ISSN 0015-4628, Fluid Dynamics, 2014, Vol. 49, No. 4, pp. 540–546. © Pleiades Publishing, Ltd., 2014. Original Russian Text © S.V. Guvernyuk, A.F. Zubkov, M.M. Simonenko, A.I. Shvetz, 2014, published in Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, 2014, Vol. 49, No. 4, pp. 136–142.
Experimental Investigation of Three-Dimensional Supersonic Flow past an Axisymmetric Body with an Annular Cavity S. V. Guvernyuk, A. F. Zubkov, M. M. Simonenko, and A. I. Shvetz Institute of Mechanics, Lomonosov Moscow State University, Michurinskii pr. 1, Moscow, 119192 Russia e-mail:
[email protected] Received July 14, 2013
Abstract—The results of an experimental investigation of supersonic Mach 2.5 flow past an axisymmetric cylindrical model body with a rectangular annular cut-out on its lateral surface are presented. The evolution of the structure of the flow over the cavity with continuous variation in the angle of attack is studied on the basis of the data of flow visualization and balance measurements on the range of the relative cavity lengths L/h from 8 to 16. Hysteresis phenomena are revealed and analyzed. Keywords: hysteresis, supersonic flow, flow separation, annular cavity, angle of attack, aerodynamic drag coefficient. DOI: 10.1134/S0015462814040140
Separation flow past cavities (cut-outs, cut-offs) in body surfaces is an important problem of aerodynamics commonly encountered in practice. In this case, the aerodynamic hysteresis characterized by a nonunique dependence of the flow on the physical and geometric parameters of the problem can occur. The hysteresis range is interesting in that an even slight dynamic or thermal action on the flow can result in the flow restructuring from one pattern to another, with considerably different aerodynamic and thermal loads on the body. The restructuring of separation flow regimes with variation in the angle of attack can be the reason for the antidamping effect in the case of angular oscillations of bodies in flight [1]. It is known that in supersonic flow past bodies with cavities the so-called open and closed flow patterns can occur depending on the flow parameters and the cavity length L and depth h [2]. If the relative cavity length L/h is small, then the open flow pattern is realized. It is characterized by the presence of subsonic circulation flow throughout the entire cavity region separated from the external supersonic flow by a mixing layer extending from the leading to the trailing edge of the cavity. At large L/h the closed flow pattern is realized, when the external supersonic flow attaches to the cavity bottom surface and forms two isolated separation zones near the forward and backward steps, while intense rarefaction and shock waves arise in the external flowfield. For plane [2] and axisymmetric (annular) [3] cavities in supersonic flow at zero incidence on the Mach number range from 2.0 to 3.5 there are known empirical estimates of the critical closure length of the cavity: Ls /h ≈ 10 to 13. A weak dependence of the ratio Ls /h on the Reynolds number can be noted. A decrease in the relative length of the cavity in the closed flow regime leads to the approach of the separation zones in the vicinities of the forward and backward steps. The interaction of these separation zones followed by their coalescence is accompanied by the formation of a return flow from the compression to the rarefaction region with the result that the open flow pattern arises. It is noticed [2] that the critical length of cavity openness Lo is somewhat smaller than the critical length of its closure Ls . The L = Ls − Lo interval determines the hysteresis region with respect to the cavity length, both open and closed flow patterns being possible on this range [4]. 540
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Fig. 1. Experimental model.
