~ u 3

l= c^d ′ (4-1)

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where cd is a constant with the value 0.37 and u’ is the velocity fluctuation.

With the integral length scale it is possible to express the turbulent time

u t_T l

= ′ (4-2)

The Kolmogorov length and time scales expressed in Favre-averaged quantities are

4 1 3

~ 



= ε

η ν ^(4-3)

2 1

~



 



= ε

η ν

t (4-4)

As an intermediate scale, the Taylor length scale

2 1 2

~ 

 ′

= νε

λ ^u ^(4-5)

may also be considered. Define the Damkhöler number as the ratio of the turbulent time to the flame time

F T

t

Da= t (4-6)

where

L F

F u

t = l (4-7)

l_F is the flame thickness and u_L is the laminar burning velocity. Define the Karlovitz number as

η ^tη

t Ka l^F = ^F





=

2

(4-8)

Based on the integral scales a turbulent Reynolds number may be defined as

ν l u

T

= ′

Re (4-9)

The turbulence Reynolds number has a more direct coupling to the structure of turbulence than the ordinary Reynolds number. It is apparent that Re_T <

Re.

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4.1 Burning Velocities

The flame velocity, also called burning velocity or laminar flame speed is defined [7] as the velocity at which unburned gases move through the combustion wave in the direction normal to the wave surface.

The first theoretical analyses for determination of the burning velocity falls into three categories: thermal theories, diffusion theories and compre- hensive theories. Only thermal theories will be discussed here. The theory of Mallard and Le Chatelier stated that the burning velocity is given by the expression

( )

^^_^







−

= −

δ ρ

λ 1

u i p u

i b c

L c T T

T

u T (4-10)

where λc is the thermal conductivity, ρu is the density of the unburned gas, cp is the specific heat and δ is the thickness of the reaction zone. The sub- scripts b, u and i stands for burned, unburned and ignition. There are two things to point out with this expression, firstly an actual ignition temperature does not exist in a flame and secondly δ is unknown. Therefore a better representation is required.

The theory of Zeldovich, Frank-Kamenetskii and Semenov is an exten- sion of the theory of Mallard and Le Chatelier. Their theory states that the burning velocity is given by

( )

2 1

2 2

















 −























=  ⁻

u b RT b

E

p u

c u

L ET T

Ze RT c

u a ^b

ρ

λ (4-11)

where a is the mass of reactant per cubic centimetre, Z is the pre- exponential factor in the Arrhenius expression, E is the activation energy. In this expression it is assumed that the number of moles is constant during the reaction. This expression can however be extended to allow the number of moles to change.

Unlike a laminar flame, which has a propagation velocity that depends uniquely on the thermal and chemical properties of the mixture, a turbulent flame has a propagation velocity that depends on the character of the flow, as well as the mixture properties. Damköhler proposed for large-scale, small-intensity turbulence that the turbulence burning velocity, uT, is given by

u u

u_T = _L + ′ ^(4-12)

where u’ is the fluctuation of the unburned gases ahead of the flame front.

Shchelkin also considered large-scale, small-intensity turbulence and he assumed that the flame surfaces are cones, formed through distortion. The bases of the cones are assumed to be proportional to l, the height to u’ and to the time t. Shchelkin's expression is given by

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2 1

2 2

1 













 ′ +

=

L L

T u

u u

u (4-13)

Clavin and Williams made a more rigorous development of a model and dropped the factor 2 in Shchelkin’s expression. Isotropic turbulence is assumed and the expression is given by

2 1 2

1 













 ′ +

=

L L

T u

u u

u (4-14)

In a laminar premixed flamelet (flamelets are discussed in chapter 4.2) structure with a coordinate system attached to the flame front, the continuity equation for this simple case [12] reduces to

T uu const

u ρ

ρ~= .= (4-15)

The above expression makes it possible to express the turbulent burning velocity as

τ ρ

ρ

= +

= 1

b b u b T

u u

u (4-16)

where ub is the velocity of the burned gases and τ is a heat release parameter defined as

−1

=

u b

T

τ T ^(4-17)

