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Effects of flap on the reentry aerodynamics of a blunt cone in the supersonic flow
International Journal of Mechanical Sciences 176 (2020) 105396
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Effects of flap on the reentry aerodynamics of a blunt cone in the supersonic flow Senthil Kumar Raman a, Wu Kexin a, Tae Ho Kim a, Abhilash Suryan b, Heuy Dong Kim a,∗ a b
Mechanical Engineering, Andong National University, Andong, South Korea Mechanical Engineering, College of Engineering, Trivandrum, Kerala
a r t i c l e Keywords: Flap Reentry capsule Supersonic speed Shock wave
i n f o
a b s t r a c t Flap over blunt asymmetric reentry vehicles improves the aerodynamic stability and control. The present study has focused on the reentry aerodynamics of blunt-nosed conical body with flap configurations. Three dimensional, steady, viscous and compressible flow over the reentry body configurations were numerically analyzed by solving Reynolds averaged Navier-Stokes equations and SST K-𝜔 turbulence model. The fundamental governing equations were discretized from the partial differential form to numerical analogue using the finite volume approach. Numerical simulations were carried out to investigate the flow characteristics of the three different reentry body configurations at different pitch angles, flap angles, the Mach numbers, and altitudes. For increment in the pitch angle from 0∘ to 6∘ , the axial force coefficient is invariant, while the normal force coefficient linearly increases. It is found that the axial force coefficient is directly proportional to the flap angle and inversely proportional to the Mach number. Presence of flap introduces streamwise vortices and increases the flow complexity after the base to a large extent.
1. Introduction Space exploration has obtained an extensive interest in the last decade with an overwhelming involvement of the private sector along with public agencies. Most of the recent space programs by these agencies are focused on human space flight. The accomplishment of these human space missions mainly depends on the successful reentry into the atmosphere. In 1951, H. Allen discovered the classical blunt body theory as a solution for reentry problem by creating a strong shock in front of the nose as in Fig. 1(a). High temperature rise occurs after the strong shock wave and most of the heat dissipated to the air around the reentry body. Since then, continuous research on blunt shaped vehicles are still going on [1–4] and as a result, some phenomenal advancement such as the blunt nose with lifting surface, control surfaces, and aerospikes were developed. The blunt body flying with hypersonic speed is studied extensively [1,4–7]. At hypersonic speed and higher altitude(stratosphere and mesosphere), the flows are usually considered as non-continuum and analyzed with DSMC approach [8,9]. Meanwhile, the understanding of continuum flow physics of supersonic flow over the blunt body is also essential [10–12]. However, the studies about aerodynamic characteris-
tics of the blunt body with flap, available in open literature are limited [13,14]. Flaps on the blunt axisymmetric reentry vehicles has aerodynamically two main advantages. First, when the vehicles flying at an incidence, it assures a pertinent aerodynamic stability by offsetting center of gravity. In other words, flap contributes enough aerodynamic trim without changing the lift-drag ratio (L/D) ratio. Second, flap alter the cross area of reentry bodies and effective angle of attack which allows the steering control at entry, descent and landing. The flap counts is defined by the control required over the number of degrees of freedom. For instance, single flap allows for control over angle of attack alone, while the two flaps at 180∘ to each other provides an additional control on side slip also. Four flaps at 90∘ each other allows to reduce its ballistic coefficient along with the control over the angle of attack and side slip. This decreased ballistic coefficient results in higher drag deceleration and enhancing the entry margin as well as the descent timeline. In general, supersonic flow past an angular obstacle such as wedge produces an oblique shock and raise the pressure [15]. Similarly, the flap attached over the blunt cone compresses the flow field and leads to the formation of oblique shock as shown in Fig. 1(b) and increases the drag. Either at a higher Mach number or a higher flap angle, the
∗
Corresponding author. E-mail addresses: [email protected] (S.K. Raman), [email protected] (T.H. Kim), [email protected] (A. Suryan), [email protected] (H.D. Kim). https://doi.org/10.1016/j.ijmecsci.2019.105396 Received 23 February 2019; Received in revised form 5 December 2019; Accepted 20 December 2019 Available online 22 February 2020 0020-7403/© 2020 Elsevier Ltd. All rights reserved.
S.K. Raman, W. Kexin and T.H. Kim et al.
International Journal of Mechanical Sciences 176 (2020) 105396
Fig. 1. Flow field around a blunt cone and flap reentry configurations. Blunt cone with flap has oblique shock in addition to the flow field of blunt cone.
oblique shock becomes a detached shock which causes the earlier onset of pressure rise and a recirculation region forms ahead of the flap; thus adversely affects the flap effectiveness. Furthermore, the finite spanned flap induces streamwise vortices which severely alters the base flow and increases the flow complexity. Hence, the understanding of high-speed flow over reentry vehicles with flap is essential. Massobrio [16] numerically analyzed the aero-thermodynamic characteristics of four EXPEerimental Reentry Testbed (EXPERT) configuration designed and developed by the European Space Agency. EXPERT has four flaps attached on ramp surfaces of a conical body with a blunt nose. At zero angle of attack, computational model with quarter domain is considered. At non-zero angle of attack, model with a half domain at symmetrical plane is considered. Kharitonov at al [17] experimentally measured the axial force coefficient (CA ) of EXPERT at M=4 and 𝛼 = -3∘ to 6∘ , and found that the CA is nearly constant. Chen et al [18] analyzed the hypersonic flow characteristics of asymmetric blunt body with flap. The monostablity features is significantly enhanced with a slight reduction in L/D on inclusion of flap. Rosa et al [19] analysed the hypersonic flow field over winged reentry vehicles with flap. A recirculation bubble forms ahead of flap and overshoots the pressure at hinge line; this recirculation bubble affects the flap effectiveness. Barrio et al. [13] presented the trajectory details of EXPERT, and numerically calculated the force coefficients at supersonic speed. The the axial and normal force coefficients were found to increase with increasing subsonic Mach number and decrease with increasing supersonic speed. Grasso et al. [13] analyzed the aerodynamic performance of the control surface of the blunt-nosed vehicle and concluded that turbulence becomes important for flap angles higher than the critical deflection angles. More work is required to understand the effect of this flap angle on surface pressure distributions, pressure peaks, and base vortical flow structures. The present study mainly focuses on the effect of the flap deflection angle on the flow field over the blunt cone at supersonic speed. Numerical simulations were carried out to investigate the effect of the flap
on surface pressure distribution, axial force coefficient, shock structure, and the base vortical flow behavior.
2. Geometry The present study considered three different reentry configurations such as a blunt cone, ramped, and flapped (similar to EXPERT) configurations as shown in Fig. 2(a), (b) and (c) respectively. Blunt cone is a body of revolution with an ellipse-clothoid-cone profile. The elliptical nose has an eccentricity of 2.5, and the conical body has a semi-cone angle of 12.50 as shown in Fig. 3(a). In the ramped configuration, four plane ramp surfaces with 9∘ inclination cut on conical surfaces are made at the perpendicular to each another as shown in Fig. 3(b). Flapped configurations consist of four flaps attached at the rear section of the ramp as illustrated in Fig. 3(c).
3. Computation Three-dimensional, steady, compressible flow over different reentry configurations was computationally simulated. The fundamental governing equations represented in Eq. (1)–(3) were discretized using the finite volume method. The discretized system of equations is solved with a coupled implicit solver with cell centered scheme which stores the flow variables data at the center of the mesh element [20]. The modeling and simulation process is similar to the methodology adopted by Raina et al. [21]. The air is considered as a calorically perfect gas, and the density was calculated using ideal gas law. For viscous and compressible flow, the viscosity is generally [22] calculated as a function of temperature as defined by Sutherland law as in Eq. 4, and hence it is used in the present work. 𝐶𝑜𝑛𝑡𝑖𝑛𝑢𝑖𝑡𝑦 𝑒𝑞 𝑢𝑎𝑡𝑖𝑜𝑛 ∶
𝜕𝜌 𝜕 + (𝜌𝑢𝑖 ) = 0 𝜕𝑡 𝜕 𝑥𝑖
(1)
S.K. Raman, W. Kexin and T.H. Kim et al.
International Journal of Mechanical Sciences 176 (2020) 105396
Fig. 2. Model of different reentry body configurations.
Fig. 3. Geometry details of three different reentry configurations considered in the present study. All dimensions are in millimeter.