A detailed investigation of axisymmetric supersonic flow past an annular cavity in a cylindrical body with a conical head was made in the experimental study [3]. In the same study the research on supersonic flow past cavities at zero incidence is reviewed and the data on the pressure distribution within the cavity and the results of flow visualization are presented on the wide range of relative cavity lengths in the both open and closed flow regimes. Along with experimental investigations. supersonic viscous flow past cavities was also numerically modeled in [5]. An example of the experimental realization of the hysteresis in the case of supersonic flow past an annular cavity is given in [4], where the possibility of controlling the flow in a plane cavity on the hysteresis range by means of applying a thermal short-duration pulse was also considered. The calculations of supersonic axisymmetric flow past an annular cavity in the case of continuous variation (both increase and decrease) in its length [6] made it possible to theoretically estimate the hysteresis range extent. The available results of experimental and theoretical investigations of supersonic flow over cavities pertain to two-dimensional problems, either plane or axisymmetric. Under actual conditions axisymmetric bodies are commonly in a flow at an angle of attack, the axial symmetry of the flow being also violated within an annular cavity. Owing to the mutual influence of the windward and leeward flows through subsonic regions of the cavity, the flow structures generated might be expected to have more complicated structures than in axisymmetric flow. The study of these structures and of the special features of three-dimensional supersonic separated flow past annular cavities is of practical importance. The corresponding experimental data are also of interest for the sake of verification of numerical models pretending on the adequate description of separation flows. In this study the model problem of three-dimensional supersonic flow past an annular cavity in a cylinder with a conical head is experimentally investigated at the Mach number M = 2.5. The purpose of the study is to reveal and analyze possible hysteresis phenomena with continuous variation in the angle of attack. 1. EXPERIMENTAL MODEL AND TECHNIQUE A cylindrical frame, 45 mm in diameter, is equipped with a cylindrical-conical head and a cylindrical tail, both D = 64 mm in diameter (Fig. 1). The cavity formed by these bodies is in the axial section a rectangular cut-out with the same heights of the forward and backward steps: h = 9.5 mm and h/D ≈ 0.15. The half-angle of the conical part of the head is 20∘ , while the length of its cylindrical part is 14 mm. The tail can be displaced along the axis of symmetry, so that the relative length of the cavity can vary on the L/h = 8–16 range. In the balance experiments a modified model of fixed length of 304 mm was used. In this case, the cavity length is varied by means of mounting additional annular inserts with the outside diameter equal to the tail body diameter ahead of the fixed tail body. The experiments were performed in the A-8 wind tunnel of the Institute of Mechanics of Moscow State University [7] at the Mach number M = 2.5 and the Reynolds number based on the diameter of the middle section of the model Re = 2.34 × 106 . The tunnel has a closed 1.5 m-long test section with a square cross section measuring 0.6 × 0.6 m2 . The upper and lower walls of the test section are inclined at an angle of 0.5∘ to the horizontal plane of symmetry of the nozzle to offset the displacement effect produced by the downstream-growing boundary layer. The working medium is the air at the stagnation temperature of about 275 K. The pressure in the plenum chamber was about 3.4 × 105 Pa. The test section obstruction with the model was not greater than 1.5%. The flow structure was visualized using the optical IAB-451 FLUID DYNAMICS
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instrument. In the course of the experiments the schlieren photos of the flow patterns were realized in the digital form. The visualization means adopted made it possible to observe the flow evolution in the vicinities of the windward and leeward sides of the cavity. The balance measurements were conducted using standard electromechanical balance of the rider type. In all the cases considered at the moment of the wind tunnel starting the model was set at zero angle of attack, α = 0. The short-duration starting process characterized by the passage of a chaotic system of shock waves was recorded by means of high-speed videofilming. After the working regime has been attained, the angle of attack α of the model was continuously varied on the range from 0 to 16∘ , both increasing and decreasing α . In the experiments the rate of the angle of attack variation was 0.5 deg/s. The error in measuring the current value of α was not greater than 15′ . 2. RESULTS OF THE FLOW STRUCTURE VISUALIZATION We will denote the relative cavity length by λ = L/h. After the tunnel starting only the open pattern of the flow past the cavity was observed for all λ < λ1 ≈ 12.0 and only the closed flow pattern was observable for λ > λ2 ≈ 13.4. For intermediate values λ1 ≤ λ < λ2 both flow patterns could be observable, the open and closed patterns occurring in the 3 : 1 proportion in the set of 20 repeated tunnel runs. The high-speed videofilming showed that for the time from the tunnel starting to the steady-state flow attainment the open and closed flow patterns chaotically interchange. A considerably greater frequency of the open flow pattern realization can indicate a greater stability of the open flow pattern against external disturbances, compared with the closed flow pattern at λ1 ≤ λ < λ2 . For the values of λ outside the (λ1 , λ2 ) interval any angular deviations of the model followed by the return to zero angle of attack did not lead to the changeover in the flow pattern, that is, the open pattern remained open at λ < λ1 , while the closed pattern remained closed at λ > λ2 . At λ1 ≤ λ < λ2 the situation is different. If the closed flow pattern arose after the tunnel starting, then an irreversible flow restructuring from the closed to the open pattern can be attained by increasing the angle of attack of the model. At the same time, if it was the open flow pattern that was formed after the tunnel starting, then the disturbances due to angular deviations of the model did not lead to the flow restructuring. Thus, at λ1 ≤ λ < λ2 both the open and the closed patterns of axisymmetric flow past the cavity in the