1 Introduction

Wall interactions of supersonic jets are commonly found in many aerospace applications like jet noise reduction methods, wall cooling of aerospace structures, control of jet mixing, and design of embedded propulsion systems. The phenomenon of wall interaction involving a supersonic jet flow field is also an interesting field of study in subject areas such as thrust vector control of high-speed jet aircraft and shock–boundary-layer interactions. In many such interactions, it can be observed that the supersonic core of the jet is affected by the presence of a solid wall through a boundary layer. Wall surface modifications also alter the structure of the interacting supersonic flow significantly. Wall interaction by non-circular geometries of supersonic jets is of great importance in aerospace applications, especially in the design of supersonic and hypersonic propulsion systems. The mixing and noise characteristics of non-circular supersonic jets are different from their circular counterparts in that the growth of the shear layer on the jet boundary varies azimuthally. As such, the wall interaction features involving the non-circular jets will also be different from those with circular jets. To understand the wall interaction effects, the study of the flow features of isolated jets is essential. One can observe from the literature that investigations were carried on isolated non-circular jets and some of these included axis switching phenomenon of low aspect ratio rectangular jets [1], scaling laws in elliptic jets [2], screech characteristics of high aspect ratio elliptic jets [3], the decay characteristics of rectangular and elliptic jets [4], flow characteristics of underexpanded non-circular jets [5], and also centreline mixing characteristics of jets from nine differently shaped nozzles [6]. The effect of co-flow on the axial flow development and mixing aspects of a triangular supersonic jet was also studied in the recent past [7]. Geometrical modifications at the exit of rectangular jets such as employing tabs [8] and diagonal expansion ramps [9] on jet development and decay characteristics were also studied in the past. The influence of jet geometry on the performance of aerospace vehicle components such as ejectors was investigated by Zare-Behtash et al. [10]. Supersonic jets emerging from both circular and non-circular ducts and interacting with after-duct wall surfaces are common in high-speed aerospace vehicle designs. These jet configurations, partially constrained by a solid surface have an asymmetry that affects several physical phenomena, such as compressible waves, shear layers, boundary layers, and shock–boundary-layer interactions. Instabilities, acoustic feedback loops, and other unsteady factors can influence these phenomena. Ibrahim et al. [11] studied screech characteristics of jet interaction with a flat plate. Roben Hortensius et al. [12] undertook a study in which an underexpanded axisymmetric jet issuing from 12.7-mm-diameter convergent nozzle at NPR = 5 interacted with a flat surface that was kept at three different separation distances from the jet. The leading edge of the surface was aligned with the exit plane of the nozzle. They observed that the jet flow field was dominated by shock–boundary-layer interaction due to the presence of the surface. This interaction altered the structure of the jet and they further noticed that the magnitude of the separation distance determined the strength and location of the shock–boundary-layer interaction. The interaction effects between jet blast deflectors and supersonic jets were investigated by Liu et al. [13], while the noise from aft-deck nozzles was studied by Bridges [14]. The important aspect such as the resonant interaction in the acoustic characteristics of rectangular supersonic jet interaction with a flat wall was investigated by Zaman et al. [15]. Pablo Mora et al. also investigated the acoustic characteristics of rectangular supersonic jet over a flat surface [16]. James et al. [17] studied plume development in the near field over a flat surface like aft-deck attached to lower wall of a convergent nozzle with rectangular exit of aspect ratio 7.35. The investigation was carried out in the range of NPRs from 1.5 to 6 using schlieren flow visualization, pitot probe, and laser Doppler anemometry measurements. It was observed from their studies that the aft-deck which was 2.37Dh long created asymmetry in the entrainment characteristics within the shear layers and also produced a net transverse force on the plume by changing the inviscid shock structure. The study of rectangular supersonic jets interacting with a flat plate also attracted the attention of other investigators in the subsequent years. Romain Gojon et al. [18] studied the response of a rectangular supersonic jet when a flat plate was located at the jet near field. Florian Baier et al. [19] undertook the flow measurements at high temperatures when a supersonic flow exhausted over a flat surface. Vaisakh and Muruganandam [20] studied the wall effect on an overexpanded rectangular supersonic jet of Mach 1.7. The aspect ratio of the rectangular exit of the nozzle was 1.2 and the wall width was more than the width of the exit. Experiments were conducted at one wall length and they found that the wall had no effect on the position of the first triple point but the height of the Mach stem of the primary shock increased along the wall direction. They observed both shock–boundary layer and shock–shear layer interactions due to the wall effect. Manikanta et al. [21] studied the effects of wall length on a square wall–jet’s shock structure and decay characteristics. The situations where supersonic jets flowing past a wall are partially constrained on the wall-bound side and free to expand on the other sides can also be found in hypersonic vehicle systems. Notable among them would be the supersonic flow over a single expansion ramp nozzle (SERN). In SERN flows, the expansion ramp is a constraint for the expanding supersonic flow over the ramp wall. The flow on the other sides is unconstrained (barring side fences if present). The expanding flow on the ramp not only influences the performance of the scramjet, but the wall pressure impressed on the expansion ramp also contributes to the aerothermal loads on the vehicle. Mousavi et al. [22] numerically investigated the overexpansion behavior of the flow in SERN. Biju and Sridhar [23, 24] studied the SERN flow interacting with a rectangular strut on the expansion ramp with application to thrust vector control of hypersonic vehicles. Experiments were conducted with pressure ratios ranging from 4 to 10, and it was observed that the changes in strut position and strut height resulted in directional changes in the exhaust plume over the single expansion ramp. An increase in strut height produced an increase in side force which was more effective at lower pressure ratios.

