Direct numerical simulation and reduced chemical schemes for combustion of perfect and real gases

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Direct numerical simulation and reduced chemical schemes for combustion of perfect and

real gases

Axel Coussement

A thesis submitted for the degree of Docteur en Science de l’Ing´ enieur

January 2012

Aero-Thermo-Mechanic Department EM2C Laboratory

Universit´e Libre de Bruxelles Ecole Centrale de Paris

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To my little Esther.

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The real problem with acknowledgements is not to forget anyone, I will try to thank everyone, if not please excuse me. First of all, this would not have been possible without the support of my girlfriend who became my wife and the mother of my little daughter during this thesis. Thank you for your patience, your kindness, your faith, your support. I love you so much. Then since, during the works in my house, my mother- and father- in-law have kindly lodged me and understood the importance of this thesis, I really wanted to tell them how grateful I am. Eric, Maryvonne thank you for everything. Loic, thank you for your humor, you are the only person who could make me laugh when my computations were everything but working.

Clementine, thank you, for ... well, for being you. Obviously I could not forgot my mother and sister, you were supportive and present, and thank you for deep, deep, deep implication, I love you. Finally, thank you dad for giving me your taste of science, wherever your are, I will always remember you.

After the family comes the colleges: thank to all the members of the”Fine

´equipe” of the ATM department: Thomas, Adrien and Nicolas. Damn, it was fun working with you. I would exchange those years for nothing in the world. Thank you for your humor and support. Thomas, a special thanks goes to you: you tought me a lot in numerical methods. I hope your thesis will soon be over. How can I forgot Matthew, you are just awesome, and thank you for all those long hours spend correcting a big part of this work.

That was for the ULB, now the EM2C Laboratory. Nasser, thank you for giving me the REGATH Library and answering my questions, this allowed me to develop the real gases version of YWC. Benoit, you are a pain in the ass when you correct papers, but when the corrections are done, the final result

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you are great. I almost forgot, Ronan, Pierre and Jean. Thank you for your help, conferences were fun. Anytime guys ! The team from the IC engine department of the ENSPM should also be thanked for the faith they put in me by giving a teaching position during this thesis.

Finally, the ones without who nothing could have been possible. Olivier, G´erard and Alessandro. Alessandro, thank you for teaching me PCA, thank you for your ideas. G´erard you are an encyclopedia of fluid dynamics, thank you for all the teaching, your presence, your faith in me (and I’m sure I forgot a lot a things ...). Finally, certainly the most important, Olivier, the best PhD thesis manager, even if you are certainly as obstinate as I am (which can yield some problems sometimes). I will not make a list of the reasons why I should thank you, I will simply say: Thank you Olivier, because without a guide there is no path . . .

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Introduction

”Turbulence is the invention of the Devil on the seventh day of creation”. This inspired description of turbulence was given by Bradshaw in 1994 (1). 15 years later, this still holds. In fact, citations like this can be found throughout the history of turbulence.

One of the most remarkable facts about turbulence is that its description is given by the relatively simple set of Navier-Stokes equations introduced nearly two centuries ago.

During this period, numerous scientists have studied turbulence and yet, it is far from being totally understood. By inventing turbulence, the Devil also gave birth to turbulent combustion. Turbulent combustion can be described by simple equations, but is even more complex, if possible, than hydrodynamic turbulence.

Not understanding turbulence, or turbulent combustion, would not represent an issue if those processes didn’t have a crucial importance in of our lives. But turbulent combustion processes take place in planes, cars, power plants, buildings, etc . . ., therefore, detailed computations are needed to allow an optimal design of those devices.

The most obvious way to study turbulence is get a full description of the turbulent flow, which requires to solve the Navier-Stokes equations. However, this can only be done numerically because, while relatively simple, Navier-Stokes equations do not have any analytical solution for turbulent flows and, in the case of laminar flow, analytical solutions can only be found for simple problems. These numerical simulations are called Direct Numerical Simulations or DNS in short, because the Navier-Stokes equations are solved directly (2).

The computing cost of such solutions can be prohibitive: the widest DNSs reported today have dimensions of the order of a few centimeters (3, 4). The result presented in

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Figure 1.1: Instantaneous image of HO2 and OH from DNS of a lifted hydrogen/air jet flame from (3).

Figure 1.1 comes from a DNS whose domain size is 6 cm³ and required 3.5 million CPU hours.

Engineers often deal with much larger problems: the cylinder in the internal combustion engine, the combustion chamber for aeronautical turbojet engines . . . . That is why approximate techniques, where complex turbulence effects are modeled, have been developed in past 30 years. The two main techniques are the Reynolds Averaged Navier- Stokes, or RANS, and the Large Eddy Simulation, or LES, techniques. For the first one, the approach is based on finding the statistical average of the flow instead of solving the instantaneous flow field. The LES approach is somewhat intermediate between RANS and DNS: one solves the instantaneous flow field for the largest structures while modeling the influence of the smallest ones.

Even though those two techniques are efficient in terms of required computing power

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LES or a RANS is able to accurately compute a combustion process without any model.

This is known as the curse of closure: the equations describing the statistical evolution of the flow cannot be obtained from the Navier-Stokes equations and therefore require modeling.

In the past 10 years, numerous works developing new or existing models have been published. Indeed, as the complexity of the configurations computed by RANS and LES increases, improvement in the statistical modeling is sought.

The search for new, more accurate, numerical models is what drives most of the research in numerical simulation of turbulence and combustion. While DNS can be used to study specific phenomena, it is also a powerful tool dedicated to model development and validation.

The DNS tool in a model development framework is generally used in the following way: the same computational case in computed by DNS and using the model and flow statistics are compared. This yields the following problem: the models must be validated on quite simple and small configurations and then applied on more complex and large configurations with no guarantee about the consistency of the model on those large configurations. Indeed, DNS simulation are limited in flow complexity and size due to their cost.

If one considers the development of combustion models, one not only wants to pro- duce a statistical closure but also to reduce the dimensionality of the problem. For example, the full methane combustion process requires 53 species and 325 reactions (5), if a much lower dimensional manifold can be found without loosing accuracy, the computing cost will be drastically reduced. Therefore combustion modeling not only suffers from the curse of closure but also from the curse of dimensionality.

