Acoustic Reynolds Stress: The Source of Coherent Structures During Combustion Instability - Novel Combustion Concepts for Sustainable Energy Development - page 359

and combustion by churning out large-scale vortical structures is presented. These
events eventually result in intermittent combustion and drive the combustor to an
unstable mode of operation.

2 Mathematical Formulation

The schematic of the problem with speci c context to a dump combustor is shown in

Fig. 1 . The field is modelled in a shorter length scale h and longer

(convective) time scale t = h/u, whereas the acoustics is modelled in a longer length
scale L (for longitudinal acoustic modes) and shorter time scale

fl flow/combustion

= L/c. h is char-
acteristic of the lateral dimensions of the combustor, such as the step height, and L is
the length of the combustor. u and c are reference

τ

fl ow and sound speeds. The fl fluid is

considered to be a calorically perfect gas and all the diffusion coef cients are

assumed equal and constant. The conservation equations are non-dimensionalised
with shorter length and time scales, h and τ , respectively (Chakravarthy et al. 2007 ).

Apart from other non-dimensional quantities, the resulting set of equations have
M = u/c and h/L, the Mach number and the ratio of length scales, respectively. Taken
together, these constitute the simultaneous multiple length and time scales. Both of
these quantities are taken to be unequally small in general, here. To handle two small
parameters simultaneously, each of them is, in turn, represented as a function of a
single small number ʵ n . Here, we choose

m = 1, and let n evolve in the solution as a marker of the compactness of

ʵ m and h/L =

ʵ through the relations M =

fl ame due to

its coupling with the acoustics.
Any dependent variable

is expanded according to the ansatz . For m = 1, this is
the conventional Mach number expansion, with M as the small parameter. Splitting
the spatial and temporal derivatives for the two scales considered, expanding the
variables as powers of

ˆ

ʵ and comparing equi-order terms in ʵ , we decompose the

conservation equations for the flow and acoustics in their respective length and time

scales. In the resulting set of equations, it is observed that both the

fl

fl ow and

acoustics affect the fl fluid velocity to leading order. Temporally averaging the base-

fl flow equations and spatially averaging the acoustic equations results in the

Fig. 1 Schematic of the backward facing step combustor