The Role of CFD in Modern Jet Engine Combustor Design

*Zhi X. Chen, Ivan Langella and Nedunchezhian Swaminathan*

### **Abstract**

Recent advances in the application of computational fluid dynamics (CFD) for turbulent combustion with the relevance for gas turbine jet engines are discussed. Large eddy simulation (LES) has emerged as a powerful approach to handle the highly turbulent, unsteady and thermochemically non-linear flows in the practical combustors, and it is a matter of time for the industry to replace the conventional Reynolds averaged Navier-Stokes (RANS) approach by LES as the main CFD tool for combustor research and development. Since combustion is a subgrid scale phenomenon in LES, appropriate modelling is required to describe the SGS combustion effects on the resolved scales. Among the various available models, the flamelet approach is seen to be a promising candidate for practical application because of its computational efficiency, robustness and accuracy. A revised flamelet formulation, FlaRe, is introduced to outline the general LES methodology for combustion modelling and then used for a range of test cases to demonstrate its capabilities for both laboratory burners and practical engine combustors. The LES results generally compare well with the experimental measurements showing that the important physical processes are captured in the simulations.

**Keywords:** jet engine combustion, computational fluid dynamics, large eddy simulation, turbulent combustion modelling, FlaRe

#### **1. Introduction**

Over the past half a century, the combustor design for jet engines has been driven to meet the increasingly higher standards of thermal efficiency, emission reduction and power-to-size/weight ratio. Consequently, the operating conditions have experienced a dramatic change, e.g., the operating pressure increasing from several bars to few tens of bars, combustor inlet temperature from about 500 to nearly 1000 K, and turbine inlet temperature from just above 1000 to almost 2000 K in today's turbo-fan engines [1]. Despite the severe change in the fluid-mechanical and thermodynamic properties of the combustor internal flow, the combustor length and frontal area are still strictly limited by the design factors for other engine components. Also, the current unprecedented demand for global travels requires a much longer lifespan for aero engines, typically many tens of thousands of hours without major maintenance. To meet these requirements, jet engine manufacturers continuously seek avenues for a reliable, efficient and economical combustor design cycle.

The emergence of computational fluid dynamics (CFD) has made computeraided design an integral part of the gas turbine (GT) combustor design process [2]. Compared to the expensive experimental tests, which provide only global information (e.g., stability, outlet properties), CFD is much cheaper to run and, most importantly it can be repeated during the design process to examine the effects of small design changes. Thus, it is attractive for practical applications. Over the recent years, there has been a significant increase in the investment from the industry for the development of CFD tools, but the challenges remain because fully resolving the turbulent reacting flows in practical jet engines using direct numerical simulation (DNS) is still far beyond our reach. The Reynolds averaged Navier-Stokes (RANS) approach has been broadly used as the main CFD tool for practical combustor design in the last few decades. Because all the scales are modelled in RANS, only mean flow quantities are computed leading to a cheap and fast calculation. For steady combusting flows, good accuracy can be achieved if the correlation between the fluctuating quantities is handled correctly. The drawback for RANS is also obvious since the transient phenomena, such as ignition, flashback, thermoacoustics and blow-off, cannot be captured by the mean flow calculation. While these phenomena are of high interest for the industry, there is an increasing need for predictive CFD tools. Large eddy simulation (LES) is recognised as a promising candidate as the energy-containing flow structures are directly resolved by the numerical grid and the subgrid scales (SGS) are modelled. In general, with respect to RANS the main advantages of LES are twofold: the capability to capture the transient phenomena and a better prediction of mixing. Thus, the LES modelling paradigm is of interest here.

controlling the computational effort at the same time. This is achieved by varying the LES filter, whose shape is implicit and depends on local mesh size, SGS model and numerical scheme. It is generally accepted that the filter size is proportional to

SGS modelling to the results, with the generally accepted rule that at least 80% of the turbulent kinetic energy should be resolved [3]. The general LES equation for a

> ¼ � *<sup>∂</sup><sup>ρ</sup> ∂xj*

where the overbar denotes a spacial filtering operation. For density varying flows the filtering operation would lead to additional unclosed terms and to avoid this a density-weighted, or Favre-filtered operator is introduced, defined as *φ*~ ¼ *ρφ=ρ*. Note that as *φ* 6¼ *φ*~, one has to be mindful when comparing CFD results with measurements in regions of strong density gradients, and this will be discussed again in Section 4. The terms on the LHS of Eq. (1) represent time variation and convection of the filtered quantity *φ*~. The pressure derivative on the RHS in present

only if *φ* is a velocity component, *uj*. The application of the LES filter to the

the effect of subgrid processes, and this term requires modelling. Well accepted models are available, e.g., the Smagorinsky model [6] for SGS stresses and the gradient transport model for unresolved scalar fluxes [2]. The filtered diffusion term, D, takes the form of a Laplacian and also may need modelling. However, this modelling is generally irrelevant for most high-turbulent conditions proper of gas turbines since the SGS turbulent diffusion is dominant. The last term in Eq. (1) is the filtered source term representing compressibility, or gravitational, or evaporation or heat release effects. For the equation of species the source term is the

