A numerical and experimental study of the high ... - Springer Link

Thermophysics and Aeromechanics, 2012, Vol. 19, No. 4

A numerical and experimental study of the high-enthalpy high-speed cavity flow* 1

1

M.A. Goldfeld , Yu.V. Zakharova , and N.N. Fedorova

1,2

1

Khristianovich Institute of Theoretical and Applied Mechanics SB RAS, Novosibirsk, Russia

2

Novosibirsk State University of Architecture and Civil Engineering, Novosibirsk, Russia

E-mail: [email protected] (Received December 7, 2011) Results of an experimental and numerical study of supersonic turbulent high-enthalpy flow in a channel with cavity are reported. On the basis of wind-tunnel tests performed in the IT-302M short duration wind tunnel, data on the flow structure and on the distribution of static pressure along the model walls were obtained. These data were subsequently used to verify the numerical algorithm. In the calculations, a parametric study of the effects of Mach number, cavity configuration, and temperature factor on flow quantities was performed. It was numerically shown that variation of the above parameters leads to a transition of the flow regimes in the vicinity of the cavity. Key words: high-speed, high-enthalpy flows; apparatus channels; cavity; turbulence; experiment; mathematical modeling.

Introduction The cavity, or cavern, is a simple geometric configuration often encountered on aircraft surfaces or in apparatus channels. Recent interest in the study of cavities was due to the fact that cavities are often used to exert control over complex flows at supersonic speeds, for instance, at boundary layer bleed [1] or as flame holders [2, 3]. Since 1950, sub- and supersonic flows in cavities have been the subject matter of many experimental and theoretical studies in which main characteristic features of cavity flows were revealed [2−7]. A comprehensive review of studies performed in the 50−60ths can be found, for instance, in [4]. The regime of cavity flow is defined by the ratio between the cavity length L and the cavity depth h (see [4]). In the case of L/h > 10, the mixing layer shedding from the leading edge of cavity reattaches at the cavity bottom to undergo subsequent detachment in front of the rear wall. Under such a flow regime, the cavity is called a closed

*

This study was supported by the Russian Foundation for Basic Research (Grant No. 09-08-01001-a) and by the AVTsP RNP VSh Program (Project No. 2.1.1/11316). © M.A. Goldfeld, Yu.V. Zakharova, and N.N. Fedorova, 2012

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M.A. Goldfeld, Yu.V. Zakharova, and N.N. Fedorova

cavity. In the case of L/h < 10, the mixing layer, as a rule, reattaches in the vicinity of the rear wall, and the cavity in that case is called an open cavity. The flow in an open cavity with a vertical rear wall is an unsteady flow. Based on experimental measurements, J.E. Rossiter [8] put forward a semi-empirical formula for dominating pulsation frequencies. This formula has later found a widespread use in verification of calculated data [9]. The authors of [10−14] examined the streamwise and cross-flow pressure pulsations in a cavity flow. It was shown that the intensity and spectrum of pressure pulsations were the functions of the ratio L/h and free-stream Mach number M∞. In the case of narrow cavities, with just one large eddy formed inside the cavity, pressure pulsations are controlled by the transverse mechanism, whereas in the case of wide cavities hosting two or a greater number of internal eddies, pressure pulsations are controlled by the longitudinal mechanism. The transition from cross-flow to streamwise pulsations occurs at L/h = 2 for Mach number М∞ = 1.5 and in the interval 2 < L/h < 3 for Mach number М∞ = 2.5. In [11, 12], it was shown that pressure pulsations are generated as the shear layer interacts with the rear wall of cavity. In calculating unsteady flows in the vicinity of cavity, the LES approach to modeling turbulence, highresolution numerical algorithms, and adaptive grids are normally used [11, 13]. For suppressing pressure pulsations in cavities, special active and passive control means were developed [15−18]. Recent studies of cavities of various types were stimulated by the use of cavities as flame holders in combustion chambers [2, 3, 19−22]. Like in the case of the backwardfacing step, the use of a cavity as a flame holder is based on the trapped-vortex concept [23] implying that the low-velocity, high-temperature recirculation region inside cavity can serve an igniter of the fuel/oxidant mixture. For efficient operation of a combustion chamber, the chamber configuration should ensure rapid mixing of the oxidant and fuel fed into the channel upstream of cavity, and also a sufficiently long time of residence of the mixture in the recirculation region at static pressure and temperature optimal for ignition. To study the physical processes in combustion chambers, because of their complexity, numerical methods are often used along with experimental methods. The simulation allows obtaining detailed data about the cavity flow, performing parametric calculations, and choosing optimal values of geometric and gas-dynamic parameters [24, 25]. A key study of supersonic cavity flow in a combustion-chamber channel was performed in [3], where experimental and numerical data for М∞ = 3 and various geometrical parameters (cavity width and rear-wall inclination angle) were reported. In the experiments, steady and unsteady flow regimes, and also regimes with the formation of various (compression and expansion) waves on the front wall of cavity, were realized. The data gained in 2D calculations performed with the k-ω turbulence model [26] and a special procedure enabling the use of a large grid step in the vicinity of walls were found to be consistent with the experimental data on the pressure distribution along the cavity walls. Yet, the authors of [3] failed to reproduce the unsteady regime and the regime featuring the formation of a shock wave at the leading edge of cavity. This failure could be a consequence of an insufficiently detailed calculation grid, and also timeaveraged governing equations. It should be emphasized here that the numerical studies cited above were performed for the case of moderate stagnation temperatures or adiabatic wall temperature conditions. Simultaneously, when using such a configuration as a model combustion chamber, it is required to examine flows with high stagnation temperature, and also investigate the effect of non-adiabatic (cold) model walls realized in the experiments performed in impulse wind tunnels. 542