To solve this dimensionality problem, one can either use:

• mechanism reduction: the chemical mechanism is modified to reduce the number of species and its stiffness.

• state-space parameterization: where it is assumed that the combustion processes are limited to a manifold. Then a small number of species is used to access the manifold allowing to recover the remaining species.

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In both of these techniques, information is lost in the process, so that one must knowa priori if that lost does or does not affect the accuracy of the model. For example, the analysis performed in the work of Chen et al (3), showed that for the DNS of Figure 1.1 the autoignition process had a crucial importance in the stabilization of the flame.

Therefore if a combustion model where the information regarding the autoingnition process is lost, is benchmarked against those DNS results the model will obviously fail.

Therefore, one must knowa priori to which extent the model can be reduced and which effects can be neglected.

In case of combustion modeling, not only the DNS is used as a benchmarking tool for dimensionality reduction and statistical closure but also as a study tool to understand which processes control the flame and therefore which effects must be included in the low dimensional model.

1.1 Contribution of this thesis

This work contributes to solve two of the questions raised in the above discussion:

• Increase the accuracy a DNS solver through a careful treatment of boundary conditions, for both perfect and real gases flows.

• The development of an optimal state-space parameterization combustion model using Principal Component Analysis (PCA), and its first application as a chemistry tabulation technique.

Ideally, the boundary conditions of a DNS solver should not induce any spurious oscillations or flow distortions in the computational domain. Indeed, since the size of a DNS computational domain remains small nowadays, if, for example, an outlet boundary condition induces flow distortions when complex structures are leaving the domain, they will quickly propagate in the domain and ruin the results. A careful treatment of boundary conditions is thus required. In this thesis, the so-called 3D-NSCBC introduced by Yoo et al. (6, 7) are improved to allow the exit of non-normal flow without any distortion.

It will be demonstrated that those improved boundary conditions are able to cleanly evacuate a flame vortex interaction. To further prove the physical soundness of those boundary conditions, a DNS of an highly turbulent hydrogen jet is performed in two

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computational domains, one being ten times smaller than the other, and penetration statistics are compared.

Since DNS of real gases flow are becoming more common nowadays, these boundary conditions are extended to real gases flow and further improved to allow a clean exit of high density gradients. The physical soundness of these conditions will be also be demonstrated for various real gases flows.

PCA is also used to find an optimal state-space parameterization for a combustion process. Indeed, using only a one dimensional steady state flame as input for the model, it is a able to accurately predict the behavior of the flame in a complex flame-vortex interaction. This is, to the author’s knowledge, the first numerical simulation using PCA data as a chemistry tabulation technique.

All those computations were performed using the YWC compressible solver which was developed in the scope of this thesis.

1.2 Structure of the Manuscript

This manuscript is divided as follow:

• Chapter 2 presents the state of the art of DNS solver and chemistry tabulation techniques.

• Chapter 3 gives a general presentation of the YWC solver and its thermo-chemical library, especially the real gases part.

• Chapter 4 present the original developments introduced in the 3-D NSCBC initially developed by Yoo et al. (6) and Lodato et al. (8). Most noticeably, the improvement proposed allows the exit of non-normal flows. Moreover the reduction of the size of the domain allowed by the proposed 3-D NSCBC is demonstrated.

• Chapter 5 presents the extension of the 3-D NSCBC in the real gases framework.

Along this, many test cases are performed to check to consistency of these boundary conditions. This extension to real gases had not, to the author’s knowledge, been performed before. Moreover, most of the test cases presented are also original in the sense that the consistency of real gases boundary conditions had never been so deeply checked.

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• Chapter 6 presents a chemistry tabulation technique deduced from the PCA and is benchmarked against the DNS. To the author’s knowledge, this is the first suc- cessful implementation of PCA as a chemistry tabulation technique.

• Chapter 7 deals with the improvement of the PCA technique used to generate the chemistry database.

• Finally, chapter 8 ends this thesis by giving the conclusions of this work and its perspectives.

1.3 Publications

The theoretical development and results that can be found in this thesis have led to the following publications (9, 10, 11, 12):

• Coussement, A., Gicquel, O., Caudal, J., Fiorina, B. and Degrez, G. Three- dimensional boundary conditions for numerical simulations of reactive compressible flows with complex thermochemistry. Submitted to Journal of Computational Physics

• Coussement, A., Gicquel, O., Fiorina, B., Degrez, G and Darabiha, N. Multicompo- nent real gas 3-D-NSCBC for direct numerical simulation of reactive compressible viscous flows. Submitted to Journal of Computational Physics

• Coussement, A., Gicquel, O. and Parente, A. Kernel Density Weighted Princi- pal Component Analysis of Combustion Processes. Submitted to Combustion and Flame

• Coussement, A., Gicquel, O. and Parente, A. Reduced-order combustion modeling using principal components: the MG-PCA approach. Submitted to Combustion and Flame

Further, contribution to conferences (13, 14, 15) have been done. Finally, this thesis also led to following publication (16), which can be found in Appendix C:

• Coussement A., Gicquel O. and Degrez G. Large Eddy Simulation of a Pulsed Jet in Crossflow. Accepted for publication in Journal of Fluid Mechanics

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Direct numerical simulation and combustion modeling

2.1 Direct Numerical Simulation

As stated in the introduction, DNS computations are performed mainly for two reasons:

• Extensive study of flow behavior, leading to a better understanding of physics.

• Validation of LES or RANS models, which can be done eithera priori ora poste- riori. In the first case, the model is tested on existing DNS data by applying it on those results and checking the error. In the latter case LES or RANS simulations are benchmarked against DNS data. The two processes are often used together at different stages of model development.

In the case of combustion, or turbulent combustion, those two points are strongly linked.

Indeed, if one does not know the relevant physics that the model should recover, it is almost impossible to build. For example, if a hydrogen flame dynamics is controlled by the H₂O₂ species, any model trying to reproduce the flame dynamics should accurately predict this species.

