2.1. Supersonic jet rig and control minijet

A schematic of the supersonic jet rig is given in figure 1. Air is compressed by a compressor and stored in three air tanks connected in series, with a total storage capacity of 8 m³ at a pressure of 12 bar. The compressed air from the storage tanks passes through pressure regulators and control valves before entering a plenum chamber. The chamber consists of a diffuser with a half-angle of 15° and a cylindrical settling chamber of 114 mm in diameter and 400 mm in length. The contraction section from 114 mm at the end of the settling chamber to 20 mm is the same as used in Perumal & Zhou (Reference Perumal and Zhou2018), whose contour is given by $D/2 = 57 - 47\,\textrm{si}{\textrm{n}^{1.5}}[{\rm \pi} /2 - 9(x+L)/8*\varPi/180]$, where L (= 30.68 mm) is the nozzle length. The required pressure in the settling chamber is achieved via a pressure regulating valve and is measured using a gauge transducer (MEAS M3234) with a pressure range of 0–7 bar. Two wire-mesh screens are placed at the diffuser and the settling chamber, respectively, to reduce the turbulence intensity. The nozzle mounted at the end of the contraction section can be replaced to obtain different design conditions.

Figure 1.

Schematics of (a) supersonic jet facility and (b) schlieren flow visualization measurement. All lengths are in mm.

An axisymmetric converging–diverging nozzle is used to generate a jet of M_d = 1.8. The converging section contracts from D ₁ = 20 mm at the end of the contraction section to D_th = 8.34 mm at the throat, following a fifth-order polynomial function, i.e. $({D(x)} - {D_{th}})/(D_{1} - {D_{th}}) = 1\unicode{x2013} 10{[(x + L)/{L_1}]^3} + 15{[(x + L)/{L_1}]^4} - 6{[(x + L)/{L_1}]^5}$ (Zhang & Fan Reference Zhang and Fan2003), where L ₁ (= 18 mm) is the length of the converging section. The diverging section is designed based on the method of characteristics presented in Zucrow & Hoffman (Reference Zucrow and Hoffman1976) from D_th to the nozzle exit diameter D = 10 mm.

A control minijet is issued from a stainless steel tube with an inner diameter d, whose axis is 0.15–0.25 mm downstream of the nozzle exit. Following Davis (Reference Davis1982), the tube exit is 2 mm away from the nozzle lip along the z direction so that the interference between the issuing minijet and the spreading shear layer of main jet can be minimized. Six different tubes are used, resulting in d/D = 1/20, 1/9.5, 1/7.7, 1/6.5, 1/5.3 and 1/4.1. The minijet comes from another chamber where the required pressure is separately maintained.

2.2. Flow conditions

The stagnation temperature T ₀ of main jet and minijet is as ambient temperature T_a (= 300 K), i.e. T ₀/T_a = 1. The NPR is from 3 to 6 for both natural and manipulated jets. The nozzle is calibrated following Akram, Perumal & Rathakrishnan (Reference Akram, Perumal and Rathakrishnan2021), and the design NPR, i.e. NPR_d, as determined from the isentropic equation is 5.75 for the jet of M_d = 1.8. Then, NPR < NPR_d and NPR > NPR_d correspond to the over- and under-expanded states, respectively. Parameters M_j and P_e/P_a are related via the isentropic relations:

(2.1)\begin{gather}{M_j} = {\left[ {({NP}{{R}^{(\gamma - 1)/\gamma }} - 1)\frac{2}{{\gamma - 1}}} \right]^{0.5}},\end{gather}

(2.2)\begin{gather}\frac{{{P_e}}}{{{P_a}}} = {\left( {\frac{{1 + \dfrac{{\gamma - 1}}{2}M_j^2}}{{1 + \dfrac{{\gamma - 1}}{2}M_d^2}}} \right)^{\gamma /(\gamma - 1)}},\end{gather}

where $\; \gamma = 1.4$ for air. From (2.1)–(2.2), the over- and under-expanded states of the jet correspond to M_j < M_d or P_e/P_a < 1 and M_j > M_d or P_e/P_a > 1, respectively. The velocity of the main jet may be calculated from the following equation:

