A three dimensional analysis for fully viscous, subsonic, compressible flow is evaluated. An approximate form of the Navier Stokes equations is solved by an implicit spatial marching technique. Calculations were made for flow in a circular S duct and in the F 16 inlet duct. The computed total pressure contours and secondary flow velocity vectors are presented. Qualitative comparisons with experiment are shown for both ducts. The analysis is used to show how the cross section transitioning in the F 16 inlet suppresses the development of a secondary flow vortex

Anderson, B. H.

Towne, C. E.

English

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NASA Technical Memorandum 81672
Numerical Simulation  of how
in Curved Diffusers with
	
i	 a
Crass-Sectional Transitioning Using a
Three-Diinensionai ,Viscous Analysis
(NASA-TM-81672)
 NUMERICA% SIMULATION OF
	 N81-152391	 i
FLOWS IN CURVED DIFFUSERS WITH
	 j
i	 CROSS-SECTIONAL TRANSITION,ING USING ,A
1
TIIRSE-DIMENSIONAL VISCOUS ANALYSIS (NASA)
	 Unclas y^
19 p HC A02/MF AU1
	 CSCL 20D G3/34 29693
C. E. Towne and B. H. Anderson
Lewis Research Center
	
x
Cleveland, Ohio
1 F
	
n	
y^
Prepared for the
	
111! 
CC42r"mT 
2aP	
1
Nineteenth_ Aerospace Sciences Meeting
f	 sponsored by the American Institute of Aeronautics
and Astronautics 	 t
St.. Louis Missouri, January 12=15, 1981
PJASA
v
	
.ct ',	 y'	 useybb.ak"dad"!'''.Wad^9'Lktl&Fdt.W4wut...ir.... v..>.n 3 	 ..	 . ..	 .
,^n
^n
a►a
NUMERICAL SIMR)IMION OF FLOWS IN CURVED DIFFUSERS
WdT,II CROSS-SECTIONAL. TRANSITIONING USING A
THREE-DIMENSIONAL VISCOUS ANALYSIS
by C. E. Tm nie* and B, H. Anderson**
National Aeronautics and Space Administration
'Lewis Research Centex
Cleveland, Ohio 44135
pI„D	 one-dimensional pressure correction
p3-D	 three-dimcnsionnl inviscid pressure field
ul,u x ,u3
	local Cartesian velocities
xl ,XV^ 3	 local mrtesi nn coordinates
ylry2vy3
	
curvilinear computntional coordinates
d	 Kronecher delta
P t	effective viscosity
P	 density
volocity potentinl for irrotntional component
of secondary flow
strer.m function for rotational component of
secondary flow
n	 stienmwise vorticity
INTRODU CTION
Three-aimensionnl subsonic diffusers are relatively
com►non in modern ni-brenthincg propulsion systems. The shape
of the dust crass--vIction cony vary in the streamwise
direction, and offset bends are often present, Strong
*Aerospace Engineer; Member, AIAA,
**dead. Aerodynamics Analysis Section; Member, AIAA.
4
isecondary flows can thus be generated. In addition, since
the flow is diffusing, the boundary layer may grow until its
thickness is comparable to the duct radius. These phenomena
have important effects on overall diffuser performance and
engine4nlet compatibility.
Conventional boundary Layer methods cannot properly
nnaly--e these complex flows. n complete three -dimensional
Navier -Stokes analysis could be used, but extremely large
Amounts of computer time and storage would be required for a
detaile3 solution, thus precluding the use of such an
analysis for routine calculation using present Computer
technology,
However, if it is assumed that a primary flow direction
can be identified, certain approximations can be made in the
Havier-$to{tes equations, resulting ill a set of equations for
fully viscous, subsonic, compressible flow that can be
solved by forward marching, one such method, designated
PEPSIG, was developed by Briley and McDonaldl , and recently
modified by Levy, McDonald, Briley, and Kreskovsky 2. This
paper presents the results of the first in a series of
studies to evaluate the PSPSIG analysis.
coy xE NING ERuAT19HS
In this analysis, the flow is computed by a spatial.
marching procedure which solves an approximate form of the
Havier-Stokes equations. Three basic assumptions are made.
First, it is assumed that the flow is primarily in the
direction of the duct centerline, with transverse secondary
flow. Second, the pressure field is represented by the sum 	 .
of a previously determined three-dimensional pressure field
and a one-dimensional pressure correction computecl as part
of the marching analysis. And finally, second derivatives
in the primary flow direction are assumed negligible.
2
^.
The derivation of the governing equations actually
solved in the PEPSTG analysis is Fairly complex, and has
been presented elsewhere 2 . For completeness, however, a
brief discussion is presented here. The equations are first
written in local Cartesian coordinates x l, x 2 , and x31 with
corresponding velocities u l, u 21 and u 3 . The x3 direction
is the primary flow direction, and is aligned with the duct
centerline at each marching step. The x l and x 2 directions
define the transverse, or secondary flow plane. The
equations are then transformed into general non-orthogonnl,
body-fitted, curvilinear coordinates y l, y2 1 and y3.
The x 3 momentum equation is then given, using tensor
notation, by:
ay i au
3 	_ aY i. ap3-D	 aY3 apl-D
"J axi ay i	 ax3 ayi
 - 
ax3 aY3
r
aYi a	 ayn 
BU 3 
aYn a'uj.
axi aYi
	 L	 J3 VBXj Yn	 3 yn
- 
2 
a- a	 ayn
 a ll
	
