Analysis of Flow in a Convering-Diverging Nozzle

Alber Douglawi Analysis of a Converging-Diverging Nozzle AERO 406

1

Analysis of a Converging-Diverging Nozzle (Final Project Report – setup and results)

ALBER DOUGLAWI, Cal Poly ID 008603136

Department of Aerospace Engineering California Polytechnic State University

San Luis Obispo, CA 93407

Nomenclature

d Normal Distance to the Nearest Wall (m)

Cb1 Spalart-Allmaras Model Constant

L Characteristic Length (m)

P Turbulence Production

Re Reynolds Number

U Flow Velocity (m/s)

Z Compressibility Factor

𝜈 Viscosity (Pa*s)

μ Dynamic Viscosity (N*s/m2)

ρ density (kg/m3)

Ω Vorticity (1/s)

1 Introduction

The purpose of this project was to model the flow of air

in a converging-diverging nozzle. The intent was to

study the flow properties such as Mach number,

pressure, and temperature. Creating a mesh that would

adequately capture flow features such as oblique or

normal shocks was also a priority during these

simulations. Various meshes and models were utilized

in Star CCM+ to model the supersonic flow. This type

of simulation is used in industry to aid in testing,

analysis, and the design process. Computational Fluid

Dynamics (CFD) simulations can save on lead time and

cost of design iterations and can also be used to study

systems where controlled experiments are difficult or

impossible to perform [1].

1.1 Project Description

The original proposal for this project contained several

consecutive stages, each with more complex aspects.

The goals outlined in the original proposal began with

analysing flow properties for flow in a converging-

diverging nozzle. Once this is complete, the following

steps included studying flow features such as shocks and

expansion waves. The flow features in the nozzle have

been modelled and grid convergence has been

demonstrated. Further analysis was also conducted

including a number of simulations to compare the results

from inviscid cases to those of two different turbulence

models.

The nozzle for this simulation was modelled based on an

area function given in a NASA CFD verification nozzle

[2]. This part was created using the integrated CAD

package in Star CCM+ because that facilitates possible

modifications to dimensions without having to import

the part once again. The nozzle was revolved to 90° and

two symmetry planes were designated at the beginning

and end planes of the revolve. The nozzle can be seen in

Figure 1. While the NASA resource provided the area

function, the specified boundary conditions included

100°R flow which proved to be problematic when

entered into the simulation. This was most likely due to

the nozzle containing two phase flow at that

temperature. As a result different boundary conditions

were chosen.

Figure 1. Geometry of the NASA CFD verification nozzle. The flow moves from left to right and the planes of interest are labelled.

The nozzle geometry parameters are given in Table 1

and the area function can be found in reference 2.

Stagnation

Inlet Symmetry planes

Pressure

Outlet


2

Table 1. Geometric dimensions of interest for the nozzle.

Parameter Value

Length 254 mm

Inlet Diameter 45.4 mm

Throat Diameter 20.0 mm

Outlet Diameter 55.8 mm

It was quickly determined that setting physical boundary

condition values that are compatible with one another

greatly affects the rate of convergence. For this reason,

a MATLAB script was created such that when given the

initial conditions for one boundary, the remaining

properties are calculated using isentropic relations.

Table 2 shows an example set of boundary conditions.

Table 2. Sample boundary conditions.

Pressure

(kPa)

Temperature

(K)

Velocity

(m/s)

Stag. Inlet 4238 700 N/A

Pres. Outlet 101 243 N/A

Initial Cond. 4033 695 100

A stagnation inlet and pressure outlet were chosen as the

boundary types because parameters for nozzles are

typically given as chamber stagnation conditions and a

back pressure specification. The values of pressure and

temperature were set such that the flow would be choked

at the nozzle throat to ensure supersonic flow.

The general approach was to address the problem in

stages. This meant that the initial simulation was a

simple case and more complex aspects were added as the

residual errors settled. This gave the effect of having an

initial condition that is close to the solution. This

approach was implemented after a number of

simulations that initialized including turbulence failed to

converge.

