Flow separation in rocket nozzles has been studied mostly under sea-level conditions, which fail to take into account changes in ambient density and ambient pressure during the flight of a rocket. In the present study, numerical analysis is conducted of flow characteristics within a truncated ideal contour (TIC) nozzle to investigate the influence of ambient density and pressure on flow separation. Six different altitudes from a typical flight are considered, from a very low altitude to a high altitude. The flow is analyzed by varying the nozzle pressure ratios corresponding to these altitudes. Both cold flow and hot flow simulations are conducted. The locations of separation positions at various altitude conditions are accurately captured and are found to be in good agreement with experimental results. The results of the study establish that for a given nozzle pressure ratio, the flow separation point is shifted upstream with increasing altitude. This clearly points to a dependence of separation position on the altitude of operation for TIC rocket nozzles.

A rocket nozzle is designed to work at its optimum performance for only one particular pressure ratio at which the gases expand without any internal shocks and associated losses. Nozzles used in launch vehicles or space shuttles have operational ranges varying from sea level to outer space. As the altitude increases, a rocket nozzle has to operate from a low-pressure ratio to a high-pressure ratio. During ascent, flow separation may occur within the nozzle owing to over-expansion. In advanced rocket nozzles,1,2 and altitude-adaptive nozzles such as aerospike nozzles,3–5 dual-bell/double divergent nozzles,6–10 and expansion deflection nozzles,11 the losses due to over- and under-expansion can be minimized. In a conventional supersonic rocket nozzle, the boundary layer separates beyond a certain degree of over-expansion. In some cases, this flow separation can be asymmetric, leading to side loading.12 Flow phenomena within one conventional type of convergent–divergent rocket nozzle, a truncated ideal contour (TIC) nozzle, are illustrated in Fig. 1. For the efficient design of a rocket engine, it is very important to have accurate information on flow phenomena in the nozzle, especially the flow separation.

FIG. 1.

Flow phenomena within a TIC nozzle.

FIG. 1.

Flow phenomena within a TIC nozzle.

