Computational investigation of circular-to-rectangular ...

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///&as’ AIAA 91-3342 Computational Investigation of Circular-ToRectangular Transition Ducts S. Paul Pao NASA Langley Research Center Hampton, VA

Khaled S. Abdol-Hamid Analytical Services and Materials, Inc. Hampton, VA and John R. Carlson NASA Langley Research Center Hampton, VA

AIAA 9th Applied Aerodynamics

Conference September 23-25, 1991 / Baltimore, MD For permission to copy or republish, contact the American Institute of Aeronautics and Astronautics 370 L Enfant Promenade, S.W., Washington, D.C. 20024

AIAA 9th Applied Aerodynamics Conference September 23-25, 1991/Baltimore. Maryland

COMPUTATIONAL INVESTIGATION OF CIRCULAR-TO-RECTANGULARTRANSITION DUCTS S. Paul Pao* NASA Langley Research Center Hampton. Virginia

P2

Khaled S. Abdol-Hamid** Analytical Services and Materials, Inc. Hampton, Virginia

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and John R Carlson*** NASA Langley Research Center Hampton. Virginia

Abstract Recent developments of a threedimensional code (PAB3D) had paved the way for a computational investigation of the internal flow details of a family of three short circular-to-rectangular transition ducts with a constant cross sectional area. The PAB3D code was developed for solving the simplified Navier-Stokes equations in a 3-dimensional multiblock/multizone structured mesh domain. Several solver strategies and numerous boundary conditions were incorporated into this code. Modeling of the Reynolds averaged stress terms included algebraic and the k-e turbulence models. The transition ducts were designed for connecting the circular cross section of a typical engine to a high-aspect-ratio rectangular supersonic Aerospace Engineer, Associate Fellow AIAA Propulsion Aerodynamics Branch, Applied Aerodynamics Division **SeniorScientist. Member AIAA ***AerospaceEngineer, Propulsion Aerodynamics Branch, Applied Aerodynamics Division. Senior Member AIAA Copyright 0 1991 by the American Institute of Aeronautics and Astronautics, Inc. No copyright is asserted in the United States under Title 17,U.S. Code. The U.S. Government has a royalty-free license to exercise all rights under the copyright claimed herein for Governmental purposes. All other rights are resewed by the copyright owner.

nozzle. The transitional shapes of the duct cross section were represented by superellipses. A supersonic nozzle was included in the flow path for the present computational investigation. Internal flow solutions were computed by using both laminar viscosity and the two-equation k-E turbulence modeling. Static pressure distributions, discharge coefficient. and thrust ratio quantities were calculated for on-design nozzle operating conditions. Nozzle performance was predicted to within experimental accuracy for both the laminar and the k-e models. Reasonable agreement between predicted surface static pressures and experimental data was observed.

Introduction Nonaxisymmetric nozzles have shown potential for improving aircraft performance and effectiveness especially with the incorporation of multifunction capabilities into the nozzle concept1. Even if these nozzles can be effectively utilized and integrated into the aircraft maintaining external drag levels similar to those of axisymmetric configurations, poor internal flow design could produce off-setting performance losses. The internal shape presents design problems when transitioning from the axisymmetric engine flow to the rectangular exhaust system. The transition should be short to minimize the duct weight. but long enough to prevent flow separation that may adversely affect surface heat transfer and nozzle perfomnce.