𝜕 𝜌𝑢𝑖 𝜕𝑝 𝜕 𝑀𝑜𝑚𝑒𝑛𝑡𝑢𝑚 𝑒𝑞 𝑢𝑎𝑡𝑖𝑜𝑛 ∶ + (𝜌𝑢𝑖 𝑢𝑗 ) = − 𝜕𝑡 𝜕 𝑥𝑗 𝜕 𝑥𝑖 [ ( )] ( ) 𝜕𝑢𝑗 𝜕𝑢𝑖 𝜕 2 𝜕𝑢𝑘 𝜕 ′′ ′′ −𝜌𝑢𝑖 𝑢𝑗 + 𝜇 + − 𝛿𝑖𝑗 + 𝜕𝑥𝑗 𝜕𝑥𝑗 𝜕𝑥𝑖 3 𝜕𝑥𝑘 𝜕𝑥𝑗
(2)
3.1. Computational model
′′ ′′
The term −𝜌𝑢𝑖 𝑢𝑗 is Reynolds stress, and it needs to be approximated for solving the closure problem of Reynolds Averaged Navier Stokes equations (RANS). Hence, the additional transport equations of turbulence models and Reynolds Stress Models (RSM) are required. 𝜕 (𝜌𝐶𝑃 𝑇 ) 𝜕 𝐸𝑛𝑒𝑟𝑔 𝑦 𝑒𝑞 𝑢𝑎𝑡𝑖𝑜𝑛 ∶ + (𝐶 𝑇 𝜌𝑢𝑖 ) 𝜕(𝑡 𝜕 𝑥𝑖 𝑃) ( ) 𝜕(𝑝𝑢𝑖 ) 𝜕𝑝 𝜕 𝜕𝑇 = 𝑘 + 𝛽𝑇 + +Φ 𝜕𝑥𝑖 𝜕𝑥𝑖 𝜕𝑇 𝜕𝑥𝑖 [ ( 𝜕𝑢 where the viscous dissipation rate, Φ = 𝜇 𝜕𝑥𝑖 + 𝑗
( 𝑆𝑢𝑡ℎ𝑒𝑟𝑙𝑎𝑛𝑑 ′ 𝑠 𝑙𝑎𝑤 ∶ 𝜇 = 𝜇0
𝑇 𝑇0
)3 2
(𝑇0 + 𝑆) (𝑇 + 𝑆)
(3)
𝜕𝑢𝑗 𝜕𝑥𝑖
At standard atmospheric temperature and pressure, 𝜇0 = 1.716 x 10−5 kg/(m-s), T0 = 273.11 K, and S = 110.56 K.
−
2 𝜕𝑢𝑘 𝛿 3 𝜕𝑥𝑘 𝑖𝑗
)]
𝜕 𝑢𝑖 𝜕 𝑥𝑗
(4)
Rectangular computational domain 110Dx60Dx30D is modeled with half reentry geometry using symmetry boundary condition as shown in Fig. 4. Freestream static pressure, temperature, and Mach number were defined at the inlet boundary condition, and the pressure was implicitly determined from the interior cells at the outlet. 3.2. Mesh independence Structured grid with hexahedron mesh elements is generated using ICEM-CFD. The mesh independence test is carried out individually for each reentry body configuration for determining the optimum computational efficiency. Fig. 5 shows the mesh independence study for the flap at 25∘ case at M=3 with four different mesh counts such as 0.45 million, 0.89 million, 1.88 million and 3.87 million. The CP along the
S.K. Raman, W. Kexin and T.H. Kim et al.
Fig. 4. Computational domain at symmetrical plane with boundary conditions. The “D” represents the base diameter of reentry configurations. The wall are considered as adiabatic and no-slip.
base is also included in the mesh independence study as it is a dominant factor in determining the CA . It is found that the grid refinement above 0.53 million elements, CP appreciably not changed. In a similar manner, numerical models with 0.53 million and 0.71 million mesh elements for blunt cone and ramped configurations respectively were selected. The close-up view of the mesh near the ramped and flapped configurations are shown in Fig. 6. 3.3. Turbulence model selection RANS equations require either a model or closure to compute the Reynolds stress term and thereby solves the time-averaged flow. The development of these turbulence models are generally based on the incom-
International Journal of Mechanical Sciences 176 (2020) 105396
pressible flow without considering the energy equation. Supersonic flow over reentry body forms several shock waves which inherits strong pressure gradient and density gradient discontinuously. Since, the standard turbulence models are developed based on incompressible flow, these models exhibit severe flaws in accurately estimating the “compressible turbulence” characteristics [23]. In particular, important flow features such as the skin friction at the reattachment point, pressure inside the recirculation region, and the size of the separated regions which determine the drag, are not accurately estimated as they do not consider the compressibility effects. Therefore, the solution highly varies with the selection of different turbulence models and modification of available turbulence models for compressible flows with large density gradients are ongoing research [24]. There is no single turbulence model is available to solve the RANS equations for different flow conditions [25,26]. It is necessary to compare the numerical results of RANS equations with different turbulence models with the experimental data for further calculations [27]. For external compressible flow, earlier study suggests that two equations turbulence models calculates flow characteristics better than one equation turbulence models [28]. Hence, estimation of turbulence parameters from five different types of two equation turbulence models were performed for the closure of the RANS equations. The transport equations of considered 𝑘 − 𝜖 and 𝑘 − 𝜔 turbulence models are described in following sub-sections. 3.3.1. Standard 𝑘 − 𝜖 model (SKE) SKE model is a robust and reasonably accurate turbulence model. It is widely-used engineering turbulence model for industrial applications. The baseline two-transport-equation for solving k and 𝜖 is as follows. [( ) ] 𝜇 𝜕 𝜕 𝜕 𝜕𝑘 (𝜌𝑘) + (𝜌𝑘𝑢𝑖 ) = 𝜇+ 𝑡 + 𝐺𝑘 − 𝐺𝑏 − 𝜌𝜖 − 𝑌𝑀 (5) 𝜕𝑡 𝜕 𝑥𝑖 𝜕 𝑥𝑗 𝜎𝑘 𝜕 𝑥 𝑗 𝜕 𝜕 (𝜌𝜖) + (𝜌𝜖𝑢𝑖 ) 𝜕𝑡 𝜕 𝑥𝑖 [( ) ] ) 𝜇 𝜕 𝜕𝜖 𝜖( 𝜖2 = 𝜇+ 𝑡 + 𝐶1𝜖 𝐺𝑘 + 𝐶3𝜖 𝐺𝑏 − 𝐶2𝜖 𝜌 𝜕 𝑥𝑗 𝜎𝜖 𝜕 𝑥 𝑗 𝑘 𝑘
(6)
where, 𝜇 t is turbulent viscosity. Coefficients are empirically derived from benchmark experiments such as 𝜎𝑘 = 1.0, 𝜎𝜖 = 1.3, 𝐶1𝜖 = 1.44, 𝐶2𝜖 =1.92 and C𝜇=0.09. Fig. 5. Mesh independence test for the flapped configurations with flap angle=25∘ . Numerical results of surface pressure distribution over the reentry configuration with flap angle=25∘ is calculated at 𝛼=00 and M=3.Sr ef represent the curve length from nose to trailing edge of the flap.
S.K. Raman, W. Kexin and T.H. Kim et al.
International Journal of Mechanical Sciences 176 (2020) 105396
Fig. 6. The grid near the reentry configurations at the symmetry plane.
3.3.2. Realizable 𝑘 − 𝜖 model (RKE) In RKE model, dssipation rate (𝜖) equation is derived from the meansquare vorticity fluctuation, which is fundamentally different from the SKE model. Also, several realizability conditions are enforced for Reynolds stresses. Its “realizability” stems from changes that allow certain mathematical constraints to be obeyed which ultimately improves the performance of this model. Hence, RKE model likely to provide superior performance for flows involving rotation, boundary layers under strong adverse pressure gradients, separation, and recirculation. The formulations for RKE model is, [( )] 𝜇 𝜕 𝜕 𝜕 𝜕𝑘 (𝜌𝑘) + (𝜌𝑘𝑢𝑗 ) = 𝜇+ 𝑡 + 𝐺𝑘 − 𝐺𝑏 − 𝜌𝜖 − 𝑌𝑀 (7) 𝜕𝑡 𝜕 𝑥𝑖 𝜕 𝑥𝑗 𝜎𝑘 𝜕 𝑥𝑗 𝜕 𝜕 (𝜌𝜖) + (𝜌𝜖𝑢𝑖 ) 𝜕𝑡 𝜕 𝑥𝑖 [( ) ] 𝜇 𝜕 𝜕𝜖 𝜖2 𝜖 = 𝜇+ 𝑡 + 𝜌𝐶1 𝑆𝜖 − 𝜌𝐶2 √ + 𝐶1𝜖 𝐶3𝜖 𝐺𝑏 𝜕 𝑥𝑗 𝜎𝜖 𝜕 𝑥 𝑗 𝑘 𝑘 + 𝜈𝜖
(8)
and
3.3.3. Renormalization group 𝑘 − 𝜖 model (RNG) In RNG model, the constants are derived from renormalization group theory instead of empiricism. RNG models inherits similar advantages as that of RKE model. A variant of the standard k-𝜖 model. Equations and coefficients are analytically derived. Significant changes in the 𝜖 equation improves the ability to model highly strained flows. The transport equations for this model is as follows, ) 𝜕𝑘 𝜕 𝜕 𝜕 ( 𝛼 .𝜇 (𝜌𝑘) + (𝜌𝑘𝑢𝑖 ) = + 𝐺𝑘 − 𝐺𝑏 − 𝜌𝜖 − 𝑌𝑀 𝜕𝑡 𝜕 𝑥𝑖 𝜕 𝑥𝑗 𝜖 𝑒𝑓 𝑓 𝜕 𝑥𝑗 𝜕 𝜕 (𝜌𝜖) + (𝜌𝜖𝑢𝑖 ) 𝜕𝑡 𝜕 𝑥𝑖 ) 𝜕𝜖 ) 𝜕 ( 𝜖( 𝜖2 = + 𝐶1𝜖 𝐺𝑘 + 𝐶3𝜖 𝐺𝑏 − 𝐶2𝜖 𝜌 − 𝑅𝜖 𝛼𝜖 .𝜇𝑒𝑓 𝑓 𝜕 𝑥𝑗 𝜕 𝑥𝑗 𝑘 𝑘
3.3.4. Standard 𝑘 − 𝜔 model In 𝑘 − 𝜔 models, the transport equation for the turbulent dissipation rate, 𝜖, is replaced with an equation for the specific dissipation rate, 𝜔. The turbulent kinetic energy transport equation is still solved in this model. For separation, transition, low Re effects, and impingement, 𝑘 − 𝜔 models are more accurate than 𝑘 − 𝜖 models. Further, it is accurate and robust for a wide range of boundary layer flows with pressure gradient. One of the advantages of the 𝑘 − 𝜔 formulation is the near wall treatment for low-Reynolds number computations. This model is designed to predict correct behavior when integrated to the wall. The 𝑘 − 𝜔 models switches between a viscous sublayer formulation (i.e. direct resolution of the boundary layer) at low y+ values and a wall function approach at higher y+ values. The formulations for Standard 𝑘 − 𝜔 model is as follows, ( ) 𝜕 𝜕 𝜕 𝜕𝑘 (𝜌𝑘) + (𝜌𝑘𝑢𝑖 ) = Γ𝑘 + 𝐺𝑘 − 𝑌𝑘 (11) 𝜕𝑡 𝜕 𝑥𝑖 𝜕 𝑥𝑗 𝜕 𝑥𝑗
(9)
𝜕 𝜕 𝜕 (𝜌𝜔) + (𝜌𝜔𝑢𝑖 ) = 𝜕𝑡 𝜕 𝑥𝑖 𝜕 𝑥𝑗
where, constants are set as C𝜇 =0.0845, C1𝜖 =1.42, C2𝜖 =1.68. ”Enhanced Wall Treatment” available as one of a sub-model for compressibility is adopted along with all 𝑘 − 𝜖 models to accurately capture viscous sublayer behavior.