For industrial gas turbine conditions the flame thickness, *δ*, is generally small, so

to keep the computational cost affordable for industrial operations, Δ is always larger than *δ* and thus the combustion processes are entirely at SGS level. Note that it is not generally a problem to satisfy the 80% rule, since the evaluation of the turbulent kinetic energy excludes the combustion dilatation effects [3, 7]. In light of the above considerations, recent development of combustion modelling has gone in two directions. One is to include the full thermochemistry into the modelling, and at

the expenses of computational cost. These types of modelling are usually

unaffordable for industrial design purposes, but together with DNS methodology they can be used to investigate complex phenomena and SGS processes in laboratory scale burners. The other approach is more industrial-oriented, where the objective is to keep the computational cost to a minimum so that the model is usable for practical combustors. The thermochemistry is included through statistical or geometrical means. Based on this distinction, these two directions can be

categorised respectively as non-flamelet and flamelet approaches. The gap in the accuracy between these two categories has reduced with time and recent advances have shown that flamelet-based models are capable of representing the complex flow features in gas turbines despite the limitations of their underlying assumptions. These models are reviewed in a number of works, see for example [2, 4, 8]. Because of the relevance for gas turbine applications, only the flamelet category is discussed here. Within the flamelet category there are geometrical and statistical approaches. Although in both cases the thermochemistry is computed *a priori*, the assumptions behind geometrical models usually lead to the need of relatively large mesh in order to achieve a significant increase of accuracy, see for example the

convective term leads to the appearance of the term *<sup>τ</sup>* <sup>¼</sup> *<sup>ρ</sup> <sup>u</sup>*f*<sup>i</sup><sup>φ</sup>* � *<sup>u</sup>*~*<sup>i</sup>*

reaction rate and its modelling is the objective of this section.

þ D þ

*∂τ ∂xi*

*∂ρu*~*iφ*~ *∂xi*

. The larger is Δ, the stronger is the contribution of the

þ *S,* (1)

*<sup>φ</sup>*~<sup>Þ</sup> � representing

the local cell volume, Δ ≈ V<sup>1</sup>*=*<sup>3</sup>

**47**

quantity *φ* takes the form, in Einstein's notation, *∂ρφ*~ *∂t* þ

*The Role of CFD in Modern Jet Engine Combustor Design*

*DOI: http://dx.doi.org/10.5772/intechopen.88267*

The level of resolved scales in LES dictates the computational cost, i.e., the more resolved the more expensive; however, it only partly determines the overall accuracy. A significant part of the accuracy is attributed to SGS modelling, which represents the influence of subgrid motion on the resolved scales. For the velocity field, this influence usually appears through the SGS eddy viscosity and in a wellresolved LES, i.e., typically over 80% of the turbulent kinetic energy is resolved [3], the SGS effect is relatively small. For the flame, however, chemical reactions occur at scales smaller than the typical LES filter size and thus are SGS phenomena requiring closure models. The major challenge is how to model the SGS interaction between turbulence and chemistry, with the latter involving a large number of species and reactions (typically many thousands for common jet fuels). Consequently, this makes it practically unfeasible to directly integrate detailed chemical kinetics into the LES. Finding a computationally efficient model with good accuracy and robustness for the SGS turbulence-chemistry interaction has been a central topic for turbulent combustion research in the last two decades, and a number of approaches are available in the literature. Extensive review of these approaches is beyond the scope of this chapter as detailed reviews are available elsewhere, see for example [2–5], and the focus here is to showcase the current capabilities of combustion LES modelling for practical applications.

The rest of this chapter is organised as follows. Section 2 describes the modelling challenges in LES of gas turbine combustion and a representative approach to tackle these challenges. The validation test cases for this approach are presented in Sections 3 and 4 for laboratory and practical burners respectively. The conclusions are summarised with a future outlook in Section 5.

#### **2. State-of-the-art LES modelling for gas turbine combustion**

With the advent of high performing computing, large eddy simulation has become increasingly popular to investigate complex and unsteady physics in gas turbines due to its versatility in capturing time-dependent phenomena and in

*The Role of CFD in Modern Jet Engine Combustor Design DOI: http://dx.doi.org/10.5772/intechopen.88267*

Compared to the expensive experimental tests, which provide only global information (e.g., stability, outlet properties), CFD is much cheaper to run and, most importantly it can be repeated during the design process to examine the effects of small design changes. Thus, it is attractive for practical applications. Over the recent years, there has been a significant increase in the investment from the industry for the development of CFD tools, but the challenges remain because fully resolving the turbulent reacting flows in practical jet engines using direct numerical simulation (DNS) is still far beyond our reach. The Reynolds averaged Navier-Stokes (RANS) approach has been broadly used as the main CFD tool for practical combustor design in the last few decades. Because all the scales are modelled in RANS, only mean flow quantities are computed leading to a cheap and fast calculation. For steady combusting flows, good accuracy can be achieved if the correlation between the fluctuating quantities is handled correctly. The drawback for RANS is also

obvious since the transient phenomena, such as ignition, flashback,

*Environmental Impact of Aviation and Sustainable Solutions*

ling paradigm is of interest here.

bustion LES modelling for practical applications.

summarised with a future outlook in Section 5.