The path followed through this thesis is the same. The first part of this manuscript is dedicated to the development of the YWC solver. The second to the modeling of combustion using the data from YWC. However, before going further, a review of DNS for combustion processes and of the main chemistry tabulation techniques is given.

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2.2 A Brief history of DNS

The DNS foundation were laid at the National Center for Atmospheric Research in 1972 by Orszag and Patterson (17): they computed an isotropic turbulence at a Reynolds number of Re = 35 ¹ on a 32 × 32 × 32 grid, however such a computation can be judged under-resolved by today (18). The next step was undertaken by Rogallo (19).

In his study, he examined the effects of mean shear, irrotational strain, and rotation on homogenous turbulence. His data were used to evaluate various turbulence models.

This was the beginning of the use of DNS as a benchmark tool for models.

In the early 80’s computing resources were ridiculous compared to today’s standards.

The Lawrence Livermore National Laboratory supercomputer was, in the 80’s, a CRAY X-MP which had a peak CPU performance of 400 Mega FLoating Operation Per Second (MFLOPS). The same laboratory will be delivered a 20 Peta FLOPS or 20 10⁹ MFLOPS supercomputer in 2012. This was the reason why, at that time, DNS of wall bounded turbulence was unachievable, those computations were done using LES.

In 1987, the first wall bounded computation of turbulence was performed by Kim et al. (20) and subsequent studies where performed to evaluate the response of wall- bounded turbulence to various factors such as rotation (21), mean three-dimensionality (22), transpiration (23) and heat transfer (24).

All those simulations were homogeneous in the stream wise direction, the streamwise boundary conditions were therefore periodic. Computing inhomogenous flow in the streamwise direction was so the next logical step. However such flow requires injection of turbulence at the inflow and it was not until 1992 that the first results were obtained for a backward facing step by Le and Moin (25).

The DNS of compressible flows took, for its part, longer to develop. A DNS of homogenous compressible turbulence was first undertaken in the early 80’s by Feiereisen et al. (26). However, it took nearly a decade before results were carefully studied (27, 28, 29, 30) and more complex homogenous flows began to be simulated in the mid- 90’s by Coleman et al. (31) for wall bounded turbulence and by Rai et al. (32) for turbulent boundary layers. As for its incompressible counterpart, improvement of the

1The Reynold’s number is defined as: Re=^{u L}_ν whereuis the flow characteristic velocity in m/s,L a characteristic dimension in m andνthe fluid kinematic viscosity in m²/s

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boundary conditions initiated by the work of Poinsot and Lele (33), allowed to perform studies of in-homogenous flows.

By comparison to incompressible flows, DNS of compressible flows has numerous advantages. First, it allows the simulation of high speed, supersonic flows and thus the computation of shock waves and their interaction with turbulence (34) like in the work of Lee et al. (35) and Mahesh et al. (36). Second, it allows the direct computation of the radiated sound (37) and third, it allows the study of flame-acoustics interaction.

However, those advantages come with a higher CPU cost.

After all those pioneer works, both DNS of compressible and incompressible turbulence have evolved the same way in the past decade:

• Wider domain: the number of computational nodes increased. In that category one should distinguish the increase of the Reynolds number associated with the simulation and the increase of the domain size, which can result in an increase of the geometric complexity. Indeed the resolution needed, and therefore the nodes density increase with the Reynolds number likeRe⁹⁴. The simulation of Yokokawa et al. (4), where an isotropic turbulence at a Reynolds number of Re= 1217 was computed on a 4 096 × 4 096 × 4 096 grid, is an example of the increase of the Reynolds number. On the other hand DNS like the DNS of transitional flow in a stenosed carotid bifurcation (38) or the DNS of a round turbulent jet in crossflow (39) are example of increased geometric complexity.

• More complex physical problems: the number of equations to solve are increased to capture more complex phenomena like combustion or transcritical flows.

The present work falls in the second category.

DNS of combustion processes appeared in the mid-90’s (40, 41). These first simulations concerned three-dimensional H₂/O₂/N₂ flame front propagation in a turbulent flow. Even though the combustion schemes used were simple, the size of these simulations were very limited: a few millimeters (40, 41). However, these simulations already allowed to develop combustion models (40, 42).

After these pioneer works, and as for the DNS of non-reactive flows, the size and complexity of the domains increased: DNS of diffusion flames were carried out (e.g by Mizobuchi et al. (43, 44)). This study still considers a hydrogen flame to limit

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the dimensionality problem associated with DNS of combustion. As the fuel gets more complex, the complexity of the chemical reaction scheme increases quasi-exponentially.

Indeed, a full hydrogen/air reaction scheme requires 9 species and 19 reactions (45), methane/air combustion needs 53 species and 325 reactions (5) and kerosene at least 220 species and 1 500 reactions (46). Namely, the reduction of the combustion process to a lower dimension manifold (42) is of such great importance. Moreover, it is for the same reason that even the most recently published DNS results concern either hydrogen-air (3) or methane-air flames (47).

All those results were achieved using the perfect gases state equation. However, for specific applications like rocket engine, diesel internal combustion engines or even gas turbine engines, the thermodynamic conditions do not allow to use the perfect gases assumption. One must thus switch to a real gas state equation like the Peng-Robinson (48) or the Soave modified Redlich-Kwong (49) equation of state, which increases the computational complexity of the simulation. In this field, first results for a two species mixing layer (50, 51) and for a mono-species jet (52) have only been published recently and no three-dimensional DNS of a transcritical of supercritical flame have been reported yet. Finally, the use of these DNS data did not yield any reported model

This history of DNS shows two important patterns. First, generally, quickly after the first DNS are computed, existing models are benchmarked against DNS data and/or new models are developed from the analysis of these results. Second, when DNS computations have reached a given level of complexity, one adds more physics in the computations: when non-reactive flows DNS were becoming common, one started combustion and since DNS of combustion are nowadays becoming more and more common, one adds real gases thermodynamics to combustion. This pattern is only controlled by the available computing power, and not because the development of DNS codes is a slow and complicated process. However, DNS still has numerical issues as it will be described in the next section.