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As a result, the effect of duct length on the internal performance of a high-aspect ratio nozzle was investigated by Burley, Banged. and Carlson2, where several circle-to-rectangle transition sections were tested in the static test facility of the 16-Foot Transonic Tunnel at NASA Langley. Duct surface static pressures and nozzle performance were measured for these ducts. Though papers in the past have dealt with calculating flows through d u c t s with superelliptic cross section^^-^. few of the investigations had used ducts as short or a s high an aspect ratio a s those tested by Burley et al. An analysis of the internal flow using a potential flow panel method8 was included in Ref. 2. In the present paper, the 3-dimensional multiblock/ multizone Navier-Stokes code, PAB3D. was applied to these ducts for a detailed internal flow analysis. The PAB3D code was originally developed by AbdolH a m i d g - l 0 for supersonic jet plume and external flow analysis. Recent developments of t h i s code by Abdol-Hamidll had incorporated several solver strategies, numerous boundary conditions, and several algebraic turbulence models for the turbulent eddy viscosity. Accurate predictions of nozzle internal flows were obtained by using this code -12 . In addition, a two-equation k-E turbulence model was added to the PAB3D code by Abdol-Hamid. Uenishi. and Turner13 and a module for performance calculation was added by Carlsonl*. In particular, the performance module has proven to be highly accurate in predicting nozzle discharge coefficients, t h r u s t , and vectoring forces for both axisymmetric and nonaxisymmetric type nozzle geometries. A compact grid generation code was developed at Langley to satisfy the unusual geometrical requirements of this investigation. Surface fitted grids containing a total of up to 24oooO grid points in two blocks were generated for each of the transition duct configurations. The volume grid covered one quadrant of t h e duct configuration by assuming flow symmetry at the vertical and horizontal quadrant boundaries. Flow calculations were performed for the nozzle flowing a t design conditions with three duct lengths. Comparison with experiment is presented for the distribution of pressure . coefficient wp/ wi. coefficient p / p t ~ discharge and internal thrust ratio F/Ft. Of particular interest in this investigation is an assessment of the flow uniformity and

the extent of flow separation in the duct. As experimental data were limited for these items, it is hoped that the computational analysis will shed some light on flow properties which are important for transition duct design.

Symbols and Abbreviations A

constant duct cross sectional area. cm2

a

semi-major axis of superellipse. cm

b

semi-minor axis of superellipse. cm

d

diameter of transition duct entrance,

cm e

total energy per unit volume, J/m3

F

gross thrust along body axis,N

Fi

ideal isentropic gross thrust, N

1

length from duct-connect station to duct-exit station, cm

tn NPR

nozzle reference length, cm

P

local static pressure. Pa

fiJ

jet total pressure, Pa

nozzle pressure ratio

TtJ

gas constant (y = 1.40).287.3 J / k g . K jet total temperature, K

U

velocity in streamwise direction,

R

m/sec. U

velocity in lateral direction, m/sec.

W

velocity in vertical direction, m/sec.

wi

ideal mass-flow rate, kg/sec

wP

calculated or measured mass-flow rate,

X

axial distance, positive downstream,

kg/sec cm

2

Y

lateral distance from model centerline.

Y+

dimensionless wall distance

Z

vertical distance from model centerline. cm

Y

ratio of specific heats, 1.40 for air

T

exponent of superellipse

P

density, kg/m3

cm

Table 1. Transition Duct Geometry

Transition Duct Model Configurations

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A single-engine nozzle model was tested in the static facility of the 16-Foot Transonic Tunnel at the Langley Research Center to investigate the effect of internal transition duct length on nozzle performance. A sketch of the duct model is presented in figure 1. The transition-duct is designed by using a sequence of constant-area, superelliptic cross sections.

Constant Area: Entrance Circle: diameter Exit Rectangle: width height aspect ratio

where a and b are defined a s the semi-major and the semi-minor axis, respectively. Both circles, for a = b and 7 = 2. and rectangles, for q+ are included in this family of curves. For practical purposes, the cross-sectional shape was considered a rectangle for q 2 50. The transition duct entrance was circular with a diameter of 12.863 cm. The rectangular transition section exit had dimensions of 28.682 cm x 4.531 cm. with an aspect ratio of 6.331. The planform and sideview profiles were cubic-curves with the second-derivative inflection occurring at mid-point of the profiles. The cubic functions were chosen to achieve zero slope and curvatures at the endpoints. These functions defined the semimajor axis, a. and the semi-minor axis, b. of the superellipse. The exponent of the superellipse was calculated from an implicit function relating the quantities a,b. and q to the constant area, A. of the superellipse: 00,

