NASA
Contractor
Report
3446
Computational Grids in Improving Accuracy and Convergence
Akin
Ecer and
GRANT NSG3294 JULY 1981
NASA
Hasan
U.
Akay
LO,IN &PY: RETURNTO /OWL TECHNICALLIBRARY KIRIIA~B A! 3, N.N\.


5
.
TECH LIBRARY KAFB, NM
NASA
Contractor
Report
3446
Finite Element Analysis of Transonic Flows in Cascades  Importance of Computational Grids in Improving Accuracy and Convergence
Akin
Ecer
and
Hasan
U.
Akay
Purdue Utziversity at Indianapolis Itldimzupolis, Illdiana
Prepared for Lewis Research Center under Grant NSG3294
NASA
National Aeronautics and Space Administration Scientific and Technical Information Branch
1981
.:%k

9
IL 


TABLE OF CONTENTS Page SUMMARY..............................
1
INTRODUCTION
2
FINITE
. . . . . . . . . . . . . . . . . . . . . . . . . . .
ELEMENT FORMULATION OF TRANSONIC FLOWS ........... ..................... Governing Equations ....................... Cascade Geometry .................. Finite Element Formulation ................. PseudoUnsteady Formulation The Numerical Integration Technique for ..................... PseudoTime Problem ...................... Modeling of Shocks .................... a) Shock Capturing ..................... b) Shock Fitting Periodic Boundaries Computational Grid, ...................... and Kutta Condition
5 5 6 8
11 13 14 15 16 18
........... CONVERGENCE CHARACTERISTICS OF THE EQUATIONS ...... Convergence of Variable Coefficients Scheme (VCS) a) Convergence in Subsonic Flow, c = 0: .......... b) Convergence in Supersonic Flow, c $ 0: ......... Convergence of Constant Coefficient Scheme (CCS) ....... ....... Choice of the Artificial Viscosity Distribution .......... a) Von Neumann Analysis of Convergence:
22 23 25 30 33
ACCURACY OF THE SOLUTIONS ..................... ......................... Introduction ......... Computational Grid for Subsonic Flow Regions ............. Designing Grids for Transonic Flows ....... Modeling of the Supersonic Pocket and the Shock Modeling of the Trailing and Leading Edges ..........
36 36 36 37 47 54
........................... CASESTUDIES Circular Cylinder Test Case I  Isolated .................. Moo=0.51 a) Case Ia:
57 57 57
b) Test NACA Test NACA Test
Case
Ib:
M, = 0.45
...........
..................
Case II  45O Staggered Low Solidity ...................... 0012 Cascade Case III  45O Staggered High Solidity ...................... 0012 Cascade ............ Case IV  8% CircularArc Cascade
iii
21 21
72 76 82
91
.l:;
Page Test Case V  Supercritical Efficiency of the Developed
Cascade . . . . . . . . . . . . Computational Procedure . . . .
96 96
CONCLUSIONS . . . . . . . . . . . . . . . . . . . . . . . . . . .
103
REFERENCES
105
. . . . . . . . . . . . . . . . . . . . . . . . . . .
iv
. 
FINITE ELEMENT ANALYSIS OF TRANSONIC FLOWS IN CASCADES  IMPORTANCEOF COMPUTATIONALGRIDS IN IMPROVING ACCURACYAND CONVERGENCE Akin Purdue
Ecer* and Hasan U. Akay** University at Indianapolis SUMMARY
The finite element method is applied for the solution of transonic Convergence characteristics potential flows through a cascade of airfoils. Accuracy of the obtained numerical of the solution scheme are discussed. solutions is studied for various flow regions in the transonic flow configuration. The design of an efficient finite element computational grid is discussed in improving accuracy and convergence. This study shows that the solution of nonlinear equations corresponds to the numerical integration of a pseudotime system in the solution of steady, inviscid potential equations. The choice of the artificial viscosity parameter, which is necessary for obtaining convergence in the supersonic pocket, and the relaxation parameter, which controls both the convergence characteristics and the accuracy of the obtained solution, are investigated. Proper choice of these parameters as a function of the computational grid is presented. The accuracy of the obtained solutions around the leading and trailing edges of the airfoil, in the subsonic and supersonic flow regions, and around the shock are discussed, and appropriate grid refinement techniques are suggested. An efficient numerical integration scheme for the solution of transonic flow equations is also presented. The numerical results include several basic test cases to demonstrate the degree of grid refinement required for accuracy and convergence. Also, included in the report are results for complicated flow configurations with choked flow conditions and cascades with high staggers and high solidities.
* Professor ** Research
of Mechanical Associate
Engineering
INTRODUCTION Analysis of transonic flows through a cascade of airfoils requires accurate modeling of the flow boundaries as well as the details of the The following basic subsonic and supersonic flow regions and the shock. properties must be considered in the analysis of this problem: change their . Equations subsonic and supersonic have irregular shapes by a discontinuity, and trailing . Leading are sources of near . Cascades The efficiency of the geometric configurations irregular computational sufficiently accurate can be obtained.
usually
characteristics in flow regions which and may be separated
edges of singularities
have
irregular
the
airfoils in the
flow,
geometries.
computational techniques in treating such irregular is directly related to the applicability of When such grids are designed properly, grids. approximations of such irregular flow configurations
Numerical solutions of steady, inviscid transonic flow problems have been treated using the finite difference method by several researchers during the last decade (refs. 24,25,28,29). More recent studies have been in the direction of improving the efficiency and accuracy of these solutions (refs. 8,10,16,19,20,21,23,26,31,32). During these investigations, the importance of a computational grid in the calculation of transonic flows The modification of computational grids has been explored was observed. as one of the main tools for improving efficiency and accuracy (refs. 21, 26,31). Application of the finite element method for the analysis of inviscid transonic flow problems has also been considered by several investigators during the last five years, e.g., (refs. l4,6,7,9,12,15,17). The importance of the computational grid in obtaining an accurate solution with finite element method was reported by the present authors in reference 15. One of the main advantages of the finite element method has been its ability to accommodate irregular grids. A less utilized advantage is the capability of adapting the computational grid to the solution either automatically or This property becomes quite important in solving by interactive means. transonic flows in cascades and in choosing the most suitable computational grid for a particular flow problem. The objectives of the present tional techniques for the solution investigate the important Eeatures 2
study are to develop efficient of transonic flows in cascades, of computational grids for this
computaand to problem.
The existing literature on the development of a computational grid for improving accuracy and convergence of numerical solutions has dealt Accuracy considerations are generally basically with elliptic systems. For improving converbased on refining regions with steep gradients. gence characteristics of nonlinear elliptic equations, multigrid methods have been developed recently to smooth the residuals with high frequency content in relaxation procedures (ref. 5). Due to the hyperbolic nature of the equations in the supersonic the development of an efficient computational grid for analyzing pocket, transonic flow through a cascade of airfoils cannot be based only on the In the computations it is observed that the conabove considerations. vergence problem is far more critical in the supersonic pocket than it convergence and accuracy is in the surrounding subsonic flow region. Also, are generally most critical around the shock. the basic considerations in the design of a finite In this report, element grid for obtaining accurate and efficient solutions to the tranEffects of the grid on sonic flow problem for cascades are presented. subsonic and supersonic flow regions and the shock are investigated sepaa computational procedure rately. For an accurate and efficient solution, should be based on consideration of each of these problems and their interaction. the finite element analysis of the transonic In the following chapter, In particular, the flow problem for a cascade of airfoils is summarized. treatment of periodic boundary conditions and the Kutta condition for sharp A combined shock fitting and shock capturing trailing edges are discussed. It is also shown that the solution of the set of nonscheme is presented. linear equations for steady transonic flows requires a numerical integration A constant coefin time of a pseudotime dependent system of equations. ficient integration scheme is developed to improve the computational efficiency of this numerical integration scheme. In the next chapter, convergence characteristics to a steady state are discussed. For the subsonic region, it is shown that the scheme basically requires the solution of an elliptic equation with a relaxation procedure. For the supersonic flow region, the convergence is both a function of the relaxation parameter w and the artificial viscosity parameter cx which is e introduced by upwinding the element density. It is emphasized that while convergence improves with increasing values of CY. the accuracy of the e' solutions is very sensitive to the amount of artificial viscosity. The relationship between the computational grid and the distribution of artificial viscosity is found to be very important and is discussed in detail. This convergence study is also applicable to finite difference and finite volumes methods based on the upwinding of local density. grids curacy
In the next chapter, basic concepts in the development of computational for the solution of transonic flow problems are related to the acof the numerical solutions. For obtaining accurate solutions in
P
subsonic flow regions, grid refinement of regions with high pressure In the supersonic pockets, where pressure gradients gradients is important. the accuracy problem is mostly related to the appliare generally small, cation of an artificial viscosity. Simple yet efficient finite element grids can be employed for the solution of this problem if the artificial It is shown that accuracy around the shock viscosity is correctly applied. is also very sensitive to the application of the artificial viscosity. Finally, the sensitivity of the flow at the trailing edge is discussed for cases where a shock is close to the trailing edge. The last chapter includes several case studies illustrating the generality of the developed computational scheme and the effect of computational grids in terms of accuracy and convergence. The test cases b) a cascade of include a) transonic flow around an isolated c linder, 8 NACA 0012 airfoils, with a stagger angle of 45 and gap to chord ratio with a stagger angle of 45' of 3.6, c) a cascade of NACA 0012 airfoils, and gap to chord ratio of 1.0, (choked flow), d) a cascade of 8% circulararc airfoil, with a stagger angle of 45O and gap to chord ratio of 1.0, and e) a cascade of Josh Sanz stator blades. The work presented in this report summarized the applicability of the finite element method for the solution of transonic flow in cascades. The ability to interactively modify the finite element grid in order to improve the accuracy in specific flow regions proves to be one of the main advantages of the method. It is demonstrated in this report that a designer can analyze transonic flows in cascades: . by starting with a rough finite refining the critical regions . by controlling bution over
the
By using obtained complex,
4
element grid necessary,
and
the artificial viscosity distrigrid and through the iterations,
. by using a combined scheme to check the . by checking comparison
if
shock fitting accuracy of
and shock capturing the shock region,
the accuracy of the solution from of results obtained using different
the grids.
the above procedure, one can always ensure the correctness of the results. It should also be noted that, as the problems get more the need for such a programmed approach becomes more important.
FINITE
ELEMENT FORMULATION OF TRANSONIC FLOWS Governing
inviscid
Equations
We consider the solution of the full and irrotational flows expressed
potential equation in the conservative
(@,x1 ,x + (P$,~>,~ where 4 is the velocity potential, &? is The general mass density of the fluid. are
@=Q
= 0
in Q
for steady, form

