Investigation on asymmetric flow over a blunt-nose slender body at high angle of attack

Figure 1 presents the schematics of the test model of a blunt-nose slender body, along with symbol designations. The test model included three parts, i.e., the blunt nose, the cylindrical slender afterbody, and the conical part connecting the former two parts. The blunt nose, which was a part of a ball of radius 27.2 mm (=0.4121D), exhibited a bluntness of 80%. Bluntness is defined as the ratio between the nose diameter and the afterbody diameter. The cylindrical slender afterbody held a diameter D = 66 mm and a full length of 1060 mm (=16.0606D). The conical part was 38.21 mm long (=0.5789D) with an 18° apex angle. The coordinate system was defined at the tip of the blunt nose (i.e., the origin) with the x-direction found along the symmetric axis of the slender body, and the y- and z-directions represent the horizontal and vertical axes, respectively, in the normal plane to the slender body axis (figure 1). The blunt-nose slender body involved an AoA α, which was the incident angle of the incoming flow to the x-direction. A tiny hemisphere with a radius r = 0.012D was used as the microperturbation and was attached onto the nose surface. Its location was determined by two angles, namely, the meridian and circumferential angles γ and θ, respectively. γ is the angle between the x-axis and the nose radius corresponding to the perturbation location (figure 1(a)), and θ is the angle between the z-axis and the sectional radius corresponding to the perturbation location (figure 1(b)). The location of the measured pressure is indicated by the angle θ_s between the negative z-axis and the sectional radius corresponding to the measured pressure location.

Zoom In Zoom Out Reset image size

A test model with pressure taps was used in the pressure measurements. Seven tapping stations were prepared (figure 1), and each station had 24 equally spaced taps along the circumference. The pressure tap showed a diameter of 1 mm, and all the curves were fitted through the data points were based on the spline fitting method.

Experiments were performed in a low-speed open-circuit wind tunnel with a 2.5 m long square test section measuring 1.5 m × 1.5 m in size. The tunnel involved a velocity range of 2–60 m s⁻¹ and turbulence intensity of ≤0.08%. The wind tunnel was equipped with a support mechanism with an automated turntable driven by an electric motor, which allows for remote model positioning and a C-strut for the change in AoA. The test model of the blunt-nose slender body was sting mounted on the support mechanism (figure 2). The sideslip angle can be varied in the range of −90° to 90° by rotating the turntable, and the AoA can be varied in the range of 0 °C–70° by sliding the model along the C-strut. The incoming freestream showed a V_∞ = 35 m s⁻¹ in all the following tests. This value corresponded with the Reynolds number Re_D = 1.54 × 10⁵ on the basis of D and V_∞.

Zoom In Zoom Out Reset image size

Pressure distributions around the test model were obtained through a pressure scanning system from the PSI Company. The system consisted of a DTC initium module and electronic scanning pressure (ESP) modules with an uncertainty within ±7 Pa. Each module possessed 64 channels. Thus, three ESP modules were entailed for the pressure measurement of the seven sections (see figure 1). Each pressure tap on the test model was connected to one channel on the ESP module with a tube of 1 mm inner diameter. The DTC initium module, which showed a maximum sampling frequency of 600 Hz, received the data measured from the ESP modules through the data cables and then sent the data to a personal computer. The sampling frequency in this paper was 50 Hz, and the record time was 12 s in the pressure measurements.

In the present work, 3D transient simulations were employed to simulate the flow around a blunt-nose slender body. The small vortex structures in the region near the body play a significant role in determining the overall flow field (Moskovitz 1990). Similar to Anwar et al 2005, we used the shear–stress transport k-ω turbulence model that considers the unsteady Reynolds-averaged Navier–Stokes equations, which are solved using the finite-volume method. This turbulence model can resolve the turbulent flow far from the wall and the boundary-layer flow close to the wall, thereby ensuring a highly accurate prediction of the flow separation with adverse pressure gradients (Menter 1993). This turbulence model has been used by Zhou et al (2010) and Liu et al (2016) in their simulations of turbulent flows around bluff or slender bodies.

The computational domain is a cylinder with a diameter of 60D and a length of 90D. The inlet boundary was at 30D upstream of the model tip (i.e., origin of the coordinate system), and the outflow boundary was located at the 60D downstream of the model tip. The pressure far field condition was applied to the computational domain, and the constant free-stream velocity (i.e., V_∞) was imposed. The AoA of the slender body model was determined by the two components of V_∞ along the x- and z-directions. The wall condition was applied to the surfaces of the blunt-nose slender body and the microperturbation, whereas the 'no-slip' condition was applied to the model surface.

