Representing Turbulence/Chemistry interaction with strained planar premixed

Representing Turbulence/Chemistry interaction with strained planar premixed flames CES Seminar Andreas Pesch Matr. Nr. 300884 Institut für technische Verbrennung Prof. H. Pitsch Rheinisch-Westfälische Technische Hochschule Aachen Supervisor: Dipl.-Ing. Konstantin Kleinheinz

Contents 1 Abstract 2 2 Fundamentals 3 2.1 Premixed combustion....................... 3 2.2 Fundamentals of turbulent premixed combustion........ 3 2.2.1 Large-Eddy simulation.................. 3 2.2.2 Flamelets......................... 4 2.2.3 Burning velocity..................... 5 2.2.4 Level-set equation..................... 6 2.2.5 Tabulated chemistry................... 7 2.2.6 Presumed PDF-approach................. 7 3 Strained flamelet modeling 9 3.1 Strain-rate fundamentals..................... 9 3.1.1 Definition of the strain-rate............... 9 3.1.2 Influence on turbulent combustion............ 9 3.2 Implementation into the LES framework............ 11 3.2.1 Strained flamelet formulation.............. 11 3.2.2 Parameterization of the strain-rate in reactive flows.. 11 3.2.3 LES framework...................... 13 3.2.4 Subfilter modeling.................... 13 3.2.5 Level set coupling..................... 14 3.3 Simulation results......................... 15 3.3.1 Validation through 2D laminar flame analysis..... 15 3.3.2 3D LES of a Sandia jet DNS............... 16 4 Conclusion and outlook 21 1

1 Abstract Due to the interaction of turbulence and chemistry, the simulation of turbulent premixed combustion is a very complex field of research. Large- Eddy-Simulations are increasingly used in practical applications because of their higher accuracy in comparison to Reynolds-averaged Navier Stokes approaches and their lower computational cost as compared with direct numerical simulations [9]. Despite the saves in computational cost, the results of the simulations need to be sufficiently accurate. This leads to new challenges in modeling the subgrid scale turbulence chemistry interaction. Strain is one of the phenomena that strongly influences the combustion and thus is the main subject of this work [3]. A new method of incorporating strain-dependence into a LES framework using a flamelet approach is presented here. This seminar work is separated into three sections. The first section briefly introduces the fundamentals needed for turbulent premixed combustion. The main part of the work focuses on strain and it s effects on the combustion process. The integration of this concept into the LES framework and results of recent computations are discussed, too. In the end everything will be summarized with a conclusion and a brief outlook. 2

2 Fundamentals 2.1 Premixed combustion Premixed combustion is used, where an intense combustion process in a small volume is needed [4]. As the name says, the educts are mixed before entering the combustion chamber. As opposed to non-premixed combustion the process generally proceeds soot free [4]. For safety reasons of only small amounts of fuel and oxidator should be premixed because a large amount of the mixture is likely to explode in case of an defect [4]. Also the combustion is not as stable as in the non-premixed case. Applications therefore are spark ignition engines, household and industrial burners or stationary gas turbines [5]. 2.2 Fundamentals of turbulent premixed combustion 2.2.1 Large-Eddy simulation Basically there are three ways to simulate turbulent reactive flows. The first way, the direct numerical simulation (DNS), fully resolves the flow field and thus leads to a detailed solution of the flow equations. This is very expensive in terms of computational cost [8]. Another simulation type are the so called Reynolds-Averaged-Navier-Stokes (RANS) simulations, which are based on averaging all flow quantities with respect to time. This is computationally cheaper but also leading to less detailed results considering the dynamics of the flow [8]. In this work we focus on models for Large-Eddy Simulations (LES). In LES, the filtered Navier-Stokes equations are solved. Therefore, large scale turbulence (large vortices, also called eddies) is computed directly but the small scales are filtered out and their effect needs to be modeled. This leads to more detailed solutions than RANS but is not as expensive as DNS [9]. Applying spatially uniform filters, the Navier-Stokes equations can be written as follows [9]. The continuity equation the momentum equation ( ) Ui x i = U i x i = 0, (1) DU j Dt = ν 2 U j τ ij r 1 x i x i x i ρ p, (2) x j 3

