Dimensions and operating condition of nozzles.
Abstract
The computational fluid dynamics analysis is carried out to analyze the shock and flow characteristics of under- and over expanded supersonic free jets emanating from convergent-divergent nozzles. Influence of exit Mach number on shock cell lengths are analyzed with the help of density contours and schlieren images. A parameter based on an exit Mach number is obtained to characterize the pressure variation along the jet axis which shows them independent of the exit to ambient pressure ratio. Impingement flow fields over axisymmetric and wedge deflector are investigated employing numerically and compared with experimental results. Effects of jet expansion ratio and distance between the nozzle to deflector apex has been studied at various expansion ratios and distances. Impingement load is computed at various conditions. Pressure distributions over a surface of wedge and axisymmetric jet deflector are computed and compared between them. Pressure load on a diaphragm of a solid motor during lift-off of a satellite launch vehicle having four liquid engines is numerically simulated.
Keywords
- CFD
- compressible flow
- convergent-divergent nozzle
- expansion
- Mach disc
- shock
- supersonic jets deflector
- supersonic free jets
1. Introduction
The phenomenon of under- and overexpanded supersonic free jets emanating from a convergent-divergent nozzle and their impingement over a deflector are employed during the lift-off of a satellite launch vehicle. When a convergent-divergent nozzle deviates from its operating conditions then semi-periodic shock diamonds [1] and Prandtl-Meyer fans appear in the jet field. However, at the design conditions, the traditional smoothly contoured convergent-divergent nozzle with exit flow is a shock-free condition.
Mehta and Prasad [2] studied the effects of various exit Mach numbers and exit-to-ambient pressure ratios on shock-cell lengths with schlieren and density contour plots. The results showed that the Mach disk moves closer to the nozzle exit plane with decreasing pressure ratio, and the length of the first shock cell decreases. Three-dimensional Navier-Stokes equations [3] were solved to estimate the heat flux on a jet deflector caused due to impingement of the rocket exhaust of a canisterized missile. Based on the theory of computational fluid dynamics [4], the flow field of a rocket engine plume is simulated and analyzed by numerical method. Allgood and Ahuja [5] investigated the plume impact characteristics of ARES V propulsion system by numerical method, and the impact shape, surface pressure, temperature and other parameters of a flow deflector were obtained, which can help the practical construction of flow deflector and improve the safety and reliability of space launch.
Zhou et al. [6] have analyzed the four-engine liquid rocket flow field during the launching phase. Jiang et al. [7] presented an overview on progresses and perspectives of the jet impingement research for rocket launching.
Numerical simulations [8] have been performed to investigate the exhaust plume impinging on the wedge-shaped and cone-shaped deflectors. By comparative analysis between the four-engine rockets impinging jet on different deflector. Compared to the wedge-shaped deflector, the cone-shaped deflector could achieve better performance for deflecting with sufficient distance of the sidewalls. The flow fields of the four-engine rocket impinging on the flame deflector under different impingement and uplift angles are simulated. They observed that high temperatures on the deflector surface mainly occur on the impingement point or the cambered surface. A large impingement angle causes the reverse flow intensity to increase whilst a small angle causes the exhaust gas to deflect a little, a suitable uplift angle can smoothly guide the exhaust gas away from the deflector that the best thermal environment of the deflector channel appears at an impingement angle of 25°and an uplift angle of 5°. Numerical solutions of the impingement of an underexpanded axisymmetric supersonic jet on a flat plate at varied angles have been carried out by Wu et al. [9] and Kim et al. [10]. Experimental investigation of underexpanded supersonic jet impingement over inclined plate is carried out by Nakai et al. [11]. Computational analysis of underexpanded jets on inclined plate is carried out by McIlroy and Fujii [12]. Supersonic gas jets in conical convergent-divergent nozzles are studied numerically using the rhoOpenFOAM software solver [13].
The present study attempts to address flow field of supersonic free jets. Phenomena of jet expansion ratio, and distance between nozzle exit plane and deflector apex has been analyzed at various stand-off distance. Overall load coefficient has been obtained and compared with experimental data [14].
2. Supersonic free jets
2.1 Supersonic free jets from Laval nozzle
A conical convergent-divergent nozzle delivers over-expanded, fully expanded and underexpanded jet depending on the nozzle pressure ratio, ratio of stagnation pressure in the settling chamber to the ambient pressure.