(1)
the solution domain, boundary conditions
and p is the of equation (1)
on Sr
0
(2)
and =f
o',n $.
and
f are
and n is
the
outward
where
tion,
By combining one obtains
some specified normal
the the
onS2
,
quantities
on the
(3) boundary
S = S1 + S2,
on S2.
isentropic relation
equation
of
state
with
the
Bernoulli
equa
1 p = const for
the
where
mass density,
y is K2
(K2  q2) the
ratio
of
''
(4) specific
= yl+ 2a2 q2,
heats,
(5)
and
In the above, out the flow,
K is a is
the the
maximum attainable speed which local speed of sound, and q is
is constant throughthe local flow speed.
sound
For the is a in
cascade geometries shown normalized then equation
M in
is
= 9inlain
the
inlet
1,
if
the
inlet
speed
of
yl2 + Mi;
K2= where
in figure (5)becomes
(7)
Mach number.
Cascade
Geometry
Either of the solution domains shown in figure 1 can be used to model boundaries totwodimensional cascades. In each case, a set of periodic gether with the blade surfaces, the inlet and exit boundaries, the stagger Since the domain in h define the cascade geometry. angle a .$ and the pitch a splitting figure lb is multiply connected, EF shown in the figure is required in order to be discontinuous across EF. This assures
boundary such as the slit line to allow the velocity potential that 4 will be single valued.
The configuration shown in figure lb is found to be more advantageous for highly staggered cascades, Firstly, it lends itself easily to the generation of less distorted grids, especially at the rounded leading and training edges. Secondly, since the periodic boundaries are furthest from the blade surfaces, the effects of approximations introduced in satisfying the periodicity conditions in this case are minimal. The boundary conditions for configuration of figure ia have been disAssuming cussed in reference 15. Only the case b will be discussed here. that the farfield inlet and exit flow conditions are uniform and M in' B in' conditions become Bex are known, the boundary i> ii) iii)
where ditions
h is the as
= pinqin
f = fex
=  fin
CosBin
on AB
airfoil
iv)
@+  $ = 1
on EF
circulation
which
(9)
surface
 qex(h
(10) (11)
may be determined
+ hySinain)
(8)
on CD
f = 0 on the
X = qin(hxCosgin
6
f = fin
X
from
CosBex
the
farfield
+ hySinBex)
con
(12)
H
F B
q ex
qin
/f
h h
Pin
G A
Figure
I L
1.
Solution
domains
for
the cascade
problem
Pcx
where = hSincl
hX
and q
is
ex
computed
iteratively
condition
v>
@ (x,y)
is
the
the
+ fex
in
= hCoscts,
Y
continuity
equation
= 0.
is:
4 (x+hx,y+hy) + qin(hxCosB in + hySin8in)
= $(x,Y) where
h
'
from f
The periodicity
s
unknown
distribution
of
Cp on AC.
on BD
(15)
Finally,
1
vi)
@= 0
on A
.
(16)
In the above, equation (9) is due to conservation of mass, (15) is due to and (16) is needed to avoid the arbitrariness of the potential perodicity, function $. In some cases the circulation Kutta condition
the
X is
exit
also
flow
angle
an unknown.
Bex is
not
known
To determine
X in
a priori; this
therefore, case,
the
q21 = cl21 E+
(17)
E
must be satisfied at the sharp trailing edge of way of satisfying equation (17) and determining is discussed on page 18.
Finite For Bateman's
Element
a finite element formulation variational functional (ref.
Formulation of equation 13)
TT = IQp dQ + js2f$ where
for
isentropic
and perfect
yl
8
the
pressure.
(1)
dS

