NPS67-88-OOICR NAVAL POSTGRADUATE SCHOOL 411 Monterey, California c1 tjD STATq~ o {%) 7e DTIC 9 xLECTE R JAN 12 1990 U Do CONTRACTOR REPORT FAR FIELD COMPUTATIONAL BOUNDARY CONDITIONS FOR INTERNAL FLOW PROBLEMS AUGUST VERHOFF MCDONNELIL A IR(;RAFT COMPANY P.O. BOX 516 ST. LOUIS, MISSOURI 63166 SEPTEMBER 1988 APPROVED FOR PUBLIC RELEASE; DISTRIBUTION UNI IMITE). PRFPARFD FOR: NAVAl. POSTGRAI)IIATE S t()iOL MONTEREY, (ALIFORNIA 93943-5000 I9 L•9 0 01 11 1i14
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California c1 o {%) · duct flow and to cascade flow, where conditions are periodic in the direction of the blade row. For duct flow, both isentropic and non-isentropic boundary conditions
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NPS67-88-OOICR
NAVAL POSTGRADUATE SCHOOL411 Monterey, California
c1
tjD STATq~o {%)7eDTIC
9 xLECTE R
JAN 12 1990 UDo
CONTRACTOR REPORT
FAR FIELD COMPUTATIONAL BOUNDARY CONDITIONS
FOR INTERNAL FLOW PROBLEMS
AUGUST VERHOFFMCDONNELIL A IR(;RAFT COMPANY
P.O. BOX 516
ST. LOUIS, MISSOURI 63166
SEPTEMBER 1988
APPROVED FOR PUBLIC RELEASE; DISTRIBUTION UNI IMITE).
PRFPARFD FOR: NAVAl. POSTGRAI)IIATE S t()iOLMONTEREY, (ALIFORNIA 93943-5000
I9
L•9 0 01 11 1i14
NAVAL POSTGRADUATE SCHOOLMonterey, California
RADM R. W. West, JR. H. SHULLSuperintendent Provost
The present study was initiated under contract N62271-84-M-1857during the period May 1984 to November 1984. The work was extend-ed to the analysis of cascade flows under contract N62271-87-M-0202 during the period December 1987 to August 1988. The workwas sponsored by the Air-Breathing Propulsion Research Programat Naval Air Systems Command under the cognizance of G. Derderian(AIR 931E).
This report was prepared by:
A. VERHOFF
Publication of the report does not constitute approval of thesponsor for the findings or conclusions. It is published forinformation and for the exchange and stimulation of ideas.
Reviewed by:
RAYMOND P. SHREEVE E. R. WOODProfessor ChairmanDepartment of Aeronautics Department of Aeronautics
19 ABSTRACT (Coninue on reve'rse if necesary and Identify by hlock numnber) I
P AR FIL)COMPUT'ATI1ONAL, BOUNDARY CONDITIONS F7OR 21) INTrERNAL FLOW PROBLEMS ARE DEVELOPED
FROM ANALYTIC SOLUATIONS OF T1IIW 1INE-ARIZED EULER EQUATIONS. TIHE EULER EQUATIONS ARE
LI NEARI ZED) ABOUT A CONSTrANT PRESSURE, RECTILINEAR FLOW WH!I CH MAY hIAVE STREANWTSE-NORMAL
VAR[AT i ONS IN TI:'MI'ERATITRE AND) VELOCITY AS A RESULJT OF ENTRO1'Y PROI)UCTION TN THE NONLINEAR
cI)mlITTAT [ONAI. REG;ION. THlE BOUNDARY l'ROCEIURF CAN BE USED WIT11 ANY NUMER ICAL EULER SOLTITON
MEFTHOD AND ALLOWS COMP'UT'ATrIONAL BOUN)AR IEs ron m.E P1 .ACE!) MUCH CLOSER TO THlE NONLINEAR
RlIC ION" OF I NTFEREST .
