We used contemporary computational fluid dynamics techniques to evaluate how the geometric parameters of a recessed nozzle affect the perfection of flow processes. We verified our numerical simulation and obtained acceptable agreement between numerical and experimental investigation results in terms of specific impulse loss. We plotted the discharge coefficient as a function of the geometrical parameters of a recessed nozzle. Our numerical investigation forms the basis of certain guidelines we developed for designing arc-based recessed nozzles.

Современными методами вычислительной гидродинамики реализована оценка влияния геометрических параметров утопленного сопла на совершенство процессов истечения. Проведена верификация численной модели, получено удовлетворительное соответствие численных и экспериментальных результатов исследований потерь удельного импульса. Установлены зависимости коэффициента расхода от геометрических параметров утопленного сопла. По результатам численных исследований сформулированы некоторые рекомендации для проектирования радиусных утопленных сопел.

Today, the main goal of design developments related to rocket engines is to increase aircraft speed, range, and manoeuvrability. The specified parameters can be improved, for example, by increasing discharge characteristics of the engine.

One of the parameters indicating the perfection of nozzle flow processes is the discharge coefficient. According to the study [

Gas-dynamic losses are the main component of all losses. As a rule, they are determined by the following geometric parameters:

The distinctive design feature of many solidpropellant rocket engines (SPRE) is that they have a recessed (flush mounted) nozzle. Using this nozzle design allows to reduce overall dimensions of the rocket engine and therefore to avoid an increase in loss due to chemical non-equilibrium and dissipation of flow. Improving the SPRE design configuration leads to extra losses [

The studies [3, 6] provide data indicating specific impulse losses caused by the SPRE recessed nozzles. In case the recess depth changes in the range of 0.3…0.5, the specific pulse loss varies between 0.22 %...1.4 % with the aluminium content in the propellant 5…21.5 %. Hereinafter the recess depth is the ratio of the length of the nozzle’s recessed section to charge length

The known studies [3, 6] lay emphasis on the trend of increasing specific impulse losses with increasing recess depth. With the fixed recess depth, the specific pulse losses are getting higher as the percentage of the condensed phase content in combustion products grows. Maximum losses correspond to a rocket engine discussed here, having the maximum recess depth = 0.75 and high content of aluminium in propellant equal to 21.5 %. However, an engine with aluminium content of 16 % at = 0,26 takes the second place in terms of the loss value. These facts question the said trend that the recess depth and the content of condensed phase in combustion products affects specific impulse losses without regard to the shape of a recessed nozzle, and in their turn, define the importance of the gas-dynamic component of the losses. Therefore, the given data indicates that we can investigate how the recess depth affects the perfection of flow processes when studying homogeneous environment.

The above-mentioned studies have been conducted with no regard to different components of losses, i.e. a complex effect of the nozzle recess depth on the perfection of flow processes has not been considered. In addition to the flow inhomogeneity and non-equilibrium at the nozzle inlet, the complex effect also implies the dependence of losses on different geometric parameters of the recessed nozzle section similarly to the specified parameters of “classic design” axis-symmetrical nozzles (Fig. 1).

There are a lot of various geometric shapes of nozzle recessed sections, but, first and foremost, when analysing how its geometric parameters affect the perfection of flow processes and comparing it with the existing dependencies, it isreasonable to consider its radial shape. Studying the effect of the recess depth and geometric parameters of the radial recessed section of the nozzle upon the gas-dynamic loss component is a process of interest intended to enhance the performance of rocket engines being developed and improved.

This paper describes contemporary computational fluid dynamics techniques used for estimating the effect of geometric parameters on the perfection of flow processes. We have used the ANSYS Fluent software for simulation in an axis-symmetrical approximation with the ideal gas-adiabatic conditions of the steady-state problem. Objects of research are axis-symmetrical supersonic recessed nozzles in the rocket engine combustion chamber.

Based on the existing experimental results of the study of specific impulse losses due to recessed nozzle design [3, 4, 6], we can verify the numerical simulation method [

The computational model has been verified according to the diagram shown in Fig. 2. We have studied the rocket engine with minimum cross-section diameter Dкр = 200 mm, with a cylindrical charge featuring a conical section near the rear bottom, the length of which is Lз = 2400 mm.

The nozzle recess depth has varied within = 0,09...0,34. The working medium is the air supplied from the charge surface at a temperature corresponding to the operating conditions.

In addition to the combustion chamber and the nozzle, the geometric layout for computations comprised an extra volume for simulation of the jet discharge into free space. This allowed to skip the determination of boundary conditions at the nozzle outlet. The total number of computational grid cells was about 180,000, while non-dimensional number y+ calculated by the near-wall grid step and the dynamic velocity was not greater than 35 at the nozzle inlet.

The following boundary conditions of simulation have been established for the study: the surface of the assumed solid propellant has uniform distribution of the working medium, its temperature and flow turbulence parameters; constant atmospheric pressure at the outlet of the additional volume; the combustion chamber walls and nozzle walls are smooth with no-slip and impermeability conditions for the working medium.

As turbulence models, we studied a twoparameter RNG k − ε model with a typical set of model constants verified for this type of problems [

Specific impulse losses ξ are determined as the ratio of the difference between the theoretical value of the specific impulse and its actual value reduced to the theoretical specific impulse [

ξ = (Iид - I)/Iид.