and the energy equation DE f Dt x i [U j ( 2νS ij τ r ij p ρ δ ij )] = ε f P r, (3) where E f 1 2 U U is the kinetiv energy of the filterd velocity field, ε f and P r are defined by ε f 2νS ij S ij (4) P τ r ijs ij (5) Because of the lower cost and higher accuracy, LES is increasingly used in practical cases. In the work regarded in this seminar, the accuracy of Large- Eddy Simulations is improved further by considering the effect of strain on the combustion processes. Typically, this effect cannot be captured by LES directly because it occurs on subgrid scales. Hence, it needs to be modeled. Another thing that typically is modeled in reactive flow simulations is the combustion chemistry. One way to handle this will be explained in the following section. 2.2.2 Flamelets Under the assumption of fast chemical reactions compared to the flow time scales, a combustion process can be described by thin laminar flames, so called flamelets. The reaction only takes place within these thin structures whereas the rest of the flow field is not reacting at all [7]. Because of the fast chemical time scales compared to flow time scales, gradients perpendicular to the flame are much larger than tangential gradients and thus the dimensions of the problem can be reduced by a coordinate transformation to only one. Flamelet models are based on a progress variable C. This progress variable can be either described by a normalized temperature difference or a normalized product mass fraction [7] C = T T u T b T u or C = Y P Y P,b. (6) For this scalar a transport equation is solved, for example like in the Bray- Moss-Libby-Model [7]: ρ C t + ρũ C i = ( ρ x i x ) u i C + ω c. (7) i The chemistry of the combustion can then be computed as a function of this variable which will be explained further in section 2.2.5. 4

2.2.3 Burning velocity One of the most important quantities in premixed combustion is the burning velocity. It is the velocity of the flame propagating through the burning chamber and is often needed as a parameter in the simulation. In the laminar case, the burning velocity s L only depends on the thermo-chemical properties of the premixed gas and thus can be measured. This is for example done by observing the burning angle of a Bunsen flame, shown in figures 1 and 2 from [5]. The laminar burning velocity can be influenced by certain phenomena such as flame curvature and strain though [7]. This will be explained in the main part of the seminar work. In the turbulent case the burning velocity s T can be described by the continuity equation and a respective turbulent surface area A T as shown in figure 3. Under the assumption of constant density, this yields equation (8).[5] ṁ = ρ u s L A T = ρ u s T A (8) Figure 1: The Bunsen burner [5] Figure 2: Kinematic balance for a steady oblique flame [5] 5

Figure 3: An idealized steady premixed flame in a duct [5] 2.2.4 Level-set equation One of the main drawbacks of LES is the description of the flame fronts. Typically the flames are thinner than the filter size of the LES [6]. Underresolving the flame may lead to a significant jump of the progress variable or temperature within just one computational cell. This leads to numerical diffusion which can affect the burning velocity [6]. Also the exact position of the flame can not be tracked. One way to deal with this under-resolution of the flame is the so called level-set method [5]. A kinematic relation is used to describe the flame position in space dx f dt = v + s L n (9) with x f being the position vector of the flame, v the velocity vector, s L the laminar burning velocity and n the normal-vector of the flame. If then a scalar field describes distances between a point in space and the flame, it s zero-entries describe the position of the flame. This is visualized in figure 4. To get the position of the flame, one can now evaluate the total time- Figure 4: Flame structure described by a level-set ansatz [7] 6

