2.1 Intake geometry

The current investigation involves conducting a two-dimensional steady-state simulation of a rectangular supersonic air intake. The study focuses on exploring the effects of varying cowl deflection across a range of Mach numbers. The cowl undergoes deflections of 0 ${{\rm{\;}}^ \circ }$ , 1 ${{\rm{\;}}^ \circ }$ , 2 ${{\rm{\;}}^ \circ }$ , 3 ${{\rm{\;}}^ \circ }$ and 4 ${{\rm{\;}}^ \circ }$ under intake conditions characterised by Mach numbers of 1.5, 2, 2.2, 2.5 and 3. Intake geometry is obtained from the experimental works conducted by Neale and Lamb [Reference Neale and Lamb42]. A detailed schematic of the intake having a cowl deflection of 4 ${{\rm{\;}}^ \circ }$ is given in Fig. 1.

Figure 1.

Schematic diagram of the intake geometry ( ${\rm{\alpha }}$ = 4 ${{\rm{\;}}^ \circ }$ ).

External compression is attained through the utilisation of two compression ramps featuring angles of 7 ${{\rm{\;}}^ \circ }$ and 14 ${{\rm{\;}}^ \circ }$ relative to the flow direction. The intake captures the incoming flow through a 15 mm opening. Subsequently, the ramp section is followed by a dual diffuser section. The separation between the diffuser section and the second ramp is 8.25 mm, incorporating both a short curved and straight structure. The first and second diffusers possess divergence angles of 2.3 ${{\rm{\;}}^ \circ }$ and 6 ${{\rm{\;}}^ \circ }$ , respectively. The channel area is maintained constant aft of the diffuser section. The cowl is positioned on the top side of the ramp, with the cowl tip offset by a distance of 22.98 mm from the ramp tip. To mitigate flow distortions, both ramp and cowl tips are kept sharp. The cowl tip can be deflected inwards from 0 ${{\rm{\;}}^ \circ }$ to 4 ${{\rm{\;}}^ \circ }$ at a distance of 14.36 mm from the tip for present study similar to the previous experimental investigation [Reference Das and Prasad9]. The cowl surface, from the point of deflection onward, is maintained horizontally to the central axis. The overall length of the geometry (L) is 119 mm. To facilitate the description of the separation bubble position in the intake, the lower section, comprising the ramps, diffusers, and constant area section, is termed lower surface (LS), while the upper surface is designated as cowl.

2.2 Governing equations

The physical phenomena of fluid flow are mathematically represented by solving equations that govern continuity, momentum, energy and the equation of state. Actual flow field is turbulent in nature. A two-dimensional compressible Reynolds-averaged Navier-Stokes (RANS) equations have been solved in the present simulation to capture the actual flow physics. The equations solved are:

1. 1. Continuity equation

   (1) \begin{align}\frac{{\partial \bar \rho }}{{\partial t}} + \frac{{\partial\! \left( {\bar \rho {{\tilde u}_j}} \right)}}{{\partial {x_j}}} = 0\end{align}

2. 2. Momentum equation

   (2) \begin{align}\frac{{\partial\! \left( {\bar \rho {{\tilde u}_j}} \right)}}{{\partial t}} + \frac{{\partial\! \left( {\bar \rho {{\tilde u}_i}{{\tilde u}_j}} \right)}}{{\partial {x_j}}} = - \frac{{\partial \bar p}}{{\partial {x_i}}} + \frac{{\partial\! \left( {{{\bar \tau }_{ji}} - \overline {\rho {u_i}{\rm{'}}{u_j}{\rm{'}}} } \right)}}{{\partial {x_j}}}\end{align}

3. 3. Energy equation

   (3) \begin{align}\frac{{\partial\! \left( {\bar \rho \tilde h} \right)}}{{\partial t}} + \frac{{\partial\! \left( {\bar \rho \tilde h{{\tilde u}_j}} \right)}}{{\partial {x_j}}} = \frac{{\partial \bar p}}{{\partial t}} + {\tilde u_j}\frac{{\partial \bar p}}{{\partial {x_j}}} + \overline {u{{\rm{'}}_j}\frac{{\partial p}}{{\partial {x_j}}}} + \frac{{\partial\! \left( { - {{\bar q}_j} - \overline {\rho h{\rm{'}}u{{\rm{'}}_j}} } \right)}}{{\partial {x_j}}} + \overline {{\tau _{ji}}\frac{{\partial u{{\rm{'}}_i}}}{{\partial {x_j}}}} \end{align}

where ${u_i}$ and ${x_i}$ are velocity and corresponding direction. $\rho $ , $h$ and $p$ are density, enthalpy and pressure. ( ${\,\bar{}\,}$ ), $(\,\tilde{}\,)$ and (^′) denotes Reynods average, favre average and turbulent fluctuation respectively. ${\tau _{ji}}$ and ${q_j}$ are molecular shear stress and instantaneous heat flux respectively and are defined by,