(2.3)\begin{equation}{U_j} = \sqrt {\frac{{2\gamma }}{{\gamma - 1}}R{T_0}\left[ {1 - {{\left( {\frac{1}{{\textit{NPR}}}} \right)}^{(\gamma - 1)/\gamma }}} \right]} ,\end{equation}

where $R = 287\;\textrm{J}\;\textrm{k}{\textrm{g}^{ - 1}}\;{\textrm{K}^{ - 1}}$ is the gas constant for air. The Reynolds numbers $Re = {\rho _j}D{U_j}/{\mu _j}$ of the jet at the minimum and maximum NPR or M_j are 4.5 × 10⁵ and 8.0 × 10⁵, respectively, where the jet density ρ_j is determined from

(2.4)\begin{equation}{\rho _j} = {\left.{\left( {\frac{{\textit{NPR} \times {P_a}}}{{R{T_0}}}} \right)} \right/ {{{\left( {1 + \frac{{\gamma - 1}}{2}M_j^2} \right)}^{1/(\gamma - 1)}}}},\end{equation}

and μ_j is the viscosity calculated based on Sutherland's formula (Anderson, Tannehill & Pletcher Reference Anderson, Tannehill and Pletcher1984). Table 1 lists NPR, M_j, U_j, P_e/P_a and other parameters of interest.

Table 1.

Flow conditions for the natural jet of design Mach number M_d = 1.8.

The IPR varies from 2 to 8 and ${C_m} = {\dot{m}_i}/{\dot{m}_j}$ is between 0.18 % and 11.87 % for various d/D (table 2), where ${\dot{m}_i}$ and ${\dot{m}_j}$ are the minijet and main jet mass flow rates, respectively. Both ${\dot{m}_i}$ and ${\dot{m}_j}$ are calculated from the isentropic equations:

(2.5)\begin{gather}{\dot{m}_i} = 0.6847\frac{{{P_{0s,i}}{A_i}}}{{\sqrt {R{T_0}} }},\end{gather}

(2.6)\begin{gather}{\dot{m}_j} = 0.6847\frac{{{P_{0s}}{A_j}}}{{\sqrt {R{T_0}} }},\end{gather}

where ${A_i} = {\rm \pi}{d^2}/4$ and ${A_j} = {\rm \pi}D_{th}^2/4$. The mass flow rate of a jet calculated from inviscid compressible flow theory would be close to its actual mass flow rate if the nozzle diameter exceeds 0.35 mm (Jindra Reference Jindra1970). The present minimum d is 0.5 mm. Furthermore, both minijet and main jet are choked (IPR > 1.89 and NPR > 1.89) so that (2.5)–(2.6) can be used to calculate ${\dot{m}_i}$ and ${\dot{m}_j}$. The velocity U_i and fully expanded Mach number M_i of the minijet can be estimated from

(2.7)\begin{gather}{U_i} = \sqrt {\frac{{2\gamma }}{{\gamma - 1}}R{T_0}\left[ {1 - {{\left( {\frac{1}{{\textit{IPR}}}} \right)}^{(\gamma - 1)/\gamma }}} \right]} ,\end{gather}

(2.8)\begin{gather}{M_i} = {\left[ {({IP}{{R}^{(\gamma - 1)/\gamma }} - 1)\frac{2}{{\gamma - 1}}} \right]^{0.5}}.\end{gather}

Table 2.

Mass flow rate ratio C_m of minijet to main jet for various IPRs at exit pressure ratio P_e/P_a = 0.70 (NPR = 4).

