Vance F. Dippold, III

Glenn Research Center, Cleveland, Ohio

Computational Simulations of Convergent

Nozzles for the AIAA 1st Propulsion

Aerodynamics Workshop

NASA/TM—2014-218329

August 2014
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Vance F. Dippold, III

Glenn Research Center, Cleveland, Ohio

Computational Simulations of Convergent

Nozzles for the AIAA 1st Propulsion

Aerodynamics Workshop

NASA/TM—2014-218329

August 2014

National Aeronautics and

Space Administration

Glenn Research Center

Cleveland, Ohio 44135
Acknowledgments

This work was supported by the Fundamental Aeronautics Program’s Subsonic Fixed-Wing and High Speed Projects.

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Level of Review: This material has been technically reviewed by technical management. 
NASA/TM—2014-218329 1

Computational Simulations of Convergent Nozzles for 

the AIAA 1st Propulsion Aerodynamics Workshop 

Vance F. Dippold, III 

National Aeronautics and Space Administration 

Glenn Research Center 

Cleveland, Ohio 44135 

Abstract 

Computational Fluid Dynamics (CFD) simulations were completed for a series of convergent nozzles 

in participation of the American Institute of Aeronautics and Astronautics (AIAA) 1st Propulsion 

Aerodynamics Workshop. The simulations were performed using the Wind-US flow solver. Discharge 

and thrust coefficients were computed for four axisymmetric nozzles with nozzle pressure ratios (NPR) 

ranging from 1.4 to 7.0. The computed discharge coefficients showed excellent agreement with available 

experimental data; the computed thrust coefficients captured trends observed in the experimental data, but 

over-predicted the thrust coefficient by 0.25 to 1.0 percent. Sonic lines were computed for cases with 

NPR 2.0 and agreed well with experimental data for NPR 2.5. Simulations were also performed for a 

25° conic nozzle bifurcated by a flat plate at NPR = 4.0. The jet plume shock structure was compared 

with and without the splitter plate to the experimental data. The Wind-US simulations predicted the shock 

structure well, though lack of grid resolution in the plume reduced the sharpness of the shock waves. 

Unsteady Reynolds-Averaged Navier-Stokes (URANS) simulations and Detached Eddy Simulations 

(DES) were performed at NPR = 1.6 for the 25° conic nozzle with splitter plate. The simulations 

predicted vortex shedding from the trailing edge of the splitter plate. However, the vortices of URANS 

and DES solutions appeared to dissipate earlier than observed experimentally. It is believed that a lack of 

grid resolution in the region of the vortex shedding may have caused the vortices to break down too soon. 

Nomenclature 

A area 

Cd nozzle discharge coefficient 

CV nozzle thrust coefficient, integrated across jet only 

CV,total total nozzle thrust coefficient, integrated across jet and base region 

D diameter 

M Mach number 

NPR nozzle pressure ratio 

p pressure 

R gas constant; 1716 (ftlb)/(slug°R) 

r radius 

T temperature 

u velocity in streamwise (x-) direction 

x,y,z orthogonal coordinate system 

y

+

 non-dimensional wall distance 

ratio of specific heats; 1.4 

 density 
NASA/TM—2014-218329 2

Subscripts: 

0 total condition 

jet jet condition 

nozzle full nozzle, jet and base area 

wall wall condition 

 freestream static condition 

Introduction 

In 2012, the Air-Breathing Propulsion Technical Committees of the American Institute of 

Aeronautics and Astronautics (AIAA) conducted the 1st Propulsion Aerodynamics Workshop. The 

objectives of the 1st Propulsion Aerodynamics Workshop were as follows: 

1. To assess the accuracy of computational fluid dynamics for air breathing propulsion applications;

2. To assess current numerical prediction capability;

3. To develop practical guidelines for 2-D and 3-D simulations.

Workshop participants from government, academia, and industry completed Computational Fluid 

Dynamics (CFD) simulations of problems relevant to air-breathing propulsion: an S-duct and a series of 

convergent nozzles. Each participant presented their results at the workshop, held prior to the 48th Joint 

Propulsion Conference, in Atlanta in July 2012. Some of the CFD studies were performed as blind trials 

and compared with the available experimental data during the workshop. After each participant had 

presented their results, the workshop committee compared the results from all of the participants and 

compared their results with experimental data. The workshop committee presented follow-up reports for 

the nozzle cases (Ref. 1) and the S-duct cases (Ref. 2) at the 49th Joint Propulsion Conference, held in 

San Jose in July 2013. Several participants also published reports with their findings (Ref. 3). This paper 

discusses computational simulations performed for the series of convergent nozzles as a part of the 

1st Propulsion Aerodynamics Workshop. 

