Fanconi Anemia/BRCA pathway and the repair of interstrand-crosslink lesions

If the G-equation is to be used as a basis for turbulence modeling, it is convenient to ignore at first its nonuniqueness outside the surface G(x, t) = G0. Then the G-equation would have properties similar to other field equations used in fluid dynamics and scalar mixing. This would allow us to define, at point x and time t in the flow field, a probability density function P(G; x, t) for the scalar G.

From P(G; x, t) the first two moments of G, the mean and the variance, can be calculated as

G(x, t) =¯

_+∞

−∞ G P(G; x, t) dG, (2.122)

G^2(x, t) =

_+∞

−∞ (G− ¯G)²P(G; x, t) dG. (2.123) If modeled equations for these two moments are formulated and solved, one could, for instance, use the presumed shape pdf approach to calculate P(G; x, t) by presuming a two-parameter shape function. However, since G is only defined at the flame front, P(G; x, t) and its moments carry the nonuniqueness of its definition outside G(x, t) = G0.

There is, nevertheless, a quantity that is well defined and of physical rele-vance, which may be derived from P(G; x, t). This is the probability density of finding the flame surface G(x, t) = G0at x and t given by

P(G0, x, t) =

_+∞

−∞ δ(G − G0)P(G; x, t) dG = P(x, t). (2.124) This quantity can be measured, for instance, by counting the number of flame crossings in a small volumeV located at x over a small time difference t.

In Figures 2.18 and 2.19 two experimental examples of this pdf are shown.

The pdf P(G^) in Figure 2.18 was obtained by Wirth et al. (1993) by evaluating photographs of the flame front in the transparent spark-ignition engine described in Section 2.1. Smoke particles, which burn out immediately in the flame front, were added to the unburnt mixture, allowing the front to be visualized by a laser sheet as the borderline of the region where Mie scattering of particles could be detected. Experimental details may be found in Wirth and Peters (1992) and

Figure 2.18. Probability density function of flame front fluctuations in an internal combustion engine. —— Gaussian distribution. Measurements by Wirth et al. (1993).

(Reprinted with permission from SAE paper 932646
^C 1993 Society of Automotive Engineers Inc.)

0 0.01 0.02 0.03 0.04

0 10 20 30 40 50 60

5.5 8.4 P (x) 13.8

[1/mm]

v^/sL

x[mm]

Figure 2.19. The probability density of finding the instantaneous flame front at the axial position x in a turbulent flame stabilized on a weak swirl burner (cf. Figure 2.7).

Measurements by Plessing et al. (2000). (Reprinted with permission by The Combustion Institute.)

Wirth et al. (1993). The pdf P(G^) represents the pdf of fluctuations around the mean flame contour of several instantaneous images. The relation between G^ and the coordinate x normal to the flame brush will be given in (2.131) below.

A comparison of the measured pdf in Figure 2.18 with a Gaussian distri-bution shows it to be slightly skewed to the unburnt gas side. This is due to

2.8 Modeling premixed turbulent combustion 111 the nonsymmetric influence of the laminar burning velocity on the shape of the flame front: There are rounded leading edges toward the unburnt mixture but sharp and narrow troughs toward the burnt gas.

This nonsymmetry is also found in the experimental pdfs shown in Figure 2.19. Plessing et al. (1999, 2000) have measured the probability density of finding the flame surface in steady turbulent premixed flames on a weak swirl burner. The flames were stabilized nearly horizontally on the burner thus representing one-dimensional steady turbulent flames. The pdfs were obtained by averaging over 300 temperature images obtained from Rayleigh scattering.

The three profiles of P(x), shown in Figure 2.19 for three velocity ratiosv^/sL, nearly coincide and are slightly skewed toward the unburnt gas side.

Without loss of generality, we now want to consider, for illustration purpose, a one-dimensional steady turbulent flame propagating in the x direction. We will analyze its structure by introducing the flame-normal coordinate x, such that all turbulent quantities are a function of this coordinate only. Then the pdf of finding the flame surface at a particular location x within the flame brush simplifies to P(G0; x), which we write as P(x). We normalize P(x) by

_+∞

−∞ P(x) d x= 1 (2.125)

and define the mean flame position xf as

xf =

_+∞

−∞ x P(x) d x. (2.126)

The turbulent flame brush thicknessF,tcan also be defined using P(x). With the definition of the variance

(x− xf)²=

_+∞

−∞ (x− xf)²P(x) d x (2.127) a plausible definition forF,t is

F,t = ((x − xf)²)^1/2. (2.128) We note that from P(x) two important properties of a premixed turbulent flame, namely the mean flame position and the flame brush thickness, can be calculated.

Now we ask the question: How can these quantities be related to the mean ¯G(x) and the variance G^2(x) of the G-field? For the present example we choose a linear dependence of ¯G on x:

G(x)¯ − G0= x − xf. (2.129)

The mean flame front position x= xf is thereby defined by ¯G(x= xf)= G0. Using (2.129), we can relate the spatial fluctuation (x− xf) of the flame front location to the scalar fluctuation G^= G − ¯G as

G^= G − G0− (x − xf). (2.130) Since G is only defined at the flame front, one may set G = G0in (2.130) to obtain the simple equivalence

G^= −(x − xf). (2.131)

This shows that spatial fluctuations of the flame front position correspond to fluctuation of the scalar G, conditioned at G(x, t) = G0. Since the variance defined by (2.127) is a property of the entire flame brush, it is by definition independent of the position x within the flame brush. This must also hold for the conditional variance derived from (2.131)