The present study was aimed at: a) an investigation of the regimes of cavity flow and of the flow structure under high-enthalpy (Т 0 = 1500÷2500 K) conditions at high external flow velocities (М∞ = 2÷3); b) a numerical study of the effects due to free-stream Mach number, cavity geometry (cavity width and rear-wall inclination angle), and temperature factor on the evolution of the flow pattern in a channel with cavity, on the static temperature, and on the pressure in cavity; c) obtaining the fundamental experimental data on the flowfields and on the pressure distribution along the model surface necessary for calculation data verification. 1. The experimental study The experimental study of the flow through a channel with cavity was carried out in the IT-302M hot-shot wind tunnel of ITAM, SB RAS [27], in the attached-pipeline regime. As the source of a high-enthalpy gas, the prechamber of the wind tunnel was used. The model comprised the following main components: a second (throttling) prechamber, a planar profiled nozzle producing a flow with Mach number М∞ = 2÷4 at the inlet to the channel, an insulator for leveling the flow, a 120-mm long test section, and a measuring system. The subsonic part of the nozzle was designed as a Vitoshinsky profile with a desired subsonic-flow contraction factor close to 6. The external components of the nozzle unit and the profiled nozzle were made of structural steel providing for sufficient stability to wear of the nozzle contour in the specified interval of test temperatures during the operating regime of the wind tunnel. The insulator section, having a constant cross section of 50×100 mm, was attached to the nozzle through a special spacer that ensured electrical insulation of the model from the discharge prechamber. The nozzles were calibrated with the help of a Pitot rake formed by five Pitot tubes spaced uniformly across the channel. Ten static-pressure gauges (five gauges on the upper channel wall and five gauges on the bottom channel wall) were installed in the same cross sections where the Pitot comb was located. For controlling the flow at the inlet to the test channel, the insulator section was provided with two static-pressure gauges and two heat-flux gages (one heat-flux gage on the upper wall of the channel and the other heat-flux gage on the bottom wall). The test section (Fig. 1) had two configurations of the upper wall, with and without a cavity. The size and the configurations of the cavity could be modified with the help of a special insert (Fig. 1b). As follows from the diagram, the depth and the width of cavity in the tests were, respectively, h = 16 mm and L = 56.4 mm, the rear-wall inclination angle was θ = 22.5°, and the channel height in the upstream region of cavity was d = 50 mm. In the downstream region, the channel was made weakly expanding (3°) to ensure its matching with the subsequent channel section.