Close modal

Flow separation has previously been studied by adjusting the nozzle pressure ratio (NPR) p0/pa to representative values at sea level.13,14 In 1948, Swan15 studied the effects of flow separation on the performance of a conical nozzle under over-expanded conditions and concluded that the separation position varied linearly with chamber pressure and that the extent increased with decreasing specific heat ratio γ. In addition, it was found that an increase in the cone angle decreased the area ratio at which jet separation occurs. In 1949, Foster and Cowles16 conducted a detailed study of flow separation in rocket nozzles. This study was primarily concerned with over-expansion when a rocket operated at an altitude lower than the design altitude for the exhaust nozzle. The separation point was observed to vary linearly with chamber pressure. A decrease in separation pressure psep was reported with increasing chamber pressure p0. The separation pressure was found to be independent of the propellant combination. In 1954, Summerfield et al.17 observed that the flow separation was symmetric and also confirmed the previous finding of a linear relation between pressure ratio and separation position. They also suggested the first separation criterion, known as the Summerfield criterion, which stated that the ratio of lowest wall pressure to ambient pressure psep/pa was approximately equal to 0.4. In 1955, Reshotko and Tucker18 suggested the Mach number ratio across the shock as a characteristic parameter for defining shock-induced separation. Also in 1955, Love19 studied laminar and turbulent boundary layer separation both experimentally and analytically and confirmed that the effect of Reynolds number on separation Mach number was negligible. In 1958, Rao20 introduced the method of characteristics (MoC) for designing the wall contour of an exhaust nozzle to obtain optimum thrust. Following this, in 1960, Parley and Campbell21 compared the flow characteristics of a 15° conical nozzle and a nozzle designed by the method of characteristics. They showed that at low operating pressure ratios, the extent of over-expansion in the designed nozzle was greater than that in a 15° conical nozzle. Although this resulted in a reduced thrust, it was beneficial in cases where flow separation was undesirable. In 1967, Lawrence22 investigated flow separation in supersonic nozzles and examined the reasons for asymmetric and unstable characteristics at low chamber pressures. He demonstrated that coupling between boundary layer separation and jet entrainment led to asymmetric and unstable separation. In a study conducted in 1973 by Nave and Coffey23 in high-area-ratio engines, the separated flow was found to be reattached to the wall. This phenomenon was called restricted shock separation (RSS). In 1998, Frey and Hagemann24 conducted experiments on the Vulcain engine for the Ariane launcher and on the Space Shuttle Main Engine (SSME) and provided an explanation for the appearance of restricted shock separation. The nozzle contour was found to have a strong influence on this phenomenon. Verma25 conducted an experimental study on a TIC nozzle in 2002 and explained the fluctuating characteristics dominated by shock-induced boundary layer separation under off-design over-expanded flow conditions. The study also showed an asymmetric flow separation at certain pressure ratios. In 2002, Deck and Guillen26 conducted a numerical simulation of a TIC nozzle. Flow separation was investigated together with side loads, and an analysis of the results suggested that unsteadiness existed downstream of the separation shock. The flow remained perfectly axisymmetric upstream. In 2004, Pilinski and Nebbache27 numerically studied flow separation in an over-expanded truncated ideal contour nozzle. Free shock separation (FSS) was observed at low and high pressures. A cap shock without reattachment of the boundary layer appeared between the two ranges of pressure ratios. In 2005, Stark28 conducted a series of cold and hot flow tests to investigate flow separation in nozzles and introduced a simple separation criterion. In an experimental study on the TIC nozzle in 2006, Stark and Wagner13 observed a convex-shaped Mach disk for NPR < 10 and a slightly concave disk for NPR > 20. They studied boundary layer separation and established a relation between forced side-loads and the shape of the shear layer. Following this, in 2009, they conducted an experimental study14 on a TIC nozzle, primarily focusing on the origin of a locally reattached flow condition at low-pressure ratios. At low nozzle pressure ratios, the convex Mach disk was tilted temporarily and redirected the flow toward the nozzle wall owing to boundary layer transition. This study introduced a separation criterion for turbulent nozzle flows for both cold and hot flows. A study conducted by Ruf et al.29 in 2010 showed that wall pressures and side-load moments exhibited the FSS behavior expected for TIC nozzles. To study flow separation under high-altitude conditions, Stark and Génin30 conducted an experiment in 2016. They observed that for a given NPR, the separation position shifted upstream under high-altitude conditions. In 2019, Liu et al.31 demonstrated a significant effect of flow separation on side-loads on nozzle walls. Also in 2019, Khobragade et al.32 studied the performance of a scaled supersonic rocket nozzle. They identified flow separation locations along the diverging section under different conditions of nozzle operation. Previous work at DLR Lampoldshausen comparing the density ratios under real flight conditions and simulated condition for Vulcain 2 and a Subscale Main Engine (SSME) had revealed a noticeable difference between the values obtained under actual and simulated conditions at high altitude.30 

There have been few studies of flow separation in TIC nozzles under high-altitude conditions, taking account of the variation in density with changes in altitude, despite the fact that flow transitions occurring under such conditions are of both theoretical interest and practical importance. The objective of the present study is to simulate the flow through a TIC nozzle at various altitudes in flight and investigate the flow separation within the nozzle at various altitudes during flight. The outlet conditions are similar to real flight conditions at a particular altitude. NPR values between 15 and 50 are considered in this study.

A two-dimensional axisymmetric TIC nozzle geometry is considered. The Reynolds-averaged Navier–Stokes (RANS) equations are solved along with the k-ω shear stress transport (SST) turbulence model. In the present study, the ideal gas equation of state is employed to close the governing equations, assuming a predominantly gaseous state within the system and that viscosity is determined according to Sutherland’s law.33 It is essential to acknowledge that this ideal gas assumption might lack complete accuracy in scenarios involving condensation. Furthermore, the effect of condensation on the separation position proves to be an interesting topic, prompting further investigation. This aspect will be explored in subsequent numerical studies to gain a more comprehensive understanding of the effect of condensation on the propulsion performance and separation characteristics of the nozzle.

The governing equations are as follows:

  • Continuity equation [Eq. (1)]
    ρv=0,
    (1)
  • Momentum equation [Eq. (2)]
    ρvv=p+μv+vT23vI,
    (2)
  • Energy equation [Eq. (3)]
    vρE+p=kT+cpμtPrtT+μeffv+vT23vIv,
    (3)

where ρ is the density, v is the velocity vector, p is the static pressure, μ is the molecular viscosity, I is the unit tensor, E is the total energy, kT is the thermal conductivity, cp is the specific heat at constant pressure, μt is the turbulent (eddy) viscosity, Prt is the turbulent Prandtl number for energy, and T is the temperature. The effective viscosity μeffis given by μeff = μ + μt.