(3)
we use the
(18)
gases
p : 2y is
the airfoil. A convenient h in an iterative fashion
const
p
Y
.
(19)
that Ritz type It can be verified functional consistent use of Bateman's (1) (ref. 1). formulation to equation To introduce the finite for an element e and sum the The stationary values of the variational statement
finite elements are equivalent
element approximations, contributions of all resulting expression
6Tr = g L/Q
P,
(@,,Qx
obtained through the to applying a Galerkin
we express equation elements in the domain. yield the following
(18)
+ 4,yQy)dQ
e + js
fe
@ds]
= 0.
2e
We then choose bilinear element the interpolation ~$~(x,y) $T are
the
shape
= Nei(x,y)$y
holds,
where
nodal
values
number obtain
of element nodal points. the following system of
functions
NP(x,y)
so that
within
each
= NT$e ,(i=l,...,m) of
velocity
Substituting equations
potentials equation
and m is
(21)
into
the
(20),
we
(22) where
(23a)
5 = g JR P,@ ,E ; + !iyE+ e ”
(23b)
fzdS
f=cr
e
S 2e
the
Fournode isoparametric bilinear shape functions
quadrilateral (ref. 35)
NT = + (1 + SSi) are employed for denote and Si,ni
this investigation, the nodal point
where coordinates
elements
shown
in
figure
2 with
(1 + mq PZ
(35)
of the nearest element at the upstream side on is the mass density eu This corresponds to the well known "upwinding" technique used in of e. In the following section, the amount of finite differences (ref. 19). artificial viscosity o e and the relaxation factor w to be used will be
where
determined integrations.
according
to
the
stability
considerations
of
the
numerical
Obtaining an accurate and efficient solution of the potential flow equations in the transonic regime has been a major concern during the recent years. A popular approach has been to extend the finite difference procedures originally developed for the solution of subsonic flows by adding artificial viscosity in the supersonic region. In reference 19, various types of iterative procedures for solving the same equations are reviewed. ADI, SOR and multigrid methods as well as higher order relaxation schemes have been applied to improve the convergence properties of transonic flow equations (refs. 8,19,20,21,23,26,31,32). The finite element method, on the other hand, has also been applied to the solution of elliptic problems by many investigators for several years. The extension of this method to transonic flows has been accomplished basically by following the same procedure, i.e., by adding artificial viscosity to the original system of equations (ref. 15).
The Numerical Integration Technique for PseudoTime Problem The numerical integration scheme based on a direct solution of the system defined in equation (31) has been employed with success for the solution of transonic flow problems in reference 15. However, since the lefthandside of equation (31) is nonlinear, a full decomposition of the coefficient matrix K is needed at each step which makes the scheme computationally somewhat inefficient. For efficiency considerations the time integrations in equation (31) are modified as follows:
g
p+l
= fn+1 + (K 0  K”‘l)&p
(36)
with n+l
42
= bJ$n+l
+ (
[email protected]
(37)
13
where
Irl) = X(P,) are constant for all equations (33) and first time step and inexpensive forward scheme in equation defined by equation
and
i,
= Pi,
n+l time steps and K is evaluated (34). Hence,the decomposition of the subsequent solutions can be and backward substitutions. We (31) as the variable coefficient (36) as the constant coefficient
Modeling
using on+' as given in X0 is needed only for the obtained with relatively shall refer to the scheme (VSC) and the one scheme (CCS).
of Shocks
Considerations for convergence and accuracy in the shock region rely strongly on the modeling of the shock. Either shock capturing or shock fitting techniques can be employed to model the shock. Shock capturing techniques have recently become more popular due to the generality of locating the shock and the simplicity of the modeling. However, to attain convergence and stability around the shock, the choice of the computational grid and the artificial viscosity parameter requires considerable effort with either shock capturing or shock fitting methods. The basic considerations in analyzing the stability and the convergence of the numerical scheme in the supersonic flow do not remain valid since the problem involves a shock discontinuity during the transition from supersonic to subsonic flow. To localize the smearing that occurs while modeling the sharp changes at the shock, the computational grid often needs refinement. Moreover, to stabilize the high frequency oscillations in the vicinity of the shock, artificial viscosity is sometimes added even to portions of the subsonic flow immediately following the shock (ref. 26). The main disadvantage of this approach, hence, is the need for judicious modeling in this area. On the other hand, if one attempts to use the shock fitting procedure directly, convergence difficulties are reduced considerably. However, one is faced with the basic problem of locating the shock accurately. In the present formulation, the potential function $J is continuous across the element interfaces but the derived quantities p,p and a are discontinuous. Supersonic elements are treated by adding artificial viscosity as discussed previously. While the flow is changing from supersonic to subsonic, the continuity of the mass flux in the form ,+ + ' ',n = PG,,
(39)
is satisfied as a natural consequence of the variational formulation, where + and  denote the upstream and downstream sides of the interface, and i; is the modified density defined by equation (34). Hence, providing that the interfaces are sufficiently aligned with the shock lines, the shocks are allowed to appear between the elements, and the continuity of the actual
14
IiI
massflux + + ' ',n = p4);; is
satisfied
to zero.
in For
the
finite
limit,
(40).
as the artificial
values
viscosity
of oe equation
(40)
parameter
can be imposed
tional constraint to the variational form, as will the effect of violating equation. (.40) is However, cant for small values of cI e'
approaches as an addi
be discussed on page 17. found to be insignifi
across the shock, the influence from downstream to upstream is Since, The scheme still present, the above is basically a shock capturing scheme. requires grid refinement at the shock in order to localize the smearing Moreover, high frequency oscillations effects due to downstream influence. appearing in the shock vicinity require relatively more artificial viscosity in this region to damp out such oscillations. The resulting solution at the shock is extremely sensitive to the way. these difficulties are handled, consequently, no unique solution is readily available. To remedy this situation, we propose to use a shock fitting technique for checking the uniqueness of the obtained results. In the shock fitting procedure, the shock is also assumed to occur at the element interfaces. However, after a shock is detected as a change from supersonic to subsonic flow between two adjoining elements, these elements are uncoupled, and appropriate boundary conditions are imposed on both sides of the shock (ref. 15). Although this technique is computationally straightforward to implement, as in all other shock fitting techniques, it relies strongly on the prior determination of the shock position. For this reason a combined version of shock capturing and shock fitting techniques is employed. a) In the case of shock capturing, no modifications are made in the basic form of finite element equations. Although shocks can still occur as discontinuities between elements, smearing is present due to the coupling between the subsonic and supersonic elements neighboring the shock. An additional constraint equation is introduced along the shock, since the conservation is no longer guaranteed at the shock interface because of upwinding. The variational functional in equation (18) is modified as follows: * 7r
where
X is
a penalty
= c lTe + xc Js f f e
constant,
Sf is
(P;L$+ ,n
the

P,
(41)
@Inj2ds
common edge of
the elements
on the
shock and the symbols (+) and (> denote each side of the shock. The implementation of this concept is simply achieved by checking the adjacent
15
~
....... ._..
._.I. I I I,
II
I
Ill1
I
The element equations flow regions. of the line integral in equation (41). are given in reference 2.
elements in subsonic and supersonic are then modified with the inclusion Details of such interface elements b)
Shock
Fitting
In the shock fitting procedure, the shock is again assumed to occur However, when a shock is detected as a change at the element interfaces. from supersonic to subsonic flow between two adjoining elements, these This results in a physical elements are uncoupled as shown in figure 3. separation of subsonic and supersonic regions of flow on the computational grid.
Figure
sonic
A natural element
3.
Shock
boundary condition is in the following form: n+l fe
fitting
imposed
= fZ  cl,as
model on the
fn e e,s
shock
line
of
the
super
(42)
where (43)
16
The value ary
of
@i at
the
shock
points
are
turn
placed
as forced
bound
conditions q
at
in
the
corresponding
nodes
is once more satisfied but this time only for as follows:
of
the
= 0;
(44)
subsonic
element
through the use of the the downstream element,
* Tr = ne + XJ, e
(P;@,;

along
Sf.
Conservation
penalty function technique, by modifying the functional
P,@,,,~
(45)
ds.
f
In equation (45) no variations are taken with respect to (+) terms since they are assumed to be known from the upstream conditions. No additional node points are created. This requires the modification of the element equations along the shocks. In this way, fundamentally different characteristics of subsonic and supersonic flows are separately modeled. as it will be shown that the scheme is uniformly convergent Since, in a wider range of initial conditions for higher artificial viscosities, shock capturing solutions are first obtained by selecting a large artificial viscosity parameter oe. Following this, a series of shock capturing solutions
are
obtained
by gradually
decreasing
c1e until
convergence
diffi
culties are detected. Using the previously converged shock capturing solutions as an initial guess, an attempt at shock fitting is subsequently made to check the shock position. If the initial shock position is correct and the artificial viscosity content is optimum, the influence from downstream to upstream at the shock is minimal in the shock capturing. Hence, both techniques yield essentially the same solution beyond which no further decrease in the artificial viscosity is possible. If the initial position of shock is incorrect, the shock moves to a different position after fitting, generally towards the downstream direction. The developed fitting scheme differs from the classical fitting procedures for which the position of the shock remains fixed. In the present case, however, when the conditions in equations (42) and (44) are applied on a particular flow configuration, a new shock position is free to develop after each iteration. The computer code automatically detects the new shock position and applies fitting once more. In that case the shock capturing is again tried by lowering the artificial viscosity, and the above process is repeated until a unique solution is obtained. A novel feature of the present a combined version of both techniques
formulation is utilized.
is
the ease with At any point
which
17
,during the iterations, scheme to another to computational grid. grid to the direction plied by using shock developed procedure.
the solution procedure can be switched from one improve efficiency and accuracy without altering the, It should also be mentioned that the adaptation of the of an oblique shock which has been successfully apfitting techniques can also be implemented in the
Computational
Grid, and Kutta
Periodic Condition
Boundaries
A typical computational grid shown in figure 4 is employed here in The broken lines on the grid denote modeling highly staggered cascades. the farfield inlet and exit boundaries of the solution domain shown in The extensions at the inlet and exit boundaries are used to figure lb. maintain a rectangular outer domain instead of a parellelogram for efficiency and ease in the grid generation. Uniform farfield flow conditions are imposed in these extended regions. The use of rectanglelike elements in the grid has the advantage of providing grid lines which are aligned better with the direction of the flow as well as the direction of the Moreover, with this grid severe element distortions can shock lines. readily be avoided at the rounded leading and trailing edges. Along the periodic boundaries,every node need not have a matching node on the opposite periodic side. The matching periodic nodes are identified with solid circles in figure 4 for which the periodicity constraint given Equation (15) is satisfied by assigning the by equation (15) is imposed. same equation number to each pair of the periodically located nodes and modifying the righthand side of the element equations on the boundary line Cosp. BD in figure lb, with the potential difference qin(h + hy SinBin). X in The nodal values of the intermediate periodic nodes are computed by interpolations from the values of the nearest pair of nodes. The final periodicity is then attained iteratively. This scheme allows a flexibility in to the generation of more efficient placing the periodic nodes, which leads grids with refinements only in the regions of interest. A symmetric frontal solver (ref. 22) is employed in solving the system In this solution procedure, of equations. only those equations that are currently required for the elimination of a specific nodal point variable As opposed to bandneed by assembled and saved in the highspeed storage. the effectiveness of the scheme does not depend on the nodal ed solvers, Hence, the periodic point numbering sequence. boundary conditions can be imposed conveniently by assigning the same equation numbers to each pair of periodically located nodes. This may be accomplished without renumbering and also with no loss of computational efficiency. the grid points, If the exit flow conditions are not known a priori, the Kutta condition at the trailing edge is iteratively satisfied through the following procedure. The circulation around the airfoil is determined from a previous 18
I
Figure
4.
Finite
element
computational
grid
I
II
I
I
I
I
I
I
I.11
I
iteration, coupled
using nodes at
the the
computed velocity potential values sharp trailing edge, as follows:
of
the
two un
h = 6$+ @)I,
(46)
where + and  denote the two sides of the slit which extends from the trailing edge to the exit boundary. This value of circulation is then imposed all along the slit as a constant discontinuity in velocity potential for the new iteration. The procedure is repeated until an overall convergence is reached and the Kutta condition
2+ q I = q21at
20
the
trailing
edge
is
satisfied.
(47)
CONVERGENCECHARACTERISTICS OF THE EQUATIONS Convergence
of
Variable
Coefficient
Scheme
(VCS)
The computational procedure outlined thus far involves appropriate of a in equation (34) and w in equation (37) in obtaining accurate e and convergent results. To investigate the convergence characteristics of the equation, to a steady state solution, a simple onedimensional model will be used, and since the convergence limits obtained from a single element model provides an upper bound to the general system, only one element will be considered. Hence, for an element e and for y = 2, equation (31) takes the form choice
{K2 If one assumes that steady state solution
(q;)'
+ UeAxe[(q~)2],
[email protected]
3
zTx$;+I ,
the estimate to the nth solution s or q, in element e so that
(48)
= ' is
close
to
the
(49) n q,
then
= qe + A;;:
denotes equation
3
heI
a suitable vector (48) becomes
'>
IAGE1
norm
and
( .
I denotes
an absolute
[K* qz+ “,Axe(qz) ,x]N,xrr’xn_$z+l , = [2qeAqz Recognizing
,I!
,
xT!;G

