Accurate and efficient numerical simulation of flows through turbomachinery blade rows depends on the topology of the computational grids. These grids must reflect the periodic nature of turbomachinery blade row geometries and conform to the blade shapes. Three types of grids can be generated that meet these minimal requirements: through-flow grids, O-type grids, and C-type grids. A procedure which can be used to generate all three types of grids is presented. The resulting grids are orthogonal and can be stretched to capture the essential physics of the flow. A discussion is also presented detailing the extension of the generation procedure to three dimensional geometries

Adamczyk, J. J.

English

NASA Technical Reports Server

· 
""-N 81-14695 
AN ELECTROSTATIC ANALOG FOR GENERATING CASCADE GRIDS 
John J. Adamczyk 
NASA Lewis Research Center 
ABSTRACT 
Accurate and efficient numerical simulation of flows through turbomachinery 
blade rows depends on the topology of the computational grids. These gdds must 
reflect the periodic nature of turbomachinery blade row geometries and conform 
to the blade shapes. Three types of grids can be generated that meet these min-
imal requirements: (1) through-flow grids, (2) a-type grids, and (3) C-type 
grids. This paper presents a procedure which can be used to generate all three 
types of grids. The resulting grids are orthogonal and can be stretched to 
capture the essential physics of the flow. In addition, a discussion is also 
presented detailing the extension of the generation procedure to three-
dimensional geometrjes. 
Preceding page blank 129 
https://ntrs.nasa.gov/search.jsp?R=19810006181 2020-03-21T14:27:22+00:00Z
BLADE GEm-fETRY AND COORDINATES 
The development of the grid generation procedure begins by considering the 
electrostatic potential field generated by an infinite linear array of point 
charges in a two-dimensional space. The density of the charges is assumed to 
alternate between plus and minus 1. The mathematical expression for the com-
plex electrostatic field is given by equation (1), where K(z - zo) is the com-
plex potential and i = J=I. For a Cartesian space z = x + iy, Zo is the 
location of the zeroth charge and s is the physical distance between charges. 
For a blade row whose geometry is given on a blade-to-blade surface of revolu-
tion, z = m + iO, where m is the meridional distance and e is the angular 
position; s then represents the angular distance between charges. 
130 
o 
o +1 
i o -1 
s 
0 -. x , +1 
0 -I 
Array of point charges 
2: (_l)n Inez - Zo - ins) 
n=-oo 
Blacle-to-bl~ sUrfiCe of revolutIon. 
snowing m - e coordinates. 
- Blade-to-
blade sur-
face 
(1) 
DEVELOPMENT OF EQUATION FOR POTENTIAL FIELD SURROUNDING THE BLADE 
The field equation for the array of point charges is expressed in closed 
form in equation (2). Note that this expression is periodic in either the y-
or 8-direction with period 2s. The potential field generated by distributing 
the fundamental solution Ceq. (2)) over the surface of a blade is given by equa-
tion (3), where y(zo) is the source density distribution on the blade surface. 
It is required that the real part of w (that is, ~) be equal to I on the 
blade surface Ceq. (4)). At large distances upstream or downstream of the 
cascade, C(z) is assumed to approach zero. This additional requirement is 
expressed by equation (5). Equation (3) evaluated on the blade surface forms a 
singular integral equation, equation (6), for the source density. Once y is 
known, the potential field surrounding the blade can be computed by direct 
integration of equation (3). 
00 
'2 (_1)n In(z - Zo - ins) 
n=-OO 
w(z) 
Real (0(Z) ~(z) - 1 
1 
I 
(z - ZO)TT 
In tanh ----2s 
(Z E L) 
(z E L) 
(2) 
(3) 
(4) 
(5) 
(6) 
131 
DEVELOPMENT OF EQUATIONS FOR SOURCE DENSITY DISTRIBUTION 
The singular integral equations for y can be solved by paneling methods 
similar to those used in solving potential flow problems in fluid mechanics. 
To employ these procedures, one first factors out of the integral equation its 
singular behavior. (For eq. (6) the factoied form is expressed by eq. (7).) 
:~::e t::g::~:: s~~:ao~e a~d i:
n
d[ivid2esd ~nt~a~h S::i:S
z 
o~ ~elme:::. ap::::i::::d o:y 
(z - zo)TI 0 2s1 
polynomials in zoo For the cases examined to date it was found that these 
terms could be approximated by a constant equal to their value at the midpoint 
of each segment. (This approximation assumes the length of the segment could 
be made smaller than the scale of the local geometric blade features.) Thus 
the singular part of equation (7) integrated over a segment is approximated by 
equation (8), while the regular part is approximated by equation (9). The 
auxiliary condition (eq. (6» is approximated over a segment by equation (10). 