Fig. 1. Test section (а) and the insert with cavity (b). 543

M.A. Goldfeld, Yu.V. Zakharova, and N.N. Fedorova Table 1 Test parameters in the IT-302M wind tunnel M∞ 2 2.33 2.8

−5

P0×10 , Pa 24 36 34

T0, K 1731 2078 2146

−5

P1×10 , Pa 2.8 2.5 1.08

T1, K 938 973 887

The model was provided with sensors that measured the static pressure and the heat fluxes on the model walls. Data were registered by a high-speed data acquisition and processing system. The model used for optical measurements had windows made of heat-resistant quartz glass. A large area of the windows (the entire channel height and the entire width of cavity) enabled rather detailed visualization of the flow pattern throughout the whole test-section volume. For measuring the static pressures on the upper and bottom walls of the channel, strain gauges rated at 1.6, 4, and 10 bars were used. According to our estimate of the measuring accuracy in the hot-shot wind tunnel, the pressure in the indicated range was measured accurate to a root-mean-square accuracy of 1−1.5 %. In the tests, the total pressures in the first and second prechambers of the wind tunnel, and also the distribution of static pressures along the model channel, were measured, and visualization of the flow was performed. The parameters of performed tests are listed in Table 1. Here, M∞ is the free-stream Mach number, P0 and T0 are the stagnation parameters in the second prechamber, and P1 and T1 are the static pressure and temperature at the entrance to the measuring channel. In the upstream region of the cavity, rather thick boundary layer is formed on the channel walls. It is a well-known fact that the thickness of boundary layer is one of the main parameters governing the flow in cavity. Measurement of boundary-layer characteristics in a hot-shot wind tunnel normally presents a difficult problem. For evaluating those characteristics at the inlet to the test section, measurements with the help of a Pitot rake formed by nine Pitot tubes were performed. Staggered arrangement of Pitot tubes (Fig. 2) has allowed minimization of the cross-flow separation between the tubes and excluding the effect due to shock waves. An analysis of velocity profiles measured at М∞ = 2.8 has showed that the degree of filling of velocity profiles in the vicinity of the wall was rather high. This means that the flow in boundary layer was turbulent. The boundary layer was approximately 10 mm thick, this value being in good agreement with calculated values obtained from the velocity profiles (Fig. 3). Some difference between the calculated and experimental data on the degree of filling of velocity profiles can be due to a high turbulence level of the flow typical of experimental facilities such as the wind tunnel used in the present study. 2. Calculation method and its verification Numerical simulations were performed using the ANSYS CFD (Fluent) program package, by solving complete non-stationary Favre-averaged Navier⎯Stokes equations supplemented with the SST modification of Wilcox turbulence Fig. 2. Pitot rake for measuring flow quantities in boundary layer. 544

Thermophysics and Aeromechanics, 2012, Vol. 19, No. 4 Fig. 3. Experimental (1) and calculated (2) velocity profiles in the upstream section of cavity non-dimensionalized by the free-stream velocity U1.

model [26, 28]. A «coupled» solver was chosen to simultaneously solve the momentum, continuity, and the energy equations, and also the equations of the turbulence model; afterwards, a check for solution convergence was performed. The problem domain treated in our calculations was bounded from the left by the inlet section; at the top and bottom, by the upper and bottom channel walls; and from the right, by the outlet boundary. The solution process involved two stages. First, we calculated a rectangular channel of 800-mm length using a calculation grid that comprised 1000 nodes in the x-direction and 300 nodes in the y-direction. The length of the channel was chosen to ensure a developed turbulent velocity profile at the channel exit with characteristics close to experimentally established ones (Fig. 3). At the channel exit, the profiles of mean velocity, static pressure, and temperature, and also profiles of turbulent parameters were registered. Then, the profiles thus obtained were adopted as boundary conditions at the inlet to the calculation domain presenting a channel with cavity. A structured grid condensed towards the solid walls was used. As a rule, the grid in that section comprised 650 cells along the x-direction. Along the y-direction, the grid comprised 300 nodes; of those nodes, approximately 50 ones were located inside the upperand bottom-wall boundary layers. The minimal step of the grid in the vicinity of the wall + was chosen such that we had y1 ~ 1 for the adopted freestream conditions while +