model under consideration are possible. Outside this interval, only one of these patterns can be realized. The above-noted interval of the variation of the parameter λ can be considered as the hysteresis range with respect to the cavity length for the experimental model under consideration and under the actual conditions of the wind-tunnel experiments. The further investigation is concentrated on the details of three-dimensional flow past models with different values of λ under continuous variation in the angle of attack. In the photos presented below the supersonic flow is directed from left to right, while the nose of the conical part of the model is beyond the watch window; clearly visible are the bow shock and the shocks ahead of the backward step of the cavity (in the case of the closed flow pattern), as well as the mixing layer extending from the leading to the trailing edge of the cavity (in the case of the open flow pattern). Visible also is the mesh of Mach lines, that is, weak disturbances in the supersonic flow generated by a small bend in the tunnel contour in the zone of the junction between the nozzle and the test section. These weak disturbances of large spatial extent are projected on the photograph plane in the form of clearly discernible rectilinear intervals which produce illusion that the external flow disturbances hit on the cavity zone. However, they are almost completely shielded by the bow shock and, as showed methodological experiments, including the experiments with downstream and upstream displacement of the model, have no effect on the flow over the cavity. As noted above, at zero incidence (α = 0) flow past long cavities (λ > λ2 ) is realized only in accordance with the closed pattern. Using the flow visualization under continuous increase and then decrease in the angle of attack α we established that on the windward side of a long cavity the flow pattern remains almost invariable (Fig. 2a–2e). FLUID DYNAMICS
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Fig. 2. Visualization of flow past the closed cavity with λ = 14 at angles of attack α = 0, 5.0, 5.25, 5.5, and 8∘ (a–e) and past the open cavity with λ = 10 at angles of attack α = 0, 4, 8, 12, and 16∘ (f–j).
On the leeward side transitions from the closed (Fig. 2a) to the open (Fig. 2e) flow pattern, and vice versa, can be observable. With increase in the angle of attack the flow restructures itself as a certain critical value of α is reached (α ≈ 5, 5.5, and 8∘ at λ = 13.4, 14, and 16, respectively). The transition process is accompanied by considerable flow fluctuations in the form of increase and decrease in the separation zone thickness behind the forward step and in the form of the appearance, followed by the vanishing, of shock in the leeward region ahead of the backward step (Fig. 2b and 2c), which can be attributed to the effect of the pressure overrun over the bottom of the leeward side of the cavity from the compression to the rarefaction region behind the forward step. With further increase in α the fluctuations stop and a combined flow structure is attained; typical of this structure is that it has some features of the closed pattern on the windward side and those of the open pattern on the leeward side (Fig. 2d and 2e). This combined pattern is observable up to α = 16∘ . Then a decrease in the angle of attack results in the inverse restructuring, the transition from the open to the closed flow pattern on the leeward side of the cavity occurring at smaller angles of attack than the preceding direct transition from the closed to the open flow pattern. The corresponding small (about 1.5∘ ) lag in the angle of attack indicates the presence of hysteresis with respect to the angle of attack as the flow restructures itself between the closed and combined flow patterns for the cavities with the aspect ratios λ > λ2 . Flow over short cavities (λ < λ1 ) at zero incidence always occurs in accordance with the open pattern. The results of the visualization of spatial flow patterns on the angle of attack range 0 < α < 16∘ allow us to assert that on the leeward side of the cavity the open flow pattern is conserved for all α , whereas on the windward side transitions between the open and closed flow patterns can occur depending on λ . At λ < 10 the flow follows the open pattern over the entire angle of attack range studied. Transition from the open to the combined flow pattern and vice versa is observable at λ ≥ 10 (Fig. 2). With increase in the angle of attack on the windward side of an initially open cavity (Fig. 2f) the mixing layer is pressed against the cavity bottom by the oncoming flow, while a shock is formed ahead of the backward step (Fig. 2g). The elevated pressure on the windward side ahead of the backward step propagates upstream, hindering the mixing layer displacement toward the cavity bottom and generating compression waves above this layer. Further upstream the compression waves coalesce to the shock ahead the backward step of the cavity (Fig. 2h and 2i). Ultimately, a flow pattern having the features of the closed pattern is formed (Fig. 2j). The formation of the closed flow pattern on the windward side of the cavity with increase in the angle of attack occurs as the values α ≈ 5.5, 8, and 15∘ are reached at λ = 13, 12, and 10, respectively, while the inverse transition to the open pattern with decrease in the angle of attack takes place at 1 to 1.5∘ smaller values of α . As in the case of long cavities, there is a small lag in the angle of attack in the flow restructuring domain, but now it occurs on the windward side of the cavity. It should be noted that in transition from the open to the combined flow pattern and vice versa the flow pattern instability was observable. The most complicated case is that of flow past cavities with aspect ratios λ from the hysteresis interval (λ1 , λ2 ). At zero incidence flow past these cavities can occur in accordance with both the open and the closed patterns. Depending on this, different transition scenarios can be realized with variation in the angle of attack. If at α = 0 the pattern of the flow past the cavity is closed, then any deviations of the model on the range of low angles of attack (0 < α < αk ≈ 3.8∘ at λ ≈ 12) do not lead to transition to another flow FLUID DYNAMICS
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Fig. 3. Evolution of the structure of the flow past an initially closed cavity on the hysteresis range (λ = 12) with increase (a–c) and subsequent decrease (d–f) in the angle of attack; (a, f), (b, e), and (c, d) correspond to α = 0, 2, and 3.8∘ .
