Calculation Method for Determining the Thrust Parameter of Perspective Nozzles of Rocket Engines

Alyona Kirshina, Baltic State Technical University “Voenmeh” named after D. F. Ustinov, St. Petersburg, Russian Federation,

Artyom LevikhinBaltic State Technical University “Voenmeh” named after D. F. Ustinov, St. Petersburg, Russian

Anton KirshinBaltic State Technical University “Voenmeh” named after D. F. Ustinov, St. Petersburg, Russian Federation,

Anton MusteykisBaltic State Technical University “Voenmeh” named after D. F. Ustinov, St. Petersburg, Russia,

Konstantin MikhailovBaltic State Technical University “Voenmeh” named after D. F. Ustinov, St. Petersburg, Russia,

Yuliya AniskevichFSUE Krylov State Research Centre, St. Petersburg, Russia,

Abstract  To assess the energy efficiency of a propulsion system with an unconventional nozzle, it is necessary to evaluate the thrust parameter realized by the nozzle device of the engine. This article describes a calculation thrust method using the example of a wide-range nozzle of a rocket engine  a Tarasov  Levin nozzle with a flat wall The results of the thrust calculation by analytical dependencies are compared with the results of numerical simulation on high-altitude sections of the trajectory and in outer space. The calculation results are compared with the experimental values of thrust obtained on a bench three-­component chamber of a rocket engine under atmospheric conditions.

This calculation method can be effectively used both to evaluate new nozzle designs and to improve existing nozzle designs of wide-range rocket engines.

Keywords: rocket engine, Tarasov-­Levin nozzle, numerical simulation

© The Authors, published by CULTURAL-EDUCATIONAL CENTER, LLC, 2020

This work is licensed under Attribution-NonCommercial 4.0 International

I. Introduction

The research of new nozzle devices for rocket engines is a complex and multifactorial task. Operating of a rocket or aerospace vehicle, which were calculated to fly over a wide-range of altitudes, the engine should operate in closed to the calculated modes, when the pressure at the nozzle exit is approximately equal to the pressure in the environment. It seems promising to use the Tarasov — Levin nozzle as part of a rocket engine, which is able to work in a wide-range of altitudes due to the free boundary of the stream and the absence of energy loss due to the lack of stream sections, where traditional nozzle (Laval nozzle) operates in deep over-expansion and under-­expansion modes.

The purpose of the research is to evaluate the energy efficiency of a nozzle of wide-range rocket engine (WRE), using the Tarasov — Levin flat wall nozzle as an example.

The object of the study is the nozzle of a wide-range rocket engine — a Tarasov — Levin nozzle with a flat wall, the model of which was tested on a bench three-­component liquid propellant rocket engine on the basis of the Department of Aircrafts Engines and Power Plants of the Military Technical University “Voenmekh” named after D. F. Ustinov. Verification of the experimental results by the thrust parameter obtained on the bench liquid propellant rocket engine (LPRE) chamber is carried out by comparing the experimental data with the calculated values and the results of numerical simulation in one mode.

The traditional Tarasov — Levin nozzle [1, 2] is a circular nozzle with a spherical cavity at the exit. The nozzle exit forms a circular gap directed to the axis of symmetry. The Tarasov — Levin nozzle in terms of flights at different altitudes is more energy efficient than traditional schemes, we suppose; and using of the Tarasov — Levin nozzle will slightly reduce the cost of the flight, including the flights of putting objects into orbit.

Historically, the development of the Tarasov — Levin nozzle direction was associated with two ideas that were different from the idea of working in a wide-range of heights and speeds implemented in this study.

