A Method for Generating Reduced Order Linear Models of. Multidimensional. Supersonic Inlets. Amy Chicatelli. Tom T. Hartley. The Department of Electrical.
NASA
/ CRm1998-207405
A Method for Generating Reduced-Order Linear Models of Multidimensional Supersonic
Inlets
Amy Chicatelli and Tom T. Hartley University of Akron, Akron, Ohio
Prepared
National
under
Grant
Aeronautics
Space
Administration
Lewis
Research
May
1998
Center
NCC3-508
and
Available NASA Center 7121 Standard
for Aerospace Drive
Hanover, MD 21076 Price Code: A05
Information
from National
Technical
Information
Service
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Road
VA 22100 Code: A05
:::_?,i:i_!:_
:! i ¸ i•
:I !i ::_
Contents 1
Introduction
2
CFD
3
4
5
............................................................................
Model
Development
...............................................................
2.1
Governing
Equations
2.2
Split Flux
Model
2.3
Boundary
Conditions
Linear
2.3.1
Wall
2.3.2
Inflow
Model
2
.............................................................
2
................................................................
and
6
............................................................
Boundary
Conditions
Outflow
Development
9
.................................................
Boundary
Conditions
9
...................................
.........................................
3.1
Small
Perturbation
3.2
System
Matrix
3.2.1
Wall Boundary
3.2.2
Inflow
3.2.3
Outflow
Model
11 ...................
.......................................................
13
................................................................. Conditions
Boundary
Conditions
Boundary
Conditions
System
for the Number
System
............................
Matrix
16 21
.........................
22
Matrix
..........................
3.4
Output
Matrix
3.4.1
Static
Pressure
..........................................................
30
3.4.2
Total
Pressure
..........................................................
30
3.4.3
Mach
Number
.......................................
.................................................................
...........................................................
......................................................................
4.1
Square
4.2
Modified
Uncertainty
Mach
Matrix
Input
Reduction
for Downstream
14
for the System for the
13
3.3
Model
Matrix
1
Root
Model
Square
Reduction
Root
Model
.................................................... Reduction
...........................................
............................................................................
5.1
Linearization
5.2
Model
5.3
Total
Error
Reduction Error
Bound
............................................................. Error
..........................................................
..............................................................
23 30
31 32 33 34 35 35 38 39
!
6 Example Application .................................................................. 6.1
6.2 7
Conclusion
8
Appendices 8.1
8.2
8.3
39
Results
........................................................................
6.1.1
2D VDC
Inlet
Model:
Downstream
Mach
Number
Perturbation
...............
39
6.1.2
3D VDC
Inlet
Model:
Downstream
Mach
Number
Perturbation
...............
40
Discussion
39
.....................................................................
41
.............................................................................
42
............................................................................ Symbols
43
.......................................................................
8.1.1
Greek
8.1.2
Subscripts
8.1.3
Superscripts
Figures
........................................................................
8.2.1
2D VDC
Inlet
Model:
Downstream
Mach
Number
Perturbation
...............
45
8.2.2
3D VDC
Inlet
Model:
Downstream
Mach
Number
Perturbation
...............
51
8.2.3
2D VDC
Inlet
Model:
Data
8.2.4
2D VDC
Inlet
Model:
1D Model
from
2D Averaged
Data
.....................
54
8.2.5
3D VDC
Inlet
Model:
1D Model
from
3D Averaged
Data
.....................
55
Matrices 8.3.1
8.3.2
Symbols
43
..........................................................
44
..............................................................
44
............................................................
for Reduced
Order Model:
Linear
45
Comparison
Models
....................................
53
........................................
Inlet
8.3.1.1
26 x 3 Grid
....................................................
56
8.3.1.2
26 x 5 Grid
....................................................
58
8.3.1.3
26 x 7 Grid
....................................................
60
8.3.1.4
48 x 3 Grid
....................................................
62
8.3.1.5
48 x 5 Grid
....................................................
64
8.3.1.6
48 x 7 Grid
....................................................
66
3D VDC
Inlet
Downstream
vi
Mach
Mach
Number
Number
Perturbation
56
2D VDC
Model:
Downstream
44
Perturbation
...............
...............
56
68
8.3.2.1 15x 3 x 3 Grid.................................................
68
8.3.2.2 20x 3 x 3 Grid.................................................
70
1DModelfrom2DAveraged Data.........................................
72
1DModelfrom3DAveraged Data.........................................
74
9 References.............................................................................
vii
76
A Method
for Generating
Reduced
Multidimensional
Order Linear Models of
Supersonic
Inlets
Amy Chicatelli Tom T. Hartley The
Department of Electrical Engineering University Of Akron Akron, OH 44325-3904
Summary
Simulation of high speed propulsion systems may be divided into two categories, nonlinear and linear. The nonlinear simulations are usually based on multidimensional computational fluid dynamics (CFD) methodologies and tend to provide high resolution results that show the fine detail of the flow. Consequently, these simulations are large, numerically intensive, and run much slower than real-time. The linear simulations are usually based on large lumping techniques that are linearized about a steady-state operating condition. These simplistic models often run at or near real-time but do not always capture the detailed dynamics of the plant. Under a grant sponsored by the NASA Lewis Research Center, Cleveland, Ohio, a new method has been developed that can be used to generate improved linear models for control design from multidimensional steady-state CFD results. This CFD-based linear modeling technique provides a small perturbation model that can be used for control applications and real-time simulations. It is important to note the utility of the modeling procedure; all that is needed to obtain a linear model of the propulsion system is the geometry and steady-state operating conditions from a multidimensional CFD simulation or experiment. This research represents a beginning step in establishing a bridge between the controls discipline and the CFD discipline so that the control engineer is able to effectively use multidimensional CFD results in control system design and analysis.