pattern (Fig. 3a–3c). However, an even small excess over the threshold value α = αk results in a jumpwise restructuring from the closed to the open flow pattern (Fig. 3d–3f). On the other hand, if at the initial moment of time at α = 0 the flow past the cavity occurs in accordance with the open pattern, then with increase in the angle of attack there occurs transition to the combined flow pattern in accordance with the scenario described above for the cavities with λ < λ1 ; the inverse transition from the combined to the open flow pattern with decrease in the angle of attack occurs in the same fashion. The stages of transition from the initially closed to the open cavity presented in Fig. 3 were obtained for λ ≈ 12. The analogous transition stages corresponding to variation in the angle of attack were fixed near the upper boundary of the hysteresis range for λ ≈ 13.2. The initially closed cavity suddenly opened as the critical angle-of-attack value αk ≈ 3.5∘ was attained, whereupon neither variations in the angle of attack could lead to the inverse restructuring from the open to the closed flow pattern. Contrariwise, at aspect ratios fairly similar in value (λ ≈ 13.4) but outside the hysteresis range the initially closed pattern turned out to be stable against angular disturbances and restored after any angular model displacements on the angle-of-attack range considered.
3. RESULTS OF BALANCE EXPERIMENTS From the results of balance experiments the drag coefficient of the model Cx was determined (Fig. 4). Curves 3 and 4 (λ = 16 and 14) were obtained for the flow patterns with a closed cavity (at low angles of attack) with transition to the combined flow pattern at large α . Curves 5 to 7 (λ = 12, 10, and 8) correspond to the flow patterns with an open cavity (at low angles of attack) with transition to the combined flow pattern at large α . It turned out that a small interval of the hysteresis with respect to the angle of attack in the vicinity of α = αk (at transition from the closed or the open to the combined flow pattern or vice versa) observed in visualizing the flow pattern does not involve any noticeable anomalies in the α -dependence of the drag coefficient. It should be noted that at subcritical values α < αk large diffferences in the value of Cx are due to the higher resistance of closed cavities, as compared with open ones. With increase in the angle of attack there occurs transition to the combined flow pattern and these differences gradually vanish, so that as the value α = 16∘ is attained, the difference in Cx is not greater than 5% for all the cases considered (Fig. 4). An anomalous behavior of Cx is observed in the case of the cavities with λ from the range of the hysteresis with respect to the length. The corresponding example is presented in Fig. 4: curve 5 (λ = 12) and data 1 and 2. On the angle-of-attack range 0 < α < 3.8∘ the α -dependence of Cx is nonunique; the regimes with both open and closed cavities can exist on this α range. In the case of the closed flow pattern the upper branch 1 is realized, whereas in the case of the open cavity it is the lower branch 2 (Fig. 4). At the end of this interval the closed flow pattern is changed jumpwise for the open pattern with a sharp decrease in the drag coefficient down to the value of about 0.55. This transition is irreversible and the further variation in the drag coefficient with increase or decrease in the angle of attack takes place only along curve 5 (λ = 12); in the case of the return to zero angle of attack the value Cx ≈ 0.49 is minimum. FLUID DYNAMICS
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Fig. 4. Angle-of-attack effect on the drag coefficient of the model with an annular cavity; (1, 2) are the regimes of the closed and open cavities, λ = 12; (3, 4) are the closed cavity regimes, λ = 16 and 14; and (5, 7) are the open cavity regimes, λ = 12, 10, and 8.