2.3. Pressure measurement and flow visualization

The total pressure of the main jet is measured using a Pitot tube with inner and outer diameters of 0.6 and 0.9 mm, respectively, which is connected to a miniature pressure transducer (MEAS XPM4-5BA-/ET1) with a measurement range of 0–5 bar and linearity up to a maximum departure of 0.25 %. During measurements, the Pitot probe induces a bow shock wave ahead of its head (figure 2). Therefore, the measured total pressure could be less than the actual total pressure due to the pressure loss associated with the shock wave. The total pressure loss is negligibly small since the shock strength is impaired as the Pitot probe is moved downstream and the flow upstream of the bow shock wave is virtually undisturbed (Katanoda et al. Reference Katanoda, Miyazato, Masuda and Matsuo2000). As such, no correction is made for the pressure measurement. Rathakrishnan (Reference Rathakrishnan2016) advocated that, when the ratio of the nozzle exit area to the probe projected area is greater than 64, the probe interference to the flow is negligibly small. This view is reinforced by numerous investigations. Miller et al. (Reference Miller, Veltin, Morris and Mclaughlin2009) compared the total pressure measured using a Pitot probe and that calculated from the Reynolds-averaged Navier–Stokes (RANS) simulation of M_j = 1.3 and 1.5 jets. Their blockage ratio was 161. The two sets of data show excellent agreement (their figure 10) even at x* = 0–2, where a strong shock occurred due to the presence of the Pitot probe (figure 2). Miller & Veltin (Reference Miller and Veltin2011) compared the velocity profiles of an M_j = 1.5 jet at various x* (up to 8), measured from a rake of five Pitot probes, with that of the RANS simulation. Their blockage ratio was about 90. The comparison was very good (their figures 3 and 4), especially when the total temperature ratio was less than 3.6 (the present ratio is about 1). A similar observation was also made by André, Castelain & Bailly (Reference André, Castelain and Bailly2013) for M_j = 1.15 and 1.5 jets where the Pitot-probe-measured velocities were compared with laser Doppler velocimetry data (their probe blockage ratio was 652). As a matter of fact, the Pitot probe blockage effects are insignificant even when the probe blockage ratio is smaller than 64. For example, Katanoda et al. (Reference Katanoda, Miyazato, Masuda and Matsuo2000) observed in an ${M_j} = 2$ jet that the total pressure, measured using a Pitot probe with a blockage ratio of 42, agreed well with numerical simulation based on the Euler equation. Phalnikar, Kumar & Alvi (Reference Phalnikar, Kumar and Alvi2008) showed that the shock-cell spacing measured using a Pitot probe with a blockage ratio of 16 was in excellent agreement with the estimate from shadowgraph images (their figures 7 and 8). The present probe blockage ratio is 123, and the probe interference to the flow should be negligibly small.

Figure 2.

Schlieren images of supersonic natural jets, NPR = 4.0 (${P_e}/{P_a} = 0.70$). Pitot tube at various x* locations.

The Pitot tube is traversed within the range of (x*, y*, z*) = (0–20, −2–2, −2–2). As noted by Perumal & Rathakrishnan (Reference Perumal and Rathakrishnan2022), the measured pressure P _0t can be used to determine $L_c^\ast $, which is currently used to quantify jet mixing. The Re based on the inner diameter of the Pitot tube is 10⁴, well above 200, and therefore the viscous effect on the measured total pressure can be safely neglected (Chue Reference Chue1975). The raw data are recorded at a sampling frequency of 2000 Hz for a duration of 2 s by a PC through a 16-bit National Instruments DAQ board (6361) and then fed into LabVIEW for data processing.

A conventional Z-type schlieren system is used for flow visualization (figure 1b). Two concave mirrors, each with a diameter of 200 mm and a focal length of 2000 mm, have a surface finish of λ/10, where λ denotes the light wavelength. Illumination is provided by a white-light source that passes through two condenser lenses before reaching one of the concave mirrors. A parallel beam from the first mirror passes through the test section before being projected on another mirror and is then focused on a knife edge before being projected on a screen. The time-averaged and instantaneous images are recorded at a sampling frequency of 300–1000 Hz with an exposure of 1000 μs and 30–90 kHz with an exposure of 1–3 μs, respectively, using a Dantec camera (2560 × 1600 pixels), and then processed using PCC 2.12 software.

4.1. Effects of the control parameters on jet mixing

It is well known that a jet manipulated using a single minijet may deflect, referred to as thrust vectoring. As a result, jet mixing quantified based on a change in the centreline velocity or pressure may be artificially exaggerated. Following Perumal & Rathakrishnan (Reference Perumal and Rathakrishnan2021), we may estimate $L_c^\ast $ from the actual position of the measured maximum P _0t/P _0s. Note that the minijet injection is along the positive z direction. Figure 7 presents the contours of time-averaged P _0t/P _0s measured in the injection and non-injection planes or xz and xy planes for various C_m (d/D = 1/9.5, P_e/P_a = 0.70) along with those of the natural jet (P_e/P_a = 0.70). The contours are symmetric about the centreline ($\kern 1.5pt {y^\ast } = {z^\ast } = 0$) for the natural jet (figure 7a) but asymmetric in the xz plane under manipulation (figure 7b–d), where the maximum P _0t/P _0s occurs at a negative ${z^\ast }$, suggesting a deflected jet. This deflection becomes more obvious with increasing C_m. In the xy plane, the contours show slight asymmetry about ${y^\mathrm{\ast }} = 0$ (figure 7e–g). Therefore, we determine $L_c^\ast $ directly from the P _0t/P _0s contours based on the cutoff level of M = 1 (Anderson Reference Anderson1982) (figure 7a–g). Note that the estimated $L_c^\ast $ in the xy plane (figure 7e–g) is shorter than in the xz plane (figure 7b–d). This observation is internally consistent with the fact that the xy plane cuts through the xz plane where the sonic Mach number is not a maximum. Similar observations are also made at other P_e/P_a.