Numerical simulations were performed using different turbulence models at an NPR of 25. The results shown in Fig. 2 suggest that the k-ω SST model34 gives the best match with the experimental results. This model has previously been shown to give superior predictions of flow characteristics featuring separation by Choudhury et al.,35 Nair et al.,36 and Paul P et al.11 for different supersonic nozzles. The k-ω SST model combines the features of the k-ω and k-ϵ models, with the k-ω model being used near the surface and the k-ϵ model in the outer region.

FIG. 2.

Wall pressure vs position for NPR = 25.

FIG. 2.

Wall pressure vs position for NPR = 25.

Close modal
The equations for the k-ω SST model are
ρvk=Γkk+GkYk,
(4)
ρvω=Γωω+GωYω+Dω.
(5)
In Eqs. (4) and (5), Gk represents the generation of turbulent kinetic energy k due to mean velocity gradients, Gω represents the generation of a specific rate of dissipation ω, and Γk and Γω are the effective diffusivities of kandω, respectively. Dω represents the cross-diffusion term, and Yk and Yω are the dissipations of k and ω, respectively.
The k-ω SST model provides a better prediction of flow separation compared with the majority of RANS models and also exhibits favorable performance in adverse pressure gradients. However, the model tends to overpredict turbulence levels in regions with intense acceleration. In response to this concern, we have meticulously defined the turbulence intensity I at the inlet using the following relationship [Eq. (6)] for a fully developed flow for a pipe of hydraulic diameter dh:
I=0.16Redh1/8.
(6)
The turbulence length scale l is set at 7% of dh. This approach has resulted in a favorable alignment with experimental results.

The divergent part of the nozzle contour is designed with a MATLAB37 code, using the discretized characteristic equations of the method of characteristics for an ideal contour. The contour was truncated based on the experimental study. An experimental study by Stark and Génin was considered for validation of the numerical code. The nozzle had a design Mach number of 5.15 and a throat radius of 0.01 m. The domain is extended by 100Rth in the horizontal direction from the inlet and by 40Rth in the vertical direction from the axis, where Rth is the radius of the throat. The computational domain with boundary conditions is shown in Fig. 3. The nozzle wall and lip thickness are taken as adiabatic no-slip walls (solid line). Everything external is specified as an outlet (pressure outlet).

FIG. 3.

Computational domain with boundary conditions.

FIG. 3.

Computational domain with boundary conditions.

Close modal
Grid was developed by using Ansys ICEM CFD38 as shown in Fig. 4. Coarse, medium, and fine grids (grids 1, 2, and 3, respectively) were constructed, and the grid independence of the solution was ascertained using Roache’s method.39 As per this method, the grid convergence index (GCI) for coarse and fine grids are given by Eqs. (7) and (8), respectively.
GCIcoarse=3εroro1,
(7)
GCIfine=3εro1,
(8)
and certainty is achieved when on satisfying the condition given by Eq. (9)
GCIcoarseroGCIfine,
(9)
where r is the ratio of the number of elements between the coarser and finer grids, o is the order of accuracy, assumed to be 2 in this study, and ε is the difference between the values of a chosen parameter computed at a particular node in the two different grids. For the present study, the Mach number at the center of the primary nozzle exit is the chosen parameter. GCIcoarse denotes the GCI between grids 2 and 1, and GCIfine denotes that between grids 3 and 2. The results obtained are presented in Table I. Further details of the grid independence study are given in Table II.
FIG. 4.

Computational grid.

FIG. 4.

Computational grid.

Close modal
TABLE I.

Grid size and Mach number.

GridNodesMach number at primary nozzle exit
72 668 0.816 
110 309 0.798 
166 173 0.806 
GridNodesMach number at primary nozzle exit
72 668 0.816 
110 309 0.798 
166 173 0.806 
TABLE II.

Details of grid-independence study.

r32r21ε32ε21GCIcoarseGCIfineroGCIfine
1.5064 1.5179 0.017 62 0.018 21 9.655 4.164 9.45 
r32r21ε32ε21GCIcoarseGCIfineroGCIfine
1.5064 1.5179 0.017 62 0.018 21 9.655 4.164 9.45 

In addition to the use of Roache’s method, the grid independence of the solution was also verified through a conventional grid-independence check by comparing the computational and experimental results as shown in Fig. 5. Based on the grid independence study, grid 2, with 110 309 nodes, was selected for subsequent simulations.

FIG. 5.