(51)
, etc.,
(52)
that $xAg+l
equation
 2aeAxe (qeAq;),
(51)
= AGz+l
can be rewritten
NTxk ..
= q,
as follows:
[K2  qz + oeAxe(qi),xlN
= PqZ
and
 aeAxe(qz),xJN
,
xA,z+l
,
aq;
 2aeAxeqz
,
xAqn e,x
(53) 21
By using
At
equation
.(26)
it
Aq;+’
 Aq:
N, ,x
At
w
may further
='
be shown
that
2q2  2a2  2aeAxe(qE) e e 2ai
+ "eAxe(qE)
,x
'x
AqnN e,X
2aeAxeqt 2aI+ aeAxe (qi),x aq:’ N ’ x
(54)
where a 2 = (K* e The above difference satisfies the following
relationship differential
09:, t where
for
small
amounts
of
 Q/2
(55)
implies equati0.n:
+ cAq;
artificial

,
that
the
error
AqE(x,t)
x = bAqz
(56)
viscosities
c = cle Axe Mz k
b = (M2  l)k e
,
(57)
,
(58)
and Me = 4,/a e is the local Mach number. This describe the basic characteristics in the different flow regions. a) Convergence In solution reduced
in
the case of this system
Subsonic
Flow,
model of
error equation the pseudotime
c = 0:
with no added artificial of subsonic flows, problem requires accurate and efficient
aq: , t = bAq2
22
will be employed to integration procedure
,
b 0.
is
Aqz+’ = Aqzexp[bAt]
(60)
The rate which represents the response of a first order damped system. convergence depends on both the relaxation factor w and the local Mach to assure the stability of the numerical integration In addition, number. the bounds for w become procedure in time, 0 < w < 2/(1
 Mz)
of
(61)
Equation (61) shows that for compressible flows, the overrelaxation can be higher as a function of local compressibility compared to incompressible Ylows. Multigrid methods or the use of overrelaxation schemes mainly aim at increasing the convergence of the iterations for small values of b, which is typical of elliptic systems. b)
Convergence
in
Supersonic
Flow,
c $ 0:
When the local flow is supersonic and no artificial viscosity is used a = 0) the solution of equation (56) becomes divergent as implied e when a positive artificial viscosity is introby equation (60). However, the characteristics of the system change and it becomes a hyperbolic duced, The solution of equation (56) for oe > 0 can be written system. as (i.e.,
Aqz+l = Aqz
(xcat)
exp
[bat]
(62)
This solution indicates that since the errors in this case cannot be in the flow direction with a wave speed c damped, they are convected which is linearly proportional to the amount of added artificial viscosity While propogating however, the errors are factor w. ct and the relaxation e magnified since b > 0. The rate of magnification depends on the values cle and w as defined
in
equation
(58).
By changing the behavior of the equations into a convective form with the addition of artificial viscosity, convergence to a steady state is guaranteed in equation (62) for a fixed point in the supersonic pocket. At a particular time step during the iterations, typical variation of errors in the supersonic pocket is as shown in figure 5. Errors at any time step have increasing magnitudes in the flow direction and are largest around the As the flow reaches a steady state at the subsonic upstream, Aq,, shock. at the subsonic depends on the
point converges to variation of Aqe at
zero. The convergence the supersonic element
at an element until all the 23
Figure
24
5.
Instantaneous distribution in the supersonic pocket
of
errors

It will be shown subsequently that this will errors are convected through. depend on how far downstream the element is located and on the number of elements placed in the supersonic pocket up to that point. The above analysis of the convection of the errors in the supersonic pocket is general for an "artificial compressibility" scheme, whether it is based on a finite difference, finite volume or a finite element method. The The solution present study provides a rigorous analysis of the problem. scheme for transonic flows should be suited to account for the propogation of the error in the flow direction. In the application of the present scheme, it was observed that the convection of the errors is the determining factor in the convergence rate of the solutions. To illustrate this, a sample case is tested for a cascade problem (NACA 0012 airfoils, with O" The variation of errors for stagger angle, h/C = 3.6, B = 0, Min = 0.8). in each time step are plotted in figure 6 for all of the supersonic elements on the airfoil surface, following the numerical integration scheme in equation (31) for different values of cx and w. In these figures me denotes the e element location at the supersonic pocket; m = 1 being the first supersonic e element past the sonic line. As can be seen, when the flow is subsonic for an element, solutions exhibit uniform convergence described by equation (60). As soon as the element becomes supersonic, an artificial viscosity is added and the errors are convected with a wave speed linearly proportional to the artificial viscosity from equation (57). Also from the same equation, the change in the relaxation parameter, w, increases both the damping factor, which is negative in this case, and the wave speed. Therefore, the errors are magnified as they are convected along the flow direction until they reach the shock. The convergence of the solution is complete when all errors generated by the upstream elements 'have been convected. This makes the shock region critical for convergence and determines the stability characteristics of the scheme.
Convergence Following onedimensional equation (36)
of
Constant
Coefficient
the same procedure presented version of equation (36). takes the form (K'
Cd + (qz)2 
 c}
Scheme
in For
this chapter, we consider a a single element e and y = 2,
N N T;6n+l , F, x*
aeAxe[(q;)z],xj
(CCS)
",xfi
=
,
;e
= 0.
(63)
25
I
M=l
I 0
o4
10

Figure
26
o =
6.
20
Convergence supersonic
n
to the pocket
30,
steady
50
40
state
in
the
ln
i
~
d
a
I
1 I f9
M=l I
e =1I,
(1L) M2 e
/‘=25 o=O*4
i

i
0
30 n 40
0 Figure
6.
Convergence supersonic
to the pocket
steady state (Continued)
60
50 in
70
the
27
cl
e
=lJ e(lL) M:
i
,
50
n Figure
28
6.
Convergence supersonic
to the pocket
steady state (Continued)
in
the
If %
the estimate to the in element e, i.e., c
nth if
solution
= c& + A$" e
is
close
with
to
lb&l1
the
steady
state
solution
(64)
>> IIA~II
and n qe = q, + Aiz then
equation
(63)
with
Iq,l>>bG;l
(65)
3
becomes
[K2 
= [2q;
Recognizing
equation
+ 4:
,,lN
,
xN >c ,
+ [2qeAq;  %,nq: 2~eAxeqen4~lN ,xE,;& .
(66)
?&+'
(67)
that
(66)
is
= Ai;+l
simplified
= [2qL equation
(37)
and
N TV ,xe
= q
, etc.,
e
as follows
[K2  q;lfi,xA:;+l
Using
 aeAxe hi)
=
+ 34:
(68)  2aeAxe(qz),x]N
and
the
2ai
= K2  qz
Aqz+’
 Aqz N 9X At
,
xAqz