1
Introduction
The development of inlet models for high speed propulsion systems is important because of the current interest in high speed air-breathing propulsion systems. Modeling of these systems is difficult, because the complex physical processes are represented by nonlinear partial differential equations (PDE). An accurate plant model is required to develop a control system for the plant; the more accurate the model, the better the control design. Typically these models are either based on traditional propulsion control models or CFD models.
Traditionalpropulsion controlmodelstypicallyutilizealargelumpingtechnique forthespatialderivatives sothat the propulsion systemis represented by a setof nonlinearordinarydifferentialequations(ODE). Theseequations areoftenlinearized abouta steady-state pointsothatthecontrolmodelis linear.Methods basedonthislinearODEapproach havebeendeveloped forpropulsion systems, someofwhichare:theColeWillohmodel[1],theMartin model[2],theBarrymodels[3],circuitmodels[4],andtheLaplacetransform ofthe Green'sfunctionmethod[5][6].Unfortunately thesemodelsareoftendifficultto implement, donot alwayscapturethenonlineardynamics ofthesystem,andarenot typicallyusedformultidimensional flows. Accuratenonlinearmodelsof complexflowsareusuallyobtainedfromCFDcodes[7][8]. Thesemodels canto somedegree predictthe behaviorof largeperturbations in the flow field,includingunstart,buzz, turbulence, boundarylayergrowth,et cetera.TypicallytheseCFDmodelsarebasedona largenumberof nodeswhichcanthenbeusedin afinitedifference methodtoproducea largesystemofnonlinearequations. However dueto theirnonlinearityandlargesize,thesemodelsrequirelargeamounts of computational time andtherefore arenot suitablefor controlsanalysisanddesign. An effectivepropulsionsystemmodelfor controlsystemdesignmustadequately capturethe dynamics of thesystembut alsobeof smallorder.CFDmodelsfulfill thefirst requirement, andtraditionalcontrols modelsfulfill thesecondrequirement.Therefore, a methodthat is basedon both ideasmightprovidea reasonable modelfor controlsapplications.This concepthasalreadybeenillustratedfor onedimensional CFDmodels[9]. In that paper,the development of a CFD-based linearmodelingmethodcombined with modelreductionis usedto modelthe inviscidflowof anaxisymmetric onedimensional mixedcompression inlet. In this paper,theCFD-based linearmodeling methodis appliedto theinviscidflowofanaxisymmetric multidimensional
mixed
compression
inlet.
It should
be noted
that
the whole
inlet
needs
to be modeled
in
order to accurately capture transient behavior. The CFD code PARC is used to obtain the steady-state and transient data. The inflow boundary condition is assumed to be supersonic, the outflow boundary condition is assumed to be subsonic, and the Mach number at the exit is used as the boundary condition input. The next section describes the multidimensional CFD model development which is the basis for the linear model of the inlet; development of the linearized methods A summary of the
this includes the governing equations, the development of the split flux model, and the boundary conditions. Then in section three, the linear model is derived by implementing of the previous section; an input and various outputs are also developed in this section. model reduction method and calculation of the associated error bounds follow in sections
four and five. In section six, an application a conclusion follows in section seven.
2
CFD
2.1
Model
Governing The dynamics
sional
Euler
on a mixed
compression
inlet,
and
Development
of an internal The
flow propulsion
conservative
form
system
are often
represented
of these
equations
is defined
by the nonviscous by Hirsch
multidimen-
[10] as:
of Mass:
O(/u) Conservation
is illustrated
Equations
equations.
Conservation
of the method
O(pv) +(p) o( )Oz - o
(2.1)
of Momentum:
o (;u____ A)+ o Ot
+ p) + o Ox
A) + o Oy
Oz
--
0
a (_) + a (_v) + a (pv_+ p) + o (_,o) Ot
Ox
Oy
Oz
o (p____A) + o (p______A) + a (pv_____2 + o (pw2+ p) &
Conservation
Ox
Oy
Oz
-
0
-
0
(2.2)
of Energy: 0 (pE) Ot
+ O [u (pE +p)] Ox
+ O Iv (pE +p)] Oy
+
a [w(pE + p)] =
0
(2.3)
Oz
The number of equations in the nonlinear system of partial differential equations is 5N1N2N3 where N1, N2, and N3 are the number of grid points in the x, y, and z-directions. The two dimensional form of the Euler equations is obtained by removing the third momentum equation and the spatial derivatives with respect to z in the remaining equations. Throughout this paper the three dimensional form of the equations will be implemented. If the partial
derivative
terms
are expanded,
a-_
aT(-w)
0---_+ _vhere the vector State
components,
u,
7
the conservative
Ox (-_),
aT(_) +
Oy
-_ (-_)
and
form
of the equations
can be rewritten
as:
o-Y(_) +
Oz
h (-_),
-- 0
are defined
(2.4) below.
vector: P U
ml m2 m3
-_
pw pE
The
total
energy Flux
energy
per unit
per unit
volume,
e, is defined
mass.