To compare the obtained results of simulation with the known data [3, 6], specific impulse losses ξ can be expressed through the effective flow velocity and calculated using the following formula

where ωидэф – effective ideal flow velocity;ωa – calculated values of flow velocity at the nozzle exit;pa – calculated values of pressure at the nozzle exit;pH – pressure inside the additional volume simulating the environment;ρa – flux density at the nozzle exit.

The share of specific impulse losses due to nozzle recess depth ξут is determined as the ratio , of the difference between losses with recess ξLут and without recess ξLут =0 to the losses for a protruded nozzle:

Fig. 3 shows a wide range of specific impulse loss variations depending on the nozzle recess depth and aluminium content in propellant. Calculated values of losses obtained for homogeneous environment are in good agreement with the experimental data at the minimum content of the condensed phase in combustion products. Satisfactory verification results allow to justify the possibility for studying the discharge coefficient (only by the gas-dynamic component with no regard to the condensed phase) of the SPRE recessed nozzle depending on geometric characteristics by the gas-dynamic component for a computational model with homogeneous working medium.

Let us proceed to study the effect of geometric characteristics of an arc-based recessed nozzle of the combustion chamber. Based on simulation results, we can calculate the gas-dynamic component of the discharge coefficient using thefollowing formula

where - actual flow rate value;A(k) - gas-dynamic function;k – thermal capacity ratio;poc – pressure;R – equilibrium value of gas constant across the minimum nozzle cross-section;Toc – stagnation temperature at nozzle inlet.

The complex effect of nozzle recess depth on the discharge coefficient is determined by the ratio

where – discharge coefficient with a protruded nozzle;– discharge coefficient with current recess depth.

The combustion chamber of the rocket engine with a cylindrical charge, a supersonic conical nozzle and a radial recessed inlet has been used as the computational geometric model (Fig. 4). Recess depth varies in the range of 0…0.35. The charge length is Lз = 400 mm, the minimum cross-section diameter is Dкр = 40 mm. The simulation boundary conditions, working medium, and turbulence model are similar to the previous computational model. The total number of computational grid cells is about 150,000; nozzle inlet y+ is not greater than 35, similarly to the previous computational model. Geometric parameters under study:

When investigating the effect of the relative radius of recessed nozzle inlet to minimum cross-section upon the discharge coefficient, we have analysed variation = 0,1 ...1,0 for differentnozzle recess depths. We have found out that has the similar effect on the discharge coefficient both for recessed nozzles discussed herein and for“classic design” nozzles [2, 4] (Fig. 5). The paper [

Fig. 6 shows the results represented in the paper regarding the recess depth. It is shown that for the same recess depth the discharge coefficient may considerably depend on . We should note that the higher , the lower the discharge coefficient for all the discussed .

The component that takes into account the effect of recess depth relative to a protruded nozzle with the relevant relative radius and inlet nozzle section angle θкр= 90° , changes nonuniformly for recessed nozzles with different (Fig. 7).

For nozzles with , a recessed nozzle in the combustion chamber has the similar effect on losses. The effect of recess depth in nozzle design is in the range of 0…0.2 %, while for nozzles with relative radius < 0,5 this parameter may reach 0.88 %. We may conclude that nozzles with > 0,5 can be recessed deeper with no risk of severe degradation of discharge characteristics with other geometric conditions being equal. When simulating nozzles with less than 0.5, we should take into account a considerable increase in loss if the nozzle recess is getting deeper.

The effect of the relative radius of the recessed nozzle inlet section on the discharge coefficient has been investigated for a geometric model, which has the nozzle inlet with constant radius R2 and at R3 = 0 (Fig. 8). When studying the effect of , radius R2 has been simulated as the quadrant, the initial and end points of which are located in horizontal and vertical planes, respectively. As the nozzle inlet section radius was becomingsmaller, the circumference sector reduced accordingly. Calculation has been conducted for maximum relative radius = 1.

Obtained results prove that the relative radius of the recessed nozzle section inlet shall be ≤ 0,75 irrespective of the recess depth (see Fig. 8). Further reduction of the inlet section relative radius severely affects discharge characteristics with other geometric conditions being equal. Degradation of the discharge coefficient with the inlet section radius reduced from 0.44 to 0.95 reaches 13.16 % at = 0.35. When the recess depth is reduced, the effect on the discharge coefficient is reduced too. The paper [

Therefore, for rocket engines with recessed nozzles, degradation of discharge characteristics mainly depends not only on the recess depth and the content of the condensed phase in combustion products with the ambiguity of their effect confirmed through the analysis of the existing experimental data and calculation results, but also on geometric parameters of the recessed nozzle inlet section. Verification of the numerical model proves that the gas-dynamic component of the discharge coefficient can be used for analysing the perfection of flow processes in rocket engines with recessed nozzles. Numerical studies prove that for nozzles with the similar recess depth the discharge coefficient may vary up to 13 % if the geometry of the inlet section changes.

Based on research data for designing radial recessed nozzles, the following recommendations may be given:

The authors declare that there are no conflicts of interest present.