Computation of the inviscid supersonic flow about cones at large angles of attack by a floating

96 Pages·2017·1.57 MB·English

by  James Edward Daywitt

Iowa State University Capstones, Theses and
Retrospective Theses and Dissertations
Dissertations
1977
Computation of the inviscid supersonic flow about
cones at large angles of attack by a floating
discontinuity approach
James Edward Daywitt
Iowa State University
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Recommended Citation
Daywitt, James Edward, "Computation of the inviscid supersonic flow about cones at large angles of attack by a floating discontinuity
approach " (1977).Retrospective Theses and Dissertations. 6064.
https://lib.dr.iastate.edu/rtd/6064
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University Microfilms International
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DAYWITT, James Edward, 1947-
COMPUTATION OF THE INVISCID SUPER­
SONIC FLOW ABOUT CONES AT LARGE ANGLES
OF ATTACK BY A FLOATING DISCONTINUITY
APPROACH.
Iowa State University,
Ph.D., 1977
Engineering, aeronautical
University Microfilms international, Ann Arbor.M ichigan 48io6
Computation of the inviscid supersonic flow about cones at
angles of attack by a floating discontinuity approach
by
James Edward Daywitt
A Dissertation Submitted to the
Graduate Faculty in Partial Fulfillment of
The Requirements for the Degree of
DOCTOR OF PHILOSOPHY
Major; Aerospace Engineering
Approved:
Signature was redacted for privacy.
In Charge of Major Work
Signature was redacted for privacy.
For the Major B^partment
Signature was redacted for privacy.
For the Grad :e College
Iowa State University
Ames, Iowa
1977
ii
TABLE OF CONTENTS
Page
NOMENCLATURE iii
CHAPTER I. INTRODUCTION 1
CHAPTER II. GOVERNING EQUATIONS—CHOICE OF VARIABLES 9
Reference Mesh 11
Floating Discontinuity Points 15
CHAPTER III. FINITE-DIFFERENCE METHOD 21
Special Discretization Formulas 21
Reference mesh points 21
Temporal derivatives 22
Spatial derivatives 22
Discontinuity mesh points 25
Stability Analysis 25
Accuracy and Convergence 26
CHAPTER IV- INITIAL CONDITIONS 28
Bow Shock Shape 30
Distribution of Flow Variables. .... 32
CHAPTER V. BOUNDARY CONDITIONS 35
One-Sided Differencing 35
Cone Surface 36
Inflow and Outflow Meridional Boundaries 42
Syirunetry Planes 43
Discontinuities 44
Detection and monitoring 44
Propagation 46
Variable area effect 51
CHAPTER VI. RESULTS AND DISCUSSION 53
CHAPTER VII. CONCLUDING REMARKS 56
ACKNOWLEDGMENTS 58
REFERENCES 59
APPENDIX A. SHOULDER REGION CALCULATION 74
Dependent and Independent Variables 74
Finite-Difference Method 75
Stability Analysis 77
Surface Boundary 78
Shock Boundary 80
APPENDIX B. INTEGRATION ALGORITHM 83
iii
NOMENCLATURE
a Speed of sound
a,b,c Least-squares coefficients for crossflow shock upstream
surface pressure
a,S,c Ratio of , and 1 to
A,B,C Ellipse shape parameters for initial bow shock estimate
A3 Vector of flow variables in the nonconservation-law form
of the Euler equations
[Aj] 0-derivative coefficient matrix in the nonconservation-law
form of the Euler equations
[Ag] ç-derivative coefficient matrix in the nonconservation-law
form of the Euler equations
b,c Coefficients in surface total energy as a function of
pressure equation
B Parameter in meridional clustering transformation
[Bj] X-derivative coefficient matrix in the discontinuity aligned
nonconservation-law form of the Euler equations
[B2] Y-derivative coefficient matrix in the discontinuity aligned
nonconservation-law form of the Euler equations
CN Courant parameter
d Vector of dependent variables in the nonconservation-law
form of the Euler equations
d,e,f Least-squares coefficients for crossflow shock downstream
surface pressure
e Total energy per unit volume
E R-dependent vector of conservation-law variables
E' z-dependent vector of conservation-law variables
F X-depende;it vector of conservation-law variables
F X-dependent vector of conservation-law variables
iv
F 0-dependent vector of conservation-law variables
F' r-dependent vector of conservation-law variables
F* Ç-dependent vector of conservation-law variables