BESTAVAILABLE COPY
20 DiIR1R1.1J10ONAVAILAIILITY OF ANSTRACT 21, ABSTRACT SFCURITY CLASSIFICATION
M UICLASSITIEDUNLIMITEO 0i SAME AS RPT L1 DTIC USERS UNCLASSI VIED
22a NAME OF RESPONSIBLE INDIVIDUAL 22b TELEP14ONE (include AraCd)a.OFFICE SYMBOLR. P. SHRE4FEVE (408) 646-2593 67Sf
DD FORM 1 473, 94 MAR 83 APR edition M,.y be used until ewhausted _SECURITY CLASSIFICATION OF T1415 PAGEAll other edition ate obsolete *..nvnn.i o t
ti
FAR FIELD COMPUTATIONAL BOUNDARY CONDITIONS
FOR INTERNAL FLOW PROBLEMS
A. Verhoff
McDonnell Aircraft CompanySt. Louis, Missouri
ACCesl1:1 For
J ,J , :,tI : ,;- d
By _
Ist' ibinio;o I-1
Avj#pbbl)t. ,
Dist L
SE-I 1
,,, SEPTEM'BER 1988
iii
FAR FIELD COMPUTATIONAL BOUNDARY CONDITIONS
FOR INTERNAL FLOW PROBLEMS
A. VERHOFF
MCD(ONNELL AIRCRAFT COMPANYST. LOUIS, MISSOURI
A OSTRACT
Far field computational boundary conditions for 2D internal flow problemsare developed from analytic solutions of the linearized Euler equations. TheEuler equations are linearized about a constant pressure, rectilinear f")wwhich nay have streamwise-normal variations in temperature and velocity as aresult of entropy production in the nonlinear computational region. The bound-ary procedure can he used with any numerical Euler solution method and allowscomputational boundaries to be placed much closer to the nonlinearregion of interest.
iv
I. INTRODUCTION
Numerical solution procedures for nonlinear fluid dynamic equations
usually use one or more artificial computational boundaries located at some
distance from the primary region of interest in order to limit the physical
domain to finite size. If the flow crossing such a boundary (either inflow or
outflow) is subsonic, then some type of computational boundary conditions must
be imposed which simulate the influence of the true far field conditions at
infinity. These boundary conditions must be such that waves crossing the
boundary do not produce erroneous reflections back into the computational
field to degrade the calculations. It is generally acknowledged that simply
imposing free stream conditions (or conditions at infinity) at computational
boundaries is usually inappropriate. Standard practice has consisted of
locating the boundaries quite far from the region of interest in an attempt to
simplify the boundary condition models and minimize any effects of
inconsistent modeling. The net effect is a significant increase in the number
of grid points required for an accurate flowfield calculation.
A boundary modeling procedure for two-dimensional internal flows is
presented which alleviates the difficulties mentioned above and also allows
the computational boundary to be located much closer to the nonlinear region
of interest. The procedure is limited to steady, inviscid flow, although the
flow can be rotational. It represents a logical extension of the so-called
characteristic boundary conditions commonly used with inviscid numerical
solution methods. Extension to axisymmetric or three-dimensional flows is
straightforward.
The analysis presented here is based on the Riemann variable formulation
of the Euler equations given in Reference 1. This represents a natural
starting point because the characteristic (or zero-order) boundary conditions
mentioned above are expressed in terms of Riemann variables. The equations
are linearized about a constant pressure, rectilinear flow condition, which
truly represents conditions at infinity. These linearized equations are
assumed applicable in the far field region beyond a computational boundary.
Within the nonlinear computational domain, strong entropy-producing (i.e.,
I
rotational) effects can exist which create variations in density, velocity,
etc. in the far field in the streamline-normal direction which are not
necessarily small perturbations. Such variations are modeled in the present
analysis.
The linearized equations are solved analytically using Fourier analysis
techniques as outlined in Reference 2. These solutions are coupled to the
nonlinear numerical solution to provide a smooth transition across the
boundary to the true far field conditions at infinity. The coupling is
accomplished by the boundary conditions. The underlying principle is that the
streamwise variations of both upstream and downstream running waves should
decay to zero at infinity. These first-order boundary conditions provide
distributions of flow quantities to be imposed along the boundary, not
constant conditions. They represent a logical extension of the zero-order (or
characteristic) boundary conditions. Furthermore, the boundary analysis can
be coupled with any inviscid numerical solution method.