Grid independence: wall pressure vs position for NPR = 25.

FIG. 5.

Grid independence: wall pressure vs position for NPR = 25.

Close modal

Criteria for determining the point of flow separation generally involve the examination of key flow parameters such as wall pressure distribution, velocity profiles, wall shear stress, or streamline behavior. For the present study, the accurate identification of the flow separation location relies on monitoring the wall static pressure distribution, with a significant jump serving as a clear indicator of separation.

Governing equations were solved in steady state. Ansys Fluent software38 was used in the study. The coupled solver was used for faster convergence.40,41 The inlet pressure was specified for the inlet boundary, and the stagnation pressure was varied to simulate the flow at the values of NPR considered in the study. The pressure outlet was specified to match the ambient pressure of a particular altitude. The experiment was conducted under six different ambient pressure conditions from pressures of 780–80 mbar (cases C1–C6), as shown in Table III. A no-slip condition was specified at the nozzle walls.

TABLE III.

Outlet pressures at different altitudes.30 

Casepa (mbar)
C1 780 
C2 750 
C3 615 
C4 303 
C5 141 
C6 80 
Casepa (mbar)
C1 780 
C2 750 
C3 615 
C4 303 
C5 141 
C6 80 

The numerical analysis was performed in two phases: the initial simulation involved a cold flow simulation, which was followed by a hot flow simulation.

A cold flow test analysis was done numerically. The computational results obtained are validated against experimental results of Stark and Génin.30 Their study used dry nitrogen gas as the working fluid. To study altitude effects on the separation phenomenon, computations were performed for six cases. Outlet pressure was varied based on altitude, as shown in Table III. The NPR was changed within a range of 13–55. The Mach contours and schlieren image obtained for an NPR of 14 in case 2, with an outlet pressure of 750 mbar at an altitude of about 2.465 km, are shown in Fig. 6. Subsequently, the NPR was increased to 26 and then to 36, and Figs. 7 and 8 illustrate the respective Mach contours and numerical schlieren images. It is clear that the point of separation shifts downstream with an increase in NPR. For all cases considered in the analysis, when NPR increased, the separation position shifted downstream of the nozzle.

FIG. 6.

NPR = 14, case 2.

FIG. 6.

NPR = 14, case 2.

Close modal
FIG. 7.

NPR = 26, case 2.

FIG. 7.

NPR = 26, case 2.

Close modal
FIG. 8.

NPR = 36, case 2.

FIG. 8.

NPR = 36, case 2.

Close modal

The shift in the point of separation with changes in altitude and NPR for case 2 is plotted in Fig. 9. At low NPR (<30), the computed values are observed to match well with the experimental results. At NPR > 30, there is a deviation from the experimental results. This is due to the temperature of the nozzle walls falling below a threshold value, causing the nitrogen to condense near the walls and leading to delayed separation.30 A computational analysis assuming perfect gas behavior fails to account for this condensation. Figures 10 and 11 show the wall pressure distribution and shift of separation position, respectively, for case 1. The same trend is observed at this altitude. As the NPR increases, the separation pressure psep decreases (Figs. 12 and 13).

FIG. 9.

Variation of separation position with NPR, case 2.

FIG. 9.

Variation of separation position with NPR, case 2.

Close modal
FIG. 10.

Variation of wall pressure with NPR, case 1.

FIG. 10.

Variation of wall pressure with NPR, case 1.

Close modal
FIG. 11.

Variation of separation position with NPR, case 1.

FIG. 11.

Variation of separation position with NPR, case 1.

Close modal
FIG. 12.

Variation of separation position with NPR, case 3.

FIG. 12.

Variation of separation position with NPR, case 3.

Close modal
FIG. 13.

Variation of separation position with NPR, case 4.

FIG. 13.

Variation of separation position with NPR, case 4.

Close modal

Numerical results for cases 5 and 6 show greater deviations from experimental values, as seen in Figs. 14 and 15, respectively. The numerical study predicts a linear trend, but the experimental values deviate from a linear trend. As NPR increases, the separation point shifts downstream, causing further expansion of flow. The higher Mach number leads to greater turbulence in the flow. Separation in experiments occurs further downstream, as the Xsep of experiments is greater than the Xsep of simulations for a particular NPR. This may be because the condensation occurring in experiments experiences higher turbulence. Condensing flow interacts with mean flow, resulting in higher turbulence and thus in delayed separation in the experiments.

FIG. 14.

