Abstract
A numerical study has been performed to characterize the nozzle flow field of secondary injection thrust vector control (SITVC) and to estimate the performance parameters of SITVC. After validating the CFD turbulence models with an experimental data, a numerical simulation has been conducted in order to investigate the influence of changing the injection location, the injection angle, and the primary nozzle divergence half angle on the SITVC nozzle flow field structure and on the SITVC performance parameters. The secondary mass flow rate was kept constant for all cases during the simulation. The results showed that downstream injection near the nozzle exit increases the high-pressure zone upstream the injection leading to an increase in the side force; also, the higher divergence half angle 15° slightly increases the side force and it provides a wide range of deflection without shock impingement on the opposite wall becoming more effective for SITVC. The injection angle in the upstream direction 135° increases the side force, and by decreasing the injection angle to downstream direction 45°, the side force decreases. However, the SITVC performance parameters and the flow field structure are more influenced by the injection location and the primary nozzle divergence half angle while being less influenced by the injection angle.
1. Introduction
Thrust vector control (TVC) is a way that controls the thrust by deflecting the main flow of a rocket motor or jet engine from the main axis to generate a specified force on the desired axis. As a result of this imbalance, a side force arises to be used in controlling the attitude of the aerial vehicle. Thrust vector control greatly enhances maneuverability, especially at low velocities or high angles of attack where conventional aerodynamic control surfaces are not effective. The thrust can be controlled mechanically by using flex joints, hinged nozzles, jetavators, and jet vanes/tabs or by fluidic thrust vector (FTV) control. In contrast to the mechanical systems that need actuators to move the mechanical parts leading to complex design and weight penalty, fluidic thrust vector control is a technology with less weight, faster dynamic response, and no mechanical movable parts and is controlled by flow regulations, which decreases the axial thrust losses during a change in the direction of the thrust [1, 2]. There are different techniques of fluidic thrust vector control such as secondary injection thrust vector control (SITVC) or shock vector control (SVC), counter flow, Coanda effect, and throat skewing [3, 4].
From these techniques, SITVC is the most straightforward and effective technique of FTV [5, 6]. Since the 1960s, SITVC has been used successfully and is achieved by the injection of secondary fluid at the divergent part of the nozzle to the supersonic flow inside. The injection creates a complicated flow field in the divergent part of the nozzle upstream and downstream the injection where the upstream part contains a strong bow shock that creates asymmetry in the nozzle flow field, a weak separation shock, and two rotating vortices between the separation and the wall known as primary upstream vortex (PUV) and secondary upstream vortex (SUV) as a result of the separation that occurs in the boundary layer; in the downstream part there are a Mach disk that recompress the under-expanded secondary fluid and a reattachment zone with recompression that contains primary downstream vortices (PDV) [7–11]. Figure 1 illustrates the SITVC nozzle flow field structure [12]. The main reason for the deflection is the side force which is generated by a combination of the interaction force (due to a rise in the pressure along the injector wall) and the jet reaction force (as a result of the momentum of the secondary injection) [13].
Many analytical models as Linearized model [14], blast-wave analogy [9], and boundary layer separation model [15], characterized the flow field that is induced due to a secondary fluid interaction with supersonic flow, but due to its complexity, these analytical models have lack in general and deal only with very low injection flow rates, so experimental tests with cold flow [16, 17] and real static firing [18, 19] were utilized to provide the SITVC main data. However, experimental tests only provide macroscopic SITVC performance predictions with high costs. On the other hand, development of numerical methods and computational power results in the development of computational fluid dynamics (CFD) to give a microscopic detailed description and analysis of the flow characteristics. Numerical simulation gives accurate results, which can be a strong complementary to experimental work [20, 21].