Simplifying assumptions were made including that the

air was an ideal gas and that the flow was inviscid for

the first simulation. The compressibility factor, Z, was

found to be 1.01 which supports the ideal gas

assumption [3]. The Reynolds number, Re, was

calculated using the following equation,

𝑅𝑒 =

𝜌𝑈𝐿

𝜇 (1)

where 𝜌 is the density of the flow, U is the velocity, 𝜇 is

the dynamic viscosity, and L is a characteristic length

which in this case is the nozzle diameter. The Reynolds

number at the nozzle exit was found to be 4.5*105. A

high Reynolds number suggests that the viscous effects

in the boundary layer are negligible [6]. This is because

Reynold’s number is a ratio of inertial to viscous forces

and a high value means that viscous forces are

dominated by inertial forces. After the residual errors

converged for an inviscid solution, the part was

remeshed with a prism layer and the Spalart-Almaras

turbulence model was included. This method of starting

the simulation in a simplified state and stepping through

stages of increasing complexity proved to be effective.

An example of this can be seen in Figure 9 in the

Appendix. This figure shows the steps in which the

simulation was brought to converge including turbulent

parameters.

A study modelling the Space Shuttle Main Engine

(SSME) was used as a resource to guide the approach in

this project. In this study a structured mesh was utilized

in a 2-D axisymmetric case. Isentropic relations were

used to determine the boundary conditions based on the

known chamber pressure and temperature [1]. CEA

(Chemical Element Analysis) was used to determine the

equilibrium composition of the combustion products [1].

Due to the flow having a high Reynold’s number, the

flow was expected to be turbulent [1]. The turbulence

model chosen in this simulation was the Baldwin-

Lomax model. This is an algebraic zero equation model

well suited for high speed flows with attached boundary

layers [7]. This is because the viscosity in this model is

calculated using the distribution of vorticity, meaning

that far away from the nozzle wall the viscosity is

negligible [1]. This model is used in the aerospace

propulsion industry and would be helpful for this

simulation but is not available in Star CCM+. A similar

method to the one used in this study was implemented

calculate the initial conditions in an attempt to model the

flow in the SSME in this project. This is discussed

further in section 2.

2 Numerical Model

The software used was CD Adapco’s Star CCM+. This

software was chosen over ANSYS due to the fact that

Cal Poly’s license to use ANSYS includes a cell count

limit. The simulations were run on a personal windows

desktop using an AMD Phenom II 6-core processor and

laptop using an Intel i7-4500U quadcore processor in

parallel. The run times ranged from 2-10 hours

depending on complexity and the number of cells.

3-D simulations were run for a converging-diverging

nozzle that was modelled after a NASA CFD

verification nozzle with air assumed to be an ideal gas.

For a solution including turbulence, the approach was to

begin with inviscid flow to achieve a baseline before

including turbulence. The turbulence models used were

the Spalart-Almaras and the K-𝜔 SST models. A more

detailed discussion of these turbulence models is located

in Section 2.2.

Initially it was intended that the numerical results be

compared to experimental data. However, it was

difficult to find sufficient information to recreate a

nozzle geometry and have data available for that nozzle.

An attempt was made to model the SSME based on a


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CFD study for that nozzle [1]. For that case the flow

would consist of combustion products from hydrogen

and oxygen. NASA CEA was used to determine the

chamber pressure, chamber temperature, and the mole

fractions of the products before those values were

entered into Star CCM+. The settings were then changed

to a non-reacting flow. After multiple attempts a

recurring error concerning mass fractions prevented this

case from running. The method used to ensure proper

convergence for these simulations was to vary the grid

size and rerun the simulation to compare the results.

These findings are presented in section 2.2.