129.941 an2 12.863 cm 28.682 cm 4.531 cm 6.331

Duct Lengths:

Duct 2 Duct 3 Duct 4

12.863 cm 9.647 cm 6.431 cm

Nozzle :

length throat height exit height throat aspect ratio

7.394 cm 0.958 cm 1.227 cm 14.976

In the experiment reported in Ref. 2. the transition duct/nozzle combinations were tested by utilizing high-pressure air exhausting into static air to simulate engine exhaust flow. Details of the nozzle and pressure orifice locations can be found in Ref. 2. In the experimental setup, the total flow path length for all the three ducts, Le. from the upstream duct connect to the nozzle exit face, was identical. The longest duct, duct 2, fitted between the duct connect at the end of the instrumentation section and the nozzle connect station. For the shorter ducts, 3 and 4. a constant cross section rectangular extension fitted between the end of the transition section and the nozzle connect station made up the difference in length. Nozzle discharge coefficient was determined from experimentally measured jet total temperature Ttj, jet total pressure ptj and mass-flow rate . The thrust ratio F/Fr was determined from experimentally measured balance axial force, F. divided by the ideal thrust, Fr. which was computed according to the measured mass flow.

where r( ) denotes the Gamma function. The transition ducts in this family had the same cross sectional shape at a given value of x/C. Their overall lengths in the flow direction were different: L/d was 1.0, .75.and .5for Duct 2, Duct 3, and Duct 4, respectively. The reference number for these ducts were chosen to be the same a s Ref. 2 in which two other ducts with dif€erent design characteristics, 1 and 5. were also included. The exhaust nozzle used with all three ducts was a convergent-divergent, twodimensional type with a circular-arc throat contour, straight divergent flaps and flat sidewalls. The nozzle had a throat aspect ratio of 15 and an expansion ratio of 1.281. The controlling dimensions of the transition ducts and nozzle are summarized in Table 1.

Computational Procedure Solver and Boundary Conditions

The 3-dimensional Navier-Stokes code, PABSD, was used for flow analysis. In the present transition duct configuration, the flow field contained both subsonic and supersonic regions, with the possibility of flow separation. The PAI33D code contains several options for solving the governing equations, including van Leer, Roe, and space marching schemes. The Roe scheme with third order

3

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accuracy was used in the current study. Since the inflow was subsonic. the total pressure ptJ. and the total temperature Tta were specifled at the inflow face. An extrapolation outflow boundary condition was used at the nozzle exit. The nozzle exit flow was supersonic except for portions of the boundary layer. Although several turbulence models were available as options in PAJ33D. only the k-E turbulence model was found suitable for this study. The algebraic turbulence models were not used in this study for two reasons. The primary reason was that the boundary layer at the extreme ends of the high-aspect-ratio cross sections of the duct may fill the full height of the duct. The length scale of the algebraic turbulence model becomes ambiguous. A secondary reason had to do with the mesh structure for this particular duct geometry. In regions of extensive flow separation, the region of separation may cross a block boundary. There was no convenient way to continue the algebraic turbulence model across the block boundary in this multiblock/ multizone structured code. In the PAJ33D code implementation, different boundary conditions or solver methods can be selected within the length of one block. For the present study, a laminar solution was obtained in the constant diameter duct segment upstream of the transition section. The solver algorithm in the transition section and the rectangular nozzle was chosen as either laminar or the k-E turbulence model in a given set of computations. This was done to minimize numerical errors which may occur in the specification of the inflow velocity profile at the starting point of the turbulent boundary layer in the computational domain.