2.2 Over expanded jets
If the pressure in the ambient medium to which is discharging is greater than the nozzle exit pressure, the jet is said to be over expanded. Oblique shock waves are formed at the edge of the nozzle exit. These oblique shocks will be reflected as expansion waves from the boundary of the jet. Figure 1a schematically diagram delineates the waves in an over expanded jet. Due to this periodic shock cell structure is generated in the jet and the wave length of this periodic structure is found to increase with Mach number [15, 16].
Flow has passed through the shock waves will be turned the centre line and the oblique shock wave directed toward the centre line of the nozzle. This process of expansion and compression wave formation continues until the pressure of the jet field becomes the same as the ambient pressure and the flow becomes parallel to the nozzle centre line. These expansion and compression waves which intact with each other lead to the formation of diamond patterns termed shock cells.
2.3 Fully expanded jets
Figure 1b shows schematic sketch of fully expanded jet. The fully expanded flow occurs when ambient pressure is equal to exit pressure then the jet is also alternatively called as correctly expanded [17]. This jet is also wave dominated as an imperfectly expanded jet.
2.4 Under expanded jets
If the pressure in the ambient medium to which is discharging is less than the nozzle exit pressure, the jet is said to be under expanded as shown in Figure 1c. Since
The Prandtl-Meyer angle is the angle through which sonic flow turns to attain a supersonic local Mach number. When supersonic flow is turned through a convex corner, the flow expands, resulting in an increase in velocity and a drop in pressure. When a
Supersonic jet emanating from a convergent-divergent nozzle may be overexpanded (
The location of Mach disk [2] moves away from the nozzle exit plane with increase in expansion pressure ratio
The noise generated from the supersonic jet is dominated due to turbulent mixing in the shear layer for
During the lift-off condition of the multi-engine rocket, the exhaust jet from the rocket nozzle impinges on the launch pad and is produced complex impingement flow field. The
3. Numerical algorithm
A numerical flow simulation is carried out to analysis supersonic free jets and jets deflector using turbulent, compressible Reynolds averaged Navier-Stokes equations. A two-equation turbulence model [18] with compressibility correction [19] is used to solve the governing fluid dynamics equations. A finite volume discretization is carried out in spatial coordinates to compute inviscid and viscous flux vectors. Time evolution is carried out by an explicit multistage Runge-Kutta method to achieve a steady state solution. The numerical algorithm is developed by taking into consideration structured grid arrangement. The numerical results are obtained for nozzle exit Mach number of 2.2, 2.6 and 3.1, exit to ambient pressure ratio of 0.6, 0.8, 1.0 and 1.2 and at different distance from nozzle exit to the apex of the axisymmetric and the double wedge deflector. The numerical scheme is computationally fast, easy to program and efficient. The centre line pressure distributions inside the free jets differs in the presence of the jet deflector. The numerical results are compared with the experimental data and are discussed in the next sections.
4. Experimental facility
All the experimental simulations of supersonic free jet and jet deflector are conducted in an open jet facility as shown in Figure 3. High pressure dry air at 4.3 × 106 Pa at ambient temperature is fed through a 15 × 10−3 m diameter pipe line to the settling chamber and nozzle assembly. A pressure regulating valve is used to control the operating pressure. The pressure in the settling chamber is continuously recorded and monitored using a Bourden pressure gauge and a pressure transducer. The experimental set up is coupled with data acquisition. The open jet facility can be operated continuously at the maximum pressure up to about 80 s.