Figure 7.

Iso-contours of time-averaged pressure ratio ${P_{0t}}/{P_{0s}}$. (a) Natural jet (${P_e}/{P_a} = 0.70$). Manipulated jet ($d/D = 1/9.5$): (b–d) injection plane; (e–g) non-injection plane. The red thick curve denotes the cutoff pressure ratio (0.472) for the sonic Mach number.

There are two factors that may contribute to the asymmetry of the pressure contours about y* = 0 in the xy plane. One is ascribed to experimental uncertainties, which could be eliminated if the sample size is infinitely large. The present sampling rate and duration of the pressure data were limited, 2000 Hz and 2 s, respectively. The choice of the 2 s sampling duration is based on a convergence test. The variation in the pressure data obtained at (x*, y*, z*) = (0.5, 0, 0) of the natural jet (NPR = 4) is converged to within 1 % once the sampling duration exceeds 1.8 s (not shown). Another factor could be linked to the part of the L-shape Pitot tube (figure 2), which is normal to the flow and located downstream of the pressure data obtained. When the Pitot tube is traversed across the jet, this part may interfere with the pressure data obtained upstream at y* = −1.5–0. This interference retreats at y* = 0–1.5, where the blockage produced by the Pitot tube is smaller. Note that the jet asymmetry is appreciable only downstream of the cutoff pressure (see the red-coloured contour in figure 7e–g) that is used to estimate $L_c^\ast $. That is, the jet asymmetry takes place downstream of the supersonic potential core, where the flow is subsonic, thus being more susceptible to downstream disturbance. It is worth pointing out that this asymmetry of the pressure data in the xy plane produces essentially no effect on the estimate of $L_c^\ast $, which is determined only from the pressure data measured in the xz plane as described before.

One can see a pressure dip at y* = 0 in figure 7(g) for ${C_m} \ge 1.98\,\%$ at (P_e/P_a, d/D) = (0.70, 1/9.5) when the IPR is quite large (= 5). This is because at very large C_m or IPR, the penetration depth of the minijet is also large, causing the cross-section of the main jet to deform into a kidney-shaped structure, as observed by Zaman et al. (Reference Zaman, Reeder and Samimy1994) in a supersonic jet manipulated by a single tab. Note that the injection is along the positive z* direction. This kidney-shaped structure produces a double peak in the pressure profile, thus causing the dip at y* = 0 in the xy plane.

Figure 8 presents the variation in $L_c^\ast $ with C_m for d/D = 1/20–1/5.3 and P_e/P_a = 0.70 and 1.05. The control is considered to be effective once $L_c^\ast $ falls below that of the natural jet (C_m = 0). Several observations can be made. First, for d/D ≥ 1/9.5, $L_c^\ast $ retreats with increasing C_m, regardless of P_e/P_a. At P_e/P_a = 0.70, $L_c^\ast $ is larger than that of the natural jet for C_m < 1.8 % but smaller for C_m ≥ 1.8 %. The minijet injection at small C_m is unable to penetrate the main jet core (figure 7b–d) and may be swept away downstream (Davis Reference Davis1982). In their manipulation of an M_d = 1.1 jet using 16 minijets at d/D = 1/33 (C_m = 0.9 %), Callender, Gutmark & Martens (Reference Callender, Gutmark and Martens2007) also observed an increased jet core length with respect to the natural jet. They argued, based on particle image velocimetry data, that the minijets acted as a shield for the main jet and impeded the shear-layer growth, resulting in a prolonged jet core. On the other hand, the minijet injection penetrates the main jet core given a larger C_m, resulting in a reduced $L_c^\ast $. Second, for d/D = 1/20, $L_c^\ast $ increases with increasing C_m until reaching a local maximum, which may vary with P_e/P_a, and then drops with a further increase in C_m. The value of $L_c^\ast $ exceeds that of the natural jet for the range of C_m examined. The present P _0t/P _0s contours (not shown) of the manipulated jet for d/D = 1/20 do not show any sign of minijet penetration into the shear layer of the main jet. Third, for a given C_m, an increase in d/D leads to a significant increase in $L_c^\ast $, suggesting that small d/D promotes jet mixing, though d/D cannot be too small, say below 1/20 in the present case. Evidently, a smaller d/D for a given C_m leads to a higher U_i/U_j, and hence deeper penetration into the main jet. Finally, given d/D and C_m, the manipulated jet at P_e/P_a = 1.05 experiences a more significant contraction in $L_c^\ast $ than at P_e/P_a = 0.70. For example, given d/D = 1/5.3 and C_m = 3.2 %, $L_c^\ast $ drops from 17.1 without control to 10.6 at P_e/P_a = 1.05, a decrease of 38 %. At P_e/P_a = 0.70, however, $L_c^\ast $ reduces by only 22 % for the same d/D and C_m. This suggests a more effective manipulation under the design condition (P_e/P_a = 1) than under off-design conditions (P_e/P_a ≠ 1). In their manipulation of a NPR_d = 4 (M_d = 1.56) jet at NPR = 2.5–4.5 using 24 minijets, Cuppoletti & Gutmark (Reference Cuppoletti and Gutmark2014) reported a more pronounced noise reduction at NPR_d = 4 than at other NPR values. Their particle image velocimetry measurements showed that the strength of normalized streamwise vortices and shear-layer thickness under the design condition exceeded those of the off-design condition due to deeper penetration, even though the design condition corresponded to smaller C_m. It may be inferred that the perfectly expanded jet (P_e/P_a = 1.05) may experience a deeper minijet penetration into the main jet, thus weakening the shock cells more substantially, than under the off-design condition (P_e/P_a = 0.70).