Given the importance of the presence of such jet flow fields in embedded propulsion systems, on aft-deck geometries and on expansion ramps in hypersonic vehicles, it is imperative that a detailed investigation of these interaction effects would be useful. Studying the effect of variation of wall length on jet–wall interaction would be especially helpful from the vehicle/propulsion system design point of view. Most interaction studies in the existing literature mainly focus on acoustic experiments. Research investigations elucidating the fluid dynamic details, especially on interacting supersonic jets with surfaces such as flat walls, are very much limited. Further in the existing literature, perhaps, there are no parameters based on the fluid dynamic data to characterize the jet–wall interactions except in ref-21 for an interacting square supersonic jet case. In the present paper, an experimental fluid dynamic study was undertaken, and an eclectic description of fluid dynamic effects involved in the jet–wall interactions is presented. The consequences of such interactions on the jet flow field and flat wall were quantified and presented in terms of jet deflection and fluid dynamic coefficients. Moreover, the role of wall length on the above phenomena was also explored. It is detailed in the literature that jet–wall interaction is influenced by key parameters like nozzle-exit Reynolds number, nozzle pressure ratio, nozzle divergent angle, compression effects, nozzle-exit shape and wall dimensions, and its orientation at the nozzle exit. Hence, the interaction is influenced by a broad range of parameters. In the present paper, the interaction effects arising from placing a flat wall at the rectangular exit of a converging–diverging nozzle (Fig. 1) were studied by varying the length of the flat wall (\({L}_{{\text{w}}}\)) and the nozzle pressure ratio (NPR) while keeping the other parameters constant. The variation in NPR corresponded to underexpansion (NPR = 8), near optimum expansion (NPR = 6), and overexpansion (NPR = 4) conditions. \({L}_{{\text{w}}}\) was varied from 0.5 \({D}_{{\text{h}}}\) to 8 \({D}_{{\text{h}}}\) where \({D}_{{\text{h}}}\) was the hydraulic diameter of the rectangular exit area of the nozzle. The flow visualization technique, i.e., the schlieren method, was used to investigate the shock structure and jet deflection over the flat wall. Wall pressure (\({p}_{{\text{w}}}\)) measurements were made along the mid-width line of the flat wall. From the \({p}_{{\text{w}}}\) data, two-dimensional normal force and moment coefficients were calculated at various wall lengths. The wall–jet interactions were interpreted in terms of (i) jet shock structure, (ii) jet deflection, (iii) wall pressure distributions, and (iv) force and moment coefficients of the wall.

Fig. 1
figure 1

Three-dimensional flow model of supersonic flow over a flat wall

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2 Experimentation

In the following paragraphs, the details of experimentation have been described. Section 2.1 describes the high-speed jet facility, and Sect. 2.2 gives details of geometrical dimensions, methods, and fabrication. \({p}_{{\text{w}}}\) measurements and schlieren flow visualizations systems have been narrated in Sect. 2.3.