(12)
3.3.5. Shear-stress transport 𝑘 − 𝜔 model SST 𝑘 − 𝜔 models use standard 𝑘 − 𝜔 near the wall and standard 𝑘 − 𝜖 away from the wall with a blending function. Offers similar benefits as standard k-𝜔. Not overly sensitive to inlet boundary conditions like the standard k-𝜔. Provides more accurate prediction of flow separation than other RANS models. Formulations for k and 𝜔 in this model is as follows, ( ) 𝜕 𝜕 𝜕 𝜕𝑘 ̃𝑘 − 𝑌𝑘 (𝜌𝑘) + (𝜌𝑘𝑢𝑖 ) = Γ𝑘 +𝐺 (13) 𝜕𝑡 𝜕 𝑥𝑖 𝜕 𝑥𝑗 𝜕 𝑥𝑗 and
(10)
( ) 𝜕𝜔 Γ𝜔 + 𝐺𝜔 − 𝑌𝜔 𝜕 𝑥𝑗
𝜕 𝜕 𝜕 (𝜌𝜔) + (𝜌𝜔𝑢𝑖 ) = 𝜕𝑡 𝜕 𝑥𝑖 𝜕 𝑥𝑗
( ) 𝜕𝜔 Γ𝜔 + 𝐺𝜔 − 𝑌𝜔 + 𝐷𝜔 𝜕 𝑥𝑗
(14)
In these equations, 𝐺̃ 𝑘 and G𝜔 represent the generation of turbulent kinetic energy due to mean velocity gradients and the generation of 𝜔 respectively. Γk and Γ𝜔 represent the effective diffusivity of k and 𝜔, respectively. The terms Yk and Y𝜔 represent the dissipation of k and 𝜔
S.K. Raman, W. Kexin and T.H. Kim et al.
International Journal of Mechanical Sciences 176 (2020) 105396
4.1. Effect of ramp and flap
Fig. 7. Axial force coefficient predicted with various two-equation turbulence models are compared with the experimental data from Barrio et al. [13]. SST 𝑘 − 𝜔 turbulence model accurately predict the experimental results than other turbulence models.
Fig. 9 (a) and (b) shows a bow shock wave formed ahead of the bluntnose section of the blunt cone and flapped configurations respectively. Flow becomes subsonic behind the normal portion of the bow shock wave as in Fig. 10. Around the convex corner of the nose-body junction, flow turns almost perpendicularly away from it and forms expansion fan. This expansion fan is marked in Fig. 9 also as a darker region. The subsonic flow in the vicinity of stagnation point accelerates to supersonic through this expansion fan. In the flapped configuration, additional acceleration happens at the body surface due to the expansion at ramp surface. Flap turns the flow into it which leads to the formation of oblique shock and flow gets decelerated. Finally, flow accelerates through the expansion fan at the base edge to Mach number higher than free-stream configurations for both configurations. The streamlines of the flow, leaving from the base edge of the flapped configuration is shown in Fig. 11. At the circular base edge, a circumferentially uniform expansion fan forms, and deflects the flow along the longitudinal axis. The finite spanned flap causes the streamwise vortices and consequently, a complex base flow field with the recirculation region is formed. In order to investigate the complex base flow, the streamlines from the different edges of the flapped configuration are shown in Fig. 12 and the streamlines are colored with velocity magnitude. Streamline flow from the intersection point of two flap edges flows with higher velocity. This streamline moves towards the axis faster than the streamlines from the flap-ramp corner as in inset Fig. 12ii and Fig. 12iii. This velocity variation introduces a spanwise velocity component in addition to streamwise eventually results in a circulation. Meanwhile, the flow from the circular edge reverses its direction without any circulation as seen in inset Fig. 12iv. The surface pressure distribution over the blunt cone and flapped configurations at a various angle of attack are shown in Fig. 13. For 𝛼 =< 00 , in the leeward side, the negative pressure bleeds along the ramp-cone conjunction For 𝛼 =40 , in the windward side Above the 𝛼 = 40 , in the windward side, an interaction between the bow shock and the oblique shock wave happens as shown in Fig. 14. On further increment in the value of 𝛼, this interaction point moves close to the windward flap. 4.2. Effect of flap deflection
Fig. 8. Definition of body axes system.
due to turbulence. D𝜔 represents the cross-diffusion term. “Compressiblity effects”, a additional compressible sub-model is adopted along with 𝑘 − 𝜔 models. The numerical results of axial force coefficient calculated with the various two-equation turbulence models were compared with the experimental results of Barrio et al. [13] in Fig. 7. The standard and SST 𝑘 − 𝜔 models estimate the axial force coefficient relatively accurate compared to other turbulence models. The SST k -𝜔 has consistency in the accuracy of numerical results at all pitch angles. Therefore, Reynolds-stress in the RANS equation is approximated with transport equations of SST 𝑘 − 𝜔 turbulence model for further calculations. 4. Results and discussions Computational results of high-speed flow over different reentry configurations are presented in this section. The reference coordinate on which aerodynamic data calculated is reported in Fig. 8. In this system, the reference point is placed at the center of gravity.
The flap angle is varied from the 200 to investigate the influence of flap deflection angle, 𝛿. Numerical simulations were done for three additional flap deflection cases such as 150 , 250 and 450 . Computed pressure distribution on the surface at zero pitch angle for different flap configurations is shown in Fig. 15. At the starting point of the ramp, a weak expansion fan occurs; hence, the pressure drop at S/Sref of 0.16. For the flapped configuration of 20∘ deflection angle, the shock forms slightly ahead of flap leading edge itself and earlier pressure rise occur. On decreasing the flap angle to 15∘ , the oblique shock forms exactly at flap leading edge. During the increment of flap angle to 25∘ and 45∘ , oblique shock becomes detached, and the pressure behind shock is nearly constant till the flap leading edge. The pressure peak in the flap region increases with increase in the flap deflection angle. Close-up view of Mach number contour near the flap region for the different flap deflection angle cases are shown in Fig. 16. On increasing 𝛿 value, the Mach number aft the shock decreases; thus, the shock wave becomes stronger. This stronger shock wave leads to higher pressure rise and higher peak pressure over the flap surface as in Fig. 15. For increasing 𝛿, the shock becomes detached ahead of flap leading edge and moves upstream. Over the surface, a low Mach region forms immediately behind shock and grows with increasing 𝛿 values. In this low-speed region, the pressure is nearly constant till the flap edge as in Fig. 15. At the flap leading edge, compression occurs, and the pressure rises. In 𝛿 = 450 , after the shock, the pressure is nearly constant for 10% total body length due to the large separation zone. Thus, 𝛿 = 450 has severe flap ineffectiveness, and it is not studied further.
S.K. Raman, W. Kexin and T.H. Kim et al.
International Journal of Mechanical Sciences 176 (2020) 105396
Fig. 9. x-density gradient for the blunt cone and flapped configurations at M=2 and 𝛼=00 . Compression waves are denoted by lighter region and the expansion waves are denoted by darker region.