**46**

thermoacoustics and blow-off, cannot be captured by the mean flow calculation. While these phenomena are of high interest for the industry, there is an increasing need for predictive CFD tools. Large eddy simulation (LES) is recognised as a promising candidate as the energy-containing flow structures are directly resolved by the numerical grid and the subgrid scales (SGS) are modelled. In general, with respect to RANS the main advantages of LES are twofold: the capability to capture the transient phenomena and a better prediction of mixing. Thus, the LES model-

The level of resolved scales in LES dictates the computational cost, i.e., the more resolved the more expensive; however, it only partly determines the overall accuracy. A significant part of the accuracy is attributed to SGS modelling, which represents the influence of subgrid motion on the resolved scales. For the velocity field, this influence usually appears through the SGS eddy viscosity and in a wellresolved LES, i.e., typically over 80% of the turbulent kinetic energy is resolved [3], the SGS effect is relatively small. For the flame, however, chemical reactions occur at scales smaller than the typical LES filter size and thus are SGS phenomena requiring closure models. The major challenge is how to model the SGS interaction between turbulence and chemistry, with the latter involving a large number of species and reactions (typically many thousands for common jet fuels). Consequently, this makes it practically unfeasible to directly integrate detailed chemical kinetics into the LES. Finding a computationally efficient model with good accuracy and robustness for the SGS turbulence-chemistry interaction has been a central topic for turbulent combustion research in the last two decades, and a number of approaches are available in the literature. Extensive review of these approaches is beyond the scope of this chapter as detailed reviews are available elsewhere, see for example [2–5], and the focus here is to showcase the current capabilities of com-

The rest of this chapter is organised as follows. Section 2 describes the modelling challenges in LES of gas turbine combustion and a representative approach to tackle these challenges. The validation test cases for this approach are presented in Sections 3 and 4 for laboratory and practical burners respectively. The conclusions are

**2. State-of-the-art LES modelling for gas turbine combustion**

With the advent of high performing computing, large eddy simulation has become increasingly popular to investigate complex and unsteady physics in gas turbines due to its versatility in capturing time-dependent phenomena and in

controlling the computational effort at the same time. This is achieved by varying the LES filter, whose shape is implicit and depends on local mesh size, SGS model and numerical scheme. It is generally accepted that the filter size is proportional to the local cell volume, Δ ≈ V<sup>1</sup>*=*<sup>3</sup> . The larger is Δ, the stronger is the contribution of the SGS modelling to the results, with the generally accepted rule that at least 80% of the turbulent kinetic energy should be resolved [3]. The general LES equation for a quantity *φ* takes the form, in Einstein's notation,

$$\frac{\partial \overline{\rho} \overline{\rho}}{\partial t} + \frac{\partial \overline{\rho} \tilde{u}\_i \tilde{\rho}}{\partial \mathbf{x}\_i} = -\frac{\partial \overline{\rho}}{\partial \mathbf{x}\_j} + \overline{\mathcal{D}} + \frac{\partial \overline{\pi}}{\partial \mathbf{x}\_i} + \overline{\mathcal{S}},\tag{1}$$

where the overbar denotes a spacial filtering operation. For density varying flows the filtering operation would lead to additional unclosed terms and to avoid this a density-weighted, or Favre-filtered operator is introduced, defined as *φ*~ ¼ *ρφ=ρ*. Note that as *φ* 6¼ *φ*~, one has to be mindful when comparing CFD results with measurements in regions of strong density gradients, and this will be discussed again in Section 4. The terms on the LHS of Eq. (1) represent time variation and convection of the filtered quantity *φ*~. The pressure derivative on the RHS in present only if *φ* is a velocity component, *uj*. The application of the LES filter to the convective term leads to the appearance of the term *<sup>τ</sup>* <sup>¼</sup> *<sup>ρ</sup> <sup>u</sup>*f*<sup>i</sup><sup>φ</sup>* � *<sup>u</sup>*~*<sup>i</sup> <sup>φ</sup>*~<sup>Þ</sup> � representing the effect of subgrid processes, and this term requires modelling. Well accepted models are available, e.g., the Smagorinsky model [6] for SGS stresses and the gradient transport model for unresolved scalar fluxes [2]. The filtered diffusion term, D, takes the form of a Laplacian and also may need modelling. However, this modelling is generally irrelevant for most high-turbulent conditions proper of gas turbines since the SGS turbulent diffusion is dominant. The last term in Eq. (1) is the filtered source term representing compressibility, or gravitational, or evaporation or heat release effects. For the equation of species the source term is the reaction rate and its modelling is the objective of this section.