description of both the central region of the duct and the boundary layer at the wall. A twoblock grid which combined a 36 x 15 Cartesian mesh in the central region and a 50 x 31 polar mesh fitted to the duct wall was used to provide a smooth discretization of the volume inside the transition duct. Sample cross sections of this grid are shown in Fig. 2. The polar-type mesh, fitted to the duct and nozzle internal surface, provided a continuous boundary layer grid for the entire configuration. The polar mesh region was matched to the H-type mesh which filled the remainder of the interior. Since the polar mesh block was relatively thin as compared to the extent of the duct surface, local mesh orthogonality was easily maintained. Similarly, the central H-type mesh responded to the rapidly increasing aspect ratio of the duct cross section by merely adjusting the aspect ratio of each individual cell in the block without sacrificing orthogonality. In the streamwise direction, the constant diameter section. the transition duct, and the nozzle contained 12. 42, and 26 grid planes, respectively. The number of grid planes in the rectangular extension varied according to the length requirement: none for duct 2. 16 for duct 3. and 36 for duct 4. The largest grid, duct 4. had a total of 240.000 grid points. The polar mesh contained 31 grids in directions normal to the wall. The first 21 grids covered approximately the estimated momentum thickness of an attached boundary layer for the given values of duct length and Reynolds number, where the unit Reynolds number per foot was approximately 6.6 x lo6 in the transition section, and 18.5 x lo6 near the nozzle throat. The distance of the first grid point from the duct wall corresponded to a value of y+ of less than 5. The first three grid intervals near the wall were made equal. while the remaining intervals were stretched exponentially away from the wall. A starting solution for each configuration was estimated by using a one-dimensional isentropic solution. The Mach number distribution along the axial distance of the duct was computed according to the cross sectional area distribution. A sonic condition was established at the nozzle throat. On-design jet total pressure and temperature values were chosen to construct the conservative quantities p , pu. pu. pw,and e a s described in Ref. 11. Furthermore. an approximation of the boundary layer velocity distribution was computed for the starting solution.

Grid Generation

For the computational mesh, flow symmetry was assumed at the horizontal and vertical planes of geometrical symmetry to reduce grid requirements. The overall mesh configuration is shown in Figure 1. There are four distinct segments: a constant diameter circular entrance section, the transition section, followed by the rectangular extension, and finally the 2-dimensional convergentdivergent nozzle. Because of the drastic changes in cross sectional shape within a short distance in the flow direction, simple mesh topologies are incapable of providing a smooth

4

Calculation of Performance

I

Since the main purpose of the transition duct is to provide an efficient flow path from the circular engine exhaust duct to the highaspect-ratio rectangular nozzle, the prediction of overall mass flow and thrust performances are of primary importance. In the performance module, the cell centered quantities of the solution were extrapolated to the outflow face of the nozzle and integrated to calculate the thrust, F, which is the sum of a momentum thrust and the net pressure force at that location. The nozzle mass flow, w p was determined by the statistical average of the mass-flow through several cross sectional grid planes in the flow volume. The standard deviation was taken as one of the indicators of the numerical convergence of the solution. The ideal mass flow, wi, is determined from the isentropic flow equations and is used to normalize the predicted mass flow of the solution. The ideal thrust, Fi, is determined from the isentropic flow equations according to the nozzle pressure ratio, jet total pressure, total temperature, and the computed mass-flow rate.

-

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I

side of the section. For x/f 2 .50. however, a dominant horizontal pressure gradient is established. There are two noteworthy features at the exit section of the transition duct where x/f = 1. The lowest pressure occurs* in a thin horizontal layer near the top center of the rectangular cross section. This flow feature persists for the remainder of the flow path but is somewhat diminished at the nozzle exit plane due to the contraction upstream of the throat. The second feature is the high pressure recovery at the extreme end of the superelliptic cross sections for x/f 2 .75. As one may expect, the increased pressure near the side wall is a clear signal for imminent flow separation. The laminar flow solution and the k-e turbulence model solution can be quite different in detail. Fig. 5 shows comparisons of u-velocity distributions in the horizontal symmetry plane of all three ducts. The duct-2 velocity distributions for the laminar and the k-e solutions are almost identical except for a small region near the end of the sidewall. The laminar solution has separated flow in this region, while the k-e solution remains attached. The comparison for duct 3 shows significant differences. Extensive regions of separated flow exist in the laminar solution. The flow field also has the appearance of unsteadiness, although the computations were conducted for a steady state solution. The k-e solution remains smooth. A thin layer of separated flow has established along the middle portion of the side wall. Extensive areas of separated flow are present in both solutions for duct 4. It is dimcult to judge from the u-velocity distribution that one solution is better than the other. It should be noted, however, that the laminar solution starts to separate a s soon a s the flow enters the transition section. This prediction of early separation is not in agreement with experimental pressure data given in Ref. 2. Computed pressure distributions for the same cases are shown in Fig. 6. Static pressure distributions of the computed solutions for duct 2 are almost identical. The duct-3 solutions are quite different. The static pressure levels for both distributions fall within the same range. The k-e solution has a smoother pressure distribution and a clearly defined low-pressure expansion region on the sidewall. The laminar solution. on the other hand, contains pressure fluctuations near the sidewall and across the transition duct exit. The k-e solution for duct 4 again shows a