Variation of separation position with NPR, case 5.

FIG. 14.

Variation of separation position with NPR, case 5.

Close modal
FIG. 15.

Variation of separation position with NPR, case 6.

FIG. 15.

Variation of separation position with NPR, case 6.

Close modal

The numerical model is consistent with the results of Stark and Génin30 up to an NPR of 30, beyond which a deviation from these experimental results is observed. This deviation is attributed to the condensation of nitrogen near the walls, with the interaction of the condensation droplets with the mean flow inducing stronger turbulence and thus leading to delayed separation. Because of the involvement of several physical processes in the formation and growth of these droplets, the modeling of this phenomenon is complex. Thus, the present understanding of supersonic condensation characteristics in a nozzle remains limited, with only a few studies having been conducted in this area. Additionally, supersonic flows may undergo nonequilibrium condensation, reaching thermodynamic states below saturation without actual liquid droplet formation. Hence, the dependability of commonly used multiphase models becomes highly uncertain when dealing with these nonideal condensation regions. As a result, the present study has not considered the real gas and condensation effects occurring under low-outlet-pressure conditions (NPR > 30). However, a more detailed analysis is planned for future studies to comprehensively address these complexities.

The experimental result for case 6 (Fig. 15) indicates a high position shift around an NPR of 25–28. This means that for a small change in NPR, there is a large difference in separation position. This strong gradient can be attributed to a laminar-to-turbulent transition of the flow.30 For sea-level conditions, as well as for case 6, the transition in separation characteristics occurs at Reynolds numbers of about 106 in the experimental results.30 Hence, the difference in separation position gradient for an NPR of 25–28 indicates a transition from laminar to turbulent flow separation. In cases 1–4, the flow is already turbulent at low NPR, and hence the transition is not observed.

The numerical results for the variation of the separation position with NPR for all six cases combined are shown in Fig. 16. For cases 1–4, the numerical results show a linear trend just like in the experimental results of Stark and Génin.30 By contrast, for cases 5 and 6, the experimental results30 exhibit a nonlinear trend, unlike the numerical ones. These factors point to a possible inadequacy of the turbulence model to accurately simulate the flow phenomena under the specified conditions. A laminar-to-turbulent transition is observed in the experimental study. Also, there is the possibility of condensation, leading to more turbulence and delaying separation.

FIG. 16.

Combined variation of separation position with NPR (numerical study).

FIG. 16.

Combined variation of separation position with NPR (numerical study).

Close modal

It can be seen from the experimental setup of Stark and Génin30 that the nitrogen flow at the nozzle inlet has high pressure and low temperature. The expansion of flow in the nozzle results in decreases in temperature and pressure, leading to induction of condensation and the formation of droplets in the flow. Modeling of this phenomenon is intricate, given the involvement of multiple physical processes in the formation and growth of the droplets. Accurately capturing the interaction between these condensed droplets and the supersonic core flow, along with the formation of multiple condensation shocks, presents significant challenges in numerical simulations. To adequately address these, a transition to time-consuming large eddy simulation (LES) is warranted for a more accurate representation of the intricacies of the flow, which is set as a target for future work.

The primary objective of the cold flow simulation was to find a relationship between ambient pressure and density on the one hand and the separation position on the other. The results obtained have shown that for a given NPR, as the altitude increases, flow separation is shifted upstream. This is illustrated in Fig. 17, where the separation position can clearly be seen to shift upstream as the density decreases. This indicates a change in separation position for a given NPR at different altitudes. This change has not usually been considered in studies of flow separation.

FIG. 17.

Variation of separation position with density.

FIG. 17.

Variation of separation position with density.

Close modal

A hot flow analysis is performed using a liquid oxygen/liquid hydrogen (LO2/LH2) combustion mixture to enable comparison between the results of cold and hot flow tests. Unlike the cold flow test, an experimental study was not conducted using hot gases, and so the comparison is made between numerical analyses of each test. The numerical simulation of the test is performed using this combustion mixture for six different cases, similar to the cold flow test. The NPR is varied from 15 to 50 for each case. An LO2/LH2 combustion mixture gas model is considered, and it is assumed that no reaction occurs inside the nozzle. The properties of the combustion mixture, namely, viscosity, thermal conductivity, and specific heat capacity, were determined using the NASA CEA code.42 Cases 5 and 6 of the cold flow test have also been included to verify whether there was any change in the results.