For industrial gas turbine conditions the flame thickness, *δ*, is generally small, so to keep the computational cost affordable for industrial operations, Δ is always larger than *δ* and thus the combustion processes are entirely at SGS level. Note that it is not generally a problem to satisfy the 80% rule, since the evaluation of the turbulent kinetic energy excludes the combustion dilatation effects [3, 7]. In light of the above considerations, recent development of combustion modelling has gone in two directions. One is to include the full thermochemistry into the modelling, and at the expenses of computational cost. These types of modelling are usually unaffordable for industrial design purposes, but together with DNS methodology they can be used to investigate complex phenomena and SGS processes in laboratory scale burners. The other approach is more industrial-oriented, where the objective is to keep the computational cost to a minimum so that the model is usable for practical combustors. The thermochemistry is included through statistical or geometrical means. Based on this distinction, these two directions can be categorised respectively as non-flamelet and flamelet approaches. The gap in the accuracy between these two categories has reduced with time and recent advances have shown that flamelet-based models are capable of representing the complex flow features in gas turbines despite the limitations of their underlying assumptions. These models are reviewed in a number of works, see for example [2, 4, 8]. Because of the relevance for gas turbine applications, only the flamelet category is discussed here. Within the flamelet category there are geometrical and statistical approaches. Although in both cases the thermochemistry is computed *a priori*, the assumptions behind geometrical models usually lead to the need of relatively large mesh in order to achieve a significant increase of accuracy, see for example the

discussion on thickened flame [9, 10] model in Section 4, or the introduction of additional complexity to smooth the G-equation in level set models [11–13]. Thus, the use of this category of models for industrial applications is still unclear. The statistical models within the flamelet approach include the turbulence-chemistry interaction using probability density functions (PDFs), which are typically presumed and thus they do not incur additional computational effort. Although additional equations are still necessary depending on the particular model, these are generally computationally cheap to solve because the filter size can be kept larger than the flame thickness, at least in principle, as long as the presumed PDF used is able to represent the statistical behaviour at scales smaller than Δ correctly. The potential of flamelet-based models for GT applications has thus to pass through a deep understanding of the SGS processes, which has been the focus of research in the last 30 years. Only in recent years, however, revised flamelets formulations for LES have demonstrated potentials to bridge the gap that separated them from models directly accounting for the thermochemistry.

of flamelets. Following the above discussion, the parameterisation of a model based on unstrained premixed flamelets at various equivalence ratios reduces to only two controlling variables: mixture fraction, *ξ*, to track the amount of mixing (thus the equivalence ratio) and a progress variable, *c*, to track the reaction progress. The first is usually defined using Bilger's expression [25], and assuming that all species have

where De*ff* is the effective molecular diffusivity (sum of filtered diffusivity and the SGS contribution due to the filtering of the non-linear terms). The source term, *ω*\_ *<sup>S</sup>*, is for the evaporation of fuel droplets [23, 26, 27]. The progress variable is usually defined as a combination of reactant or product species and varies monotonically from 0 in the reactants to 1 in the products when it is normalised appropriately, although unscaled formulations are often used. A good choice for lean combustion is to define the progress variable as a linear combination of CO2 and CO [28] mass fractions (normalised by their burnt value). The transport equation for

*Dt* <sup>¼</sup> <sup>∇</sup> � *<sup>ρ</sup>Deff* <sup>∇</sup>~*c*Þ þ *<sup>ω</sup>*\_

accessed using the controlling variables themselves during the simulation runtime.

contribution (including mixture stratification). The additional terms represent nonpremixed mode contribution and mixed mode due to the interaction of *ξ* and *c* gradients, and they appear only for normalised definitions of the progress variable, see additional details for example in [22, 25, 29]. At this point the set of equations would be closed if the above reaction rates depend only on the two controlling variables, as long as a thermodynamic model is provided. However, in this form the effect of wrinkling of the flame by turbulence at the subgrid level is not accounted. This effect is introduced in a statistical way using a presumed subgrid PDF<sup>2</sup>

where *ω*\_ is the laminar reaction rate from the flamelet and *ζ* and *η* are the sample space variables for *c* and *ξ* respectively. This type of closure was first introduced by Bradley for RANS and non-premixed combustion [30]. The SGS joint PDF can be written as the product of two PDFs as *P*ð Þ¼ *ζ; η P*ð Þ*ζ P*ð Þ *η*j*ζ* . There are various possible choices for the shape of these two SGS PDFs, each with its own advantages and disadvantages, the most common being the Beta PDF and laminar flamelet PDF [31], and it is commonly accepted that these shapes need to be dependent at least on first and second moments. However, for cases involving large turbulence and filter

sizes larger than the flame thickness, the Beta PDF was shown to be more

<sup>2</sup> Note that the term PDF in this case does not strictly relates to a probability density function in a statistical sense, since in the LES this operator is used to account for events in space at one time. The use of the term subgrid PDF is thus made for simplicity and analogy to the statistical PDF operator; other

*<sup>c</sup>* is the modified filtered reaction rate, which is tabulated and thus is

*Dt* <sup>¼</sup> <sup>∇</sup> � *<sup>ρ</sup>*D*eff* <sup>∇</sup>~*ξ*Þ þ *<sup>ω</sup>*\_ *S,* � (2)