CFD simulations were performed for a series of four single-stream, axisymmetric, convergent nozzles 

at nozzle pressure ratios (NPR)—the ratio of nozzle total pressure to ambient static pressure—ranging 

from 1.4 to 7.0. The nozzle geometries consisted of a reference nozzle, with a circular-arc wall contour, 

and three conical nozzles, with half angles of 15°, 25°, and 40°. The internal nozzle geometries are 

illustrated in Figure 1. Each nozzle had an exit diameter of 3.0 in. Experimental data was collected for 

these nozzles by Thornock and Brown (Ref. 4). The experimental nozzles consisted of two halves, 

between which a splitter plate was inserted. The splitter plate included an array of pressure taps that were 

used to compute local Mach number inside the nozzle. Thornock and Brown reported the discharge and 

thrust coefficients for each nozzle and NPR combination. Using the splitter plate pressure taps, the sonic 

line was also determined. Spark shadowgraph images, while not presented in the original report, were 

compared to CFD solutions at the Propulsion Aerodynamics Workshop for select nozzle and NPR 

combinations in the follow-up report (Ref. 1). The shadowgraph images showed the plume shock 

structure for a nozzle at supersonic NPR and unsteady vortex shedding for one nozzle at a subsonic NPR. 

The Propulsion Aerodynamics Workshop had three Instances for the nozzle cases. Instance 1 

consisted of simulating all four nozzle geometries, without splitter plates, with NPRs ranging from 1.4 to 

4.0. The predicted discharge coefficient, thrust coefficient, and sonic line would be compared to 

experimental data. Instance 2 consisted of simulating the 25° conic nozzle with NPR = 4.0, with and 

without the splitter plate. The plume flowfield—specifically, the series of shock waves and expansion 

waves—would be compared to shadowgraph images. Finally, Instance 3 consisted of a time-accurate 

simulation of the 25° conic nozzle with splitter plate for NPR = 1.6. The simulated vortex shedding from 

the trailing edge of the splitter plate would be compared to the vortex shedding observed in a spark 

shadowgraph image. 
NASA/TM—2014-218329 3

a) Reference nozzle. b) 15° conic nozzle.

c) 25° conic nozzle. d) 40° conic nozzle.

Figure 1.—Flow lines of nozzles. 

This series of nozzle simulations tests a broad range of a CFD code’s capabilities: the ability to 

predict mass flow and thrust, the ability to predict and resolve plume shock structures, and the ability to 

predict and resolve unsteady phenomena. Successfully simulating and predicting these elements is key to 

using CFD simulations to help design and improve modern aircraft propulsion systems. 

Nozzle Cases and Flow Conditions 

The 1st Propulsion Aerodynamics Workshop nozzle problem consisted of three Instances, which are 

described below. 

Instance 1 

Instance 1 consisted of 40 total CFD simulations of four axisymmetric, convergent nozzles with NPR 

ranging from 1.4 to 7.0. The four nozzle geometries consisted of a reference nozzle, with a circular-arc 

wall contour radius of 5.449 in. (Ref. 1), and three conic nozzles, with half angles of 15°, 25°, and 40°. 

The nozzle exit diameter was 3.0 in. for each nozzle. Each nozzle also had a rather thick base region, 

giving the nozzle a total diameter of 7.5 in. at the exit plane. The axisymmetric nozzle geometries are 

shown in Figure 1. Table 1 shows the distinct NPRs for which each nozzle was simulated. The discharge 
NASA/TM—2014-218329 4

TABLE 1.—INSTANCE 1 NOZZLES AND NOZZLE PRESSURE RATIOS 

Nozzle Nozzle Pressure Ratio (NPR) 

1.4 1.6 1.8 2.0 2.5 3.0 3.5 4.0 5.0 6.0 7.0 

Reference X --- X X --- X --- X X X X 

15° Conic X X X X X X --- X X X X 

25° Conic X X X X X X X X X X X 

40° Conic X X X X X X X X X X X 

coefficient and thrust coefficient from all simulations were reported for the Propulsion Aerodynamics 

Workshop and were compared with experimental data. Additionally, the static pressure was sampled at 

specific locations along the interior surface of the nozzle and through the jet plume that corresponded to 

pressure taps along the nozzle wall and splitter plate in the experiment. The wall equivalent Mach number 

was calculated using these local static pressures. The wall equivalent Mach number was reported to the 

Propulsion Aerodynamics Workshop and used to compute the sonic line for each jet. The sonic line data 

were compared to experimental data. The details of the discharge coefficient, thrust coefficient, and sonic 

line calculations will be more fully discussed in the Results section. 