2aeAxeq2
xAqne,x ,
*
identities: ;
?a:
= K2 _ q2 co
(69)
we obtain At co w
a2
= [qi
 ai
 a,Ax,(sz),
xlAqz
,
x  “eAxeqzAq;
,
$J x 9
.
(70)
29
The above difference relationship the following differential.equation
implies
that
the
aq: , t + cAqE , x = bAqE where
for
small
amounts
of
artificial c = aeAxe
error
(71)
'
(72)
viscosities a2 M2 eW eaoo2At
.
for are
Comparison of the above stability analysis VCS in equations (5658) shows that in this It both magnified with a factor of a2e Ia:.
ing
section
the
choice
of
artificial
of
the
Artificial
with case will
viscosity,
(57,58) or from (72,73) provides the same bounds be concluded that both CCS and VCS exhibit similar but CCS is computationally more efficient. istics,
Choice
satisfies
,
a2 b=P.(Mzl)k aco2
that
again
Viscosity
(73)
the results obtained the coefficients b and c be shown in the followae from
for
equations
convergence. It can convergence character
Distribution
One of the most important factors in determining the efficiency of the solution procedure has been the choice of the artificial viscosity distriThe finite difference schemes have traditionally employed a fixed bution. artificial viscosity distribution for a particular grid. For each grid this distribution is determined only as a function of the local Mach point, Thus, while for rough grids the convergence is fast, for finer number. grids it slows down due to the reduction in the artificial viscosity. The basic steps of the finite element scheme developed here are as i) choose a relatively high artificial viscosity, ii) obtain follows: iii) reduce the artificial viscosity, convergence to a steadystate, iv) repeat the same steps until a solution with sufficiently small artiThis procedure has proven to be very reficial viscosity is obtained. liable; one can always converge to a stable solution which is also checked It is similar to the applifor accuracy by the shock fitting technique. cation of successive mesh refinement techniques where one is increasing the number of mesh points for reducing the artificial viscosity.
30
For a better understanding equation (56) is distribution, equation can be written in the
Aq;= If
an initial
solution,
is
Aqz,
of the further following Aq", (xct) assumed for
Aq," = 0
Aq" = q e where
L is
TIESX
choice of studied. form:
of
viscosity this
exp(bt) in
the
x < 0
(74) form
and
of
a ramp
function,
x>L
Odx
Case
the
farfield
Ia:
boundaries (i)
Two
M, = 0.51.
Type
to
types
simulate
1  Velocity
of the
potentials
boundary uniform are
conditions flow
field
are for
imposed this
at
case.
specified: (99)
4 = 4,x
57
Ill
I, I
11111II III I I II
Figure
23.
Grid B (Intermediate (with the sonic line
 30x15) superimposed
for
M, = 0.51
case)
Figure
24.
Grid
C (Fine
 40x20)
(ii)
Type
2  Fluxes
are
specified:
The surface distribution of the pressure coefficients obtained Results of the three grids are as given in figures 25 and 26. and fine grids are in close agreement whereas the coarse grid Flux boundary significantly from the other two at the shock. yield slightly stronger shocks and the shock positions are one ther downstream of what was predicted by the velocity potential conditions.
using all intermediate results deviate conditions element furboundary
In order to determine the effects of replacing an unbounded domain by a large bounded domain, the above analyses were repeatedaby taking the farfield boundaries of grid B also at 5R and 14R radial distances from the center. The grids thus obtained are shown respectively in figures 27 and As it is seen from the obtained results in figure 29, the solutions of 28. 10R and 14R grids are in agreement for the velocity potential boundary condition whereas the 5R grid predicts considerable weaker shocks. The comparison of the results for flux boundary conditions in figure 30 indicates an opposite trend and the shock in the 5R grid is stronger. The above results for the cylinder were obtained by employing the shock capturing scheme. Shock fitting applied to the radial grids shown in figures 22 and 24 failed. This is attributed to the misalignment of the element interfaces with the shock line because, for the shock fitting to be successful, the element interfaces must be reasonably aligned with the predicted shock directions. The degree of misalignment of element interfaces can be determined by computing the direction of the shock line on the surface of the cylinder using isentropic shockjump relations. For instance, from the computed quantities on two sides of the shock for the grid shown in figure 23 with velocity potential boundary conditions, the shock direction at the surface of the cylinder is predicted to be 85O with respect to the xaxis (positive counterclockwise) whereas the interface of the element at that location makes an angle of 60' with the xaxis. For this reason the radial grid B is modified so that the element interfaces in the supersonic pocket are more in line with the shock direction as shown in figure 31. With the new results in this case, the shock direction is computed to be 92' while the element interfaces make an angle of 86O which indicates a better alignment. The comparison of shock fitting and shock capturing results for this case is presented in figure 32. As it is observed, both the shock fitting and the shock capturing results of the modified grid are in good agreement, indicating the accuracy of the solutions. Moreover, slight smearing present at the upstream shock region of the capturing solution in the radial grid disappears with the modification of the grid. For visualization of the supersonic pocket geometry with the computational grid, the sonic line is superimposed both for the regular and the modified grids as shown respectively in figures 23 and 31. Also, isoMach and isopotential lines of this problem are shown in figures 33 and 34. Plach contours in figure 33 are plotted with increments of AM = 0.1. 61
A
GRID A GRID B GRID C ( R,= 10 R 1
l

cp*
I
I .5
1
X/C
Figure
62
25.
Effect of grid refinement (Mm = 0.51, @ specified)
for
cylinder
* GRID A l GRID B GRID C (R,=l!O R 1

A

cp*
i
Figure
26.
Effect of grid refinements (M, = 0.51, flux specified)
63
Figure
27.
Grid
B with
5R outer
radius
(30x15)
Figure
28.
Grid
B with
14R outer
radius
(30x15)
R, . *
=14R R, =lOR R,=5R (GRID B 1
Figure
66
29.
Effect of finite boundaries (Mm = 0.51, @ specified)
.
P m I
R,= 14 R h=lOR
A R,=5R (GRID B)
9 dI
0 rri I P cu I n v
9 
I
2
0,’ P cu Figure
30.
Effect of finite (M, = 0.51, flux
boundaries specified)
67
Figure
31.
Grid B modified at the supersonic superimposed for Mco = 0.51 case)
region
(with
the
sonic
line
l
A 
CAPTURING 1 MODIFIED 1 CAPTURING (REGULAR) FITTING ( Ro= 10R)
f
1
.5 x/c
Figure
32.
Comparison of shock fitting capturing solutions (Moo = 0.51, @ specified)
and shock
69
Figure
33.
IsoMach lines for the (Moo = 0.51, C$ specified)
modified
grid
1
Figure
34.
IsoPotential (Moo = 0.51,
lines for C$ specified)
the
modified
grid
It may be observed with the constant The final
from figure 34 that the modified potential lines in the supersonic
results
obtained
for
Shown summarized in figure 35. by Hafez, et.al., (ref. 19) who compressibility method. For the uniform radial grid of 5R outer the farfield boundaries using a As may be seen, the finite boundary conditions, including source term to the uniform flow there close agreement. However, present results and the result The application
of
this
M, = 0.51
grid is pocket. farfield
better
aligned
condition
are
in the same figure is the result reported employed a finite differenceartificial same problem they have used 60x30 radius and specified velocity potentials farfield formula.
at
element results obtained for the various the result obtained with the addition of a expression in equation (99), are all in is a considerable difference between the reported in Reference 19.
an additional
source
term
$a = qco x R’/R:, obtained from the not seem to affect but increased the
b)
Case Ib:
incompressible the results strength of
M, = 0.45.
The
of the
potential solution 10R and 14R grids shock considerably
effect
of
grid
for the farfield to any noticeable in the 5R grid.
refinement
on the
results
did degree
is
summarized in figure 36 for this flow condition. Flux boundary conditions and 10R radial grids were used in each case. As can be seen, both intermediate and fine grids predict essentially the same solution whereas the coarse grid again does not have the required accuracy due to poor resolution near the shock. Shown in figure 37 are the surface distribution of pressure coefficients obtained using 5R, 10R and 14R intermediate size grids. Also shown in the same figure for comparison is the solution reported by Bristeau, et al., (ref. 6) who employed a conjugate gradient solution algorithm in conjunction with finite elements having a grid of 3456 triangular elements and 1813 nodal points. They have used flux boundary conditions on a large bounded polygonal domain; however, the size of the outer boundary was not indicated. As it may 5R grid solution agrees well with the solution of be seen, the present reference 6.
72
TYPE 2 o TYPE 1 l TYPE 1 (FITTING A 
1
TYPE 1 (SOURCE 1 REF. 19
  cp*
i /
/I I
1
l5 X/C
Figure
35.
Comparison MO3= 0.51
of final condition
results for (R = 10R) 0
73
A l

GRID ‘A GRID 6 GRID C (Ro=lOR
Figure
74
36.
Effect of grid refinements (M, = 0.45, flux specified)
1