Balu et al. [7] solved the Euler equations for the estimation of the secondary injection performance parameters. Dhinagram and Bose [22] solved both Navier-Stokes and Euler equations in two-dimensional nozzles concluding that solving Navier-Stokes equations gives more accuracy to the results. Ko and Yoon [23] present a flow analysis of SITVC by solving three-dimensional Navier stokes equations with the Baldwin-Lomax turbulence model and turbulence model. Sriram and Mathew [24] used the fluent code to simulate the flow field of the nozzle utilized by Masuya et al. [16] and Ko and Yoon [23] with different solution parameters. Vishnu et al. [25] used the realizable turbulence model with enhanced wall treatment for the design optimization of SVC. Zmijanovic et al. [26] investigated experimentally and numerically the effects of FTV within the framework of a small satellite launcher. Prince Raj et al. [27] used a fluent code to solve the three-dimensional governing equations to analyze the flow field produced by secondary hot gas injection in a nozzle rocket motor. Rajendran et al. [28] carried out numerical studies using the two-dimensional RNG turbulence model to compare between injecting sonic and supersonic jets for thrust vector control. Sellam et al. [5] analyzed the thermodynamic characteristics of the injected gas on FTV performance. Deng et al. [29] established an analytical model for transverse injection and performed a numerical model for secondary injection to study the thrust vectoring efficiency and system thrust ratio. Erdem and Kontis [30] carried out numerical simulations and experiments to analyze the transverse injection, then Erdem et al. [31] investigated experimentally the penetration characteristics of the transverse sonic jets in Mach 5 cross flows for air, carbon dioxide, and helium. Forghany et al. [32] studied numerically the effect of the fluidic injection angle on TVC system efficiency. Chandra Sekar et al. [33] carried out an experimental work to achieve FTV in the yaw direction by transverse injection in a converging nozzle with an elliptic exit and triangular after.
Although there are several researches on SITVC, there are rare investigations on the influence of the injection angle (downstream, perpendicular, and upstream inclination) and the primary nozzle divergence half angle on both the flow field structure and the performance parameters of SITVC. As a result, the current study is aimed at presenting a detailed qualitative analysis (without changing the secondary mass flow rate) on the influence of injection angle and primary nozzle divergence half angle on the SITVC flow field structure and performance parameters; also, the influence of injection location was investigated in order to investigate all the geometric parameters in the present study.
2. Numerical Methodology
Numerical simulation of the nozzle flow field in the presence of secondary injection has been performed. For this objective, Reynolds-averaged Navier–Stokes (RANS) equations, compressible, two-dimensional equations discretized in a finite volume form on each of the quadrilateral control volumes, were solved. An implicit density-based algorithm was utilized to solve the equations; for the flux type, Roe-averaged flux difference splitting (Roe FDS) was selected, and for the spatial discretization at the beginning, a first-order upwind scheme was chosen until a convergence behavior is shown, then a second-order upwind scheme was selected to increase the accuracy of the solution. The Courant-Friedrichs-Lewy (CFL) number starts with a small value of 0.5 to ensure the solution stability until a convergence behavior appears. Then, the CFL number increased to 1 to speed up the convergence, and the underrelaxation factor remains at 0.25 to ensure the solution stability. For material type, air (ideal gas) was used in all the cases.
In this specific problem, the injection of sonic jet into supersonic flow is inherently turbulent [24, 34]. Thus, the selection of the turbulence model was critical for the solution accuracy. In this study, the experimental data of Gushe [17] were used in the turbulence model validation.
The realizable model, model, and shear stress transport (SST) model were used in the validation against the experimental data.
The two transport equations for the realizable model are [35] where
Generation of turbulence kinetic energy due to mean velocity gradients
Generation of turbulence kinetic energy due to buoyancy
Contribution of the fluctuating dilatation in compressible turbulence to the overall dissipation rate
and Constants
and Turbulent Prandtl number for and .
The two transport equations for the model are [35] where
Generation of turbulence kinetic energy due to mean velocity gradients
Generation of
and Dissipation of and due to turbulence
and Dissipation of and due to turbulence
The two transport equations for the shear stress transport (SST) model are [35–37] where
Generation of turbulence kinetic energy due to mean velocity gradients
Generation of
and Dissipation of and due to turbulence
and Dissipation of and due to turbulence
Effective diffusivity of and
The wall -plus about 1 in the boundary layer was achieved as shown in Figure 2; for this purpose, the first cell height in the nozzle wall is predicted by using the skin friction coefficient () for the flat plate. where the suffix is related to the distance in which the boundary layer separation occurs and it is calculated from the inlet of the nozzle to the point of injection, and utilizing the wall -plus and equations, the first cell height is predicted. Then by numerical simulation, it was validated.
A typical structured grid system is selected to be utilized in the computational domain for grid independence analysis. Table 1 shows the three grids with different aspects used for the independence analysis.
Upon grid independence analysis, the primary axial thrust was almost identical for the three grids; the maximum difference between the three grids for this value was less than 0.5%. As a result, in order to achieve more accuracy in the numerical simulation, grid 2 was selected for the turbulence model validation study. In the computational grid 2 shown in Figure 3, the mesh near upstream, downstream the injection slot and the nozzle walls were refined carefully as shown in Figure 4.