The ambient pressure for a particular altitude is specified to vary the NPR in each of the six different cases considered. As expected, for all six cases, the separation position shifts downstream when the NPR is increased. The trend of the results obtained for a hot flow is similar to that obtained in the cold flow test, showing a linear relation between NPR and separation position. The separation pressure psep decreased as the NPR increases, and the separation position shifts downstream, as shown in Figs. 18, 19, and 20, respectively. The trend is the same for all cases, as can be seen in Fig. 21.

FIG. 18.

Variation of wall pressure with NPR, case 1.

FIG. 18.

Variation of wall pressure with NPR, case 1.

Close modal
FIG. 19.

Variation of separation position with NPR, case 1.

FIG. 19.

Variation of separation position with NPR, case 1.

Close modal
FIG. 20.

Combined variation of separation position with NPR (numerical study).

FIG. 20.

Combined variation of separation position with NPR (numerical study).

Close modal
FIG. 21.

Variation of separation position with pressure.

FIG. 21.

Variation of separation position with pressure.

Close modal

As expected, the numerical analysis is contrary to the experimental results of the cold flow test for cases 5 and 6. To study altitude effects, a comparison is made between the six different cases. In the present numerical study, we have accounted for the ambient pressure variation to study flow separation at high altitudes. As the altitude increases, the ambient pressure decreases, which in turn affects the flow separation. The results show that for a given NPR, as the pressure decreases, the location of separation moves upstream, as shown in Fig. 21. This represents the change of separation position with the ambient pressure during a rocket ascent. For an NPR of 30, it can be seen that above 300 mbar, the separation point changes very little with ambient pressure, whereas below 300 mbar, i.e., at higher altitudes, there is a marked variation in Xsep. From the results, it is clear that the difference in change of separation position is higher at higher altitudes with higher NPR, and we can see that the slope varies considerably for NPR of 45 and 50 for an ambient pressure in the range from 303 to 80 mbar. Similar to the cold flow analysis, for a given NPR, there is a difference in separation position between different altitudes. As the NPR decreases, there is a gradual change in separation position. These differences in separation position have not been considered in previous studies. From this analysis, we can establish that there is a significant dependence of separation position on the altitude of operation.

The flow characteristics in a TIC nozzle at various altitudes have been analyzed through cold and hot flow simulations. The significance of ambient pressure and density for rocket nozzle design have been considered. Analysis of separation positions at various NPRs at different altitudes have confirmed a variation in flow separation with altitude compared with the results of conventional studies where changes in ambient density and ambient pressure have not been considered. For the cold flow simulation at low NPRs, the separation positions at various altitudes are nearly the same as the positions observed in experiments. At higher NPRs, the numerical scheme does not capture the condensation of nitrogen occurring in the experimental study, resulting in a significant difference in the results. The numerical studies of both tests show that as the altitude increases, flow separation is shifted upstream for the same NPR, with the separation position shifting upstream with decreasing density. The shift is larger at higher altitudes with lower NPR. The present numerical scheme has to be improved to accurately capture nozzle flow features beyond 13 km in the atmosphere. This will be taken up in further studies.

The authors have no conflicts to disclose.

Ijas Muhammed V V: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Validation (equal); Visualization (equal); Writing – original draft (equal). Shamsia Banu N: Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Validation (equal); Visualization (equal); Writing – original draft (equal). Abhilash Suryan: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Project administration (equal); Resources (equal); Software (equal); Supervision (equal); Writing – review & editing (equal). Vincent Lijo: Data curation (equal); Methodology (equal); Supervision (equal); Writing – review & editing (equal). David Simurda: Data curation (equal); Methodology (equal); Visualization (equal); Writing – review & editing (equal). Heuy Dong Kim: Conceptualization (equal); Methodology (equal); Resources (equal); Supervision (equal); Writing – review & editing (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.