Results and Discussion

-

Solutions were obtained for all three transition ducts using the three-dimensional Navier-Stokes code PAB3D with the laminar flow option and the 2-equation k-e turbulence model. Several numerical and performance criteria were used to monitor convergence of the numerical solutions. Fig. 3 shows the static pressure distribution on the wall of duct 2 to illustrate basic flow properties of the transition section. As the transition duct height decreases, the pressure increases initially at the top of the cross sections. However, a s the flow accelerates downstream, the static pressure decreases to a value below its average value in the initial circular section. On the side of the duct. the flow accelerates rapidly as soon a s the duct width begins to expand. A strong local low pressure region can be seen in Fig. 3. The static pressure does not recover to a value above the initial average until a location beyond x / f = .75. Typical cross sectional pressure distributions for the same duct are shown in Fig. 4. At x / f = 0 and .25,the pressure distributions exhibit a polar asymmetric pattern: a pressure increase at the top is matched by a corresponding decrease at the

5

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small but clearly defined low-pressure expansion region on the sidewall. Near the end of the transition section, recompression of the flow is established near the upper right hand comer. By comparison, neither of these features is found in the laminar solution. In general, the k-e solutions for both ducts 3 and 4 seem to agree better with the experimental data in Ref. 2. If the trends of the k-E solutions were qualitatively accurate, duct 3 would be near the lower limit for duct length with acceptable structural cooling characteristics. This type of design related details would have been difficult to obtain through experimental measurements. Fig. 7 shows detailed comparison of the k-E solution (part a) with experimental data (part b) for the static pressure distribution on the duct-4 bottom wall. There are five identifiable regions: a high pressure region near the center of the duct entrance, a narrow low pressure region extending more than half way across the rectangular duct exit. a very low pressure region near the entrance comer of the side wall. a mild recompression region near the exit comer of the sidewall, and a plateau area in between the four areas mentioned above. In comparison with experimental data, there are good agreements in these qualitative features. Quantatively, the agreement in static pressure coefficients are within 3 percent for three of the five regions. In the other two regions, the discrepancies are as high as 6 percent. Figure 8 shows the comparison of sidewall pressures from the experimental data2, and the computed solutions. The static pressure orifices in Ref. 2 were located at various x/C within the transition section along the sidewall intersection with the horizontal plane of symmetry. In all cases, the trends of pressure distribution are predicted adequately. For duct 2. the computed results are between 1 to 3 percent above the measured data. For duct 3, the maximum difference was 4 percent. Although the k-E solution is relatively smooth, the laminar solution shows several cycles of pressure fluctuation along the x-direction. The agreement in pressure level has improved to within 2 percent for the most part in the duct-4 comparisons. Both the laminar and the k-E solutions show some degree of pressure fluctuation. Static pressure comparisons along the center of the bottom wall are shown in Fig. 9. The trends of computed pressure distribution are in good agreement with the experimental data for all three ducts. However, The computed static pressure values are