2.1 Grid Description and Refinement

A polyhedral mesh was chosen for this assignment. This

mesh offered an ability to better conform to the shape of

the nozzle than other mesher choices. The base size used

was 0.001m and the meshers ultimately included the

polyhedral, surface wrapper, surface remesher, and

prism layer meshers. Initially, one volumetric control

was used to refine the mesh downstream of the nozzle

throat. The size in this volume was set to be 25% of the

base size. This was not an efficient use of cells as it

covered the entire second half of the nozzle and resulted

in a large number of cells, but it was used to locate the

flow features of interest. Prior to this, the residual errors

were not converging and it was suspected that it due to

flow features such as shocks were beginning to form and

the mesh was not sufficiently fine in those regions. By

refining the mesh downstream of the throat, it was

ensured that the shock formation is captured. Figure 2

shows this mesh with the volumetric control.

After running this mesh, the shocks developing in the

diverging section were located. The next step was to

refine the mesh only around these flow features. Figure

3 shows an oblique shock forming past the throat of the

nozzle and the shock reflecting at the intersection of the

symmetry planes. To see a larger figure of this plot with

the Mach color legend, see Figure 11 in the Appendix.

Figure 3. Visualization of the shock location using a contour of the Mach number.

Another feature resembling flow separation can be seen

forming immediately before the nozzle outlet. Figure 4

shows the mesh refined around these flow features.

These volumetric controls were created as cones to make

efficient use of cells and still follow along the

formations of the shocks. The number of cells was cut

from over 2.5 million to 1.06 million. Table 3 shows the

settings used for the meshers. Note that size parameters

are given as percentages of the base size.

Figure 2. Mesh with block volumetric control set to 25% of base size. This volumetric control engulfs the entire diverging section of the nozzle.

Figure 4. Various views of mesh refinement results with volumetric controls. Volumetric controls are in place along the first shock and its reflection as well as the second shock forming just before the outlet.


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Table 3. Mesher settings used in this simulation.

Setting Value

Base Size 0.001m

Number of prism layers 7

Prism layer stretching ratio 2

Prism layer thickness 40%

Surface growth rate 1.3

Optimization cycles 3

Quality Threshold 0.8

The base size was set to 0.001m because it was fitting

for a 0.254m long nozzle. The prism layer was created

to capture the boundary layer effects. The values for

number of prism layers, stretching ratio and prism layer

thickness were varied to obtain a smooth transition from

the prism layer to the polyhedral cells with the last cell

in the prism layer having a similar volume to that of the

polyhedral cells. Figure 5 shows this gradual transition

along with the boundary layer representation.

(a)

(b)

Figure 5. a) Close-up view of the transition from the prism layer to the polyhedral mesh. b) Velocity vector representation of the boundary layer.

The surface growth rate was set for similar reasons. As

stated earlier, a number of volumetric controls were used

to refine the mesh in areas of interest. The surface

growth rate was set to prevent sudden shifts in cell sizes

near the volumetric controls. The optimization cycles

and mesh quality threshold were set to 3 and 0.8,

respectively. While this increased the mesh generation

time significantly, it was found to increase the resulting

cell quality. Figure 14 in the Appendix shows a

histogram plot of cell quality for one of the simulations.

It can also be seen from Figure 5a that the boundary

layer was successfully captured.

2.2 Discussion of Results

For the purposes of comparing solutions for grid

independence and the effects of turbulence model

selection, four simulations were run for an inviscid case,

five were run using the Spalart-Allmaras model, and

four were run using K-𝜔 SST. For each of the inviscid

cases, the base size was changed and the simulation

reinitialized and run. The first viscous case was the

original simulation that allowed capture of the flow

features. This simulation was run initially as inviscid

and refined with volumetric controls twice before the

Spalart-Allmaras turbulence model was enabled. After

enabling the turbulence model and allowing the

residuals to settle, the base size was changed and the

simulation was resumed. The same process was repeated

for the K-𝜔 SST model. The residuals plot for the

Spalart-Allmaras cases is shown in Appendix Figure 9

and the residuals for the K-𝜔 SST cases are shown in

Appendix Figure 10. In total, nine solutions were

obtained for viscous cases. Thrust was chosen as the

value for comparison across all cases because this

application is dependent on thrust. This also facilitates

the comparison because it is one value to be compared

rather than a contour plot. Table 4 in the Appendix

shows the resulting thrust for each case along with the

number of cells. These values were plotted in Figure 6

for comparison.