The first idea was connected with the possibility of organizing the afterburning of fuel in the cavity by the compressed detonation waves generated during spherical focusing of perturbations. The pulsating process in the nozzle should be carried out due to the excitation of resonant high-frequency fluctuations in a gas-dynamic resonator periodically filled with a fuel-air mixture, and heat is generated in the overcompressed detonation waves formed in the resonator [3]. This idea finds application in the concept of a jet engine equipped with a traction module [4, 5], in the variant of a dual-circuit turbojet engine and a single-­circuit turbo-­compressor engine. In the first case, the traction module in the outer contour of the dual-circuit turbojet engine plays the role of an afterburner: due to the afterburning of fuel in overcompressed detonation waves, the efficiency of such afterburner will be much higher than that of a conventional front-end device. In the second case, the main chambers are used to generate activated gas, burned in the traction modules. The staff of the gas dynamics laboratory of the Institute of Mechanics of Moscow State University under the guidance of academician V. A. Levin, in collaboration with the Moscow Aviation Institute and PJSC UEC-Saturn (Rybinsk), performed detailed numerical and experimental studies of an air-jet engine and a rocket engine with a circular nozzle, including studying the influence of physicochemical transformations [6, 7] and shock-wave processes [8] on the amplitude-­frequency characteristics of the flow [9, 10]. A sufficiently detailed review of the problem is given in the work [11].

In the work [12], it was doubted that a detonation process takes place in the engine cavity; it is assumed that the fluctuations occur according to the “Gavro whistle” scheme, their frequency being determined only by the volume of the cavity. The fluctuations can be enhanced by adding a sharp-­edged disc on the axis of symmetry.

The second idea is associated with increasing of main stream ejection ability and connection of the external stream mass to it due to the generation of nonlinear acoustic and shock waves. This direction is developing mainly through the efforts of V. I. Bogdanov (UEC-Saturn, Rybinsk) [13]. NASA’s works are also known [14], in which the ejector is combined with the Hartmann resonator. The basic information about the Hartmann resonator and calculation methods are given in the work [15].

NASA’s experiments showed ejection coefficient increasing in the presence of nonlinear acoustic waves by 20%. These works caused a series of works at the Moscow Aviation Institute and the Russian State Aviation Technical University (Rybinsk) on the subject of pulsating air-jet engines (PuAJE). These results are most fully generalized in open monograph by V. I. Bogdanov [16]. It was noted [17] that the specific propellant flow rate of the PuAJE decreases when the pipe-resonator length increases. In the work [18] the authors give a methodology of PuAJE thrust calculation. It is noted that when a pipe length is more than 20 calibers, the specific propellant flow rate (and process efficiency) of the PuAJE is practically compared with the small gas turbine setting indicators. Work on the PuAJE with a cavity and the Tarasov — Levin nozzle was continued for high supersonic and hypersonic speeds [19]: an external supersonic or hypersonic flow creates peculiar channel in which compression waves and vortices are spread, they are generated by spherical cavity [20].

II. Experiment Description

Comparative tests of the energy efficiency of a wide-range rocket engine nozzle and a traditional nozzle were carried out at the test complex at the Department of Engines and Power Plants of the Military Technical University “Voenmekh” named after D. F. Ustinov. The object of study is the nozzle of a wide-range rocket engine — the Tarasov-­Levin nozzle with a flat wall. The object of comparison is the conical Laval nozzle.

The test complex includes: a system of fuel components supplying and a cooling system, a rocket engine chamber, a test object. The scheme of ignition fuel feed and the scheme of main fuel feed are displacing. Nitrogen gas is used as a component for tanks pressurization. Nitrogen is also used as a control gas for pneumatically controlled valves and for mixing head blowing in the main fuel line. The oxidizing agent (ignition and main) goes from monoblocks to the corresponding fittings of the mixing head of the tested liquid propellant rocket engine (LPRE) chamber. The cooling water enters the inter-­shell space of the combustion chamber and cools it. Water is also entered into the injection sector to cool the combustion products of the fuel components. The Figure 1 shows the main components of the test complex: the rocket engine chamber and the tested object, the control and monitoring system console, the mnemonic diagram. During the stand operation the automated registration and recording of parameters are performed. These parameters ensure the test conditions formation and determine the object characteristics.