,r 2
L_A/T /
a
A
L .I :I I
cp*
A
Figure
I .A
37.
Comparison (Mm = 0.45,
of final results flux specified)
REF. 6 R,= 5R Ro= 10R
45O Staggered
h/C
Kutta
Test Case II Low Solidity NACA 0012
Cascade
A cascade of NACA 0012 airfoils with a 45O stagger angle = 3.6 is analyzed for the following inlet flow conditions: Case IIa:
Min = 0.8
,
Bin
= lo
Case IIb:
Min = 0.8
,
Bin
= 0'
Case 11~:
Min = 0.8
,
6. in
= lo
condition
was applied
at
the
trailing
edge
for
the
above
and
cases.
The chordwise distribution of the pressure coefficients obtained for each case using the finite element grid shown in figure 4 on page 19 are There are only 20 elements over respectively given in figures 3840. The plotted results are those each surface of the airfoil in this grid. With the combined shock capturing computed at the centroid of each element. and shock fitting scheme, convergent unique solutions were reached at For these = 1.25, 1.5 and 4.2 respectively for Cases IIa, IIb, and 11~. 'e cases the maximum computed local Mach numbers were respectively Me = 1.25, 1.16 the
and 1.07 lower
over
surface
the of
upper
the
surface,
and Me
= 1.11,
1.26
and 1.39
over
airfoil.
The need for large artificial viscosity in Case IIc is attributed to the size of the supersonic pocket, as has been implied by the von Neumann convergence analysis. The greater the size of the supersonic pocket, the higher the amount of artificial viscosity required for uniform convergence. This general trend is evident in all three cases analyzed herein. The development satisfaction of the cases. In addition, artificial viscosity upper surface be attributed direction.
of to
of sharp shock fronts with almost no smearing and the trailing edge Kutta conditions are apparent in all figures 3840 show that, with the above set of multipliers, is produced along the 1I more smearing e' the airfoil regardless of the local Mach number. This may the slight misalignment of the grid with the local flow
IsoMach line plots of the above three cases are presented in figures On these figures sonic lines are denoted as darkened contours. 4143. The size of the supersonic pocket and the extension of the shock beyond the midchannel of the cascade in Case IIc are of interest. In this case, with the higher values of artificial viscosity, convergent solutions were obtained initially with the shock placed close to the midchord of the airfoil. As the artificial viscosity was reduced, the shock moved downstream and the size of the supersonic pocket was increased. Each time the artificial viscosity was reduced, shock fitting was applied to the convergent shock 76
aLOWER r.UPPER
1
J

a
CPl”
‘“;,
Figure
38.
Variation over the ("in
of pressure airfoil for
= 0.8,
M lower
B
in
= lo,
coefficients Case IIa M upper
max = 1.25,
max = 1.15)
77
Figure
39.
Variation the airfoil
of
pressure for Case
coefficients IIb (Min
Bin = O”, Mupper max = 1.16,
78
over = 0.8,
Mlower max = 1.26)
J .
 cp*
Figure
40.
Variation over the ("in
of pressure airfoil for
= 0.8,
M lower
max
Bin
= lo,
coefficients Case IIc M upper
max
= 1.07,
= 1.39)
79
80
Figure
41.
IsoMach
lines
for
Case
IIa
(Min
= 0.8,
Bin
= lo)
Figure
42.
IsoMach
lines
for
Case
IIb
(Kin
= 0.8,
Bin
= 0')
I \
Figure
43.
IsoMach
lines
for
Case
IIc
(Min
= 0.8,
Bin
= lo>
81
This was repeated until the shock moved downstream capturing solution. far enough to satisfy the shock jump conditions; in which case both the shock fitting and the shock capturing yielded the same solution. the chordwise pressure Finally, additional test case are presented airfoil respectively in figures 44 of Case IIb were used, but instead the trailing edge Kutta condition,
distributions obtained for an for upper and lower surface of the Here the inlet flow conditions and 45. of determining theoexit flow angle from along = 0 was specified a value of 8 ex
Both the grid shown in figure the exit stations. airfoil) and a fine grid shown in figure 46 (30 The differences were employed for this analysis. butions obtained from these grids are too small are the results Also shown in thesame figures, ficial compressibility scheme employed by Farrell The results obtained by both methods are within
45'
Staggered
High
4 (20 elements over the elements over the airfoil) in the pressure distrito show in figures 44 and 45. of a finite differenceartiand Adamczyk (ref. 16). modeling accuracy.
Test Case III Solidity NACA 0012
Cascade
A high solidity cascade of NACA 0012 airfoils with a stagger angle of The computational grid em45' and h/C = 1, is considered in this case. ployed in the analysis is given in figure 47. The flow through this Mach numbers and angles of
cascade attack:
is
analyzed
for
the
following
inlet
Case IIIa:
Min
= 0.80,
Bin
= O",
Bex = 4.01°
(Computed)
Case IIIb:
Min = 0.85,
Bin
= O",
Bex = 4.70'
(Computed)
Case IIIC:
Min = 0.85,
Bin
= lo,
Bex = 4*60°
(Computed)
Case IIId:
Min = 0.87,
Bin
= O",
Bex = 4.87O
(Computed)
The obtained numerical results for the pressure coefficients and isoMach lines are presented in figures 48 and 49 respectively. Mach contours in figure 49 are plotted with increments of AM = 0.05 and the sonic lines on these figures are denoted by darkened lines.
82
CD
PRESENT REF.16
i
W i m 0
‘A,
 c P* l.
m i
0.
m.
W. J Figure
44.
Variation of upper surface exit
angle
pressure of the (Min
= 0.8,
coefficients airfoil for Bin
= 8,x
over the a specified = O">
83
c
PRESENT
mm REF.16

.:
cp*
I .1
Figure
45.
I
Variation of lower surface exit
84
angle
pressure on the
(Min
= 0.8,
coefficients airfoil for B. m
over the a specified
= B = o"> ex
m 
Figure
46.
Refined ("in
grid
= 0.8,
employed for Case II with B = hex = 0') in
a
specified
exit
angle
Figure
47.
Finite element of NACA 0012
grid
for
45' staggered,
high
solidity
cascade
l
LOWER
*UPPER
a) Case IIIa
03 II
Figure
48.
(Min = 0.80,
Pressure
b) Case IIIb
Bin = 0')
coefficients
over the airfoil
surfaces
for
Case 111
(Min = 0.85,
Bin = 9')
x/c
m, 9O,
c) Case IIIc
Figure
48.
(Min = 0.85,
Pressure
coefficient
Bin = lo)
over
d) Case IIId
the airfoil
surfaces
for
Case III
(Min = 0.87,
(Continued)
ain = 0')
a) Case IIIa
(Min = 0.80,
Bin = 0')
b) Case IIIb
(Min=
Choked flow
Fig&e 49.
IsoMach lines
for Case III
( AM = 0.5 )
0.85,
Bin = 0')
a) Case 111~ (Min = 0.85,
6. = lo> m
b) Case IIId
Choked flow
Figure
49.
IsoMach
Choked
lines
for
Case III
(Continued)
(M = 0.87, in flow
(3 = o"> in
the solution consists of embedded shocks, one on each For Case IIIa, As the inlet the lower shock being much stronger. surface of the airfoil, the strength and the size Mach number is increased to 0.85 in Case IIIb, a new supersonic pocket deof both shocks grow considerably. Meanwhile, a subsonic to supersonic sonic velops on the upper surface. At this point, It is interesting line crosses the entire channel and the flow is choked. to note the increased turning effect of the Kutta condition at the trailing edge as seen in figure 49b. When the inlet Mach number'is further increased to 0.87, a new set of stable solutions are obtained. As can be observed from figures 49b, 49d, and 50, the position of the sonic line crossing the channel does not change. In fact, the flow field past the sonic line in the channel, remains virtuthe outlet mass ally the same except in the vicinity of the shocks. Since, flow is increased for Mex = 0.87, the flow turns to produce an oblique The change in circulation, in shock to satisfy the exit flow conditions. The senturn, affects the flow configuration around the trailing edge. sitivity of flow conditions around the trailing edge will further be discussed. In the above test cases, no additional difficulties were encountered due to the developed complex shock patterns. The obtained results illustrate the generality of the computational method. Multiple shocks, supersonic pockets with irregular geometries, interaction of the trailing edge conditions with the shocks, and the effect of large stagger angles can all be accounted for. the accuracy of the shock capturing solutions In all cases, were tested by comparing each with the shock fitting results (ref. 15) and found to be satisfactory. it should be noted that the Mach Furthermore, contour patterns obtained for the choked flow cases are in close agreement with those reported by Kozel, et.al. (ref. 27). Their results include both finite difference solutions of small disturbance equations and experimental measurements for a cascade of 8%, symmetric, circulararc airfoils with a
S
= 45'
and h/C
= 1.
Test
Case IV  8% CircularArc
Cascade
Using the grid shown in figure 51, the 45' staggered circulararc airfoil of Kozel, et.al. (ref. 27) is also the inlet flow conditions M = 0.876 and 6 = 0.5O. in in are the experimental and numerical results reported in well as the finite element results of the same problem. the numerically determined isoMach lines with that of obtained isodensity lines is excellent. The calculated bution over the airfoil surface is shown in figure 53. for this case was computed to be 2.88O.
cascade of 8% analyzed here for Shown in figure 52 reference 27, as The agreement of the experimentally pressure distriThe exit flow angle
91

Mm=
35
tn 9.