4.2 Laminar Flamelets

Requirements for the applicability of thin laminar flamelet models can be identified by plotting for example, the logarithm of the ratio of the fluctuating velocity to the laminar burning velocity versus the logarithm of the ratio of the integral length scale to the flame thickness. Different burning regimes may be identified in the diagram in terms of Damköhler, Karlovitz and turbulence Reynolds dimensionless numbers, figure 4.1 [1]. The laminar flamelet regime is defined to lie below the line Ka = 1 and above and to the right of the line Re = 1 to avoid low Reynolds number effects. The line Ka = 1 is called the Klimov-Williams line. The flamelet regime is where Ka < 1 so l < η and the smallest scales of turbulence can not enter the laminar flame structure. The Damkhöler number is always Da > 1 (fast chemistry) and this means that the time scales associated with turbulence are always greater than those associated with combustion.

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Figure 4.1. Burning regimes in premixed turbulent combustion. Here s_L = u_L

The structure of a turbulent premixed flame can be viewed by superposition of instantaneous reaction fronts at different times, figure 4.2 [6].

Figure 4.2. Superposition of instantaneous reaction fronts at different times.

The instantaneous view shows however that the reaction fronts are rela- tively thin as in a laminar premixed flame. These thin reaction fronts are sometimes referred to as laminar flamelets. The laminar flamelet concept views a turbulent flame as an ensemble of thin locally one-dimensional structures within the turbulent flow field. In the flamelet approach, the chemical time scales are much smaller than the time scales associated with the turbulence. The eddies connected to the turbulence can not change the local structure of the flame and this is the reason why a turbulent flame can be viewed as an ensemble of locally laminar flames, called flamelets.

5 Direct Numerical Simulation (DNS)

In DNS all equations are solved directly for a particular geometry and thereby obtaining all the detailed and statistical information without further approximation. All the length scales and the time scales are resolved so there is no need for a turbulence model. The disadvantage with DNS is that

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it requires a lot of computational power and therefore it can only be applied to flows with low Reynolds numbers and simple geometries. Computations with high Reynolds numbers, so that the flow is fully turbulent is presently beyond the capabilities of available computational power.

It can be shown that the range of length scales increases rapidly with the Reynolds number because

4 3

∝Re η

l (5-1)

In a three-dimensional calculation, the number of required grid points (N_g) for adequate resolution increases with the Reynolds number as

4 3 9

∝Re





∝ η

N_g l (5-2)

It is this situation which limits DNS not only to simple flow geometries but to flows with Reynolds numbers smaller than those of interest. Despite the limitations with DNS it provides important information to the analysis of turbulence. This discussion clearly indicates the need for methods for the treatment of flows with more general geometries and higher Reynolds numbers.

6 Computational Fluid Dynamics (CFD)

Analytical solutions of Navier-Stokes equations exist only for very simple idealised flow situations. As discussed in chapter 5, DNS is only appli- cable to low-Reynolds number flows with simple geometry. In CFD, all equations are averaged which leads to additional, unknown terms – the Reynolds stress and the turbulent scalar flux, the second moments. The second moments appear as unknowns in the first moment equations (the averaged equations). The new unknowns have to be modelled and there are four main categories of turbulence models [10], viz., Algebraic (zero-equation) models, One-Equation models, Two-Equation models and Second Moment Closure models. The Standard k-ε Model is an example of a Two-Equation model and it is well known that this model have serious limitations. Clo- sures at the second moment level represent a reasonably detailed description of turbulence at an acceptable level of complexity. The Reynolds Stress Model is an example of closure at the second moment level.

The equations are then solved numerically. To solve the equations numerically, the equations have to be discretized. The discretization is achieved by approximating the governing equations with algebraic expressions. Interpolation is then used to get values at the surface of the control volume and there are several ways to do this. The Upwind Interpolation Scheme and the Central Differencing Scheme are discussed here.

6.1 Averaging procedure

For an arbitrary quantity φ the decomposition into a mean and fluctuating part can be written [14] as

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)*

, ( ) , ( ) ,

(x_i t φ x_i t ϕ x_i t

φ = + ^with ϕ^* =⁰ ^(6-1)

The term on the left-hand side is the instantaneous value, the first term on the right-hand side is the mean value and the second term on the right-hand side is the fluctuating value. The mean value in Eq. (6-1) can be obtained using a ensemble averaging procedure

∑

=

∞

= → ^N

n

n N i

i W x t

t N x

1

) , 1 (

lim )

,

( φ

φ ^(6-2)

where N is the number of ensembles and W denotes a weighting function.