the laminar sublayer y < 10 comprised a few nodes permitting a sufficient resolution of this sublayer. For checking the grid convergence, we performed calculations of a channel flow with a cavity of depth h = 16 mm, width L/h = 4.5, and rear-wall inclination angle θ = 45° at М∞ = 2, stagnation pressure P0 = 30 bar, and stagnation temperature T0 = 1500 K on the grids refined over the y-coordinate. The grid parameters are indicated in Table 2. The calculation data obtained on the three grids are shown in Fig. 4. It is seen from the figure that the pressure distributions calculated on the medium and fine grids are coincident. The 5-% difference between the skin friction distributions obtained on the coarse and medium grids can be the consequence of insufficient resolution of the boundary layer in the vicinity of the wall. The calculated data presented below were obtained on the medium grid. The numerical algorithm and the turbulence model used were verified through comparison of calculated data with experimental results obtained in the IT-302M wind Table 2 Parameters of calculation grids Grid

Total cell number

Minimal y-step, m

Maximal y-step, m

Coarse

450500

2⋅10−5

7⋅10−4

Medium

503000

5⋅10−6

7⋅10−4

Fine

694000

1.3⋅10−6

1⋅10−4

545


Fig. 4. Distributions of non-dimensionalized static pressure P/P1 (а) and skin friction coefficient Cf (b) along the bottom wall of the channel with cavity calculated with different grids: 1 ⎯ coarse grid, 2 ⎯ medium grid, 3 ⎯ fine grid.

tunnel for the following geometric sizes of cavity: h = 16 mm, L/h = 3.5, d = 50 mm, and θ = 22.5°, and for the freestream parameters indicated in Table 1. Figures 5 and 6 show the experimental and calculated static pressure distributions along the upper and bottom channel walls, respectively, for M∞ = 2 and 2.3. Here, the pressure is non-dimensionalized by the value of static pressure in the free stream P1 (see Table 1), the x axis being directed along the bottom wall (the vertical wall is not shown). The calculated static pressure distributions on the channel upper and bottom walls agree with the experimental data. Behind the leading edge of cavity (x = 0), a small decrease of static-pressure is observed, and then the static pressure starts growing. At the location of the bottom-wall compression ramp, x = 0.052 m, a small local pressure maximum is observed. The main pressure maximum on the bottom wall is observed on the inclined cavity rear wall; this pressure maximum is followed by a sharp pressure drop giving rise to a narrow flow overexpansion zone. A local minimum is attained at the cavity rear wall, near the point x = 0.1 m. A characteristic saw-tooth pressure profile behind the cavity (x > 0.1 m) is due to the shock waves and expansion fans formed in the cavity and propagating along the channel. As will be shown below, the calculated data allow one to comprehend all characteristic features of the shock-wave pattern of the flow in the channel with cavity at different Mach numbers. 3. Flow structure For gaining a better insight into the distributions of pressure in Figs. 4-6, an analysis of flow fields at different Mach numbers was performed. The calculated Mach-number contours for M∞ = 2 (Fig. 7a) are indicative of the presence of a thick boundary layer. At the cavity forward wall (x = 0), the boundary layer separates to form a mixing layer, which then reattaches to the surface in the vicinity of the rear cavity wall.

Fig. 5. The pressure distributions on the bottom (а) and upper (b) channel walls. M∞ = 2. 1 — experimental and 2 — calculated data.

546


Fig. 6. Pressure distributions on the bottom (а) and upper (b) channel walls. M∞ = 2.33. Experimental (1) and calculated (2) data.

