Computational Grids in Improving Accuracy and Convergence

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CONVERGENCE CHARACTERISTICS OF THE EQUATIONS ........... 21 ..... scheme to check the accuracy of the shock ..... On the other hand, if one attempts.

NASA

Contractor

Report

3446

Computational Grids in Improving Accuracy and Convergence

Akin

Ecer and

GRANT NSG-3294 JULY 1981

NASA

Hasan

U.

Akay

LO,IN &PY: RETURNTO /OWL TECHNICALLIBRARY KIR-IIA~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 NSG-3294

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 ................. Pseudo-Unsteady Formulation The Numerical Integration Technique for ..................... Pseudo-Time 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% Circular-Arc 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 pseudo-time 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. l-4,6,7,9,12,15,17). 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 pseudo-time 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% circular-arc 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

= y-l+ 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

y-l2 + 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 totwo-dimensional 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, E-F shown in the figure is required in order to be discontinuous across E-F. 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 A-B

airfoil

iv)

@+ - $- = 1

on E-F

circulation

which

(9)

surface

- qex(h

(10) (11)

may be determined

+ hySinain)

(8)

on C-D

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 A-C.

on B-D

(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

y-l

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

Four-node 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 + m-q 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 multi-grid 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 Pseudo-Time 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 left-hand-side 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 = P-G,,

(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

mass-flux + + ' ',n = p-4);; 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 straight-forward 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 rectangle-like 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 right-hand side of the element equations on the boundary line Cosp. B-D 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 high-speed 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 one-dimensional 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 pseudo-time

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 over-relaxation can be higher as a function of local compressibility compared to incompressible Ylows. Multi-grid methods or the use of over-relaxation 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

(x-cat)

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 one-dimensional 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

o-4

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 =1-I,

(1-L) M2 e

/‘=2-5 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(l-L) 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 -,x-e

= 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 (56-58) 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.(Mz-l)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 steady-state, 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", (x-ct) 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 shock-jump 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 x-axis (positive counter-clockwise) whereas the interface of the element at that location makes an angle of 60' with the x-axis. 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, iso-Mach and iso-potential 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.

Iso-Mach lines for the (Moo = 0.51, C$ specified)

modified

grid

1

Figure

34.

Iso-Potential (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 difference-artificial 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 38-40. 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 38-40 show that, with the above set of multipliers, is produced along the 1-I more smearing e' the airfoil regardless of the local Mach number. This may the slight misalignment of the grid with the local flow

Iso-Mach line plots of the above three cases are presented in figures On these figures sonic lines are denoted as darkened contours. 41-43. The size of the supersonic pocket and the extension of the shock beyond the mid-channel 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 mid-chord 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.

Iso-Mach

lines

for

Case

IIa

(Min

= 0.8,

Bin

= lo)

Figure

42.

Iso-Mach

lines

for

Case

IIb

(Kin

= 0.8,

Bin

= 0')

I \

Figure

43.

Iso-Mach

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 the-same 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 iso-Mach 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

m-m 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.

Iso-Mach 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.

Iso-Mach

-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, circular-arc airfoils with a

S

= 45'

and h/C

= 1.

Test

Case IV - 8% Circular-Arc

Cascade

Using the grid shown in figure 51, the 45' staggered circular-arc 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 iso-Mach lines with that of obtained iso-density 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% circular-arc

airfoil

(a)

(b)

Figure

52.

Iso-Mach and iso-density circular-arc 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 shock-free supercritical blades designed by Sanz (ref. 30) from the hodoThe cascade was analyzed using the following shock-free 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 full-Euler

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 shock-free 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 shock-free

*

REF. 32

+

HODOGRAPH

-

Figure

98

55.

Comparison shock-free

of velocity Sanz blade

distribution

PRESENT

over

; I I I I I I I I I I

I I A

Figure



56.