For steady-state flows, the mean value can also be obtained using a time averaging procedure

dt t x t W x

t t i

i =∆→∞∆ ^∆

∫

0

) , 1 (

lim )

( φ

φ ^(6-3)

6.1.1 Conventional Averaging or Reynolds Averaging

Setting the weighting factor in Eqs. (6-2) and (6-3) to unity, results in the conventional Reynolds averaging procedure, used mainly for incompressible flows. In this case Eq. (6-1) is written as

φ φ

φ = + ^′ with φ′=0 (6-4)

where the bar denotes the Reynolds average. If the averaging is applied to the continuity equation for an incompressible flow field it becomes

=0

∂

i i

x

u (6-5)

For the same flow field, the momentum equations are

) ( )

( _i _j

j j

i j j j

i i j

u x u

x u x

x p x

u u t

u ′ ′

∂

− ∂

∂

∂ + ∂

∂

− ∂

∂ = + ∂

∂

∂ ρ µ ρ

ρ ^(6-6)

An additional term is introduced into the momentum equations, the last term in Eq. (6-6), and it is called the Reynolds stresses. This term cannot be expressed in terms of the averaged quantities and has to be modelled. If the density is varying and the decompositions is applied, the continuity equation becomes

=0

∂

′

∂ ′

∂ + +∂

∂

i i i

i

x u x

u t

ρ ρ

ρ (6-7)

Note that an additional term appears, the last term in the continuity equation.

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6.1.2 Density-weighted Averaging or Favre Averaging

To avoid this additional term, a density weighted averaging procedure, called Favre averaging, is used by setting the weighting factor [14] in Eqs.

(6-2) and (6-3) to

ρ

= ρ

W (6-8)

With this decomposition into a mean and fluctuating part, Eq. (6-1) can be written as

φ φ ρ φ

φ = ρφ + ′′= ~+ ^′′

with ~ ′′ =0

′′= ρ

φ

φ ρ ^(6-9)

where the tilde denotes the Favre average. The Favre average is applied to the continuity equation and it becomes

0

~ =

∂ +∂

∂

=∂

∂ + ∂

∂

i i i

i

x u t

ρ ρ ρ ρ

(6-10)

By applying the density weighted average, the problem with an additional term, as in Eq. (6-7) is avoided. This is the reason for using Favre averaging on density varying flows. The Favre averaged momentum equations are

) ( )

~ 3

~ 2

~ (

~

~ ~

j i j k

k ij i

j j i j

j

j i i j

u x u

x u x

u x u x

x p

x u u t

u

′′

∂ ′′

− ∂













∂

− ∂

∂ +∂

∂

∂ + ∂

∂

− ∂

∂ = +∂

∂

ρ δ

µ ρ ρ

(6-11)

where the Reynolds stresses are still present and have to be modelled. But applying the Favre averaging procedure has not increased the number of the terms that have to be modelled. In a density varying flow field, such as a reacting flow field, all the governing equations are averaged using the Favre averaging procedure. All quantities except, the pressure and the density, are averaged using Favre average.

6.2 Turbulence Modelling

There are a few ways possible for numerical prediction of a turbulent flow. These ways still range from a solution of the steady Reynolds- Averaged Navier-Stokes (RANS) equations to a Direct Numerical Simula- tion (DNS), with Large-Eddy Simulation (LES) in between. Recent years have added other strategies, URANS (Unsteady RANS) and DES (Detached Eddy-Simulation) for example. Advantages and disadvantages with DNS are discussed in chapter 5. In LES, the governing equations are averaged using a space average. The space average also introduces unknown terms that have to be modelled. Very fine grids are required to resolve all length scales. URANS calculations are based on RANS models but are unsteady,

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even with steady boundary conditions. DES is a mixture of RANS and LES.

In [11], a summary of the strategies is made, see table 1.