We carried out comparative tests and obtained experimental values of thrust and specific impulse for a conical Laval nozzle and for Tarasov-­Levin nozzle (WRE nozzle) with flat wall at pressure of 506625 Pa, 607950 Pa, 709275 Pa, 810600 Pa, 861263 Pa in the combustion chamber. The total test time was 1080 seconds (3 test cycles of nozzle devices, 360 seconds each cycle).

III. Calculation Methods

Numerical modeling of the WRE nozzle and comparable Laval nozzle processes is carried out in ANSYS Fluid. The same numerical model and boundary conditions are used in the calculations: averaged by Reynolds the Navier-­Stokes equations are solved using the k-ω SST turbulence model.

The k-ω SST turbulence model shows good convergence with the experimental results [21], it is a combination of k-ε and k-ω turbulence models. In free turbulent flow the SST model behaves like a k-ε turbulence model, and it leads to decreasing of initial conditions sensitivity and increasing of solution stability. In the area near the walls equations of the k-ω model are used for calculation.

The following assumptions were made in the calculations: the ideal gas model is used as a fuel component, the processes at the combustion chamber exit are stationary. For the correct simulation of the possibility of flow separation from the nozzle walls the size of the grid elements near the wall provides y+ ≈1.

Figure 1. Components of the test complex: a — LPRE chamber and test object; b — control and monitoring system console; c — mnemonic diagram.

A. Boundary Conditions

At the nozzle entry temperature and pressure values are set; these values correspond to the experimental values. The calculation is carried out for an experimental point where the pressure at the nozzle entry is 810600 Pa, temperature is 1086 K, total propellant flow rate is 0.286 kg / s. Static pressure and temperature at the nozzle exit vary from 101325 Pa / 288.15 K to 0.0319 Pa / 196 K, and that corresponds to atmospheric parameters in the ground conditions and at an altitude of 100000 m above the sea level.

The nozzle traction force calculation in the software package is carried out by the method of numerically integrating of pressure that is distributed on the nozzle walls (excluding aerodynamic drag forces). The resulting value is added to the pressure value distributed over the butt of the combustion chamber.

B. Laval Cone Nozzle Thrust Calculation

To refine the thrust calculation method in software a numerical simulation of the comparable nozzle was performed; this nozzle involved in the full-scale tests of the Laval conical nozzle. It was made a comparison of the numerical simulation results with the calculated thrust value made by formula (1), under condition that pch = 810600 Pa, p= 50000 Pa, where pchpare the chamber pressure and the nozzle exit pressure, respectively. Then thrust was calculated at off-balance mode at atmospheric pressure according to formula (2) at ambient pressure pam =101325 Pa. The results of the analytical calculation are compared with the result of numerical simulation and with the value obtained as a result of the experiment (Table 1).



By comparing the experimental value and the result of numerical simulation, we see the thrust of the rocket engine chamber with the Laval nozzle, calculated using software, exceeds the experimental value by 10%. The difference between the thrust values calculated by the calculation method using formulas (1), (2) and those calculated using software is negligible. It is assumed that the experimental value is underestimated because of losses occurring in the nozzle due to nonequilibrium processes. Thus the calculation method by numerical simulation can be considered as reliable and applicable for rocket engine nozzle thrust calculating.

Table 1. Comparison of Laval Nozzle Thrust

Flow conditionThrust value, N
of numerical experiment
Result of analytical dependencies calculation
pch = 810600 Pа,pam = 101325 Pа240268267
pch = 810600 Pа,pam pа = 101325 Pа307306

C. Calculation of Wide-Range Rocket Engine Nozzle Thrust

To derive the thrust formula for a WRE with a flat wall nozzle the definition of thrust is used: the resultant of all forces acting on the inner and outer walls of the chamber, with the exception of aerodynamic drag forces. The geometry of the chamber is divided into two components: the outer shell, which consists of the geometry of the combustion chamber and the nozzle outer casing, and the central body. For each component the resultant pressure forces are found. For the outer shell it is a resultant of the forces applied to the combustion chamber mixing head, to the tapering part and to the exit section of the nozzle. For the central body it is a resultant pressure force acting on the input and output sections of the central body. Figure 2 shows the distribution of forces acting on the walls of the chamber with a flat section nozzle.

