Page 47 - Basic Structured Grid Generation
P. 47
36 Basic Structured Grid Generation
in terms of the covariant derivatives, by eqns (1.119) and (1.121). Since du/ds is
a vector quantity, we can define the intrinsic derivatives of the contravariant and
covariant components of u by:
δu i i dx j i j δu i dx j j
= u ,j = u t , = u i,j = u i,j t , (2.31)
,j
δs ds δs ds
so that
du δu i δu i i
= g i = g .
ds δs δs
Another equation which will be useful later is
δu i i dx j ∂u i i k dx j du i i k dx j
= u ,j = + u = + u . (2.32)
kj
kj
δs ds ∂x j ds ds ds
Similarly, intrinsic derivatives of second-order tensors can be defined in terms of
covariant derivatives. For example, the intrinsic derivatives of contravariant second-
order tensor components T ij are
δT ij ij dx k
= T ,k , (2.33)
δs ds
where the covariant derivatives of T ij are given by eqn (1.129). Thus it follows, from
eqn (1.128) that
δg ij
= 0.
δs
ij
Now rewriting eqn (2.29) as g t i t j = 1 and taking intrinsic derivatives (with the
usual product rule) gives
δg ij ij δt i ij δt j ij δt j
t i t j + g t j + g t i = 0 + 2g t i = 0,
δs δs δs δs
ij
using the symmetry of g . Thus
ij δt j
g t i = 0. (2.34)
δs
From eqn (1.54) it follows that δt j /δs represent the covariant components of a vector
orthogonal to t, and we can put
δt i
= κn i , (2.35)
δs
where n i are the covariant components of the unit vector n which satisfies
ij
g n i n j = 1 (2.36)
and
ij
g t i n j = 0. (2.37)
We shall assume here that κ is non-negative.