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CFD Analysis of Steady, Inviscid Flow in a Converging-Diverging (CD) Nozzle
Presentation · June 2020
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MSK 600: CFD Project
CFD Analysis of Steady, Inviscid Flow in a
Converging-Diverging (CD) Nozzle
Under the guidance of
Dr. Knut Erik Giljarhus
Submitted By:
Pravesh Kumar Yadav (254143)
Bishal Bhandari (254142)
Submitted on: May 2020
Abstract
This project aims to simulate flow through a de Laval nozzle using OpenFOAM software.
Assuming quasi one-dimensional flow, the governing equations for flow through a
convergent-divergent nozzle can be easily obtained. The project also analyzes results such as
shock location and pressure distribution along the length of the nozzle, obtained from the
simulation with the analytical results.
Three different flow; subsonic isentropic flow (Pexit/Pin = 0.89), supersonic flow with a
normal shock in the diffusing section (Pexit/Pin = 0.75), supersonic isentropic flow (Pexit/Pin
= 0.16) is analyzed using two different solvers rhoCentralFoam and rhoPrimpleFoam. The
data for analysis part has to be taken reference from Experimental data of the 'convergingDiverging Nozzle. Then the results are compared with the Experimental data of the “CD
Nozzle-Study” which was done by John W. Slater.
While comparing with the experimental results, we conclude that when rhoCentralFoam is
used as a solver, they match with the experimental values of the “CD Nozzle-study”. In the
case with rhoPimpleFoam, the results are almost the same as the rhoCentralFoam because
there are no differences in the computed equations.
ii
Contents
Abstract…………………………………………………………………………………….….ii
1.
Introduction ................................................................................................................... 1
2. Governing Equations ....................................................................................................... 1
2.1 Quasi-One-Dimensional Flow: .................................................................................... 2
2.1.1. Isentropic Expansion from Subsonic to Supersonic Speeds: ..................................... 3
2.1.2. Isentropic Subsonic Flow: ........................................................................................ 4
2.1.3 Supersonic Flow with a Normal Shock: .................................................................... 5
3.
Computational Setup ..................................................................................................... 6
3.1 Problem Statement: ..................................................................................................... 6
3.2 Geometry and Mesh: ................................................................................................... 7
3.3 Initial and Boundary Conditions: ................................................................................. 8
3.4 Solvers: ....................................................................................................................... 9
4.
Results and Discussions ................................................................................................. 9
4.1Solver Results: ........................................................................................................... 10
4.1.1 RhoCentralFOAM Results: ..................................................................................... 10
Case 1: Isentropic Subsonic Flow: ............................................................................... 10
Case 2: Isentropic Expansion from Subsonic to Supersonic Speeds ............................. 12
Case 3: Supersonic Flow with Normal Shock: ............................................................. 14
4.1.2 RhoPimpleFOAM Results: ..................................................................................... 16
Case 1: Isentropic Subsonic Flow: ............................................................................... 16
Case 2: Isentropic Expansion from Subsonic to Supersonic Speeds ............................. 18
Case 3: Supersonic Flow with Normal Shock: ............................................................. 20
4.2
5.
Solver Comparison:............................................................................................... 22
Conclusion .................................................................................................................. 22
References .......................................................................................................................... 23
Appendix ............................................................................................................................ 24
iii
1. Introduction
Consider a subsonic flow through a convergent nozzle. The flow in quasi-onedimensional equations speeds up along the nozzle, reaches sonic speed, and then slows down.
Therefore, to expand a gas isentropically from subsonic to supersonic speeds, it must pass
through a convergent-divergent nozzle or de Laval nozzle. The minimum area, called the
throat that divides the convergent and divergent sections of the duct has sonic flow. The flowthrough CD nozzle has different characteristics based on the geometry and exit conditions of
the nozzle.
2. Governing Equations
The Navier-Stokes equations for an inviscid compressible flow in an arbitrary domain is as
.......................(1)
.................................(1)
where „u’ is the fluid velocity vector, „P’ is the fluid pressure, „ρ‟ is the fluid density, and
„∇’ is the Laplacian operator.
The Navier-Stokes equation is supplemented with the
conservation of mass.