Instance 2 

Instance 2 involved comparing the plume shock structure of the 25° conic nozzle with and without the 

splitter plate for a NPR of 4.0. The axisymmetric solution of the 25° conic nozzle without the splitter 

plate from Instance 1 was used. Therefore, only a 3-D simulation of the 25° conic nozzle with splitter 

plate was run for Instance 2. The splitter plate was 0.0625 in. thick and extended 12.6 in. upstream into 

the nozzle and 3.0 in. downstream of the nozzle. The splitter plate was 7.5 in. wide—the width of the 

entire nozzle, including the base region. The experimental splitter plate contained an array of 128 static 

pressure taps, which were used to calculate Mach number and the sonic line. The addition of the splitter 

plate increased the diameter of the nozzle to 3.0625 in. along the axis perpendicular to the plate. The 

location of the first shock wave downstream of the nozzle was compared between the two solutions. 

Additionally, the CFD solutions were compared with experimentally-obtained spark shadowgraph images 

for their respective case. 

Instance 3 

The goal of Instance 3 was to test the CFD solver’s ability to predict vortex shedding from the trailing 

end of the splitter plate for the 25° conic nozzle with splitter plate with a NPR of 1.6. The vortex shedding 

was a surprising phenomenon that was observed experimentally and recorded in a spark shadowgraph 

image (Ref. 1). A time-accurate simulation was needed to obtain the vortex shedding. 

Numerical Methods 

Wind-US Flow Solver 

Wind-US, version 2.0 (Refs. 5 and 6), was used to run the computational fluid dynamics (CFD) 

simulations of the convergent nozzles. The Wind-US code is a general-purpose, Reynolds-Averaged 

Navier-Stokes (RANS) CFD solver for both structured and unstructured grids. The code is developed and 

managed by the NPARC Alliance, a partnership between NASA GRC, USAF Arnold Engineering 

Development Complex, and The Boeing Company. The Wind-US code offers several numerical schemes, 

as well as several zero-, one-, and two-equation turbulence models. The Wind-US code includes the 

second-order Roe numerical scheme and the Menter Shear Stress Transport (SST) turbulence model 

(Ref. 7), both of which were used for these simulations. 
NASA/TM—2014-218329 5

For all simulations, the jet exhausted the nozzle into a quiescent freestream, with a static temperature 

of 500 °R, and static pressure of 14.7 psi. While the freestream was quiescent, the freestream 

Mach number was 0.01. It has been found that compressible flow solvers, like Wind-US, struggle with 

convergent nozzle flows when the freestream velocity is set to zeroeven a small freestream velocity helps 

the solver converge the solution more quickly. At the nozzle inflow, the total temperature was set to 

500 °R; the total pressure was set according to the NPR (see Table 1) and the freestream static pressure: 

p0  NPR  p (1)

The downstream outflow was set to the freestream static pressure, 14.7 psi. 

Grid Strategy 

A total of five structured, point-matched grids were constructed with the commercially-available 

Pointwise (Pointwise, Inc.) grid generation software: four axisymmetric grids for the reference nozzle and 

three conic nozzles; one 3-D grid for the 25° conic nozzle with splitter plate. Grids for the four 

axisymmetric nozzles are shown in Figure 2. Inside the nozzle, the grids had between 161 and 193 grid 

points in the streamwise direction and 65 grid points in the radial direction, between the centerline and the 

interior wall of the nozzle. Beyond the nozzle exit, the grids continued for 50 jet diameters downstream of 

a) Reference nozzle. b) 15° conic nozzle.

c) 25° conic nozzle. d) 40° conic nozzle.

Figure 2.—Grids for axisymmetric nozzles. 
NASA/TM—2014-218329 6

a) Full grid. b) Grid perpendicular to jet plume.