Figure 8.

Dependence of the supersonic core length $L_c^\ast $ on mass flow rate ratio ${C_m}$ and correlation between velocity ratio ${U_i}/{U_j}$ and ${C_m}$: (a) ${P_e}/{P_a} = 0.70$, (b) 1.05. The dotted line represents $L_c^\ast $ of natural jet.

4.2. Scaling law of control

As discussed in § 4.1, $L_c^\ast $ depends strongly on C_m, d/D and P_e/P_a, and jet mixing is correlated with the penetration depth of the minijet into the main jet (Perumal & Zhou Reference Perumal and Zhou2018). The ratios U_i/U_j and d/D are related via C_m, where U_j and U_i are calculated from (2.3) and (2.7), respectively (figure 8). Apparently, for a given C_m, U_i/U_j increases with decreasing d/D, which is consistent with our earlier discussion that jet mixing is enhanced as d/D reduces. Further, C_m required to achieve the same U_i/U_j drops given a smaller d/D. Note that either U_i/U_j or C_m may affect directly the penetration depth (Davis Reference Davis1982; Semlitsch et al. Reference Semlitsch, Cuppoletti, Gutmark and Mihăescu2019) that scales with MR_N (Henderson Reference Henderson2010). Khan et al. (Reference Khan, Nageswara Rao, Baghel, Perumal and Kumar2022) and Perumal & Rathakrishnan (Reference Perumal and Rathakrishnan2022) performed empirical scaling analysis based on experimental data of jets of M_d = 1.5 and 2.0, manipulated using N (= 2–6) minijets, and found indeed that jet mixing scales with the square root of MR_N (= C_{m ,N}U_i/U_j), where C_{m ,N} is the mass flow rate ratio of an individual minijet to the main jet. Their scaling law involves only a fixed d/D. Then one question arises: can we find a scaling law that incorporates varying d/D?

Let us consider $L_c^\ast = {f_1}({C_m},d/D,{P_e}/{P_a},{M_j})$. A supersonic jet is characterized by P_e/P_a and M_j (Driftmyer Reference Driftmyer1972). Thus, a developed scaling law with both P_e/P_a and M_j incorporated may be applied for varying M_d (Werle, Shaffer & Driftmyer Reference Werle, Shaffer and Driftmyer1970). To further accommodate different gas species, we may include γ in the scaling law to be developed. Then we have $L_c^\ast = {f_2}({C_m},d/D,{P_e}/{P_a},{M_j},\gamma )$. Tam, Seiner & Yu (Reference Tam, Seiner and Yu1986) introduced a new parameter D_j, i.e. the fully expanded diameter, accounting for the jet contracting and expanding at the nozzle exit due to off-design conditions, and D_j is related to D through mass conservation:

(4.1)\begin{equation}\frac{{{D_j}}}{D} = {\left[ {\frac{{1 + {\textstyle{1 \over 2}}(\gamma - 1)M_j^2}}{{1 + {\textstyle{1 \over 2}}(\gamma - 1)M_d^2}}} \right]^{(\gamma + 1)/4(\gamma - 1)}}{\left( {\frac{{{M_d}}}{{{M_j}}}} \right)^{1/2}}.\end{equation}