(2.5)
C
/ U2 "_-V2 -_-W2_ as, e = pE = p (e + 2 ]
where
e is the internal
vectors: pu pu 2 +p puv
(2.6)
puw
u (_+ p)
pv
puv
7(7)
=
pv 2 + P pvw
v(_+ p)
(2.7)
pw puw
-Y (-¢) =
pvw
(2.s)
pw 2 +p
For a perfect
gas the static
pressure
can be defined P
where
7
flux vectors
= ui" + v] + wk, are homogenous
and
=
(7-1)(e,
=
(7-
=
(7-1)
-_vP--*2N)
1) (e-
_P (7
.7))
e-p
the flux vectors
functions
as:
may
of degree
(2.9)
2
be rewritten
one in _-_, they
-7(-_)
=
9
=
in terms
of the state
can be written
variables.
Since
the
as:
A-_
(2.1o)
____+
h (7) Now the derivative following,
of the first
flux vector
with
=
respect
O_-o(-U) where
A is the Jacobian
of the
C_' to x can be represented
in quasi-linear
form
by the
-- A ,0-_ 0x
(2.11)
0___
(2.12)
flux vector,
A_ The Jacobians of the other flux vectors may be computed Jacobians into equation (2.4) results in the following partial 0-_ 0_' O'---[+ A ox where
the Jacobians
are defined
A
-u[_
+ _--_-b
+ B 0-_ 0-_ "-_y + C oz
The
substitution
= 0
(2.13)
1
-.2
(3-
0
7)u
-(7-
0
1)v
-(7-
0
1)w
7-1
-uv
v
u
0
0
-uw
w
0
u
0
- (7-
1) 72]
of the
as:
0
-u2
in the same manner. differential equation,
7ep
7- 2u2)2 1 (_2
+
- (7-
1)uv
- (7-
1) uw
7u
(2.14)
!
B
0
0
1
0
0
--UV
V
u
0
0
-v2 + 2-_-_
2
-(3"-
(3 - 3") v
1)u
--VW
0
3"-1 2
0
--UW
(3" - 1) 7 2
characteristics,
or local
]
of the second
1)u
characteristics
of the
Jacobian
of the third
(2.15)
3"v
1
0
P
2
are equal
to:
----
U
A4
=
U-_-C
A5
---- U -- C
(72 + 2w2)
3'w
(2.17)
to:
3
_--
V
A4
=
v +c
A5
=
v--
are equal )tl,2,3
(2.16)
(3"- 1)
(3 - 3") w
first Jacobian
are equal
Jacobian
1)v
- (3" - 1) vw
3
0
V
-(3"-
)tl,2,
and the
:
0
(-_2"4-2v2)-(3"-i)vw
W
)tl,2,
the characteristics
'
0
- (3" - 1) uw
eigenvalues,
' , ,
3'-1
W
-(3"-
The
1) w
v
0
0
-
- (3"-
w
0
3'_
-w
::':_i ¸'',.''LI
(2.18) c
to:
_
W
A4
=
w+c
_5
=
w -- C
(2.19)
Note
that c is the speed of sound. A system of ordinary differential equations that approximate equation (2.13) can be obtained by replacing the spatial derivative terms with finite difference expressions; then the system of equations may be integrated numerically to obtain the flow field solution. In order for the overall system to be numerically stable, the direction of the characteristics must be taken into account when
the
spatial
derivatives
are replaced.
For example
in the
axial
direction
when
the
flow is supersonic,
07 the
characteristics
the
flow is subsonic,
07
are all positive, the
signs
and
one finite
of the characteristics
difference
expression
are mixed,
can be used
and a single
finite
for the
difference
_----- term. 0x . expression
If for
--_. will create an unstable set of ordinary differential equations. If the Jacobians of equation (2.13) are s_l]t according to the signs of the characteristics, then different finite difference expressions for the spatial derivatives can be used for each of the positive and negative terms. The next section illustrates how to split the system into its positive and negative parts.
:
_: i/?_)
2.2
Split Flux Model
Thesplit flux methoddetailedin references [10]and[11]is summarized in this section.Thesplit flux methodseparates a fluxvectorintosubvectors whichcorrespond to thepositiveandnegative characteristics of theJacobian.Thesplit fluxmodelcanbewrittenasthefollowingequation, 07 Ot and
the positive
0F(7) +
0F(7)
Ox
+
and negative
0J(7)
Ox
subvectors
+
0g- (7)
0-_
+
can be cMculated f-_(-_)
Substitution
of equation
__0 -_ Ot The positive
(2.21)
+ A. + _0 _
and negative
and are calculated
into
the split
+ A-
Jacobians
Oy from
=
A+_
(-_)
=
B+_ -_
(7)
= c+_ *
flux model
oh+(-_) +
equation
O-_ _ Ox + B + O-_ Oy + B - OOy
Oz
oh- (7) +
Oz
-- 0
(2.20)
the following:
(2.21)
(2.20)
+ C+ _z
produces
+ C_
the
following
___ -_zO = 0
result.