Figure 6. Plot of thrust versus number of cells to demonstrate grid independence.

The largest percent difference in the calculated thrust for

the inviscid cases was 0.1%. These cases were run in

completely independent solutions starting from zero

iterations unlike the turbulent cases which were

sequential. Figure 11 in the Appendix shows a sample

residuals plot for an inviscid case. It can be seen for the

viscous cases that the residuals for continuity, energy,

and momentum consistently dropped with each mesh

refinement starting from the order of magnitude of 10-5

and decreasing to a minimum on the order of 10-8. The

largest percent difference for the thrust values from the

viscous cases was 0.02%. The percent difference

between the average of the viscous and inviscid cases

was 0.4%. This was expected because viscous effects are

dominated by inertial forces for flows with a high


5

Reynold’s number such as this. The contour plots for the

inviscid and viscous cases were nearly identical with the

exception of the boundary layer and two other regions

that are discussed further in this report.

Another topic worth noting is that the viscous cases had

reversed flow on a number of faces on the outlet plane.

The number of faces with reversed flow ranged between

400 and 900 for the different cases of varying mesh size.

This issue was not present in the inviscid cases. This was

most likely due to what appears to be flow separation

just before the outlet of the nozzle. Figure 7 illustrates

this flow feature.

Figure 7. Close-up view of the separated flow just before the outlet of the nozzle.

To investigate whether or not this issue is mesh related,

the mesh size was modified using volumetric controls.

The size in this region was set to be one quarter of the

base size and the simulation was rerun, yet the feature

remained. By creating a velocity vector scene, it was

found that the flow separated and that there was

recirculation immediately before the outlet of the

nozzle. This explains the reversed flow along the outer

rim of the nozzle outlet. Flow separation occurs when

the boundary layer travels a sufficient distance along an

adverse pressure gradient that the speed of the boundary

layer relative to the surface falls to near zero. This seems

to be in agreement with the results shown here. The

pressure contours show a highly adverse pressure

gradient just before the outlet and the velocity vector

scene shows the recirculation.

Running multiple cases also allowed for the comparison

of two turbulence models. The first turbulence model

used was the Spalart-Allmaras model. This model has

proven to be more numerically well behaved and

consistent in a variety of cases [5]. This is a one equation

model that is suited for flows with a thin boundary layer

and is better suited for supersonic flow than the K-𝜖 or

K-𝜔 models [7]. The Spalart-Allmaras model was

beneficial to use in this simulation for its stability and

also because the flow is mainly along one axis and the

exhaust plume is not included. One drawback of the

Spalart-Allmaras model is that it requires a calculation

to the nearest wall for every field point which is

computationally expensive [7]. For a comparison an

attempt was made to use the K-𝜖 model. This failed a

number of attempts including some with a considerably

smaller base size. The residuals would diverge until a

floating point exception stopped the simulation. This is

most likely due to the weakness of this model near the

wall and in separated regions. The K-𝜖 model assumes

that the turbulent viscosity is isotropic which introduces

error in regions with highly adverse pressure gradients

or separated flow [7]. This likely explains why the K-𝜖

model failed to converge as there was a region of

separated flow just prior to the outlet of the nozzle.

Another option was the K-𝜔 model which works well

for separated flows. The downside of this model is that

it is highly dependent on the ratio of the turbulent kinetic

energy to dissipation rate in the free stream. This led to

the use of the K-𝜔 SST (Shear Stress Transport) model.

The K-𝜔 SST is a combination of the K-𝜖 and K-𝜔

models that uses a blending function that is dependent

on the normal distance to the wall, y [7]. The blending

function places emphasis on the K-𝜖 model when far

from the wall and on the K-𝜔 model when near the wall.