The boundary condition analysis has been applied to two-dimensional
duct flow and to cascade flow, where conditions are periodic in the direction
of the blade row. For duct flow, both isentropic and non-isentropic boundary
conditions are derived. Only isentropic results are given for cascades.
Extension to non-isentropic cascade flow can be carried out by following the
procedure used for duct flow.
Numerical results are presented for both isentropic and non-isentropic
duct flow. Results obtained using the first-order boundary conditions are
compared with those using the zero-order boundary conditions. Numerical
results for cascade flow will be presented in Reference 3. It was found that
the size of the computational field and associated number of grid points
needed for the nonlinear numerical solution could be reduced significantly by
using the new first-order boundary condition procedure with no loss in
numerical solution accuracy. The reduction in number of grid points was as
much as 50 percent in some cases. The additional computational effort
required by the new boundary procedure is small (less than 10 percent) so that
a significant saving in overall computational effort was realized. A large
portion of the gain is due to the sizeable reduction in the physical extent of
2
the computational field and the fact that fewer solution iterations are
required for information to propagate between the upstream and downstream
computational boundaries leading to more rapid solution convergence.
3
11. PERTURBATION EQUATION DERIVATION
The system of two-dimensional, steady, linearized Euler equations which
describe first-order spatial perturbations from a constant pressure state will
be derived in this section. A Riemann variable formulation taken from
Reference 1 will be used because of its close relationship with the character-
istic (or zero-order) boundary conditions commonly used in numerical solution
of the nonlinear Euler equations.
The two-dimensional form of the Euler equations is (Reference 1)
+ (q + a) i z -1 a (S -2 ) [ - a - qaS (1)at as 2 -I as y- as] 2 an
aR-a) = + a (S 2-2_) [ + 2 aa )a-t (2as 2 -1 a -i-s] 2 qaS (2)
a8 ae a2 aPt + as yq an (3)
as+ as 0 (4)
Velocity magnitude and speed of sound are denoted by q and a, respectively,
and P is the logarithm of pressure. The Extended Riemann Variables Q and R
are defined as
Q = q + aS (5)
R = q - aS
while the modified entropy is defined in terms of pressure p and density p as
_ 1 [2r log (p/pY)] (6)S (y-1)
The flow angle is 0, time is denoted by t, and local distances along and
normal to the streamline direction are denoted by s and n, respectively.
4
For steady flow the analysis can be greatly simplified by defining a new
dependent variable
T = Q - R (7)
and replacing equations (1) and (2) by
(M2 - 1) 2- + (y-1 ) q M S 38 = 0 (8)
as an
a2 + Y-1 q2 1 (9)2
The local Mach number is denoted by M. Equation (8) is obtained by
subtracting equations (1) and (2). Equation (9) is obtained by adding
equations (1) and (2) and integrating. The constant of integration, which is
proportional to stagnation temperature, can be set to unity by proper choice
of non-dimensionalizing quantities. The simplified form of the steady Euler
equations is then
(M2 - 1) 21 + (1-1) q M S 2- 0 (10)as an
M2 aO + 2 1 aT + 1 - 2 = 0as y-l T an S (S y-1 an
asT- 0 (12)
a2 + Y-1 q2 1 (13)2
According to equation (12) entropy remains constant along streamlines.
In regions of the flowfield where nonlinear effects are weak, the flow can be
treated as a perturbation to a constant pressure, rectilinear flow. Such
regions occur near and beyond far field computational boundaries. The depend-
ent variables in equations (10)-(12) can then be expanded in asymptotic series
T =To + T1 + T2 +
S :S + S1 + S2 + ... (14)
S=0 + e1 + 02 +
~ 1 2
The flow direction at infinity is assumed constant and denoted by 0.; the
perturbation quantities Ti, Si and 8 vanish at infinity. Entropy variation
is not excluded so that the flow can be rotational. Furthermore, entropy
variations can be strong (i.e., not small perturbations) so that S and T. are
not necessarily constant, but may vary normal to the streamline direction.
Note that T. depends only on S (because p. is constant).