(4) \begin{align}{{\bar \tau }_{ji}} &= \bar \mu \left( {\frac{{\partial {{\bar u}_i}}}{{\partial {x_j}}} + \frac{{\partial {{\bar u}_j}}}{{\partial {x_i}}}} \right) + \frac{1}{3}\frac{{\partial {{\bar u}_j}}}{{\partial {x_j}}}{\delta _{ij}} \end{align}

(5) \begin{align}{{\bar q}_j} &= - \bar \lambda \frac{{\partial \bar T}}{\partial {x_j}} \\[0pt]\nonumber\end{align}

T is the temperature and $\bar \lambda $ = $\frac{{{C_p}\bar \mu }}{{{P_r}}}$ is the thermal conductivity with Pr as molecular Prandtl number.

Compressible RANS equations exhibit striking similarity to their incompressible counterparts, with the notable distinction lying in the Reynolds stress component. Here, the Reynolds stress ( $\overline {\rho {u_i}{\rm{'}}{u_j}{\rm{'}}} $ ) includes a fluctuating density component. The simultaneous solution of these three equations, coupled with the incorporation of turbulent kinetic energy (k) and various Reynolds stress equations, constitutes a comprehensive approach in addressing compressible flows in turbulent regimes [Reference Marvin43]. Governing equations are closed using standard k- $\omega $ turbulence model where k is the turbulent kinetic energy and $\omega $ is its specific dissipation rate as proposed by previous studies [Reference Das and Prasad9].

(6) \begin{align}\frac{\partial }{\partial t }\left( {\rho k} \right) + \frac{\partial }{{\partial {x_i}}}\left( {\rho k{u_i}} \right) = \frac{\partial }{{\partial {x_j}}}\left( {{{\rm{\Gamma }}_k}\frac{{\partial k}}{{\partial {x_j}}}} \right) + {G_k} - {Y_k} + {S_k}\nonumber\\\frac{\partial }{{\partial t}}\left( {\rho \omega } \right) + \frac{\partial }{{\partial {x_i}}}\left( {\rho \omega {u_i}} \right) = \frac{\partial }{{\partial {x_j}}}\left( {{{\rm{\Gamma }}_\omega }\frac{{\partial \omega }}{{\partial {x_j}}}} \right) + {G_\omega } - {Y_\omega } + {S_\omega }\end{align}

In these equations, ${G_k}$ and ${G_\omega }$ represents the generation of turbulence kinetic energy and $\omega $ respectively. Its effective diffusivity is represented by ${{\rm{\Gamma }}_k}$ and ${{\rm{\Gamma }}_\omega }$ . ${Y_k}$ and ${Y_\omega }$ represent the dissipation of $k$ and $\omega $ due to turbulence. ${S_k}$ and ${S_\omega }$ are user-defined source terms. Further discussions on the calculation of each term is available in the reference [Reference Wilcox44]. Adaptive time stepping based on the CFL number 0.8 is used with an initial step size of 1 $ \times {10^{{\rm{\;\;}} - 6}}$ . Simulation is run for 0.45 ms till it attained steady state.

2.3 Pertinent performance parameters

Current study focuses on correlating intake performance with local flow separations occurring inside the intake. Main performance parameters analysed here are:

1. 1. Total pressure recovery (TPR)

TPR is a key performance parameter index for supersonic intakes. It quantifies how effectively the intake system recovers total pressure after series of shock deceleration. It is defined as the ratio of total pressure at the outlet to the free-stream total pressure. A perfect value of TPR = 1 means no losses indicating perfect recovery. Any value less than TPR = 1 denotes losses inside the intake resulting from flow separation, improper handling of shock waves, etc.