derivative of the scalar field evaluated at zero DG = G x + G = 0. (10) Dt t t G=G0 G=G0 Inserting the kinematic relation (9) and the definition of the normal vector n = G G (11) leads to the so called level-set equation which can then be solved to determine the flame position in space [5]. 2.2.5 Tabulated chemistry G t + v G = s L G (12) As discussed in section 2.2.2, the chemistry of the combustion process has to be computed, too. The direct computation of complex reaction mechanisms is very expensive in terms of computational costs [1]. The reduction of the mechanism to only a small number of slow reactions that determine the overall kinetics is typically used. Another technique used to significantly reduce the computational cost of a reactive flow simulation is tabulated chemistry. The essence of this technique is that the chemistry of the reacting flow is not computed directly during runtime but accessed from tables that are previously computed [1]. Based on the progress variable introduced in section 2.2.2 (and maybe also a second or a third variable), fluid properties dependent on the chemical behavior of the flow can be looked up there (e.g. viscosity, temperature and density). 2.2.6 Presumed PDF-approach Finally, there is need of a sub-filter description of the quantities that are used as lookup-variables of the tabulated chemistry. Sub-filter scale modeling can be handled by using a presumed PDF-approach. Because of the stochastic character of the sub-filter quantities a probability density function is commonly used to describe them. One could solve a transport equation to get the PDF but this is very costly in terms of computational resources [7]. Instead, the PDF can also be presumed in advance. Therefore the shape of the PDF has to be known. There are different common types 7

of PDFs used like the beta-pdf which can be modified by two parameters (e.g. α and β) P (Z) = Zα 1 (1 Z) β 1 Γ(α + β). (13) Γ(α)Γ(β) Parameters can for example be related to the Favre mean Z and the variance Z 2. The filtered quantities of the LES can then be determined by integrating the product of the flamelet solution and the presumed distribution [3]. 8

3 Strained flamelet modeling 3.1 Strain-rate fundamentals 3.1.1 Definition of the strain-rate Strain is well known as a measure of deformation of a solid body. A fluid element can be deformed in the same manner if it is exposed to significant velocity variations. These velocity variations in space can be described by the strain-rate. It is defined as a tensor quantity (e.g. in [9]): a ij = 1 2 ( Ui x j + U j x i ). (14) In 1-D this reduces to the derivative of the velocity in the corresponding direction (e.g. x) [3] ( ) du a =. (15) dx 3.1.2 Influence on turbulent combustion Turbulence and chemistry interaction takes place on subgrid scales when small eddies deform the flame structures. This enhances molecular mixing and thus affects the combustion process. The flame stretch-rate, caused by curvature and by strain determines many aspects of the flame behavior. According to Steinberg and Driscoll [11] it describes the rate of change in flame surface area and wrinkling due to turbulence flame interactions. This influences local properties such as propagation speed and flame thickness. It also may change the overall dynamics of a flame, for example leading to extinction. Influence on the flame surface area Several experimental studies (e.g.[11] and [12]) showed that the fluid-dynamic strain-rate is the crucial measure for flame surface area generation as opposed to the vortical structures that cause wrinkling of the flame. Figure 5 shows the effect of surface area generation due to strain. The flame surface change can be described by the following equation from [11]: with the tangential strain-rate κ = 1 D(δA f ) δa f Dt = a t + κ c, (16) a t = ˆn (ˆn ) u f + u f (17) 9

Figure 5: Flame surface area generation due to strain [12] and the curvature κ c = s l ˆn = s l C. (18) u f denotes the velocity field at the flame surface, ˆn is the flame surface normal pointing into the reactants, s l is the laminar burning velocity and δa f is a flame surface area element. Influence on the burning velocity As mentioned before, strain can also have an impact on the burning velocity. A modified burning velocity equation is stated in [5] which includes contributions of flame curvature and strain for a one-step large activation energy reaction and the assumption of constant properties s L = s 0 L s 0 }{{} LLκ + Ln }{{}} {{ v n}, (19) unstretched burning velocity curvature term strain term where L denotes the Markstein length which is of the same order of magnitude and proportional to the laminar flame thickness. The changes in laminar burning velocity and flame surface area influence the turbulent burning velocity according to equation (8) in section 2.2.3 s T = s L A T A. (20) 10