Consistent with (14), spatial derivatives in equations (10)-(12) can be
approximated by
a_ = cos + a + co ('+8j)i-. +
asaxa(15)
a a 8aan -cos W= (1+01)- + Cos 8= (1-16I)y +
where x and y are reference Cartesian coordinates, 8 is measured from the x
axis, and
i tan 8 (16)
If expansions (14) and (15) are introduced into equations (10)-(12), the
resulting first-order perturbed Euler equations are
2 _BaT aT1 aT, aT0
(M 0-)[-+ y 81 ('x -y )] +
(17)
(y-l) q. M S - 0
y12 [ + 1 1 aT1 aT1 T 1 [ aT.. aT +
t M; S.,, a t -]y-] + i-- C- T i -y -- +
(18)
- .(s 2 S1 _S S1 Si a- as= 02 ( Y-0 y) ay . a] ax
as1 + -L + 0, as . as ., = 0 (19 )
6
Velocity, speed of sound, and Mach number at infinity are denoted by q., a
and M., respectively, and may vary normal to the streamline direction. The
fact that all dependent quantities are convected unchanged along rectilinear
streamlines at infinity has been incorporated into the above equations.
Asymptotic expansions of the Riemann variables Q and R can also be
defined as
Q =Q + QI + Q2 +
(20)
R R_ + R1 + R2 +
Using the definition (7) and the expansion (14) for T, it follows that
T Q -R.
(21)
Ti = - 1
Introducing the expansions (14) and (20) into the algebraic equation (13)
gives the first-order relationship
M. (Qj-RI) + II 2 (22)2 M (QI+Rl) - 2q. S I = 0 (22)
This will be used later in Section IV where the boundary conditions are
derived.
7
III. SOLUTION OF FIRST-ORDER EQUATIONS
Solutions of the first-order equations (17)-(19) are developed in this
section for two-dimensional duct flows confined by parallel walls and for
infinite cascade flows. Duct flow solutions are obtained for both isentropic
and non-isentropic conditions. The cascade flow solution is provided for
isentropic conditions.
Duct Flow
For two-dimensional duct flow between parallel walls, the far field flow
angle 0 (and consequently T) can be set to zero by aligning the coordinate x
with the duct axis (see Figure 1). The width of the duct can be taken as
unity without loss of generality. Equations (17)-(19) then reduce to
(Moa-1) [a-+ I q-] + (,-i)ql M S ay = 0 (23)
I E T1 I1 IT-3 I S
2 ax T2 ayy T.S B
(24)+ 1-i a (S. 2_2_) =S 0
2 Y-1 ay
-X + 01 o(25)
Isentropic Conditions - According to the scaled entropy definition (6), the
far field entropy S is -_L. Equations (23)-(25) then simplify for isentropicf-1.
flow to
- 0 (26)
2q M, ax ay
ae1 1 aT1I+ I IT' -0 (27)
ax 2q" ay
where
0 1-M, (28)
Note that q. and M are constant.
Equations (26) and (27) can be solved by separation of variables by
assuming
1= 2 q M e F(y)(29)
= e x H(y)
where X is the unknown separation constant. Equations (26) and (27) then
reduce to the ordinary differential equation system
02 A F- H' = 0
(30)XH + F' = 0
with boundary conditions
H(O) = H(1) = 0 (31)
Primes denote differentiation with respect to y. Eliminating F gives the
single equation
HII + A2 2 H = 0 (32)
which has the solution
H = kI sin xBy + k2 cos xBy (33)
To satisfy the boundary conditions (31), the constant k2 must be zero and
A = + n_ (n = 1, 2, ...)(34)
9
Using these results, the general solution of the system (26) and (27) can
be written as
STI 00 [ 2q. cos R lI =i ~ s n y nliX/P
IKn Re +K E sin ny(35)
-2 . .cos niiy -nnx/ p3
K2n e
sin nry
where Kin and K2n are arbitrary coefficients.
Non-Isentropic Conditions - Although the system (23)-(25) is linear, it has
non-constant coefficients since S is independent and varies with the
streamline-normal direction y. The thermodynamic variables are related by
r- 1 -1 ( 2 S.)2y 2 i
a. = pm e (36)
where p. is the constant pressure at infinity.