(2.22)
satisfy, A
=
A++A
-
B
=.
B++B
-
C
--
C++C
-
A+
=
K1AilK11
B :t:
=
K2A2:t:K21
(2.23)
from:
(2.24)
c ± = K3A_K¢ _ The right
eigenvectors
of A, B, and C are defined
K1
y
as: P 2c
P 2c
1
0
0
u
0
0
pv
pv
v
o
-p
2_
2_
w
p
0
pw 2c
pw 2c
pw
-pv
P (u + 2"-c
+-_+cu
c)
P (_ 2"-_c
)
_c
Jr 3'-1
c) (2.25)
cu)
o
1
o
0
u
p
0
v
0
-p
w
0
_ 2c
2c
pu 2c
pu 2c
P (v+ c)
I(2
P (v- c)
2"_
(2.26)
2"--_
pw 2c
pw 2c cv I
7-1 0
0
1
P 2c
P 2c
0
-p
u
pu 2c
pu 2c
p
0
v
pv 2c
pv 2c
o
o
w
P (w + c) 2-_
P (w - _) 2_
pv
- pu
2
A_,
and
K3
and
the matrices
A_,
2c
+ _
A3i can be defined
+ cw
)
(2.27)
+---"7-1
as:
½(_ ± lul)
o
o
o
o
o
½(_ ± }ul)
o
o
o
o
o
½(_± lu{)
o
o
o
o
o
0
0
0
0
½(v ± Ivl)
o
o
o
o
o
½(v _ Ivl)
o
o
o
o
o
½(v± Iv})
o
o
o
o
o
½(v+_±lv+_l)
o
0
0
0
0
1(_÷_±1_+cl)
(2.28)
o 1 (_ _ _± I_ - _1)
1 (V --
C 4-
(2.29)
Iv -
½(w ± I_,1)
o
o
o
o
o
1 (_ ± }wl)
o
o
o
0
0
0
0
0
0
0
0
0
0
1 (W -[-{_1)
l(w+c_lw+cl) 0
cl)
(2.30)
0 1 (w
-- c 4- ]w -- El)
There are a variety of splittings that can be used for A. As long as the characteristics of A + and A- satisfy A = A + + A-, the splitting is valid. Once the system, equation (2.22), is split into its positive and negative
Jacobians, a differentfirfitedifference expression canbeusedto approximate thespatialderivatives foreach Jacobian.Thespatialderivatives associated with the positiveJacobians arediscretized with a backward difference operator: 0
i,j,l_
_
U i,j,k
--
Xi,j,k
--
U i,j,k
--
Yi,j,k
--
OX
U i,j,k
__
O--_,j,k
_
associated
U i,j--l,k
Zi,j,
with
the negative
0Wi,j,k
_
k --
Zi,j,k_
-
l,j,
k --
Ylj+l,k
Ouis,k
_
Xi,j,
Zi,j,k+
with
--
_
_-
._-Ci+,j,k
k
Xi--l,J,
(2.32)
-- YiS,k ui,j,k 1 --
Zi,j,
k
\
Zi,J,
k
_x
Yij-l,k
/ + B_j'k
Zi,J,
]
(._i'j'lo---_i'j'k--l._
'
can be rewritten
Xi,J,
Yi,i,k
'
This
i,j,k
\
.__C_.,k
k
k-1
difference
k
The grid point is denoted by the subscript i,j, k. The approximations for the spatial tuted into equation (2.22) which results in the following equation at each grid point
0
a forward
u _,j,_
uid,k+l--
OZ
are discretized u i,j,_
u iS+1,k -
Oy
1
Jacobians
XiT
_
l,k
u i,j,k-1
-_i+l,j,k
OX
O--_i,j,_
Yi,j--
u i,j,k -
OZ
derivatives
Xi--l,j,k
(2.31)
Oy
and the spatial operator:
U i--l,j,k
Xi+l,j,k
-Yi,j+l,k
derivatives are substiof the system.
Xi,j,k
Yi,j,k
(2.33)
]
(-Ui,j,k+_______l---_i,j,k_ k
Zi,J,k+
1
Zi,J,
Ic
]
as:
85, 0
--
Ot
Xi,j,k
+ xi,j,k
--
Xi--l,j,k
i,j,k A+ -- Xi--l,j,lc
--
U i--l,j,k
_
Yi,j,k
A_j,_: Xi+l,j,k
--
--
+ xi,j,k
Yi,j,k
Yi,j--l,k
U i,j--l,lc
--
Zi,j,k
--
Zi,j,k--1
U
i,j,k--1
_,i,k B-- +Yi,j--l,k (2.34)
Bi'J'k Yi,j+l,lo
--
4 Yi,j,k
A_J'k Xi+l,j,k
Equation conditions
--
¢,j,k zi,j,lc
-_i+l,j,k Xi,j,k
--
_
zi,j,k-1
_-
C_,j,k zi,j,k+l
Bi"-J'k Yi,j+l,k
--
--
-_,.i,k zi,j,l_
-Ui,j+l,k Yi,j,k
(2.34) represents the dynamics of the internal grid points that must be satisfied at the following locations: i
---- j=k=l
i
=
N1
j
=
N2
k
=
N3
-]-
Ci'j'k Zi,j,k-F1
_i,j,l:+l --
Zi,j,k
of the CFD model;
there
are still boundary
where N1, N2, and N3 are the total number conditions are developed in the next section. 2.3
Boundary
of grid points in the x, y and z-directions.