6

approximately 3 percent higher than the measured data. Static pressure comparisons between the computed k-E solutions and data along the bottom wall centerline of the 2-dimensional nozzle for all three duct configurations are shown in Fig. 10. Nearly perfect agreements are obtained for all three ducts. In addition to the centerline pressure distributions shown in Fig. 10. the solution also indicated a small volume of high pressure air along the downstream sidewall region of the duct and through the nozzle as discussed previously. This is a mild departure from the 2dimensional flow field typically observed for this family of nozzles when uniform inflow is supplied to the nozzle plenum. Finally, comparisons of overall mass flow and thrust performance are shown in Figures 11 and 12. The predicted parameters agree with the measured data within 0.5 percent, which is less than or equal to the margin of experimental error. In general, the thrust is overpredicted by the laminar solution. The k-E solution provided better agreement with the experimental data than the laminar solution in all cases. Good computational and experimental agreements of mass flow and thrust performance are certainly related to the agreements in static pressure distributions in the nozzle section. However, the reason for the disagreement between the computed and the measured pressure distributions in the transition duct section is not clear. Grid effect and turbulence modeling can be partially responsible for these differences. Concluding Remarks

This investigation has demonstrated that flows inside transition ducts with unusual geometry can be analyzed by using proper selections of a Navier-Stokes code. turbulence modeling, and grid topology. Based on comparison between the computed results and existing experimental data for the same configurations. reasonable agreements were obtained for wall static pressure in the transition duct. Static pressure comparisons in the supersonic nozzle section were excellent. Excellent agreement between computed and measured mass flow and thrust performance were also obtained. The computed solutions provided significant insight for flow details such as the extent and location of flow separation.

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internal flow velocity and pressure distributions, and the possibility of unsteady internal flow. Detailed comparison of laminar and k-E solutions seemed to indicate that the solution quality was better for the k-E solutions. However, one should bear in mind that the laminar flow calculations required much less computer resources. Further investigation will be necessary to clarify the disagreement between the computed solution and the measurements. Attention will be focused on several areas: turbulence modeling, flow unsteadiness, grid effects, and additional experimental measurements.

Transitional Ducts. Report No. 87-41, United Technology Research Center, East Hartford CI'.June 1988.

6. Reichert. B. A.; Hingst. W. R; and Okiishi, T. H.:An Experimental Trace Gas Investigation of Fluid Transport and Mixing in a Circular'to-Rectangular Transition Duct. AIAA Paper 91-2370. June 1991. 7. Sirbaugh, J. R.; and Reichert. B. A.: Computation of a Circular-toRectangular Transition Duct Flow Field. AIAA Paper 91-1741, June 1991. 8. Hawk, J. Dennis; and Bristow, Dean R: Development of MCAERO Wing Design Panel Method with Interactive Graphics Module. NASA CR-3775. 1984.

Acknowledgement The authors would like to acknowledge Kenji Uenishi and William M. Turner of the General Electric Aircraft Engines for making the k-E turbulence model available for t h i s investigation, a n d for their helpful suggestions.

9. Abdol-Hamid. K.S. and Compton. W.B. 111: Supersonic Navier-Stokes Simulations of Turbulent Afterbody Flows. AIAA Paper 89-2194,1989. 10. Abdol-Hamid. K.S.: The Application of 3 D Space Marching Scheme for the Prediction of Supersonic Free Jets. AIAA Paper 89-2897.1989.

References 1.

Capone, Francis, J.: The Nonaxisymmetric Nozzle - It is for Real. AIAA Paper 79-1810. August 1979.

11.

2. Burley. James R II; Bangert. Linda S.; and Carlson. John R.: Static Investigation of Circular-to-Rectangular Transition Ducts for H i g h -A s p e c t -R a t i o Nonaxisymmetric Nozzles, NASA TP2534.March 1986.

Abdol-Hamid. Khaled S.: Application of a Multiblock/Multizone Code (PAB3D) for the Three-Dimensional Navier-Stokes Equations. AIAA Paper 91-2155,June 1991.