Static pressure distribution along the upper wall (with injector) and the lower wall (without injector) was used for the comparison between the numerical results and experimental data by Gushe [17]. The realizable model, model, and shear stress transport (SST) model were used in the comparison.
Figure 5 shows that the three models behave similarly, having an excellent accuracy in the unaffected zone and good accuracy with error than less 10% in the high- and low-pressure zones compared to the experimental data; also for the boundary layer separation point, there is little difference (slightly overpredicted) between the numerical results and experimental data.
Figure 6 shows that all the models behave in an excellent accuracy with the experimental data. Finally, the three turbulence models behave similarly, so the realizable turbulence model with enhanced wall treatment was selected, as it is advised for flows containing a boundary layer under strong adverse pressure gradients [35], and it converges faster than the other two models do, whereas the model and the shear stress transport (SST) model take about 8 hours (about 35000 iterations) to converge, while the realizable model takes about 6 hours (about 25000 iterations) for convergence, so the calculation time can be reduced.
Also, in order to verify the current model, the amplification factor calculated by Gushe [17] compared to amplification factor calculated from the numerical simulation shows good accuracy.
After validating and choosing the turbulence model, the solution of the two-dimensional flow field in a conical convergent divergent nozzle with SITVC has been carried out. The injector used for the secondary injection was a two-dimensional slot extended throughout the depth of the primary nozzle in the -axis with different injection locations (in terms of axial Mach number of primary flow corresponding to the injection point located on the primary nozzle ) = 2, 2.5, and 2.75; injection angles 45°, 80°, 90°, 100°, and 135° as in Figure 7 (when the secondary injection is in the direction of the primary flow, the injection angle is 0°); and primary nozzle divergence half angles 5°, 10°, and 15°; the geometric model of the two-dimensional convergent divergent nozzle used in the numerical investigation is shown in Figure 7.
Also, for a grid independence analysis, a typical structured grid system has been utilized in the computational domain as shown in Table 2.
The grid independence analysis results showed that the primary axial thrust was almost identical and the maximum difference between the three grids for this value was less than 0.2%. Based on the grid independence study, grid 2 was used for more accuracy and lower computational time.
The computational structured grid 2 is shown in Figure 8; the mesh refined around the injection slot and the nozzle walls is shown in Figure 9 to achieve a wall -plus about 2 at the nozzle critical section and less than 1 at the injection slot as shown in Figure 10.
It has been supposed that the numerical solution converged for each case when the mass flow rate monitor at the inlet control volume displays no change (constant watching is kept at a considerable number of digits) with more than 100 sequential iterations [38].
2.1. Boundary Conditions
A pressure inlet boundary was chosen for inlet and outlet of the primary nozzle, and the mass flow inlet boundary was used for the secondary inlet.
Inlet: primary stagnation pressure kPa, primary stagnation temperature K
Outlet: primary nozzle exit pressure, kPa, exit Mach
Secondary inlet: mass flow , secondary to primary mass flow rate , secondary to primary stagnation pressure ratio , secondary injection temperature K, and ratio between secondary injection slot area and primary critical area
2.2. SITVC Performance Parameters
SITVC performance parameters investigated in the current study include [13, 39] (1)Axial thrust augmentation ATA% (2)Thrust ratio: the ratio between the side force and the axial thrust force where (i)Side force is the interaction force (side force-pressure component). where means that the summation is taken over all the cells of the computational grid is the jet reaction force (side force-momentum component). (ii)Primary axial thrust : (3)System-specific impulse loss (4)Specific impulse ratio (amplification factor) AK is the ratio of the secondary injection-specific impulse to the primary flow-specific impulse, and it determines the amount of secondary fluid to be injected to achieve a specified side force [10]. where
3. Results and Discussion
3.1. Flow Field Structure
For a constant secondary mass flow rate influence of the injection location, the injection angle and primary nozzle divergence half angle on the nozzle flow field structure will be discussed in the following part.
3.1.1. Components of the Flow Field Structure
The SITVC creates a complex flow field in the nozzle divergent part; Figure 11 illustrates the main components of the SITVC flow field.
3.1.2. Influence of Injection Location
For a given secondary mass flow rate and injection angle, the change in the flow field structure as a function of injection location will be investigated in this part. The strength of the primary bow shock decreases as the injection location moved farther downstream due to the increase in the primary flow Mach number decreasing the angle of the primary bow shock also in case of downstream injection ; the relatively shorter wall length available on the opposite wall for shock impact minimizes the possibility of shock impingement as it can be seen in Figure 12.