The distance between the model surface and the nearest grid points was y⁺ < 1. A progression ratio of approximately 1.1 was used to cluster the grid points radially outward from the body surface. A total of 200 grids were presented around the cylinder circumference and yielded 5 million cells in the entire computational domain. The finite volume discretization of a second-order upwind scheme was applied for the momentum and temporal discretization. The velocity–pressure coupling was based on the algorithm of Semi-Implicit Method for Pressure-Linked Equations-Consistent.

Figure 3 presents the variations of C_y with x/D when the microperturbation was placed at different circumferential angles θ (α = 50°, γ = 10°). C_y is defined as ${C}_{y}=\displaystyle \sum _{i=1}^{24}p(i)/(0.5\rho {V}_{\infty }^{2}),$ where p(i) is the time-averaged static pressure measured at the ith tap. The data from the wind tunnel tests were included. The C_y–x/D curves were divided into two families that are nearly symmetric about C_y = 0 within the entire x/D range. In particular, the two families of curves revealed that the side forces were stabilized into two regular states when the microperturbation was induced on the nose at θ = 0° to θ = 180° or θ = 180° to θ = 360°. The special C_y–x/D curve of θ = 180° showed that the critical perturbation locations of the transform between both regular states produced a weak side force. The behavior of C_y for these different θ indicated the sectional side force. Thus, the overall side force acting on the slender body was determined in the presence of perturbation.

Zoom In Zoom Out Reset image size

Figure 4 presents the variation with θ of the sectional side force coefficient (C_y) at different sections (x/D) in the presence of microhemispherical perturbation (γ = 10°, α = 50°, and r/D = 0.012). The square-wave shape was observed for any an x/D when θ was increased from 0° to 360°. This result implies that only two values of the side forces, either positive or negative, existed. For example, given x/D = 3, the switch from the negative C_y to the positive C_y occurred at θ ≈ 180°, where the symmetrical xz-plane was located. The magnitudes of C_y were almost uniform at 0° < θ < 180° or 180° <θ < 360°. These observations indicated that the sectional side force acting on the slender body was manageable in the presence of the perturbation. Notably, the magnitudes of C_y were slightly different at 0° < θ < 180° and 180° < θ < 360°. This difference may be due to the effects of the uncertain natural perturbations over the slender body (Pick 1971).

Zoom In Zoom Out Reset image size

Given the predominant role of microperturbation in asymmetric vortex patterns, a singly periodic square wave was observed for the side-force variation with the circumferential angle (θ) of microperturbation set from 0° to 360° around the body axis (figure 4). However, the characteristic of a single period differed from that of a double period in a pointed-nose slender body (Chen et al 2002). The flow structure over a pointed-nose body at a high AoA was described in detail by Deng and Wang (2004), but no study has described the flow structure over a blunt-nose body at a high AoA. The detailed asymmetric vortex structure over a blunt-nose body at a high AoA (α = 50°) was discussed in this section.

Figure 5 presents the variations of C_y with x/D when the microperturbation was placed at the circumferential angle θ = 90° and the meridian angle γ = 10° (α = 50°). Data from both numerical simulations and wind tunnel tests were included. The numerical data were in good agreement with the experimental data and hence validated the present numerical simulations. The C_y variation with x/D was sinusoidal with reducing amplitudes. This trend suggested that the curve was inherently related to the evolution of the complicated asymmetric vortices along x/D. In particular, figure 6 presents the distributions of instantaneous pressure and the streamlines of asymmetric vortices at high AoA values (α = 50°, θ = 90°, and γ = 10°). These distributions were closely related to the curve shown in figure 5. Therefore, the C_y variation with x/D (figure 5) was divided into different regions along x/D based on the development of flow structure over the blunt-nose slender body. These regions included the inception region at x/D ≤ 3.0, the triple-vortex region at 3.0 ≤ x/D ≤ 6.0, the four-vortex region at 6.0 ≤ x/D ≤ 8.5, and the five-vortex region at 8.5 ≤ x/D ≤ 12. The evolution of asymmetric vortices along the body axis was described in detail through the pressure distributions of sections (figure 7).