(1)3 3j (1 k3) ark By 
Here P3-D '_ p3-D (xl,x.,,x3) is the known three-dimensional
pressure field and P1-D G PI-D (x3 ) is the correction computed
during the marching solution. In the present study, P3-D was
computed using a three-dimensional potential flow analysis .
The differential continuity equation is written in
terms of a scalar potential ^ as:
a
ri a	
ayk	 ayk 2 a 2
ax  ayj P axi ayk + P axi By 
k
aYl ay 2
	 3 2	 _ aYn aPu3
	
+ 2P ax  axi aylay "
	ax  ay 
3
F
r
(2)
.....^_.,... ,..^.....rY:,..,^,:_w
	
...:>._..^....,..,^. v;-,,.,..x^*se;^,r,^^^,.,?s-r.v'a+-^+ar,^:«^w..rev:aa:-..w..^,,r^?art^tul.n^^a*?"^d:W.^w 3>^x,w^d+r..^c«-^^w&t^.st.r:x...,^.:..._..r....
The x l and xx momentum equations are combined to form
an equation for vorticity transport in the x 3 direction and
n cross-plane stream function equation. After some
simplifications, the vorticity equation is given by:
	
an	
au "t^3 	3 R 3
	
u^ 
axJ	 S2^ axi 	P ax2	 C3)
i
and the stream function equation by:
2
8.	 a 1 ayk., 1 ayk
ax  ay3 p axi
 2yk prIX 	 ay2
k
2 By, By
	 a2J^
P ax q axi ayya y2
 . 
-^3
	