(a)
(b)
(c)
The momentum flux ratio is an important parameter in determining the separation distance and the depth of penetration of a jet in cross flow as in eqn. (13) [11, 29]. increases as the injection location moves downstream, as shown in Table 3, leading to increasing penetration depth and the separation distance upstream the injection
As well, the injection location moving downstream the boundary layer is more developed which decreases the slope of velocity gradient () and increases the possibility of boundary layer separation [40, 41] as it can be observed in Figure 13.
It was noticed that the strength of the separation decreases as the injection location goes downstream; this is due to increase in the primary flow Mach number decreasing the separation shock angle, and this in turn results into a relatively lower pressure in the upstream higher pressure zone. Downstream the injector, the lower pressure zone expands, as the injection location moved farther downstream, as shown in Figure 13.
The chance of shock impingement is very high in case of upstream injection but only if the secondary mass flow rate was small, and as the injection location moved farther downstream and 2.7, the chance of the primary bow shock impingement decreases, as shown in Figure 14. Finally, as the injection location moved downstream, the interaction force increases by increasing the primary flow velocity and the jet reaction force increases by decreasing the primary flow pressure.
3.1.3. Influence of Injection Angle
For a given secondary mass flow rate and injection location, the influence of the injection angle will be investigated in this section. The momentum of the secondary injection decreases as the injection angle inclined downstream or upstream, so the momentum flux decreases as shown in Table 4, but in the case of upstream inclination due to the existence of the portion of the injection in the upstream region, the interaction increases as it appears as an additional “block” that pushes the separated boundary layer leading to an increase in the separation as shown in Figure 15(c). As a result of the previous reason, as the angle of injection is increased (towards perpendicular or upstream inclination), the strength of primary bow shock increases and its originating point moves upstream the injector, and the strength of separation increases as the angle of injection increases (upstream inclination) which leads to an extension of the higher-pressure zone upstream the injector; thus, a stronger separation shock results in higher pressure in the high-pressure zone upstream the injector as shown in Figures 15 and 16. Also, the strength of primary bow shock decreases as the injection angle decreases (downstream inclination) due to the decrease in the momentum flux ratio , as shown in Figures 16 and 17. Thus, in case of shock impingement, the downstream injection angles diminish the effect of shock impingement on the opposite wall as shown in Figure 18, and by increasing the injection angle (upstream inclination) to , the chance of primary bow shock impingement on the opposite wall increases, and the impact point of the shock moves further upstream on the opposite wall with the increase in the angle of injection (upstream inclination). Injection at upstream angles of injection produces a stronger shock impact as it can be observed in Figure 18.
(a) °
(b)
(c) °
3.1.4. Influence of Primary Nozzle Divergence Half Angle
(1) At Injection Location . For a given secondary mass flow rate, the strength of the primary bow shock increases by decreasing the divergence half angle, as the rate of increase in Mach number contour is lower due to the greater wall length with the same expansion ratio.
Momentum flux ratio decreased in smaller divergence half angles as in Table 5, due to an increase in the velocity of the primary nozzle flow (decreasing cosine effect), but a smaller divergence half angle results in a slight expansion of the pressure zones upstream and downstream the injector as depicted in Figure 19, as a result of the development of the boundary layer and lower pressure contour upstream the injector (the rate of decrease in the pressure contour is lower). The strength of the recompression shock increases, as the divergence half angle decreases; in the 5° divergence half angle, the recompression shock increases till it reaches the lower wall and is reflected as can be observed in Figure 20. It can be observed that the shock impingement increases and moves downstream as the divergence half angle decreases as in Figure 21; furthermore, upstream injection accompanied with smaller divergence half angles results into multiple primary bow shock impingement on the lower and upper walls due to the increase in shock impingement on the lower wall and reflected to the upper wall as shown in Figures 19–21.
(a) Div-half angle °
(b) Div-half angle °
(c) Div-half angle °
(2) At Injection Location . For a given secondary mass flow rate, the strength of the primary bow shock increases by the decrease in the divergence half angle; also, the secondary bow shock increases by the decrease in the divergence half angle as shown in Figure 22. Although is more in the higher divergence half angles as in Table 6, due to the lower rate of decrease in the pressure contour in smaller divergence half angles and the more developed boundary layer, the separation distance in the upstream high-pressure zone is almost identical as shown in Figure 23. Smaller divergence half angles extend the lower pressure zone as depicted in Figure 23. For a given secondary mass flow rate, injection location downstream injection with smaller divergence half angle results into no shock impingement as can be observed in Figure 22.