∗

*<sup>c</sup>* ¼ *ω*\_ *<sup>c</sup>* þ *ω*\_ <sup>n</sup>*<sup>p</sup>* þ *ω*\_ <sup>c</sup>*<sup>t</sup>*, where *ω*\_ *<sup>c</sup>* is the premixed flame

*<sup>c</sup> ,* � (3)

*ω ζ* \_ð Þ *; η P*ð Þ *ζ; η dζdη,* (4)

:

the same mass diffusivity, its transport equation is:

*The Role of CFD in Modern Jet Engine Combustor Design*

*DOI: http://dx.doi.org/10.5772/intechopen.88267*

the filtered progress variable is:

where *ω*\_

**49**

∗

This term is expressed as *ω*\_

*ρ D*~*ξ*

*ρ D*~*c*

∗

*ω*\_ *<sup>c</sup>* ¼ ð1 0 ð1 0

authors prefer the term 'filtered density function' to make this distinction.

In flamelet models, the turbulent flame is seen as an ensemble of thin, onedimensional structures (flamelets) which are wrinkled by turbulence; turbulent eddies are either not small enough to penetrate into the flame and alter its internal structure, or they do not last long enough. Therefore, the thermochemistry can be computed *a priori* through one-dimensional computations and then parameterised using a set of control variables. For partially premixed combustion these are usually a variable to track the rate of mixing, and another to track the reaction progress. Other parameters can be introduced to include additional effects such as pressure variation, non-adiabaticity, strain, etc. The 1D laminar flames can be of any type, e.g. they can be premixed or diffusion flame. Premixed flamelets are, however, more versatile for partially premixed combustion as strain, reaction progress and mixing can be controlled independently; thus they can, in principle, be used for situations involving local extinctions. The big challenge for using flamelet models for GT combustion conditions is that turbulence is extremely high and the smallest eddies can penetrate and affect the internal flame structure, thus invalidating the flamelet hypothesis. This can happen when the Karlovitz number, *Ka* ¼ *τc=τη* ≫ 1, where *τ<sup>c</sup>* and *τη* are chemical and the Kolmogorov time scales respectively. Nevertheless, a number of relatively recent works (see for example [14–16]) have shown that flamelet structures are present at GT combustion conditions, but distributed over a wider region yielding a thicker flame brush rather than a thicker flame.<sup>1</sup> This is because small eddies may not have enough energy to impart significant changes to the flame structures [17, 18] and thus the limits of applicability of premixed flamelets are unclear [2, 19].

For stable GT combustion conditions the pressure across the flame does not vary significantly and thus different flamelets for different pressures are not typically computed. The effect of heat losses is also generally taken to be small for combustion modelling purposes. The effect of strain on a premixed flamelet is well accepted to be important in the case of RANS simulations, however its relevance for LES is more controversial. Recent findings [20, 21] show that, since part of the strain is resolved in the LES, its effect on the flame is implicitly captured as long as the local mesh size is appropriate. These preliminary findings have been confirmed by GT calculations [22–24] and show that strained flamelets are unnecessary at least for the conditions considered. For an industrial perspective where the reduction of computational cost is essential, these recent findings open the way to effective use

<sup>1</sup> The flame brush is the time-averaged high temperature region. Hence, it can be thick despite the flame being thin when the flame moves or is distributed spatially.

*The Role of CFD in Modern Jet Engine Combustor Design DOI: http://dx.doi.org/10.5772/intechopen.88267*

discussion on thickened flame [9, 10] model in Section 4, or the introduction of additional complexity to smooth the G-equation in level set models [11–13]. Thus, the use of this category of models for industrial applications is still unclear. The statistical models within the flamelet approach include the turbulence-chemistry interaction using probability density functions (PDFs), which are typically presumed and thus they do not incur additional computational effort. Although additional equations are still necessary depending on the particular model, these are generally computationally cheap to solve because the filter size can be kept larger than the flame thickness, at least in principle, as long as the presumed PDF used is able to represent the statistical behaviour at scales smaller than Δ correctly. The potential of flamelet-based models for GT applications has thus to pass through a deep understanding of the SGS processes, which has been the focus of research in the last 30 years. Only in recent years, however, revised flamelets formulations for LES have demonstrated potentials to bridge the gap that separated them from

In flamelet models, the turbulent flame is seen as an ensemble of thin, onedimensional structures (flamelets) which are wrinkled by turbulence; turbulent eddies are either not small enough to penetrate into the flame and alter its internal structure, or they do not last long enough. Therefore, the thermochemistry can be computed *a priori* through one-dimensional computations and then parameterised using a set of control variables. For partially premixed combustion these are usually a variable to track the rate of mixing, and another to track the reaction progress. Other parameters can be introduced to include additional effects such as pressure variation, non-adiabaticity, strain, etc. The 1D laminar flames can be of any type, e.g. they can be premixed or diffusion flame. Premixed flamelets are, however, more versatile for partially premixed combustion as strain, reaction progress and mixing can be controlled independently; thus they can, in principle, be used for situations involving local extinctions. The big challenge for using flamelet models for GT combustion conditions is that turbulence is extremely high and the smallest eddies can penetrate and affect the internal flame structure, thus invalidating the flamelet hypothesis. This can happen when the Karlovitz number, *Ka* ¼ *τc=τη* ≫ 1, where *τ<sup>c</sup>* and *τη* are chemical and the Kolmogorov time scales respectively. Nevertheless, a number of relatively recent works (see for example [14–16]) have shown that flamelet structures are present at GT combustion conditions, but distributed over a wider region yielding a thicker flame brush rather than a thicker flame.<sup>1</sup> This is because small eddies may not have enough energy to impart significant changes to the flame structures [17, 18] and thus the limits of applicability of premixed