Figure 3.—Grid for 25° conic nozzle with splitter plate. 

the nozzle exit, containing 221 grid points. The grids extended 23 jet diameters radially outward of the 

nozzle, containing 233 grid points. The axisymmetric reference nozzle grid consisted of 74,000 grid 

points. Each axisymmetric conic nozzle consisted of 71,000 grid points. In order to properly resolve the 

boundary layers along the viscous walls inside the nozzle, grid points were clustered at the walls, with an 

initial grid point spacing of 1.0e-4 in., resulting in a nominal y

+

 value of 2. It has been shown that the 

non-dimensional wall spacing, y

+

, of the first grid point off the wall needs to be on the order of 1.0 

(Refs. 8 and 9) for RANS solvers to adequately resolve the boundary layer. The axisymmetric grids were 

composed of 6 zones. 

The 3-D grid for the 25° conic nozzle with splitter plate is shown in Figure 3. The grid took 

advantage of symmetry: only a 180° sector was modeled, with the symmetry plane being perpendicular to 

the splitter plate. The 3-D grid was constructed in a manner similar to the axisymmetric 25° conic nozzle. 

Figure 3(b) shows the grid perpendicular to the jet plume at the trailing edge of the splitter plate. Grid 

points were also clustered radially towards the splitter plate (1.0e-4 in.) and clustered axially at the splitter 

plate trailing edge (1.0e-4 in.). There were 121 points along the splitter plate in the streamwise direction. 

The 3-D grid for the 25° conic nozzle with splitter plate consisted of 14 million grid points. The grid was 

split into 68 zones to improve performance on multi-processor computer systems. 

Convergence 

To speed up convergence of each solution, each simulation was solved by sequencing through three 

grid resolutions: first, the solution was solved using every fourth grid point, then using every second grid 

point, and finally using every grid point. Solving on the coarser grids first allowed the overall flowfield to 

set up rather quickly before solving the turbulence details in the finest grid. This grid sequencing 

approach is standard practice for structured RANS calculations. Furthermore, converging the solution on 

multiple grid sequence levels helped ensure that the finest grid solution was not dependent on the grid 

(i.e., the solution would not change significantly if minor changes to the grid were made). For all cases, 

discharge coefficient and thrust coefficient were monitored for convergence. Additionally, the streamwise 

velocity and the turbulent kinetic energy along the nozzle centerline were monitored for Instance 1 and 

Instance 2. The solutions were considered converged when the discharge and thrust coefficients varied by 

less than 0.01 percent over 10,000 iterations and the centerline velocity and turbulent kinetic energy 

varied by less than 0.1 percent over 10,000 iterations. 
NASA/TM—2014-218329 7

Due to the large number of cases that had to be completed for the workshop, a “set it and forget it” 

approach was applied to the steady-state solutions in Instance 1 and Instance 2: the final Courant-FreidrichsLevy (CFL) number was set to 0.10 to ensure stability; each solution was run for 70,000 to 120,000 

iterations to ensure convergence. For cases with NPR  2.0, the jet plumes exhibited behavior that seemed 

reminiscent of unsteady behavior: the velocity and turbulent kinetic energy along the centerline fluctuated 

and were not converging. Similar behavior has been observed before, for simulations of high-subsonic and 

transonic flows through convergent nozzles. The procedure to obtain a converged solution was to run these 

simulations with constant time-steps. Time-steps of 1.0e-8 sec were used for a few of the 15° conic nozzle 

solutions; these cases required up to 800,000 iterations to fully converge. Later constant time-step solutions 

(including all the 25° and 40° conic nozzles with NPR  2.0) used time-steps of 2.0e-8 sec and needed 

300,000 iterations to converge. For the unsteady solution in Instance 3, two analyses were performed: an 

unsteady RANS (URANS) simulation and a detached eddy simulation (DES). For both unsteady 

simulations, a constant time step of 1.0e-8 sec was used. The unsteady simulations were run long enough for 

the vortex shedding to set up sufficiently, 400,000 to 500,000 iterations. All Wind-US simulations were 

performed on Linux (Linus Torvalds) computers, primarily the multi-node, multi-core NASA Advanced 

Supercomputing (NAS) Pleiades supercomputer (Ref. 10). 

Results 

CFD simulations were performed as previously mentioned. The typical convergence of the discharge 

coefficient and the thrust coefficient is illustrated in Figure 4. Based on the discharge and thrust 

coefficients, most solutions were adequately converged by 50,000 iterations. Figure 5 shows the 

streamwise velocity and turbulent kinetic energy along the centerline for the typical Instance 1 nozzle 

solution. Again, the solution was adequately converged much earlier than the full run (120,000 iterations), 

often by 60,000 to 70,000 iterations. Nonetheless, allowing the simulations to run 120,000 iterations 

guaranteed that most solutions were fully converged. As mentioned previously, the conic nozzles with 

low NPR 1.4 to 2.0 did exhibit unsteady-like behaviors and required a constant time-step simulation with 

300,000 or more iterations to fully converge. 

a) Discharge coefficient. b) Thrust coefficient.