...................................(2)
.......................................(2)
Conservation of total energy for an inviscid compressible flow gives:
....................(3)
........................(3)
where the total energy density, 𝐸 = 𝑒 + 𝑢 ∕ 2, with „e‟ specific internal energy.
These three equations are supplemented with an equation of state which is the isentropic
relation
𝑑𝑝
𝑑𝜌
=
𝜕𝑝
𝜕𝜌
= 𝑎2 .................(4)
Where„a‟is the speed of sound.
1
2.1 Quasi One-Dimensional Flow:
The flow through the duct is 3-dimensional in reality. However, as we assume flow is quasione-dimensional, the flow through the area-variable duct varies only as a function of
distance, i.e., u=u(x), p=p(x). The flow properties are uniform across any given section and
this represents the mean actual flow distributed across the cross-section of the duct, which is
approximate to the actual fluid flow [4]. Consider an incremental volume, as shown in fig. 1.
Figure 1 Incremental Volume
Considering this infinitesimal control volume for the conservation of mass,
momentum, and energy equation, and after simplification we have
𝑑 𝜌𝑢𝐴 = 0 … … … … … … (5)
𝑑𝑝 = −𝜌𝑢 𝑑𝑢, … … … … . . (6)
𝑑ℎ + 𝑢 𝑑𝑢 = 0 … … … … … (7)
Where „h‟ is the specific enthalpy.
Equation (5) and (6) along with the isentropic relation gives
.........(8)
..............(8)
Equation (8) is called the area-velocity relation. It can be implied from Equation (8)
that for gas to expand isentropically from subsonic to supersonic speeds, it must flow through
a convergent-divergent duct.
2
The flow through a convergent-divergent nozzle can be categorised in three broad
cases depending on the exit condition. The results from each case are discussed without
proof. The derivation is beyond the scope of this case study.
2.1.1. Isentropic Expansion from Subsonic to Supersonic Speeds:
When the inlet pressure is specified, no other details about exit conditions are known
except,pexit<pin = p0. The subsonic flow expands isentropically at the inlet to a sonic speed at
the throat and supersonic speed at the exit.The isentropic relation governs the flow properties.
The variation of pressure and Mach number along the nozzle shown in fig. 2. In the
throat, the flow is sonic. An asterisk denotes the sonic conditions.
Figure 2: Isentropic supersonic nozzle flow [2].
3
2.1.2. Isentropic Subsonic Flow:
When the inlet and exit pressure are specified, and the pressure ratio (pexit/ pin) is
weak, that is less than or close to one. The subsonic flow increases till the throat, and the flow
is still subsonic at the throat. The flow then slows down along the divergent section until exit.
It means that the flow is subsonic throughout the nozzle.
The variation of pressure and Mach number along the nozzle are shown in fig. 3. The
Mach number is maximum at the minimum cross-section area that is at the throat (At) and
hence, there are no asterisk properties there.
Figure 3: Subsonic flow in a convergent-divergent nozzle [2].
4
2.1.3 Supersonic Flow with a Normal Shock:
When the inlet and exit pressure are specified, and the pressure ratio is high ,i.e.
greater than one. A normal shock wave occurs inside the divergent duct when the subsonic
inlet flow expands to the sonic speed at the throat isentropically, as shown in figure 4.
Isentropic relations are not valid in this section [3].
Figure 4: Flow with a shock wave inside a convergent-divergent nozzle [2].
5
The location of the normal shock wave in the duct is determined by the requirement that
the increase of static pressure across the wave plus that in the divergent portion of the subsonic
flow behind the shock be just right to achieve
Pe4
(as in fig. 4), at the exit. As the exit pressure is
reduced further, the normal shock wave will move downstream, closer to the nozzle exit.
3. Computational Setup
3.1 Problem Statement:
In this study, CFD simulation of a convergent-divergent nozzle is performed using to
OpenFOAM solvers, namely rhoCentralFOAM and rhoPimpleFOAM using 2D geometry. The
inlet pressure (𝑝𝑖𝑛) and the temperatures are 10000 Pa and 298 K respectively. The geometry of the
nozzle was created using an equation defining the cross-section area. The equation is given
below:
1.75 − 0.75𝑐𝑜𝑠 (0.2𝑥 − 1)𝜋,
0 < 𝑥 ≤5
𝐴 (𝑥) = {1.25 − 0.25𝑐𝑜𝑠 (0.2𝑥 − 1)𝜋, 5 < 𝑥 ≤ 10
The geometry obtained using the above equation can be visualised in figure 1.
