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```the only differences lie in two aspects : the terms associated with those forces are
multiplied by some correction factors, and corrections for curvatures are added to the
nonhomogeneous terms, as if they were virtual lateral inflows or additional body
forces.
When q., qy and h are taken as dependent variables, the continuity equation and
momentum equation in the x-direction can be written as
n2 Jut + vi + B C U ~ ,
K = g h1/3
(1. 5. 71)
(1. 5. 72)
X U . FORM IN A GENERAL CURVILINEAR COORDINATE SYSTEM
1. Fixed curvilinear coordinate system
The chief characteristics expressing the local behavior of a curvilinear coordinate
system are covariant and contravariant basis vectors. The former are defined as vec-
tors tangential to the coordinate curves <=const and q=const passing through a given
point P ( x , y).
The latter and defined as the vectors normal to those curves at P
a l = <ax/a<, ay/a<>',
a1 = (a</ax, a 5 / a d T ,
a2 = (ads?, ay/aq>T
a2 = (aV/ax, aq/ay)T
(1. 5. 73)
(1. 5. 7 4 )
Obviously, for z f j , a' is perpendicular to a,. Only in an OCCS do the two
groups of tensors coincide ; meanwhile, the two vectors in the same group are orthog-
onal to each other. For the purpose of expressing the relations between arc elements
(or area elements) in the neighborhood of point P(on the x-y plane) and its image
point Q (on the 4-7 plane), introduce an order-2 covariant measuring tensor, defined
by
g., = a, a, = gjt (1. 5 .75)
where
(1. 5. 76)
(1. 5. 76a)
51
(1. 5. 766)
It is noted that formulas for gI1 and g22 are the same as those for gCe and grm in an
OCCS. However, in the latter case, from the orthogonality condition Eq. (1. 5. 39),
we have g l2=gZ1 = O ; moreover, all other formulas are special forms of those given
in this section.
For brevity, change the notation (5, q ) to (<', <2). A transformation formula
for the length d s of an arc element in the 2-y plane is
(1. 5. 77)
As a special case, the length ds of an arc element on the 5"-coordinate curve is
ds' = I a, Id? = &df
A transformation formula for the area d o of an area element in the x-y plane is
and this is used in the transformation of area integrals between the two coordinate
systems
(1. 5. 78)
do = m d { d q (1.5. 79)
SZ' is the image of SZ in the <-q plane. ,@ is the transformation Jacobian , given by
0 = vq%J = J911922 - g:z (1. 5. 81)
or
Notice that as glz#O, Eq. (1. 5. 81) is different from 6 in an OCCS.
Similarly, with the definition of contravariant basis vectors
(1. 5. 82)
(1. 5. 83)
(1. 5. 84)
an order-2 contravariant measuring tensor can be defined by
g'J = aJ = gV (1. 5. 85)
which is the inverse of the symmetric covariant measuring tensor, with components
9" =922/g (1. 5. 86)
g22=gll/g (1. 5. 87)
g'Z=g21= -g,z/g* (1. 5. 88)
Now we are in a position to be able to write down the relations between various dif-
ferential operators defined on the x-y and <-q planes. Each formula has both conser-
vative and non-conservative forms.
( 1) Gradient (f denotes a scalar function)
Conservative form
52
Nonconservative form
(2) Divergence v A ( A denotes a vector function)
Conservative form
(1. 5. 89)
(1. 5.90)
(1.5.91)
(1. 5. 92)
(1. 5. 94)
(3) Vorticity V X A
Conservative form
k is a unit vector perpendicular to the x-y plane.
Nonconservative form
(1. 5. 96)
( 4 ) Laplacian operator v f
Conservative form
(1. 5.97)
where
(1. 5.98)
53
Nonconservative form
(1. 5. 99)
(5 ) Order-2 derivatives
Nonconservative form
(1.5.102)
where subscripts denote arguments of partial derivatives.