For open cavities an increase in Cx with increase in λ can be noted. In the case of zero angle of attack Cx ≈ 0.43, 0.46, and 0.49 at λ = 8, 10, and 12. This result is in qualitative agreement with the well-known data [2] on the effect of λ on the drag coefficient of an open cavity in a supersonic flow (α = 0, M = 2 to 3.5). In open annular cavities (λ > 8) with increase in λ the pressure ahead of the backward step increases, while that behind the forward step decreases [3]. The cavity drag is determined by the pressure on its lateral walls, since the friction on the cavity bottom is negligible. With increase in the angle of attack the above-noted tendency in the λ -dependence of Cx for open cavities is conserved. In the case of the closed flow pattern (λ = 12, 14, and 16), the drag coefficient Cx is almost independent of λ ; a weak dependence of Cx on α is also observable on the α = 0–8∘ range (Fig. 4). On this angle-of-attack range the deviation of Cx from the mean value 0.74 is not greater than 3%. Summary. The angle-of-attack effect on supersonic flow over axisymmetric cavities with different aspect ratios L/h is experimentally investigated. The flow patterns with open and closed cavities are reproduced. The range of the hysteresis with respect to the cavity length is established; at low angles of attack the flow patterns with both closed and open cavities can exist on this range. In this case, the flow restructuring from the closed to the open flow pattern occurs in irreversible fashion and is accompanied by a sharp decrease in the cavity drag. At large angles of attack in all the cases there occurs transition to the combined flow pattern which possesses the features of the closed pattern on the windward side and of the open pattern on the leeward side of the cavity. The flow restructuring from the closed or the open pattern to the combined pattern of flow past the cavity and vice versa is characterized by the presence of a delay in the angle of attack and is accompanied by unsteady processes within the cavity. The results obtained can be a test example for computational technologies for three-dimensional separation flows. At the same time, a detailed analysis of the structure of the combined regime of flow past an annular cavity found in the physical experiment is hardly possible without involving the numerical experiment. The authors wish to thank S.N. Barannikov and A.F. Mosin for assistance in conducting the experiments. The study was carried out with the support of the Russian Foundation for Basic Research (project No. 1201-00985). REFERENCES 1. A.N. Lyubimov, N.M. Tyumnev, and G.I. Khut, Methods for Studying Gas Flows and Determining the Aerodynamic Characteristics of Axisymmetric Bodies [in Russian], Nauka, Moscow (1995). 2. P.K. Chang, Separation of Flow. Vol. 2, Pergamon, Oxford (1970). 3. A.I. Shvetz,“Investigation of the Flow in an Annular Cavity in a Cylindrical Body in a Supersonic Stream,” Fluid Dynamics 37 (1), 109 (2002). FLUID DYNAMICS
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4. S.V. Guvernyuk and A.A. Sinyavin, “Controlling the Hysteresis in Supersonic Flow over a Rectangular Cavity by Means of a Thermal Pulse,” in Advances in Continuum Mechanics. On the 70th Anniversary of Academician V.A. Levin [in Russian], Dalnauka, Vladivostok (2009), p. 196. 5. I.A. Graur, T.G. Elizarova, and B.N. Chetverushkin, “Numerical Modeling of Supersonic Viscous Compressible Flow over Cavities,” Inzh.-Fiz. Zh. 61, 570 (1991). 6. A.A. Aksenov, S.V. Guvernyuk, Yu.N. Deryugin, S.V. Zhluktov, A.F. Zubkov, A.S. Kozelkov, M.M. Simonenko, and A.S. Shishaeva, “Numerical Investigation of Hysteresis in Supersonic Turbulent Flow past a Body with an Annular Cavity Using the LOGOS Software Package,” in Proceedings of the 14th International Conference ‘Supercomputations and Mathematical Modeling’, Sarov, 2012 [in Russian] (2012), p. 164. 7. G.G. Chernyi et al. (eds.), Aerodynamic Setups of the Institute of Mechanics of Moscow State University [in Russian], Moscow Univ. Press, Moscow (1985).
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