(a) Div-half angle °
(b) Div-half angle °
(c) Div-half angle °
3.2. Performance Analysis
SITVC performance was investigated for injection location, angle of injection, and primary nozzle divergence half angle for the same secondary mass flow rate for all cases.
3.2.1. Influence of Injection Location
For a given secondary mass flow rate, it was observed as shown in Table 7 that upstream injection increases the axial thrust augmentation; this is due to the efficient adiabatic expansion of the injected gases. The amplification factor increases as the injection location moved downstream as the chance of shock impingement decreases; also, the thrust ratio increases as the injection location moved downstream due to the increase in the side force. The system-specific impulse loss increases as the injection location moved farther downstream due to inefficient adiabatic expansion and the increase in the side force.
3.2.2. Influence of Injection Angle
The given secondary mass flow rate and injection location are shown in Tables 8 and 9. The axial thrust augmentation increases as the injection angle decreases to 45°; this is due to the relatively higher parallel velocity (to nozzle axis) component of the injection gas. The thrust ratio increases (negative due to shock impingement or positive no impingement) by the increase in the injection angle as the side force increases (negative or positive); as the injection angle increases to 135°, the rate of increase in interaction force is more than the rate of decrease in the jet reaction force (due to the inclination). The amplification factor and specific impulse loss increases (negative in case of shock impingement or positive) as the injection angle increases to 135°.
3.2.3. Influence of Primary Nozzle Divergence Half Angle
For a given secondary mass flow rate, the upstream injection is as shown in Table 10 and the downstream injection is as shown in Table 11; the axial thrust augmentation increases by the increase in the divergence angle, but the system-specific impulse loss increases by the decrease in the divergence half angle due to a decrease in the primary axial thrust. The thrust ratio and amplification factor increase by the increase in the divergence half angle due to the increase in the side force as a result of the increase in the value of the interaction force; this is at the downstream injection, but at the upstream injection, this changes due to the increase in shock impingement in the 10° divergence half angle and the multiple shock impingements in case of the 5° divergence half angle.
4. Conclusions
Numerical simulation has been carried out with the help of the realizable turbulence model accompanied with enhanced wall treatment and showed a good accuracy with the experimental data in order to analyze the influence of changing the injection angle, injection location, and primary nozzle divergence half angle on the SITVC nozzle flow field structure and the SITVC performance parameters for a given secondary mass flow rate. It was found that as the injection location moved downstream , the chance of shock impingement decreases and the side force increases, but the axial thrust augmentation decreases due to the inefficient expansion of the secondary injection, also with increasing from 45° to 135°; the side force increases, but axial thrust augmentation slightly decreases, and it was concluded that small divergence half angles 5° are not efficient from the point of view of SITVC as the chance of the shock impingement increases. Finally, results showed that downstream injection, increasing the injection angle (towards upstream inclination) and higher divergence half angle, improves the performance of SITVC.
Nomenclature
| As: | Secondary injection area (m2) |
| Fp0: | Primary axial thrust without injection (N) |
| Hcr: | Nozzle throat height |
| Mp: | Injection location (in terms of axial Mach number of primary flow corresponding to the injection point located on the primary nozzle wall) |
| : | Total exit mass flow rate (kg/s) |
| : | Primary mass flow rate (kg/s) |
| : | Secondary mass flow rate (kg/s) |
| P: | Pressure (Pa) |
| Pes: | Secondary exit pressure (Pa) |
| Pas: | Ambient pressure at the secondary port before injection (Pa) |
| Rex: | Reynolds number at the distance x |
| Uτ: | Friction velocity (m/s) |
| Vx: | Velocity along the x-axis (m/s) |
| Vsy: | Secondary gas velocity along the y-direction (m/s) |
| X: | Distance along the nozzle wall |
| Isps: | Secondary specific impulse (sec) |
| : | System-specific impulse (sec) |
| Isppo: | Primary specific impulse without injection (sec) |
| δIsp: | System-specific impulse loss |
| θ: | Divergence half angle (deg) |
| : | Injection angle (deg) |
| βinj: | Wall angle at point of injection (deg) |
| γ: | Specific heat ratio. |
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that there is no conflict of interests regarding the publication of this paper.