Inside the cavity, the flow is subsonic. The sonic line in the mixing layer forms an effective surface streamlined by the external supersonic flow. The shape of this surface governs, among other things, the pressure distribution in the supersonic flow over the cavity. With the sonic line «sagging» downward, an expansion fan (EF) is observed near the leading edge of the cavity. If, on the contrary, the boundary layer «swells» upward, then a shock wave is formed. Although a compression shock formed at the leading edge of cavity was detected in the experiments of [3], the authors of [3] failed to reproduce it in their calculations. In the present study, a regime with a compression wave was obtained in calculations performed for М∞ = 2, L/h = 4.5, and θ = 45° (Fig. 4). In the case of a smaller rear-wall inclination angle, θ = 22.5° (Fig. 5), at the same Mach number, the sonic line turns out to be sagging downward after the cavity leading edge, this leading to the formation of the weak expansion fan. In the vicinity of the trailing edge, the sonic line forms an effective compression ramp. The compression waves produced in the zone where the mixing layer gets attached to the surface form a shock wave that extends downstream experiencing multiple reflections from channel walls. The intensity of this shock decreases behind the cavity because of the channel expansion, and after reflection from the bottom wall (x ≈ 0.23 m), the shock becomes completely degenerated. Figure 7b shows the static pressure field for the case of M∞ = 2.3. With the growth of Mach number, the inclination angles of the shocks become smaller, that can be detected considering the downstream shift of the regions of interaction of shocks with channel walls. A slight pressure reduction above the cavity and a high-pressure region in the zone of mixing-layer attachment are noteworthy. The pattern of the cavity flow can be grasped in more detail considering Fig. 7с, which shows density contours for the case of M∞ = 2.8. The labeled flow features in the figure are the following: an expansion fan that forms in the vicinity of the cavity leading edge (EF1); a recirculation zone (RZ) with two counter-rotating eddies observed inside the cavity; an expansion fan formed at the breakpoint of the upper wall (EF2); a shock wave (SW) formed in the mixing layer reattachment zone; and a subsequent expansion fan (EF3). For the latter Mach-number, the compression shock, having reflected from the upper wall of the channel, leaves the calculation domain. A comparison between Figs. 4, 6, and 7 explains the behavior of pressure distributions along the channel upper and bottom walls by the position of expansion fans and shocks indicated in Fig. 7. For all the Mach numbers, the pressure at the leading edge shows a reduction due to the downward bending of the mixing layer in the region above the cavity. The maximum in the pressure distributions over the bottom wall observed in the vicinity of the cavity rear wall (x = 0.1) is due to the formation of compression waves in the region of the mixing layer reattachment at the trailing edge of cavity. Further 547


Fig. 7. Calculated Mach number contours (а), static pressure field (b), and density contours (с). а — М∞ = 2, b — М∞ = 2.3, с — М∞ = 2.8.

downstream, the compression waves form a shock that propagates downstream and gives rise to a characteristic sawtooth profile in the pressure distribution. 4. Effect due to Mach number For the various freestream Mach numbers and stagnation temperatures, we performed calculations of the flows in the channel with cavity for the following values of geometric parameters: h = 16 mm, L/h = 4.5 mm, θ = 22.5°, and d = 50 mm. This configuration differed from the configuration considered in the previous section by its larger length and by the absence of additional channel expansion behind the cavity. Calculations were performed for conditions typical of the IT-302 M wind tunnel; those conditions are listed in Table 3. Here, Re1 is the dimensional Reynolds number, and ρ1 and U1 are the freestream density and velocity. Numerical modeling was performed under the cold-wall conditions (Tw = 300 K) with the use of the SST modification of the k-ω turbulence model [28]. For all the Mach numbers under consideration, М∞ = 2, 2.5, and 3, at the cavity leading edge (x = 0) there an expansion fan is formed which is more intensive than that in the case of the shorter cavity. It is seen from Fig. 8a that, with increasing the freestream Mach number, the intensity of the expansion fan and the angle of flow turning Table 3 Parameters of calculated flows

548

No.

M∞

1 2 3

2 2.5 3

−6

3

P0, bar

T0, K

Re1⋅10 /m

P1, bar

ρ1,kg/m

T1, K

U1, m/s

30

1500 1800 2000

49.3 29.7 18.97

3.834 1.755 0.817

1.66 0.765 0.398

833 800 714

1157 1417 1600


Fig. 8. Calculated static pressure (а) and skin friction (b) distributions on the cavity bottom. М∞ = 2 (1), 2.5 (2), 3 (3).

