A rapid procedure for generating C-type cascade grids suitable for viscous flow computations in turbomachinery blade rows is presented. The resulting mesh is periodic from one blade passage to the next, nearly orthogonal, and continuous across the wake downstream of a blade. The procedure employs a pair of conformal mappings that take the exterior of the cascade into the interior of an infinite strip with curved boundaries. The final transformation to a rectangular computational domain is accomplished numerically. The boundary values are obtained from a panel solution of an integral equation and the interior values by a rapid ADI solution of Laplace's equation. Examples of C-type grids are presented for both compressor and turbine blades and the extension of the procedure to three dimensions is briefly outlined

Sockol, P. M.

English

NASA Technical Reports Server

4 • • 
"'N 81 - 147 2 1 
GENERATION OF C-TYPE CASCADE GRIDS 
FOR VISCOUS F'L(fi'J COMPUTATION 
Peter H. Sockol 
NASA Lewis Research Center 
Cleveland, Ohio 
ABSTRAcr 
This paper presents a rapid procedure for generating C-type cascade grids 
suitable for viscous flow computations in turborr~chinery blade rows. The 
resulting mesh is periodic from one blade passage to the next, nearly 
orthogonal, and continuous across the wake downstream of a blade. The 
procedure employs a pair of conformal mappings that take the exterior of the 
~ cascade into the interior of an infinite strip with curved boundaries. The 
final transformation to a rectangular computational domain is accomplished 
numerically. The boundary values are obtained from a panel solution of an 
integral equation and the interior values by a rapid AnI solution of Laplace's 
equation. Examples of C-type grids are presented for both compressor and 
turbine blades and the extension of the procedure to three dimensions is 
briefly outlined. 
-' ' 437 
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~31 https://ntrs.nasa.gov/search.jsp?R=19810006207 2020-03-21T14:25:35+00:00Z
Most of the coordinate systems in current use for turbornachinery flow 
computations are of one of three types. The channel grio has one family of 
lines starting upstream, passing through the blade rows, and continuing on 
downstream. The o-type grid has one family of lines that form closed loops 
around the blades. Finally, the C-type grid has one family of lines that wrap 
around the blade leading edge and continue on downstream. While the channel 
grid can be aligned with the flow and is fairly easy to generate, the 
resolution around the leading edge is usually poor and a choice usually has to 
be made between periodicity or near orthogonality for highly staggerea 
cascades. Although the o-type grid provides excellent resolution around 
leading and trailing edges and may be both periodic and orthogonal, in general 
there is no mesh line aligned with the downstream flow and, hence, it is 
unsuitable tor viscous computation. The C-type grid, on the other hand, 
appears to be a good choice for viscous flows. It provides good leading edge 
resolution, it can be both periodic and orthogonal, and it can be aligned with 
the downstream flow. This paper presents a rapid procedure for generating 
such c-type grids. 
0-6RID 
CHANNEL GRID 
438 
~ The procedure starts with a conformal mapping which takes the exterior of a 
cascade of semi-infinite flat plates in the Z-plane into the interior of the 
unit circle in the W-plane. Upstream infinity maps to the origin and 
downstream infinity to +1 on the real axis. This mapping is a limiting form 
of the standard mapping for a cascade of finite-chord flat plates (1). When 
this mapping is applied to a real geometry, such as the turbine cascade in the 
figure, the semi-infinite flat plate is taken to run tram a point Zl 
inside the leading edge through the downstream end of the wake. 
T 
5 
1 
CONTOUR IN CASCADE PLANE 
\ 
\ 
\ 
\ 
\ 
\ 
-B 
z = Zo + A [e- iY (log W - inl - 2 cos Y log(l - WU 
A = 251f e iY , Zo = Zz + 2A[Y sin Y + cos Y log(2 cos Yl] 
439 
The mapping of the turbine cascade and wakes of the preceding figure produces '--
the highly 'iistorted "circle" in the adjacent figure. Note that the contour 
actually crosses the real axis twice between zero and one. The next mapping 
takes the interior of the unit circle in the W plane, with a branch cut from 
zero to one along the real axis, to the interior of the infinite strip between 
the real axis and (- i ~ ) in the l; -plane. The upper and lower sides of 
the wake at downstream infinity are mapped to plus and minus infinity, 
respectively, while upstream infinity maps to the origin. Since W is a 
function of l;; 2 , reflection of Z;; through the origin leaves W unchanged. 
CONTOUR IN W PLANE 
440 
V The irrage of the cascade of turbine blades and wakes in the Z; -plane is a 
pair of parallel straight lines cvlmected by a roughly S-shaped curve. In 
actual practice W is eliminated between the two functions and the transfor-
mabon from Z to s is obtained by Newton iteration proceeding from point to 
point around the contour. To insure that the branch cuts of the logarithms 
are never crossed, the imaginary parts of these logarithms are saved. 
Whenever the change in either of these quantities between adjacent points 
exceeds ± 1T , the computed value of the logarithm at the new point is 
incremented by +2 1T i, i.e. in the opposite direction. 
CONTOUR IN ~ PLANE 
--~ ----------------------------------- -----+~- --~~-~s 
441 
The final mapping transforms the infinite strip in the 7:; -plane, bounced \. 
by the blade-wake contour and its reflection through the origin, into a 