Figure 2. Scheme: forces acting on the chamber walls.

The thrust can be calculated by the formula (2) derived from the thrust definition.

P = P1 – P2 – P3 – P4+P5, where (3)

cut ch

P(pch – pam)Fch +mΣ Wch (4)

cut kk-vx

Ppch Fch – pvx Fvx+ mΣ (Wch – Wvx) – pam(Fch – Fvx) (5)

cut out

P(pout – pam) Fout + mΣ Wout (6)

cut in_ct1-in_ct2

Ppin_ct2 Fin_ct2 – pin_ct1 Fin_ct1 + mΣ (Wct_2 – Wct_1) (7)

cut ct

P5= pct Fct + mΣ Wct + pam Fa (8)

where Fi is the surface area on which the pressure pi and velocities Wi act in the corresponding sections, mΣ is the total propellant flow rate, Fa is the area of the nozzle exit section, pam is the ambient pressure.

Figure 3 and Figure 4 show diagrams of the velocity and pressure parameters distribution in the exit section of the nozzle.

Figure 3. Diagram of the velocity distribution in the WRE nozzle.

In order to calculate according to the above formulas, the areas of the output section of the shell Fout and the area of the output section of the central body Fct are divided into zones: high-speed Fout2, Fct1 and low-speed Fout1Fct2. The pressure and velocity parameters in the rarefaction region behind the central body corresponds to the low-speed zone Fct2. Also an assumption is made about the constancy of pressure and velocity in sections ct_1 and ct_2.

Figure 4. Diagram of the preassure distribution
in the WRE nozzle.

Given the above, formulas (7), (6), (8) are written as follows:

P4= pch Fct + mΣ Wch (9)

P= pout1 Fout1 + pout2 Fout2 +mΣ (Wout1 + Wout2) – pam Fout (10)

Ppct1 Fct1 pct2 Fct2+ mΣ (Wct1+ Wct2) – pam (Fout – Fa) (11)

Here: Fct1 = 0.64Fct;

Fct2 = 0.36Fct;

Fout1 = 0.79Fout;

Fout2 = 0.21Fout.

The values of the pressure and velocity parameters: pout1pout2Wout1Wout2pct1pct2Wct1Wct2 are found by numerical integration of the desired parameters over the cross section. For the convenience of calculations the dependence of the parameters changing through the pressure in the combustion chamber was found — it is a known value. The relationship between the desired parameters and the pressure in the combustion chamber is found through the coefficients that were derived from the pressure and velocity diagrams depending on the flight altitude (12) — (17).

Wout1 = cc Wch (12)

Wout2 = c Wch (13)

Wct1 = gg Wch (14)

Wct2 = g Wch (15)

pct1 = ff Wch (16)

pct2 = f Wch (17)

pout1 = aa pch (18)

pout2 = a pch (19)

Thus, the parameters of velocity and pressure in the exit sections of the researching nozzle can be found as a product of the pressure (velocity) in the combustion chamber and the coefficients calculated from the diagrams of the distribution of parameters at the nozzle exit for different heights.

The coefficients variation intervals depending on the flight altitude change in the following ranges: сс = 1.8…2.3, с = 5.3…5.7, gg = 5.2…6.4, g = 1.1…1.2, ff = 0.63…0.66, f = 0.5…0.6, aa = 1, a = 0.63…0.66. The dependence of their changes on the flight altitude is presented in Figure 5.

Figure 5. A graph of the dependence of the velocity (pressure) coefficients on the flight altitude.