2.3 Numerical issues of DNS

The main idea of DNS is, as already said, to directly solve the Navier-Stokes equations, which seems a priori less complicated than LES or RANS. Indeed, RANS (Reynolds Averaged Navier-Stokes) simulations relies on a complete modeling of the turbulence,

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in the numerical solution only the statistical average of the flow field is computed. All the turbulent energy spectrum is thus modeled as indicated by Figure 2.1. LES (Large Eddy Simulation) relies on the fact that one only wishes to capture the most energetic turbulent structures and model the remaining ones, the turbulent energy spectrum can thus be decomposed as indicated by Figure 2.2.

Figure 2.1: Turbulent energy spectrum decomposition for RANS (53).

Figure 2.2: Turbulent energy spectrum decomposition for LES (53).

Moreover if one wishes to add combustion processes in an LES or RANS solver one should also add a combustion model. Indeed, if a flame front is present in the computed flow, the solver must be able to capture it. However, for most LES and nearly all RANS computational cases, using a computational grid able to capture the flame front is impossible. For example, the LES computation performed by Moureauet al. (54, 55) was a DNS from a turbulent flow point of view but was unable to fully capture the flame front.

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From the above discussion, one can conclude that the RANS or LES computations require a high level of model complexity with respect to DNS, and as such are numerically more complex than DNS. While it is true that the quality of modeling has a crucial importance in both LES or RANS results, DNS has its own constraints, mainly due to the absence of a turbulence model. Indeed turbulent models (and turbulent combustion models) are generally dissipative. They mimic the turbulent energy spectrum decrease by dissipating the turbulent structures below a given size, and therefore they tend to stabilize the computation.

Not using those dissipative models in DNS solver yields, principally, the following concerns (18):

• Spatial resolution: since all the turbulent energy spectrum must be computed, the solver must be able to capture all turbulent scale. Moreover the DNS solver should also be able to capture the flame front.

• Time advancement: not only the smallest turbulent scale must be resolved in space but also in time. Again, the same is true for combustion processes.

• Boundary conditions: rigorous treatment of the boundary conditions is necessary to avoid inducing spurious oscillations in the computational domain. Indeed, since no artificial dissipation is (explicitly) present in DNS solvers, those oscillations will propagate in the flow and could result, if amplified, to a numerical failure of the computation.

These problems will now be reviewed.

2.3.1 Spatial resolution

The problem of spatial resolution in DNS can be stated the following way: the grid used to perform the DNS determines the scales that are represented, and the accuracy with which they are represented is determined by numerical method. However the scales that must be resolved is dictated by the physics, which raises the question of knowing what are the smallest and largest scales to be resolved.

DNS theory (18) answers this question in a quite unsatisfactory way: the Kolmogorov length scale,ηk, gives the smallest length to be resolved, and the integral length scale, lt, gives the largest scales to the resolved. In most practical DNS configurations the

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integral length scale is related to the geometric characteristics of the flow. For example in a ducted flow, the integral length scale is the order of the duct size. In the case of homogenous flow, that is when periodic boundary conditions are used, the integral length scale is such that a two points correlation in the solution is required to vanish within half the domain. The Kolmogorov length scale is defined as (56):

ηk= ν³

¹₄

(2.1) whereν is the fluid kinematic viscosityis the dissipation of the kinetic turbulent energy defined as:

= u⁰(r)³

r (2.2)

where u⁰(r) is the turbulent velocity associated with a scale of size r. Note that the Reynolds number associated with ηk is defined as:

Rek=Re(ηk) = u⁰(ηk)νk

ν = ¹^/³η_k⁴^/³

ν = 1 (2.3)

and is thus equal to one.

While the integral length scale resolution requirement is easily met in most DNS, the Kolmogorov length scale is generally not resolved, which is why this requirement is quite unsatisfactory. Indeed in most DNS of homogenous turbulence, the smallest revolved scales were of O(ηk) and not equal to ηk. Table 2.1 gives the resolution in terms of Kolmogorov scales for early incompressible DNS, which shows that despite their excellent agreement with the turbulence theory or experiments, their spatial resolution was not equal to ηk. In fact the smallest length scale to be resolved depends on the energy spectrum; for example, in their work Moser and Moin (57) show that most of the dissipation occurred at a length scale of 15ηk and therefore the smallest resolved scale should be just inferior to this value and not equal to ηk.

However the required resolution depends also on the numerical schemes: even if the smallest length scale can be resolved, the numerical scheme could still induce an error on those scales so large that they will not be accurately resolved. Indeed numerical schemes can induce three types of errors:

• Differentiation error: this error is characterized by the difference between the exact derivative and the computed derivative. It could be quantified by the modified

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Flow Type Resolution in terms ofηk

Plane channel (20) ∆x= 7.5, ∆y = 0.03, ∆z= 4.4 Boundary Layer (58) ∆x= 14.3, ∆y = 0.33, ∆z= 4.8 Homogeneous shear (59) ∆x= 7.8, ∆y = 3.9, ∆z= 3.9

Table 2.1: Spacial resolution of DNS of turbulence in terms of the Kolmogorov length scale η_k.

wavenumber computed by a Fourier analysis. For example, if a single Fourier mode, f = eîkx is considered on a uniform one dimensional mesh of length 2π containing N nodes with a spacing of h = ²_N^π, the exact first derivative at point j is ik eîkx^j. The computed derivative will be equal to ik⁰eîkx^j with k⁰ being the modified wavenumber. The differentiation error is thus quantified by the difference between the real wavenumber k and the modified one k⁰. For example a second- order finite difference scheme givesk⁰ = ^sin_h⁽^kh⁾. Figure 2.3 gives the differentiation error for several centered difference schemes.

• Triadic interaction resolution: the non-linear terms produces a triadic interaction between the scales. The smallest resolved scale in the DNS is therefore influenced by scales that are bigger, which are resolved, but also smaller ones, which are not resolved. This error should be small enough to be negligible. However, generally this issue is not critical if the grid resolution is ofO(ηk) (18)

• Aliasing: this is the second error induced by the non-linear terms. In a numerical computation, the flow field can be projected in a basis of Fourier modes. The non-linear terms in the Navier-Stokes equations can generate modes that are not included in the Fourier mode basis. Therefore, these modes cannot be represented and their contribution will be mistakably added to the lower order modes, this is called the aliasing error. This error has been considered for a long time as only a source of numerical instability, but the work of Kimet al. (20) demonstrated that if the aliasing error was not removed it generates erroneous results.