M.= a87 In
s.
0.
Figure
92
50.
Airfoil surface velocity for choked flows
distributions
Figure
51.
Finite
element
grid
for
45' cascade of 8% circulararc
airfoil
(a)
(b)
Figure
52.
IsoMach and isodensity circulararc airfoil a) experimental
(ref.
lines
94
element
8%
27),
b) finite difference solution perturbation equation (ref. c) finite
for
solution
of small 27),
( AM = 0.5
)
Figure
53.
l
LOWER
A
UPPER
Pressure distribution over the 8% circulararc airfoil of 45O staggered cascade ("in
= 0.876,
Bin = 0.5',
(3 = 2.88O) ex
95
Case V  Supercritical
Test
Cascade
Shown in figure 54 is the computational grid used for a cascade of shockfree supercritical blades designed by Sanz (ref. 30) from the hodoThe cascade was analyzed using the following shockfree graph solution. design conditions: M in
= 0.711
13in = 30.81° Bex = 0.35O a
S
h/C
= 9.33O = 1.034
.
The same cascade was also implicit finite difference
analyzed by Steger, code for fullEuler
et.al., (ref. equations.
32)
using
an
Mach number distributions obtained from the three methods are presented in figure 55 for comparison, showing the present solution to be in excellent agreement with the hodograph solution. The Mach number contours of the flow, determined by the present finite element method, are displayed in figure 56 using AM = 0.05 increments. No particular difficulty was encountered in obtaining steady state solutions to this problem. A small "wiggle" appears in the finite element solution of the pressure distribution at the upper surface of the blade which does not exist in the hodograph solution. This is attributed to the sensitivity of the shockfree solution to the accurate definition of the geometry at this point of high curvature. A refinement of the grid at this region seems necessary for better accuracy.
Efficiency
of
the
Developed
The solution of the transonic developed procedure involves the . generate . start iterate . reduce
96
a finite
Computational
flow problem following basic
element
artificial
in a cascade steps:
by using
grid,
with a high artificial until a steady state the
Procedure
viscosity
viscosity solution
parameter, is obtained,
and repeat
the
procedure,
the
Figure
54.
Computational grid blade of Sanz
used for
a
cascade of shockfree
*
REF. 32
+
HODOGRAPH