2.1 High-speed jet test facility

The schematic diagram of the experimental facility and its components are shown in Fig. 2. High-pressure air was supplied from a storage tank with a volume of \({1{\text{m}}}^{3}\) capacity, in which the air was stored at a pressure of 2 MPa. The high-pressure air to the tank was delivered by a two-cylinder reciprocating compressor with 35 HP rating with a delivery pressure of 2 MPa. An air dryer between the compressor and reservoir tank ensured a major percentage of moisture removal in the air and storage of dry air in the tank. This tank was connected to a settling chamber via a pipeline. A very high-precision pressure regulator and a ball valve were used to control the pressure inside the settling chamber. Between the pressure regulator and high-pressure air reservoir tank, a moisture remover was provided to remove the moisture in the high-pressure air. The settling chamber was of circular cross-section which was connected to high-pressure line from the ball valve through a diffuser. The required value of settling chamber pressure could be set using the pressure regulator. The blowdown at constant settling chamber pressure was carried out using the quick opening ball valve.

Fig. 2
figure 2

Schematic layout of the supersonic jet facility used for experiments

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2.2 Details of experimental model

The experimental model used in the present study was a convergent–divergent(C-D) nozzle of rectangular diverging cross-section with aspect ratio (AR) = 2. The cross-section of the nozzle transforms from the round inlet of 34-mm internal diameter (attached to the settling chamber of the jet test facility shown in Fig. 2) to rectangular at the throat in the converging section. The cross-sectional dimensions of the rectangular nozzle throat are 17.70 mm for the major axis and 8.85 mm for the minor axis. In the diverging segment of the nozzle, the cross-section remains rectangular with increasing area with major and minor axis dimensions of 21.2 mm × 10.6 mm (AR = 2) at the exit, as shown in Fig. 3a. The design exit Mach number \(\left({M}_{{\text{e}}}\right)\) was 1.8 and area ratio was 1.44. The nozzle-exit hydraulic diameter \(\left({D}_{{\text{h}}}\right)\) was 14.13 mm, and this was used as a reference dimension to make all length variables non-dimensional. The flat wall had a constant thickness of 6 mm. The length of the flat wall was varied as \({0.5D}_{{\text{h}}},1{D}_{{\text{h}}},{2D}_{{\text{h}}},{4D}_{{\text{h}}} {\text{and}} {8D}_{{\text{h}}}\) in the experiments to assess the wall length (\({L}_{{\text{w}}}\)) effect. The nozzle and flat walls were fabricated using an electrical discharge machining procedure, and the material used was stainless steel SS304.

Fig. 3
figure 3

Details of laboratory model hardware for experiments

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2.3 Wall pressure measurement and flow visualization

A total of 15 wall pressure ports (P1–P15) were employed in the \({8D}_{{\text{h}}}\) long flat wall, as shown in Fig. 3b and c. The wall pressure along the mid-width line was acquired using a 16-channel intelligent pressure scanner model-9116 supplied by Measurement Specialties, USA. The measurement repeatability was evaluated by repeating the measurement several times and estimating the difference in terms of variance. The pressure scanner used in this study has a good accuracy of \(\pm\) 0.05% of full scale pressure (1700 kPa). The wall pressures were measured at various locations along the flat wall mid-width line to study the jet flow interaction with the flat wall. Flow visualization of the supersonic wall–jet was carried out by a twin mirror Z-type schlieren system (Fig. 4), which was utilized to qualitatively observe the jet flow field for various flat wall lengths. The schlieren setup included a 150W halogen light source, a beam cutter/expander, and a knife-edge in combination with an arrangement of two parabolic mirrors \(\left(\mathrm{\varnothing } 150 {\text{mm}}\right)\). The images on the screen were captured using a digital camera.