12. Carlson, John R.; and Abdol-Hamid. Khaled S . : Prediction of Internal Performance for Two-Dimensional Convergent-Divergent Nozzles. AIAA Paper 91-2369, June 1991.

3. Mayer. E.: Effect of Transition in CrossSectional Shape on the Development of the Velocity and Pressure Distribution of the Turbulence Flow in Pipes. NACA TM903.1939

13. Abdol-Hamid. Khaled S.; Uenishi. Kenji; and Turner. William M.: ThreeDimensional Upwinding Navier-Stokes Code with K-E Model for Supersonic Flows. AIAA Paper 91-1669, June 1991.

4. Patrick, W. P.; and McCormick, D. C.: Circular to Rectangular Duct Flows: A Benchmark Experimental Study. SAE Technical Paper 871776,1987.

14. Carlson, John R: Validation of a Nozzle Internal Performance Prediction Method, Proposed NASATP , 1991.

5. Patrick, W. P.. and McCormick. D. C.: Laser Velocimeter and Total Pressure Measurements in Circular to Rectangular

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Straight Circular Section

Transition sectio

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Rectangular extension

Convergwt-divergent nozrle

Figure 1. Grid configuration of transition duct and nozzle combination.

x / l =050 x/l

=o.o x / l =0.75

x / l =O.E

x / l =1.00

Figure 2. Transition duct grid cross sections.

8

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Duct 2

Figure 3. Overall static pressure distribution on the wall of duct 2. p/pt,j.

L x / 1 = 0.0

X/l

=050

7 %

1 X I1

= 0.75

m !.

x / l =O.E

x / l =1.00

Figure 4. Static pressure distribution on selected cross sections in Duct 2. p/pw.

9

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(a)Duct 2

lb)Duct 3

0

Figure 5. Axial velocity distribution in the horizontal plane of symmetry for the laminar and the k-E solutions, m/sec.

10

95

95

969798

96 .9l 98

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(a) Duct 2

I

94

-

Figure 6. Static pressure distribution in the horizontal plane of symmetry for the laminar and the k-E solution. p/ ptj.

11

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Duct 4 - Bottom Wall

[a) Computed solution

Flow Direction Ib) Measured data Figure 7. Computed and measured static pressure distribution on the Duct 4 bottom wall, p/pt,j.

12

1 .o

1.0

6 'a

sn

0.9

S i d e Wall

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0.8

0.8

-

0.8

!

- Duct I

I

i

I

- Duct 2

,

0.8

I

I

1

I

i

1.0 1

BP

0.9

I 0.00

Experimental Data

3

"1 0.8-L:

Laminar

k-e

Bottom Wall

- Duct 2

,

Side Wall

---

0.9

, 0.2

I

I

0.4

0.6

0.8 0.8

1.0

- Duct 4

0 0

f

0.00

x/ I

Bottom Wall

0.2

I

I

I

0.4

0.6

0.8

x/ I

Figure 8. Computed and measured static pressure on the duct sidewall along the hoiizontal plane of symmetry.

Figure 9. Computed and measured static pressure along the duct bottom wall centerline.

13

i

1.0

- 9

-

9

-

11 7

~i

-

1.0

L

k-c Duct 2 k--E Duct 3 k-.5 Duct 4 exp. Duct 2 exp. Duct 3 exp. Duct 4

k-e ~aminar

"ool

o n Design N P R

-

0.8

0.99

0.6-

-

P,

0.98

LL

\

h

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Experimental ~ a t a

0.2

-

0.0

0.97

0.96 i

I

i

I

i

0.95

Duct 2

Figure 12. Figure 10.

Computed and measured static pressure along the nozzle bottom wall centerline.

Experimentd D8t8

k-e ~aminar

On Design N P R 0.99

\

0.97

0.96 0.95

Duct 2

Figure 11.

Duct 3

Duct 4

Computed and measured massflow discharge coefficient.

14

Duct 3

Duct 4

Computed and measured nozzle thrust ratio.