at the leading edge both grow in value; hence, the angle through which the mixing layer turns in the vicinity of the cavity rear wall also increases. As a result, the strength of the shock formed in the region of flow reattachment grows, as well as the local pressure maxima at the bottom corner point (x = 0.072 m) and behind the zone of flow overexpansion (x = 0.11 m). The growth of the Mach number leads to a decrease of the skin friction coefficient upstream of the cavity (Fig. 8b); yet, at the cavity bottom and downstream the cavity, the skin friction coefficient exhibits values roughly identical for all the Mach numbers. 5. Effect due to cavity geometry At M∞ = 2 and 3, for the flow parameters given in Table 3 we examined the effect due to cavity geometry (relative cavity length L/h and rear-wall inclination angle θ) on the variation of the flow pattern and on the distribution of flow quantities in the channel with cavity. The geometric characteristics of the considered configurations are presented in Table 4. Configurations 1−3 were calculated at M∞ = 2, and configurations 4−7, at M∞ = 3. The simulation was performed under the cold-wall conditions (Tw = 300 K) using the SST turbulence model. Figure 9, which shows the pressure and skin friction distributions along the bottom channel wall for cases 1−3 of Table 4, during variation of rear-wall inclination angle demonstrate considerable changes in the pressure inside cavity, caused by modification of the structure of cavity flow. In the case of the rectangular cavity (curve 1),

Fig. 9. Static pressure (а) and relative skin friction (b) distributions on the bottom wall of the channel. М∞ = 2, L/h = 4.5, θ = 90° (1), 45° (2), and 22.5° (3). Table 4 Geometric parameters of cavity No. L/h

1

2 4.5

3

θ°

45

22.5

90

4 4.5

5 8

6 10

7 12

22.5

549


the wave formed near the leading edge of cavity is a shock; this is evident from the pressure distribution in Fig. 9а. Here, the shock-wave pressure rise amounts to about 10 % of the upstream static-pressure level, and the high-pressure region occupies a predominant part of the cavity. With decreasing the angle (θ = 45°, curve 2), the shock wave in the vicinity of the leading edge becomes weaker while the strength of the shock wave in the flow reattachment region near the cavity rear wall increases. On further decrease of the angle (θ = 22.5°), near the leading edge there forms an expansion fan, the latter being evident from the pressure distribution (Fig. 9а, curve 3). With growth of θ, the pressure local maximum in the inner corner in the vicinity of the rear wall increases. The surface of the rear wall for θ = 90° is not shown in the graph; that is why the second local maximum due to mixing-layer attachment, and also the local overexpansion minimum lie close to each other. With θ decreasing, the length of the inclined facet increases; that is why the position of the second local maximum shifts downstream, and the width of the region with enhanced pressure on the inclined wall increases in value. The amplitude of the maximum itself varies insignificantly. At the locations of mixinglayer reattachment, the distributions of skin friction coefficient (Fig. 9b) also display peaks. Minimal values of the skin friction coefficient on the cavity bottom are observed at θ = 90° (curve 1 in Fig. 9b); at this angle θ, the skin friction-coefficient level is maximal behind the cavity. At Mach number М∞ = 3 for θ = 22.5°, we performed calculations with variation of cavity length in accord with the data of Table 4 (cases 4−7). At this Mach number, for all considered L‘s in the vicinity of the leading edge, an expansion fan was observed (Fig. 10а). For L/h = 4.5, 8 and 10 at М∞ = 3, we have an open cavity. The curves for L/h =10 in Fig. 10 are not shown since those curves qualitatively resemble the curves for L/h = 8. It is seen that the pressure level at the bottom of cavity depends on the relative cavity length. At increasing the cavity length. it becomes possible for the mixing layer to turn through a greater angle at the leading edge of cavity. As a result, a more pronounced reduction of pressure at the cavity bottom becomes possible. The level of pressure and its behavior behind the open cavity is roughly identical for the various values of L/h. Note that the curves of skin friction coefficient (Fig. 10b) for open cavities exhibit a tendency towards the emergence of a secondary separation at the breaking point of the bottom, х/h = 4.5 and 8. In the static pressure distributions (Fig. 10а), a local maximum is observed at the same points. In the case of L/h = 12 (Table 4, case 7), we have a regime of closed cavity; here, the mixing layer reattaches at the bottom of cavity to subsequently separate in front of the inclined rear wall with the formation of a separation shock, a pressure plateau, and an attachment shock (curve 3 in Fig. 10а). The occurrence of two separation regions is also evident from the distribution of skin friction coefficient (Fig. 10b), which shows that, here, a profound portion of the cavity bottom is occupied by a region with positive skin friction value.

