OCR Text |
Show from basic principles (Candel and Poinsot [5], and Pope [18]). It takes its origin from the basic equation for a material surface per unit volume E . Replacing the flame surface density E (flame area per unit volume, m2/m3) by ^Sf and introducing the mass averaged decomposition of the different variables, one gets : djpSj) | djpujSf) _ d / p€ dSf\ | p p (3) di dxi dxi V <TSJ dxi J where Sf is the flame surface per unit mass (m2/kg), P is the flame surface production term which describes the increase of flame area due to the local suain rate, whereas the destruction term D represents the flame surface dissipation or flame shortening due to mutual interaction of adjacent flame elements. All the actual efforts in turbulent combustion modelling using the flamelet approach and the F S D balance equation are focused on the modelling of these source and sink terms. Expressions for these are given in Duclos et al. [8]. This reference also contains a comparison of a set of F S D models in the case of one-dimensional premixed turbulent flame propagation in frozen turbulence. In all these models the mean consumption rate of fuel (the mass of fuel burnt per unit time per unit volume) is expressed as : uju = POUL^ , where po is the density of the fuel in the fresh gases and U L the mean consumption speed along the flame front. The quantities U L and E have to be modelled, and this leads to various degrees of complexity in flamelet models. In earliers models [8], the production term in Eq. (3) is modelled as P - pE3Sf, where E* is a mean strain rate. This term describes the augmentation of material surface by the strain rate. One of the central difficulties in flamelet modelling is the determination of the stretch induced by the turbulent eddies. The suain rate may be determined from velocity gradients of the mean flow (as proposed by Marble and Broadwell [12]). Another way consists in the determination of a small scale strain (Cant et al., [7]) or large scale strain which can be estimated from turbulent characteristics like the turbulent kinetic energy and its dissipation. Improved methods of evaluation of the effective strain rate producing the flame area have been also devised by Meneveau and Poinsot [13]. New Description of the FSD Equation The present study draws its originality in the reformulation of the source terms in the F S D equation (surface production and destruction) and of the equation for fuel conservation. In such models, the F S D prediction represents the main feature. A good prediction of the flame surface shape and position gives a good prediction of fuel consumption rate, temperature and heat release. These new formulations are based on the flamelet characteristics (laminar flame speed and flame width). Hence this model no longer needs a flamelet library. The F S D production term is modelled as the sum of the contributions of chemistry and turbulence : P = Pc + Pt = pEe sSj (4) The first term of Eq. (4) represents the flame surface production due to chemistry effects. This term is modelled as where UL is the laminar flame speed, UL0 the same quantity for stoichiometric flame and /; is the flame thickness. The contribution of turbulence to the flame surface production mechanism is represented by the second term of Eq. (4). This term is modelled as follows : «"* (s)(fe)* Note that this term does not allow turbulence to create flame surface everywhere. In other words, it prevents flame surface production where chemistry forbids it, i.e, at any point of the chamber where the equivalence ratio is respectively lower/upper than the lean/rich flammability limits of the fuel, since U L is equal to zero in these cases. The constant of the model a is equal to 0.5. By combining Equations (4), (5) and (6), an equivalent stretch representing the sum of the inverse of chemical and turbulent characteristic times can be written as : 4 |