3.2 Implementation into the LES framework In this section it is described how the influence of strain is considered in a LES framework. There are many different ways to realize this goal. In particular the implementation of Knudsen et al. [3] is presented here. A strained flamelet formulation is used where the flamelet solutions are tabulated with two parameters, the progress variable and a parameter describing strain. A low-mach flow solver is used to compute the reactive flow using the tabulated solutions to update the state variables in the flow field. Subgrid scales are modeled through a presumed PDF approach. 3.2.1 Strained flamelet formulation Chemistry is described by a premixed counterflow reaction that is solved with the FlameMaster program 1. The counterflow reaction is suitable to describe a pure strain-rate flow field according to Steinberg et al.[11]. In order to be able to solve the equations for the reaction with the FlameMaster program, they have to be expressed as a function of one variable. This is achieved through introducing a similarity coordinate η which is defined as ( ) 1/2 a y η = v 0 0 ρ(s) ρ 0 ds, (21) where a is the strain-rate and defined as the derivative of the velocity in x direction evaluated in the unburnt gas a = (du/dx) 0. The x direction is perpendicular to the y coordinate that is used as the physical space coordinate perpendicular to the counterflow flame fromts [3]. The configuration of the counterflow burner is shown in figure 6. 3.2.2 Parameterization of the strain-rate in reactive flows In order to take strain into account, the lookup tables of the flamelet solutions need a second parameter as a measure for the strain-rate. Several different approaches have been considered in fully resolved contexts [3]. Candidates for the second parameter therefore are the strain-rate itself, elemental mass fractions, the CO radical, the OH radical, temperature or the dissipation rate of the progress variable [3]. The strain-rate itself was not taken into account because it describes a parameter imposed on a flame while the others describe the flame structure s internal response to strain which was desired 1 The FlameMaster program is available at the ITV homepage by request http://www. itv.rwth-aachen.de/downloads/flamemaster/ 11

Figure 6: Symmetric back-to-back stagnation-point flow configuration [10] Figure 7: The s-curve describing the effect of strain-rate change on the maximum flamelet temperature [3] in this work. A comparison of the remaining quantities is done by Knudsen et al.[3] to evaluate by which parameter the effect of strain-rate change is captured the best. It was found out that both the OH and the H mass fraction vary significantly as a function of the imposed strain and decrease monotonically moving down the s-curve. Other parameters like the dissipation rate of the progress variable or the CO mass fraction are not monotonic functions moving down the s-curve leading to non-unique parameterizations. Temperature s sensitivity to strain is too little whereas the H2 mass fraction leads to overlapping flamelet solutions for temperatures below 1600 K. Because of it s slightly larger sensitivity to strain and due to the possibility of capturing differential diffusion effects, the H species mass fraction is chosen as the second parameter. The flamelet solutions therefore are tabulated as function of the progress variable and the H species mass fraction: Y i = Y i (C, Y H ). (22) Also, a value of the laminar burning velocity is needed for modeling purposes. Since strain influences the burning velocity, a representative value has to be computed for each strained flamelet and is parameterized in the same manner as the flamelet solutions [3] s k,u = s k,u (C, Y H ) (23) 12

3.2.3 LES framework The equations that have to be solved in order to determine the flow variables are described shortly in this section. To reduce computational cost, the filtered Navier-Stokes equations can be solved in their low Mach limit if acoustic effects do not strongly influence the flow [2]. Thus, there is no need to solve the energy equation. t ( ρũ i) + t ( ρ) + ( ρũ j ) = 0 (24) x j ( ρũ i ũ j ) = ( p) + x j x j x j ( σ ij + σ ij,sfs ) (25) Spatial filtering is denoted by the ( ) operator while the ( ) operator denotes spatial density weighted filtering. The stress tensor is defined as [ σ ij = 2 ρṽ ã ij 1 ] 3 δ ũ k ij (26) x k with the strain-rate tensor ã ij defined like in equation (14) with spatial filtering. In addition, transport equations for the progress variable C and the H species mass fraction Y H are solved: ( ρ t C ) + ( ) ρũ j C = x j x j ( ) ρỹh + ( ) ρũ j Ỹ H = ( ρ t x j x D H j ( ) ρ D C ( x C) + Ξ jc,sfs + ω C (27) j ) (ỸH) + x Ξ jh,sfs + ω H. (28) j D is the filtered molecular diffusivity while Ξ is the scalar flux. The index ( sfs ) denotes subfilter scale values. These values have to be modeled in order to close the equation system [3]. 3.2.4 Subfilter modeling Subfilter scale values of the equations described in section 3.2.3 are modeled using a Smagorinsky-type model which will not be explained in further detail. However, all the filtered values of the equations also require a subfilter model, which will be described briefly. The subfilter model is realized using a presumed PDF approach. A beta-pdf is used to describe the subfilter progress variable fields. The independent coordinate of the PDF is named C. In 13

