Parametric Study of Cantilever Plates Exposed to Supersonic and Hypersonic Flows

Analysis of hypersonic flows associated with re-entry vehicles has gained a lot of significance due to the advancements in Aerospace Engineering. An area that is studied extensively by researchers is the simultaneous reduction aerodynamic drag and aero heating in re-entry vehicles. Out of the many strategies being studied, the use of aerospikes at the stagnation point of the vehicle is found to give favourable results. The structural stability of the aerospike becomes important as it is exposed to very high pressures and temperatures. Keeping this in view, the deflection and vibration of an inclined cantilever plate in hypersonic flow is carried out using ANSYS. Steady state pressure distribution obtained from Fluent is applied as load to the transient structural module for analysis. After due validation of the methods, the effects of parameters like flow Mach number, plate inclination and plate thickness on the deflection and vibration are studied.


Introduction
A lot of emphasis has been placed on the design of hypersonic re-entry vehicles, especially with regards to drag reduction and heat flux reduction. The fact that these are conflicting constraints makes the design process difficult. On one hand, streamlining the vehicles would result in reduced drag forces, but very high values of heat flux are experienced owing to attached shock formation and a very thin shock layer. On the other hand, blunt bodies provide better heat dissipation at the expense of increased drag forces. Attaching slender bodies (aerospikes) to the stagnation point of re-entry vehicles has proved to be an effective way of countering both drag and heat flux problems.
Fluid-structural interaction problems are usually analysed by numerical simulation as they are too complex to solve analytically. It was observed that under the action of a hypersonic flow, various structural parameters vary [1][2][3]. P. Le Tallec et al [4], Ergin and Ugurlu [5] have discussed these effects on cantilever beams. Fluid-Structure Interaction (FSI) problems are widely studied by researchers all over the world. The conditions under which a one-way coupling solution is appropriate and the differences between one way and two way FSI methods and solutions was discussed in literature [6,7].The differences in various methods of formulation for the fluid flow was discussed by Piperno [8]. The literature survey has helped the authors in understanding the scope of structural cantilevers exposed to hypersonic flows, and has given a proper direction in which to proceed. A parametric study of hypersonic flow over inclined cantilever plates is done with an intention to extend the study to model Fluid-Structure Interaction (FSI) in hypersonic re describes the details of numerical modelling and validation, and the results obtained using these models is described in Chapter 3. A conclusion and scope for further studies are

Modelling and validation
The vibrationand deflection of an inclined cantilever plate is carried out using the Fluent and Structural modules of ANSYS. A 2 distribution around the plate. This pressure distribution is assumed to be constant in the perpendicular (z-) direction. For the Fluent analysis, the domain and the mesh The left (inlet), right (outlet) and top sides of the domain are modelled as pressure far all other edges are treated as no-slip wall. which is considered as an ideal gas (C Sutherland's viscosity model). A 3 The use of aerospikes was first suggested by Alexander in 1947 [9].It was recorded certain Mach number the surface pressures and aero-thermodynamic heating rates may collectively be severe enough to cause failure of the aerospike [10].Gopala Krishnan et.al [11] and Sreekanth et.al [12] are among the researchers who studied various configurations of aerospikes numerically. done in literature which is similar to the present study [13 , and one of them is validated in section 2.
by Gaetano M. D. Currao et al. [16]discussed the fluid structure in Mach 5.8 air flow. Analytical and FEM numerical estimation of p oscillation and plastic deformation was done. The plate displacement as well as flow features like ary layer were measured using high-speed Schlieren video, whilef sensitive paint was used to study the pressure distribution history on the surface of Numerical results showed reasonable agreement in predict and suggestions for improving the experiment were discussed.
[17] is a wind-tunnel study of an unsteady flow of Mach num mounted wedges, double-wedge and flat-plate air-foil models with three edge radii were subjected to this flow. The data was obtained by taking high schlieren motion pictures of the decaying motion of the model as it was released from an initial , 19] have studied shock structure interactions used for forming of The literature survey has helped the authors in understanding the scope of structural cantilevers exposed to hypersonic flows, and has given a proper direction in which to proceed. A parametric study of hypersonic flow over inclined cantilever plates is done with an intention to extend Structure Interaction (FSI) in hypersonic re-entry vehicles. Chapter 2 describes the details of numerical modelling and validation, and the results obtained using these models is described in Chapter 3. A conclusion and scope for further studies are given in Chapter 4. and deflection of an inclined cantilever plate is carried out using the Fluent and Structural modules of ANSYS. A 2-D analysis is done in Fluent to get the flow field and pressure nd the plate. This pressure distribution is assumed to be constant in the perpendicular direction. For the Fluent analysis, the domain and the mesh is as shown in Figure 1. The left (inlet), right (outlet) and top sides of the domain are modelled as pressure far slip wall. A density based solver is used to simulate the flow of air, which is considered as an ideal gas (C p = 1006.43 J/kg-K, Thermal conductivity = 0.0242w/m A 3-D model of the plate with a mesh that is consistent with edge It was recorded that after a thermodynamic heating rates may collectively be Gopala Krishnan et.al [11] and Sreekanth et.al [12] are among the researchers who studied various configurations of aerospikes numerically. . Two of e fluid structure interaction of a Analytical and FEM numerical estimation of plate The plate displacement as well as flow features like speed Schlieren video, whilefast on the surface of in predicting the plastic tunnel study of an unsteady flow of Mach number of foil models with three edge radii were subjected to this flow. The data was obtained by taking high-speed f the model as it was released from an initial ] have studied shock structure interactions used for forming of The literature survey has helped the authors in understanding the scope of structural analysis of cantilevers exposed to hypersonic flows, and has given a proper direction in which to proceed. A parametric study of hypersonic flow over inclined cantilever plates is done with an intention to extend entry vehicles. Chapter 2 describes the details of numerical modelling and validation, and the results obtained using these given in Chapter 4. and deflection of an inclined cantilever plate is carried out using the Fluent and D analysis is done in Fluent to get the flow field and pressure nd the plate. This pressure distribution is assumed to be constant in the perpendicular . The left (inlet), right (outlet) and top sides of the domain are modelled as pressure far-field and to simulate the flow of air, K, Thermal conductivity = 0.0242w/m-K and D model of the plate with a mesh that is consistent with edge sizing used in Fluent is generated using the Structural module. yield strength = 2.5E+08 Pa, Young's modulus strength =4.6E+08 Pa) was taken as the material for the plate condition (with Young Modulus (E) = 200GPa, Tangent Modulus (E 205MPa) was used to account plastic deformation Fluent, a transient analysis is done in the Structural mod size of 1 ms. A major assumption used here is that the effect of vibration and deflection of the plate on the flow field is negligible. Thus, there is a one modules, wherein the flow field is causing the deflection, but field.
Validation of the numerical model was [16]in which experimental and numerical study was inclinationcantilever exposed to Mach K. The experimental model used was a 1 mm thick and 230 mm long steel plate that was mounted on a rigid 20 o wedgeand the numerical analysis was carried out considering only the cantilevered part of the structure with transient fluid and structural  The same approach is followed in the subsequent section and the structural behaviour of the inclined cantilever plate is studied under different Mach The pressure acting on the sides of the plate obtained from 2-D FLUENT analysis was used as D transient structural analysis, assuming uniform pressure distribution in the z The deflection of the first cycle compared with the results from the paper, as shown in validating the modelling methodology undertaken.
: Comparison of deflection for the first cycle.
the subsequent section and the structural behaviour of the inclined cantilever plate is studied under different Mach numbers, plate thicknesses and inclinations.

