In the present study, the base pressure variations induced by the presence of a cavity, known to have a strong influence of the behaviour of supersonic projectiles, are investigated through numerical solution of the balance equations for mass, momentum, and energy. An area ratio of four is considered and numerical simulations are carried out at Mach M = 1.2, 1.4, 1.6, and 1.8 assuming no cavity or cavity locations 0.5D, 1D, 1.5D, and 2D. The inlet pressure of the nozzle is considered as a flow variable. The Taguchi method is also used, and the considered cases are then analyzed using a full factorial experimental design. The results show that the cavity is effective in increasing the base pressure for the conditions examined. For other nozzle pressure ratios, cavities do not lead to passive control due the change in the reattachment length. The distribution of wall pressure reveals that, in general, a cavity used to implement passive control of the base pressure does not adversely influence the flow pattern in the domain.

The study of base pressure in a high-speed supersonic flow is an important topic, as the resulting base drag contributes to nearly seventy per cent of the total drag of missiles, rockets, and projectiles. However, even a slight rise in pressure will significantly reduce the base drag and this decrease in the base drag will cause a significant improvement in the overall performance of the projectiles. Hence, many researchers are currently working on ways to control base pressure. A search of the literature reveals that there are two ways to regulate base pressure. One is a dynamic/active control, and the other is passive control. Passive control of the base pressure can be achieved through changes to the geometry of the flow field. Whereas, for dynamic/active control, an additional source of energy is needed to achieve control. In dynamic conditions, arranging external energy sources may not always be feasible.

Pandey et al. [

On the other hand, control of supersonic compression corner flow using a plasma actuator has been recognized as a practical approach [

Additionally, the flow of control outside the system/object has been determined very well using the CFD approach, from which results show adequate information and significant appropriateness for the fluid flow analysis. In recent years, ANSYS fluent study was found for the sharp wedge flow formation, and shock observations were determined with the CFD approach and compared with the theoretical results [

The objective of the current work is focused on passive control using a cavity in a CD expansion duct of the nozzle. The waves dominate supersonic flows, and the combination of each parameter will give different outcomes and variations in flow and geometrical parameters provide a different flow pattern was determined. The cavity in the duct is employed to determine its effectiveness on the base pressure control. While simulating the flow, the nozzle inlet pressure has been considered as a flow variable. The duct diameter considered for the analysis is 20 mm, and the cavity used for the flow control has dimensions 3:3. Therefore, the current work with the finite volume method designed and modelled the three-dimensional CD nozzle with a circular duct and the duct controlled by the cavities to reduce the base drag and determine the pressure rates. Additionally, a Taguchi method was used to optimize the flow parameters and nozzle dimensions for the optimum solution of base drag control and increase the flow rate of the duct.

The parameters considered for the current work are Mach number (M), length-to-diameter ratio (L), Cavity location (C), and Nozzle pressure ratio (NPR). The enlarged duct diameter (D) is constant at 20 mm and the exit diameter of the nozzle is 10 mm. The nozzle dimensions are calculated for various Mach numbers and presented in

Parameters | Mach no. | |||
---|---|---|---|---|

1.2 | 1.4 | 1.6 | 1.8 | |

Inlet diameter (D_{i}) |
20.57 | 20.19 | 22.34 | 21.73 |

Throat diameter (D_{t}) |
9.85 | 9.47 | 8.94 | 8.34 |

Exit diameter (D_{e}) |
10.00 | 10.00 | 10.00 | 10.00 |

Converging length (L_{c}) |
20.00 | 20.00 | 25.00 | 25.00 |

Diverging length (L_{d}) |
8.53 | 10.11 | 10.08 | 9.51 |

Since the duct is axisymmetric, this results in a reduction of the computation time. Using 2D or 3D analysis with the half model, numerical simulations can be done easily by considering the symmetric model. We may consider symmetric about one plane and 3D analysis with the quarter model, i.e., symmetric about two planes.

The CFD analysis is done using ANSYS fluent for all the groupings of constraints by considering a complete factorial design. The K-epsilon turbulent model is utilized during computation, giving more accurate results in a reasonable time [

The results for the base pressure are extracted from the fluent software. The pressure values are gauge pressure, converted into absolute pressure, and then normalized by ambient pressure. The purpose of converting the base pressure into dimensionless base pressure is to better visualize and understand the results. The pressure contours are also extracted from the fluent software.

