Page 12 - kursus eBook

The same conditions that lead to the conservation of B along streamlines for steady flow

also imply that the entropy is conserved , and for steady flow, is constant too along
streamlines so both B and s are constant along streamlines.

Now for a general variation of entropy, the state equation (6.1.20) yields,

# 1 &
T!s = !e + p! % ( (10.1.15)
$
"'

If we take the gradient of the Bernoulli function,

!
u 2 !p # 1 &
!B = ! + !e + + p! % ( + !)
2 " $ "'

(10.1.16)
!
u 2 !p
= ! + T!s + + !)
2 "

whereas the steady momentum equation in the absence of friction is,

! ! !p
! 2
! u / 2 + " # u = $ $ !& (10.1.17)
a
%
which when added to (10.1.16) yields an unexpected relation between the spatial

variation of the entropy, the Bernoulli function and the absolute vorticity,

! !
T!s = !B + " # u (Crocco’s theorem) (10.1.18)
a
also note, that from (10.1.15)

!# " !p
!T " !s = (10.1.19)
# 2

which relates the baroclinic production of vorticity to the misalignment of the

temperature and entropy surfaces in space.

Chapter 10 6