The boundary
Conditions
Boundary conditions can be categorized as either physical or numerical. Numerical boundary conditions correspond to characteristics leaving the domain; therefore they are determined from the interior grid points. The physical boundary conditions correspond to characteristics entering the domain and cannot be determined from the interior grid points; therefore, they must be specified. The numerical treatment for the boundary conditions follows in the next two sections.
2.3.1
Wall
Boundary
Conditions
The boundary conditions at the y and z-planes are implemented using the method of non-reflective boundary conditions [10]. When using this method, the physical boundary conditions are set equal to zero, and the numerical boundary conditions are determined from the interior grid points of the computational grid. Since the characteristics at the boundary are propagating in one direction, one finite difference equation can be used to replace the spatial derivative. The following is a general equation that can be used at these boundaries (3"= 1,j = N2, k = 1, and k = N3) for the boundary conditions. 0
Ou*_,j,k _t
A+j,k Xi,j,k
_
(
Xi-l,j,k
U
i-l,j,k
_-
_
A+ xi'j'k
_
Xi-l,J,
k
--
Xi+l,j,k
_
Xi,J,k
]
d_z_j,k
Xi+l,j,k
At the boundaries, A- are determined
For
-- Xi,j,k
U i+l,j,k
"J- K2A2bc
K21
O--_i,j,kO______ + K3A3bc
K-13
O--_i,j,kOz
A3bc are determined from the numerical boundary conditions; from the positive and negative Jacobians from the split flux method.
A2bc and
j = 1, the spatial derivative
0-_. -_y is replaced with a forward finite difference,
A2b_ must have all negative characteristics.
Therefore, 0W Oy
--_-y, A2bc, B +, and B-
W_,j+l,k Yi,j+l,k
where as, A + and
equation
(2.32),
hence
become the following:
u i,j,k
-- Yi,j,k
A2bc
0 0 0 0 0 0 0 0 0 0 0 0 O000v-c
0 0 0 0
0 0 0 0
A2bc
v O 0 0 O v O 0 OOv 0 0 0 0 0 O000v-c
0 0 0 0
,v>0
(2.36)
,v0
(2.37) W 0 0 0 0
0 W 0 0 0
0 0 W 0 0
C+
=
0
C-
=
K3A3_K31
A3bc
'
0 0 0 0 0
0 0 0 0 W--C
,W0
(2.38)
A2b
c
B+ B-
0 0 0 0 0 = =
K2A2,,cK_ -1 0
10
,v
u N_,l,k-1
-}- fNl,l,k6-_Nl,l,k+l
_ aNl,l,k6
U N_,l,k
, k = 2,''',
N3 -
1
(3.25b)
05-_ NI,_,N_ ---- dNl,l,N36
Ot
U NI--I,I,N3
+ gNI,I,N36
U NI,I,N3--1
+ aNl,l,N36
U NI,I,N3
-}- CN,,1,N36
u N1,2,N3
(3.25c) 06"_N1
,j,1 =
&
dN,,j,16
u NI--I,j,1
-{- bNl,j,16
+CN15,15-_Nl,j+l,1
05 u Nt,j,k &
dN_,j,k6_ -{-aNl,j,
Nt--l,j,k k6 u Nl,j,k
u NI,j-I,1
+ aNx,j,16
u N,,j,1
+ fg1,j,15-_Nt j,2, j = 2,''',
-F- bNt,j,k6-_ -_- CNl,j,k6
N_,j_l,
N2 - 1
k -_- gNt,j,k6-_
u Nl,j+l,k
_- fNl,j,k6
(3.25d)
Nt,j,k_
1
u Nl,j,k+l
(3.25e)
j=2,...,N2-1,k=2,...,N3-1
-..+
06 U Nt,j,Na --
Ot
dNt,j,N36
u N_-I,j,Na
-]-aN,,j,N36-_
Nl,j,N
"-[-bNx,j,N36 3 + CNl,j,N36-?_
U NI,j--I,N3 Nt,j+l,N3,
"{- gNt,j,Na6 j =
2,...,
U NI,j,N3--1 W2 -
(3.25f)
1
06 U N1,No,1 -- dN_,N2,16
Ot
U NI-I,N2,1
"_ bN_,N2,16
U N1,N2-1,1
-_- aNI,N2,16
u Nt,N2,1
-{- fN1,N2,16
U Nt,N2,2 (3.25g)
05-_ N_,N_,k --
Ot
dNt,N2,k6-?_
Nt-l,N2,k
-[bN_,N2,k6--_
Nt,N2--1,k
-5 gN1,N2,k6_
+aN_,N2,kg U N_,N2,_ + fgt,N_,_5 U N1,N_,k+I, k = 2,--.,
06"_
Nt ,N2 ,N3 Ot
=
dNt,N2,N36 +aNt
,N2
U N_-I,N2,N3 ,g3
6
"U'_NI
,N2
+ b N_,N_,N36
-+ U Nx,N2--1,N3
N1,N2,k-1
N3 - 1
+ gNt,N2,N36
(3.25h)
u N1,N2,N3--1 (3.25i)
,N3
22
•
j ..... _
':.
