Lecture 14. Turbulent Combustion. We know what a turbulent flow is, when we see it! it is characterized by disorder, vorticity and mixing.

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1 Lecture 14 Turbulent Combustion 1 We know what a turbulent flow is, when we see it! it is characterized by disorder, vorticity and mixing. In a fluid flow, turbulence is characterized by fluctuations of velocities and state variables in space and time. This occurs when the Reynolds number Re = UL/ν 1 L is a characteristic dimension of the vessel U is a characteristic velocity of the flow ν is the kinematic viscosity µ/ρ (cm 2 /s). Re 1 = L/U L 2 /ν = t a overall time to advect a fluid particle a distance L = 1 t d viscous dissipation time over distance L Disturbances in the flow, due to instabilities, may develop large velocity gradients. At Re 1, viscous dissipation prevents the development of these large velocity gradients. But at Re 1, viscous dissipation is unable to smear out the large developing gradients. 2 permission of the owner, M. Matalon. 1

2 Statistical description The aim is to describe the fluctuating velocity and scalar fields in terms of their statistical distributions. 1. Averaging time averaging ū(x) = lim T T 0 u(t, x)dt u(t) ū ensemble averaging compiled from many different realizations where u(t n, x) isthen th realization 1 N u(t, x) = lim u(t n, x) N N n=1 good also if ū depends on time ū t 1 t 2 t n measurements taken at a fixed point t t If we decompose u(t, x) into a mean and a fluctuation u (t, x) then, by definition, u (t, x) =u (t, x) = Probability Density Functions (pdf) The distribution function F u (U) of u is defined by the probability of finding a value of u<u F u (U) =p(u <U) U is the sample space of the random variable u consists of all possible realizations of u. The probability of finding a value of u in an interval U<u<U+ U is given by p(u<u<u+ U) =F u (U + U) F u (U) The probability density function (pdf) of u is defined as Clearly P u (U) = df u(u) du P u (U)dU =1 4 permission of the owner, M. Matalon. 2

3 In turbulent flows, the pdf of any stochastic variable depends, in principle, on the position x and on time t, or P u (U; x, t) is the probability density in U-space and a function of x and t. The simplified notation P (u; x, t) will be used from now on. Once the pdf of a variable is known one can define the average or mean value the mean of a function g(u) ū(x, t) = ḡ(x, t) = σ 2 = up(u; x, t)du g(u) P (u; x, t)du and the variance (a measure of the width of the pdf) (u ū) 2 P (u; x, t))du σ 2 = u 2 2uū +ū 2 = u 2 ū 2 σ 2 = u 2 u =ū + u u 2 ū 2 = ū 2 +2u ū u 2 ū 2 = u 2 mean of the fluctuations squared 5 u(t) ū u bimodal pdf P (u) ū u normal pdf t P (u) 6 permission of the owner, M. Matalon. 3

4 u 2 is the variance - the extent of variation from the mean the root mean square u 2 is the intensity the correlation of the fluctuations of two variables is often of interest uv= (ū + u )( v + v )=ū v + ū v +ū v + u v =ū v + u v correlation R uv R uv indicates the extent of the interdependence of the two variables the correlation coefficient R uv is often normalized such that 1 <R uv < 1 R uv = u v u 2 v 2 R uv 0 R uv ±1 R uv > 0 u and v are poorly correlated u and v are nearly perfectly correlated (or anticorrelated) u and v are correlated (and have the same sign most of the time) 7 Turbulent Scales Turbulent flows are said to be made up of small eddies with a multitude of sizes and vorticities. A fluid eddy is a macroscopic fluid element in which the microscopic elements comprising the eddy behave as a unit. An eddy is identified by a characteristic size and a characteristic velocity. A number of small eddies may be embedded in a large eddy. A characteristic of a turbulent flow is the existence of a wide range of length scales, or eddy sizes. The largest size L, is the macroscale or the flow scale. Eddies of size L are characterized by the mean flow velocity U. The Reynolds number of these eddies Re = UL/ν is large so that the effect of viscosity is negligibly small. The large eddies, however, break up as a result of instabilities, transferring their energy to smaller eddies, characterized by smaller Reynolds numbers. This process continues until the Reynolds number is sufficiently small; now viscosity acts to smear out large velocity gradients or dissipate the available kinetic energy. This is basically the eddy cascade hypothesis, proposed by Kolmogorov for homogeneous isotropic turbulence. 8 permission of the owner, M. Matalon. 4