To achieve an analytic solution, a second level of linearization can be
introduced by defining a new variable o(y) by
S. 2 -i (1-) (37)
Since a is generally small, equation (36) can be approximated by
Y-1a. = p. 2y (1+o) (38)
10
An approximation for q. is then provided by equation (9). Reference
quantities defined by
2y-
q -Y17(1a.) (39)
Mo= q._a
can be introduced to simplify the notation. The approximate thermodynamic
relations then become
a,= a. (1 + a)
M. 2 0) (40)
4-= -I
All quadratic and higher terms in a have been neglected.
Introducing the above approximations (40) into the governing equations
(23)-(25) gives
(0 2 + -L o) aT1 4 o) (41)
2q0Y10 (1 - 4 -2) !-0 + Tl 2a. (42)y 2 a ay a(oS)
BSI 2 do;-x T 0 (43)
11
The parameter B is defined by equation (28) with M substituted for M.
Equation (43) suggests that S is of higher order than T1 or 81. If so, then
the RHS of equation (42) can be set to zero and equation (43) becomes
decoupled. Validity of this assumption can be verified later by evaluating
numerical results obtained from this analysis. The final system of governing
equations is therefore
4 -L .-a) !T1 - (-4 4 )) 1y-1 -2 ax 1-1-2 ay 0
2qM. (1 _- - ax ay (45)
After these equations are solved for T1 and 81, the perturbation variable S1
can be obtained from equation (43) by quadrature.
As for the isentropic analysis presented above, a separation of variables
solution approach is again appropriate. Analagous to equations (29) assume
T= 2q. M e x F(y) (46)
81 =eX H(y)
Equations (44) and (45) then reduce to the ordinary differential equation
system
H' - Ao2 F -[XF + H']
y-1 a2 A2
(47)
F' + AH = 4 o AH
with boundary conditions
H(O) = H(1) = 0 (48) o
12
Eliminating F from equations (47) gives the single equation
H' x H 4 1 [H" X H+ 2 + + 2H" + X2 2 H [o(H" +H F') o'(H' + AM, F)] (49)
y-1 -2+k 2 F ' H
This equation is in a form suitable for solution by iteration. The lowest
order solution (i.e., first approximation) satisfies
Ht+X2 2H=0IH +A2B2H=O
(50)
F - 1 H'X0
2
Using these results in the RHS (denoted by f) of equation (49) produces
H" 4 12 H2 - __H"1
2 2 H - [o' H + a H"] = (51)
The solution of this equation provides an improved second approximation for
the solution of equation (49). Repetition of this iterative process generates
ever-improving approximations. Only the solution for the second approximation
will be given here. Whether or not it is sufficiently accurate can be
assessed by evaluating the numerical results presented later.
The solution for the first-approximation equations (50) was obtained in
the above isentropic analysis as
H = An sin nwy +n (52)
where the An are arbitrary coefficients. Note that the iterative solution
procedure used here is consistent in that the isentropic solution is recovered
for vanishingly small a. Using the approximation (52) the RHS of equation
(51) can be written
j - [' ' n An cos nry - w a o n2 An sin nry] (53)
13
Equation (51) can be separated into component equations
Hn + X2 H = n (n 1, 2,...) (54)n n n n
where 0 n represents the nth component of the source term 0 defined as
4 fnAn I-j *- -2_ [a cos nry - w M2 o n sin niy] (55)
For consistency, the separation constant An must also be expanded as
n 6+A( +..) (56)An _- +6n •
so that equation (54) becomes
Hn + n T H n- 2van 6X A sin niy (57)n n n n n
The solution of this equation which satisfies the boundary conditions (48) is
1 iyHn = Kn sin nry + P 6Xn An Y cos nry + -j On sin nr(y-) dn (58)
where K is an arbitrary coefficient andn
= I A S n sin nrq d9 (59)6 n rno An o
This solution is valid regardless of the choice of sign in equation (56).
The solution for the nth component of F can be obtained from the system
(47) as
Fn + [ (1 - ) cos nry - 6An An y sin nry
(60)P I- o2 -q.
+ 1 0 n cos nr(y-n) dnj
14
The choice of sign corresponds to that of equation (56).