Table 1. Summary of strategies

Name Unsteady Re-depend-

ence

Empiricism Grid Ready

3DRANS No Weak Strong 10⁷ 1990

3DURANS Yes Weak Strong 10⁷ 1995

DES Yes Weak Strong 10⁸ 2000

LES Yes Weak Weak 10^11.5 2045

DNS Yes Strong None 10¹⁶ 2080

Column 4 refers to, if the strategy depends on empirical relations, column 5 represents the number of grid points for a certain case and the readiness es- timates are based on that computer power increases by a factor 5 every five years. Reynolds (Favre) averaged equations will be discussed here.

The fundamental problem of turbulence for the engineer is the appear- ance of the Reynolds stress tensor

j i ij =−ρu′u′

σ ^(6-12)

which is a symmetric tensor and thus has six independent components. All six components are unknown but it is possible to derive a differential equation for the Reynolds stress tensor [10], the Reynolds stress equation (incompressible) is



 



 +

∂

∂ + ∂

−

∂ +

− ∂

∂

− ∂

∂ = + ∂

∂

ijk k

ij k

ij ij k

i jk k

j ik

k ij k ij

x C x

x u x

u u x t

ν σ π

ε σ

σ

σ σ

(6-13)

where

ik j jk

i k j i ijk

i j j i ij

k j k

i ij

u p u

p u u u C

x u x p u

x u x u

δ δ

ρ π

µ ε

′ + ′

′

= ′













∂

∂ ′

∂ +

∂ ′

= ′

∂

∂ ′

∂

∂ ′

=2

(6-14)

This exercise illustrates the closure problem of turbulence. Eq. (6-13) is a equation at the second moment level and in Cijk third moments appear. As higher and higher moments are taken, additional unknowns are introduced.

The derivation of Eq. (6-13) is strictly mathematical and to obtain closure, one tries to introduce physical properties to approximate the unknown correlations.

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6.2.1 The Standard k-ε Model

The Standard k-ε Model is an example of a Two-Equation model and this means that closure is obtained through two extra partial differential equa- tions, one for the turbulence kinetic energy, k, and one for the turbulence dissipation rate, ε^{. k and}ε are defined [10] as

i iu u k = ′ ′

2

1 (6-15)

k i k

i

x u x u

∂

∂ ′

∂

∂ ′

=ν

ε ^(6-16)

An equation for k is obtained by taking the trace of Eq. (6-13), that is σii = - 2ρk. The third moments are still present and they are modelled by an eddy viscosity, µT. The eddy viscosity is defined as

ρ ε

µT = ^C_µ ^k² (6-17)

where C_µ is a constant. In the Standard k-ε Model, Boussinesq eddy- viscosity approximation assumes to be valid and therefore, the Reynolds stress tensor given by

ij ij

T ij i

j j i T

ij k S k

x u x

u ρ δ µ ρ δ

µ

σ 3

2 2 3

2 = −

−













∂ +∂

∂

= ∂ (6-18)

The exact equation for ε is more complicated to derive and will not be discussed here. See appendix 6 to see the exact ε-equation.

6.2.2 Reynolds Stress Model

The Boussinesq eddy-viscosity approximation assumes that the principal axes of the Reynolds stress tensor, σij, are coincident with those of the mean strain-rate tensor, Sij, at all points in a turbulent flow. Experimental evidence shows that flow history, effects σij for long distances in a turbulent flow.

This is why there is a doubt about the simple linear relation (6-18). It is often capable of providing good results, but it is incapable of predicting swirling flows, the effects of strong streamline curvature and flows with boundary-layer separation, just to mention some of its limitations.

A model, based on closures at the second moment level naturally include effects of streamline curvature, swirling flows and effects of flow history. A second moment closure model is based on the exact Reynolds stress equation, Eq. (6-13). This equation gives no reason for the normal stresses to be equal (but they can of course be equal), i.e. it takes anistropy into account.

To close Eq. (6-13), models for the dissipation tensor, εij, the turbulent transport tensor, Cijk, and the pressure-strain correlation tensor, πij, are nec- essary.

The disadvantage with the Reynolds stress model is that it increases the number of averaged equations which leads to an increased number of terms

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to model. The price to be paid is in complexity and computational difficulty for the gains above.