Figure 5: This is the geometry of the convergent-divergent nozzle obtained using equation 1.
The CFD analysis is performed for three different exit conditions. These are:

Subsonic, isentropic flow, (pexit/pin = 0.89)

Isentropic Expansion from Subsonic to Supersonic Speeds, (pexit/pin =0.16)

Supersonic flow with normal shock in divergent section, (pexit/pin = 0.75)
6
3.2 Geometry and Mesh:
For CFD analysis, the geometry of the convergent-divergent nozzle is simplified. As the
geometry is symmetrical about the central line (longitudinal axis) of the nozzle, CFD analysis is
performed on only one side of the convergent-divergent nozzle for the shake of simplicity.
The mesh size is 51 31 having coarse elements with the wall spacing of 0.01. The length
of the nozzle is about 10.6 inch. The BlockMesh tool is used to create the mesh. It is a structured
mesh. The mesh is generated by the vertices and faces of the domain. The mesh grid can be
visualized in the figure.
Figure 6(a): Mesh grid for 2D analysis of a convergent-divergent nozzle. Only the upper half of the
geometry is considered for analysis because it is symmetrical about the longitudinal axis.
Figure 6 (b): Zoomed view of Mesh Grid
7
3.3 Initial and Boundary Conditions:
Table 1-3 below briefly explains all the boundary conditions being used in the three
cases.
Table 1: Initial &Boundary Conditions for Subsonic flow in a convergent-divergent nozzle.
Case 1. Isentropic Subsonic Flow
Pressure
Velocity
Temperature
Inlet
10000 Pa
Nozzle
Zero Gradient
Outlet
8900 Pa
Inlet
Zero Gradient
Nozzle
Slip
Outlet
Zero Gradient
Inlet
298 K
Nozzle
Zero Gradient
Outlet
Zero Gradient
Table 2: Initial &Boundary Conditions for Supersonic flow in a convergent-divergent nozzle.
Case 2. Isentropic Expansion from Subsonic to Supersonic Speeds
Pressure
Velocity
Temperature
Inlet
10000 Pa
Nozzle
Zero Gradient
Outlet
1600 Pa
Inlet
Zero Gradient
Nozzle
Slip
Outlet
Zero Gradient
Inlet
298 K
Nozzle
Zero Gradient
Outlet
Zero Gradient
8
Table 3: Initial &Boundary Conditions for Supersonic flow with normal shock in a convergent-divergent
nozzle.
Case 3. Supersonic flow with normal Shock.
Pressure
Velocity
Temperature
Inlet
10000 Pa
Nozzle
Zero Gradient
Outlet
7500 Pa
Inlet
Zero Gradient
Nozzle
Slip
Outlet
Zero Gradient
Inlet
298 K
Nozzle
Zero Gradient
Outlet
Zero Gradient
3.4 Solvers:
OpenFOAM is an open-source CFD software. It can solve a vast range of engineering
problems, including complex fluid flow involving chemical reactions, turbulence, and heat
transfer.
A CFD analysis of the convergent-divergent nozzle is performed using these two
solvers,rhoCentralFOAM and rhoPimpleFOAM, with the same mesh configuration and
boundary conditions. These two solvers are compressible fluid flow solvers performing transient
analysis. Heat transfer, turbulence, and shock-capturing analysis can also be done using these
solvers. Inlet pressure, velocity, and temperature are required input fields to perform an analysis
in these two solvers. The thermophysical properties of air, assuming perfect gas, is used. The
simulation type is laminar. The comparison results will be explained in the results and discussion
section.
4. Results and Discussions
The simulations are run on OpenFOAM and the post-processing is done using ParaView. In
this section, the pressure distribution for all three cases is briefly described. The comparison
between the results of the selected OpenFOAM solvers is also described in detail concerning the
9
respective figures. These results are then validated using experimental results obtained from
previous research [1].