1.
G.
Hagemann
,
H.
Immich
,
T.
Van Nguyen
, and
G. E.
Dumnov
, “
Advanced rocket nozzles
,”
J. Propul. Power
14
(
5
),
620
634
(
1998
).
2.
P. P.
Nair
,
V.
Narayanan
,
A.
Suryan
, and
H. D.
Kim
, “
Prediction and visualization of supersonic nozzle flows using OpenFOAM
,”
J. Visualization
25
(
6
),
1227
1247
(
2022
).
3.
P. P.
Nair
,
A.
Suryan
, and
H. D.
Kim
, “
Computational study of performance characteristics for truncated conical aerospike nozzles
,”
J. Therm. Sci.
26
,
483
489
(
2017
).
4.
S.
Soman
,
A.
Suryan
,
P. P.
Nair
, and
H.
Dong Kim
, “
Numerical analysis of flowfield in linear plug nozzle with base bleed
,”
J. Spacecr. Rockets
58
(
6
),
1786
1798
(
2021
).
5.
S.
Soman
,
A.
Suryan
,
P. P.
Nair
,
M.
Ritter
, and
H.
Dong Kim
, “
Study on control of unsteadiness and flow asymmetry in linear aerospike nozzles
,”
J. Spacecr. Rockets
60
(
1
),
326
338
(
2023
).
6.
M.
Raju
,
V. I.
Muhammed
,
A.
Suryan
, and
H. D.
Kim
, “
Computational study on the flow characteristics in a film cooled dual-bell nozzle
,” in
Proceedings of the 16th Asian Congress of Fluid Mechanics
(
Springer
,
Singapore
,
2021
), pp.
225
232
.
7.
M.
Raju
,
A.
Suryan
, and
D.
Šimurda
, “
Computational investigation of cooling effectiveness for film cooled dual-bell exhaust nozzle for LO2/LH2 liquid rocket engines
,”
Energy Sources, Part A
(published online)
(
2021
)..
8.
P. P.
Nair
,
A.
Suryan
, and
H. D.
Kim
, “
Computational study on reducing flow asymmetry in over-expanded planar nozzle by incorporating double divergence
,”
Aerosp. Sci. Technol.
100
,
105790
(
2020
).
9.
S.
Soman
,
J.
George
,
P. P.
Nair
, and
A.
Suryan
, “
Numerical study of flow through planar double divergent nozzles
,”
AIP Conf. Proc.
2134
(
1
),
020006
(
2019
).
10.
J.
George
,
P. P.
Nair
,
S.
Soman
,
A.
Suryan
, and
H. D.
Kim
, “
Visualization of flow through planar double divergent nozzles by computational method
,”
J. Visualization
24
,
711
732
(
2021
).
11.
J.
Paul P
,
P. P.
Nair
,
A.
Suryan
,
J. P.
Martin M
, and
H.
Dong Kim
, “
Numerical simulation on optimization of pintle base shape in planar expansion-deflection nozzles
,”
J. Spacecr. Rockets
57
(
3
),
539
548
(
2020
).
12.
P. P.
Nair
,
A.
Suryan
, and
R.
Chandran
, “
A numerical study on planar nozzles with different divergence angles
,” in
Recent Asian Research on Thermal and Fluid Sciences: Proceedings of the AJWTF7
(
Springer
,
2020
), pp.
133
146
.
13.
R.
Stark
and
B.
Wagner
, “
Experimental flow investigation of a truncated ideal contour nozzle
,” in
Proceedings of the AIAA/ASME/SAE/ASEE Joint Propulsion Conference and Exhibit
(
AIAA
,
2006
), p.
5208
.
14.
R.
Stark
and
B.
Wagner
, “
Experimental study of boundary layer separation in truncated ideal contour nozzles
,”
Shock Waves
19
,
185
191
(
2009
).
15.
W. C.
Swan
, “
The influence of nozzle design on the flight performance of rocket vehicles, with an analysis of the results of jet separation
,” M.S. thesis,
California Institute of Technology
,
Pasadena, CA
,
1948
.
16.
C. R.
Foster
and
F. B.
Cowles
, “
Experimental study of gas flow separation in over-expanded exhaust nozzles for rocket motors
,”
Technical Report No. JPL-PR-4-103
,
Jet Propulsion Laboratory, California Institute of Technology
,
Pasadena, 18 CA
,
1949
.
17.
M.
Summerfield
,
C. R.
Foster
, and
W. C.
Swan
, “
Flow separation in over-expanded supersonic exhaust nozzles
,”
Jet Propul.
24
(
5
),
319
321
(
1954
).
18.
E.
Reshotko
and
M.
Tucker
, “
Effect of discontinuity on turbulent boundary layer thickness parameters with application to shock induced separation