6.3 Combustion Modelling

The chemical source term q_α in Eq. (2-6) is the connection between fluid mechanical and chemical behaviour of (turbulent) reacting flows. When Eq.

(2-6) is averaged, an additional term appears, similar to the Reynolds stress tensor (Eq. (6-12)). This term is called the turbulent scalar flux and is given by

α ρ α

θ_i⁽ ⁾ =− u_i′′Y′′ ^(6-19)

and it also requires modelling.

6.3.1 Finite Rate Formulation

This approach is based on the solution of the species transport equation (Eq. (2-6)). The turbulent scalar flux (Eq. (6-19)) is modelled using a turbulent version of Fick’s Law (Eq. (2-5)) called gradient diffusion

( )

i T T

i x

Y

∂

− ∂

= ^α

α µ

θ Pr ^(6-20)

where PrT is the turbulent Prandtl number.

The chemical source term is computed from Arrhenius rate expressions or by using the eddy-dissipation concept. q_α is a consequence of the creation and destruction of species α due to each reaction step [3], i.e.

∑

= =

=

= ^R

j j R

j

j W q

q q

1 1

ˆ_α

α α

α (6-21)

where qαj = 0 if species α does not participate in reaction step j. qαj is the reaction rate and qˆ is the molar rate of creation/destruction of species _α_j αⁱⁿ reaction j. The Arrhenius molar rate expression is

(

⁻

)

_^

^{[ ]}

⁻

^{[ ]}

_^

Ψ

=

∏ ∏

= =

c c p

kj r

kj

N

k

N

k k bk k

fk r

j p

j

j k C k C

q

1 1

ˆ_α υ_α υ_α ^η ^η ^(6-22)

where Ψ represents the net effect of third bodies on the reaction rate, υ^p^and υ^r are stoichiometric coefficients for products and reactants respectively, k_f and kb are forward and backward (if the reaction is reversible) rate con- stants. Ck is the molar concentration of each reactant or product and η^p^and η^r are rate exponents for products and reactants respectively. kfk is computed using the Arrhenius expression

RT E k

fk

k ke T Z

k = ^β ⁻ (6-23)

where βk is a temperature exponent.

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The influence of turbulence on the reaction rate is taken into account by using the Eddy-Dissipation model, In this model, the rate of reaction is given by the limiting value of the two expressions below

∑

=

∑

=

M

k k p kj p

p r

j j

r r j r r

j j

W W AB k

W q

W Y A k

W q

υ ρε υ

α α α

(6-24)

where A and B are empirical constants equal to 4 and 0.5 respectively, index r and p stands for reactants and products and M is the number of chemical species. In FLUENT, there is a possibility to use both models at the same time. The slowest rate, given by Eq. (6-22) or Eq. (6-24) is used as the reaction rate and the contributions to the source terms in the species conservation and energy equations are calculated from this reaction rate.

6.3.2 The Bray-Moss-Libby (BML) Model

In the BML Model a reaction progress variable, c(x,y,z,t), is introduced.

This approach is based on the solution of the reaction progress variable transport equation (compare with Eq. (2-4))

c i c i i

i q

x J x

c u t

c +

∂

−∂

∂ = +∂

∂

∂ρ ρ ⁽ ⁾

(6-25)

The reaction progress variable is assumed to determine completely the en- tire state of the gas. This variable has the value zero in reactants, the value one in products and intermediate values in gas which may be undergoing chemical reaction. The reaction progress variable can be defined for example as

fb fu

f fu u b

u

Y Y

Y Y T T

T c T

−

= −

−

= − (6-26)

where Y_f is the mass fraction of fuel. Subscripts u and b stands for unburned and burned respectively. Delta functions at c = 0, 1 is describing the distri- bution of c. They denote the probability of finding reactants and products in a point.

When Eq. (6-25) is averaged, a term similar to (6-19) appears (but, it is expressed in c’’). It is possible to derive a differential equation in terms of the turbulent scalar flux similar to Eq. (6-13) and this equation also requires modelling. In the BML Model they have obtained closure of the second moment equations in the flamelet (flamelets are discussed in chapter 4.2) regime of combustion.