>..
J
i
•
Outflow
Boundary
J J
f
Conditions
Equation
3.25a
Equation
3.25b
Equation
3.25c
Equation
3.25d
Equation
3.25e
Equation
3.25f
Equation
3.25g
Equation
3.25h
Equation
3.25i
These boundary conditions are included as modifications to the system matrix x-direction. They will be discussed at length in section 3.3 where the coefficients and the input matrix, B, is derived.
3.3
Input
Matrix
for
Downstream
Mach
at the last grid of the equations
point in the are defined,
Number
When the flow at the compressor face is subsonic, there are four numerical boundary conditions and one physical boundary condition [10]. The numerical boundary conditions are associated with the positive characteristics, and the physical boundary condition is associated with the negative characteristic. The implementation of the downstream Mach number as a boundary condition input is derived below. To begin with, section 2.3.2.
take
the
inverse
of (2.25),
and
then
23
partition
the matrix
following
the
procedure
from
i _, i/, ,¸ -_:_,,_*_:•
-1
(Kll)
N
K1 1 =
(3.26) (Kll)
P
Here, (K_-I)N is the _st fourrowsof K_-1 , and (K_-I)P is the last row of Ki-1. The physicalboundary condition
equation
is: Bbc = M - Minput
where
Minpu
t
is the prescribed
boundary
condition
input
(3.27)
or set point,
and
M is defined
as:
M = x/u2 + v2 + w2 c If M is rewritten
in terms
of the state
M
Taking
the partial
derivative
OM Oral'
OM Ore2'
OM
of Bbc(--_)
with respect [ OM
-- [
Op
Oe
OM
OM
Oral
0rn2
0rn3
v_/ml
2 + m2 + m 2 (¢_
p-_2)
_/7
OMoe ]
(2¢ (7-
1)-
(3.30)
/7p-v -_2) +
p--6.2
7 (2_- p-_)2 (1- 7) Ipl
-
(2e (7 - 1) - 7p-_ 2) + p-_'2
V
};
sg.(p)
7 (2¢ - p-F 2) (7 - 1) vim 2 + m 2 + m 2
-
(2¢ (7 - 1) - 7p-_ 2) + p-F 2
V 7 (2¢ - p-_2)
OM
OM
below.
x/_m2/7
Om3
the following: (3.29)
I
Oral
OM
becomes
to _-* yields,
OM "-_-c are shown
vF2ml/7
OM
(3.28)
+ m22 + m2
OM Ore----3'and
Op
OM
m2, m3, and e, equation
v/2X/ml2
0-_ OM 0-7'
p, ml,
=
OBb_
where
variables
(3.28)
};
sgn(p)
(3.31)
(7 - 1) v/m 2 + m 2 + m]
v_m3_/7 (2¢(7- i)-pTp7 _) + pV 2 s_(p) 7 (2¢- j_)
(7- 1)_/._ + ._ + m]
v_fm_ + ._ + m_ 7 (2¢(7 - i)-pTp_ _) + p_ 7 (2¢ - p_2) 24
2 (1 - 7)
sgn (p)
Therefore,
for
the
downstream
Mach
number
boundary
condition,
(Kll) nl
and
L2 is defined
0
U
N1 ,j,le
(3.32)
as:
(2.48),
.q_
the
L71L2
compressor
face
ANI,j,k
+
Ot
N"J'k
q-
condition
Lll
i/igl / AOt k,
L2
to
be
(3.33)
boundary
BN1
condition
,j,k
may
be
written
,j,k
in the
+
BN1
= L-_ 1 Oz
small
model,
CN_ ,j,k
+ O_--_
O_-_
N,,j,koy
(3.34) --Bb_
perturbation
,j,k
Nl,j,k_x
as follows.
CNl,j,k
q-
Oy
implemented
06-_
g
--_
Ox
For this boundary shown below.
O_-_
as:
--
L2
equation
is defined
N
(Kll)
From
L1
it
NI'j'k
must
)oz
be
linearized
= L11 [
as
--SBb 35)
6Bbc
is calculated
as:
5Bb_and
can
be
rewritten
(3.36)
as:
5Bbc
Now substituting
OB_ 5-_ OBb_ _ + OMi,_putSM_n_,_t
equation 06-_N
OM Op
=
OM Oral
OM Ore2
(3.37) into equation
1 ,j,k O_
--
L11L2
-L[
1L2BNI
OM 06-
,j,k
( \
_ _ U Nl XNi
'j'Ie ,j,k
--
O6-'_ Ni,j,k Oy
6Mi,_p_t
] 6--_--
(3.35) and replacing " 'J'k
ANl
OM Ore3
06_ Nl,_,k Ox with equation
-_--U Nl-1'J'k XN i --1,j,k
Lll
(3.37)
(2.31) yields:
_ /
L2CN_,j,k
06_
g_,j,k Oz
04x5
-L11
+L{ -1
0
aM
aM
OM
OM
OM
Op
Oml
Om2
Om3
OC
0
0
0
25
1
_Minput
_--_
N1 ,j,k
(3.38)
this
can be rewritten
05"-_Nl,j,k (_t
_
as the following:
LllL2
(
ANI,j,k -- XNI_I,j,
XNI,j,k
k
5_gl,j,k
ANx ,j,k
+
XNI,j,k
--
_--*_ q-'?N1
XNI--I,j,k
+ BN_'J'k YNI,j,k
--
5 u Nl,j-l,k
....
