Bray–Moss–Libby model
In premixed turbulent combustion, Bray–Moss–Libby (BML) model is a closure model for a scalar field, built on the assumption that the reaction sheet is infinitely thin compared with the turbulent scales, so that the scalar can be found either at the state of burnt gas or unburnt gas. The model is named after Kenneth Bray, J. B. Moss and Paul A. Libby. Mathematical description. Let us define a non-dimensional scalar variable or progress variable formula_1 such that formula_2 at the unburnt mixture and formula_3 at the burnt gas side. For example, if formula_4  is the unburnt gas temperature and formula_5 is the burnt gas temperature, then the non-dimensional temperature can be defined as The progress variable could be any scalar, i.e., we could have chosen the concentration of a reactant as a progress variable. Since the reaction sheet is infinitely thin, at any point in the flow field, we can find the value of formula_1 to be either unity or zero. The transition from zero to unity occurs instantaneously at the reaction sheet. Therefore, the probability density function (suppressing the time variable formula_8) for the progress variable is given by where formula_10  and formula_11 are the probability of finding unburnt and burnt mixture, respectively and formula_12 is the Dirac delta function. By definition, the normalization condition leads to It can be seen that the mean progress variable, is nothing but the probability of finding burnt gas at location formula_15. The density function is completely described by the mean progress variable. Assuming constant pressure and constant molecular weight, ideal gas law can be shown to reduce to where formula_17 is the heat release parameter. Using the above relation, the mean density can be calculated as follows The Favre averaging of the progress variable  is given by Combining the two expressions, we find and hence The density average is General density function. If reaction sheet is not assumed to be thin, then there is a chance that one can find a value for formula_1 in between zero and unity, although in reality, the reaction sheet is mostly thin compared to turbulent scales. Nevertheless, the general form the density function can be written as where formula_25 is the probability of finding the progress variable which is undergoing reaction (where transition from zero to unity is effected). Here, we have where formula_27 is negligible in most  regions.