Figure 4.—Typical convergence of nozzle discharge and thrust coefficients (shown for 40° conic nozzle with 

NPR = 4.0). 
NASA/TM—2014-218329 8

a) Streamwise velocity. b) Turbulent kinetic energy.

Figure 5.—Typical convergence of nozzle centerline streamwise velocity and turbulent kinetic energy (shown for 40

conic nozzle with NPR = 4.0). 

Instance 1 

A total of 40 axisymmetric nozzle solutions were computed for Instance 1. The nozzle pressure ratios 

(ranging from 1.4 to 7.0) were applied to four convergent nozzle geometries according to Table 1. The 

discharge coefficient, Cd, and thrust coefficient, CV, were computed for each solution as follows: 

jet jet jet

0

jet 2

U A

u r dr

C

r

d

	  


  	   

  (2)

  
 





 	   


  	     

 

jet

jet

0 jet

0

2 2

r

r

V

U u r dr

u p p r dr

C (3)

The standard discharge coefficient, Cd, and thrust coefficient, CV, are computed over the radius of the 

nozzle exit, rjet, using the local density (), local streamwise velocity (u), and local radius from the 

centerline (r). The resultant nozzle exit area, Ajet, was 7.0686 in.2

. The ideal jet velocity, Ujet, is computed 

from the Mach number of the ideally expanded jet, Mjet, and the temperature of the jet, Tjet, all found in 

the following equations: 

U jet M jet R Tjet      (4)





















  





 







   



















1

1

2 1

M jet NPR (5)

1

2

0 jet 2

1 1







 



 



 



   Tjet  T   M (6)
NASA/TM—2014-218329 9

TABLE 2.—IDEAL JET MACH NUMBER AND 

VELOCITY FOR EACH NOZZLE PRESSURE RATIO 

NPR Mjet Ujet [ft/s] 

1.4 0.710 742.0

1.6 0.848 868.7

1.8 0.956 963.6

2.0 1.046 1038.8

2.5 1.223 1176.2

3.0 1.358 1272.0

3.5 1.467 1344.3

6.0 1.559 1401.5

5.0 1.709 1487.9

6.0 1.828 1551.3

7.0 1.928 1600.5

The values of Mjet and Ujet are shown for each NPR in Table 2. The discharge coefficient from each 

nozzle solution is plotted in Figure 6 along with the experimental data (available for the 15°, 25°, and 40° 

conic nozzle cases). The Wind-US solution discharge coefficients show excellent agreement with the 

experimental data. For each nozzle, the discharge coefficient increases with NPR before becoming 

relatively constant at larger NPRs. The reference nozzle has the largest discharge coefficient (0.995), 

followed by the 15° conic nozzle (0.969), the 25° conic nozzle (0.946), and the 40° conic nozzle (0.914). 

The CFD results show that the nozzle is less efficient at passing flow as the nozzle half-angle increases. 

The thrust coefficient from each nozzle solution is plotted in Figure 7 along with the experimental data. In 

contrast to the discharge coefficient, the thrust coefficient increases as the nozzle half-angle increases. 

Additionally, the thrust coefficient is relatively constant at smaller NPRs, but decreases substantially 

(4.0 to 5.0 percent) as NPR increases. Finally, the thrust coefficients from Wind-US are generally 0.5 to 

1.5 percent larger than the experimental thrust coefficients. 

There was concern as to why such a large disparity exists between the CFD and experimental thrust 

coefficients. It is clear from the definition of the thrust coefficient equation (Eq. (3)), that only the force 

across the jet exit is considered. However, the nozzles had a large base region—525 percent the size of 

the actual nozzle exit area—and surely there was a significant force on the base region that could have 

affected the experimental thrust measurements. Therefore a new total thrust coefficient was computed: 

  
 





 	   


  	     

 

nozzle

nozzle

0

0

2

Total

2

, r

jet

r

V

U u r dr

u p p r dr

C (7)

In this definition of total thrust coefficient, rnozzle is the radius to the outer edge of the nozzle exterior, 

which includes the large base area. The total thrust coefficient is plotted in Figure 8. The Wind-US total 

thrust coefficients for each nozzle solution are 0.25 to 0.5 percent smaller than the original thrust 

coefficient in Figure 7. However, there is still a substantial difference between the CFD total thrust 

coefficients and the experimental thrust coefficients. It is interesting to note that the total thrust coefficient 

for the 15° conic nozzle is now the largest thrust coefficient for NPR  4.0; this does agree with the 

experimental results. 
NASA/TM—2014-218329 10

Figure 6.—Discharge coefficient, plotted for all solutions in Instance 1. 