The two main numerical methods used to perform DNS can be classified on the basis of those errors. The spectral method, on which are based most of the homogenous turbulence computations, display a low differentiation error and high errors due to the

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Figure 2.3: Differentiation error of several centered difference schemes from the work of Kennedy and Carpenter (60). The digits refers to the order of the scheme and the letters stands for: P pentadiagonal, T tridiagonal and E explicit.

non-linear terms (triadic and aliasing errors). On the other hand, central difference methods displays a high differentiation error and low non-linear errors. This is why the spectral methods are generally used to perform DNS of incompressible flows: its low differentiation error allows to increase the grid spacing and since non-linear coupling are limited in incompressible flows, the non-linear error can be managed. However the strong non-linear coupling in the compressible form of the Navier-Stokes equations nearly forbids the use of spectral methods, so that central difference schemes are preferred.

Finally if one deals with combustion, the inner flame structure must also be resolved.

For premixed flame using simple chemistry description (i.e. one or two step(s) irreversible reactions), the number of points required in the flame front is ≈20. In other word, if one considers a laminar flame thickness ofδLthe grid spacing should be at least ^δ₂₀^L (61).

However, if a detailed scheme like the GRI mechanism (5) is used, all the species gradients

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should be resolved which could require a much higher resolution. The resolution required will therefore depend on the chemical scheme used (61).

2.3.2 Time resolution

The wide range of scales found in DNS computations, does not allow to use large time steps in DNS computations. Indeed, even if the largest turbulent scales can be accurately tracked using large time steps, the smallest ones will be erroneously computed.

Von Neumann analysis can be used to quantify this error: consider the following one dimensional advection equation:

df dt +cdf

dx = 0 (2.4)

on a periodic domain. As for the differentiation error, the solution can represented in a base of Fourier modes, ˆf, which yields the following differential equation:

dfˆ

dt =−ik⁰fˆ (2.5)

wherek⁰ is the modified wave number. It is then possible to obtain a complex amplification factor of each mode between the two time stepsnandn+ 1, defined as σ= ^f^ˆⁿ⁺¹_f_ˆ if equation 2.5 is numerically integrated. This amplification factor is a function of then

CF Lnumber (CF L= ^c^∆_h^t, with ∆tbeing the time step) and ofk⁰h, and could therefore be expressed also as a function of kh for a given spatial derivative scheme. Figure 2.4 shows the norm of the amplification factor as a function of the CF L and kh using a fourth order Runge-Kutta temporal scheme and a sixth-order Pad´e scheme. From this Figure, it can be concluded that, if one uses aCF Lof 1, a wide range of scales will be erroneously revolved in time.

However, if these scales do not require an accurate modeling, one can use aCF L of 1 (or higher). It is for example of common practice in DNS of wall bounded flow to use an explicit time advancement algorithm for the convective terms and an implicit time advancement for the viscous terms. Some works, like the one of Choi and Moin (62), have examined the possibility of using an implicit time advancement for the convective terms and concluded that a higher CF L could be used with respect to explicit time advancement. However, the computational overhead associated with the implicit method made it uncompetitive with respect to explicit time advancement techniques. Moreover,

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Figure 2.4: Norm of the amplification ratio using a fourth order Runge-Kutta temporal scheme and a sixth-order Pad´e scheme as a function of kh. SolidCF L = 0.5 and dashed CF L= 1.

since supercomputers rely nowadays on massive parallel architectures, the use of implicit methods for the viscous terms yields programming issues. Indeed, inverting a matrix simultaneously on a small number of processor requires a very careful treatment if one does not want to induce bottlenecks in the computation processes, and consequently a decrease in the computational performance of the solver. The complexity of this problem is drastically increased if one considers a large number of processors (>10 000).

Chemical time scales present in combustion processes can also constrain the time step used to perform the DNS and as for the spatial resolution it is strongly dependent on the scheme used and the flow configuration. For example, in their work Caudal et al. (63) computed a DNS of methane reforming. Due the flow characteristics (i.e high temperature and pressure) the spatial resolution was assured with a grid spacing of nearly 10⁻³m and by using a explicit Runge-Kutta temporal scheme the convective and diffusive terms were accurately resolved with a time step of 10⁻⁷s. However, the chemical characteristic time scale was two orders of magnitude lower and was thus not fully resolved. In that case, one can either use a low CFL number to resolve the chemical time scale or assume that the chemical reactions are at equilibrium. However in most

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reactive DNS computation, the chemical time scales are close enough to the convective time scale to avoid a specific treatment.

2.3.3 Boundary conditions

Boundary conditions are essential in DNS: the size of the computational domains used are so small that if the boundary conditions are physically wrong, they will induce erroneous results in the domain. Moreover, the numerical stability of the DNS is also strongly dependent on the boundary conditions, especially when real gases are involved.

For incompressible flow solvers, the specification of accurate boundary conditions (i.e. inflow and outflows) is not an issue (18), however turbulence injection is much more complicated. Indeed, for a long time, turbulence injection had relied on Taylor’s hypothesis: Le and Moin (25) generated a divergence free field of turbulence and added a convection speed to specify the inflow velocities. However, this techniques requires fairly long domains. The turbulence field enforced on the inflow generates a ”realistic”

turbulence field after a fairly long distance (i.e. 50 displacement thicknesses in the channel used by Le and Moin (25)). Nowadays, the Taylor hypothesis is no more used, instead the enforced flow conditions on the inflow are generated using a separate DNS reducing the length required to recover a ”realistic” turbulence field (64, 65, 66).

Physically sound boundary conditions for compressible flow are much more complicated to achieve: it must be handled using characteristic wave decomposition. Initially developed for the Euler equations (67, 68), these approaches have been extended to viscous flows with the introduction of the Navier-Stokes Charateristic Boundary Conditions (NSCBC) by Poinsot and Lele (33). This technique consists of imposing the boundary conditions in the form of waves entering or leaving the computational domain with a specific treatment of the viscous terms. Initially developed for ideal single-component gases, this approach has been extended to more complex situations such as multi-component reactive flows (69) or real gas mixtures (70). Improvement of the method for low frequency oscillating waves or for low Mach number expansion flows have also been discussed (71).