order to ensure that the beta-pdf is bounded between zero and one, it is rescaled by the maximum expected progress variable. The mean value C and the variance C 2 of the beta distribution serve as parameters, leading to the following equation describing LES filtered quantities in the unstrained model: Ỹ i ( C, C 2 ) = 1 0 Y i (C C max )β (C : C, C 2 ) dc. (29) In the case of the strained model another parameterization coordinate is added in form of the hydrogen mass fraction. However the distribution of Y H and C are statistically dependent which leads to a challenge in describing the subfilter distribution of Y H. The joint distribution of C and Y H would have to be written as P (C, Y H ) = P (C) P (Y H C). As a workaround a coordinate ψ is introduced that temporarily replaces Y H [3]. It is defined as the maximum hydrogen mass fraction that can be found in a flamelet and therefore is unique for every flamelet while being statistically independent of C: P (C, ψ) = P (C) P (ψ). The resulting equation describing the LES-filtered quantities is then (C ; C, C ) 2 Ỹ i ( C, C 2, ỸH) = Y i (C C max, ψ ) β ( δ ψ ψ(ỹh) ) dc dψ ( = (Ŷi (ψ ) C, C )) 2 ( δ ψ ψ(ỹh) ) dψ. (30) The shape of P (ψ ) is chosen to be a delta function P (ψ ) = δ(ψ ψ) with the implicit definition of ψ: ( Ỹ H = ŶH( ψ) C, C ) 2. (31) Equations (29) and (30) are therefore used to access chemical solutions in the LES. 3.2.5 Level set coupling Another challenge in modeling premixed combustion with LES is the resolution of the flame fronts. Typically the flame fronts are much smaller than the filter sizes which results in errors in the flame front speed. Special treatment is needed to ensure that correct flame front speeds are computed. One way 14

to achieve this goal is the level set coupling approach. The level set equation is used to describe a progress variable field [3]: t (Ĝ) + ũ j x (Ĝ) = ρ u ST,u Ĝ. (32) j ρ This field is not under resolved opposed to the progress variable field resulting from the scalar transport equation. At every point, the exact distance of that point to the flame front is stored. A blended value C m is computed by combining the level set and the transport equation progress variable so that the level set only influences the progress variable immediately at the flame front: C m = (ξ) C G + (1 ξ) C, (33) where C G is the level set progress variable and ξ is defined between zero (far away from the flame front) and unity (immediately at the flame front). Only the progress variable source term is affected by the level set coupling as it is accessed in the strained tabulated results with this blended progress variable. The hydrogen mass fraction source term is not coupled with the level set because the hydrogen concentration is not fixed at the flame front and thus no value can be safely assumed there. Also the burning velocity is a function of hydrogen which would result in deviations in the propagation speed. 3.3 Simulation results Two different kinds of simulations were performed using the strained premixed flamelet model. First, to validate the model, a 2D laminar premixed flow simulation is performed. A Sandia slot-jet flame is then computed via LES and compared to DNS data. 3.3.1 Validation through 2D laminar flame analysis A 2D laminar premixed flow as shown in figure 8 (a) is computed using both fully resolved finite rate chemistry and the strained flamelet model. The premixed stream enters on the outer radius on the left side of the cylindrical mesh flowing against a centerbody wall. Another wall section is defined on the outer radius top and bottom. The outlet is placed on the rightmost side of the mesh. The mesh consists 160 points in radial and 264 points in azimuthal direction. As strain is a direct function of the velocity of the flow in this case, the simulation is performed with different inlet velocities. The ability of the flamelet model to capture strain-effects can thus be investigated. 15