(7) \begin{align}TPR = \frac{{{{\bar P}_{{0_O}}}}}{{{P_{{0_\infty }}}}}\end{align}

1. 2. Flow distortion (FD)

Flow distortion can be considered as another important metric to analyse the effectiveness of the intake. It defines the uniformity of flow supplied by the intake to the engine. It is defined as the ratio of the difference between maximum and minimum total pressure at exit to average total pressure at exit. For smooth operation, flow from the intake to the engine should be uniform and steady. Distorted flow at the exit can lead to issues such as non-uniform pressure distribution, fluctuating mass flow rates or sudden flow separations, all of which can disrupt engine operation.

(8) \begin{align}FD = \frac{{{P_{{0_{O,max}}}} - {P_{{0_{O,min}}}}}}{{{{\bar P}_{{0_O}}}}}\end{align}

2.4 Numerical approach

Detailed studies analysing the effect of various modeling decisions like turbulence models, two-dimensional vs three-dimensional formulations etc., are available in the literature [Reference Gnani, Zare-Behtash, White and Kontis45]. Current simulation is carried out in finite volume solver ANSYS-FLUENT(2021 R2). Flowing fluid is air and its density is obtained from ideal gas equations. Viscosity is calculated from Sutherland’s law. Simulation is done using implicit formulation since it is unconditionally stable. Convective flux vector is calculated using the Advection Upstream Splitting Method (AUSM) by Liou and Steffen [Reference Liou and Steffen46]. AUSM method is better at resolving the shockwaves since it uses neighbouring cell velocities to calculate interface Mach number. All terms are discretised using a second-order upwind scheme except in some off-design conditions, where the order of discretisation is reduced to first order. A CFL number of 0.5 is used to reach the steady-state solution.

2.4.1 Grid generation

Two-dimensional structured meshing is done with high refinement near the intake inlet, throat and near all walls. Biasing is done near all junctions (Fig. 2(a)). The first cell thickness is given in such a way as to keep the Y+ value near 1. A low Y+ value resolves the boundary layer more realistically than using other near wall functions. Domain surrounding the intake is extended in downstream, top and bottom directions to develop the flow and to make sure that the deflected shock waves escape the domain as shown in Fig. 2(b). Downstream expansion of the computational domain is to ensure that the flow is fully developed and stable after interacting with fixed walls. Meshing inside the intake is fine with a low aspect ratio and skewness values. But outside the intake cell spacing gradually increases.

Figure 2.

(a) Zoomed view of the grid near the intake inlet, and (b) computational domain with boundary conditions (not in scale).

2.4.2 Boundary conditions

Boundary conditions are defined based on previous research [Reference Das and Prasad9]. Inlet of the domain is modeled as a pressure-inlet. Stagnation pressure is calculated from isentropic flow equations corresponding to simulated Mach number. Total temperature of the flow is kept at 300K. Outlet is modeled as a supersonic outlet where flow properties are extrapolated from the interior. The flow is allowed to expand freely in all the cases simulated. Walls are adiabatic and satisfy the no-slip condition. Turbulent intensity at inlet and outlet are kept low to a value of 0.5% and the turbulent viscosity ratio is kept at 10.5 (Table 1).

Table 1.

Boundary conditions

2.4.3 Simulation validation

The numerical setup, boundary conditions and turbulence model is first validated against already established results to make sure the setup will be able to reproduce the flow physics accurately. Das and Prasad [Reference Das and Prasad9] conducted both experimental and two-dimensional numerical simulation on a rectangular intake with free-stream conditions at the outlet for understanding the cowl angle at which best performance is shown. Validation of the current setup is achieved by comparing static pressure (P) along the lower surface of the geometry for a cowl angle of 4 ${{\rm{\;}}^ \circ }$ at M = 2.2 with the results from the experiments and numerical simulations done by Das and Prasad. Static pressure and downstream distance are normalised using the free stream pressure ( ${P_\infty }$ ) and total length of the intake (L), respectively. A good agreement is obtained in the results between the lower surface pressure of the current study with the existing experimental and numerical results presented in the literature as shown in Fig. 3(a). The momentary rise and fall in static pressure corresponding the encounter of flow with shock waves and expansion fans are clearly captured. Slight under prediction in the pressure is observed at the opening of the intake between experimental and numerical results, which is also present in the previous studies [Reference Das and Prasad9]. This may be due to the inability of the solver to predict properties at the immediate interaction of supersonic flow with a sharp object. But further along the intake numerical results are matching with experimental values with average deviation of 10%.

Figure 3.