Fig. 10. Mach contour around the blunt cone and flapped reentry configurations at M=2 and 𝛼=00 . Presence of ramp in the flapped configuration additionally accelerates to higher Mach number of 1.79. Stronger expansion fan arise from the flap trailing edge leads to higher Mach number in base region.
Fig. 11. Isometric view of streamlines from the base edge of flapped configuration reentry configurations at M=2.
At varying 𝛼, the CP distribution over the surface of the blunt cone, flap deflection angle of 200 , and 250 are shown in Figs. 17–19 respectively. The pressure increment at the windward side is equivalent to the reduction in surface pressure at the leeward side. However, at 𝛼 = 60 ,
Fig. 12. Streamlines from the base edge of flapped configuration. Streamlines are colored with velocity magnitude. The streamlines leaving from the flapflap edge has higher velocity than the other regions this results in a vortical structures.
the pressure peak over the windward flap surface is increasing at a high rate while in the leeward side flap, the pressure peak is not increasing.
S.K. Raman, W. Kexin and T.H. Kim et al.
International Journal of Mechanical Sciences 176 (2020) 105396
Fig. 13. Surface pressure distribution for blunt cone and flapped configuration at 𝛼 = 00 , 𝛼 = 20 , and 𝛼 = 60 at M=2. In flapped configuration, the pressure decrement in the leeward side is more pronounced through the rampbody conical junction. In blunt cone, the pressure on the leeward side gradually decreases.
Fig. 14. The flow field around 200 configuration at various angle of attack. At 𝛼= 8∘ , the bow shock from the nose region and oblique shock from the flap region interacts. This interaction point moves towards the flap at 𝛼= 16∘ .
The calculated CA values for different reentry configuration at varying 𝛼 are shown in Fig. 20. Flattening of a blunt cone with ramp surface decrease the pressure distribution at the ramp starting location as seen in Fig. 15. This reduction in pressure distribution reduces the difference between upstream and downstream momentum. Therefore, the axial force of the ramped configuration is reduced from the blunt cone. The peak pressure rise in the flap region behind the shock increases the momentum difference in axial direction, and hence, CA linearly increases with increase in flap angle. As seen earlier, for varying 𝛼, the surface pressure variation in the windward and the leeward side is equivalent and hence, the momentum difference in axial direction is constant. Therefore, CA is invariant with a changing 𝛼 up to 60 . The CN /CA linearly increases with pitch angle as shown in Fig. 21. The static margin is a measure of the location of the center of pressure, X𝐶𝑃 aft the center of gravity, X𝐶𝐺 and is calculated as a percentage of ”D”. Positive static margin implies that the vehicle makes a restoring moment, C𝑀𝑍 for any change in pitch angle from the trim position. The values of static margin for the different reentry configurations are tabulated in Table 1. Ramp surface makes the blunt cone more unstable. The static margin increases for increasing flap angle indicates the increase in the restoring moment.
Table 1 Static Margin of different reentry configurations at an angle of attack of 60 . Reentry configuration
Static Margin,
Blunt cone 𝛿= 00 𝛿= 150 𝛿= 200 𝛿= 250
-6.7 -7.8 3.1 7.2 11.4
𝑋𝐶𝑃 𝐷
x100 [%]
4.3. Effect of mach number At varying M and 𝛼, the axial force coefficient of different reentry configurations is calculated as shown in Fig. 22. The values of CA of flapped configurations of different flap deflections are nondimensionalized by dividing with CA of 200 flapped configurations at M=2 and 𝛼 = 00 . At constant flap angle, on increasing Mach number, the CA is linearly decreased. At the same altitude, the density and speed
S.K. Raman, W. Kexin and T.H. Kim et al.
International Journal of Mechanical Sciences 176 (2020) 105396
Fig. 15. Surface pressure distribution for various flap deflection angle at M=2 and 𝛼=0. For increasing the flap deflection angle, the earlier onset of pressure rise occurs. Pressure remains constant till the flap leading edge indicates the existence of recirculation region.
Fig. 16. Mach contour near flap for various flap deflection angle at Mach 2. The low velocity region in front of the flap grows as the flap deflection angle increases indicates the flap ineffectiveness.
Fig. 17. Variation of CP with angle of attack for blunt cone reentry configuration. The rise in pressure along the windward direction is equivalent to the reduction in the leeward direction.
Fig. 18. Variation of CP with angle of attack for 200 flapped reentry configuration. The pressure rise on the windward flap region is slightly higher than the pressure reduction in leeward side flap.
of sound are constant, any increase in Mach number is analogous to the increase in velocity; thus, decreases the axial force coefficient. For an
increasing Mach number, the stand-off distance decreases as shown in Table 2 which indicates the movement of the shock towards the body.
S.K. Raman, W. Kexin and T.H. Kim et al.
International Journal of Mechanical Sciences 176 (2020) 105396
Fig. 19. Variation of CP with angle of attack for 250 flapped reentry configuration. The difference between the pressure rise and pressure decrements on the windward and leeward side flaps are higher.
Fig. 21. Normal-to-Axial force coefficients at varying pitch angle for different reentry configurations at M=2. CN /CA slightly decreases for 𝛿=15∘ and 20∘ and the difference increases with the increase in pitch angle.
Fig. 20. Axial force coefficient at varying pitch angle for different reentry configurations at M=2. The CA increases with the flap deflection angle increases. Table 2 Variation of standoff distance with Mach number. All the values are nondimensionalized with the value of 200 flapped configuration at M=2 (28.467 x 10−3 m). Flap angle
M=2 M=3 M=4
150
200
250
1.000 0.972 0.963
1.000 0.972 0.963
1.000 0.972 0.963
Fig. 22. Axial force coefficient at varying Mach number for different flapped configurations. The CA values of flapped configurations of different flap deflections are non-dimensionalized with CA of 200 flapped configurations at M=2. CA increases with flap deflection angle and decreases with Mach number. Table 3 Freestream conditions and numerical results along the trajectory. M
H [Km]
P∞ [Pa]
𝜌∞ [kg/m3 ]
a∞ [m/s]
FA [N]
CA
2 4 5
16.3 19.51 20.6
9812.13 5914.69 4981.29
0.1577 0.095107 0.079877
295.07 295.07 295.478
6.269514 10.4781 13.98811
0.661317 0.458162 0.4648
4.4. Effect of altitude The trajectory details of EXPERT, freestream conditions and calculated aerodynamic results at 𝛼 = 0 are tabulated in Table 3. Despite the fact that the axial force is linearly increasing with increase in Mach number, the CA is not following the trend. The main reason for this discrepancy is the non-linear variation in freestream density along the alti-
tude. Therefore, the influence of altitude is analyzed at constant M = 2 and with freestream conditions of the corresponding altitude tabulated in Table 4. The axial force coefficient of different flap angle configurations is non-dimensionalized with CA of 200 flaps at M=2 and 𝛼=00 . The results indicate that the CA is slightly increased at the lower altitude and increase highly at a higher altitude as shown in Fig. 23. The
S.K. Raman, W. Kexin and T.H. Kim et al.
International Journal of Mechanical Sciences 176 (2020) 105396
Table 4 Atmospheric free-stream conditions at varying altitude. H [km]
P∞ [Pa]
𝜌∞ [kg/m3 ]
a∞ [m/s]
0 10 20 30 40 50
101,325 26,436 5474.89 1171.87 277.522 75.9448
1.225 0.412707 0.0880349 0.0180119 0.00385101 0.000977525
340.294 299.463 295.070 301.803 317.633 329.799
𝜌∞,𝐻=0 𝜌∞
[-]
1 2.97 13.91 68.01 318.09 1253.16
𝑎∞,𝐻=0 𝑎∞
[-]
1 1.14 1.15 1.13 1.07 1.03
while the ramp surfaces decrease the stability. On keeping Mach number constant and increasing the altitude, the axial force coefficient is increased. This increment rate is low at the lower altitude and high at higher altitude. Acknowledgments This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea Government (MSIP) (No. NRF2016R1A2B3016436). References
Fig. 23. Axial force coefficient with different altitude at Mach number ’2’. CA increases with altitude at constant Mach number.