Upon introducing these approximations into equation (7) and restricting the 
value of Zo to the midpoint of each segment, a system of linear algebraic 
equations is obtained from which y(zo) can be determined. 
1 
132 
Real [/L Y(zo) 
(z 
- zo)TI 
In 2s 
+ i '{(zo) In (z 28 
- zo)1T 
2s 
y(zo) In (z _ z )IT tanh 
o 
dz o 
(2 - Z ) iT dZ~ tanh 0 2s 
y(Zo) [(z - b) In (z - b) 
- (z - a) In (z - a) + b - all =b+a ~J Zo 2 
(z - Z )IT 
o 
2s 
~ 2s (z - zo)IT~ ~ y(zo) (b - a) In tanh ---------(z - zo)TI 2s 
b+a 
Z =--
o 2 
(7) 
(8) 
(9) 
(10) 
\. 
BOUNDARIES OF COMPLEX ELECTROSTATIC FIELD SURROU}'"'DING THE BLADE 
With y known, the complex electrostatic field surrounding the blade sec-
tions can be found. The real part ~ and the imaginary part n of the field 
form a periodic orthogonal body-fitted coordinate system. The contours 
~ = Constant enclose the blade, while the curves n Constant project [rom 
the blade (i.e., ~ = 1) to the periodic boundary (~ = 0). The curves extend-
ing to upstream and downstream infinity are denoted n = 0, n = ±L, respec-
tively. The locations of these bounding coordinate curves are found using a 
Newton-Raphson scheme with equation (3) to numerically generate the inverse 
mapping function. This procedure, however, because of jts slow computational 
speed, was not used to construct the interior grid contonrs. 
ORlGIN.\'C PA,'S IS 
OF POOR QUALITY 
133 
U,T'fERIOR GRID 
The interior grid points are constructed from the solution of the inverse 
electrostatic problem in which (x,y) or (m,e) are specified as functions of ~ 
or n on the boundaries. The field equation for this is Laplace's equation in 
terms of (~,n) (eq. (11). The solution of equation (11) which satisfies 
boundary conditions consistent with the coordinates of the bounding curvilinear 
curves yields the interior grid geometry. This solution can be obtained by 
either numerical or analytical procedures. 
134 
a2(x zm) 
a~2 
a2(x:,m) 
a~2 
n -L 
i 
+ 
a2 (x,m) a :: 
3n2 
+ 
a2(y,e) 0 
(x ,m) 
(y,e) 
an
2 
fen) / 
g (n) 
L __ 
(11) 
1 
(x,m) h(O 
-. (y,8) reO 
n L 
I- n ~ 0 
GRID CLUSTERING 
Grid clustering to capture the physics of the flow field can be introduced 
prior to or after the solution of the inverse problem is obtained. To insure 
orthogonality, the stretching functions used for clustering must be one-
dimensional (i.e., eqs. (12). The ~ transformation can be quite arbitrary. 
For potential flow computations, a linear transformation is generally used. 
For viscous flows, one attempts to cluster grid points near the blade surface. 
An example of a transformation which can be used for this purpose is given by 
equation (13). The parameters mo,ml control the degree of stretching in the 
transformation. The clustering of grid points in the n-direction requires 
special consideration to insure that grid point periodicity is maintained. A 
grid point located on the periodic boundary at n = no has an image at 
n = -no' In order to maintain this property, the transformation in n must be 
an odd function of ~ over the interval -L to L. A simple transformation 
which exhibits this behavior is a polynomial in odd powers of ~, an example 
being given by equation (14). The parameters mo and ml in this transforma-
tion are again used to control the degree of clustering. 
(0 S E, 5 1) } (12) (-L S. n ::; L) 
(13) 
n (14) 
L 
1 
L~------:::."""'O::-_-----I 
-L 
135 
GENERATION OF C-TYPE AND THRODGH-FLm.;r GRIDS 
Thus far the application of the grid generation procedure has been 
restricted to orthogonal a-type grids. To generate through-flow and C-type 
grids by the current procedure, the blade contours must be modified by append-
ing slits of zero thickness to their surfaces. For C-type grids, one slit is 
used. Its origin is generally taken to be the trailing edge of the blade. For 
through-flow grids, two slits are appended, their origins being the leading and 
trailing edges of the blade. The shape of these appendages can be quite gen-
eral. The generation of the grids associated with these modified blade pro-
files proceeds as in the a-grid procedure outlined above. The generated grids 
are orthogonal and periodic. In the case of blunt blades, however, they 
exhibit a singular behavior at the slit attachment point. 
136 
--
.....,...! GRIDS FOR CASCADES 
For cascades of nonzero stagger, the C-type or through-flow grids generated 
by the current procedure are discontinuous across the slit. This undesirable 
property can be corrected by a simple construction. First the two grid lines 
connecting the upper periodic boundary and the trailing edge and the lower 
periodic boundary and the trailing edge are found. Next the location and value 
of n corresponding to the periodic image of A on the lower boundary (i.e., 