Iso-Mach blade of

lines Sanz

for cascade of ( AM = 0.5 )

shock-free

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 wave-front solver is employed. After the first time step, back-substitution 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*(WF-1)/Z

~~~:(NuMEL + NODES)

where

100

NODES = total

number

of nodal

NLJMEL = total

number

of

points,

elements,

+ 10) + 5"NUMEL + 35,500

WF = maximum wave-front 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 man-hours and amount of computer time required to obtain an accurate solution to the problem, the above approach is practical.

104

REFERENCES

Element Ecer, A., and Utku, M., "Finite Flow", Proc. Symp. on Appl. Comp. Meth. California, August 1977, pp. 799-809.

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 81-0210, 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 75-875, Dynamics Conference, Hartford, Connecticut,

Element Analysis of 8th Fluid and Plasma 1975.

8.

Caughey, D. A., and Jameson, A., "Numerical Potential Flow About Wing-Body Combinations", No. 2, February 1979, pp. 175-214.

Calculation of AIAA Journal,

9.

Deconinck, H., and Hirsch, Ch., of Cascade Flows", Proc. IV Int. Sciences and Engineering, IRIA,

A., "Multi-Level Math. Comp. Vol.

Adaptive 31, 1977,

Shocks in the Computation Proc. 3rd Int. Symp. on Canada, June 1980.

Solution to Boundary pp. 333-391.

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 FDA-80-03, 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", Messerschmitt-Bolkow-Blohm-

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. 191-201.

14.

Ecer, Mixed Conf.

A., and Akay, Elliptic-Hyperbolic 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. 315-322.

15.

"Investigation of Transonic Flow in a Ecer, A., and Akay, H.U., Cascade Using an Adaptive Mesh", AIAA, paper 80-1430, 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. 249-259.

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. 838-844.

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. 1038-1045.

22.

Irons, B.M., "A Frontal Solution Program Int. J. Num. Meth. Engng., 2, No. 1, pp.

23.

Ives, D-C., and Liutermoza, J.F., Transonic Flow Over Turbomachinery No. 8, August 1979, pp. 870-876.

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. 148-161.

106

and Adamczyk, J.J., "Solution of the Transonic Quasi-3D 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. 145-152.

Schemes for the AIAA Journal,

the Conservative Transonic Mesh", AIAA Journal, Vol.

for Finite Element 5-32, 1970.

Full 17,

Anaylsis",

"Second-Order-Accurate Calculation Cascades", AIAA Journal, Vol. 17,

Transonic Flows Over Airfoils Vol. 2, 1974, pp. 283-309. Using Conservation Fluid Dynamics,

and

of

26.

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. 122-146.

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. 114-121.

No.

29.

of Embedded Shock Waves Calculated by Murman, E.M., "Analysis Kelaxation Methods", AIAA Journal, Vol. 12, No. 5, May 1973, pp. 626-633.

30 .

Sanz,

31.

South, J.C., and Brandt, A., "Application Method to Trnnsonic Flow Calculations", Turbomachinery (Edited by T.C. Adamson Publishing, 1977, pp. 180-207.

of a Multi-Level 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 80-1427, 13th Fluid and Plasma Dynamics Colorado, July 1980.

R.V., "An Implicit Finite Cascade Flow", AIAA paper Conference, Snowmass,

33.

Strang, G., Prentice-Hall,

34.

SUPFR'TAB - "Interactive Finite Element Model Generation SDRC - Structural Dynamics Research Corporation, Milford,

35 .

J.,

"Private

Communications",

Lewis

Research

and Fix, G.J., An -- Analysis --__ ____of Englewood Cliffs, 1973.

Zienkiewicz, O.C., London, 1977.

The --- Finite __-

Element

the Finite .- .___

Method,

3rd

Center,

NASA, 1980.

Element

Edition,

Method,

System", Ohio,

1980.

McGraw-Hill,

107

.1. Report

2. Government

No.

Accession

3.

No.

Recipient’s

Catalog

No.

NASA CR-3446 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

505-32-52

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 NASA-Langley,

1961

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