The grid element size plays a significant role in numerical analysis. The element sizing should be optimized to get accurate results in minimal computation time. To optimize the grid size, the grid independence test is carried out for grid element sizes of 0.1 to 5 mm. ^{−3} which is small, but the base pressure was found very less when it is 0.5 or 1.0 mm. Indeed, the use of 0.5 or 1.0 mm has approximately the same results we choose 0.5 mm because of less computational time and the grid element size of 0.5 mm is used for further CFD analysis of the current work.

Mesh element size in mm | Mesh nodes | Mesh elements | Base pressure |
---|---|---|---|

5 | 2804 | 652 | 0.610057 |

4 | 2786 | 632 | 0.635878 |

3 | 3055 | 720 | 0.612463 |

2 | 6241 | 1398 | 0.477677 |

1 | 34164 | 7884 | 0.33163 |

0.5 | 228791 | 55650 | 0.332748 |

0.25 | 1719427 | 442519 | 0.333865 |

0.1 | 25345765 | 6587205 | 0.334983 |

In order to validate or confirm the CFD model the benchmark work of Pandey et al. [

The per cent variation in CFD analysis results and the experimental outcomes are shown in

L/D ratio | ST | ASR1 | ASR2 |
---|---|---|---|

1 | 1% | 2% | 3% |

2 | 4% | 5% | 3% |

3 | 5% | 1% | 5% |

4 | 2% | 0% | 8% |

5 | 4% | 1% | 9% |

6 | 4% | 0% | 9% |

8 | 3% | 0% | 10% |

10 | 3% | 1% | 10% |

The Taguchi approach optimizes the design parameters to minimize variation before optimizing the design to achieve the target value of the output parameters. The Taguchi approach employs special orthogonal arrays to investigate all design factors with minimal investigation [

Parameters | Level 1 | Level 2 | Level 3 | Level 4 |
---|---|---|---|---|

Mach no. (M) | 1.2 | 1.4 | 1.6 | 1.8 |

L/D ratio | 3 | 4 | 5 | 6 |

Nozzle pressure ratio (NPR) | 2 | 4 | 6 | 8 |

Location of cavity (C) | 0.5D | 1D | 1.5D | 2D |

10 mm | 20 mm | 30 mm | 40 mm |

The Taguchi orthogonal array L_{16} is used in the CFD analysis’ initial stage to identify the various parameters’ main and interaction effects. The input parameters used in Taguchi orthogonal array are duct diameter (D) with sudden expansion, Mach number (M), length-to-diameter-ratio (L), cavity location (C), and nozzle pressure ratio (NPR). The base pressure (Pb) is considered an output parameter. _{16} orthogonal array.

Trial no. | Input parameters | Output parameter | ||||
---|---|---|---|---|---|---|

D | M | L | C | NPR | Pb | |

1 | 14.14 | 1.2 | 3 | 0.5 | 2 | 0.5608 |

2 | 14.14 | 1.4 | 4 | 1 | 4 | 0.4187 |

3 | 14.14 | 1.6 | 5 | 1.5 | 6 | 0.4506 |

4 | 14.14 | 1.8 | 6 | 2 | 8 | 0.4556 |

5 | 20 | 1.2 | 4 | 1.5 | 8 | 0.3152 |

6 | 20 | 1.4 | 3 | 2 | 6 | 0.1818 |

7 | 20 | 1.6 | 6 | 0.5 | 4 | 0.4882 |

8 | 20 | 1.8 | 5 | 1 | 2 | 0.7744 |

9 | 24.49 | 1.2 | 5 | 2 | 4 | 0.2476 |

10 | 24.49 | 1.4 | 6 | 1.5 | 2 | 0.7561 |

11 | 24.49 | 1.6 | 3 | 1 | 8 | 0.1006 |

12 | 24.49 | 1.8 | 4 | 0.5 | 6 | 0.5725 |

13 | 28.28 | 1.2 | 6 | 1 | 6 | 0.0943 |

14 | 28.28 | 1.4 | 5 | 0.5 | 8 | 0.4008 |

15 | 28.28 | 1.6 | 4 | 2 | 2 | 0.8659 |

16 | 28.28 | 1.8 | 3 | 1.5 | 4 | 0.7712 |

Fluent software extracts provide the total pressure contours to aid in understanding the pressure variations in the CD nozzle and enlarged duct. The pressure contours for L/D = 6, Mach 1.2–1.8, and NPR’s 2–8 are extracted.