• (1)  
  Inception region (A–B) and fully developed region (B–C) of asymmetric vortices

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Figure 8 presents the variations of the pressure coefficient C_p with the sectional angle θ_s and distributions of instantaneous vorticity at x/D = 1, 1.5, and 2 (α = 50°, θ = 90°, and γ = 10°). C_p is defined as ${C}_{{\rm{p}}}={\rm{p}}({\theta }_{{\rm{s}}})/(0.5\rho A{V}_{\infty }^{2}),$ where p(θ_s) is the time-averaged static pressure obtained at the location corresponding to θ_s. Numerical simulation data were included. Figure 8(a) presents the pressure distributions at x/D = 1, 1.5, 2 located in the inception region (A–B). The reattachment pressure peak on the lee side θ_s = 180°, which is a typical feature of twin vortices, indicated that asymmetric twin vortices occurred in this region. This occurrence can be verified in figure 8(b), which presents the instantaneous vorticity at x/D = 1, 1.5, 2. The magnitude of attachment peak, which decreases with the increasing x/D, implied the increase of vortex structure asymmetry. The magnitude of suction peak at θ_s = 160°, which was increased, indicated formation of the left vortex (VL1). However, the decreasing suction peak at θ_s = 200° suggested that the right vortex (VR1) was lifted from the surface. The separation points, which are indicated by the separation locations where the wall shear stress vanishes, moved to the sides (θ_s = 90°, 270°) with x/D. These phenomena indicate that VR1and VR2 developed in this region (A–B). However, VR1 grew faster than VL1, and this difference increased the C_y magnitude with x/D in the region (A–B) (figure 5). The (A–B) region was therefore named the inception region of asymmetric vortices.

Zoom In Zoom Out Reset image size

Figure 9 presents the variations of the pressure coefficient C_p with the sectional angle θ_s and the distributions of instantaneous vorticity at x/D = 2.5, 2.8, 3.3 (α = 50°, θ = 90°, and γ = 10°). These distributions corresponded to the fully developed region of asymmetric twin vortices. This result can be further confirmed by the disappearance of the reattachment pressure peak and the emergence of a strong suction peak near the lee side of θ_s = 175° (figure 9(a)). Such disappearance and emergence can be verified in figure 9(b), which presents the instantaneous vorticity at x/D = 2.5, 2.8, 3.3, and the breaking away of VR1 from the surface. Thus, the C_y–x/D curves (figure 5) collapsed at x/D = 3.3. This collapse indicated that the maximum asymmetry was achieved. Thus, this region (B–C) was named as the fully developed region of asymmetric vortices.

• (2)  
  Asymmetric triple vortex region (C–E) and peculiar points C and D.

Zoom In Zoom Out Reset image size

In figure 5, the sectional side force continuously increased with x/D in the twin-vortex region and decreased after the maximal point C. Figure 10 shows the variations of the pressure coefficient C_p with the sectional angle θ_s and the distributions of instantaneous vorticity at x/D = 3.3, 3.8, 4, 4.3 (α = 50°, θ = 90°, and γ = 10°) to explain the collapse at point C. Figure 10(b) reveals that VR1, which was the high vortex (further away from the surface), broke away from the surface at x/D = 3.3, and a new vortex VR2 was generated because of the separated shear layer downstream of point C. VR2, which grew along the body axis (x/D), increased the negative pressure and thus reduced the sectional side force. Therefore, the maximal point C in the curves of C_y–x/D signified the breakaways of the right vortex VR1 from the surface and the generation of a new vortex VR2. Figure 10(b) presents the flow structure, which was an asymmetric triple-vortex system that included three vortices, i.e., VL1, VR1, and VR2. VR2 was at the low position, whereas VL1 was at the high position near the body surface. However, VR1 was shed off from the surface and was positioned higher than its previous one. The induction of this triple-vortex system explains why the pressure distribution at the downstream of point C (x/D = 3.8) differed from that in the twin-vortex region (B–C) (figure 10(a)).