(a) Validation of normalised static pressure against distance along the lower surface, and (b) grid-independence test for cowl pressure aganist distance along the cowl surface.

3.1 Flow physics inside the intake

A detailed insight into SBLI inside a supersonic intake is possible only after a proper understanding of various flow phenomena occurring inside the intake. For this pressure contour inside the intake with 3 ${{\rm{\;}}^ \circ }$ cowl deflection at the design Mach number is reviewed in Fig. 4. Shocks inside the intake can be visualised by the abrupt change in color. Gradual change is denoting a gradual change in static pressure, which depicts expansion fans. The presence of an expansion fan is prominent near the throat shown in between the two dotted lines in the figure.

Figure 4.

Static pressure contour inside the intake at ${\rm{\alpha }}$ = 3 ${{\rm{\;}}^ \circ }$ , M 2.2.

Incoming air is guided inside the intake through the dual ramp structure. The ramp shocks are meeting at the cowl tip. An oblique shock from the cowl tip is falling on the lower surface near the throat. Air is expanding near the throat region forming the expansion fans. Here the intake is operating in the design Mach number of 2.2. The strength of the oblique shock is high enough to impart flow separation at the point of impact. The compression of the supersonic stream tube by the separation bubble leads to the development of separation shock. The cowl shock gets reflected from the LS as reflection shock. Another shock is forming from point A (Fig. 1), which also moves downstream through further reflection. The oblique shocks progress towards the exit by reflecting off at lower and cowl surfaces, creating the structure known as shock train [Reference Matsuo, Miyazato and Kim2]. Local flow separation can form at the point of impact of the shock wave at either surfaces. The size of the separation bubble varies as SBLIs strengthens. Compression and deceleration of the incoming supersonic flow is achieved through these shock trains.

3.2 Shock structure inside the intake

Shock waves inside the intake can be better visualised through numerical schlieren contour plots on the intake. Modulus of density gradient plots on the intake is used to compare the shock structure at various Mach numbers for the intake with different cowl deflections.

A detached bow shock is present at M = 1.5 in front of the ramp tip in Fig. 5(a). Flow immediately right to the shock is in a subsonic state and later accelerates inside the intake owing to the exit boundary condition. An entirely different flow structure is observed at M = 2 (Fig. 5(b)). The ramp tip oblique shocks meet together at a point upstream from the cowl tip. The interaction between the cowl shock and LS boundary layer causes a separation bubble to form at the point of impact. A feeble separation shock originating from the point of separation is fading out through further reflections and the expansion fan near the throat is not prominent due to the low Mach number. Both the ramp shocks are meeting the cowl tip and the cowl shock is falling on the LS farther downstream near the throat at M = 2.2 (Fig. 5(c)). Here also the oblique shock is strong enough to make a small separation on the throat. The reflected and separation shocks originating from the point of impact is evident and advance towards the nozzle exit by reflecting off from the LS and cowl surfaces. As airflow speed is increased to M = 2.5 and M = 3, the shock structure inside the intake changes dramatically (Fig. 5(d), (e)). Ramp shocks are converging earlier and imparting on the cowl at a more downstream location from the tip, resulting in significant separations in the cowl. The location where the cowl shock impacts the LS is also shifting deeper inward. The existence of robust expansion fans in the vicinity of the throat causes oblique shocks to curve, with the curvature intensifying as the Mach number increase. These irregular shock structures can be the possibility of deterioration in the intake performance. Numerical schlieren plots inside the intake for cowl angles of 0 ${{\rm{\;}}^ \circ }$ –3 ${{\rm{\;}}^ \circ }$ are shown in Figs. 6 and 7.

Figure 5.

Numerical schlieren contour plot on the intake with ${\rm{\alpha }}$ = 4 ${{\rm{\;}}^ \circ }$ for Mach numbers: (a) M = 1.5, (b) M = 2, (c) M = 2.2, (d) M = 2.5, and (e) M = 3. Contour levels used are between 0 and 150, with 100 levels (frame(a)), and in all other cases, they are between 0 and 500, with 100 levels.

Figure 6.

Numerical schlieren contour plots for ${\rm{\alpha }}$ = 0 ${{\rm{\;}}^ \circ }$ and 1 ${{\rm{\;}}^ \circ }$ at different Mach numbers. Similar contour levels are used as mentioned in Fig. 5.

Figure 7.