Results and analysis
Our study is aimed at analysing the variation in deflection with Mach number, plate thickness and plate inclination. Plates of widths 5 mm, 10 mm and 15 mm, inclined at 20 o and 30 o , for free stream Mach numbers from 3 to 8 are considered. For each case, the deflection is found to oscillate initially and then reach a steady value. The results will be presented in three sections; each section depicting the variation with respect to one of the mentioned parameters, while the others remain constant.

Mach number
Tostudy the effect of Mach number, a 5 mm thick plate inclined at 20 o is considered. It can be seen fromFigure 4that the amplitude of vibration and the final steady state value of deflection are increasing with Mach number, with Mach 8 having the largest amplitude and final deflection.

Inclination
For the same thickness, same length of the plate (100mm) and same Mach number, it is also observed that the deflection of the plate with 30 o inclination is more than that with 20 o inclination (

Thickness
For the same angle of inclination and same Mach number, it is observed that the deflection of the plate increases with decrease in thickness( Figure 8). Plates of varying thickness, inclined at 30 o and exposed to Mach 8 air flow are considered. Same M ach number and inclination result in almost uniform pressure distribution on the left side of the plate ( Figure 9). As the thickness increases, the stiffness and thus the resistance to deformation increases, thus resulting in lower amplitudes of vibration and magnitudes of deflection.

Conclusion
A One way FSI was carried on a cantilevered plate with varying thickness of 5mm, 10mm and 15mm and at angles of attack of 20 and 30 degrees. The pressure profile and deflection were analysed for Mach numbers varying from 3 to 8.
Constant free-stream conditions of P ∞ = 680 Pa, T ∞ =73 K, were taken for the steady state Fluent analysis. The pressure acting on the sides of the plate was used as the input to the 3-D transient structural analysis, assuming uniform pressure distribution in the z-direction. The transient deflection was observed and reported for the cases mentioned. It was found that the magnitude of deflection oscillated during the initial stages and eventually reached a steady value.
The maximum steady state deflection and amplitude of the vibration of the plate is found to be increasing with increase in Mach number, owing to larger values of pressure on the left side of the plate, and the shift in location of the peak pressure away from the fixed point. A reduction in deflection as the plate thickness increases, because of increased stiffness, is also observed. It is also observed that the deflection of the plate with 30 o inclination is more than that with20 o inclination which can be explained by the increased shock strength and pressure ratio at higher inclination. However, the angles chosen were less than the limiting value for the Mach numbers used, and thus, only attached oblique shocks are considered in the simulations. For higher angles, detached and curved shock waves will be formed, and the deformation of the plate under these conditions should be studied. A two-way coupling between the Fluent and Structural modules can be considered for analysis and the results should be compared with the current results in order to arrive at an effective analysis procedure, before applying it to actual re-entry applications.