rectangular domain with coordinates F = ~ + in. If we let F be the 
complex potential for flow through the strip and require F{ s = -F(- s ) 
together with n = -1 along the contour, then we can write F as a contour 
integral. Here C,S , and h are, respectively, the complex velocity, flow 
angle, and normal channel width in the far fiela. The figure shows the 
formation of an integral equation for the unknown vortex density ~. The 
source density qn is set to cancel the normal component of the velocity 
C. Here s is distance along contour. A simple panel method, with flat 
panels and locally constant qt and qn' is used to solve for 
then to find 
442 
as a function of s along the contour. 
FOP~TrON OF INTEGRAL EQUATION 
CO~lPLEX POTENT! AL 
F(c) = -.lc f~ ~(t) log tt :-f dt + c~ 
21'fl _«> ..., 
WITH F=(+i~, 2 - is C = Fi e 
WITH qn (t) = 1m [c ~J 
Or-; CONTOUR ~ = -1 AND 
( = Re(C,) - in i: [qn Re 1 log ~ : ~ ! 
- qt 1m 1109 ~ : ! !}s 
and 
J Generation of the grid in the rectangular l; , T) space proceeas in tVtD 
stages. First points are located on the boundaries such that the grid in the 
cascade plane is periodic and continuous across the wake. Periodicity is 
enforced by distributing points symmetrically about the origin along the 
s -axis. Continuity across the wake, away from the trailing edge, is 
achieved by selecting a constant mesh size !::.E, for this region such that the 
spacing in the cascade plane is an integer fraction of the staggered distance 
s sin I Bw I , where s is the blade pitch and Bw is the wake angle. The 
values of ~ along the boundary are then found by inverse interpolation in 
the solution of l; vs ~ In order to enforce continuity near the 
trailing edge, a local straining is first introduced that places a point 
directly at the trailing edge. Then pairs of neighl:x:>r ing points across the 
wake are adjusted until their images in the cascade plane coincide. The 
distribution of points with at the two ends of the region is arbitrary. 
Uniform spacing can be used for inviscid flows, and boundary layer stretching 
can be used to cluster points near the blade surface ru1d wake for viscous 
flows. Once the boundary values of ~ are specified, interior values are 
found by solving the complex Laplace equation by a cyclic ADI relaxation 
s:::::heme which has the symmetry properties of ~ built in. Estimates of the 
maximum and minimum eigenvalues of the matrix are used to obtain near optimurrl 
values of the acceleration parameters (2). Convergence to the round-off error 
limit with seven place arithmetic is typically obtained in six to twelve 
iterations, even for cases where the maximum and minimum eigenvalues differ by 
five orders of magnitude. 
443 
GRID GENERATION 
• BOUNDARY VALUES IN ( ~ In) SPACE 
~ VALUES SYMMETRIC ABOUT ORIGIN 
~~ CONSTANT ALONG WAKE WITH 
I~z I = (s sin I Bw l ) / integer 
LOCAL STRAINING UEAR TRAILING EDGE 
[ 
~ - ~ 1 ~ - ~ 2J2 
~' = ~ + °t ~t - ~l . ~t - ~~ 
~l""""'" t" ' ~ ,I I I , , I I , , I , , I I t" St t s2 
ARBITP~RY DISTRIBUTION ALONG n 
INTERIOR VALUES IN ( t;, In) SPACE 
a2 i'; a2 i'; 
-+-= 0 
at;,2 an 2 
SOLVED BY CYCLIC ADI RELAXATION 
444 
U This figure shows the grid distribution in the l;; -plane. Note that the 
upper boundary in the plot, which is found by the syrnrretric ADI solution of 
Laplace's equation, maps into the upper and lower periodic lines in the 
cascade plane. 
GRID IN l;; PLANE 
445 
The final grid in the cascade plane is obtained by conformal mapping of the 
solution in the s -plane using the two analytic functions previously 
introduced. This figure shows the grid distribution for the cascade of 
turbine blades. Note that the continuity across the wake was obtained at the 
exp=nse of a small arrount of nonorthogonali ty • The rounded cap at the 
upstream boundary was obtained by extrarx>lation fran the next two inner loops. 
Generation of this grid (99 x 7 points) required about 1.4 sec. of CPU time on 
an IBM 3033 computer. 
GRID IN CASCADE PLANE 
446 
~ The last figure presents C grids for a compressor stator and a turbine rotor. 
The stator was Jesigned to turn the flow to the axial direction, hence, there 
is zero stagger in the downstream boundary. The turbine rotor is a 
particularly difficult case as it was designed to produce 126 degrees of 
turning. In this case the imposition of continuity across the wake resulted 
in a significant change in slope. 
The extension of this procedure to the generation of three-dimensional 
turbornachinery grids should be relatively straightforward. First the spanwise 
direction is discretized by a number of coaxial, axisymmetric surfaces. Next, 
ana most difficult, the intersection of the blade with each of these surfaces 
is obtained in meridional ( m ) and tangential ( e ) coordinates. Since the 
geometry is per iOOic in e , these ( m , e ) coordinates can be fed into the 
current program to generate a C-grid on each of the axi-symmetric surfaces. 
--" For O-grids this has already been done by Dulikravich (3). 
COf'\PRESSOP. STATOR 
TURBINE ROTa? 
447 
1. Kober, H.: Dictionary of Conformal Representations, Dover Publications, 
Inc., Second ed. with corrections, 1957, p. 131. 
2. Varga, R. S.: Matrix Iterative Analysis r Prentice-Hall, Inc., 1962. 
3. Du1ikravich, D. S.: Fast Generation of Body Conforming Grids for 3-D 
Axial TUrbamachinery Flow Calculations. Numerical Grid Generation 
Techniques, NASA CP-2166, 1980. (Paper 16 of this compilation.) 
448 


Generation of C-type cascade grids for viscous flow computation

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