Conservative form
(6) Normal derivatives to <-and q-isolines in the x-y plane
(1.5. 103)
(1.5. 104)
54
Nonconservative form
(7 ) Tangential derivatives
Conservative form
Nonconservative form
fT< = f q /
(1.5. 105)
(1. 5. 106)
(1.5.107)
(1.5. 108)
(1.5. 109)
(1.5. 110)
By using the above formulas, the 2-D SSWE in rectangular coordinate systems
can be transformed into a form used in a general curvilinear coordinate system, but
they will not be derived or listed here on account of their great complexity.
2. Variable curvilinear coordinate system
When a flow field varies significantly with time, we may adopt a curvilinear
coordinate system which changes continuously
5 = 5 ( 1 , . . , Y ) , r l = r l ( t t 2 t Y ) (1.5. 111)
In consideration of the variation of the coordinate system, it is necessary to transform
the time-derivatives in the original system, i. e. , take derivatives of
f ( t , r ( t , 5 , vj ) , y ( t , 5 , 7 ) ) at fixed < and vj by using the differentiation rule for
compound function
(1.5.112)
where D f can be obtained from Eqs. (1. 5. 89)-(1. 5. 92); f denotes a velocity vec-
tor of a moving point P ( r , y) , whose image is &(<, vj >. By making an appropriate
55
transformation, the problem can be solved adaptively on a fixed domain in the x-y
plane.
However, the general approach is so tedious that we would rather adopt a special
technique for writing conservation laws in a concise form. We first derive from Eq.
(1. 5 . 35) differential equations holding on the 5-7 plane by using the total differen-
tial formula, yielding
(1. 5. 113)
With respect to t , < and 7 it is no longer in conservative form ; however, as the sym-
bols in the parentheses may be formally viewed as differential operators, it is said to
be in quasi-conservative form.
Another form of the above formula is
where
- a4 a< a<
at ax
h a7 av
a t & ?i
A, = -I + -A, + -A,
A, = -I + -A, + -A,
The system can also be written in a fully conservative form
where
-
w = w/J
J denotes a Jacobian of the plane coordinate transformation. Introducing
(1.5.114)
a4 a< a< a7 a7 av
at ax ay at ax ay 11 = - + u - + v - , and V = - + u - + v -
(1.5. 115)
(1.5.116)
(1.5. 117)
(1.5. 118)
(1.5.119)
(1.5.120)
and substituting w , G and H in Eq. (1. 5. 35a) into Eqs. (1. 5. 118) and ( 1 . 5.
1 1 9 ) , leads to
(1.5.121)
56
R = huV f p - 271 , ~ I J V f p - ” , h V I T i ax all (1 .5 .122)
where p = pgh2/2. These two expressions can be used in flow computations with a
general curvilinear coordinate system, for which all measuring coefficients can be es-
timated in the course of the numerical mesh generation (cf. Section 8. 1 ) , and
should satisfy
(1 . 5. 123)
3. General mathematical properties of a coordinate transformation
Sufficient conditions for a coordinate transformation include : ( 1 ) Functions 5
and are single-valued , continuous and have continuous order- 1 partial derivatives
in the domain considered. (2) At any point in the domain, the Jacobian of the trans-
formation I J I f o . If a mapping associated with a coordinate transformation is con-
tinuous and one-to-one, the above two conditions are satisfied. A transformation
with these two properties is said to be admissible. When a tensor equation holds in a
coordinate system, it certainly holds in all systems obtained by making admissible
transformations. If the Jacobian is positive everywhere, a right-handed coordinate
system must preserve right-handedness, and we call it a normal transformation. For
a normal orthogonal transformation, 1 ,&, I = 1 in the tensor transformation formula
Eq. ( 1. 2. 13). Conversely, if the Jacobian is negative everywhere, a right-handed
system would be transformed into a left-handed one. Such a transformation is abnor-
mal. In this book we shall always assume the transformation to be admissible and
normal.
When taking a coordinate transformation to obtain a general curvilinear coordi-
nate system, we may perform a dependent variable transformation simultaneously.
Related formulas will not be listed here. We only state the associated sufficient condi-
tions that the water body under study be bounded by a piecewise smooth boundary,
that the coordinate transformation is diffeomorphism```