Now, a model for the source term, q_c, is required. Lindstedt and Váos [12] proposed a model based on the assumption that the flamelet geometry is fractal, and is given by

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(

^c

)

c v k C u q

K L u R c

1 ~

~~

~

0

−

= ρ ε ^(6-27)

where C_R is a constant, u_L⁰ is the unstrained laminar burning velocity and v_K is the Kolmogorov velocity (ratio of Eq. (4-3) and Eq. (4-4)). A version of (6-27) is used in the GENUS code.

6.4 Discretization

The Finite Volume Method (usual method in CFD codes) uses the integral form of the conservation equations. The grid is the spatial discretization of the domain and it divides the domain into a finite number of control volumes. The usual approach is that the grid defines the control volume boundaries and the nodes are in the centre of the control volumes. The conservation equations are applied to each control volume and the values of the variables are to be calculated in the cell centre, at the node. All the integrals are approximated by an algebraic equation. Interpolation is used to get values at the surface of the control volume in terms of the surrounding node values.

6.4.1 Upwind Interpolation, UDS

A typical control volume and the notation used for a one-dimensional grid is shown in figure 6.1 [4].

Figure 6.1. One-dimensional grid

Depending on the flow direction, φe (φ is an arbitrary variable) is ap- proximated by its value at the node upstream of e (at the face e). In UDS φe

is approximated by



=

E P

e φ

φ φ

if

( ) ( )

e ^<^>⁰⁰

e

u u ρ

ρ (6-28)

This is shown graphically in figure 6.2.

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Figure 6.2. Graphic representation of UDS interpolation

The value of φe comes from the upstream side of the face, hence the name Upwind Differencing Scheme (UDS) for the approximation. The Upwind Differencing Scheme is the only approximation that satiesfies the boundedness criteria (Numerical solutions should lie within proper bounds. Physi- cally non-negative quantities should always be positive, concentrations must lie between 0% and 100%.) unconditionally and that means that it will never give oscillatory solutions. A disadvantage with UDS is that it is numerically diffusive. Taylor series expansion about P for a cartesian grid and (ρ^u)e > 0 gives

x H x

x x x

x

P P

e P P

e P

e +





∂

− ∂

 +



 





∂

− ∂ +

= ² ²₂

! 2

) ) (

( φ φ

φ

φ ^(6-29)

where H is higher order terms. The UDS approximation consists only of the first term on the right-hand side so the truncation error term is diffusive

e e

t x

e 



 





∂ Γ ∂

= φ

(6-30)

The coefficient of numerical, artificial or false diffusion is

2 ) ( u _e x

num e

= ∆

Γ ρ

(6-31)

where ∆x is the length of the cell.

Very fine grids are required to obtain accurate solutions.

6.4.2 Linear Interpolation, CDS

Another approach to approximate the control volumes face centre is with linear interpolation between the two nearest nodes. If the nodes are at equal distance from the face then φe is approximated by

φP

φE

(ρ^u)e > 0 e

φP

φE

(ρ^u)e < 0 e

φe = φP

φe = φE

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2

E P e

φ

φ =φ ⁺ (6-32)

In a more general case φe is approximated by )

1

( _e

P e E

e φ λ φ λ

φ = + − (6-33)

where

P E

P e

e x x

x x

−

= −

λ ^(6-34)

With the above expression (6-33) one can show that the leading truncation error term is proportional to the square of the grid spacing, on uniform and non-uniform grids. This scheme does not fulfil the boundedness criterion unconditionally and it may produce oscillatory solutions.

6.5 Under-relaxation

If steady problems are to be solved, one can use under-relaxation to de- crease the difference between iterations. This might be helpful when insta- bility problems are present. Allow φⁿ to change a fraction γ of the real change

(

¹

)

1 −

− + −

= ⁿ ^new ⁿ

n φ γ φ φ

φ ^(6-35)

The under-relaxation factor satisfies 0 < γ < 1. It is possible to derive a relation between the under-relaxation γ and the time step ∆t, this means that a time step is used even when steady problems are solved. Use of a constant under-relaxation factor is equivalent to applying a different time step to each control volume.

Note: When the pressure-velocity coupling SIMPLE is used, the sum of the under-relaxation factors for pressure and velocity should be close to unity [4]

=1 + _u

p γ

γ (6-36)

which has found to be nearly optimum.