In the initial NSCBC framework, the flow at the boundary is locally considered as one-dimensional and aligned with the normal at the boundary. This yields the so-called LODI system that is solved in the characteristic space in order to compute the boundary condition in the physical space. However, in their work Yoo et al. (6) indicate that the consideration of the transverse term is necessary to obtain a consistent boundary

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condition. The formulation used by Yoo et al. (6) is therefore no longer locally one- dimensional but fully three-dimensional, which gives the so called 3-D-NSCBC. In a further work Yoo and Im (7) focused on the relaxation coefficient used in the transverse terms in order to achieve a more consistent 3-D-NSCBC formulation at the boundary.

Partial treatment of edges and corners have been addressed by Lodatoet al. (8).

Compared with the original 1-D-NSCBC formulation, the 3-D extension and the specific treatment of edges and corners dramatically improve the flow prediction. However, as demonstrated further in the present work, flow distortions are still observed when complex patterns exit the domain. Also the 3D-NSCBC formulation must be extended to multicomponent reactive flows with a detailed treatment of all kinds of edges and corners.

Finally, following the work of Lee et al. (72), the Taylor’s hypothesis cannot be used for compressible flows, which requires the computation of a turbulence field using a separate DNS computation.

This section has given the main constraints of DNS solvers, from this the following conclusion can be drawn: the main issue of DNS is an accurate resolution in both space and time of all the scales. As stated, the key to this problem lies mainly in determining the grid size needed by the numerical scheme. For non-reactive flows, past experience have provided some good practice rules (see for example Table 2.1). Reactive flow grid spacing is, on the other hand, more complicated to determine because it strongly depends on the chemical scheme characteristics. Moreover, one must also pay a particular attention to boundary conditions especially for compressible flows. Indeed any unphysi- cal behavior of the boundary conditions could induce numerical instability or erroneous results.

2.4 DNS solvers for combustion

In the scope of this thesis, the compressible flow solver YWC was developed to achieve DNS of reactive real or perfect gases flows. Before describing the YWC solver in the next chapter a review of state of the art reactive flow solvers designed for DNS and major tabulation techniques for combustion will be given here.

Obviously, numerous reactive flows solvers purposely developed for DNS, both compressible and incompressible, have been developed in the last decades. Four of them will

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be reviewed:

• S3D developed at Sandia National Laboratories in the United-States of America

• π³ developed at the Laboratory of Fluid Dynamics and Technical Flows in the University of Magdeburg Otto von Guericke in Germany

• H-Allegro and SiTCom both developed at the CORIA Laboratory in France The incompressible LES solver YALES2 developed at the CORIA Laboratory in France will also be reviewed because it is the only unstructured LES solver who came close to a DNS. Finally, since the YWC solver handles also real gases thermodynamics a description of the real gases flow solvers will also be done.

2.4.1 S3D

The S3D solver is developed to perform large DNS of reactive flow (3). It solves the compressible Navier-Stokes equations on a structured 3D Cartesian grid. Spatial differencing is achieve using an eighth-order central differencing scheme (60). However the nature of compressible Navier-Stokes equations induces aliasing in the solutions, to overcome this problem the spurious high-frequency fluctuations are removed using a tenth-order filter(60). Temporal integration of Navier-Stokes equations is achieved using a six-stages fourth-order fully explicit RungeKutta method (73). The 3D-NSCBC of Yoo et al. (6, 7) are used for the boundary conditions. S3D handles only perfect gases, and their thermo-chemical properties are computed using the CHEMKIN libraries (74).

S3D is, as for nearly all DNS solvers, designed as a massively parallel solver. Paral- lelism is achieved with the MPI library and each MPI process is in charge of one piece of the 3D domain. To achieve a scaling of the code on a high number of processors (>10 000) domain decomposition should minimize the communication time between the MPI processes. A relatively simple, yet very efficient technique to do so is to decompose the domain in a way such that if two pieces of the 3-D are next to each other, the processors handling them must be able to achieve a direct communication. In most cases this means that the processors should be located physically close to each other. This technique is used in S3D.

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S3D has been used to perform various large DNS ranging from premixed flame problems (200 10⁶ grid points, 13 species) (75), non-premixed flames (500 10⁶ grid points, 11 species) (76), to lifted jet flames (10⁹, 9 species) (77).

2.4.2 π³

π³solves the incompressible Navier-Stokes equations, while this formulation restrains the physical phenomena that can be studied (i.e high speed flow and acoustic-flame interaction cannot be studied), it is very efficient for low Mach flows (47, 78). Indeed, many combustion applications display a very small maximum Mach number and therefore can be modeled using an incompressible formulation. Second, π³ does not handle complete reaction schemes but use the FPI technique described below (79) instead. Note that the thermodynamics properties of the fluid is computed using CHEMKIN-like routines (74).

From a numerical point of view,π³ uses a sixth-order central difference scheme for spatial derivatives and a fully explicit four step fourth order Runge-Kutta method. Since it uses an incompressible formulation it requires also solving the Poisson equation for the pressure, which is done using a spectral method (78). Finally as for S3D, π³ uses 3-D Cartesian structured grids.

2.4.3 H-Allegro and SiTCom

The H-Allegro DNS code is a parallel compressible Navier-Stokes solver which has the particularity to use a hybrid grid arrangement. This grid arrangement relies on staggered grid arrangement (80) in the domain which degenerates into a collocated grid arrangement on the boundaries. This method has the main advantage to be remarkably stable and avoids the complicated treatment of the boundary conditions on a staggered grid (81). Moreover it allows a reduction of the grid density without loosing accuracy (82).

This code uses 3-D structured Cartesian grids and a sixth-order hybrid staggered- collocated explicit finite difference scheme. Temporal integration is achieved using a third order explicit Runge-Kutta method (83). Finally, the H-Allegro uses the perfect gases assumption and thermo-chemical properties are computed using a CHEMKIN-like formalism.