5 In order to characterize the distribution of eddy length scales at any position in the flow field, we use the correlation of the axial velocity between two points. The correlation of the axial velocity u at time t between two points separated by a distance r, i.e., at x and x + r, is R(x, r,t)=u (x,t) u (x + r,t) it shows how related, or coherent, are the velocities at these two points in space For homogeneous isotropic turbulence the location x is arbitrary and r may be replaced by its absolute value r = r. The normalized correlation f(r, t) =R(r, t)/u 2 (t) defines a length scale (t) = 0 f(r, t)dr called the integral length scale f(r, t) 1 f(0,t) = 1 due to the normalization and goes to zero as r, because very large eddies occur less frequent. r 9 If v is a typical velocity scale, i.e., a measure of the velocity fluctuations around the mean, it can be defined from the kinetic energy k = 1 2 v i v i = 3 2 v2 2k v = 3 v represents the turnover time of the integral scale eddies. The turnover time of these eddies, t = /v is then proportional to the integral time scale. 10 permission of the owner, M. Matalon. 5

6 For very small values of r only very small eddies fit into the distance between x and x + r. The motion of these small eddies is influenced by viscosity which provides an additional dimensional quantity for scaling. We denote the dissipation of kinetic energy, per unit mass, by. [ ]= 1 s (m/s)2 =m 2 /s 3 [ ν ]=m 2 /s Dimensional analysis then yields the Kolmogorov length scale η ν 3 η = 1/4 We can also define a Kolmogorov time scale ν 1/2 t η = f(r, t) 1 and velocity scale v η =(ν) 1/4 η r inertial range 11 According to Kolmogorov s theory the energy transfer from the large eddies at the integral scale (i.e, of length scale ) is equal to the dissipation of energy at the Kolmogorov scale. Since at the integral scale the energy transfer rate = k3/2 = v2 /v = v3 = k3/2 If we define the turbulent Reynolds number as Re T = v /ν, the ratios of the smallest to largest length, time and velocity scales are η = (ν3 /) 1/4 ν 3/4 3/4 = = Re v T t η t = (ν/)1/2 v / ν 1/2 1/2 = = Re v T v η v = (ν)1/4 ν 1/4 1/4 v = = Re v T 12 permission of the owner, M. Matalon. 6

7 Various approaches Direct Numerical Simulation (DNS). Resolves the entire range of turbulent length scales, without recourse to any modeling assumptions. All the turbulence scales must be resolved in the computational mesh, from the smallest Kolmogorov scale up to the integral scale. DNS is being used for fundamental turbulent studies in a limited way. Because of the high computational cost, however, it is not a feasible tool for wide parametric studies and, at the present, is not a useful approach that can be used to simulate practical engineering problems. 13 DNS of a nonpremixed methane-air jet flame with a 4-step chemistry model and 8 species. Computations used 100 millions grid points, 340, 000 processor hours (4 months) to cover a region of approximately 1 cm 3 in physical space. Pantano, JFM 2004 methane oxygen Temperature isocontours water CO H 2 H Formation of holes Species mass fraction at a given instant in time 14 permission of the owner, M. Matalon. 7

8 Reynolds-Averaged Navier-Stokes equations (RANS). In this approach any instantaneous quantity is decomposed into a time or ensemble average and a fluctuating component, and introduced into the governing equations. The equations for the mean quantities, however, contain unknown correlation terms that call for the equations for the fluctuations. These however, contain higher order correlations and the system does not close. The closure problem is particularly critical in combustion problems because of the numerous quantities arising from the chemical source terms, the species and energy equations that need to be modeled, in addition to the usual Reynolds stresses encountered in ordinary fluid mechanics. 15 Consider first the incompressible NS equations (we will be using index notation) vi ρ t + v j v j x j =0 v i x j = σ ij x j Averaging the equations, using that the fluctuations have zero mean, yields v j =0 x j vi ρ t + v v i j = σ ij ρ(vi x j x v j ) j the extra term on the RHS resulting from the fact that Dv/Dt = D v / Dt The mean velocity field must also satisfy the continuity equation. However in the momentum equations there are additional terms or effective stresses, called Reynolds stresses that give the flow an added viscosity. 16 permission of the owner, M. Matalon. 8