For stable GT combustion conditions the pressure across the flame does not vary significantly and thus different flamelets for different pressures are not typically computed. The effect of heat losses is also generally taken to be small for combustion modelling purposes. The effect of strain on a premixed flamelet is well accepted to be important in the case of RANS simulations, however its relevance for LES is more controversial. Recent findings [20, 21] show that, since part of the strain is resolved in the LES, its effect on the flame is implicitly captured as long as the local mesh size is appropriate. These preliminary findings have been confirmed by GT calculations [22–24] and show that strained flamelets are unnecessary at least for the conditions considered. For an industrial perspective where the reduction of computational cost is essential, these recent findings open the way to effective use

<sup>1</sup> The flame brush is the time-averaged high temperature region. Hence, it can be thick despite the flame

models directly accounting for the thermochemistry.

*Environmental Impact of Aviation and Sustainable Solutions*

flamelets are unclear [2, 19].

**48**

being thin when the flame moves or is distributed spatially.

of flamelets. Following the above discussion, the parameterisation of a model based on unstrained premixed flamelets at various equivalence ratios reduces to only two controlling variables: mixture fraction, *ξ*, to track the amount of mixing (thus the equivalence ratio) and a progress variable, *c*, to track the reaction progress. The first is usually defined using Bilger's expression [25], and assuming that all species have the same mass diffusivity, its transport equation is:

$$
\overline{\rho}\frac{D\tilde{\xi}}{Dt} = \nabla \cdot \left(\overline{\rho}\mathcal{D}\_{\text{eff}}\nabla\tilde{\xi}\right) + \overline{\dot{\phi}}\_{\text{St}}\tag{2}
$$

where De*ff* is the effective molecular diffusivity (sum of filtered diffusivity and the SGS contribution due to the filtering of the non-linear terms). The source term, *ω*\_ *<sup>S</sup>*, is for the evaporation of fuel droplets [23, 26, 27]. The progress variable is usually defined as a combination of reactant or product species and varies monotonically from 0 in the reactants to 1 in the products when it is normalised appropriately, although unscaled formulations are often used. A good choice for lean combustion is to define the progress variable as a linear combination of CO2 and CO [28] mass fractions (normalised by their burnt value). The transport equation for the filtered progress variable is:

$$
\overline{\rho}\frac{D\tilde{c}}{Dt} = \nabla \cdot \left(\overline{\rho}D\_{\text{eff}}\nabla\tilde{c}\right) + \overline{\dot{\phi}}\_{c}^{\*},\tag{3}
$$

where *ω*\_ ∗ *<sup>c</sup>* is the modified filtered reaction rate, which is tabulated and thus is accessed using the controlling variables themselves during the simulation runtime. This term is expressed as *ω*\_ ∗ *<sup>c</sup>* ¼ *ω*\_ *<sup>c</sup>* þ *ω*\_ <sup>n</sup>*<sup>p</sup>* þ *ω*\_ <sup>c</sup>*<sup>t</sup>*, where *ω*\_ *<sup>c</sup>* is the premixed flame contribution (including mixture stratification). The additional terms represent nonpremixed mode contribution and mixed mode due to the interaction of *ξ* and *c* gradients, and they appear only for normalised definitions of the progress variable, see additional details for example in [22, 25, 29]. At this point the set of equations would be closed if the above reaction rates depend only on the two controlling variables, as long as a thermodynamic model is provided. However, in this form the effect of wrinkling of the flame by turbulence at the subgrid level is not accounted. This effect is introduced in a statistical way using a presumed subgrid PDF<sup>2</sup> :

$$\overline{\dot{\boldsymbol{\alpha}}}\_{\boldsymbol{c}} = \int\_{0}^{1} \int\_{0}^{1} \dot{\boldsymbol{\alpha}}(\zeta, \boldsymbol{\eta}) P(\zeta, \boldsymbol{\eta}) d\zeta d\boldsymbol{\eta},\tag{4}$$

where *ω*\_ is the laminar reaction rate from the flamelet and *ζ* and *η* are the sample space variables for *c* and *ξ* respectively. This type of closure was first introduced by Bradley for RANS and non-premixed combustion [30]. The SGS joint PDF can be written as the product of two PDFs as *P*ð Þ¼ *ζ; η P*ð Þ*ζ P*ð Þ *η*j*ζ* . There are various possible choices for the shape of these two SGS PDFs, each with its own advantages and disadvantages, the most common being the Beta PDF and laminar flamelet PDF [31], and it is commonly accepted that these shapes need to be dependent at least on first and second moments. However, for cases involving large turbulence and filter sizes larger than the flame thickness, the Beta PDF was shown to be more