,”
No. NACA TN 3454
,
1955
.
19.
E. S.
Love
, “
Pressure rise associated with shock-induced boundary layer separation
,”
No. NACA TN 3601
,
1955
.
20.
G. V. R.
Rao
, “
Exhaust nozzle contour for optimum thrust
,”
J. Jet Propul.
28
(
6
),
377
382
(
1958
).
21.
J. M.
Parley
and
C. E.
Campbell
, “
Performance of several method of characteristics exhaust nozzles
,”
NASA Technical Note D-293
,
1960
.
22.
R. A.
Lawrence
, “
Symmetrical and unsymmetrical flow separation in supersonic nozzles
,”
No. NASA CR 92587
,
1967
.
23.
L. H.
Nave
and
G. A.
Coffey
, “
Sea level side loads in high area ratio rocket engines
,”
AIAA Paper No. 73-1284
,
1973
.
24.
M.
Frey
and
G.
Hagemann
, “
Status of flow separation prediction in rocket nozzles
,”
AIAA Paper No. 98-3619
,
1998
.
25.
S. B.
Verma
, “
Study of flow separation in truncated ideal contour nozzle
,”
J. Propul. Power
18
(
5
),
1112
(
2002
).
26.
S.
Deck
and
P.
Guillen
, “
Numerical simulation of side loads in an ideal truncated nozzle
,”
J. Propul. Power
18
(
2
),
261
269
(
2002
).
27.
C.
Pilinski
and
A.
Nebbache
, “
Flow separation in a truncated ideal contour nozzle
,”
J. Turbul.
5
,
014
(
2004
).
28.
R.
Stark
, “
Flow separation in rocket nozzles, a simple criteria
,” in
Proceedings of the AIAA/ASME/SAE/ASEE Joint Propulsion Conference and Exhibit
(AIAA,
2005
), p.
3940
.
29.
J.
Ruf
,
D.
McDaniels
, and
A.
Brown
, “
Details of side load test data and analysis for a truncated ideal contour nozzle and a parabolic contour nozzle
,” in
46th AIAA/ASME/SAE/ASEE Joint Propulsion Conference and Exhibit
(
AIAA
,
2010
), p.
6813
.
30.
R.
Stark
and
C.
Génin
, “
Flow separation in rocket nozzles under high altitude condition
,”
Shock Waves
27
,
63
68
(
2017
).
31.
H.
Liu
,
L.
Zhang
,
X.
Yu
,
H.
Liu
,
B.
Wang
, and
Y.
Luo
, “
The influence of flow separation mode on side-loads in a rocket nozzle
,”
J. Phys.: Conf. Ser.
1300
,
012090
(
2019
).
32.
N.
Khobragade
,
J.
Wylie
,
J.
Gustavsson
, and
R.
Kumar
, “
Control of flow separation in a rocket nozzle using microjets
,”
New Space
7
(
1
),
31
42
(
2019
).
33.
W.
Sutherland
, “
The viscosity of gases and molecular force
,”
Philos. Mag.
5
(
36
),
507
531
(
1893
).
34.
F. R.
Menter
, “
Two-equation eddy-viscosity turbulence models for engineering applications
,”
AIAA J.
32
(
8
),
1598
1605
(
1994
).
35.
S. P.
Choudhury
,
A.
Suryan
,
J. C.
Pisharady
,
A.
Jayashree
, and
K.
Rashid
, “
Parametric study of supersonic film cooling in dual bell nozzle for an experimental air–kerosene engine
,”
Aerosp. Sci. Technol.
78
,
364
376
(
2018
).
36.
P. P.
Nair
,
A.
Suryan
, and
H. D.
Kim
, “
Study of conical aerospike nozzles with base-bleed and freestream effects
,”
J. Spacecr. Rockets
56
(
4
),
990
1005
(
2019
).
37.
The Math Works, Inc.
,
MATLAB Programming Fundamentals
(
2016
).
38.
Fluent User’s Guide,
ANSYS, Inc.
,
Canonsburg, PA
, pp.
1
1146
.
39.
P. J.
Roache
, “
Perspective: A method for uniform reporting of grid refinement studies
,”
J. Fluids Eng.
116
(
3
),
405
413
(
1994
).
40.
K.
Kurbatskii
and
F.
Montanari
, “
Application of pressure-based coupled solver to the problem of hypersonic missiles with aerospikes
,” AIAA Paper No. 2007-462,
2007
.
41.
P. P.
Nair
,
A.
Suryan
, and
H. D.
Kim
, “
Computational study on flow through truncated conical plug nozzle with base bleed
,”
Propul. Power Res.
8
(
2
),
108
120
(
2019
).
42.
NASA chemical equilibrium with applications (CEA) code, https://cearun.grc.nasa.gov/intro.html.