C4)
The energy equation is eliminated by assuming constant
total. enthalpy. A mixing length turbulence model is used,
with the distribution of mixing length given by the Buleev
formula 3 . The boundary conditions used 'result in no-slip at
the walls, and symmetry at the symmetry plane.
The above equations are solved by forward marching from
an initial station, where the flow profiles are known, using
an implicit numerical technique. Details of the solution
procedure are given in references 1 and 2.
COMPUTED RESULTS
gircular S- Duct
As an initial test of the ability of the PEPSIG
analysis to correctly predict the basic physical phenomena
present in curved ducts, a simple circular S-duct was run.
The duet had a diameter of .61 m. (2 ft.) and the centerline
was offset one diameter in a streamwise distance of five
diameters. The flow was laminar, with an entrance Mach
t"	 4
number of R2 and a Reynolds number based on diameter of
2000. The initial boundary Layer thickness was 3.0 cm. (.1
ft,.). This case was run with a 20920 mesh in the
cross-plane, and 41 streamwise stations. The calculation
required 16 minutes on an IBM 370/3033.
computed total pressure contours are shown at several
stations along the duct in figure 1. The boundary layer
growth along the top of the duct in the first bend can be
clearly seen. In the second half of the duct, this region
of low energy flow expands greatly, forming a pocket of low
total pressure at the top of the duct. This phenomenon has
been seen in several experiments 4y5 . A comparison between the
analytical results and the results from one of these
experiments  is presented in figure 2. Here total pressure
Contours at the --nit Plane are plotted for both cases.
Although only a qualitative: comparison can be made since the
tested case was for a different geometry and flow conditions
than the computed cue, the basic physics of the flow appear
to have been computed correctly.
The mechanism by which the pocket of low energy flow is
formed has been presented by nansod and Bradshaw 
4. 
This
mechanism is present in the analytical results, and can be
explained by examining the cross-flow velocity vectors, as
shown in figure 3. Except for the results at station 5 ►
which is 1/4 of the way through the first bend, the vector
plots are shown at the same stations as the total pressure
plots in figure 1. In the first half of the duct (stations
1-21), the classic secondary flow pattern for flow through a
curved pipe is set up. The low speed boundary la yer fluid
migrates circumferentially from the pressure surface toward
the suction surface, and the essentially inviscid core flow
responds to centrifugal forces. A pair of contra-rotating
vortices is thus established. In the second bend (stations
21-41), these vortices are intensified, and become centered
5
,near the top of the duct. As the low energy fluid
approaches the symmetry plane from both sides, it erupts
from the surface forming the pocket of low total pressure
chnrnoteristic of S-duct flows,
n
To evaluate the ability of the analysis to compute the
flow thro ', wh a realistic diffuser configuration,
onlculations ware made for subsonic flow in the F-16 inlet',
A schematic diagram of this inlet is given in figure 4,
cross -sections are shown at various stations along the
inlet. The stviti.on numbers are fuselage stations from
reference 6. The inlet has n generally 5-shaped centerline,
Faith. the fir st and second bends separated by a straight
section. The total offset is .616 m, (2.02 ft,) and the
length from the throat to the engine face is 4.74 m. (15.55
ft.). The cross-section shape transitions from slightly
smile-shaped nt the throat to circular at the engine face.
The overall area ratio is 1. 35.
The non-orthogonnl, body fitted mesh generntor
currently used in the PSpSIG analysis requires
super-olliptin cross -section shtipes 7 . meshes can thus be
generated for ducts with cross-sections that vary from
nearly rectangular or square (with a large super -ellipse
exponent) to elliptical or circular (with an exponent of 2).
The actual. F-16 cross
-
sections could therefore not be
modeled exactly . Instend, the following procedure was used.
At each streamwise station, the major axis of the
super-ellipse was chosen to match the maximum spanwise
dimension of the actual cross-section. The manor axis used
was the gnpwise distance in the symmetry plane of the inlet.
The super-ellipse exponent was then computed: to give the
correct wren, but was not allowed to fall below 2,
corresponding to an ellipse. The result was the baseline
configuration shown in figure 5a.
6
i
*+^u^e+,.ie.
	 1.n-•w,,,.*aiat-Pd ^Fv;ri:^.^!^3a-s+itsr.•'y-,,,`.,^e.Cn+.r^s^?!wr"+^`1i'i^t°.`?ra^-
.A calculation was made for turbulent flow in this