Numerical schlieren contour plots for ${\rm{\alpha }}$ = 2 ${{\rm{\;}}^ \circ }$ and 3 ${{\rm{\;}}^ \circ }$ at different Mach numbers. Similar contour levels are used as mentioned in Fig. 5.

A detached bow shock is observed in front of the ramp at M = 1.5 for all cowl angles. This deflects the main flow from the intake. Flow immediate to this bow shock is subsonic and later accelerate to supersonic velocities after the diffuser. At M = 2, a large separation bubble is seen on the ramp surface for the cowl angles 0 ${{\rm{\;}}^ \circ }$ –3 ${{\rm{\;}}^ \circ }$ which results in unstart of the intake. The incident shock from the cowl tip interacts with the boundary layer on the ramp surface creating a large separation bubble. This thickens the net boundary layer thickness and reduces the net area inside the throat. Thus shock gets out from the intake and forms the bow shock which results in subsonic flow inside the intake. Van Wie et al. [Reference Van Wie, Kwok and Walsh12] studied these types of unstart and described them as soft unstart. Start of the separation bubble seems to shift inward towards the throat when angle increases (0 ${{\rm{\;}}^ \circ }$ –3 ${{\rm{\;}}^ \circ }$ ) at M = 2. For Mach number 2.2, intake is in unstart condition at cowl angle 0 ${{\rm{\;}}^ \circ }$ but restarts when cowl is deflected by 1 ${{\rm{\;}}^ \circ }$ –4 ${{\rm{\;}}^ \circ }$ . Here separation between separated shock and reflected shocks reduces as the angle increases and is minimum at the cowl angle 4 ${{\rm{\;}}^ \circ }$ , which shows reduction in separation bubble size at that location. Quantitative analysis on the reduction in this bubble size is discussed in the following sections. It has been observed that the curvature of the cowl shock increases with an increase of both the Mach number and cowl angle. This could be attributed to the combined effect of the inward movement of the separation bubble with an increase in cowl angle and the formation of stronger expansion fans with increase in the Mach number.

3.3 Effect of cowl angle and operating condition on separation bubbles

The primary drawback of SBLIs is the occurrence of local separations on surfaces caused by the adverse pressure gradient induced by shock waves. The performance of the intake is adversely impacted by these separations. A qualitative study is done by plotting the velocity vector on the intake along with Mach contour in Fig. 8.

Figure 8.

Mach contour and velocity vector (enlarged at the separation points) inside the intake with ${\rm{\alpha }}$ = 4 ${{\rm{\;}}^ \circ }$ for Mach numbers: (a) M = 2, (b) M = 2.2, (c) M = 2.5, (d) M = 3.

Mach contour inside the intake with cowl angle 4 ${{\rm{\;}}^ \circ }$ is compared for different Mach numbers. The exclusion of plots for M = 1.5 is due to the absence of separation bubbles, as the primary goal is to visualise these bubble. Only separations with a minimum size of 1 mm are included in the plots. Additionally, M = 3 has a fourth separation bubble of size 0.5 mm, which is shown in Fig. 8(d) as point 4. Correlation between performance of the intake and SBLIs cannot be attempted without quantifying these separations. The extent of the interaction between shock wave and boundary layer is quantified using separation bubble width formed at different locations. A better understanding of the position and extend of these bubbles are obtained by plotting the wall shear stress along LS and cowl surfaces. Wall shear stress at a point depends on the gradient of flow velocity along the perpendicular direction. Wall shear stress along the surfaces becomes negative when separation bubbles are formed.

The point where wall shear stress becomes negative from positive is considered to be the starting of separation bubble and point where the negative value again becomes positive is considered as the end of separation bubble. The number of separation bubbles also quantifies the extent of SBLIs.

Figure 9 depicts the wall shear stress along the LS and cowl surfaces for a cowl without any deflection (0 ${{\rm{\;}}^ \circ }$ ) at various Mach numbers. At M = 1.5, there are no SBLIs occurring within the intake, regardless of the changes in cowl angle. Therefore, in the current study, any further discussions regarding the position and size of the separation bubble exclusively pertain to Mach numbers of 2 and above. The largest separation bubble for $\alpha $ = 0 ${{\rm{\;}}^ \circ }$ is observed at M = 2 on the ramp surface (Fig. 9(a)). In Fig. 9(b) wall shear stress at M = 2.2 is depicted. Similar to M = 2, M = 2.2 also has a large separation on the ramp associated with the intake unstart. The role of these separation bubbles on intake unstart will be discussed in subsequent section. Further increase of Mach number is associated with a started intake with separation size on the LS suddenly dropping (Fig. 9(c) and (d)). Cowl separation bubbles are observed only at M = 2.5 and M = 3. The bubbles are moving in the direction of the exit as Mach number increases from 2.5 to 3. This is due to the overall shift of the shock structure further downstream with an increase in the Mach number. When compared with the LS separations, cowl separation bubbles are smaller in size.