Figure
98
55.
Comparison shockfree
of velocity Sanz blade
distribution
PRESENT
over
; I I I I I I I I I I
I I A
Figure
’
56.
IsoMach blade of
lines Sanz
for cascade of ( AM = 0.5 )
shockfree
99
shock . perform shock region,
fitting
the procedure . repeat the accuracy of the The computational performing the following
when
"wiggles"
occur
with a more refined obtained solution if
efficiency tasks:
for
each
of
the
at
the
grid to check necessary.
above
steps
are
based
on
of a finite element grid requires the . the generation definition of nodal coordinates for each node and nodal In the applications connectivities for each element. summarized in this report, the grid was prepared by coding a simple special purpose mesh generation program for each case. In practice, this can be achieved interactively by using general purpose finite element mesh generation programs and computer graphics terminals, e.g., (ref. 34), solution of the equations by using the constant . iterative coefficient scheme described in page 13 requires decomposition of a symetric matrix only once at the beginning of the iterations. This matrix is banded for which a wavefront solver is employed. After the first time step, backsubstitution procedure is performed on this system of equations, . the computer cost is proportional to the number of nodal points and elements on the computational grid, and the number of iterations required for final convergence. These depend on the complexity of the problem and the users knowledge of ihe problem. For example when one solves a problem for 0 angle of attack for a cascade, the solution of lo angle of attack case can be obtained by choosing better initial conditions and a more efficient variation of artificial viscosity and relaxation factor during the time steps. The computer program named TFUNS4 which has evolved during this research has been coded and tested for both CDC 6600 and IBM 4341 machines. The central and auxillary memory requirements of the program, in number of words, for CDC 6600 can be determined by using the expressions: CM = Max {8"NODES,
Ms=
6"NODES + WF*(WF1)/Z
~~~:(NuMEL + NODES)
where
100
NODES = total
number
of nodal
NLJMEL = total
number
of
points,
elements,
+ 10) + 5"NUMEL + 35,500
WF = maximum wavefront CM = central MS = information
of
equations,
memory requirement, to be stored
on auxillary
memory devices.
In order to give a detailed description of the computer cost, computer The total requirements of four sample cases are presented in Table 1. CPU times are quoted based on the assumption that uniform flow conditions are used as initial solutions in each case and that no additional flow information is available.
101
I
Table
1  Computer computer
time and auxillary memory requirements of program on CDC 6600 of Indiana University
TRANS4
Problem
NACA 0012 Cascade 0' Stagger Low Solidity NACA 0012 Cascade 45' Stagger Low Solidity (Refined Grid)
36,517
1332
1500
35
400
24.8
'
1 50,096
54,160 c
NACA 0012 Cascade 45' Stagger High Solidity
1056
1206
34
900
45.1
119,886
50,428 c
Sanz Cascade Supercritical Blade
of 1152
1314 
31
400
21.4
51,772
130,698 
,
CONCLUSIONS Application of the finite element flow problem in cascades is presented tional grid for obtaining accurate and The following basic conclusions, which summarize the importance considerations problems in cascades:
method to the solution of transonic and the importance of the computaefficient solutions is illustrated. were reached through this study, in the solution of transonic flow
. The most critical regions in the modeling of transonic flows The accuracy and are the supersonic pocket and the shock. convergence on the solution in this region primarily depend on the artificial viscosity distribution. The artificial viscosity distribution must be chosen in terms of the local Mach number, the local grid size, and the number of elements in the supersonic pocket. . The shock is a critical region since errors in the supersonic pocket are magnified as they are convected in the direction of the flow till they reach the shock. The solution is most sensitive to the artificial viscosity distribution at this region. A combined usage of shock fitting and shock capturing techniques is very useful for the solution of the problem. Finite element method is very efficient in applying these techniques. . The leading edge of the airfoil requires a refined grid for better accuracy. However, if the flow is subsonic, the smearing of the solution in this region with a coarse grid remains local and does not generally affect the overall flow configuration, since the equations are elliptic in this region. . Application of the Kutta condition illustrates the sensitivity of the flow around the trailing edge region and the importance of proper exit conditions. The need for accurate modeling on the trailing edge is apparent in the sample cases in which interaction with the supersonic pocket develops. . Development of a finite element grid for the problem requires consideration of several different flow structures. If one uniformly refines the grid for accuracy, the efficiency suffers considerably, since the convergence requirements at the supersonic pocket become more stringent with the increasing number of elements. Since the pressure gradients are generally low, one should stay with a reasonable number of elements in this region to obtain accurate yet efficient results. Leading and trailing edges are local regions of high pressure gradients and require local refinements in the grid. The modeling of the 103
shock requires This suggests ment.
a grid fitting a modification
rather on the
than a grid refinement. grid rather than a refine
The obtained numerical results demonstrate that the finite element method can be employed quite effectively for the analysis of rather complex transonic flow problems in cascades. Simple grids that will provide accurate results as well as fast convergence can be generated quite efficiently. The above discussion also describes the basic advantage of a finite element method for the solution of transonic flow problems. In addition to the computational grids which fit the boundaries of the overall flow domain, one should also be able to generate grids which are capably of accurately describing the irregular geometries of different flow regions. The grid generation should be interactive to allow modifications at critical regions, such as around the shock or at the leading and trailing edges. The current finite element mesh generation techniques provide such capabilities resulting with considerable computational efficiencies in the solution of transonic flow problems in cascades, e.g., (ref. 34). for the designer Finally, procedure can be recommended: . Start with a coarse to the problem. . Check critical
the
of
a new cascade,
grid
the
and calculate
accuracy of the solution regions and comparing the
following
a basic
by refining results.
analysis
solution
the
grid
at
Once an efficient grid is defined for a particular set of problems, the designer can approach the problem with more confidence. In terms of practical efficiency in the solution of transonic flow problems, i.e., in terms of the number of manhours and amount of computer time required to obtain an accurate solution to the problem, the above approach is practical.
104
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Element Ecer, A., and Utku, M., "Finite Flow", Proc. Symp. on Appl. Comp. Meth. California, August 1977, pp. 799809.
Analysis of in Engng.,
1.
Akay, H. U., Compressible Los Angeles,
2.
of Akay, H. U., and Ecer, A., "Treatment of Transonic Flows Using Finite Elements", Finite Elements in Flow Problems, Banff,
3.
Element Analysis of Transonic Akay, H. U., and Ecer, A., "Finite Flows in Highly Staggered Cascades", AIAA, paper 810210, 19th Aerospace Sciences Meeting, St. Louis, Missouri, January 1981.
4.
Flow Computations in Cascades Akay, H. U., and Ecer, A., "Transonic ASME 26th International Gas Turbine Using Finite Element Method", in Conference, Houston, Texas, March 1981 (also to be published ASME Journal of Engineering for Power, 1981).
5.
Brandt, lems",
6.
Bristeau, M. O., Glowinski, R., Periaux, J., Perrier, P., Pironneau, of Optimal Control and Finite Element and Poirier, G., "Application Methods to the Calculation of Transonic Flows and Incompressible Viscous Flows", IRA Laboria, Report No. 294, April 1978.
7.
Chan, S. T. K., and Brashear, M. R., "Finite Unsteady Transonic Flow", AIAA paper 75875, Dynamics Conference, Hartford, Connecticut,
Element Analysis of 8th Fluid and Plasma 1975.
8.
Caughey, D. A., and Jameson, A., "Numerical Potential Flow About WingBody Combinations", No. 2, February 1979, pp. 175214.
Calculation of AIAA Journal,
9.
Deconinck, H., and Hirsch, Ch., of Cascade Flows", Proc. IV Int. Sciences and Engineering, IRIA,
A., "MultiLevel Math. Comp. Vol.
Adaptive 31, 1977,
Shocks in the Computation Proc. 3rd Int. Symp. on Canada, June 1980.
Solution to Boundary pp. 333391.
Value
Prob
Transonic Vol. 17,
"Subsonic and Transonic Computation Methods in Applied Symp. on Computing Paris, France, December 1979.
10.
Dulikravich, D. S., and Caughey, D. A., "Finite Volume Calculation of Transonic Potential Flow Through Rotors and Fans", Cornell UniFluid Dynamics and Aerodynamics Program, Report FDA8003, versity, Ithaca, New York, March 1980.
11.
Dulikravich, NASA, 1980.
12.
Eberle, A., "Eine Methode sonischen Potentialstromung UFE, 1352, (0), 1977.
D. S.,
"Private
O.,
Communications",
Lewis
Research
Center,
Finiter Elemente Zur Berechnung der Transurn Profile", MesserschmittBolkowBlohm
105
13.
"Application of Finite Ecer, A., and Akay, H.U., the Solution of Transonic Flow", Proc. 2nd Int. Element Methods in Flow Problems, S. Margherita, pp. 191201.
14.
Ecer, Mixed Conf.
A., and Akay, EllipticHyperbolic on Num. Meth.
H.U., for
Element Method for Symp. on Finite Italy, June 1976,
"On the Finite Element Formulation of Problems in Fluid Dynamics", Proc. Int. Engng., GAMNI, Paris, 1978, pp. 315322.
15.
"Investigation of Transonic Flow in a Ecer, A., and Akay, H.U., Cascade Using an Adaptive Mesh", AIAA, paper 801430, 13th Fluid and Plasma Dynamics Conference, Snowmass, Colorado, July 1980 (also to be published in AIAA Journal, 1981).
16.
Farrell, C., Flow T'nrough International
17.
Glowinski, R., Periaux, J., and Peronneau, O., "Transonic Flow Simulation by the Finite Element Method via Optimal Control", Proc. 2nd Int. Symp. on Finite Element Methods in Flow Problems, S. Margherita, Italy, June 1976, pp. 249259.
18.
Gostelow, J.P., "Potential Flow Through Between Exact and Approximate Solutions",
19.
Hafez, M., South, J., and Murman, E., "Artificial Compressibility Methods for Numerical Solutions of Transonic Full Potential Equation", AIAA Journal, Vol. 17, No. 8, August 1979, pp. 838844.
20.
Holst, T.L., Full Potential Vol. 17, No.
21.
Holst, T.L., "Implicit Algorithm for Potential Equation Using an Arbitrary No. 10, October 1979, pp. 10381045.
22.
Irons, B.M., "A Frontal Solution Program Int. J. Num. Meth. Engng., 2, No. 1, pp.
23.
Ives, DC., and Liutermoza, J.F., Transonic Flow Over Turbomachinery No. 8, August 1979, pp. 870876.
24.
Jameson, A., "Iterative Solution of Wings", Comm. Pure and Appl. Math.,
25.
Jameson, A., "Transonic Potential Flow Calculation Form", Proc. AIAA Second Conference on Computational Hartford, Connecticut, June 1975, pp. 148161.
106
and Adamczyk, J.J., "Solution of the Transonic Quasi3D a Cascade Using Artificial Compressibility", ASME 26th Gas Turbine Conference, Houston, Texas, March 1981.
Cascades  A Comparison A.R.C.C.P. No. 807, 1965.
and Ballhaus, W.F., "Fast Conservative Equation Applied to Transonic Flows", 2, February 1979, pp. 145152.
Schemes for the AIAA Journal,
the Conservative Transonic Mesh", AIAA Journal, Vol.
for Finite Element 532, 1970.
Full 17,
Anaylsis",
"SecondOrderAccurate Calculation Cascades", AIAA Journal, Vol. 17,
Transonic Flows Over Airfoils Vol. 2, 1974, pp. 283309. Using Conservation Fluid Dynamics,
and
of
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of Transonic Potential Flow Calculations Jameson, A., "Acceleration on Arbitrary Meshes by the Multiple Grid Method", Proceedings of AIAA Computational Fluid Dynamics Conference, Williamsburg, Virginia, July 1979, pp. 122146.
27.
Kozel, K., Polasek, J., Vavrincova, M., "Numerical Solution of Transonic Flow Through a Cascade with Slender Profiles", Sixth Conf. on Numerical Methods in Fluid Dynamics, Tbilisi, USSR, Int. June 1978.
28.
Murman, E.M., and Cole, J.U., AIAA Journal, Vol. 9, Flow",
"Calculation of Plane Steady Transonic 1, January 1971, pp. 114121.
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29.
of Embedded Shock Waves Calculated by Murman, E.M., "Analysis Kelaxation Methods", AIAA Journal, Vol. 12, No. 5, May 1973, pp. 626633.
30 .
Sanz,
31.
South, J.C., and Brandt, A., "Application Method to Trnnsonic Flow Calculations", Turbomachinery (Edited by T.C. Adamson Publishing, 1977, pp. 180207.
of a MultiLevel Grid Transonic Flow Problems in and M.F. Platxer), Hemisphere
32.
Steger, J.L., Pulliam, T.H., and Chima, Difference Code for Inviscid and Viscous 801427, 13th Fluid and Plasma Dynamics Colorado, July 1980.
R.V., "An Implicit Finite Cascade Flow", AIAA paper Conference, Snowmass,
33.
Strang, G., PrenticeHall,
34.
SUPFR'TAB  "Interactive Finite Element Model Generation SDRC  Structural Dynamics Research Corporation, Milford,
35 .
J.,
"Private
Communications",
Lewis
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and Fix, G.J., An  Analysis __ ____of Englewood Cliffs, 1973.
Zienkiewicz, O.C., London, 1977.
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Edition,
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System", Ohio,
1980.
McGrawHill,
107
.1. Report
2. Government
No.
Accession
3.
No.
Recipient’s
Catalog
No.
NASA CR3446 4. Title
and
5. Report Date July 1981
Subtitle
FINITE ELEMENT mALYSIS OF TRANSONIC FLOWS IN CASCADES  IMPORTANCE OF COMPUTATIONAL GRIDS IN IMPROVING ACCURACY AND CONVERGENCE 7. Author(s)
6. Performing
Organization
Code
6.
Organization
Report
Akin Ecer and Hasan U. Akay _ 9. Performing
Organization
Name
and
Agency
Name
and
10.
11.
Work
Unit
No.
Contract
or Grant
No.
NSG 3294 13.
Type
of
Report
Contractor
Address
National Aeronautics and Space Administration Washington, D. C. 20546 5. Supplementary
No.
Address
Purdue University at Indianapolis 1201 E. 38th Street Indianapolis, Indiana 46205 2. Sponsoring
Performing
None
14.
Sponsoring
and
Period
Covered
Report
Agency
Code
5053252
Notes
Final report. Project Manager, Rodrick V. Chima, Fluid Mechanics and Acoustics NASA Lewis Research Center, Cleveland, Ohio 44135.
Division,
6. Abstract
The finite element method is applied for the solution of transonic potential flows through a cascade of airfoils. Convergence characteristics of the solution scheme are discussed. Accuracy of the numerical solutions is investigated for various flow regions in the transonic flow configuration. The design of an efficient finite element computational grid is discussed for improving accuracy and convergence.
7. Key
Words
(Suggested
Transonic
by Author(s))
flows
Finite element method Computational grids 9. Security
Classif.
(of this report)
Unclassified
16.
Cascades Potential flows
20.
Security
Distribution
Statement
Unclassified  unlimited STAR Category 02
Classif.
(of this
page)
Unclassified
21.
No.
of
Pages
108
22. Price‘ A06
* For sale by the National Technical InformationService, Springfield, Virginia 22161 NASALangley,
1961