YN_,j--I,k
\YNI,j,k
-- YN_,j--I,k
YNt,j+l,k
--
YNI,j,k
]
5-_t
Nl'j'k
+ BNI'J'k "5-?_ Nl,j+l,k YN_ ,j+l,k -- YN1 ,j,k
"_-
C_Ii'J'lz -- ZN1 ,j,k--1
ZN_,j,k
5-_
(3.39)
Nl,j,k-1
_( ZN1 ,j,k
ZNI,j,k+I CNI'J'k--
-- ZN1 ,j,k--1
04x
-L71
OBbc
[
5--_NI,J,k+L11
0
0
The terms for the input matrix, B, are obtained from the (3.48) are used to modify the system matrix, (3.10). 05
)
ZNl,j,k
0
5"_
Nl,j,k
0
1
coefficient
CNI'j'k-ZN1 ,j,k+l
5-U
ZNI,j,k
NI,j,k+I
5Mi,_p_t
on 5Mi_t.
Equations
(3.40)
through
U N1,1,1
Ot
--
dNi,1,15
dNl,l,1
----
L11L2(
u
NI-1,1,1
-{- aNi,1,15
u
NI,I,I
-[- CNI,1,15
)
U NI,2,1
-_- fNl,l,15
(3.40)
"U, NI,1,2
-ANI'I'-----_I k XNl,l,1
-- XNI_I,1,1
I
AN_,I,1 -- XN1--1,1,1
.]
04×5 aNl,l,1
:
L11L2
XN1,l,1
_}_ YN1,2,1BNI,I,1 -- YNI,I,1
-{- ZNI,I,2
-- ZNI,I,1
]
--
L{1
OB b______
o-_ CNI,I,1
_-
-nlln2
_ B----N1-L'I'I \YN1,2,1 -- YNI,I,1
)
_kZNI,I,2
]
fNl,l,1
05-_
-- ZNI,I,1
N1,1,1z
Ot
--
dN1,1,/¢5
u Nl-l,l,k
-[-CNl,l,kS-_yl,2,1c
dNl,1,]c
:
51152(
-{- gNl,l,k5 _-fNl,l,kS_Nl,l,k+l
A---N1 1'----_' _ \ XNI,I,k
-- XNI_I,I,k
( ggl,l,k
=
L11L2
u
\ zg_,l,k
-- ZN_,l,k-1
26
/
Nl,1,k--1
-{- aN_,l,k5 , k =
2,...,
u
NI,I,H
g 3 -
1
(3.41)
:
L-_I L2 {
aN_,l,k
AN_,I,k
\
_}
+
CTv,,1,k
-L{I
CNl,l,k
--
L2 (
fNl,l,k
--nll
[o4x5]
_ L11
1
ZNI,I,k+I
B_l--'l'k
]
--
]
\YN1,2,k
YNI,I,k
--
ZNl,l,k
/
OBbc
n2
k ZNI,I,k+I
--
Zgl,l,k'/
O5-_ N_,I,N_ dN_,l,N36
u NI--I,I,N3
-_- ZNt,l,N36
J-aN_,I,N3__NI,I,N3
LllL2
__).
--_ CNI,I,N3(_
( _,XNI,I,N
(3.42)
U NI,I,N3--1
•
dN_,I,N3
U N1,2,N3
JN....!-I'I'N3 3 -- XN__I,I,N3
)
gNt,l,N3 _kZNI,I,N3
--
ZNI,I,N3_
1
BN_,I,Na aN_,I,N3
Lll L2 (
ANt'I'Na 3 -- XN__I,I,N3
XNI,I,N
q_ YN1,2,N3
--
YNI,1,N3
04x5
) -- n_-i
ZNI,I,N3 CN_I__,I,N3 ZN_,I,N3--1
-L11L2
CNI,1,N3
\ YN1,2,N3
05 u N15,1 &
--
dNl,j,16
u
-{-aNI,j,15
dN1 ,j,
1
=
51152(
=
LllL2{
NI-I,j,1
-- YN1,1,N3
"4- bNI,j,16"_
U NI,j+I,1
\ XNI,j,1
OBb_
BNI'I'N3
{
]
NI,j--1,1
-_- fN_,j,15-_NI,j,2,
JNI,j,1 -- XNI_I,j,
1
-J- aNl,j,16
u
j = 2,...,
N2
(3.43)
NI,j,1 -
1
)
B+ bg_,j,1
ant
NI,j,1
]
/ ,j, 1
=
illL2
(
+ BNI
ANI,j,1
\
XNI,j,1
--
XNI_I,j,
,j+l,1
-- YN15,1
1
YNI,j,1
--
,j,1 YNI,j-I,1
04×5
BN1,5,1 YNI
..... /
YN1,2,k--YN_,I,k
+
ZNl,l,k_
.'