<sup>2</sup> Note that the term PDF in this case does not strictly relates to a probability density function in a statistical sense, since in the LES this operator is used to account for events in space at one time. The use of the term subgrid PDF is thus made for simplicity and analogy to the statistical PDF operator; other authors prefer the term 'filtered density function' to make this distinction.

appropriate in several works (see for example [20, 32–36]). The beta function requires the first and second moments, thus the SGS joint PDF is expressed as *<sup>P</sup>*ð Þ¼ *<sup>ζ</sup>; <sup>η</sup> β ζ*;~*c; <sup>σ</sup>*<sup>2</sup> *c β η*j*ζ*; <sup>~</sup>*ξ; <sup>σ</sup>*<sup>2</sup> *ξ* , where *<sup>σ</sup>*<sup>2</sup> *<sup>c</sup>* and *σ*<sup>2</sup> *<sup>ξ</sup>* are the SGS variances of the progress variable and mixture fraction respectively. Note that here the two PDFs are treated independently, which is usually acceptable in LES with an appropriate grid size [37]. The SGS variances obtained using their transport equations are better than using algebraic expressions since convective and diffusive processes are important at subgrid scales [20]. These equations are written as:

$$\overline{\rho}\frac{D\sigma\_{\xi}^{2}}{Dt} = \nabla \cdot \left(\mathcal{D}\_{\text{eff}}\nabla\sigma\_{\xi}^{2}\right) - 2\overline{\rho}\,\overline{\epsilon}\_{\xi} + 2\overline{\rho}\,\frac{\nu\_{l}}{\text{Sc}\_{l}}\left(\nabla\bar{\xi}\cdot\nabla\tilde{\xi}\right) + \overline{\rho}\,\text{S},\tag{5}$$

effects, an energy equation is often solved. For low-Mach conditions it is convenient to use a total specific enthalpy (sum of formation and sensible enthalpies), which is a

Eq. (2) (except for thermal diffusivity in place of mass diffusivity and no source term).

is the heat capacity at constant pressure. This inversion can be performed numerically in different ways, see for example [20, 23, 42], and requires the integration of enthalpy of formation and heat capacity using an equation consistent to Eq. (4). The density is computed via the state equation, where the pressure is often assumed to be constant in low-Mach formulations for numerical stability, except for cases where compressibility effects are important, e.g. thermoacoustic instabilities. The above equations describe the general flamelet formulation with specific details for the FlaRe approach. In the next sections specific test cases relevant for gas turbine engines will be discussed using both laboratory and practical flames, and the FlaRe approach is compared to other combustion model results where they are available.

For practical jet engines, it is technically difficult and very costly to conduct measurements inside the combustion chamber due to the extremely hostile conditions and complex geometry. Therefore, laboratory burners not only play a crucial role in experimental combustion research but also serve as a main source for CFD model validation. In the past, the majority of the modelling efforts were devoted to flames in simple geometry such as jet flames and bluff-body or swirl stabilised flames in an open environment. Many of these flames have been benchmarked as standard model validation cases documented in the well-known TNF Workshop [43]. Over the last few decades, a large number of combustion models including most of those discussed earlier in Section 2 have been tested using the TNF benchmark flames [44]. Despite the different models used, the computational results converge to a similar level of very satisfactory accuracy when compared with measurements for the main flow and flame statistics. For example, the transient ignition of a lifted methane-air jet flame [45] was simulated using FlaRe [46], conditional moment closure (CMC) [47], thickened flame (TF) [48] and

transported PDF with Eulerian stochastic fields (TPDF/ESF) [49] approaches, all showing comparably good agreement with the measurements for the flame upstream propagation. However, this level of general agreement among different models is yet to be achieved for more complex engine-relevant geometry and conditions. Therefore, this section focuses on the state-of-the-art laboratory gas turbine model combustors (GTMCs). In order to demonstrate the current CFD capabilities of tackling the various issues in these combustors, two cases, for single

The dual-swirl GTMC experimentally investigated by Meier et al. [50, 51] at the

German Aerospace Center (DLR) is of interest. The schematic of this GTMC

and multiple burner configurations, are considered.