9 One can write equations for the fluctuating quantities, but these produce additional unknown terms, of the form v i v j v k. This is known as the closure problem of turbulence. Turbulence modeling is an attempt to write empirical expressions for the Reynolds stress. The level of closure depends on the complexity of the model. The simplest closure approach is to model the additional stresses ρ(vi v j ) in an analogous way to the stress-strain relation for a Newtonian fluid - the turbulentviscosity hypothesis. ρ(vi v j )+ 2 3 ρkδ ij = ρν vi T + v j x j x i deviatoric stress where k = 1 2 v i v i is half the kinetic energy ν T is the turbulent viscosity (also referred to as eddy viscosity) 17 Reynolds Averaged Navier-Stokes equations (RANS) v j x j =0 v i t + v v i j = 1 p + 2 x j ρ x 3 ρkδ ij + j x j ūi ν eff + ū j x j x i with ν eff = ν + ν T the effective viscosity Unlike ν, the turbulence viscosity is not a material property; it is a function of space and time; i.e. ν T = ν T (x,t). 18 permission of the owner, M. Matalon. 9

10 In combustion problems, the continuity equation must include density variations. When averaging the continuity equation ρ t + (ρv j) =0 x j the additional term ρ vj arises. Similarly, triple correlations arise of the form ρ vi v j arise from the momentum equations. It is therefore convenient to introduce a density-weighted average velocity, called the Favre average 1 average, ṽ i = ρv i ρ and the fluctuations about the Favre-average, denoted by v i = v i ṽ i. When averaging the continuity equation, Favre-averaging ensures that ρvi vanishes and the averaged continuity equation retains the same form as the continuity equation itself, namely ρ t + ( ρṽ j) =0 x j 1 Introduced by Favre in Note that the mean of the velocity fluctuation, in the Favre decomposition v i =ṽ i + vi, does not vanish, i.e., v i = 0 rather ρv i =0 More impressive is the simplification obtained in the momentum equation, because ρv i v j = ρṽ i ṽ j + ρ v i v j instead of four terms in the more conventional approach. (Note that it has also the same structure as the conventional average.) There are of course questions as to whether ṽ is readily measurable experimentally, and in extending the Reynolds-stress idea to ρ v i v j (instead of the conventional ρvi v j ), using ρ( v i v j )+ 2 3 ρ kδ ṽi ij = ρν T + ṽ j with k = 1 2v x j x i v i i but the mathematical simplification is quite significant. 20 permission of the owner, M. Matalon. 10

11 There are of course additional terms that need to be modeled, when averaging the combustion equations. These include ρ v i ψ for any scalar ψ, such as mass fraction and temperature, arising from Dψ/Dt and other arising from the highly nonlinear chemical source term ω. To show the difficulties associated with the chemical source term, consider the simple form ω = F (T ), where F (T )=B(T a T ) e E/RT F = ρb(t a T ) 1 T T a T F = F ( T ) 1 T T a T E/R T e 1+ ET exp ET R T R T 2 Because of the large Zel dovich number, even small temperature fluctuations in the reaction zone cause significant fluctuations of the chemical source term. 21 PDF Transport Equation. This approach consists of developing and solving model transport equations for the pdf of a set of variables that describe the hydrodynamics and/or thermochemical state of the mixture. An advantage of this approach is that the pdf of a certain variable contains more information than just the mean and the correlation. Another advantage is that some aspects of turbulent transport are in closed form, which limits the modeling requirements. The main drawback, however, is the large number of independent variables that characterizes the pdf making it computationally costly. 22 permission of the owner, M. Matalon. 11