Under the design condition, M_j = M_d, implying D_j = D; under the off-design condition, M_j ≠ M_d and D_j ≠ D. Similarly, we may introduce the fully expanded diameter d_i for the minijet:

(4.2)\begin{equation}\frac{{{d_i}}}{d} = {\left[ {\frac{{1 + {\textstyle{1 \over 2}}(\gamma - 1)M_i^2}}{{1 + {\textstyle{1 \over 2}}(\gamma - 1)}}} \right]^{(\gamma + 1)/4(\gamma - 1)}}{\left( {\frac{1}{{{M_i}}}} \right)^{1/2}},\end{equation}

where M_d is 1 since the nozzle of the minijet is a simple constant-diameter tube. As such, we have $L_c^\ast = {f_3}({C_m},{d_i}/{D_j},{P_e}/{P_a},{M_j},\gamma )$, which accounts for the contracting and expanding of both minijet and main jet.

Since C_m is related to d/D, d_i/D_j is implicitly included in C_m. To explicitly retain d_i/D_j, following Muppidi & Mahesh (Reference Muppidi and Mahesh2005) one may rewrite $L_c^\ast = {f_3}({C_m},{d_i}/{D_j},{P_e}/{P_a},{M_j},\gamma )$ as $L_c^\ast = {f_4}(J,{d_i}/{D_j},{P_e}/{P_a},{M_j},\gamma )$, where $J = \sqrt {{\rho _i}U_i^2/({\rho _j}U_j^2)} $ is the momentum flux ratio of the minijet to the main jet, ${\rho _i}$ and ${\rho _j}$ are calculated from (4.3) and (2.4), respectively, and U_i/U_j is implicitly included in C_m:

(4.3)\begin{equation}{\rho _i} = {\left.{\left( {\frac{{\textit{IPR} \times {P_a}}}{{R{T_0}}}} \right)}\right/ {{{\left( {1 + \frac{{\gamma - 1}}{2}M_i^2} \right)}^{1/(\gamma - 1)}}}}.\end{equation}

Following Werle et al. (Reference Werle, Shaffer and Driftmyer1970), we may further simplify $L_c^\ast = {f_4}(J,{d_i}/{D_j},{P_e}/{P_a},{M_j},\gamma )$ as $L_c^\ast = {f_5}(J{d_i}/{D_j},\gamma M_j^2{P_e}/{P_a})$, where $\gamma M_j^2{P_e}/{P_a}$ is a standard dimensionless pressure ratio in compressible flow theory and in the present case is the non-dimensional exit pressure ratio. Those authors and also Driftmyer (Reference Driftmyer1972) developed a scaling law to predict the terminal shock position h with respect to the nozzle exit of a highly under-expanded jet and postulated that h/b_e scales with $\gamma M_j^2{P_e}/{P_a}$, where b_e is the width of the nozzle exit. Then following Perumal & Rathakrishnan (Reference Perumal and Rathakrishnan2022), we may rewrite $L_c^\ast = {f_5}(J{d_i}/{D_j},\gamma M_j^2{P_e}/{P_a})$ as

(4.4)\begin{equation}L_c^\ast \propto \frac{{J{d_i}/{D_j}}}{{\gamma M_j^2{P_e}/{P_a}}}.\end{equation}

As a matter of fact, the data in figure 8 collapse reasonably well provided $(J{d_i}/{D_j})/(\gamma M_j^2{P_e}/{P_a})$ is used as the abscissa (figure 9), which may be least-squares-fitted to

(4.5)\begin{equation}L_c^\ast = 20.9\,{\textrm{e}^{ - 34\xi }} + 5.1.\end{equation}

That is, $L_c^\ast = {f_2}({C_m},d/D,{P_e}/{P_a},{M_j},\gamma )$ may be reduced to $L_c^\ast = {f_6}(\xi )$, where the scaling factor $\xi = (J{d_i}/{D_j})/(\gamma M_j^2{P_e}/{P_a})$. The data above the horizontal dashed line in figure 8 are not included in figure 9 since the scaling law is developed for jet mixing enhancement.

Figure 9.

Dependence of the supersonic core length $L_c^\ast $ on the scaling factor $\xi = \sqrt {{C_m}{U_i}/{U_j}} /(\gamma M_j^2{P_e}/{P_a}) = J({d_i}/{D_j})/(\gamma M_j^2{P_e}/{P_a})$, where the momentum flux ratio $J = \sqrt {{\rho _i}U_i^2/({\rho _j}U_j^2)} $. The solid curve is the least-squares fitting to experimental data and the broken curves indicate the confidence levels of +10 % and −10 %.