4.1 Solver Results:
In this section, the results obtained using the two selected solvers are briefly explained. A
comparison will be made at the last of this section comparing the validity of results obtained
using these solvers. These results are compared with the experimental results to check the
validity.
4.1.1 RhoCentralFOAM Results:
Case 1: Isentropic Subsonic Flow:
Figure 7: The pressure distribution of Isentropic Subsonic flow for Pexit / Pin = 0.89
10
Mach number (M)
Mach Number Distribution Graph
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
2
4
6
8
10
12
x (in)
Figure 8:Mach number along the centerline for pexit / pin = 0.89.
Pressure Distribution Graph
Pressure x104 (MPa)
1.2
1
0.8
0.6
0.4
0.2
0
0
2
4
6
8
10
12
x (in)
Figure 9: Static pressure along the centerline for pexit / pin = 0.89.
The Mach number and pressure variation along the nozzle is shown in fig. 8 and fig. 9
respectively. From the fig 7 and 9, it is observed that the cross-sectional area is maximum at the
inlet, so the pressure is maximum while the velocity of flow is minimum. As the cross-sectional
area decreases in the converging section, then pressure also decreases gradually whereas the
velocity of flow increases. At the throat of CD nozzle, the area is minimum; therefore pressure is
lowest and velocity is the highest. Afterwards, in the diverging section, the area starts increasing
whereas velocity decreases gradually but pressure increases. The exit pressure is low in
11
comparison to inlet pressure. The velocity of air is within the subsonic region, i.e. Mach number
is less than one throughout the flow, as shown in figure 8. The pressure and Mach number
profiles match with the theoretical observation discussed in section 2.1.2. The flow, as indicated
in fig. 8, remains subsonic throughout the nozzle
Case 2: Isentropic Expansion from Subsonic to Supersonic Speeds
Figure 10: The pressure distribution of Supersonic flow for pexit / pin = 0.16 using rhoCentralFOAM
Mach Number (M)
Mach Number Distribution Graph
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
0
2
4
6
8
10
x (in)
Figure 11:Mach number along the centerline for pexit / pin = 0.16.
12
12
Pressure Distribution Graph
Pressure x 104 (MPa)
1.2
1
0.8
0.6
0.4
0.2
0
0
2
4
6
8
10
12
x (in)
Figure 12: Static pressure along the centerline for pexit / pin = 0.16.
At the inlet, Pressure is maximum, whereas the velocity is minimum as shown in fig. 12 but the
flow is subsonic because the Mach number is less than 1. In the converging section, Pressure
starts decreasing gradually. At the same time, the velocity of flow increases isentropically, and
the flow in the converging section is subsonic because the Mach number is less than one as
observed in fig. 11. At the throat of the nozzle, Mach number is equal to 1 i.e. flow is sonic. As
the ratio of Pexit to Pin is 0.16<0.5283, the pressure ratio lies in the supersonic region. So at the
diverging section, Pressure decreases whereas the velocity of flow increase isentropically, and
Mach number increases greater than 1 (Supersonic flow at the diverging region). The pressure
and Mach number profiles match with the theoretical observation discussed in section 2.1.1. The
pressure variation, as indicated in fig. 12, also follows the isentropic relation throughout the
nozzle.
13
Case 3: Supersonic Flow with Normal Shock:
Figure 13:The pressure distribution of Supersonic flow with normal shock for pexit / pin = 0.75
Mach Number Distribution Graph
1.4
Mach Number (M)
1.2
1
0.8
0.6
0.4
0.2
0
0
2
4
6
8
10
x (in)
Figure 14: Mach number along the centerline for pexit / pin = 0.75.
14
12
Pressure Distribution Graph
Pressure x104 (MPa)
1.2
1
0.8
0.6
0.4
0.2
0
0
2
4
6
8
10
12
x (in)
Figure 15: Static pressure along the centerline for pexit / pin = 0.75.
The pressure field at steady-state in the 𝑥𝑦-plane is shown in fig. 13. The Mach number and
pressure variation along the nozzle is shown in fig. 14 and fig. 15 respectively.
