Acoustic Reynolds Stress: The Source of Coherent Structures During Combustion Instability - Novel Combustion Concepts for Sustainable Energy Development - page 360

de nition of new velocity variables that can be identi ed distinctly with the base

fl flow and the acoustics. The time-averaging over the slower of the two time scales

formally ensures that the slower process would pose to be rate-limiting in the
coupling between the two, and that the evolution of the coupled system would
occur at the slower time scale only. The averaging also results in the appearance of
explicit interaction terms between the base-

fl ow and acoustics. The final sets of

conservation equations for the fl flow and acoustics are given below (Chakravarthy

et al. 2007 ).
Flow:

¼

@q 0

@ t þ @

q 0 u 0j s 0 ð 1 Þ

@ x j

þ

@q 2
@ x i

@
@ t

Þþ @
@ x j

1
c

q 0 u 0i s q 0 u 0i s u 0j s ð

!

ð 2 Þ

u 0i s
@ x

2

@ u 0k s
@ x k

1
Re

@ 1

3

@
@ x i

¼ @
@ x j

s

q 0 u 0 0i u 0 0j

þ

2
j

dt þ cq 0 @ u 0 j s

2

d q 0 c

Pe

@ T 0

@ x @ x j ¼ðc

1 Þ Da Q R0 þ ð 3 Þ

2
j

2

@q K 0
@ t

þ @
@ x j

1
ReSc

@ q K0

@ x

q K0 u 0j s Da K x K ¼ 0 ð 4 Þ

2
i

p 0 ¼ q 0 T 0 ð 5 Þ

Acoustics:

@q 1

@s þ @

¼ @
@n j

þ q 0 0 u 0j s 0 x x

q 0 x u 0j s x q 0 x u 0 0j

ð 6 Þ

@n j

q 0 x @ u 0 0 i

1
c

@ p 1
@n i ¼ @s þ

0 ð 7 Þ

@n i ¼c p 0 @ u 0 j s x

@s þ c p 0 @ u 0 0 j

@ p 1

ð 8 Þ

@n i

@q K 1 x
@s

þ @
@n i

q K0 x u 0 0j

ð 9 Þ

¼ @
@n i

þ q 0 K0 u 0j s 0 x x

q K0 x u 0j s x

p 1 ¼ q 1 T 0 x

þ q 0 x T 1 ð 10 Þ