Fig. 10. Static pressure (а) and skin friction (b) distributions on the channel bottom wall. М∞ = 3, θ = 22.5°, L/h = 4.5 (1), 8 (2), and 12 (3).

550


A change in the relative cavity length, L/h = 4.5 and 8, at fixed rear-wall inclination angle exerts no substantial influence on the vortex structure (Fig. 11а). In the interior of the cavity, two eddies are formed: a primary eddy, rotating in clockwise direction, and a small near-wall eddy, rotating in the opposite direction. At L/h = 8 (Fig. 11b), a tendency towards splitting of the primary eddy and the formation of an additional counter-directed eddy is observed. As was noted above, on further increase of cavity length to L/h = 12 (Fig. 11с), a change of flow mode occurs. Here, the configuration should be considered as a combination of the backward-facing and forward-facing steps. In the vicinity of the backwardfacing step, there forms an intense expansion fan and a recirculation zone, and at the attachment point, there forms a closing shock. In the recirculation region behind the backward-facing step, two counter-rotating eddies are observed. One more separation region is formed as the flow moves past the cavity rear wall (x = 0.15 m) with the formation of a separation and reattachment shocks. The boundary layer reattaches on the cavity inclined wall rather far from the upper corner point. On approach to the upper wall, the tail and separation shocks merge together to form the shock wave that causes a separation of boundary layer on the upper wall. Apart from streamlines, Figure 11 shows the fields of static temperature. For the open cavity cases, the static temperature in external flow varies insignificantly. Simultaneously, in the case of a closed cavity, in the separation region over the backward-facing step, a considerable reduction of static temperature due to a strong flow expansion is observed. Inside the recirculation region, the temperature behind the backward-facing step turns out to be well in excess of the temperature in the external flow. A substantial growth of static temperature is observed in the mixing-layer reattachment region on the cavity bottom. The maximal values of temperature for closed cavities are observed in the separation regions formed on the upper and bottom channel walls due to the interaction of the boundary layer with shock waves (Fig. 11с).

Fig. 11. Stagnation temperature fields and streamlines. М∞ = 3 for θ = 22.5°, L/h = 4.5 (а), 8 (b), and 12 (c).

551


6. Effect due to the temperature factor It is a well-known fact that a change in wall temperature may have a profound effect on the pattern of cavity flow and on the distribution of flow quantities in the channel. Such an effect may be a determining one in choosing a cavity as a flame holder. Under the “cold-wall” conditions, the level of static temperature in cavity decreases markedly. To evaluate the effect due to the temperature factor, we performed calculations of the flow in a channel with cavity for М ∞ = 2, 2.5, and 3, and for a cavity length L/h = 4.5 mm, θ = 22.5° under the flow conditions indicated in Table 3. At М∞ = 2.5, the temperature factor Tw /T0 ranges from 0.15 to 1, which values corresponded to wall-temperature values Tw = 270 (1), 900 (2) K, and to adiabatic conditions (3). The distributions of pressure and skin friction over the channel bottom shown in Fig. 12 prove that, with decreasing the wall temperature, the pressure in cavity shows a monotonic reduction and the strength of the shock wave formed in the region of mixinglayer attachment that is accompanied by a 25-% pressure increase. At this Mach number for all Tw values in the vicinity of the leading cavity edge, there forms an expansion fan whose strength decreases with increasing the wall temperature (Fig. 12а). For cold wall, maximal skin friction coefficient values are observed upstream of the interaction zone and in the zone of mixing-layer attachment in the vicinity of the rear wall (Fig. 12b). Similar results were also obtained for М∞ = 3 and T0 = 2000 K during variation of wall temperature from the cold one (Tw = 300 K) to adiabatic. At Mach number М∞ = 2 and temperature T0 = 1500 K, the temperature factor was varied from 0.15 to 1, which values corresponded to the wall-temperature values Tw = 225 (1), 1125 (2) K, and to adiabatic conditions (3). The calculated distributions of pressure and skin friction along the bottom channel wall shown in Fig. 13 are indicative of the fact that, at the low values of the temperature factor, as well as at the higher Mach number, high values of static pressure in the mixing-layer reattachment region were obtained. However, here the dependence of pressure on the temperature factor becomes non-monotonic: the pressure values for Тw = 1125 K on the inclined rear wall (0.072 m < x < 0.11 m) is lower than those for adiabatic conditions (Fig. 13а). A similar effect is also observed for the skin friction distributions. In the region x < 0 (upstream of the cavity), a decrease in wall temperature leads to increase of the skin friction coefficient. For instance, for the coldest wall, the coefficient Сf is twice the value of Сf for the higher values of Tw, which monotonically decrease with increasing the temperature Tw. Inside the cavity (0 < x < 0.11 m), the skin friction coefficient behaves non-monotonically versus the temperature factor. The curve for Tw = 1125 K lies somewhat higher than the curve for adiabatic temperature (Fig. 13b).