This utilizes the performance of the K-𝜖 model in the far

field and the numerical stability of the K-𝜔 model near

the wall. The K-𝜔 SST model converged to stable

residuals.

Using the K-𝜔 SST model it was found that the

separation point occurs further upstream than the

simulation using the Spalart-Allmaras model. The K-𝜔

SST simulation had the separation point occur at 5.1mm

upstream of the outlet and the Spalart-Allmaras

simulation had the separation point at 2.5mm upstream

of the outlet. The region with significant vorticity behind

the separation point extended further into the flow for

the K-𝜔 SST simulation. To investigate the cause of this

difference further, the turbulence production term for

the Spalart-Allmaras model is displayed below,

𝑃 = 𝐶𝑏1𝑣 (Ω + (𝑣

𝑘2𝑑2) 𝑓𝑣2)

(2)

where P is the turbulence production, Cb1 is a constant,

𝑣 is the turbulence variable that has the dimensions of

viscosity, Ω is the magnitude of vorticity, d is the

distance to the wall, and fv2 is a function of a turbulent

viscosity ratio [7]. The turbulence production term for

K-𝜔 SST can be written as,

𝑃𝑘 = 𝑣𝑡𝑆𝑖𝑗Ω𝑖𝑗 (3)

where Sij is the strain rate and Ω𝑖𝑗 is the vorticity [7]. It

can be seen from comparing equations 2 and 3 that the

production term for Spalart-Allmaras will decrease as

distance to the wall increases. This may explain why the

turbulent effects seen in the simulation using the K-𝜔

SST model extended further into the flow. The Spalart-

Allmaras model is also known to overdamp the flow in

the core of a vortex which causes the damp out

prematurely [7].

Nozzle

Outlet


6

Another interesting flow feature occurred near the

intersection of the symmetry planes. The oblique shocks

developed after the nozzle throat were expected to meet

at the nozzle centreline and reflect off of one another to

form diamond shapes. Prior to the oblique shocks

meeting, a feature resembling a normal shock developed

in between. This feature is called a Mach stem and is

shown in Figure 13 in the Appendix to allow for more

detailed visualization. Figure 8 shows experimental

results with a Mach stem in which shocks were

generated in a supersonic flow using wedges. Similar

flow features to those found in the numerical simulation

can be observed. This feature has been observed in

nozzles previously but is perhaps more important for

applications that involve initiating explosives using a

wave front [8].

Figure 8. Experimental results for supersonic flow showing a Mach stem formation. [9]

According to Ivanov, a triple point forms from which an

oblique shock, the Mach stem, and a slip surface all

emanate [9]. This can be seen clearly in the experimental

results above. In addition, the slip surfaces emanating

from the triple points essentially form the shape of a

converging-diverging nozzle [9]. The flow velocity

behind the stem is nearly zero and begins to accelerate

through the converging portion of the slip surfaces until

it becomes choked once again and becomes supersonic.

All of these features can be seen in the numerical

simulation results presented in Figure 13 in the

Appendix. The lower limit of the scale has been set to

Mach 1 in order to easily distinguish between the

subsonic and supersonic regions.

The region behind the Mach stem has a flow velocity

that is nearly equal to zero. This region was identified as

a possible location that would display significant

difference in the results when using various turbulence

models. Figure 15 in the Appendix shows a comparison

of this region between the Spalart-Allmaras and K-𝜔

SST turbulence model cases. When an inviscid flow

case was inspected for the same location, it was found

that the results were nearly identical to that of the

Spalart-Allmaras case. This is expected because the

Spalart-Allmaras model is designed for supersonic

flows and includes a dependency on the distance to the

wall. Under this model when the distance to the wall is

large, turbulence is negligible. The Mach stem occurs at

the intersection of the symmetry planes and hence the

distance to the wall from this location is maximized. The

Spalart-Allmaras model resulted in higher pressure in

the region behind the Mach stem than the K-𝜔 SST

model. Figure 15 in the Appendix allows for a

comparison of these two cases. The scale was set to

match for both simulations and also set to a range that

would allow us to distinguish between contours easily.

