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.
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