Several points can be made from the scaling equation (4.5). Firstly, to our surprise, Perumal & Rathakrishnan's (Reference Perumal and Rathakrishnan2022) data from a jet of M_d = 2.0 manipulated with two oppositely placed minijets also collapse reasonably well about the scaling law (not shown). Since the two investigations differ in M_d as well as N, it seems plausible that the scaling law is valid for varying M_d. Secondly, Perumal & Rathakrishnan (Reference Perumal and Rathakrishnan2022) found that $L_c^\ast $ scales with $\sqrt {{M}{{R}_N}} /(\gamma M_j^2{P_e}/{P_a})$, implying an analogy between $(J{d_i}/{D_j})/(\gamma M_j^2{P_e}/{P_a})$ and $\sqrt {{M}{{R}_N}} /(\gamma M_j^2{P_e}/{P_a})$ due to $J{d_i}/{D_j} = \sqrt {{M}{{R}_N}} $. Apparently, the present scaling law includes one additional parameter d_i/D_j. It may be inferred that $J{d_i}/{D_j}$ is the square root of the momentum ratio of minijet to main jet. Lastly, in order to confirm the robustness of the scaling law, we have performed additional experiments for d/D = 1/4.1 and C_m = 2.88 %–7.91 % (M_j = 1.56, P_e/P_a = 0.70). Figure 10 shows that the data obtained at d/D = 1/4.1 agree reasonably well with the prediction from scaling law (4.5) for the same C_m, d/D and P_e/P_a. Similar observation has also been made for other P_e/P_a. The results suggest the robustness of the scaling law.

Figure 10.

Dependence of predicted supersonic core length $L_c^\ast $ on mass flow rate ratio ${C_m}$ from the scaling law (4.5) (${P_e}/{P_a} = 1.05$, $d/D = 4.1$), as compared with measured $L_c^\ast $ under identical conditions.

4.3. Interesting aspects of the scaling law

An attempt is made to understand the physical meaning of $\xi $. Perumal & Zhou (Reference Perumal and Zhou2018, Reference Perumal and Zhou2021) quantified the penetration depth of the minijet into the main jet based on the root-mean-square values of the streamwise velocity at the nozzle exit and observed a strong correlation between the penetration depth and C_m. This depth has never been quantified in the context of a supersonic jet. As shown in figure 11, the minijet injection induces a bow shock. The bow shock angle β with respect to the streamwise direction and the distance L_p between the nozzle lip (z* = 0.5) and the intersection point of the bow shock and minijet centreline, as defined in figure 11(h), are apparently correlated with C_m. An increase in C_m is associated with an increase in β and also L_p. It seems plausible that either L_p or β may provide a measure for the minijet penetration into the main jet. Figure 12 presents the dependences of $L_c^\ast $ and $\xi $ on $L_p^\mathrm{\ast }$ and $\beta $, respectively, for various C_m, P_e/P_a and d/D. Obviously, $L_c^\mathrm{\ast }$ decreases with increasing $L_p^\mathrm{\ast }$ and $\beta $, corresponding to enhanced jet mixing. Furthermore, $\xi $ is positively correlated with $L_p^\mathrm{\ast }$ and $\beta $. The results suggest that the scaling factor $\xi $ may be physically interpreted as the penetration depth of the minijet into the main jet, which may determine the level of jet mixing measured through $L_c^\ast $.

Figure 11.

(a–g,i–n) Time-averaged schlieren images of manipulated jet at ${P_e}/{P_a} = 0.70$ for various ${C_m}$ along with (e,h) the definitions of the shock wave angle β and the distance ${L_p}$ from the nozzle lip (z* = 0.5) to the point where the induced bow shock and the minijet centreline intersect.

Figure 12.

Dependence of the scaling factor $\xi$ and supersonic core length $L_c^\ast $ on (a) the penetration depth $L_p^\ast $ and (b) shock wave angle $\beta $.

One of the important applications of the scaling law is to predict the optimal d/D at which the required jet mixing is achieved with a minimal consumption of C_m and IPR < NPR. In practice, there are two factors to be considered when we employ minijets for jet manipulation. One is to have an IPR less than the operating NPR. The other is to make C_m as small as possible, achieving the desired performance with the minimum consumption of injection mass flow rate. It is not uncommon to bleed a maximum of 5 % of air flow entering an engine for the purpose of minijet injection as the bleed is associated with an engine thrust loss (Smith, Cain & Chenault Reference Smith, Cain and Chenault2001).