Using the above results, the approximate solution of the system (44) and
(45) can be written as
2 q " ( 1 P- 2 c o s n y
0-Y-1 q. ( + 6 n)X[~ = Kin [ 2qysin niy
-2q" (1 - cos nry
+ K2n e'(-p + 6 n)x (61)
1 sin nny
OD M fn nn +I6 fn nS+ h e- (P + 6An)X
11 hn I hn
where
fn we 6n An Y sin nry On cos nn(y-9) dq (62)
hn B k An Y osif nliy + 1 SyAn sin nn(y-q) dn (63)
~ 6A AnYcos nii + o
This solution reduces to the isentropic result (35) when a is zero. Using
this solution, equation (43) can be integrated to obtain the entropy
perturbation S1.
Cascade Flow
For isentropic cascade flow, equations (17)-(19) reduce to
aT1a 1 B I ae1e [ - + I -I 8 + a 0 = 0 (64)
2q" ax ay ay ax
15
81 a1 1 T aTBI+ T -- + 0 (65)T-)=
ax ay 2qM. y X (y5)
If the (x,y) coordinates are chosen such that the flow is periodic in the y
direction (see Figure 2), then a separation of variables defined as
T, = 2q.M F(&) eiy
01 = H(&) e inry (66)
C inrx
can be used. The blade spacing has been taken as unity. Equations (64) and
(65) are thereby transformed to the ordinary differential equation system
S2(F + TF) + tH' H = 0
(67)
H' + TH - TF' + F = 0
Primes denote differentiation with respect to k.
The solution of the system (67) has the form
F = ae H = e (68)
where the eigenvalues are
M2 ± io(1 + T2)
= +(69)
and
- + - (70)
Using these results, the general solution of the system (64) and (65) can be
written as
16
0 e +[21.. = K n[ Je (1~ + Y) (71)
-~K2n[ 0~MD enr~x+
where
T M2 + iO(1 + T2)
T M - io(l + 12) (2
Note that XIand A2 are complex conjugates.
17
IV. BOUNDARY CONDITION DEVELOPMENT
Examination of equations (1)-(4) shows that at a subsonic far field
computational boundary there are three downstream-running waves and one
upstream running wave. Therefore, the information available from the
numerical solution is not complete and differs at upstream and downstream
boundaries. The information lacking is provided by the boundary conditions.
If the flow is supersonic, all waves are downstream-running and specification
of boundary conditions is straightforward.
Far field computational boundary conditions (subsonic) are developed in
this section based on the linearized Euler solutions obtained in the previous
section. These solutions are assumed valid in the region beyond the computa-
tional boundaries where nonlinear effects are small. Within the computational
boundaries the full nonlinear Euler equations must be solved numerically. The
boundary conditions provide for a smooth coupling of the nonlinear and linear
solutions so that the true conditions at infinity can be imposed.
The three specific cases analyzed in the previous section will be
addressed in this section.
Duct Flow
Both upstream and downstream duct flow boundary conditions will be
derived for isentropic conditions. The upstream boundary analysis is valid
even if there is nonlinear entropy production downstream of the boundary
within the computational domain. For non-isentropic conditions only
downstream boundary conditions will be derived, since non-isentropic
downstream flow is the more common situation.
For isentropic, constant pressure flow in the upstream or downstream
regions, the far field Mach number M is determined implicitly from the mass
flow per unit area w by the relation
18
M[1 + :1 M2] 2 1-l = w (73)2
The mass flow is usually known (or can be calculated) for a given duct flow
problem. The associated speed of sound and velocity are
M 1+ya. = (_.) q. = a, MO (74)
w
Therefore, the far field quantities Q_ and R appearing in expansions (20) are
Q. = q. + a R2 : -I a. (75)
The far field entropy has been set to Equations (21) relate Q and R to
the variable T.
For non-isentropic, constant pressure flow in the downstream region,
equations (37), (39), and (40) give the approximate far field relationships
q_2 a,
R~ o a. -:i, a~ (76)Y - MToo 4- 1 a-(76.
Quadratic and higher terms in a have been neglected.