In the
converging section, Pressure starts decreasing isentropically. At the same time, the velocity of
flow increases isentropically, and the flow in the converging section is subsonic because the
Mach number is less than one as observed in fig. 14. At the throat of the nozzle, Mach number is
equal to one, i.e. flow is sonic. At the diverging section, Pressure decreases whereas the velocity
of flow increases isentropically and Mach number increases greater than one(Supersonic flow at
the diverging region). Due to the normal shock at the midway at the diverging zone, there is a
sudden rise in pressure. At the same time, a sudden fall in velocity and afterwards, the pressure
increases isentropically, and velocity decreases isentopically. After the normal shock, the Mach
number becomes less than 1 (Subsonic flow after the shock). The shock is located at 𝑥=7.5. This
result agrees well with the analytical solution (𝑥=7.5623).
15
4.1.2 RhoPimpleFOAM Results:
Case 1: Isentropic Subsonic Flow:
Figure 16:The pressure distribution of Subsonic flow for pexit / pin = 0.89 using PimpleFOAM solver
Mach Number Distribution Graph
0.9
Mach number (M)
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
2
4
6
8
10
x (in)
Figure 17: Mach number along the centerline for pexit / pin = 0.89.
16
12
Pressure Distribution Graph
Pressure x104 (MPa)
1.2
1
0.8
0.6
0.4
0.2
0
0
2
4
6
8
10
12
x (in)
Figure 18: Static pressure along the centerline for pexit / pin = 0.89.
The pressure field at steady-state in the 𝑥𝑦-plane is shown in fig. 16. The Mach number and
pressure variation along the nozzle is shown in fig. 17 and fig. 18 respectively. It is observed
from fig, 18 that the cross-sectional area at the inlet is maximum so that the pressure is maximum
while the flow velocity is minimum. As the cross-sectional area decreases in the converging
segment, then pressure also decreases isentropically while the flow velocity increases. The area
is minimum at the throat of the CD nozzle; thus, the pressure is the lowest and the highest
velocity. Afterwards, the region begins to increase in the diverging segment as the velocity
slowly decreases but the pressure increases. Compared with inlet pressure, the exit pressure is
low. The air velocity is in the subsonic region during the entire flow, as shown in fig 17. The
pressure and Mach number profiles match with the theoretical observation discussed in section
2.1.2. The flow, as indicated in fig. 17, remains subsonic throughout the nozzle.
17
Case 2: Isentropic Expansion from Subsonic to Supersonic Speeds
Figure 19:The pressure distribution of Supersonic flow for pexit / pin = 0.16 using rhoPimpleFOAM solver.
Mach Number Distribution Graph
1.6
Mach Number (M)
1.4
1.2
1
0.8
0.6
0.4
0.2
0
0
2
4
6
8
10
x (in)
Figure 20: Mach number along the centerline for pexit / pin = 0.16.
18
12
Pressure Distribution Graph
Pressure x 104 (MPa)
1.2
1
0.8
0.6
0.4
0.2
0
0
2
4
6
8
10
12
x (in)
Figure 21: Static pressure along the centerline for pexit / pin = 0.16.
The Mach number and pressure variation along the nozzle is shown in fig. 20 and fig. 21
respectively. The pressure is highest at the inlet while velocity is minimum as shown in fig. 21 ,
But the flow is subsonic since the Mach number is less than 1. Pressure starts to decrease slowly
in the converging section as the velocity of flow increases isentropically, and the flow in the
converging section is subsonic since the Mach number is less than one as seen in fig. 20. Mach
number is equal to 1 at the throat of the nozzle i.e. flow, is Sonic. The pressure ratio lies in the
supersonic zone, as the ratio of Pexit to Pin is 0.16<0.5283 . Pressure decreases in the diverging
area, while the flow velocity increases isentropically, and Mach number also increases above
one, i.e. Supersonic flow in the diverging zone. The pressure and Mach number profiles match
with the theoretical observation discussed in section 2.1.1. The flow, as indicated in fig. 20,
reaches sonic speed at the throat, from where it expands to supersonic speeds at the exit. The
pressure variation, as indicated in fig. 21, also follows the isentropic relation throughout the
nozzle.
19
Case 3: Supersonic Flow with Normal Shock:
Figure 22: The pressure distribution of Supersonic flow with normal shock for pexit / pin = 0.75 using
rhoPimpleFOAM solver.