speed of sound does not change much at the altitude of H=50 km from H=0K=km. Hence, at constant Mach number, the freestream velocity does not varies. At H=50 km, the density decreases 1253 times than at the H=0 km. Since, the CA is inversely proportional to the term 𝜌∞ 𝑉∞ 2 , the abrupt decrements in the density abruptly increases the CA value at higher altitude. 5. Conclusion Computational simulations have been made to understand the effect of the flap on surface pressure distribution, axial force coefficient and base vortical structures of the atmospheric re-entry vehicle. The flow field characteristics over different reentry configurations are studied at different flap deflection angles, pitch angles, Mach numbers, and altitudes. Flapped has higher Mach number behind the base than the blunt cone due to stronger expansion fan arising from flap edge. The existence of flap introduces streamwise vortices and spanwise vortices which increases the complexity of base flow. At higher pitch angle, bow shock due to the nose section and the oblique shock due to windward flap interacts and the interaction point moves closer to flap. On increasing the flap deflection angle, a separation zone forms in front of the flap and moves upstream along with shock. The growth of this separation zone affects the flap effectiveness and peak pressure rise. This recirculation region ahead of flap limits the flap angle increment. The Axial force is nearly constant with angle of attack up to 60 , and the normal force increases linearly. The axial force increases with increase in flap deflection angle and decreases with the Mach number. The stand-off distance linearly decreases with an increase in Mach number. The increasing flap deflection increases the stability of the blunt cone
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Effects of flap on the reentry aerodynamics of a blunt cone in the supersonic flow Senthil Kumar Raman a, Wu Kexin a, Tae Ho Kim a, Abhilash Suryan b, Heuy Dong Kim a,∗ a b
Mechanical Engineering, Andong National University, Andong, South Korea Mechanical Engineering, College of Engineering, Trivandrum, Kerala
a r t i c l e Keywords: Flap Reentry capsule Supersonic speed Shock wave
i n f o
a b s t r a c t Flap over blunt asymmetric reentry vehicles improves the aerodynamic stability and control. The present study has focused on the reentry aerodynamics of blunt-nosed conical body with flap configurations. Three dimensional, steady, viscous and compressible flow over the reentry body configurations were numerically analyzed by solving Reynolds averaged Navier-Stokes equations and SST K-𝜔 turbulence model. The fundamental governing equations were discretized from the partial differential form to numerical analogue using the finite volume approach. Numerical simulations were carried out to investigate the flow characteristics of the three different reentry body configurations at different pitch angles, flap angles, the Mach numbers, and altitudes. For increment in the pitch angle from 0∘ to 6∘ , the axial force coefficient is invariant, while the normal force coefficient linearly increases. It is found that the axial force coefficient is directly proportional to the flap angle and inversely proportional to the Mach number. Presence of flap introduces streamwise vortices and increases the flow complexity after the base to a large extent.
1. Introduction Space exploration has obtained an extensive interest in the last decade with an overwhelming involvement of the private sector along with public agencies. Most of the recent space programs by these agencies are focused on human space flight. The accomplishment of these human space missions mainly depends on the successful reentry into the atmosphere. In 1951, H. Allen discovered the classical blunt body theory as a solution for reentry problem by creating a strong shock in front of the nose as in Fig. 1(a). High temperature rise occurs after the strong shock wave and most of the heat dissipated to the air around the reentry body. Since then, continuous research on blunt shaped vehicles are still going on [1–4] and as a result, some phenomenal advancement such as the blunt nose with lifting surface, control surfaces, and aerospikes were developed. The blunt body flying with hypersonic speed is studied extensively [1,4–7]. At hypersonic speed and higher altitude(stratosphere and mesosphere), the flows are usually considered as non-continuum and analyzed with DSMC approach [8,9]. Meanwhile, the understanding of continuum flow physics of supersonic flow over the blunt body is also essential [10–12]. However, the studies about aerodynamic characteris-
tics of the blunt body with flap, available in open literature are limited [13,14]. Flaps on the blunt axisymmetric reentry vehicles has aerodynamically two main advantages. First, when the vehicles flying at an incidence, it assures a pertinent aerodynamic stability by offsetting center of gravity. In other words, flap contributes enough aerodynamic trim without changing the lift-drag ratio (L/D) ratio. Second, flap alter the cross area of reentry bodies and effective angle of attack which allows the steering control at entry, descent and landing. The flap counts is defined by the control required over the number of degrees of freedom. For instance, single flap allows for control over angle of attack alone, while the two flaps at 180∘ to each other provides an additional control on side slip also. Four flaps at 90∘ each other allows to reduce its ballistic coefficient along with the control over the angle of attack and side slip. This decreased ballistic coefficient results in higher drag deceleration and enhancing the entry margin as well as the descent timeline. In general, supersonic flow past an angular obstacle such as wedge produces an oblique shock and raise the pressure [15]. Similarly, the flap attached over the blunt cone compresses the flow field and leads to the formation of oblique shock as shown in Fig. 1(b) and increases the drag. Either at a higher Mach number or a higher flap angle, the
∗
Corresponding author. E-mail addresses: [email protected] (S.K. Raman), [email protected] (T.H. Kim), [email protected] (A. Suryan), [email protected] (H.D. Kim). https://doi.org/10.1016/j.ijmecsci.2019.105396 Received 23 February 2019; Received in revised form 5 December 2019; Accepted 20 December 2019 Available online 22 February 2020 0020-7403/© 2020 Elsevier Ltd. All rights reserved.
S.K. Raman, W. Kexin and T.H. Kim et al.
International Journal of Mechanical Sciences 176 (2020) 105396
Fig. 1. Flow field around a blunt cone and flap reentry configurations. Blunt cone with flap has oblique shock in addition to the flow field of blunt cone.
oblique shock becomes a detached shock which causes the earlier onset of pressure rise and a recirculation region forms ahead of the flap; thus adversely affects the flap effectiveness. Furthermore, the finite spanned flap induces streamwise vortices which severely alters the base flow and increases the flow complexity. Hence, the understanding of high-speed flow over reentry vehicles with flap is essential. Massobrio [16] numerically analyzed the aero-thermodynamic characteristics of four EXPEerimental Reentry Testbed (EXPERT) configuration designed and developed by the European Space Agency. EXPERT has four flaps attached on ramp surfaces of a conical body with a blunt nose. At zero angle of attack, computational model with quarter domain is considered. At non-zero angle of attack, model with a half domain at symmetrical plane is considered. Kharitonov at al [17] experimentally measured the axial force coefficient (CA ) of EXPERT at M=4 and 𝛼 = -3∘ to 6∘ , and found that the CA is nearly constant. Chen et al [18] analyzed the hypersonic flow characteristics of asymmetric blunt body with flap. The monostablity features is significantly enhanced with a slight reduction in L/D on inclusion of flap. Rosa et al [19] analysed the hypersonic flow field over winged reentry vehicles with flap. A recirculation bubble forms ahead of flap and overshoots the pressure at hinge line; this recirculation bubble affects the flap effectiveness. Barrio et al. [13] presented the trajectory details of EXPERT, and numerically calculated the force coefficients at supersonic speed. The the axial and normal force coefficients were found to increase with increasing subsonic Mach number and decrease with increasing supersonic speed. Grasso et al. [13] analyzed the aerodynamic performance of the control surface of the blunt-nosed vehicle and concluded that turbulence becomes important for flap angles higher than the critical deflection angles. More work is required to understand the effect of this flap angle on surface pressure distributions, pressure peaks, and base vortical flow structures. The present study mainly focuses on the effect of the flap deflection angle on the flow field over the blunt cone at supersonic speed. Numerical simulations were carried out to investigate the effect of the flap
on surface pressure distribution, axial force coefficient, shock structure, and the base vortical flow behavior.
2. Geometry The present study considered three different reentry configurations such as a blunt cone, ramped, and flapped (similar to EXPERT) configurations as shown in Fig. 2(a), (b) and (c) respectively. Blunt cone is a body of revolution with an ellipse-clothoid-cone profile. The elliptical nose has an eccentricity of 2.5, and the conical body has a semi-cone angle of 12.50 as shown in Fig. 3(a). In the ramped configuration, four plane ramp surfaces with 9∘ inclination cut on conical surfaces are made at the perpendicular to each another as shown in Fig. 3(b). Flapped configurations consist of four flaps attached at the rear section of the ramp as illustrated in Fig. 3(c).
3. Computation Three-dimensional, steady, compressible flow over different reentry configurations was computationally simulated. The fundamental governing equations represented in Eq. (1)–(3) were discretized using the finite volume method. The discretized system of equations is solved with a coupled implicit solver with cell centered scheme which stores the flow variables data at the center of the mesh element [20]. The modeling and simulation process is similar to the methodology adopted by Raina et al. [21]. The air is considered as a calorically perfect gas, and the density was calculated using ideal gas law. For viscous and compressible flow, the viscosity is generally [22] calculated as a function of temperature as defined by Sutherland law as in Eq. 4, and hence it is used in the present work. 𝐶𝑜𝑛𝑡𝑖𝑛𝑢𝑖𝑡𝑦 𝑒𝑞 𝑢𝑎𝑡𝑖𝑜𝑛 ∶
𝜕𝜌 𝜕 + (𝜌𝑢𝑖 ) = 0 𝜕𝑡 𝜕 𝑥𝑖
(1)
S.K. Raman, W. Kexin and T.H. Kim et al.
International Journal of Mechanical Sciences 176 (2020) 105396
Fig. 2. Model of different reentry body configurations.
Fig. 3. Geometry details of three different reentry configurations considered in the present study. All dimensions are in millimeter.