As noted earlier, jet mixing benefits from small d/D, at which the required C_m is small in achieving a given $L_c^\ast $ as long as d/D > 1/20. For example, to achieve a predefined $L_c^\ast = 8.0$ at P_e/P_a = 0.70 (NPR = 4), the required C_m and IPR for d/D = 1/9.5 are 3 % (figure 8) and 7.5 (table 2), respectively. In this case, IPR > NPR. When d/D is increased to 1/5.3, the IPR drops to 3.25 (table 2) so that IPR < NPR, whereas the required C_m increases to 4 % (figure 8). The result suggests the presence of an optimal d/D at which IPR < NPR and C_m ≤ 5 %. The optimal d/D may be predicted based on the scaling law for a given M_j. Assume IPR = NPR and the predefined $\Delta L_c^\ast = 15\,\%$ and 25 %, where $\Delta L_c^\ast $ is the desired jet mixing enhancement defined by

(4.6)\begin{equation}\Delta L_c^\ast = \frac{{{{(L_c^\ast )}_{natural}} - {{(L_c^\ast )}_{manipulated}}}}{{{{(L_c^\ast )}_{natural}}}} \times 100.\end{equation}

In (4.6), subscripts ‘manipulated’ and ‘natural’ denote the manipulated and natural jets, respectively. To predict the optimal d/D, we first determine the required ${(L_c^\ast )_{manipulated}}$ for given $\Delta L_c^\ast $ and M_j from (3.2) and (4.6) and then calculate d/D from (4.1)–(4.5). The obtained optimal d/D is presented in figure 13(a), from which several observations can be made. The optimal d/D decreases with increasing M_j, which is not feasible for practical applications. To avoid this, we may choose the optimal d/D based on the minimum operating M_j. If the choice of the optimal d/D is based on the maximum operating M_j, then the required IPR at the other operating M_j would exceed NPR. For example, if the optimal d/D = 0.08 is chosen based on M_j = 1.83, then for M_j = 1.69 (NPR = 4.9) the required IPR is about 7 (figure 13a), greater than the operating NPR = 4.9. On the other hand, the required IPR at the other operating M_j will be less than NPR provided the optimal d/D is determined from the minimum operating M_j.

Figure 13.

(a) Dependence of $\textit{IPR}$ on diameter ratio $d/D$ and fully expanded jet Mach number ${M_j}$ ($\Delta L_c^\ast = 15\,\%$), where the square symbols correspond to $d/D$ at which $\textit{IPR} = \textit{NPR}$ ($\textit{IPR} < \textit{NPR}$ and $\textit{IPR} > \textit{NPR}$ occur above and below the symbols, respectively). (b) Dependence of ${C_m}$ and $d/D$ on ${M_j}$.

With the optimal d/D known, we may calculate the required C_m from (2.5)–(2.6). As shown in figure 13(b), the required C_m drops with increasing M_j and reaches the minimum at the design Mach number, suggesting an efficient jet manipulation, which is internally consistent with the observation from figure 8. Obviously, the choice of the optimal d/D depends on the minimum operating M_j as well as available C_m. One may also wonder how the optimal d/D would vary if $\Delta L_c^\ast = 15\,\%$ increases to 25 %. As shown in figure 13(b), the optimal d/D becomes in general larger at $\mathrm{\Delta }L_c^\ast = 25\,\%$ than at 15 %. For example, at M_j = 1.56, the optimal d/D is 0.11 for $\mathrm{\Delta }L_c^\ast = 15\,\%$ and 0.13 for $\mathrm{\Delta }L_c^\ast = 25\,\%$. Further, the optimal d/D may increase with increasing $\mathrm{\Delta }L_c^\ast $, as is evident in figure 14 (M_j = 1.56). The required C_m exceeds 5 % once $\mathrm{\Delta }L_c^\ast > 40\,\%$. Therefore, it can be inferred from the above discussion that the optimal d/D is a trade-off between $\mathrm{\Delta }L_c^\ast $, the minimum operating M_j or NPR and available C_m.

Figure 14.

Dependence of the optimal $d/D$ and ${C_m}$ on $\Delta L_c^\ast $ at ${M_j} = 1.56$. The curve is the least-squares fitting to the optimal diameter ratios.