Fig. 4
figure 4

A schematic arrangement of the Z-type schlieren setup used in experiments

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3 Results and discussion

3.1 Effect of wall length on shock structure

All the experiments were conducted at three expansion conditions of the jet. These expansion conditions corresponded to NPR = 4 (overexpansion), NPR = 6 (near optimum expansion), and NPR = 8 (underexpansion). The NPR for correct/optimum expansion for the rectangular nozzle was 5.77. Before the discussion on the changes in shock structure induced by wall effect, it is inquisitive to dwell on the flow physics behind it. Figure 5 illustrates the typical features in the jet flow field induced by a wall on an overexpanding supersonic jet. Figure 5a shows the characteristic features of shock structure in an overexpanded isolated/free supersonic jet exiting from a convergent–divergent nozzle. The oblique shocks at the nozzle exit are followed by expansion waves which get reflected from the free jet boundary as compression waves that may coalesce to form shock waves. The pattern of alternate shock and expansion wave structure is repeated till the wave system is reduced to Mach waves in the terminal region of supersonic core of the jet flow field. The shear layer at the jet boundaries is also shown in the figure. The wave structure is symmetric about the jet axis that coincides with nozzle axis. However, when an external flat wall is attached parallel to the nozzle lower wall at the exit, the symmetry of the wave structure is altered. Figure 5b shows that the wall–jet now has two different conditions at the jet boundaries. The jet is constrained at the wall side but free to expand on the counter wall side. The differing impact of shear layers, i.e., the wall-bounded shear layer (boundary layer on the wall) and the free shear layer (at the jet boundary) determines the pattern of shock structure in the wall–jet. This is because with the incident shock waves, the wall always reflects shock waves and the free shear layer at the counter wall side jet boundary reflects expansion waves. Further, with the incident expansion waves, the free shear layer reflects compression waves and the wall reflects expansion waves. In addition, the jet flow at the wall is prone to separation giving rise to additional complexities like separation shocks, expansion waves near the separation bubble, and reattachment shocks. The wave structure with the interference of shock–shock and shock–expansion waves becomes more complex and asymmetric about the extended nozzle axis in the flow field. Some of these features can be observed in the schlieren images obtained in the present experiments and the wave structure in the interacting jet flow field varies depending upon the expansion conditions. The diameter of the concave mirrors used in the schlieren imaging facility was 150 mm and hence all observations pertaining to visualization of shock–expansion structure was limited to 150 mm of image length. At NPRs 6 and 8, it was observed that the shock–expansion train persisted even beyond 150 mm, making the observations on the full length of the supersonic region difficult. However, at NPR = 4 (overexpansion), observations on the extent of supersonic core region and on wave structure were possible.

Fig. 5
figure 5

Effect of an adjacent flat wall on flow features of a supersonic jet. a Isolated overexpanded supersonic jet, and b supersonic jet with an adjacent flat wall

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3.1.1 Effect of flat wall length at NPR = 4