6.6 Convergence Criterion

There is always a problem to know when a numerical simulation is good enough, to know when to interrupt. One way of doing this is to use residuals. Residuals are basically the difference between iteration number n+1 and n. When the residuals has reached a certain value then the iteration process will be interrupted because the difference for a number of certain parameters are so small that the solution is almost unaffected by further iteration.

7 Definition of test cases

The GENUS code will be used for turbulent reacting flows in combustors. In combustors, there are often swirling flows and recirculation zones,

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and of course combustion. It is important that the numerical tool that is used can predict these phenomenon accurately if CFD is to be used as a design tool. The first test case contains swirl and recirculation, the second one combustion.

7.1 Test Case 1: Solid body Rotation with Central Air Injec- tion

Swirling flows are frequently used in combustors. The computation of swirling flows is difficult due to large velocity gradients, resulting from vortices in the near-burner region. In the International Flame Research Foundation (IFRF) Near Field Aerodynamics Program, phase 2 isothermal, swirling flows are examined. Swirling flows are representative of gas tur- bines. The present test case considers a flow with isothermal conditions (no combustion) in a swirling combustor with central air injection. The fluid that was used is air with a density of 1.225 kg/m³ and a viscosity of 1.79e- 05 Ns/m².

Comparison of predicted axial and swirl velocities with IFRF data [15,16] is made to evaluate the performance of FLUENT and GENUS. At the same time it is possible to evaluate the performance of the different turbulence models. This is a test case that FLUENT used in February 1999 for validation of their turbulence models.

7.2 Test Case 2: One-dimensional Planar Flame Propagation

Turbulent premixed flames are of tremendous importance. Some appli- cations are spark-ignition engines, gas-turbine engines and industrial gas burners. In [5] turbulent burning velocities, uT, are measured for different RMS turbulent velocities, uRMS, in an explosion bomb. Four fans are used to create a uniform and isotropic turbulence field with little mean flow. u_RMS varies linearly with the fan speed.

The present test case is a semi-closed one-dimensional flame tube with a planar flame propagating towards the closed end. The turbulence field is frozen and is characterized by a specified RMS turbulent velocity and length scale of 10 mm. Calculations should be performed for a range of RMS turbulent velocities to see how the turbulent burning velocity varies. The mixture is stoichiometric methane/air at atmospheric conditions. The case is transient and computationally trivial, but it is well suited for evaluation of combustion models.

8 Simulation of test case 1

The purpose of this test case is to investigate how the codes can handle swirling flow that is commonly used in combustors. Experimental data from IFRF [15,16] makes it possible to check the accuracy of different turbulence models.

8.1 Geometry

The flow is modelled as 2D axisymmetric and as a 3D sector of 35°. The idea with this is twofold. Firstly, to see if there is any difference between a 2D axisymmetric model and a 3D sector. Secondly, it is only possible to solve the case in 3D with GENUS, the comparison between the codes is performed with the 3D models.

Predictive Tools for Turbulent Reacting Flows

MASTER’S THESIS

Predictive Tools for Turbulent Reacting Flows

A comparison between FLUENT and GENUS

MAGNUS PERSSON

Abstract

Preface

Table of Contents

Nomenclature

1 Introduction

2 The Conservation Equations

∑

∑

3 Turbulence

3.1 Scales of Turbulence

∫

4 Combustion

4.1 Burning Velocities

( )

( )

( )

4.2 Laminar Flamelets

5 Direct Numerical Simulation (DNS)

6 Computational Fluid Dynamics (CFD)

6.1 Averaging procedure

∑

∫

6.2 Turbulence Modelling

6.3 Combustion Modelling

∑

∑

(

)

[ ]

[ ]

∏ ∏

∑

∑

(

)

6.4 Discretization

( ) ( )

6.5 Under-relaxation

(

)

6.6 Convergence Criterion

7 Definition of test cases

7.1 Test Case 1: Solid body Rotation with Central Air Injec- tion

7.2 Test Case 2: One-dimensional Planar Flame Propagation

8 Simulation of test case 1

8.1 Geometry

^{[ ]}

^{[ ]}