Figure 9 shows the results of both computations in comparison. The values were recorded along the sampling vector shown in figure 8 (e) and (f). Both a simulation with and without level set coupling are compared to the finite rate chemistry simulation. The results show that the model is able to capture the strain dependencies of the flame. However, the level set slightly underpredicts the flame speed and thus transports the flame front toward the centerbody wall. This results from a different tabulated burning velocity than directly solved from the transport equations and different Y H values near the flame front leading to diverse progress variable source terms [3]. Figure 8: Scetch of the 2D flow simulation (a) and results of the computations (b-f) [3] 3.3.2 3D LES of a Sandia jet DNS Finally, a three-dimensional Large-Eddy simulation of a Sandia premixed jet is performed and compared to a direct numerical simulation. First, the 16

Figure 9: Comparison between finite rate chemistry and strained flamelet model data [3]. presumed PDFs are validated by comparing results from the DNS with the beta-pdfs. In order to do that, the mean and the variance of a normalized, filtered progress variable is computed from the DNS data set. A subset of these values is then compared to the presumed PDFs. Figure 10 shows the comparison and indicates that the presumed PDFs reproduce most of the 17

shapes adequately. This implies that the beta-pdf is a suitable choice to describe the sub-filter quantities. The LES of the Sandia jet is performed with two different computational meshes, one consisting of 1.2 million cells and another consisting of 9.3 million cells, resulting in filter width ratios /l F = 0.5 and /l F = 0.25 respectively. l F describes the flame thickness here. Simulations with the strained as well as the unstrained model are performed and compared to the DNS data which were computed on a 194 million cell mesh. In comparison to the unstrained model, Y H, C and ω C agree well with the DNS results. Also, refining the LES mesh lets the solution converge to the DNS results. Figure 11 shows the most conspicious result of the comparison between strained and unstrained model. One can see the underprediction of the flame length of the unstrained model while the strained model adequately predicts the length of the flame compared to the DNS results shown in figure 12. 18

Figure 10: Comparison of presumed PDFs(dashed lines) and DNS data(lines with dots) for different progress variable values. [3] 19

Figure 11: LES simulation results from left to right: unstrained with 1.2m cells, unstrained with 9.3m cells, strained with 1.2m cells, strained with 9.3m cells. The e = 0.065 isosurface is plotted with contours of the YH scalar (value from transport C equation in strained model, accessed from a database in unstrained model). [3] e = 0.065 isosurface on the Figure 12: Results from the Sandia jet DNS. The C left and the density weighted and time averaged C field on the right ranging from e = 0.065. [3] 0.00 (blue) to 0.19 (red) and a black line showing the C 20

4 Conclusion and outlook In this work, a strained flamelet model for premixed combustion is presented. A second species mass fraction besides the progress variable transport equation is solved to parameterize and access tabulated solutions of strained premixed chemistry solutions. Sub-filter values are modeled with a presumed PDF approach while level-set coupling ensures a detailed resolution of the flame front leading to accurate burning velocities. The comparison to a Sandia jet DNS showed that the strained model is capable of accurately describing the combustion process while the unstrained model fails in reproducing the DNS results [3]. The model therefore can be used to optimize industrial burners that operate at high turbulent intensities. Further improvements on the model could be made by tabulating transient solutions of premixed flamelets. This would enable the analysis of transient flame behavior but requires time as a third parameter in the tabulation [3]. Also, the effect of changing combustion parameters of the premixed flamelet solutions could be investigated in further studies to ensure variability of the proposed model. 21

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