Figure 7.—Thrust coefficient, plotted for all solutions in Instance 1. 
NASA/TM—2014-218329 11

Figure 8.—Total thrust coefficient, plotted for all solutions in Instance 1. 

Static pressure data was collected for each nozzle solution at specific points along the nozzle interior 

wall and at specific points on the nozzle splitter plate. From the wall static pressure, pwall, and the nozzle 

total pressure (assuming there are no loses), p0, the wall equivalent Mach number was computed at each 

specific point: 





















 













  





 







  





















1

1

2

1

wall

0

wall

p

p M (8)

Interpolation was performed using the wall equivalent Mach number and the location of each static 

pressure tap to determine the location of the sonic line (Mwall = 1.0) for the conic nozzle solutions with 

NPR 2.0. Sonic lines are plotted in Figure 9 for the conic nozzles with NPR = 2.5. As the half-angle of 

the nozzle increases, the sonic line is pushed further downstream. The Wind-US solutions show excellent 

agreement with the experimental data. Figure 10 shows the sonic lines for the 25° conic nozzle for NPR 

ranging from 2.0 to 5.0. The jet flow becomes supersonic earlier for the larger NPR. The Wind-US 

solutions show the best agreement with the experimental data for the larger NPR. 
NASA/TM—2014-218329 12

Figure 9.—Comparison of sonic lines for conic nozzles with NPR = 2.5.

Figure 10.—Comparison of sonic lines for 25 conic nozzle for range of NPRs. 
NASA/TM—2014-218329 13

Instance 2 

A new 3-D nozzle solution was computed for Instance 2: the 25° conic nozzle with the splitter plate 

was run with NPR = 4.0. The 3-D solution was compared to the solution of the corresponding 

axisymmetric, 25° conic nozzle without the splitter plate (obtained in Instance 1). Additionally, both CFD 

solutions were compared to experimental shadowgraph images. In Figure 11, simulated shadowgraph 

images of the two nozzle solutions are compared. These simulated shadowgraph images plot the gradient 

of the density: 

2 2 2





 





 

 	  





 







 

 	   

 





 

 	 !	  x y z

(9)

The blue line indicates the beginning of the shock train for the nozzle without the splitter plate, 

approximately x/Djet = 1.23. The red line indicates the beginning of the oblique shock coming off the 

trailing edge of the splitter plate, x/Djet = 1.0. The presence of the plate forces the initial oblique shock 

wave to move upstream. The splitter plate solution (Fig. 11(b)) has noticeably sharper shock and 

expansion waves than the non-splitter plate solution (Fig. 11(a)). This is a direct result of the techniques 

and priorities used when constructing the two grids. For the non-splitter plate grids, the priority was to 

correctly compute the discharge and thrust coefficients while reducing computational time. The grid was 

constructed with sufficient resolution in the streamwise direction inside the nozzle and near the exit. In 

the jet plume, however, the grid was rapidly stretched in the streamwise direction to reduce the number of 

a) Without splitter plate.

b) With splitter plate.

Figure 11.—Comparison of shock and expansion wave pattern for 25 conic nozzle at NPR = 4.0. Simulated 

Shadowgraph images (using gradient of density) are shown for Wind-US solutions. Red line indicates location of 

splitter plate trailing edge and induced shockwave; blue line indicates location of shock wave without splitter plate. 
NASA/TM—2014-218329 14

grid points and the computational costs. Therefore, the shock and expansion waves appear smeared in the 

streamwise direction for the non-splitter plate case (Fig. 11(a)). The streamwise grid spacing in the region 

of the oblique shock reflection is about 0.31 in. for the non-splitter plate grid. The grid for the splitter 

plate case was constructed with a priority to resolve the flow at the trailing edge of the splitter plate, 

including the shock and expansion waves for NPR = 4.0 and the vortex shedding for NPR = 1.6 

(discussed more in Instance 3). The splitter plate grid has a streamwise grid spacing of about 0.14 in. in 

the region of the oblique shock reflection. Therefore, the shock and expansion waves for the splitter plate 

solution (Fig. 11(b)) appear much sharper and better defined than the non-splitter plate case. In both 

cases, the shock wave is spread across two to three grid cells. A much sharper shockwave would be 

expected if the streamwise and radial grid spacing were no larger than 0.05 to 0.1 in. in this region of the 

jet plume. 