Using the Spalart-Allmaras model resulted in a pressure

behind the Mach stem that is about 22% higher than the

K-𝜔 SST result for the same location. The K-𝜔 SST

model uses a correction when solving in areas with

rotational flows that is similar to the Spalart-Allmaras

model [7]. This correction has a much smaller effect on

the K-𝜔 SST model because it uses strain rather than

vorticity to calculate the turbulence production term as

seen in equations 2 and 3 above. The result is that the K-

𝜔 SST model may have obtained a lower pressure at this

location. Experimental results have shown that this

model tends to underpredict pressure in regions with

highly adverse pressure gradients and rotational flow

[7].

The choice of turbulence model was found to have no

significant effect on shock thickness or flow properties

such as pressure and temperature in the free stream with

the exception of the differences discussed above.

3 Conclusion

The results of these simulations accomplished the initial

goals of modelling flow features of interest in a

converging-diverging nozzle. Using an overexpanded

nozzle allowed the flow features of interest to form.

Results were obtained for values such as thrust,

pressure, Mach number, and other flow properties for

each simulation. One valuable lesson learned is the idea

of simplifying the simulation to obtain a stable solution

before adding in more complex models. This allows the

solver to initialize closer to the new desired solution.

Utilizing this method is what allowed the simulations in

this project to include a turbulence model and still

converge. Running the simulation as inviscid and

allowing the residuals to settle before including a

turbulence model proved to be effective.

Utilizing advanced initialization methods also proved to

be useful. One issue with this simulation was that the

inlet pressure was significantly different from the

pressure set throughout the medium under the physics

initial conditions node. The properties set here are the

initial conditions in all cells. When the simulation was

run, there was a very large pressure gradient near the

inlet that would result in supersonic flow at an incorrect

location. Using a linear Courant ramp was an effective

means of overcoming this issue. Another possibility

Oblique

Shocks

Mach Stem

Triple Point

Slip Surfaces


7

would be to use field functions to define the initial

conditions with respect to position downstream from the

inlet. This could be used to set the pressure and

temperature along the nozzle using isentropic relations.

Setting these initial conditions would allow the solver to

begin closer to the desired solution which provides more

numerical stability and a faster settling time.

A number of inviscid and viscous flow cases were run

to show grid independence and the solutions were well

in agreement. It can be concluded that the mesh is

capturing the appropriate flow features because further

refinements yield the same results. The inviscid and

viscous results for flow properties were closely

matched. This supports the hypothesis that the viscous

effects can be neglected. The Spalart-Allmaras and the

K-𝜔 SST models were also compared across a number

of cases. It was found that the turbulence model has no

discernible effect on the thrust results as discussed in

section 2. The choice of turbulence model did have an

effect on the flow separation point just before the outlet

of the nozzle. It was found that the flow separation using

the K-𝜔 SST model occurred earlier in the flow and the

region with significant vorticity extended further into

the flow. The shock location and thickness were not

affected. Using the K-𝜖 turbulence model proved to be

difficult. This turbulence residual errors using this

model model did not converge after numerous attempts

with various grid sizes. This was attributed to a region

with flow separation which the K-𝜖 model is known to

have difficulties with.

This analysis resulted in a deeper understanding of the

software and various methods of approaching the same

problem. Another product of this product is the

experience gained with using some of the advanced

tools for initializing the solution and also gained

experience with post processing.


8

Appendix:

Figure 9. Residuals plot for the viscous flow simulation using Spalart-Allmaras. The simulation was started as inviscid flow and refined before turbulence was included. The mesh was then refined to obtain results to show grid convergence.

grid independence.