BN_,I,k
XN_,I,k--XNI--I,I,k
ZNl,l,k
'
CN1 +
27
ZNI,j,"-'_-- -
,j,1 Z%l,j,1
]
--
L11
OBb_
i:i
_. ×.__.>x._
:
E
-2000
x
............... _................ ................ i................ i* ...... xx_/,¢_l !
.&
_
q
i
i
i
xx x xxXx_xX_
I J
i :: i i x x× X×o× / ............... !................ ................ i................ i__×i:x_i° ...........,1 ROi=l_x
i
-6000 -6
-5
-4
-3
-2
-1
Real
Figure 14: x FOM Eigenvalues/o
51
0 x 104
ROM Eigenvalues
Output Response 0.33
oooo_oooo_
to a -3% !
Step in Compressor
!
Face Mach Number
!
I........ i........................................ ......... .......... i......:......... i......... 0
0.005
0.01
0.015
0.02
13.5
_ _ :: i 13t ........i......... i....oo_
125 / 0
i 0.005
i 0.01
14.2
i 0.02
_ i
_
:: oOy_
/ °°°°0°°°°_°'"
136 r 0
i 0.005
i 0.025
i 0.03
i 0.035
0.04
0.045
0.05
i 0.04
i 0.045
.J 0.05
:
I
....... i.......... ::......... i......... i........ -1 / ......i......... _......... ::......... ::........ 1
statid Pressu!e
!
f 0.01
0.035
_ _^'OOOO6OOOO6OOOQ6OOnnhnnnnrh : OO oU'+'- ' '. :
4- .................. :;...... _o_ 13.,r ........ _......... ;-./-._ /
0.03
_ _ _ _ _ /' _o_oooo6oooo_,o_,_,_,_, .... _.... _, ....._.......... i...... : t
i 0.015
_ :
I
0.025
!
_ 0.015
:
@ X/Bc=4.9262
:
_ = _ 0.02 0.025 0.03 Time (seconds)
Figure 15: o PARC, 103 x 15 x 8 Grid/Linear System
::
_
_
_
i 0.035
i 0.04
i 0.045
/
/ 0.05
Linear ROM 20 x 3 x 3 Grid Eigenvalues
:
X
X
6ooo x. ! ............ x..i..x............................ 8000 r |/ ............................................. x"_ x! x i 4000 ................
2000 ................
!
i
i
i
:................
: ..............
:................
i
i
!x
:
•
•
_ "_,
..........
x
i................
i
ix x
_< xX.x_
Xx
.............
X" IX_X x_x_
_ .......
x :x xX._ _Cx : .........
× xx
.Xx_.,_.._,...:!_
.. -
xx ,'_
-6ooo ................ !................ :............ ×i ............. x._:x.-_.-_ ....:................ ROM=17_ states
x
i i
i x I
-80006
-5
I
I
-4
-3
I
-2
-'
Real
Figure 16: x FOM Eigenvalues/o
52
0 x 104
ROM Eigenvalues
8.2.3 2D VDC Inlet Model: Data Comparison
2D VDC Inlet Linear Model Results 0.435
o 0 0
0.43 ....................................................................................................
.._
0.425 .....................................................................................................
0 m
W
0.42 .....................................................................................................
0.41_................ :.............. o! ................ i............ o................... ::........... o
0.41 200
i 400
i 600
i 800 Number of States
1000
i 1200
Figure 17: 2D VDC Inlet Linear Model Comparison
2D VDC Grid 26 26 26 48 48 48
x x x x x x
Number 3 5 7 3 5 5
Inlet
of States
312 520 728 576 960 1344
Data Error Bound 0.43461 0.43382 0.43342 0.41472 0.41449 0.41424
53
1400
• "i _" '" .i ..¸
8.2.4
2D VDC
Inlet
Model:
ID Model
Output
!
Response
from
to a -3%
2D Averaged
Step in Compressor
•
Data
Face Mach
Number
!
.......... i................... [................... i................... i................... i................... i................... i................. .i i i i i i i i I 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16
/ 0 291 0
13'5 /
0
0.02
146 /
0.04
i
'
i
:
144r ...... _i |
(_
148
i
0
0.08
0.1
0.12
0.14
0.16
,;................... !................... !.............. !................... !...................
........................ _........................ i............ Static Pressure
:,
0.02
Figure
0.06
i
0.04
@:X/Rc=5
i
0.06
0.08 Time (seconds)
01
i
i
i
i
0.1
0.12
0.14
18: ID Linear Model Based on 2D Averaged Linear
2.5 x 104 I
System
I
0.16
Data
Eigenvalues I
I
I
I
............................. x ×x]x_×_xx×X×x×××_x_......i................. i
i
x!
!
i
!
i
i
! x :
: :
x i :
i
i
x×
i
:
=
1.5 :
ix
i
x!
X
: _"_Xxy_, :X "
/
........ _,_ ................................. _,x_-×_.......... _x: ..... -_.-t !x : xx . x i _!xx : XxOl :
0.5
:
:
x:
,,
:
0
:
,
:
X
:
:
:
:
X
,x , x,
...................... .
X ..............
: ............................... ROM=13
-2.5 -10
'
:
X:
X X :x
X
X,_,_ : _
X
'
: xi >1 x x :x_ ,_ : : .xx!xx_!,_ ..... x.x..i ..... _...i.. x. : : x _"