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The SiTCom solver also developed at the CORIA (8), is based on a more classical finite volume formalism used on 3-D cartesian grids. This solver approximates the convective terms resorting to the fourth-order centered skew-symmetric-like scheme and the diffusive terms with a fourth-order centered scheme. Time integration is performed using the third-order Runge-Kutta scheme. This solver was used by Lodatoet al. (8) for the development of the 3-D-NSCBC. This solver embeds also an LES formalism. Finally SiTCom solves the compressible Navier-Stokes equations and handles reactive flows.

2.4.4 YALES2

YALES2 is an incompressible LES solver developed to be massively parallel and to solves gaseous and liquid phase combustion on massive unstructured meshes (54, 55). Like the π³ solver it uses the incompressible set of Navier-Stokes equations for constant and variable density flows and thus uses a Poisson equation for the pressure. Grids are obviously non structured and the solver use the finite-volume framework in conjunction with a fourth order Runge-Kutta like time integration scheme. YALES2 was not developed for DNS, however it is able to compute nearly DNS on very complex configurations. For example Figure 2.5 shows results of a ”nearly-DNS” of the PRECCINSTA burner (84) performed by Moureau et al. (54). While the flow field is claimed to be accurately resolved in both space and time, the mesh size used is not able to handle a full chemical scheme. Indeed this simulation was performed using a FPI model to account for combustion.

Codes like YALES2 are showing the way for future DNS solvers: the used of unstructured grid. However, a real DNS of the burner like the PRECCINSTA using compressible Navier-Stokes equations and detailed chemistry scheme like the GRI 3.0 mechanism (5) are out of reach for the next 10 to 20 years.

2.4.5 Real Gases flow solver

Real gases DNS solvers are much more uncommon than perfect gases DNS solvers.

Indeed, only, four major groups performing numerical simulation involving real gases transcritical flow, reactive or not, can be identified:

• The group of Bellan, which have introduced the 1D-NSCBC for compressible real gases flow and performed several DNS of mixing layers (50, 51, 70).

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Figure 2.5: Iso-Q criterion in the swirler and in the combustor (left) and temperature field in the median plane (right) from (54).

• The group of Yang, which have performed LES computation of cryogenic jet (52, 85) and transcritical combustion (86) .

• The group of Oefelein which have developed a numerical framework to compute reactive transcritical flow and applied it to a LES of a LOX/H2 flame (86).

• The group of Candel and Darabiha which is known for various studies of transcritical flames (87, 88) and more recently for the first LES of a transcritical diffusion flame (89).

The only large DNS were performed by Bellan’s group on a two species mixing layer.

The code characteristic are close to those of S3D: a Cartesian structured grid is used with sixth-order spatial difference schemes, a fourth order Runge-Kutta method for time advancement, and numerical stability is achieved using sixth-order filters.

The computational cost of the real gases thermodynamics prevents a wide usage of DNS to study real gases flows. Most of the above works have thus used various methods to overcomes this problem. Those methods can be classified in two categories:

thermochemical properties tabulations as a function of the temperature and pressure which can only be done if one or two species are involved and hybrid thermochemical properties computations where the real gases formalism is used only if necessary. For example, the works of Okong’oet al. (70), Milleret al. (51) and Zong and Yang (52, 85) have used the first technique. Oefelein (86) for his LOX/H2 combustion case used the

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real gases formalism in the cold zone and the perfect gases formalism in the flame front and in the hot zones.

2.5 Tabulation techniques for combustion

As stressed in the previous sections the cost of detailed chemical schemes is often prohibitive. However, in many applications (90), several chemical time scales are smaller than the fluid dynamic time scales, which allows to reduce the dimensionality of the problem. This reduction is of critical importance for DNS (and even computational fluid dynamics in general): if one reduces the number of equations needed by detailed chemical mechanisms, the computing power needed will decrease accordingly. This is of primary importance, considering that even simple fuels such as methane are characterized by chemical mechanisms involving a large number of species and chemical reactions (5). As said, theπ³ solver uses a FPI technique to reduce the computing power required without loosing accuracy (47).

To perform this reduction, two methods can be used (see Chapter 1):

• mechanism reduction

• state-space parameterization

In this section, only the second type of scheme reduction will be considered, the PCA model proposed in this thesis being a manifold identification technique. Numerous models have been proposed to identify this manifold, of which only the three major ones will be reviewed : Steady Laminar Flamelet Method (SLFM) (91, 92, 93), Flamelet- Generated Manifold (FGM) (94) and Flamelet-Prolongation of ILDM model (FPI) (79, 95, 96), which rely on the same ideas, and the technique introduced by Sutherland and Parente (97) based on a transport equation for the principal component (PC) scores.

The goal here is not to give a detailed review of those techniques but rather to describe the general ideas of the techniques, especially for the common SLFM and FPI techniques.

2.5.1 SLFM

The SLFM technique (91, 92, 93) relies on the physical assumption that a turbulent diffusion flame is a collection of steady laminar diffusion flames. Before going further

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the mixture fraction, Z, must be introduced:

Z= s YF −(YO−Y_O^∞)

sY_F^∞+Y_O^∞ s= νOWO

νFWF (2.6)

whereYF is the fuel mass fraction,YOis oxidizer mass fraction, the∞superscript denotes the mass fraction in the pure fuel or oxidizer flow, νF and νO are the stoechiometric chemical reaction coefficients for the fuel and oxidizer, respectively andWF andWOare the fuel and oxidizer molar mass, respectively.

Then let’s consider the classical temperature mixture fraction plot in Figure 2.6. The physical assumption behind the SLFM model is that the structure presented in Figure 2.6 is not impacted by the flow, that is: if one knowsZ, one can recover the temperature and all the species mass fraction, which is convenient because Z is conserved in the flow and can be modeled by a conservation equation which does not involve source terms.

The database is generated using 1-D steady diffusion flames.

Figure 2.6: Temperature,T, as a function of the mixture fraction,Z.