Figures 12 and 13 compare the Wind-US solutions and their respective experimental results for the 

two nozzle cases. Care was taken to align the Wind-US solutions with the experimental spark 

shadowgraphs, using the apparent jet exit diameter and location as the basis for scaling and alignment. 

However, there is still the possibility that the images may not completely align. Care was also taken to 

use contour coloring and levels in the Wind-US plots that best match the shadowgraph images. Due to the 

a) Wind-US solution; simulated shadowgraph image.

b) Experimental spark shadowgraph (Ref. 2).

Figure 12.—Comparison of shock and expansion wave pattern for 25° conic nozzle without splitter plate at 

NPR = 4.0. 
NASA/TM—2014-218329 15

a) Wind-US solution; simulated shadowgraph image.

b) Experimental spark shadowgraph (Ref. 2).

Figure 13.—Comparison of shock and expansion wave pattern for 25° conic nozzle with splitter plate at NPR = 4.0. 

uncertainties that exist in the comparisons of the shadowgraphs, these comparison figures (Figs. 12 and 

13) should only be used to determine trends rather than absolute differences. The Wind-US solution shock

locations for the non-splitter plate case show reasonable agreement with the experimental shock locations 

in Figure 12 for both first and second oblique shocks. As previously mentioned, the grid for this solution 

was constructed without adequate resolution to produce sharp shock and expansion waves in the jet 

plume. Therefore, it is not possible to infer more from the comparison. In Figure 13, the Wind-US 

solution predicts an oblique shock from the trailing edge of the splitter plate similar to what is observed in 

the experimental shadowgraph image. However, the Wind-US solution oblique shock appears to occur 

slightly further downstream than that of the experiment. Since it is difficult to discern the exact location 

of the splitter plate trailing edge in Figure 13(b), it is uncertain whether the Wind-US solution actually 

moves the oblique shock downstream or whether the discrepancy is due to a misalignment in images. 

While the grid of the splitter plate case shows the oblique shock much sharper than the non-splitter plate 

case, the shock wave is still somewhat smeared compared to the experimental photograph; the grid could 

be further refined to resolve the shockwave better. 
NASA/TM—2014-218329 16

Instance 3 

An Unsteady Reynold-Averaged Navier-Stokes (URANS) simulation and a Detached Eddy Simulation 

(DES) were performed for Instance 3 for the 25° conic nozzle with splitter plate for NPR = 1.6. The goal 

was to produce unsteady vortex shedding that was observed experimentally. Figure 14 compares simulated 

shadowgraph images for the URANS solution and DES solution with the experimental shadowgraph 

images. The gradient of density from Equation (9) is plotted. As mentioned previously, care has been taken 

to scale and align the computational solutions with the experimental photograph, but uncertainty exists as 

the trailing edge in the experimental photograph was not discernable. The URANS and DES solutions both 

show vortex shedding, similar to the experimental shadowgraph, however the URANS and DES vortices 

appear to dissipate earlier than the experimental vortices (x/Djet1.11 versus x/Djet1.16). In the URANS and 

DES solutions, there appears to be a stronger wake downstream of the vortices, observed as streamwise 

streaks in the gradient of density. It is also in this wake that we see the only real difference—though small—

between the URANS and DES solutions: the wake in the DES solution pinches in towards the centerline 

near x/Djet = 1.6. The cause of the difference is unknown, but it could merely be a sign that too few time 

steps were run and the fluctuation in the DES solution did not have time to dissipate or convect downstream. 