𝜕 𝜌𝑢𝑖 𝜕𝑝 𝜕 𝑀𝑜𝑚𝑒𝑛𝑡𝑢𝑚 𝑒𝑞 𝑢𝑎𝑡𝑖𝑜𝑛 ∶ + (𝜌𝑢𝑖 𝑢𝑗 ) = − 𝜕𝑡 𝜕 𝑥𝑗 𝜕 𝑥𝑖 [ ( )] ( ) 𝜕𝑢𝑗 𝜕𝑢𝑖 𝜕 2 𝜕𝑢𝑘 𝜕 ′′ ′′ −𝜌𝑢𝑖 𝑢𝑗 + 𝜇 + − 𝛿𝑖𝑗 + 𝜕𝑥𝑗 𝜕𝑥𝑗 𝜕𝑥𝑖 3 𝜕𝑥𝑘 𝜕𝑥𝑗
(2)
3.1. Computational model
′′ ′′
The term −𝜌𝑢𝑖 𝑢𝑗 is Reynolds stress, and it needs to be approximated for solving the closure problem of Reynolds Averaged Navier Stokes equations (RANS). Hence, the additional transport equations of turbulence models and Reynolds Stress Models (RSM) are required. 𝜕 (𝜌𝐶𝑃 𝑇 ) 𝜕 𝐸𝑛𝑒𝑟𝑔 𝑦 𝑒𝑞 𝑢𝑎𝑡𝑖𝑜𝑛 ∶ + (𝐶 𝑇 𝜌𝑢𝑖 ) 𝜕(𝑡 𝜕 𝑥𝑖 𝑃) ( ) 𝜕(𝑝𝑢𝑖 ) 𝜕𝑝 𝜕 𝜕𝑇 = 𝑘 + 𝛽𝑇 + +Φ 𝜕𝑥𝑖 𝜕𝑥𝑖 𝜕𝑇 𝜕𝑥𝑖 [ ( 𝜕𝑢 where the viscous dissipation rate, Φ = 𝜇 𝜕𝑥𝑖 + 𝑗
( 𝑆𝑢𝑡ℎ𝑒𝑟𝑙𝑎𝑛𝑑 ′ 𝑠 𝑙𝑎𝑤 ∶ 𝜇 = 𝜇0
𝑇 𝑇0
)3 2
(𝑇0 + 𝑆) (𝑇 + 𝑆)
(3)
𝜕𝑢𝑗 𝜕𝑥𝑖
At standard atmospheric temperature and pressure, 𝜇0 = 1.716 x 10−5 kg/(m-s), T0 = 273.11 K, and S = 110.56 K.
−
2 𝜕𝑢𝑘 𝛿 3 𝜕𝑥𝑘 𝑖𝑗
)]
𝜕 𝑢𝑖 𝜕 𝑥𝑗
(4)
Rectangular computational domain 110Dx60Dx30D is modeled with half reentry geometry using symmetry boundary condition as shown in Fig. 4. Freestream static pressure, temperature, and Mach number were defined at the inlet boundary condition, and the pressure was implicitly determined from the interior cells at the outlet. 3.2. Mesh independence Structured grid with hexahedron mesh elements is generated using ICEM-CFD. The mesh independence test is carried out individually for each reentry body configuration for determining the optimum computational efficiency. Fig. 5 shows the mesh independence study for the flap at 25∘ case at M=3 with four different mesh counts such as 0.45 million, 0.89 million, 1.88 million and 3.87 million. The CP along the
S.K. Raman, W. Kexin and T.H. Kim et al.
Fig. 4. Computational domain at symmetrical plane with boundary conditions. The “D” represents the base diameter of reentry configurations. The wall are considered as adiabatic and no-slip.
base is also included in the mesh independence study as it is a dominant factor in determining the CA . It is found that the grid refinement above 0.53 million elements, CP appreciably not changed. In a similar manner, numerical models with 0.53 million and 0.71 million mesh elements for blunt cone and ramped configurations respectively were selected. The close-up view of the mesh near the ramped and flapped configurations are shown in Fig. 6. 3.3. Turbulence model selection RANS equations require either a model or closure to compute the Reynolds stress term and thereby solves the time-averaged flow. The development of these turbulence models are generally based on the incom-
International Journal of Mechanical Sciences 176 (2020) 105396
pressible flow without considering the energy equation. Supersonic flow over reentry body forms several shock waves which inherits strong pressure gradient and density gradient discontinuously. Since, the standard turbulence models are developed based on incompressible flow, these models exhibit severe flaws in accurately estimating the “compressible turbulence” characteristics [23]. In particular, important flow features such as the skin friction at the reattachment point, pressure inside the recirculation region, and the size of the separated regions which determine the drag, are not accurately estimated as they do not consider the compressibility effects. Therefore, the solution highly varies with the selection of different turbulence models and modification of available turbulence models for compressible flows with large density gradients are ongoing research [24]. There is no single turbulence model is available to solve the RANS equations for different flow conditions [25,26]. It is necessary to compare the numerical results of RANS equations with different turbulence models with the experimental data for further calculations [27]. For external compressible flow, earlier study suggests that two equations turbulence models calculates flow characteristics better than one equation turbulence models [28]. Hence, estimation of turbulence parameters from five different types of two equation turbulence models were performed for the closure of the RANS equations. The transport equations of considered 𝑘 − 𝜖 and 𝑘 − 𝜔 turbulence models are described in following sub-sections. 3.3.1. Standard 𝑘 − 𝜖 model (SKE) SKE model is a robust and reasonably accurate turbulence model. It is widely-used engineering turbulence model for industrial applications. The baseline two-transport-equation for solving k and 𝜖 is as follows. [( ) ] 𝜇 𝜕 𝜕 𝜕 𝜕𝑘 (𝜌𝑘) + (𝜌𝑘𝑢𝑖 ) = 𝜇+ 𝑡 + 𝐺𝑘 − 𝐺𝑏 − 𝜌𝜖 − 𝑌𝑀 (5) 𝜕𝑡 𝜕 𝑥𝑖 𝜕 𝑥𝑗 𝜎𝑘 𝜕 𝑥 𝑗 𝜕 𝜕 (𝜌𝜖) + (𝜌𝜖𝑢𝑖 ) 𝜕𝑡 𝜕 𝑥𝑖 [( ) ] ) 𝜇 𝜕 𝜕𝜖 𝜖( 𝜖2 = 𝜇+ 𝑡 + 𝐶1𝜖 𝐺𝑘 + 𝐶3𝜖 𝐺𝑏 − 𝐶2𝜖 𝜌 𝜕 𝑥𝑗 𝜎𝜖 𝜕 𝑥 𝑗 𝑘 𝑘
(6)
where, 𝜇 t is turbulent viscosity. Coefficients are empirically derived from benchmark experiments such as 𝜎𝑘 = 1.0, 𝜎𝜖 = 1.3, 𝐶1𝜖 = 1.44, 𝐶2𝜖 =1.92 and C𝜇=0.09. Fig. 5. Mesh independence test for the flapped configurations with flap angle=25∘ . Numerical results of surface pressure distribution over the reentry configuration with flap angle=25∘ is calculated at 𝛼=00 and M=3.Sr ef represent the curve length from nose to trailing edge of the flap.
S.K. Raman, W. Kexin and T.H. Kim et al.
International Journal of Mechanical Sciences 176 (2020) 105396
Fig. 6. The grid near the reentry configurations at the symmetry plane.
3.3.2. Realizable 𝑘 − 𝜖 model (RKE) In RKE model, dssipation rate (𝜖) equation is derived from the meansquare vorticity fluctuation, which is fundamentally different from the SKE model. Also, several realizability conditions are enforced for Reynolds stresses. Its “realizability” stems from changes that allow certain mathematical constraints to be obeyed which ultimately improves the performance of this model. Hence, RKE model likely to provide superior performance for flows involving rotation, boundary layers under strong adverse pressure gradients, separation, and recirculation. The formulations for RKE model is, [( )] 𝜇 𝜕 𝜕 𝜕 𝜕𝑘 (𝜌𝑘) + (𝜌𝑘𝑢𝑗 ) = 𝜇+ 𝑡 + 𝐺𝑘 − 𝐺𝑏 − 𝜌𝜖 − 𝑌𝑀 (7) 𝜕𝑡 𝜕 𝑥𝑖 𝜕 𝑥𝑗 𝜎𝑘 𝜕 𝑥𝑗 𝜕 𝜕 (𝜌𝜖) + (𝜌𝜖𝑢𝑖 ) 𝜕𝑡 𝜕 𝑥𝑖 [( ) ] 𝜇 𝜕 𝜕𝜖 𝜖2 𝜖 = 𝜇+ 𝑡 + 𝜌𝐶1 𝑆𝜖 − 𝜌𝐶2 √ + 𝐶1𝜖 𝐶3𝜖 𝐺𝑏 𝜕 𝑥𝑗 𝜎𝜖 𝜕 𝑥 𝑗 𝑘 𝑘 + 𝜈𝜖
(8)
and
3.3.3. Renormalization group 𝑘 − 𝜖 model (RNG) In RNG model, the constants are derived from renormalization group theory instead of empiricism. RNG models inherits similar advantages as that of RKE model. A variant of the standard k-𝜖 model. Equations and coefficients are analytically derived. Significant changes in the 𝜖 equation improves the ability to model highly strained flows. The transport equations for this model is as follows, ) 𝜕𝑘 𝜕 𝜕 𝜕 ( 𝛼 .𝜇 (𝜌𝑘) + (𝜌𝑘𝑢𝑖 ) = + 𝐺𝑘 − 𝐺𝑏 − 𝜌𝜖 − 𝑌𝑀 𝜕𝑡 𝜕 𝑥𝑖 𝜕 𝑥𝑗 𝜖 𝑒𝑓 𝑓 𝜕 𝑥𝑗 𝜕 𝜕 (𝜌𝜖) + (𝜌𝜖𝑢𝑖 ) 𝜕𝑡 𝜕 𝑥𝑖 ) 𝜕𝜖 ) 𝜕 ( 𝜖( 𝜖2 = + 𝐶1𝜖 𝐺𝑘 + 𝐶3𝜖 𝐺𝑏 − 𝐶2𝜖 𝜌 − 𝑅𝜖 𝛼𝜖 .𝜇𝑒𝑓 𝑓 𝜕 𝑥𝑗 𝜕 𝑥𝑗 𝑘 𝑘
3.3.4. Standard 𝑘 − 𝜔 model In 𝑘 − 𝜔 models, the transport equation for the turbulent dissipation rate, 𝜖, is replaced with an equation for the specific dissipation rate, 𝜔. The turbulent kinetic energy transport equation is still solved in this model. For separation, transition, low Re effects, and impingement, 𝑘 − 𝜔 models are more accurate than 𝑘 − 𝜖 models. Further, it is accurate and robust for a wide range of boundary layer flows with pressure gradient. One of the advantages of the 𝑘 − 𝜔 formulation is the near wall treatment for low-Reynolds number computations. This model is designed to predict correct behavior when integrated to the wall. The 𝑘 − 𝜔 models switches between a viscous sublayer formulation (i.e. direct resolution of the boundary layer) at low y+ values and a wall function approach at higher y+ values. The formulations for Standard 𝑘 − 𝜔 model is as follows, ( ) 𝜕 𝜕 𝜕 𝜕𝑘 (𝜌𝑘) + (𝜌𝑘𝑢𝑖 ) = Γ𝑘 + 𝐺𝑘 − 𝑌𝑘 (11) 𝜕𝑡 𝜕 𝑥𝑖 𝜕 𝑥𝑗 𝜕 𝑥𝑗
(9)
𝜕 𝜕 𝜕 (𝜌𝜔) + (𝜌𝜔𝑢𝑖 ) = 𝜕𝑡 𝜕 𝑥𝑖 𝜕 𝑥𝑗
where, constants are set as C𝜇 =0.0845, C1𝜖 =1.42, C2𝜖 =1.68. ”Enhanced Wall Treatment” available as one of a sub-model for compressibility is adopted along with all 𝑘 − 𝜖 models to accurately capture viscous sublayer behavior.