Zoom In Zoom Out Reset image size

Figure 11 shows the variations of the pressure coefficient C_p with the sectional angle θ_s and the distributions of instantaneous vorticity at x/D = 4.3, 4.5, 5, 5.2 (α = 50°, θ = 90°, and γ = 10°). Figure 5 reveals that the sectional side force continued to decrease to a positive value at these sections because of the further development of VR2 in the downstream direction. The D point (x/D = 4.5), which is shown in figure 5, was a peculiar point marking a sectional side force of zero. Figure 11(a) illustrates the curve of C_y–x/D at x/D = 4.5, and the sectional pressure distribution was nearly symmetric about θ_s = 180°. However, the symmetric pressure distribution at x/D = 4.5, which was in the asymmetric triple-vortex region, differed from that of the symmetric twin vortices (pressure coefficient C_p variation with the sectional angle θ_s at x/D = 1, as presented in figure 8(a)). The former exhibited the suction peak near the symmetric plane on the lee side (figure 11(a)), whereas the latter attained a pressure peak because of flow reattachment.

Zoom In Zoom Out Reset image size

With the further development of the triple vortices, especially the new vortex VR2, the right negative pressure increased and resulted in the continual increase in the sectional side force at the positive direction (figure 5). Finally, a positive maximum was reached at point E (x/D = 6). Figure 11(a) depicts the curves of C_y–x/D at x/D = 4.3, 5.2. These curves displayed different features and were separated in the inception region (C–D) and the fully developed region (D–E) of the asymmetric triple-vortex region. VR2 was weak in the inception region (C–D) of the asymmetric triple-vortex region, and the pressure distribution was mainly controlled by the vortices VR1 and VL1. As a result, a typical step shape was attained (figure 10(a)). By contrast, vortex VR2 began strong enough in the fully developed region (D–E) of the asymmetric triple-vortex region, and the pressure distribution in this region varied smoothly under the control of the triple vortices at x/D = 5.2 (figure 11(a)).

• (3)  
  Asymmetric four vortex region (E–G), five vortex region (G–I) and peculiar points.

Figure 12 presents the variations of the pressure coefficient C_p with the sectional angle θ_s and the distributions of instantaneous vorticity at x/D = 5.5, 6, 6.5 (α = 50°, θ = 90°, and γ = 10°). The sectional side force reached the positive maximal at point E (x/D = 6) (figure 5). The high-position vortex broke away, and a new vortex was generated. These findings were similar to those at x/D = 3.3 in the asymmetric triple-vortex region. Figure 12(b) shows that the high left vortex (VL1) began to break away at the E point (x/D = 6), and the generation of a new vortex (VL2) was initiated. These observations indicated the formation of the inception region of the asymmetric four vortices. The asymmetric four-vortex system included four vortices, i.e., VL1, VR1, VR2, and VL2. Figure 12(a) presents that the profile of the curves of C_y–x/D in this inception region (E–F) was similar to that in the fully developed triple-vortex region (D–E).

Zoom In Zoom Out Reset image size

The evolution of flow structures in the four-vortex region was similar to that in the triple-vortex region. Figure 13 presents the variations of the pressure coefficient C_p with the sectional angle θ_s and the distributions of instantaneous vorticity at x/D = 7, 7.5, 7.8 (α = 50°, θ = 90°, and γ = 10°). With the development of the left vortex VL2 from the E point (x/D = 6) to the downstream, the negative pressure on the left side of the model increased, but the negative pressure on the right side decreased. These effects resulted in the variation of the sectional side force from a positive value down to zero at the F point (x/D = 7) (figure 5). Furthermore, the pressure distribution of the zero sectional side force at station F point (x/D = 7) in the four-vortex region exhibited symmetric distribution similar to that at station D point (x/D = 4.5) in the triple-vortex region.

Zoom In Zoom Out Reset image size

Figure 14 displays the variations of the pressure coefficient C_p with the sectional angle θ_s and distributions of instantaneous vorticity at x/D = 8, 8.5, 9 (α = 50°, θ = 90°, and γ = 10°). A new right vortex (VR3) was generated at G point (x/D = 8.5), wherein the sectional side force of the fully developed four-vortex system reached the negative maximum, and the inception region of the asymmetric five vortices was formed.

Zoom In Zoom Out Reset image size

Figure 15 reveals the variations of the pressure coefficient C_p with the sectional angle θ_s and the distributions of instantaneous vorticity at x/D = 9.5, 10, 10.5 (α = 50°, θ = 90°, and γ = 10°). The sectional side force of the asymmetric five-vortex system reached zero at the H point (x/D = 10) in region (G–I) with VR3 development. Six or more asymmetric vortices were possibly generated with the development of the vortex system downstream after the five-vortex region (G–I). This phenomenon will occur at a low (up to 0) magnitude of sectional side force.

Zoom In Zoom Out Reset image size