12 Large Eddy Simulations (LES). LES operates on the governing equations in order to reduce the range of length scales that need to be computed, thus lowering the computational cost. A low-pass filtering is used to remove the small scales, which are replaced by subgrid models. The range of length scale filtered out, and reduction in the computational cost compared to DNS, depend on the computational resources. 23 Regimes of Premixed Turbulent Combustion We assume equal diffusivities for all reactive scalars, and a Schmidt number equal to one D th = D F = D O D; ν/d =1 The flame thickness and flame (residence) time are then given by l f = D/S L t f = D/S 2 L As a result, the turbulent Reynolds number, may be expressed as Re T = v ν Re T = v D = v S L l f The Damköhler number number (flow time/reaction time) may be defined as D = /v = S L l f /S L l f v 24 permission of the owner, M. Matalon. 12

13 The various regimes of turbulent combustion will be presented on a log-log graph depicting v /S L versus /l f. The first characterization is the line Re T =1separating laminar and turbulent combustion. Turbulent combustion corresponds to large Reynolds numbers, i.e., the region to the right, sufficiently removed from the line Re T = 1. Another distinction is between small and large Damköhler numbers. Flames are typically characterized by fast chemistry, or D 1, while the region D 1 corresponds to well stirred reactors where all turbulent scales are smaller than the chemical time scale. v /S L Re T =1 well-stirred reactor D =1 turbulent flames 1 laminar flames /l f 25 Further refinement of this diagram involves the Karlovitz number Ka, which relates the flame scales to the Kolmogorov scales. Two such numbers can be defined, one based on the flame thickness l f and the other on the thickness of the reaction zone l R. They represents the ratio of the residence time in the flame zone, or reaction zone, relative to the smallest turbulent turnover time, and thus whether turbulence penetrate and distort the flame or reaction zone structures. Ka = t f = l2 f /ν t η η 2 /ν = l2 f η 2 or Ka = t f = ν2 /lf 2 t η ν 2 /vη 2 = v2 η SL 2 and using δ = l R /l f Ka δ = l2 R η 2 = δ2 Ka Re T = v SL v 2 v 2 vη = S L l f v = D l f S L v η v η v = (ν)1/4 v = Re 1/4 T S L 2 Re T = D Re 1/2 Ka T Re T = D 2 Ka 2 It should be noted that equalities were set here for scaling purpose only. 26 permission of the owner, M. Matalon. 13

14 The relation Re T = D 2 Ka 2 implies that v = Re S T = D 2 Ka 2 = Ka 2 SL L l f l f v l f l f v 1/3 = Ka 2/3 S L l f and may be used to identify, within the regime of large D, conditions under which the smallest eddies are or are not able to penetrate the flame zone. This depends on whether Ka is smaller/larger than one (say). 1 The boundary Ka = 1 implies v 1/3 = S L l f 27 When Ka < 1 the flame residence time is much shorter than any turbulent time scales and the flame thickness is smaller than the smallest turbulent scale. The flame retains the laminar structure but is wrinkled by the turbulence motions. (a) In the sub-region where v < S L, known as the wrinkled flamelet regime, theturnovervelocityv of even the large eddies is not sufficient to compete with the advancement of the flame front at the laminar speed S L. The only effect of turbulence is to disturb the flame front, resulting in wrinkled flames. (b) In the sub-region where v >S L, referred to as the corrugated flamelet regime, the entire flame is embedded in eddies of the size of the Kolmogorov scale with velocities exceeding S L, which are able to wrinkle the flame and form pockets of fresh and burned gas. When Ka > 1 the flame residence time is on the order of the turbulent times (based on integral scale) but much larger then the Kolmogorov turnover times, and the Kolmogorov scales are smaller than the laminar flame thickness. Here the small eddies can presumably interact and modify the flame internal structure. 28 permission of the owner, M. Matalon. 14

15 v /S L distributed reaction zones Ka δ =1 thin reaction zones Ka =1 (δ =0.1) 1 Re T =1 laminar flames corrugated flamelets wrinkled flames /l f Regime diagram of premixed turbulent combustion on a log-log plot 29 When Ka > 1 the small eddies penetrate the flame zone, but may or may not modify the much thinner reaction zone. The distinction is between what has been referred to as thin reaction zones and broken reaction zones. I the latter, the laminar structure could no longer be identified. The boundary between these two regions depends on the relative thickness of the reaction zone δ. Typically δ 10 1,whichimpliesKa = 100, or v 1/3 = 10 4/3 S L l f 30 permission of the owner, M. Matalon. 15