(12)
3.3.5. Shear-stress transport 𝑘 − 𝜔 model SST 𝑘 − 𝜔 models use standard 𝑘 − 𝜔 near the wall and standard 𝑘 − 𝜖 away from the wall with a blending function. Offers similar benefits as standard k-𝜔. Not overly sensitive to inlet boundary conditions like the standard k-𝜔. Provides more accurate prediction of flow separation than other RANS models. Formulations for k and 𝜔 in this model is as follows, ( ) 𝜕 𝜕 𝜕 𝜕𝑘 ̃𝑘 − 𝑌𝑘 (𝜌𝑘) + (𝜌𝑘𝑢𝑖 ) = Γ𝑘 +𝐺 (13) 𝜕𝑡 𝜕 𝑥𝑖 𝜕 𝑥𝑗 𝜕 𝑥𝑗 and
(10)
( ) 𝜕𝜔 Γ𝜔 + 𝐺𝜔 − 𝑌𝜔 𝜕 𝑥𝑗
𝜕 𝜕 𝜕 (𝜌𝜔) + (𝜌𝜔𝑢𝑖 ) = 𝜕𝑡 𝜕 𝑥𝑖 𝜕 𝑥𝑗
( ) 𝜕𝜔 Γ𝜔 + 𝐺𝜔 − 𝑌𝜔 + 𝐷𝜔 𝜕 𝑥𝑗
(14)
In these equations, 𝐺̃ 𝑘 and G𝜔 represent the generation of turbulent kinetic energy due to mean velocity gradients and the generation of 𝜔 respectively. Γk and Γ𝜔 represent the effective diffusivity of k and 𝜔, respectively. The terms Yk and Y𝜔 represent the dissipation of k and 𝜔
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International Journal of Mechanical Sciences 176 (2020) 105396
4.1. Effect of ramp and flap
Fig. 7. Axial force coefficient predicted with various two-equation turbulence models are compared with the experimental data from Barrio et al. [13]. SST 𝑘 − 𝜔 turbulence model accurately predict the experimental results than other turbulence models.
Fig. 9 (a) and (b) shows a bow shock wave formed ahead of the bluntnose section of the blunt cone and flapped configurations respectively. Flow becomes subsonic behind the normal portion of the bow shock wave as in Fig. 10. Around the convex corner of the nose-body junction, flow turns almost perpendicularly away from it and forms expansion fan. This expansion fan is marked in Fig. 9 also as a darker region. The subsonic flow in the vicinity of stagnation point accelerates to supersonic through this expansion fan. In the flapped configuration, additional acceleration happens at the body surface due to the expansion at ramp surface. Flap turns the flow into it which leads to the formation of oblique shock and flow gets decelerated. Finally, flow accelerates through the expansion fan at the base edge to Mach number higher than free-stream configurations for both configurations. The streamlines of the flow, leaving from the base edge of the flapped configuration is shown in Fig. 11. At the circular base edge, a circumferentially uniform expansion fan forms, and deflects the flow along the longitudinal axis. The finite spanned flap causes the streamwise vortices and consequently, a complex base flow field with the recirculation region is formed. In order to investigate the complex base flow, the streamlines from the different edges of the flapped configuration are shown in Fig. 12 and the streamlines are colored with velocity magnitude. Streamline flow from the intersection point of two flap edges flows with higher velocity. This streamline moves towards the axis faster than the streamlines from the flap-ramp corner as in inset Fig. 12ii and Fig. 12iii. This velocity variation introduces a spanwise velocity component in addition to streamwise eventually results in a circulation. Meanwhile, the flow from the circular edge reverses its direction without any circulation as seen in inset Fig. 12iv. The surface pressure distribution over the blunt cone and flapped configurations at a various angle of attack are shown in Fig. 13. For 𝛼 =< 00 , in the leeward side, the negative pressure bleeds along the ramp-cone conjunction For 𝛼 =40 , in the windward side Above the 𝛼 = 40 , in the windward side, an interaction between the bow shock and the oblique shock wave happens as shown in Fig. 14. On further increment in the value of 𝛼, this interaction point moves close to the windward flap. 4.2. Effect of flap deflection
Fig. 8. Definition of body axes system.
due to turbulence. D𝜔 represents the cross-diffusion term. “Compressiblity effects”, a additional compressible sub-model is adopted along with 𝑘 − 𝜔 models. The numerical results of axial force coefficient calculated with the various two-equation turbulence models were compared with the experimental results of Barrio et al. [13] in Fig. 7. The standard and SST 𝑘 − 𝜔 models estimate the axial force coefficient relatively accurate compared to other turbulence models. The SST k -𝜔 has consistency in the accuracy of numerical results at all pitch angles. Therefore, Reynolds-stress in the RANS equation is approximated with transport equations of SST 𝑘 − 𝜔 turbulence model for further calculations. 4. Results and discussions Computational results of high-speed flow over different reentry configurations are presented in this section. The reference coordinate on which aerodynamic data calculated is reported in Fig. 8. In this system, the reference point is placed at the center of gravity.
The flap angle is varied from the 200 to investigate the influence of flap deflection angle, 𝛿. Numerical simulations were done for three additional flap deflection cases such as 150 , 250 and 450 . Computed pressure distribution on the surface at zero pitch angle for different flap configurations is shown in Fig. 15. At the starting point of the ramp, a weak expansion fan occurs; hence, the pressure drop at S/Sref of 0.16. For the flapped configuration of 20∘ deflection angle, the shock forms slightly ahead of flap leading edge itself and earlier pressure rise occur. On decreasing the flap angle to 15∘ , the oblique shock forms exactly at flap leading edge. During the increment of flap angle to 25∘ and 45∘ , oblique shock becomes detached, and the pressure behind shock is nearly constant till the flap leading edge. The pressure peak in the flap region increases with increase in the flap deflection angle. Close-up view of Mach number contour near the flap region for the different flap deflection angle cases are shown in Fig. 16. On increasing 𝛿 value, the Mach number aft the shock decreases; thus, the shock wave becomes stronger. This stronger shock wave leads to higher pressure rise and higher peak pressure over the flap surface as in Fig. 15. For increasing 𝛿, the shock becomes detached ahead of flap leading edge and moves upstream. Over the surface, a low Mach region forms immediately behind shock and grows with increasing 𝛿 values. In this low-speed region, the pressure is nearly constant till the flap edge as in Fig. 15. At the flap leading edge, compression occurs, and the pressure rises. In 𝛿 = 450 , after the shock, the pressure is nearly constant for 10% total body length due to the large separation zone. Thus, 𝛿 = 450 has severe flap ineffectiveness, and it is not studied further.
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International Journal of Mechanical Sciences 176 (2020) 105396
Fig. 9. x-density gradient for the blunt cone and flapped configurations at M=2 and 𝛼=00 . Compression waves are denoted by lighter region and the expansion waves are denoted by darker region.
Fig. 10. Mach contour around the blunt cone and flapped reentry configurations at M=2 and 𝛼=00 . Presence of ramp in the flapped configuration additionally accelerates to higher Mach number of 1.79. Stronger expansion fan arise from the flap trailing edge leads to higher Mach number in base region.