configuration at a Reynolds number per meter of 1,44x107
(4.4x10 6 per foot) and n free stream Mach number of .9. The
initial boundary layer thickness specified was .91 cm, (,36
in.). Thee flow conditions correspond to one of the test
conditions of reference 6, A 24x24 mesh was used in the
ccross"pinne, with 250 streamwise stations. The calculation
required about 2 1/2 hours on an IBM 370/3033.
Computed total pressure contours are shown at several
stations along the duct in figure 6. Boundary Layer growth
around the entire circumference is evident through the first
half of the duct, About halfway through the duct, the
boundary layers at the top and bottom start growing more
rapidly, while near the sides the boundary layer becomes
thinner.
The surprising aspect of this calculation was the
degree of symmetry between the contours in the top and
bottom halves of the duct. Based on the S-duct calculations
presented earlier, as well as experimental results for
S-ducts, it had been expected that secondary flow vortices
would be generated by the centerline curvature and would
cause a shift in the total pressure contours. A comparison
of computed and experimental exit plane total pressure
contours is presented in figure 7. The data are from
reference 8, and are for a somewhat different geometry and
incoming flow than wns used in the anal ysis. (In the test,
the inlet had an external ramp forebody and ra compressor
face hub, and was run at a 1 degree angle of attack. The
Reynolds number was 4.9x10 6 per meter (1.5x10 6 per foot) and
the free stream Mach number was .8). The experimental
results, however, are similar to the computed results in the
degree of symmetry between the upper and lower halves of the
inlet. It should be noted that the computed boundary layer
grows very quickly during the first few marching steps.
,
t
7'
Fs	 is
r	
Ii
R
This is believed to be caused by appreciable numerical
diffusion in this thin boundary Layer calculation as a
result of inadequate mesh .asolution. This resulted in
!
larger calculated total pressure losses than were measured
experimentally, With proper mesh resolution# ti^;Ls problem 	 f
should be alleviated.
It was then hypothesized that the effect of the change
in cross-section shape was much more important in this flow
than the effect of the centerline curvature. To verify this
hypothesis, two additional geometries were run. The first,
shown in figure rkb, had the some distribution of
cross-section shape as the baseline configuration, but with
a straight centerline, The second, shown in figure 5c, had
the same centerline shape and area distribution as the
baseline configuration, but with circular cross-sections.
These configurations were chosen to isolate the effects of
shape change and centerline curvatu re. The flow conditions
and grid were the some as for the baseline calculation.
Tile total pressure contours for the straight centerline
configuration are shown in figure 8, and are virtually
identical to those for the baseline configuration. The
results for the circular cross-section case, shown in figure
9, do show a slight shift in the total. pressure contours,
indicating that the centerline curvature is strong enough to
generate at least a weak secondary flow vortex. This is
confirmed when the cross-flow velocities are examined. The
computed secondary flow velocity vectors at the exit plane
for all three F-16 configurations are shown in figure 10.
The secondary flow patterns for the baseline and straight
centerline configurations are basically similar. Although
some; ,influence of centerline curvature can be seen in the y
baseline results, no vortex pattern is set up. The circular
cross -section configuration, however, 
h
as an obvious
secondary flow vortex at the exit plane.
8
Additional insight can be gained by examining the
circumferential pressure gradient in the baseline F-16
configuration. rn figure ila the inviscid static pressure
is plotted for a crass-section at about the mid-point of the
duct. The pressure is higher in the region of the
converging sides of the cross-section than at the diverging
top and bottom. The low energy boundary la yer fluid is thus
driven in hn-t-11 directions away from the sides of the inlet,
causing the boundary layer thickness to decrease there and
to increase at the top and bottom of tho ;.islet. By the time
the second bend is reached, most of the cross-section
transitioninq has been completed. The circumferential
pressure distribution in the second bend, shown in 'figure
11b, is similar to that found in the second bend of n
constant area S-duct However, the boundary layer has
become so thin near the converging sides of the duct that
the boundary layers in the top and bottom halves of the