Figure 6a shows the characteristic features of shock structure in an overexpanded free supersonic jet with oblique shocks at the exit followed by expansion waves. The wave pattern was symmetric about the jet axis. However, when a short wall was introduced at the exit of the nozzle (Fig. 6b), it can be seen that the jet deflected downward and there was a change in shock structure caused by the presence of the short wall (\({L}_{{\text{w}}}={0.5D}_{{\text{h}}}\)). From Fig. 6a and b, it can be observed that there was an asymmetry in the shock pattern at the nozzle exit due to the wall effect and the shock angles from the free jet boundary side and the wall side were not the same. It appears from the schlieren image that the wall side shock angle is smaller than that on free jet boundary side in Fig. 6b. When the flat wall was introduced, a low-pressure region on the wall near the wall's leading edge was observed which indicated that the wall side shock could be a separation shock. This can also be verified from the wall pressure distribution plot in Fig. 10 which shows low-pressure region near the wall leading edge. The oblique shock at the nozzle exit on free jet boundary side reached the jet boundary on the wall side with shock–shear-layer interaction (SSLI) and was reflected as expansion waves which can be seen in Fig. 6b. The separation shock from the wall side reached the shear layer at the free jet boundary and got reflected as expansion waves. Except the separation shock- and boundary-layer interaction at the wall no other SBLI was observed on the wall side due to the small length. Low-pressure region was pervasive all over the wall and the jet deflection (2.6°) was downward due to the wall effect. When the wall length was increased further, i.e., \({L}_{{\text{w}}}={1D}_{{\text{h}}}\), the supersonic jet deflection was somewhat reduced (Fig. 6c). Close to the wall trailing edge (WTE), a second shock, i.e., a reflection shock by the incident shock from the free jet boundary can also be noticed as the wall length was large enough for the impingement. A steep rise in wall pressure is testimony for this which can be observed in Fig. 10 for \({L}_{{\text{w}}}={1D}_{{\text{h}}}\). This results in shock–boundary-layer interaction (SBLI) on the wall. The first shock at the nozzle exit and the reflected shock from the wall side reached the free jet boundary giving rise to SSLI. This interaction reflects expansion waves from the jet boundary. The downward deflection of the jet at shorter wall lengths is of practical importance. In case of aerospace vehicles employing a propulsion nozzle with rectangular exit, a modification at the exit like employing an aft-deck would be potentially advantageous. Because, the nozzle at sea-level operates at overexpansion conditions, as the nozzle design NPR is optimum at high altitudes. A downward deflection of the jet at sea-level take-off would assist the vehicle by providing additional upward normal force to the vehicle airframe. At larger wall lengths (\({L}_{{\text{w}}}={2D}_{{\text{h}}}\), Fig. 6d), one could observe an upward jet deflection which was in contrast to the downward deflection in the case of smaller wall lengths, i.e., \({L}_{{\text{w}}}={0.5D}_{{\text{h}}}\) and \({1D}_{{\text{h}}}\). A further increment in wall length, i.e., \({L}_{{\text{w}}}={4D}_{{\text{h}}}\) also had the same effect (Fig. 6e). More reflected shocks in the flow were noticed and the length of the shock cells also increased due to the wall effect when compared to the shock structure in free jet. The shock–expansion cells generated a fluctuating pressure distribution on the flat wall. From Fig. 6d and e, an oblique shock at the WTE can be noticed. At large wall length like \({L}_{{\text{w}}}={8D}_{{\text{h}}}\), the deflection of the jet was substantially reduced (Fig. 6f), as the jet flow field was bound to a larger wall length. It can also be noticed from the schlieren images that changes in wall length beyond \({L}_{{\text{w}}}{=2D}_{{\text{h}}}\) had caused little change in the shock structure in the upstream regions. For example, when \({L}_{{\text{w}}}\) was increased from \({2D}_{{\text{h}}}\) to \({4D}_{{\text{h}}}\), the shock structure that spanned a length of \({2D}_{{\text{h}}}\) from the nozzle exit did not suffer any significant change. The positions of first and second shocks on the wall and the expansion fans from the jet boundary remained more or less the same.

Fig. 6
figure 6

Schlieren images of shock structures in the rectangular jet flow fields over flat walls of various lengths at NPR = 4: a free rectangular jet, b \({L}_{{\text{w}}}={0.5D}_{{\text{h}}}\), c \({L}_{{\text{w}}}={1D}_{{\text{h}}}\), d \({L}_{{\text{w}}}={2D}_{{\text{h}}}\), e \({L}_{{\text{w}}}={4D}_{{\text{h}}}\), and f \({L}_{{\text{w}}}={8D}_{{\text{h}}}\). (1) Expansion wave, (2) oblique shock, (3) compression waves, (4) shear layer at jet boundary edge, (5) separation point, (6) separation shock, (7) jet trajectory downward, and (8) jet trajectory upward

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3.1.2 Effect of flat wall length at NPR = 6