Figure 9.

Quantification of wall shear stress along the lower and cowl surfaces for = 0 ${{\rm{\;}}^ \circ }$ and different Mach numbers: (a) M = 2, (b) M = 2.2, (c) M = 2.5, (d) M = 3.

Figures 10–13 are associated with the wall shear stresses for cowl angle 1 ${{\rm{\;}}^ \circ }$ , 2 ${{\rm{\;}}^ \circ }$ , 3 ${{\rm{\;}}^ \circ }$ and 4 ${{\rm{\;}}^ \circ }$ , respectively. Similar to Fig. 9, large separation on the ramp surface is observed at M = 2 for all the cowl configurations in Figs (10–12(a)). Figure 13(a) has small separation size near the throat when compared with other cowl angles at M = 2, which suggest intake is in started condition on 4 ${{\rm{\;}}^ \circ }$ cowl configuration from M = 2. Compared with Fig. 9 of $\alpha $ = 0 ${{\rm{\;}}^ \circ }$ , all other cowl angles also have two separation bubbles on the cowl surface only at M = 2.5 and M = 3. The position of the two cowl shocks are not changing with $\alpha $ for a particular Mach number.

Figure 10.

Quantification of wall shear stress along the lower and cowl surfaces for ${\rm{\alpha }}$ = 1 ${{\rm{\;}}^ \circ }$ and different Mach numbers: (a) M = 2, (b) M = 2.2, (c) M = 2.5, (d) M = 3.

Figure 11.

Quantification of wall shear stress along the lower and cowl surfaces for ${\rm{\alpha }}$ = 2 ${{\rm{\;}}^ \circ }$ and different Mach numbers: (a) M = 2, (b) M = 2.2, (c) M = 2.5, (d) M = 3.

Figure 12.

Quantification of wall shear stress along the lower and cowl surfaces for = 3 ${{\rm{\;}}^ \circ }$ and different Mach numbers: (a) M = 2, (b) M = 2.2, (c) M = 2.5, (d) M = 3.

Figure 13.

Quantification of wall shear stress along the lower and cowl surfaces for ${\rm{\alpha }}$ = 4 ${{\rm{\;}}^ \circ }$ and different Mach numbers: (a) M = 2, (b) M = 2.2, (c) M = 2.5, (d) M = 3.

Table 2 summarises all the cases analysed during the current study. It encompasses the number of separation bubbles on the cowl and lower surfaces (LS), the corresponding position where it starts, size of the separation for various cowl angles ( $\alpha $ ), and Mach numbers along with the total pressure recovery (TPR) and flow distortion (FD) for each of these cases. Intake unstart is observed at M = 2 for cowl configurations 0 ${{\rm{\;}}^ \circ }$ –3 ${{\rm{\;}}^ \circ }$ . Soft unstart is characterised with the large separation on the ramp surface (Figs. 6 and 7). Separation is diminishing and moving towards the throat section as cowl deflection increase. At $\alpha $ = 4 ${{\rm{\;}}^ \circ }$ separation size is 1.55 mm for M = 2 and the intake restart is observed. Intake unstart at M = 2.2 is observed only for $\alpha $ = 0 ${{\rm{\;}}^ \circ }$ , with a separation bubble of size 8.34 mm on the ramp. However intake starts with a small deflection of 1 ${{\rm{\;}}^ \circ }$ where separation bubble size reduces to 3.16 mm and position is shifted to the throat from ramp. (Further increase in $\alpha $ is also associated with started intake.)

Table 2.