g,h,i Least-squares coefficients for crossflow shock upstream
surface Mach number
G Y-dependent vector of conservation-law variables
G Y-dependent vector of conservation-law variables
G ^-dependent vector of conservation-law variables, spherical
polar coordinate system
G' dependent vector of conservation-law variables, cylindri­
cal polar coordinate system
G* ^-dependent vector of conservation-law variables
H (T,X,Y) source term vector of conservation-law variables
H (T,X,Y) source term vector of conservation-law variables
K (t,R,0,(>) source term vector of conservation-law variables
H' (z,r,(p) source term vector of conservation-law variables
H* ($,Ç) source term vector of conservation-law variables
î_,î.,î. Unit vectors in spherical polar (R,6,*) directions
K u 9
î^,î^,î^ Unit vectors in cylindrical polar (z,r,^) directions
[I] Identity matrix
k Nondimensior.alization parameter
L ^-function in Newton-Raphson iteration for (})~
min Minimum
max Maximum
M ^ Crossflow Mach number
cf
Mi^^^ Crossflow shock upstream normal relative Mach number
M Free-stream Mach number
V
n Unit vector normal to cone surface
c
fU, Unit vector normal to crossflow shock
s
S 8-component of unit vector normal to crossflow shock
u
n ij>-coirponent of unit vector normal to crossflow shock
9
N Scalar in the equation for the cone surface tangent vector,
$
p Pressure
PrP/UjVjWjq Surface flow variables in the corrector step following the
nonisentropic Abbett scheme turn
q Velocity modulus
q~ Crossflow shock speed normal to the shock
q^_ Crossflow shock acceleration normal to the shock
q^g Vortical singularity speed
q^g- Vortical singularity acceleration
r^ Cone cylindrical radius
r^ Bow shock cylindrical radius
r^^ Slope of the bow shock in the ^-direction
R,9,4) Spherical polar coordinates
R Terms in the crossflow shock acceleration equation
S Entropy
t Time
t^ Unit vector tangent to cone surface in z,r-plane
T Vector tangent to cone surface in z,r-plane
u,v,w Velocity components in spherical polar (R,8,^) directions
u,v,w Velocity components in cylindrical polar (z,r,ç) directions
uj Upstream velocity component normal to a shock
vi
U2 Downstream velocity component normal to a shock
U T-dependent vector of conservation-law variables
U T-dependent vector of conservation-law variables
U t-dependent vector of conservation-law variables
X,Y,T Reference mesh transformed independent variables
X,Y,T Clustered reference mesh transformed independent variables
z,r,<p Cylindrical polar coordinates
a Angle of attack
Y Ratio of specific heats
6^ Dimensionless bow shock standoff distance at the windward
symmetry plane
Dimensionless bow shock standoff distance at the cone
shoulder
5 Dimensionless bow shock standoff distance at the leeward
symmetry plane
Ax,AY Computational reference mesh spacing
Ax,AY Computational clustered reference mesh spacing
A8,A6 Physical plane mesh spacing
AO Shoulder region integration marching step size
Av Incremental turning angle
AÇ Shoulder region computational mesh spacing
AT Time step for reference mesh integration
AT Time step for clustered reference mesh integration
e Fraction of computational mesh spacing
4,^,0 Bow shock aligned, cylindrical polar transformed independent
variables
0^ Cone half-angle
vii
6^ Bow shock shape
6^^ Vortical singularity position
K Parameter in r_, equation
p Magnitude of the gradient of the crossflow shock equation
p Density
a Projection of the slope of the characteristics
Lower meridional boundary
(p~ Crossflow shock shape
Crossflow shock speed in the ^-direction
4)^ Upper meridional boundary
0 Meridional independent variable in reference mesh clustering
transformation
jp Meridional clustering parameter
w Theta clustering parameter
Subscripts:
c Cone surface
cf Crossflow component
1 Index for discontinuity alignment transformation
J MacCormack algorithm index for transformed 8-direction
differencing in reference mesh
K MacCormack algorithm index for transformed ^-direction
differencing in reference mesh
L Lower meridional boundary
max Maximum
o Circular theta outer boundary
o Meridional plane at center of meridional clustering
rel Relative to a moving discontinuity