(G^2)0= (x − xf)². (2.132) This requirement distinguishes the conditional variance (G^2)0from the uncon-ditional variance G^2(x, t) introduced in (2.123). Since it is easier to derive an equation for the unconditional variance, Equation (2.140) below, the additional requirement imposed on the conditional variance may be used to calculate the latter from the former. There are different ways to satisfy this requirement: In numerical simulations where G^2(x, t) is calculated in the entire flow field the requirement can be satisfied by using a reinitialization technique to assign the value of G^2calculated at the mean flame position ¯G(x, t) = G0to the adjacent values in normal direction. Alternatively, if gradients of G^2in the normal direc-tion are small, one may simply use the value of G^2calculated at ¯G(x, t) = G0

as a first approximation for (G^2)0. While (G^2)0 is independent of the flame normal coordinate, it may vary, however, in the tangential direction along the mean flame front and with time.

Equation (2.131) leads to an useful interpretation of the scalar G as a fluctu-ating quantity in turbulent combustion: It represents the scalar distance between the mean and the instantaneous flame front measured in the direction normal to the mean turbulent flame. This also holds for multiple crossings since multiple flame front locations also enter into the pdf P(x) shown in Figure 2.19. This interpretation shall guide the subsequent modeling of equations for the mean and the variance of G.

This analysis may be generalized to any shape of a turbulent flame front G(x¯ , t) = G0. Once the ¯G-field is calculated and the reinitialization condition

2.8 Modeling premixed turbulent combustion 113

|∇ ¯G| = 1 is applied outside of the mean flame surface, the coordinate normal to iso-surfaces of ¯G(x, t) is defined by

x= G(x, t) − G¯ 0

|∇ ¯G| + xf. (2.133)

Integrations across the flame brush then must be performed along those direc-tions.

In recent years, the G-equation has been used in a number of studies to investigate quantities relevant to premixed turbulent combustion. An early re-view was given by Ashurst (1994). Kerstein et al. (1988) have performed direct numerical simulations of (2.57) in a cubic box, assigning a stationary turbulent flow field and constant density. The constant density assumption has the ad-vantage that the flow field is not altered by gas expansion effects. The gradient

∂ ¯G/∂x in the direction of mean flame propagation was fixed at unity and cyclic boundary conditions in the two other directions were imposed. In this formu-lation all instantaneous G-levels can be interpreted as representing different flame fronts. Therefore G0was considered as a variable and averages over all G-levels were taken to show that for large times the mean gradient ¯σ can be interpreted as the flame surface area ratio. A similar approach has been used by Ulitsky and Collins (1997) to determine the effect of large coherent structures on the turbulent burning velocity.

Peters (1992) considered turbulent modeling of the G-equation in the corru-gated flamelets regime and derived Reynolds-averaged equations for the mean and the variance of G. Constant density was assumed and G and the velocityv were each split into a mean and a fluctuation. The main sink term in the variance equation resulted from the propagation term s_L⁰|∇G| = s⁰Lσ in (2.67) and was defined as

ω = −2 s¯ ⁰LG^σ^. (2.134) The quantity ¯ω was called kinematic restoration in order to emphasize the effect of local laminar flame propagation in restoring the G-field and thereby the flame surface. Corrugations produced by turbulence, which would exponentially in-crease the area of a nondiffusive iso-scalar surface with time (cf. Batchelor, 1952), are restored by this kinematic effect. Closure of the kinematic restora-tion term was achieved by deriving the scalar spectrum funcrestora-tion of two-point correlations of G in the limit of large Reynolds numbers. That analysis resulted in a closure assumption relating ¯ω to the variance G^2and the integral time scale k/ε:

ω = c¯ ωε

kG^2, (2.135)

where c_ω= 1.62 is a modeling constant. This expression shows that kinematic restoration plays a similar role in reducing fluctuations of the flame front as scalar dissipation does in reducing fluctuations of diffusive scalars.

It was also shown by Peters (1992) that kinematic restoration is active at the Gibson scaleG, since the cutoff of the inertial range in the scalar spectrum function occurs at that scale. A dissipation term involving a positive Markstein diffusivity D_L was shown to be effective at the Obukhov–Corrsin scale C

and a term called scalar-strain covariance was shown to be most effective at the Markstein lengthL. In the corrugated flamelets regime the Gibson scale

G is larger than bothC andL . Therefore these additional terms are higher order corrections, which, in view of the order of magnitude assumptions used in turbulence modeling, should be neglected.

A similar analysis was performed by Peters (1999) for the thin reaction zones regime. In that regime the diffusion term in (2.116) is dominant as shown by the order of magnitude analysis of (2.118). This leads to a dissipation term replacing kinematic restoration as the leading order sink term in the variance equation. It is defined as

χ = 2D(∇G¯ ^)². (2.136)

Closure of that term is obtained in a similar way as for nonreacting diffusive scalars and leads to

χ = c¯ _χε

kG^2. (2.137)

We will use the two closure relations (2.135) and (2.136) as the basis for the modeling of the turbulent burning velocity in the two different regimes in Section 2.10.

As was the conditional variance (G^2)0, the kinematic restoration ¯ω and the scalar dissipation ¯χ are conditional quantities defined for the entire flame brush and therefore must be independent of the normal coordinate x within the flame brush. Using the conditional variance in (2.135) and (2.137), respectively, they are to be calculated at the mean flame front. For simplicity of notation we will not distinguish between conditional and unconditional quantities in the following. From the balance equations that will be derived in the next section for unconditional quantities only those calculated at the mean flame front are of physical significance. It is assumed that they represent the respective conditional quantities.