A') and H' on the upper boundary (i.e., B) are determined. The spacing of 
grid points along the slit between EF is specified, whlch in turn determines 
the distribution of n along both sides of the slit. This in turn determines 
the distribution of n along the periodic boundaries AH and B'C'. The grid 
points along A'R' are required to be periodic images of the points along AB. 
This determines the distribution of n along A'B' and the grid geometry up 
to EB'. The construction of the grid downstream of BrC' proceeds in the 
same manner as outlined above, with the grid spacing along B'C' defining the 
clustering pattern. The resulting grids will remain orthogonal and periodic 
under this construction procedure. 
E 
" 
" , , " 
, , , ", 
, , , " 
, , , " 
, , , '" c' , , , 
, , , '\ 
, '\ , 
, 
B' 
137 
DEVELOPMENT OF THREE-DIMENSIONAL GRIDS 
Three-dimensional grids can also be developed by the current procedure. 
The geometry of a typical blade assembly as viewed in the meridional plane is 
shown in the accompanying figure. Let the surfaces of revolution describing 
the hub and shroud be denoted by r = fH(x), r = fsex). A surface of revolu-
tion bounded by these limits is given by equation (15). Similarly, let m 
denote the percentage of distance measured from the leading edge of the blade 
along the surfaces of revolution (eq. (16». On a surface of revolution the 
blade section geometry is given in terms of m and e, where e is the 
angular location around the wheel. On the blade surface of revolution m 
and 8 can be expressed in terms of the coordinates ~,n of the current 
orthogonal system. The resulting coordinate system (s,~,n) will be orthogonal 
on a s-plane and conform to the blade hub and shroud surfaCe. 
m 
138 
r - fH(x) 
~(~) - fH(x) 
JLl;; ds o 
J------l-x 
z 
(15) 
(16) 
EXA~LE O-TYPE GRID FOR TURBINE STATOR BLADE 
An example of an O-type grid generated by the current procedure is shown 
on the accompanying figure. The blade is a turbine stator with approximately 
900 of turning. No stretching was introduced ~n developing the grid. In 
regions of high surface curvature there is a high concentration of grid points, 
thus permitting accurate resolution of the local flow physics. Far removed 
from the blade, the concentration of grid points becomes sparse. This results 
in an economical distribution of grid points in regions of uniform flow. The 
grid as shown was generated to solve a potential flow problem. For viscous 
flow, a stretching of the ~ contours would have to be introduced to capture 
the boundary-layer region. In addition, the n lines would have to be 
clustered in the trailing-edge region to resolve the viscous wake. 
139 
EXAMPLE THROUGH-FLOW GRID FOR A TURBINE ROTOR BLADE 
The next example is a through-flow grid for a high-reaction turbine rotor. 
This grid, as described earlier in the paper, was generated by appending two 
slits to the blade surfaces. The grid geometry is rectangular and periodic 
upstream and downstream of the cascade. Across the wake the grid geometry 1s 
seen to be discontinuous. The use of such a grid would require special care 
in transferring flow variables across the slit. 
140 
~-
EXAMPLE C-TYPE GRID FOR A TURBINE ROTOR BLADE 
The accompanying figure shows a C-type grid for the preceding turbine 
blade. Upstream of the blade the grid geometry is similar to an O-type grid, 
while downstream it resembles a through-flow grid. For this grid the con-
struction procedure outlined earlier was used to insure continuous grid geom-
etry across the slit. For potential flow analysis the grid as shown is quite 
suitable. For viscous analysis a clustering of the grid lines near the blade 
surface and trailing-edge region would have to be introduced to accurately 
resolve the flow physics. 
1 41 
EXAMPLE C-TYPE GRID FOR CASCADE OF NACA 0012 AIRFOILS 
The final example is a C-type grid for a cascade of NACA 0012 airfoils. 
This grid was generated by When-Huei Jou of Flow Research. The grid generation 
procedure as modified by him produces grids which are nearly orthogonal. The 
slight nonorthogonality is due to stretching functions used to maintain con-
tinuous grid geometry across the slit and cascade periodicity. This grid was 
generated to solve a two-dimensional potential flow problem. It can also be 
used to solve a three-dimensional potential flow problem, provided the inviscid 
wake is convected downstream along the slit. For viscous flow calculation a 
clustering of grid points near the body and wake region would have to be 
introduced. 
142 
I 
NACA 0012 airfoil. Stagger angle 30.00; pitch 0.794; 
cascade grid 128 x 16. 
l 


An electrostatic analog for generating cascade grids

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