Number of separations, separation bubble position, size, TPR and FD for various parameters under study

Figure 14 plots net separation size against Mach number and cowl angles, respectively. Since there are a number of local separations on lower and cowl surfaces, net separation bubble size is calculated as the sum of local separation bubble sizes on cowl and LS for a particular configuration and operating condition. At M = 1.5 no separation is present due to the absence of SBLI owing to the bow shock in front of the ramp tip. The correlation between separation size and performance of the intake in terms of TPR can be understood by comparing Fig. 14 with Fig. 15. Maximum and minimum separation sizes are characterised by minimum and maximum TPR except at M = 1.5. Maximum separation size is for $\alpha $ = 0 ${{\rm{\;}}^ \circ }$ at M = 3. For all Mach numbers, intake with 4 ${{\rm{\;}}^ \circ }$ cowl deflection has the least separation bubble size except at M = 3 (Fig. 14(b)). For M = 3 minimum separation is for $\alpha $ = 1 ${{\rm{\;}}^ \circ }$ . Figure 14(a) shows that for each cowl angle, minimum net separation sizes are observed at M = 2.2 which is the design Mach number. Only deviation in this trend is observed at $\alpha $ = 0 ${{\rm{\;}}^ \circ }$ where separation size is minimum at M = 2.5. As Mach number deviates from M = 2.2, separation size is increasing. A general trend of reduction in separation size can be seen when cowl angle increases for a particular Mach number (Fig. 14(b)). Higher Mach numbers lead to the formation of stronger shock waves, causing more instances of separation at various points within the intake system. This phenomenon is contributing to the increase in net separation size.

Figure 14.

Variation of net separation bubble size with respect to (a) Mach number and (b) cowl angle.

Figure 15.

Variation of TPR aganist net separation bubble size for Mach numbers: (a) M = 2, (b) M = 2.2, (c) M = 2.5, (d) M = 3.

Correlating separation bubble with TPR provides insight into the dependency of performance on separation bubble size. A general trend is found to be a decrease in TPR with increase in separation size except in some cases (Fig. 15(a)–(d)). At M = 2, intake is in soft unstart for cowl configurations 1 ${{\rm{\;}}^ \circ }$ –3 ${{\rm{\;}}^ \circ }$ , hence TPR for these angles are comparable (Fig. 15(a)). A prominent improvement in TPR is observed when cowl angle changes from 3 ${{\rm{\;}}^ \circ }$ to 4 ${{\rm{\;}}^ \circ }$ , where intake starts. In Fig. 15(b) TPR of $\alpha $ = 1 ${{\rm{\;}}^ \circ }$ is suddenly rising from $\alpha $ = 0 ${{\rm{\;}}^ \circ }$ which is denoting the restart of intake when cowl deflection increase from $\alpha $ = 0 ${{\rm{\;}}^ \circ }$ . TPR at M = 2.2 is larger than that of M = 2. Gradual change in TPR with net separation size is again observed in Fig. 15(c) at M = 2.5 where intake is in started condition for all cowl configurations. Deviation in the general trend (higher $\alpha $ having higher TPR) is observed in Fig. 15(d) at M = 3. Here $\alpha $ = 1 ${{\rm{\;}}^ \circ }$ has the minimum net separation size. There is a 12% increase in net separation size between $\alpha $ = 1 ${{\rm{\;}}^ \circ }$ and $\alpha $ = 4 ${{\rm{\;}}^ \circ }$ accompanied by less than 1% change in TPR. Influence of net separation size on TPR is observed to be less significant at higher Mach number (M = 3).

Figure 16(a) and (b) plots FD against Mach number and cowl angle. FD is an important parameter to consider while analysing the performance of an intake. Intake exit should provide uniform, non-distorted flow to the combustion chamber for maximum efficiency [Reference Sepahi-Younsi and Forouzi Feshalami13]. FD is minimum at M = 2.2 for most of the cowl configurations except for cowl angles 0 ${{\rm{\;}}^ \circ }$ and 4 ${{\rm{\;}}^ \circ }$ (Fig. 16(a)). These configurations has minimum FD at M = 2 (For $\alpha $ = 4 ${{\rm{\;}}^ \circ }$ FD between M = 2 and M = 2.2 are almost same). Variations in the position and number of separation bubbles may be the reason for obtaining minimum FD at other Mach numbers other than the design Mach number. Observations from Fig. 16(b) show minimum FD for all cowl angles at M = 2.2 with slight deviation for $\alpha $ = 0 ${{\rm{\;}}^ \circ }$ due to intake unstart. Small deviations in FD from expected results may be attributed to the unsteady effects of SBLIs.

Figure 16.

Variation of FD with respect to (a) Mach number and (b) cowl angle.