inlet are essentially isolated from each other. Low energy
boundary layer fluid cannot migrate through this thin
boundary layer region, and therefore no secondary flow
vortex is formed. In the circular cross-section
configuration, of course, the boundary layer does not thin
out at the sides of the duct, ai.d a vortex pattern is
generated, 'Thus, although the centerline curvature in the
baseline F-16 configuration is strong enough to produce a
secondary flow vortex, the changing cross-section shape
suppresses its formation.
GQNC LUnING MARRS
1. The PEPSIG analysis gives at least qualitatively
correct predictions of the development of secondary flow
vortices and total pressure distortion in S-shaped ducts.
2. More detailed quantitative comparisons with
experiment are needed to determine the accuracy of the
analysis,
9
x^.c..,A «. d:,»L- .^m:+:,`^v..wawa::vtxw^w :x'us"°itVl!^^^ro+iN:':,,'`.,e^`F°iz;6K:^^"vt^.ax.ms, m,,. -^cx..,.	
.,.-_. •».,	 , .... ....	 .... .	
,.....	 ..:
5. Mash studies are needed to determine the resolution
required for thin boundary Layer calculations,
4. The analysis ours be used to, help understand the
development of secondary flow phenomena in complex
configurations.
5. There is a critical need for detailed experimental
data on secondary flocs d p velopment in ducts with canterl ine
curvature and/or cross-section transitioning for proper code
evaluation.
REF£RF CF,S
1, Briley, W. A. and McDonald, H., "Analysis and
Computation of Viscous Subsonic PrimAry and Secondary
Flows," AIAA Paper 79-1453, July 1979.
2. Levy, R., McDonald, H. ► Briley, W. A. ► and Kreskovsky,
4: 0. t "A Three-Dimensional Turbulent, CortipresS3.ble
Subsonic Duct Flow Analysis for Use with Constructed
Coordinate Systems," AIAA Paper,' 80 . 1398 ► July 1980.
3. Buleev, H. I., "Theoretical Model of the Mechanisms of
Turbulent Exchange in Fluid Flow," Atomic Energy
Research Establishment, U. K., ARRE-Trans-957, 1963.
4. Bansod, P. and Bradshaw, R., "The Flow in S-Shaped
Ducts," Aeronautical Qu xter . Yly, Vol. 23, Pt. 2 ► May
1972, pp. 131-140,
5. Rowe, M., "Measurements and Computations of flow in Pipe
Bends," Journal. of Fluid Mechanics, Vol. 43 ► Pt. 4, pp.
771-785.
6. Leamer, P. C. and Kennon, 1. G., IlExper mental
Investigation of a 0.15 Scale Model of an Onderfuselage 	
.
Norval-Shock inlet," NASA CR-3049, 1978.
10
7, Eisemah, F. R., McDonald, H., and Briley, W. R., ",A
Method for Computing Three-Dimensional Viscous Diffuser
Flaws, ,' United Technologies Research Centex, East Hartford,
CT, R75-911737-1, ,duly 1975.
D
5. Howkin► . J. E, ► "Experimental Investigation of a
0.15-Scale Modal of a Conformal Variable-Romp Inlet for
the P-16 Airplane," General Dynamics, Fort Worth, TX,
ERR-F'W-2014, Mar. 1980, (NASA CR-159640)
11
.f
Figure 1. w Computed total pressure contours for circular S-duct.
ANALYSIS	 EXPERINOT (REF. 4)
Figure 2. - Qualitative comparison between com-
puted and erparlmental total pressure contours
atexitof circular S-duct.
rhh ^ ^ t / RREi
1 1,) 111 ,/fll l (l 1111! 11l \ RR ^ 1
r^^ !ff f f	 11 l l / RRI
^i l ^ i l
 . f ^ ff f I f111.^11 ^^^IR
STATION 5 STATION 20STATION 10
ice: °' ^ ^ • ± ^ '.^^
•^.'^,f'^,IIrn2p 1,^,^,
I ,	 •, 11
F.S.
L;--
STATION 30	 STATION 35	 STATION 40
Figure 3, - Computed secondary flow development for circular S-duct,
Figure 4. - F-16 inlet geometry.
r
tt
t	 '
I	 I
P
t
ib1 Straight centerline configuration,
Figure 5, - F-lb Inlet geometry broaktimyn,
'(41 t iNAL PAGJ^ IS
Figure 6. - Computed total pressure contours for F-16 Inlet, baseline configuration,
fs
3, u
{{	 j
1
C	 i
1
f	
u\,l1
rFj/.(	 ..	 ^.	 i^
ANALYSIS
	 WERIMENI' (RIF. 8)
Figure 7, -Qualitative comparlson between computed and
exparinental total pressure contours at exit of r-16 in-
let.
Figure & - Computed total pressure contours for F-16 Inlet, straight cantarllne configuration,
it
^,^^ 
lljN
"
,
Figure 4. - Computed total pressure contours for F-16 inlet, circular cross-section
configuration,
BASELINE
	
STRAIGHT CENTERLINE	 CIRCULAR CROSS-SECTION
Figure M - Computed secondary flaw at exit plane of F-16 inlet 'configurations.
i
r
4'^.* .s ,,.	 ......,-_	 ,e.<,,..^«.•.- ... «czi.^xu,.. 	 `	 .Wldrpat^+^a^a^.^xw:'.,waymws^4+tvwrr+'+rr.,N`a^-%sr^vv-^ti?!^.`re^S^^waB*u,.rir..ra:n.-a:..^w.cet,.^,>._....r^ . .. s.v_. _ ..., i".._. 	 ._ _	 ^Mi
-55
.50
.,5
-40
v
iV^1tA
tA
a
U
d
O
VN
la! NEAR MIDPOINT.
n _ no
x
30	 60	 40	 120	 150
	
1B0
ANGULAR LOCATION, 0 deg
lbl IN SErOND SEND,
Figure 11. - Circumferential Inviscid static pressure distribution of baseline F-16
Inlet configuration.


Numerical simulation of flows in curved diffusers with cross-sectional transitioning using a three-dimensional viscous analysis

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