**3.1 Single burner with dual swirlers**

**51**

*<sup>f</sup>* is the enthalpy of formation of the mixture at temperature *T*<sup>0</sup> and *Cp*

*h* by inverting the following expression:

*<sup>C</sup>*~*p*ð Þ *<sup>T</sup> dT* (7)

*h*, has the same form of

conserved quantity. The equation for the filtered total enthalpy, ~

~ *h* ¼ Δ g*h*<sup>0</sup> *f* þ ð*T T*<sup>0</sup>

The temperature field is obtained using ~

*DOI: http://dx.doi.org/10.5772/intechopen.88267*

*The Role of CFD in Modern Jet Engine Combustor Design*

where Δ*h*<sup>0</sup>

**3. Laboratory burners**

for the mixture fraction variance, and

$$\overline{\rho}\frac{D\sigma\_{\varepsilon}^{2}}{Dt} \approx \nabla \cdot \left(\mathcal{D}\_{\varepsilon\overline{\mathcal{U}}}\nabla\sigma\_{\varepsilon}^{2}\right) - 2\overline{\rho}\,\overline{\varepsilon}\_{\varepsilon} + 2\overline{\rho}\,\frac{\nu\_{l}}{\mathrm{Sc}\_{l}}\left(\nabla\overline{\varepsilon}\cdot\nabla\overline{\varepsilon}\right) + 2\left(\overline{c\,\dot{\omega}}\,-\widetilde{\varepsilon}\,\overline{\dot{\omega}}\right) \tag{6}$$

for the SGS variance of progress variable. From left to right the various terms in the above equations represent total derivative, effective diffusion, scalar dissipation, turbulent production and source term. The evaporation of droplets contributes to *σ*<sup>2</sup> *<sup>ξ</sup>* and the subgrid reaction processes contribute to *σ*<sup>2</sup> *<sup>c</sup>*. The latter is closed with an expression consistent with Eq. (4). The evaporation and the spray model in general are out of the scope of this section. For the specific simulations to be presented in Section 4, a Lagrangian approach is used for the two-phase flow, with parcel sampled using a Rosin-Rammler distribution and the Sattelmayer correlation for the initial Sauter mean diameter (SMD) [38]. Only secondary breakup is considered using the process described in [39], and a rapid mixing formulation for the droplet evaporation. More details can be found in [23, 27, 40].

The scalar dissipation rate (SDR) terms in Eqs. (5) and (6) need closure. The linear relaxation model, <sup>~</sup>*εξ* <sup>¼</sup> *<sup>C</sup><sup>ξ</sup> <sup>ν</sup>t=*Δ<sup>2</sup> *<sup>σ</sup>*<sup>2</sup> *<sup>ξ</sup>*, with *C<sup>ξ</sup>* ≈ 2, is well accepted [5]. Recent works have suggested that this constant is to be revised in case of liquid fuel due to the evaporation source term in Eq. (5) [27]. For the progress variable SGS variance, it is shown in [20] that the reactive term in this equation is of leading order at least for Δ of sizes comparable or larger than the flame thickness, and thus the SDR has to balance the sources coming from reaction and turbulence. Hence, the linear relaxation model is unsuitable on physical grounds. To justify the use of linear relaxation model, a delta or three-delta function is used sometimes instead of the Beta function in Eq. (4) so that *<sup>c</sup>ω*\_ � <sup>~</sup>*cω*\_ <sup>¼</sup> 0 in Eq. (6). This, however, creates a conflict since the meaning of *σ*<sup>2</sup> *<sup>c</sup>* changes without the reactive term. More recently, revision of the flamelet modelling in the context of LES to take into account the correct reactionturbulence-diffusion balance led to appearance of more sophisticated, yet simple, model for the SDR of progress variable. One recent development in this sense is the SDR closure of Dunstan et al. [41], which approaches the linear relaxation concept in the limiting behaviour of non-reactive mixture and is thus more recommended. This model has been used in many past studies for different combustion regimes [20, 22, 24, 36] (see also Sections 3 and 4).

The set of equations shown above is used in conjunction with the LES equations for mass and momentum, which are the same for reacting and non-reacting flows, see for example [2, 3] for a more detailed explanation. In principle, the temperature and density fields can be also computed *a priori* using an equation consistent to (4) and accessed in runtime using the controlling variables. In order to account for possible non-adiabatic

effects, an energy equation is often solved. For low-Mach conditions it is convenient to use a total specific enthalpy (sum of formation and sensible enthalpies), which is a conserved quantity. The equation for the filtered total enthalpy, ~ *h*, has the same form of Eq. (2) (except for thermal diffusivity in place of mass diffusivity and no source term). The temperature field is obtained using ~ *h* by inverting the following expression:

$$
\tilde{h} = \widehat{\Delta h\_f^0} + \int\_{T\_0}^T \bar{\mathcal{C}}\_p(T) \, \, dT \, \tag{7}
$$

where Δ*h*<sup>0</sup> *<sup>f</sup>* is the enthalpy of formation of the mixture at temperature *T*<sup>0</sup> and *Cp* is the heat capacity at constant pressure. This inversion can be performed numerically in different ways, see for example [20, 23, 42], and requires the integration of enthalpy of formation and heat capacity using an equation consistent to Eq. (4). The density is computed via the state equation, where the pressure is often assumed to be constant in low-Mach formulations for numerical stability, except for cases where compressibility effects are important, e.g. thermoacoustic instabilities. The above equations describe the general flamelet formulation with specific details for the FlaRe approach. In the next sections specific test cases relevant for gas turbine engines will be discussed using both laboratory and practical flames, and the FlaRe approach is compared to other combustion model results where they are available.