The manifold is thus represented by the curve plotted on Figure 2.6. However, the- oretically all the states below this curve can be accessed, so that the manifold must be extended. Indeed, using only the curve on Figure 2.6 yields an infinitely fast chemistry, because no data are available below the curve. To account for non-infinitely fast chemistry a second parameter can be included in the database, the strain-rate of the flame, χ(91), yielding the database plotted on Figure 2.7, which is generated using 1-D steady diffusion flames at various strain rates. Even if accounting for the strain rate, a large

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region of the state space cannot be accessed with the database because if the strain rate increases too much the 1-D steady diffusion flame extinguishes.

Figure 2.7: Temperature,T, as a function of the mixture fraction, Z for different strain rates,χ.

While this approach is known to yield very good results for diffusion flames, the database is missing important information for modeling premixed flames. This stresses one of the main weak points of state space parametrization technique: it is as good as the database. For example, if one deals with a partially premixed diffusion flame using the SLFM, results will be inaccurate. Moreover, it also suffers from the curse of closure if the model is applied in LES. Indeed the conservation equation forZ involves diffusion terms which must be modeled in LES and this process is not straightforward.

2.5.2 FPI

The FPI technique is the premixed counterpart of the SLFM technique: the database is generated using 1-D premixed flames. In the FPI framework a conservation equation for a reaction advancement variable is solved (79), which requires a source term, and is then used to access the database allowing to recover the species. This advancement variable should be bijective throughout the flames: one value of the advancement variable must correspond to only one entry in the database, which can cause issues when choosing the advancement variable. For example, if one considers hydrogen/air flame the advance-

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ment variable is Y_H₂_O (98), but if one deals with methane combustion the advancement variable is Y_CO₂+Y_CO (96)

The state space accessible via the FPI technique is given on Figure 2.8 and, as for the SLFM, part of the state space can not be accessed through the database. This limitation is due to the fact that premixed flame have flammability limits (Zmin and Zmax) which prevents the inclusion of those states in the database. As for the SLFM method, the

Figure 2.8: Temperature,T, as a function of the mixture fraction,Zfor the FPI technique.

FPI method is as good as the database and also suffers from a closure problem, however this last point has been addressed recently (98). Finally note that this method had yield very accurate results for partially premixed flames (96).

2.5.3 Principal component scores

Sutherland and Parente introduced in their work (97) a tabulation technique which uses the results of a PCA as a database. The exact PCA process will be described in Chapter 6, therefore only the general process will be given here. PCA offers the possibility to automate the parameter identification process (97, 99), controlling at the same time the error induced by the reduction. In other words it allows an optimal manifold identification for a given number of parameters. The tabulation technique using PCA relies on the following idea: using PCA results one can compute the chemical

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state space anywhere in the computational domain using a sum ofq linear correlations of the state space parameter,ai, like:

[Y₁, . . . , YN_sp, T, p]≈[Y₁, . . . , YN_sp, T, p]q = Xq i=1

ziai (2.7)

whereYk is the mass fraction of speciesk,Nsp is the number of species andT and pare the temperature and pressure respectively. Theai vectors are the PCs obtained using a PCA, each of them has the same size as the state space variables (i.eNsp+ 2) and thezi

are the PCs scores. Since the ai are constant, if the zi can be computed in the domain, the full chemical state space can be recovered.

The idea behind the model of Sutherland and Parente (97) is to build a transport equation for eachzi. However, no results using this technique has been published yet, mainly because of the closure problem. Indeed transport equations for the scores require a diffusion term and a source term which are not straightforward to obtain since thezi

are the multiplicative coefficients of linear combination of the species mass fraction and even temperature and pressure in some cases. However this technique has the main advantage of reducing the size of the database: only theq vectors ai must be stored.

At this point, one can say that both the SLFM and the FPI techniques are known to generate accurate results and the computational cost reduction allowed by those method is very high with respect to detailed chemistry (up to 1 000). However their closure in LES is not straightforward and they impose to store huge a database which can be prohibitive when working on machines like the IBM Bluegenes which only allow a few hundreds of mega octets of memory for each node.

Using a transport equation for the PCs scores seems on the other hand to reduce drastically the memory requirement but the accuracy of the model is still to demonstrate.

Moreover this technique requires obviously to transport more variables than the SLFM or FPI techniques. Indeed combustion processes are strongly non-linear, thus representing them with linear combinations of vectors will require more input variables than the FPI or SLFM methods where the non linearities are naturally embedded in the database. For example a premixed hydrogen/air 1D flame will requires only one advancement variable to be computed using the FPI technique but 4 or 5 PCs scores will be needed using PCA (97).

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2.6 Conclusions

This chapter gave the general background information needed for this thesis. A brief history of DNS has been given along with a description of the DNS issues to better understand the contribution of this thesis to DNS: the accuracy of the boundary conditions. Indeed the cost of DNS is so high, especially when complex state equations like for real gases are involved, that the boundary conditions are of crucial importance.

Since the YWC solver was developed in the scope of this thesis a review of modern DNS solver dedicated for combustion was given. The use of compressible Navier-Stokes equations for DNS is clearly not a standard, and neither is the use of detailed chemical schemes. However, as available computational power is increasing, they will become standard while the use of unstructured grids will be, to my opinion, introduced in two or three generations of codes.

A description of tabulation techniques for chemistry was also given and the strength and weakness of those methods were stressed.

All the background information being presented, the description of the YWC solver will be given in the next chapter.

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The YWC Solver

The YWC solver was developed to be a state of the art, yet easy to use, DNS solver.

From the discussion of Chapter 2, a state of the art DNS solver should have the following characteristics:

• Solve the compressible Navier-Stokes equations.

• Use at least sixth order spatial schemes and third order Runge-Kutta temporal scheme.

• Handle detailed thermo-chemistery with detailed chemical schemes and differential diffusion.

• Capable of running on a large number of processors without losing performance, that is displaying a near perfect scaling law.

• Capable of dealing with perfect or real gases.

• Capable of using unstructured grids.

YWC fulfills nearly all those requirements, it is however not able to handle unstructured grid or multiphase flows. These features were not included in the code because of the particular treatment they require, which would drastically complicate the code structure and increase its learning curve. For example, using unstructured grid implies embedding a partitioner, using finite volume schemes and increase the amount of memory required by the solver.

Direct numerical simulation and reduced chemical schemes for combustion of perfect and real gases