The dependences presented above make it possible to calculate the WRE nozzle thrust: Tarasov-­Levin nozzle with a flat wall at different altitudes without numerical simulation. Such approach greatly simplifies the calculation of the thrust of a rocket engine chamber with a Tarasov-­Levin flat wall nozzle.

IV. The Results

We compared the nozzle thrust counted by a method of numerical simulation and by calculation method (according to the dependencies presented above) with the experimental thrust value. Table 2 presents the data for comparative analysis.

The table shows that the results of the numerical experiment at atmospheric pressure are higher than the full-scale experiment results (by 27%) and higher than the value calculated from the analytical dependences (by 7%). It is suggested that it may be due to the following factors:

1) Numerical simulation describes the ideal process conditions in the nozzle.

2) The experimental value is underestimated because of losses occurring in the nozzle due to non equilibrium of the processes.

Table 2. WRE Thrust Nozzle Comparison

Flow conditionThrust value, N
of numerical experiment
of analytical dependencies calculation
pch = 810600 Pa,pam = 101325 Pa210267250
pch = 810600 Pa,pam = 50000 Pa280273
pch = 810600 Pa,pam = 5000 Pa308296
pch = 810600 Pa,pam pа = 0.0319 Pa312309

3) The difference between the analytical thrust value and the numerical simulation value is explained by the accuracy of the calculated coefficients (Figure 5).

V. Discussion

The proposed calculation method allows to evaluate numerically the value of the WRE thrust at various altitudes of the engine. This method was verified by comparing the results of numerical simulation, analytical calculation and experimental determination of conical Laval nozzle thrust. The difference between the thrust value calculated from the analytical dependence and that have taken from numerical simulation is not more than 1%. It was noted that the thrust value of the rocket engine chamber with the Laval nozzle, calculated in the software, exceeds the experiment thrust value by 10%, which may be due to losses arising from the nonequilibrium of processes. In the proposed analytical dependence of the WRE thrust calculation the calculated coefficients were determined based on the results of numerical simulation. The results of analytical assessment of the thrust parameter at various altitudes in comparison with the results of numerical simulation differ by 7%.

It is noted that the experiment value differs from the results of numerical modeling by 27%. The difference between the experimental value of the thrust for the Laval nozzle and the WRE nozzle (10% and 27%) is significant, it is necessary to repeat the experiment after improving the conditions of the processes on the test complex.

Thus, the proposed calculation method allows numerically evaluate the value of the WRE thrust at various altitudes of the engine. It should be noted that in the future it is planned to evaluate the calculated coefficients for other nozzle operating modes and for another rocket engine chamber in order to compare these values and provide recommendations for further analytical calculations.

VI. Conclusion

As a result of the research the following was performed:

1) The methodology for calculating thrust was verified by numerical simulation using the example of a conical Laval nozzle.

2) The analytical formula was derived for calculating the thrust of a nozzle of a wide-range rocket engine — a Tarasov-­Levin nozzle with a flat wall.

3) Based on the numerical simulation the coefficients for calculating the speed and pressure at the exit of the WRE chamber nozzle depending on the flight altitude are calculated.

4) The calculation and comparison of the results of the WRE nozzle thrust with the experimental value and with the numerical simulation result are made.

The proposed calculation method allows numerically evaluate the WRE thrust value at various altitudes of the engine.

A comparison of the results of the WRE nozzle numerical simulation with the calculated thrust values on the high-altitude sections of the flight path shows that the difference between the thrust values is 7%. The difference between the results of thrust calculation for a conical Laval nozzle calculated from the analytical dependence and that of numerical simulation is not more than 1%.


This work has been supported through financing of the Innovation Promotion Fund “Development-­NTI-2018” within the framework of Agreement No. 378GRNTIS5/42597 of 08/23/2018.

The authors would like to acknowledge all members of engineering staff, especially the head of the Department Laboratory of Aircrafts Engines and Power Plants of BSTU “VOENMEH” named after D. F. Ustinova A. A. Galadzhun for their fruitful cooperation and for help in full-scale experiment organizing.


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