The effect of \({L}_{w}\) on the shock structure in the jet flow field at NPR = 6 is shown in Fig. 7. The nozzle expansion conditions corresponded to the near optimum. Figure 7a shows the shock structure in the jet flow field when the wall was absent. The jet flow field was characterized by expansion waves at the nozzle exit, followed by shock waves. The wave structure in the supersonic core of the jet was longer than the one in the jet at NPR = 4 (overexpansion). Unlike two oblique shocks as in the case of overexpansion, here two expansion waves can be seen at the nozzle exit. As a result, the wall effect at NPR = 6 is different from the one at NPR = 4. The wall effect introduced a greater asymmetry about the extended nozzle axis. This was because at the wall side, a shock induced by the wall was produced, whereas at the counter wall side, the expansion wave remained at the nozzle exit. When a short wall, i.e., with a wall length \({L}_{{\text{w}}}={0.5D}_{{\text{h}}}\) was introduced at the nozzle exit was introduced (Fig. 7b), unlike in overexpansion case, an upward deflection (3.4°) of the jet was observed. From the schlieren images, it can also be noticed that the wall-induced oblique shock reached the free jet boundary resulting in SSLI after which the reflection from the boundary was an expansion wave. As the length of the wall was not large enough, the wall did not receive this wave. Even the expansion wave at the nozzle exit did not impinge on the wall as the wave angle was very oblique. As a result, unlike in overexpansion case, the region close to the wall leading edge felt positive wall pressure (above ambient pressure, see Fig. 11). When the length of the wall was increased to \({L}_{{\text{w}}}={D}_{{\text{h}}},\) a somewhat reduced upward deflection (3.1°) of the jet was noticed (Fig. 7c). Strong interference between expansion waves and shock waves generated by SSLI at the jet boundaries can be noticed in the jet flow field immediately after the WTE. Similar interference could also be noticed for the previous case, i.e., \({L}_{{\text{w}}}={0.5D}_{{\text{h}}}\), however, for this larger length, the wave pattern during the interference was more symmetric. As the \({L}_{{\text{w}}}\) was increased to \({2D}_{{\text{h}}}\), the jet deflection (1.2°) switched to downward direction (Fig. 7d). It can be noticed from the schlieren image that two expansion waves from the jet boundary on the counter wall side reach the wall and in the rear part of the wall a negative pressure region exists. This can also be verified from Fig. 11. When the \({L}_{{\text{w}}}\) was extended further to \({4D}_{{\text{h}}}\) (see Fig. 7e), the jet deflection again exhibited an upward deflection (3°). It appears that the expansion waves from the wall reached the jet boundary and again got reflected as shock waves. Now that the length of the wall was large enough, these waves could impinge the wall rising the wall static pressure. The rear part of the wall as a consequence experienced positive pressures (see Fig. 11) encouraging an upward deflection of the jet. When a large wall length like \({L}_{{\text{w}}}={8D}_{{\text{h}}}\) (Fig. 7f) the jet flow aligned with the wall and had negligible deflection. However, the length of the shock train in the jet flow field was longer when compared to the cases \({L}_{{\text{w}}}\) \(<{8D}_{{\text{h}}}\). It can also be observed from the schlieren images that at this expansion level also, the increase in wall length did not fundamentally alter the wave structure in the upstream. As mentioned earlier, an increase in wall length, for example, from \({2D}_{{\text{h}}}\) to \({4D}_{{\text{h}}}\) hardly had an effect on the shock pattern in the supersonic core length of \({2D}_{{\text{h}}}\) from the nozzle exit.

Fig. 7
figure 7

Schlieren images of shock structures in the jet flow fields over flat walls of various lengths at NPR = 6: a free rectangular jet, b \({L}_{{\text{w}}}={0.5D}_{{\text{h}}}\), c \({L}_{{\text{w}}}={1D}_{{\text{h}}}\), d \({L}_{w}={2D}_{{\text{h}}}\), e \({L}_{{\text{w}}}={4D}_{{\text{h}}}\), and f \({L}_{{\text{w}}}={8D}_{{\text{h}}}\). (1) Expansion wave, (2) oblique shock, (4) shear layer at jet boundary edge, (7) jet trajectory downward, (8) jet trajectory upward, and (9) shock at WTE

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3.1.3 Effect of flat wall length at NPR = 8

The shock structure in the jet flow field at NPR = 8 (underexpansion condition) at various wall lengths is shown in Fig. 8. The shock structure in the underexpanded jet in the absence of a wall is shown in Fig. 8a. The figure shows the characteristic features of a typical underexpanded supersonic jet with expansion waves at the nozzle exit followed by shock waves after the reflection of expansion waves from the free jet boundary. When a short wall (\({L}_{{\text{w}}}={0.5D}_{{\text{h}}}\)) was introduced at the lip of the nozzle (Fig. 8b), the wave pattern in the jet flow field underwent a significant change and the jet deflected upward (4.6°). However, as discussed earlier, when the flat wall was introduced at the nozzle, a wall-induced oblique shock was observed at the leading edge of the wall. The asymmetric shock pattern can be noticed from the schlieren image Fig. 8b. The extension of wall length \(L_{{\text{w}}} {-}D_{{\text{h}}}\) made the jet deflect further upward (7.1°) which can be seen in Fig. 8c. When the wall length was increased further, (\({L}_{{\text{w}}}={2D}_{{\text{h}}})\) the jet deflection decreased (4.3°) but still maintained an upward direction (Fig. 8d). A strong oblique shock can be seen on the wall near the WTE, and from the wall pressure plot (Fig. 12), it can be deciphered that a negative pressure region exists behind this shock. Strong interference between oblique shock and expansion waves can be seen above the wall. The wall length was large enough to receive the expansion waves, and this must have caused a significant drop in wall pressure behind the oblique shock. However, the upward jet deflection (3.8°) persisted even with the further extension of wall length to \({L}_{{\text{w}}}={4D}_{{\text{h}}}\) (Fig. 8e). It can be observed from all schlieren images that at underexpansion condition (NPR = 8) the jet always deflected upward, irrespective of the wall length \({\text{till}} {L}_{{\text{w}}}={4D}_{{\text{h}}}\). This was not the case with other expansion levels, where \({L}_{{\text{w}}}\) had a role in the direction of jet deflection. Only at large wall lengths like \({L}_{{\text{w}}}={8D}_{{\text{h}}}\) (Fig. 7f) was the jet aligned more or less to the wall surface and exhibited very little deflection beyond the WTE as in the previous expansion cases, i.e., NPR = 4 and 6.