Mach Number Distribution Graph
1.4
Mach Number (M)
1.2
1
0.8
0.6
0.4
0.2
0
0
2
4
6
8
10
x (in)
Figure 23: Mach number along the centerline for pexit / pin = 0.75.
20
12
Pressure Distribution Graph
Pressure x104 (MPa)
1.2
1
0.8
0.6
0.4
0.2
0
0
2
4
6
8
10
12
x (in)
Figure 24: Static pressure along the centerline for pexit / pin = 0.75.
The Mach number and pressure variation along the nozzle is shown in fig. 23 and fig. 24
respectively. Pressure decreases isentropically in the converging section. At the same time, the
velocity of flow rises isentropically, and the flow in the converging section is subsonic since the
Mach number is less than one as seen in fig. 23. Mach number is equal to 1 at the throat of the
nozzle i.e. flow, is sonic. Pressure decreases in the diverging area, while flow velocity increases
isentropically, and Mach number also increases greater than one, i.e. Supersonic flow in the
diverging zone. And there is a sudden rise in pressure due to the sudden shock at the midway in
the diverging region; thus a sudden fall in velocity and consequently the pressure increases and
the velocity decreases isotopically. The pressure and Mach number profiles match with the
theoretical observation discussed in section 2.1.3. The flow, as indicated in fig. 23, reaches sonic
speed at the throat, from where it expands to supersonic speed till the location of normal shock.
After the shock the flow slows down and remains subsonic for the latter section of the nozzle.
The shock is located at 𝑥=7.5. This result agrees well with the analytical solution (𝑥=7.5623).
21
4.2 Solver Comparison:
In figures 7-24, Mach number and pressure distributions for subsonic, supersonic with
normal shock and supersonic cases are shown. The results using rhoCentralFOAM solver are
shown in figure 7-15, while the results using rhoPimpleFOAM are shown in figures 16-24. The
Mach number and pressure distributions are almost identical to the analytical result (section 2.1)
and experimental results [1] for each case. Hence, the results generated from rhoCentralFOAM
and rhoPimpleFOAM are valid.
Table 4: Comparison between Selected Solvers with Experimental Results
Rho CentralFOAM
Rho PimpleFOAM
Valid Results
Valid Results
Valid Results
Valid Results
Valid Results
Valid Results
Case 1: Isentropic Subsonic Nozzle
Case 2: Isentropic Expansion from
Subsonic to Supersonic Speeds
Case 3: Supersonic Flow with Shock
Nozzle
5. Conclusion
The results, as mentioned above, conclude that rhoCentralFOAM and rhoPimpleFOAM
gave the best-matched solution for the convergent-divergent nozzle in this research. The results
of both solvers are identical to that of the experimental results [1]. Thus, both of the solvers can
be considered for performing CFD analysis of Convergent-Divergent nozzle. The results,
especially in the supersonic flow with normal shock, could have been captured better with a more
refined mesh. Adaptive meshing could also be used to capture the shock better. The simulated results
match well with the experimental result of the NASA CFD benchmark case [1].
22
References
[1]. NPARC Alliance Verification and Validation Archive. Steady, Inviscid Flow in a
Converging-Diverging Verification (CDV) Nozzle (NASA CFD benchmark case )
 https://www.grc.nasa.gov/WWW/wind/valid/cdv/cdv.html

https://www.grc.nasa.gov/WWW/wind/valid/cdv/cdv01/cdv01.html
[2]. Anderson, J.D., Modern Compressible Flow, McGraw Hill Inc., New York, 1984.
[3]. H. K. Versteeg and W. Malalasekera, An Introduction to Computational Fluid Dynamics The
Finite Volume Method, 2 ed. Pearson, 2007.
[4]. B P, Madhu & Sameer, S. & Kumar, M. & Mani, G.. (2017). CFD analysis of convergentdivergent and contour nozzle. International Journal of Mechanical Engineering and Technology.
8. 670-677.
23
Appendix
The Simulation case files can be downloaded from Google drive:
https://drive.google.com/drive/u/0/folders/1sEhxZlluXYCPZlet-HEt8XDZ0LNUu93M
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