The conical convergent-divergent nozzles are used for getting supersonic free jets. The nozzles are having semi-divergent angle of 15o. The dimensions of convergent-divergent nozzle and operating conditions are tabulated in Table 1. The nozzles
Nozzle | Pressure ratio | ||||
---|---|---|---|---|---|
A | 2.18 | 30.0 | 2.005 | 2.2 | 0.36 < |
B | 9.2 | 15.72 | 2.923 | 2.6 | 0.39 < |
C | 10.66 | 23.0 | 4.657 | 3.1 | 0.47 < |
Axisymmetric and wedge deflector model consisted of 70o apex angle having tip bluntness of 0.13
5. Numerical results
5.1 Analysis of supersonic free jets
For the sake of brevity, we are displaying only density contour of
A schematic sketch of based on the computed density contour and schlieren picture is illustrated in Figure 7. Mach disc location
For the sake of brevity, we are presenting density contours in Figure 8 for
Figure 9 shows inviscid [2], present viscous and measured static pressure distribution [14] along the centre line of supersonic free jets
Figure 10 shows the variation of measured and computed pressure distribution along the jet axis for
From the density contour measurement of
The
12.88 | 1.2 | 4.36 | 5.36 | 2.15 | 5.5 | 8.8 | |
10.69 | 1.0 | 2.84 | 4.84 | 1.75 | 4.7 | 7.7 | |
8.55 | 0.8 | 3.23 | 4.23 | 1.4 | 3.0 | 6.5 | |
6.47 | 0.6 | 2.50 | 3.50 | 1.0 | 2.9 | 5.0 | |
23.95 | 1.2 | 6.38 | 7.38 | 2.6 | 6.65 | 10.95 | |
19.95 | 1.0 | 5.76 | 6.76 | 2.2 | 5.8 | 9.5 | |
15.55 | 0.8 | 5.03 | 6.03 | 1.76 | 4.95 | 8.1 | |
11.97 | 0.6 | 4.16 | 5.16 | 1.35 | 3.9 | 7.1 | |
19.12 | 0.8 | 7.70 | 8.7 | 2.2 | 6.2 | 10.1 | |
25.59 | 0.6 | 6.62 | 7.62 | 1.75 | 5.0 | 8.3 | |
17.09 | 0.4 | 5.70 | 6.24 | 1.2 | 3.6 | 6.18 |
5.2 Numerical analysis of jet deflector
The impingement flow field in a jet deflector produced due to an over expanded supersonic jet is schematically illustrated in Figure 11. It can be observed from the diagram that the flow field consists of a jet shock, Mach disk, reflected shock, wedge shock and compression shock [22, 23]. The impingement flow field data are necessary for the design of the jet deflector.
Mach density contours of the impinging jets have been shown over the axisymmetric deflector in Figure 12a and b for
Figure 13 shows pressure distribution over the axisymmetric and the wedge deflectors for
A quantity, which is of design interest is the overall impingement load,
Pressure distribution over wedge deflector for
Figure 16 shows the pressure distribution over different section of the wedge deflector for for
6. Flow field over deflector at lift-off time
During the lift-off condition of the rocket, the exhaust jet from the rocket nozzle impinges on the launch pad and is produced complex impingement flow field. The
A three-dimensional numerical simulation is carried using compressible Euler equations. The governing fluid dynamics equations are solved using finite volume method. Time integration is performed using an explicit multistage Runge-Kutta method.
This problem is solved when the L-40 engine is operated during the lift-off time as shown in Figure 17. The boundary conditions are enforced by using the idea of image cells on the plane of symmetry. On the nozzle exit plane of L-40 engine, the following conditions are taken for the computational purposes:
At the nozzle diaphragm of the core motor and solid wall of the jet deflector no normal flow conditions are applied. For quiescent external condition, the ambient pressure is imposed on the remaining sides of the computational domain (Figure 18).
Figures 19 and 20 show the velocity field and density contour over the jet deflector. The velocity vector shows complicated flow field and also having recirculating zones. This can also be noticed in the density contour plot. Static pressure distribution along the deflector surface has been shown in Figure 21. A higher pressure is observed at a location downstream of the apex of the deflector which is impingement region of the jets. Figure 22 shows pressure on the base of the launch vehicle. It can be seen from the pressure distribution that the diaphragm is having pressure above the ambient pressure. It is important to mention that the present numerical analysis may predict higher conservative pressure distribution.
7. Conclusion
Numerical simulations are carried out to obtain supersonic free jets emanating from convergent-divergent nozzles at different exit Mach number and operating nozzle pressure ratio. Shock cells are obtained using density contours and a least square straight line is obtained from the numerical and experimental data. Effect of jet expansion ratio and distance between nozzle exit plane to deflector apex has been analyzed for various stand-off distance. Overall load coefficient has been calculated for the axisymmetric and wedge deflector. Pressure distributions on the jet deflector and on the base of rocket have been computed using Euler flow solver.
Nomenclature
effective jet diameter | |
local Mach number | |
Mach number before expansion | |
Mach number after expansion | |
nozzle design Mach number | |
fully expanded jet Mach number corresponding to isentropic expansion of jet from | |
( | |
stagnation pressure | |
exit pressure | |
ambient pressure | |
Pitot pressure along jet | |
Mach disc location | |
First shock cell length | |
Second shock cell length | |
Third shock cell length | |
Axial distance from the nozzle exit plane | |
distance from nozzle exit plane to deflector apex | |
deflection angle | |
ratio of specific heats | |
Prandtl–Meyer angle |
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