Looking closely at the URANS and DES solutions very near the trailing edge of the splitter plate, in Figure 

15, it is observed that the vortices beyond x/Djet = 1.05 are only resolved by several grid points in the 

streamwise direction. The streamwise grid spacing is approximately 0.02 in. in this region—on the order of 

30 percent of the plate trailing edge thickness. Ideally, grid spacing in this region would be on the order of 

3 to 5 percent of the plate trailing edge thickness in order to resolve the vortices and not introduce numerical 

viscosity (Ref. 11). The lack of sufficient resolution is the probable cause of the vortices dissipating earlier 

than what was observed experimentally. In this case, using a grid spacing of about 2.5e-3 in. in the 

streamwise and radial directions through a distance of 1.25 to 1.56 in. downstream of the splitter plate 

(20 to 25 times the splitter plate trailing edge thickness) would ensure that the shedding, convection, and 

dissipation of vortices would be captured by many more grid cells. 

a) Unsteady Reynolds-Averaged Navier-Stokes (URANS); instantaneous simulated shadowgraph image.

b) Detached eddy simulation (DES); instantaneous simulated shadowgraph image.

c) Experimental spark shadowgraph (Ref. 2).

Figure 14.—Vortex shedding at trailing edge of splitter plate for 25° conic nozzle with splitter plate at NPR = 1.6. 
NASA/TM—2014-218329 17

a) Unsteady Reynolds-Averaged Navier-Stokes (URANS).

b) Detached eddy simulation (DES).

Figure 15.—Detail of vortex shedding at trailing edge of splitter plate for 25° conic nozzle with splitter plate at 

NPR = 1.6. Grid lines shown and colored by vorticity; contour lines denote areas of equal vorticity. 

Conclusions 

Wind-US was used to model nozzle flows for the American Institute of Aeronautics and Astronautics 

(AIAA) 1st Propulsion Aerodynamics Workshop. Reynolds-Averaged Navier-Stokes (RANS) 

simulations were performed for 40 single-stream, axisymmetric, convergent nozzles for a range in nozzle 

pressure ratios (NPR) of 1.4 to 7.0. The four nozzles consisted of a circular arc reference nozzle and three 

conic nozzles with half angles of 15°, 25°, and 40°. The discharge coefficient, thrust coefficient, and 

sonic line were computed for each solution. The discharge coefficients showed excellent agreement with 

experimental data. The total thrust coefficient followed the general trends of the experimental data (thrust 

coefficients decreased as NPR increased), but were still 0.5 to 1.0 percent greater than the experimental 

thrust coefficients. The Wind-US sonic lines showed excellent agreement with the experimental data for 

NPR 2.5. For NPR = 2.0, the RANS simulations predicted the sonic lines 5 to 10 percent upstream of 

the experimental sonic lines. 

RANS simulations were performed for the 25° conic nozzle with a splitter plate extending one 

diameter downstream of the exit for NPR = 4.0. The resulting patterns of shock and expansion waves 

were compared to a solution without the splitter plate and to experimental data. The presence of the 

splitter plate moved the initial oblique shockwave upstream in the flow to the trailing edge of the splitter 

plate. The Wind-US solution for the nozzle without the splitter plate showed reasonable agreement with 
NASA/TM—2014-218329 18

the experiment, but the lack of streamwise grid resolution in the jet plume was found to smear the shock 

waves. The solution with the splitter plate had sharper shock waves due to increased streamwise grid 

resolution in the jet plume; it showed good agreement with the experimental data. Uncertainty existed in 

the comparison of the experimental shadowgraph images with the RANS shadowgraph images due to 

difficulty in precisely aligning the location of the splitter plate trailing edges. 

Unsteady Reynolds-Averaged Navier-Stokes (URANS) simulations and Detached Eddy Simulations 

(DES) were performed with Wind-US for the 25° conic nozzle with splitter plate for NPR = 1.6. As 

observed experimentally, the URANS and DES solutions produced vortex shedding from the splitter plate 

trailing edge. The URANS and DES vortices seemed to dissipate earlier than what was observed in the 

experimental shadowgraph image. A lack of grid resolution just downstream of the splitter plate trailing 

edge may have contributed to the numerical vortices dissipating earlier than what was observed in the 

experiment. 

Although Wind-US successfully completed all three nozzle problem Instances of the AIAA 1st 

Propulsion Aerodynamics Workshop, several recommendations can be gleaned from the study. First, in 

order to best predict thrust, the calculations must be as representative to the experiment or hardware as 

possible. Future simulations must consider base regions and external surfaces in addition to simply the jet. 

Second, fine grid spacing [define] is necessary to properly resolve shock and expansion waves in jet 

plumes. Third, very fine grid resolution [define] is required to resolve unsteady phenomena—for its initial 

occurrence and for its propagation downstream. As propulsion systems become more complex, great care 

will be required to fully resolve the flow in the CFD simulations to produce better predictions. 

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