Isentropic Conditions - At a computational boundary (assumed located at x=O)
the perturbation flow variables can be represented by the Fourier series
19
1 An sin niy (77)1
Q1 = E Bn cos nny (78)1
RI = Cn cos nvy (79)1
TI = En cos nry = (Bn-Cn) cos ny (80)
1 1
The boundary conditions will be developed from relationships between the
Fourier coefficients An, Bn' and Cn* The absence of modes corresponding to
n=O for Q1, R1 and T is related to the fact that these first-order
perturbations must vanish at infinity. Further discussion of this topic is
presented below in conjunction with the boundary condition development for
cascade flows.
The general solution for linearized isentropic flow is given by equation
(35). Applying this solution at x=O and using the series expansions (77)-(80)
it follows that
An Kn + K2n(81)
En 2 M' (Kin K2n)
Therefore,
K rn = [An +- (Bn - Cn)] (82)
K2n = [An - 0 (Bn - Cn)] (83)K 2 [ 2q"1
For the region upstream of the computational boundary (i.e., x<O), the
exponentially growing component of the solution (35) can be suppressed by
forcing K2n to be zero. This requires the Fourier coefficients to be related
by
20
An On (B - Cn ) (84)2q0 M(
[he linearized upstream solution is then
2q M.[TI] = A P cs nvY e n1X/P
= An (85)I1 I sin ny I
For isentropic flow there are two downstream-running waves propagating
information to the upstream boundary from outside the computational domain and
one upstream-running wave propagating information from the numerical solution.
Equation (84) provides one of the lacking pieces of information from outside
the computational domain; the remaining information is provided by combining
equations (22), (78) and (79) to give
1 - M.Bn - + -M Cn (86)
Using the expansions (20) and the Fourier representation (79), the
coefficients C are determined from
RI = Cn cos nry = Rnum - R, (87)1
where R num is the boundary distribution of R obtained from the nonlinearnumerical solution and R is given by equation (75). The coefficients An and
Bn are then obtained from equations (84) and (86). Using the Fourier
representations (77) and (78), the distributions of e and Q on the boundary
(i.e., the boundary conditions) are calculated according to
Ob = E An sin nry1
(88)
Qb = Q. + Bn cos nry1
21
For the region downstream of the computational boundary (i.e., x>O), the
exponentially growing component of the solution (35) can be suppressed by
forcing Kin defined by equation (82) to be zero. This requires the Fourier
coefficients to be related by
Cn = Bn + 2qM_ An (89)
The linearized downstream solution is then
: E An c n ey nwx/o (90)e1 in ny I
For isentropic flow there are two downstream-running waves propagating
information to the downstream boundary from the numerical solution and one
upstream-running wave propagating information from outside the computational
dorrain. Equation (89) provides the lacking information. Using the expansions
(20) and the Fourier representations (77) and (78), the coefficients A and Bnn nare determined from
1= An sin niy = Onum1
(91)
Q1 : Bn cos nry = Qnum - Q-1
where e num and Qnum are the boundary distributions of e and Q obtained fromthe nonlinear numerical solution and Q. is given by equations (75). The
coefficients Cn are then obtained from equation (B9). Using the Fourier
representation (79), the distribution of R on the boundary (i.e., the boundary
condition) is calculated according to
Rb = R. + Z Cn cos nry
(92)
Qnum -1 a, + B Y An cos nry
22
The second form of this relation requires only the calculation of the
coefficients A .n
Many numerical solution algorithms for the Euler equations use so-called
characteristic far field boundary conditions in which . and Q. are specified
at inflow boundaries and R is imposed at outflow boundaries. The boundary
conditions (88) and (92) therefore represent a logical first-order extension
of the widely-used characteristic (or zero-order) boundary conditions.
Non-Isentropic Conditions - For the case of non-isentropic flow crossing a
downstream computational boundary, the Fourier series representation (77) for
I at the boundary is still valid because e is zero at the duct walls. The
variable a which characterizes the entropy distribution at infinity can be
represented by
S:o w + ) Dk sin(k-i/2)ny (93)
1
where ow is the wall value at y = 0. This Fourier series representation
assumes an even extension of a for 1 < y L 2. The source term component 0n