9

Figure 10. Residual errors plot for the cases using the K-𝝎 SST turbulence model. These cases were run after the Spalart-Allmaras cases seen in Figure 9. The Sdr residual can be seen spiking between 11,000 and 11,400 iterations. This was a mesh size related issue that was solved in the following simulations.


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Figure 11. Example residual plot for an inviscid case. This is the residual plot for case 4 in Table 4.

Figure 12. Mach number contour plot. This shows an oblique shock forming after the nozzle throat.


11

Figure 13. Mach contour showing the oblique shocks and Mach stem. The lower limit of the scale is set to 1.0 to aid in distinguishing between the subsonic and supersonic regions.

Figure 14. Example cell quality histogram plot for case 9 in Table 4.

Detailed View

“CDV Nozzle” Slip Surfaces

Triple Point


12

Figure 15. Pressure contour plot comparison for Spalart-Allmaras and K-Omega SST turbulence models. The scale on both contours was set to match and detailed views are presented.

Table 4. Thrust results for various grid sizes for viscous and inviscid flow cases

Case Turbulence Number of Cells Thrust (N)

1 Inviscid 273,128 1680.06

2 Inviscid 585,239 1681.39

3 Inviscid 657,041 1681.56

4 Inviscid 875,911 1681.84

5 Spalart-Allmaras 721,459 1673.88

6 Spalart-Allmaras 929,053 1673.83

7 Spalart-Allmaras 1,142,834 1673.82



10 K-Omega SST 1,162,506 1674.79

11 K-Omega SST 1,217,300 1674.77

12 K-Omega SST 1,445,052 1674.95

13 K-Omega SST 1,535,860 1674.95

Detailed Region

Spalart-Allmaras

K-𝝎 SST


13

References

[1] Chan, John S., and Jon A. Freeman. "Thrust Chamber Performance Using Navier-Stokes Solution." National

Aeronautics and Space Administration Marshall Space Flight Center (1984)

[2] "Converging-Diverging Verification (CDV) Nozzle." Converging-Diverging Verification (CDV) Nozzle. N.p., n.d.

Web. 09 Nov. 2015. <http://www.grc.nasa.gov/WWW/wind/valid/cdv/cdv.html>.

[3] Michael J. Moran, Howard N. Shapiro, Daisie D. Boettner, Margaret B. Bailey. Fundamentals of Engineering

Thermodynamics. 7th edition, Wiley, 2011. Print.

[4] Allmaras, Steven R., Forrester T. Johnson, and Philippe R. Spalart. "Modifications and Clarifications for the

Implementation of the Spalart-Allmaras Turbulence Model." Seventh International Conference on Computational

Fluid Dynamics (2012)

[5] Cummings, Russell M., Scott A. Morton, William H. Mason, and David R. McDaniel. Applied Computational

Aerodynamics: A Modern Engineering Approach. Print.

[6] Shariatzadeh, Omid, Afshin Abrishamakar, and Aliakbar Joneidi Jafari. "Computational Modeling of a Typical

Supersonic Converging-Diverging Nozzle and Validation by Real Measured Data."

[7] Nichols, R. H. "Turbulence Models and Their Application to Complex Flows." University of Alabama at Birmingham

(n.d.): n. pag. <http://people.nas.nasa.gov/~pulliam/Turbulence/Turbulence_Guide_v4.01.pdf>.

[8] Ruban, Anatoly, and Mat Hunt. "The Shape of a Mach Stem." Society for Experimental Mechanics (2008): n. pag.

Web. <http://sem-proceedings.com/08s/sem.org-SEM-XI-Int-Cong-s070p01-The-Shape-Mach-Stem.pdf>.

[9] Ivanov, Mikhail S. "Transition Between Regular and Mach Reflections of Shock Waves: New Numerical and

Experimental Results."European Congress on Computational Methods in Applied Sciences and

Engineering (2000): n. pag. Web. <http://congress.cimne.com/eccomas/eccomas2000/pdf/611.pdf>.

Analysis of Flow in a Convering-Diverging Nozzle

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