Fig. 12. Distributions of static pressure (а) and surface-friction coefficient (b) for different values of wall temperature. М∞ = 2.5, Tw = 270 (1), 900 (2) K, 3 ⎯ adiabatic conditions.

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Fig. 13. Calculated distributions of pressure (а) and surface-friction coefficient (b) along the bottom wall of the channel. М∞ = 2, Tw = 225 (1), 1125 (2) K, 3 ⎯ adiabatic wall.

Calculations performed at various Mach number and temperature factor values showed that, with variation of wall temperature, reconstruction of flow in the vicinity of the front and rear walls of cavity may occur. At М∞ = 2.5 and 3, in the vicinity of the front wall, an expansion fan was always observed, whose strength increased with decreasing Tw. At those Mach numbers, the strength of the compression wave formed in the mixing-layer reattachment region also monotonically increases with decreasing the wall temperature. At Mach number М∞ = 2, both the strength of the shock formed in the mixing-layer reattachment region and the skin friction coefficient on the cavity bottom were found to exhibit a non-monotonic behavior versus the temperature factor. This effect can be attributed to modification of the flow pattern in the vicinity of the cavity front wall. Under the adiabatic conditions, in the vicinity of the front wall, a compression wave was observed, leading to a rise of pressure in this region, whereas at lower values of wall temperature in the vicinity of the leading edge, as well as at higher Mach numbers, a weak compression wave was formed. Conclusions 1. We examined, both experimentally and numerically, the flow in a cavity at Mach numbers М∞ = 2, 2.33, and 2.8 under conditions of the IT-302М high-enthalpy wind tunnel tests. Data calculated for М∞ = 2 and 2.33 were found to be in a reasonable agreement with the experimental data on the distribution of static pressure on channel walls. 2. On variation of free-stream Mach number, cavity configuration, and temperature factor, both closed-cavity and open-cavity flow modes can be realized. For an open cavity, either a shock wave or expansion fan can be formed at the leading edge of cavity. In the calculations, an open-cavity flow regime with a shock wave was reproduced for the first time, this flow regime being intermediate between the steady and unsteady regimes of cavity flow. Calculations performed on refined grids proved the effect to be independent of a particular grid. 3. At increasing the Mach number and the relative cavity length the pressure in the vicinity of the cavity leading edge monotonically decreases, and it grows in magnitude on increasing the cavity rear-wall inclination angle. Hence, the regime with shock wave formed on the cavity leading edge is observed in the case of supersonic flow past a relatively narrow cavity with a large rear-wall inclination angle at moderate Mach numbers. 4. In our calculations, we showed, for the first time, that a transition from one flow regime to another also can occur during the temperature factor variation. Under low temperature factors, the pressure inside cavity decreases, whereas under adiabatic conditions, this pressure increases, leading in some cases to the formation of a shock wave on the leading edge of cavity. This effect, revealed in the calculations for М∞ = 2, was the cause for non-monotonic of pressure and skin friction behavior versus the temperature factor. 553


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