Before analyzing the base pressure results due to the cavities, it is necessary to explain the physics of the flow when the shear layer is exhausted in a duct with a larger area. When the Mach number is less than unity, the boundary layer will get separated, expanded, and reattached to the enlarged duct after exiting from the nozzle. The separated region will contain one or more vortices as the first vortex will be close to the base and relatively strong. This vortex is named the central vortex. It works like a pump and transfers fluids from the base region to the main jet, which is on the side of the edge of the boundary layer. Due to this pushing activity, low pressure will be created in the recirculation zone. However, as is known, this vortex spread is a periodic phenomenon, making pushing activity too irregular. This irregular pattern causes fluctuations in the base pressure. However, while conducting the tests, it is observed that these variations in the base pressure are negligible. Hence, we take the mean base pressure values while analyzing the results. Owing to the cyclicity of the vortex desquamation, the complete flow pattern in the enlarged duct may turn out to be oscillatory. These oscillations may become very severe for a set of geometrical and inertia parameters. The intensity of the central vortex positioned at the base mainly depends on the level of expansion, reattachment length, the Mach number, and the area ratio.

We theorize that in the flow-through of the CD nozzle, the exiting jet may result from either of three conditions, i.e., the flow may be ideally expanded, under-expanded, or over-expanded. In the case of ideal expansion, the exiting shear layer dominated by waves across the stream flow will be isentropic. A strong shock wave will be located where the nozzle experiences adverse pressure at the nozzle exit. This shock will make the flow move to the main flow, resulting in a delay of reattachment and more significant reattachment length, which will significantly influence the strength of the primary vortex and hence the base pressure values. Finally, for under-expanded nozzles, an expansion fan accelerates the flow as the flow is expanded, and flow will turn away towards the base, resulting in early reattachment and smaller reattachment length. When cavities exist in the duct, these grooves in the form of a cavity will generate additional vortices. These other vortices will act as promoters of mixing despite being small in size thereby, resulting in higher base pressure.

Whenever a control is employed to regulate the flow in a duct with sudden expansion, it is mandatory to examine the impact of the flow management methods on the flow field of the tube. On the researcher’s part, it is necessary to confirm that the passive control does not adversely influence the flow inside the tube. Hence, in this case, we have also considered the effect of the cavities as flow regulators on the wall pressure.

Based on our results, we can conclude that a cavity can be used to control the base flow. The cavity is adequate to regulate base pressure at NPR = 4. At NPR 2, 6, and 8, the changes in base pressure due to the cavity are insignificant. The reasons for the ineffectiveness of the cavity at these locations are that, at NPR = 2, the nozzle is over-expanded, and the reattachment length is considerable, far away from the base region. Hence there is no interaction between the flow and cavity. Therefore, the cavity is unproductive at a nozzle pressure ratio of 2. On the other hand, the nozzle is highly under-expanded at NPR 6 and 8, and the reattachment length is less than 0.5D. The flow is reattached to the tube before 0.5 times the diameter. The flow is reattached to the duct before 0.5D, and all cavities are placed after 0.5D. Hence, there is no interaction between the base region and the cavity. Thus, the cavities are unsuccessful in increasing base pressure at nozzle pressure ratios 6 and 8. From Taguchi’s main effect plots, it is seen that the parameters, Duct diameter, Mach M, L/D ratio, NPR, and cavity and its location have a substantial effect on controlling base pressure.

Enlarged duct diameter

Mach number

Location of the cavity from the nozzle exit

Convergent-Divergent

Computational fluid dynamics

Nozzle pressure ratio

Length of duct

Base pressure

_{e}

Nozzle exit diameter

_{i}

Nozzle inlet diameter

_{t}

Nozzle throat diameter

_{c}

Nozzle converging length

_{d}

Nozzle diverging length

Models with cavity aspect ratio 1 (3:3)

Models with cavity aspect ratio 2 (6:3)

Duct without cavity

Duct without cavity

Cavity location at 10 mm in the duct

Cavity location at 20 mm in the duct

Cavity location at 30 mm in the duct

Cavity location at 40 mm in the duct

Instantaneous velocity

Velocity modulus

Gas density

Gas pressure

_{j}

Heat flux

_{ij}

Viscous stress tensor

The authors received no specific funding for this study.

The authors declare that they have no conflicts of interest to report regarding the present study.