Fig. 11. Isometric view of streamlines from the base edge of flapped configuration reentry configurations at M=2.
At varying 𝛼, the CP distribution over the surface of the blunt cone, flap deflection angle of 200 , and 250 are shown in Figs. 17–19 respectively. The pressure increment at the windward side is equivalent to the reduction in surface pressure at the leeward side. However, at 𝛼 = 60 ,
Fig. 12. Streamlines from the base edge of flapped configuration. Streamlines are colored with velocity magnitude. The streamlines leaving from the flapflap edge has higher velocity than the other regions this results in a vortical structures.
the pressure peak over the windward flap surface is increasing at a high rate while in the leeward side flap, the pressure peak is not increasing.
S.K. Raman, W. Kexin and T.H. Kim et al.
International Journal of Mechanical Sciences 176 (2020) 105396
Fig. 13. Surface pressure distribution for blunt cone and flapped configuration at 𝛼 = 00 , 𝛼 = 20 , and 𝛼 = 60 at M=2. In flapped configuration, the pressure decrement in the leeward side is more pronounced through the rampbody conical junction. In blunt cone, the pressure on the leeward side gradually decreases.
Fig. 14. The flow field around 200 configuration at various angle of attack. At 𝛼= 8∘ , the bow shock from the nose region and oblique shock from the flap region interacts. This interaction point moves towards the flap at 𝛼= 16∘ .
The calculated CA values for different reentry configuration at varying 𝛼 are shown in Fig. 20. Flattening of a blunt cone with ramp surface decrease the pressure distribution at the ramp starting location as seen in Fig. 15. This reduction in pressure distribution reduces the difference between upstream and downstream momentum. Therefore, the axial force of the ramped configuration is reduced from the blunt cone. The peak pressure rise in the flap region behind the shock increases the momentum difference in axial direction, and hence, CA linearly increases with increase in flap angle. As seen earlier, for varying 𝛼, the surface pressure variation in the windward and the leeward side is equivalent and hence, the momentum difference in axial direction is constant. Therefore, CA is invariant with a changing 𝛼 up to 60 . The CN /CA linearly increases with pitch angle as shown in Fig. 21. The static margin is a measure of the location of the center of pressure, X𝐶𝑃 aft the center of gravity, X𝐶𝐺 and is calculated as a percentage of ”D”. Positive static margin implies that the vehicle makes a restoring moment, C𝑀𝑍 for any change in pitch angle from the trim position. The values of static margin for the different reentry configurations are tabulated in Table 1. Ramp surface makes the blunt cone more unstable. The static margin increases for increasing flap angle indicates the increase in the restoring moment.
Table 1 Static Margin of different reentry configurations at an angle of attack of 60 . Reentry configuration
Static Margin,
Blunt cone 𝛿= 00 𝛿= 150 𝛿= 200 𝛿= 250
-6.7 -7.8 3.1 7.2 11.4
𝑋𝐶𝑃 𝐷
x100 [%]
4.3. Effect of mach number At varying M and 𝛼, the axial force coefficient of different reentry configurations is calculated as shown in Fig. 22. The values of CA of flapped configurations of different flap deflections are nondimensionalized by dividing with CA of 200 flapped configurations at M=2 and 𝛼 = 00 . At constant flap angle, on increasing Mach number, the CA is linearly decreased. At the same altitude, the density and speed
S.K. Raman, W. Kexin and T.H. Kim et al.
International Journal of Mechanical Sciences 176 (2020) 105396
Fig. 15. Surface pressure distribution for various flap deflection angle at M=2 and 𝛼=0. For increasing the flap deflection angle, the earlier onset of pressure rise occurs. Pressure remains constant till the flap leading edge indicates the existence of recirculation region.
Fig. 16. Mach contour near flap for various flap deflection angle at Mach 2. The low velocity region in front of the flap grows as the flap deflection angle increases indicates the flap ineffectiveness.
Fig. 17. Variation of CP with angle of attack for blunt cone reentry configuration. The rise in pressure along the windward direction is equivalent to the reduction in the leeward direction.
Fig. 18. Variation of CP with angle of attack for 200 flapped reentry configuration. The pressure rise on the windward flap region is slightly higher than the pressure reduction in leeward side flap.
of sound are constant, any increase in Mach number is analogous to the increase in velocity; thus, decreases the axial force coefficient. For an
increasing Mach number, the stand-off distance decreases as shown in Table 2 which indicates the movement of the shock towards the body.
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International Journal of Mechanical Sciences 176 (2020) 105396
Fig. 19. Variation of CP with angle of attack for 250 flapped reentry configuration. The difference between the pressure rise and pressure decrements on the windward and leeward side flaps are higher.
Fig. 21. Normal-to-Axial force coefficients at varying pitch angle for different reentry configurations at M=2. CN /CA slightly decreases for 𝛿=15∘ and 20∘ and the difference increases with the increase in pitch angle.
Fig. 20. Axial force coefficient at varying pitch angle for different reentry configurations at M=2. The CA increases with the flap deflection angle increases. Table 2 Variation of standoff distance with Mach number. All the values are nondimensionalized with the value of 200 flapped configuration at M=2 (28.467 x 10−3 m). Flap angle
M=2 M=3 M=4
150
200
250
1.000 0.972 0.963
1.000 0.972 0.963
1.000 0.972 0.963
Fig. 22. Axial force coefficient at varying Mach number for different flapped configurations. The CA values of flapped configurations of different flap deflections are non-dimensionalized with CA of 200 flapped configurations at M=2. CA increases with flap deflection angle and decreases with Mach number. Table 3 Freestream conditions and numerical results along the trajectory. M
H [Km]
P∞ [Pa]
𝜌∞ [kg/m3 ]
a∞ [m/s]
FA [N]
CA
2 4 5
16.3 19.51 20.6
9812.13 5914.69 4981.29
0.1577 0.095107 0.079877
295.07 295.07 295.478
6.269514 10.4781 13.98811
0.661317 0.458162 0.4648
4.4. Effect of altitude The trajectory details of EXPERT, freestream conditions and calculated aerodynamic results at 𝛼 = 0 are tabulated in Table 3. Despite the fact that the axial force is linearly increasing with increase in Mach number, the CA is not following the trend. The main reason for this discrepancy is the non-linear variation in freestream density along the alti-
tude. Therefore, the influence of altitude is analyzed at constant M = 2 and with freestream conditions of the corresponding altitude tabulated in Table 4. The axial force coefficient of different flap angle configurations is non-dimensionalized with CA of 200 flaps at M=2 and 𝛼=00 . The results indicate that the CA is slightly increased at the lower altitude and increase highly at a higher altitude as shown in Fig. 23. The
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International Journal of Mechanical Sciences 176 (2020) 105396
Table 4 Atmospheric free-stream conditions at varying altitude. H [km]
P∞ [Pa]
𝜌∞ [kg/m3 ]
a∞ [m/s]
0 10 20 30 40 50
101,325 26,436 5474.89 1171.87 277.522 75.9448
1.225 0.412707 0.0880349 0.0180119 0.00385101 0.000977525
340.294 299.463 295.070 301.803 317.633 329.799
𝜌∞,𝐻=0 𝜌∞
[-]
1 2.97 13.91 68.01 318.09 1253.16
𝑎∞,𝐻=0 𝑎∞
[-]
1 1.14 1.15 1.13 1.07 1.03
while the ramp surfaces decrease the stability. On keeping Mach number constant and increasing the altitude, the axial force coefficient is increased. This increment rate is low at the lower altitude and high at higher altitude. Acknowledgments This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea Government (MSIP) (No. NRF2016R1A2B3016436). References
Fig. 23. Axial force coefficient with different altitude at Mach number ’2’. CA increases with altitude at constant Mach number.
speed of sound does not change much at the altitude of H=50 km from H=0K=km. Hence, at constant Mach number, the freestream velocity does not varies. At H=50 km, the density decreases 1253 times than at the H=0 km. Since, the CA is inversely proportional to the term 𝜌∞ 𝑉∞ 2 , the abrupt decrements in the density abruptly increases the CA value at higher altitude. 5. Conclusion Computational simulations have been made to understand the effect of the flap on surface pressure distribution, axial force coefficient and base vortical structures of the atmospheric re-entry vehicle. The flow field characteristics over different reentry configurations are studied at different flap deflection angles, pitch angles, Mach numbers, and altitudes. Flapped has higher Mach number behind the base than the blunt cone due to stronger expansion fan arising from flap edge. The existence of flap introduces streamwise vortices and spanwise vortices which increases the complexity of base flow. At higher pitch angle, bow shock due to the nose section and the oblique shock due to windward flap interacts and the interaction point moves closer to flap. On increasing the flap deflection angle, a separation zone forms in front of the flap and moves upstream along with shock. The growth of this separation zone affects the flap effectiveness and peak pressure rise. This recirculation region ahead of flap limits the flap angle increment. The Axial force is nearly constant with angle of attack up to 60 , and the normal force increases linearly. The axial force increases with increase in flap deflection angle and decreases with the Mach number. The stand-off distance linearly decreases with an increase in Mach number. The increasing flap deflection increases the stability of the blunt cone
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