Fig. 8
figure 8

Schlieren images of shock structures in the jet flow fields over flat walls of various lengths at NPR = 8: a free rectangular jet, b \({L}_{{\text{w}}}={0.5D}_{{\text{h}}}\), c \({L}_{{\text{w}}}={1D}_{{\text{h}}}\), d \({L}_{{\text{w}}}={2D}_{{\text{h}}}\), e \({L}_{{\text{w}}}={4D}_{{\text{h}}}\), and f \({L}_{{\text{w}}}={8D}_{{\text{h}}}\). (1) Expansion wave, (2) oblique shock, (3) compression waves, (4) shear layer at jet boundary edge, and (8) jet trajectory upward

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Before concluding this section, it is pointed out that shock–expansion wave structure in the jet flow field above the wall leaves its footprint as the pressure distribution on the wall. The jet deflection either upward or downward is governed by the direction of net force on the wall because of the \({p}_{{\text{w}}}\) distribution. It was observed in this investigation that if this net force was downward, the jet suffered an upward deflection and if this force was upward, then the jet made a downward deflection. This net force per unit span was determined by integrating the \({p}_{{\text{w}}}\) distribution (both top and bottom surfaces of the wall, of course the bottom surface pressure was nothing but constant ambient pressure) along the mid-width line of the wall from wall leading edge to trailing edge. Alternatively, it can also be interpreted that the net downward force on the wall is a manifestation of the vertical component of the reactive force acting opposite to the direction of the upwardly deflected jet and so is the net upward force on the wall for the downwardly deflected jet.

3.2 Effect of wall length on jet deflection

Figure 9 depicts the measured jet deflection angles (\(\theta\)) from schlieren images for various wall lengths (\({L}_{{\text{w}}}\)) at different expansion conditions of the jet flow. The variation in wall length on the x-axis is shown from \({L}_{{\text{w}}}={0.5D}_{{\text{h}}}\) to \({4D}_{{\text{h}}}.\) An upward deflection of the jet trajectory indicates a positive \(\theta\) value, and a downward deflection indicates a negative \(\theta\). The deflection angles in the schlieren images were measured using the open source ImageJ software. The uncertainty depended upon the measurement of dimensions of nozzle block thickness as reference dimension. The least count of the instrument used for measuring this thickness was 0.01 mm. The scale was 24.9327 pixels/mm while processing the schlieren image using the software. The error in the scale was 0.2493 pixels. The identification of jet boundary was done by importing the image to the software to get to know the jet boundary edges. Identification of jet boundary was also due partly to eyeballing. The maximum error involved in the measurement of \(\theta\) due to uncertainty in pixels per mm was ± 0.13°. As the size of the error bar was almost equal to the size of the symbols, use of error bas was avoided in Fig. 9. The figure shows that the jet deflected upward at all wall lengths in underexpansion conditions (NPR = 8). The maximum deflection (7\(.1^\circ\)) occurred at \({L}_{{\text{w}}}={D}_{{\text{h}}}\). In the case of near optimum expansion conditions (NPR = 6), the wall effect caused upward jet deflection at all wall lengths except at \({L}_{{\text{w}}}={2D}_{{\text{h}}}\) where the deflection was downward. Overexpanded jet (NPR = 4) experienced downward deflection due to the wall effect till \({L}_{{\text{w}}}<{2D}_{{\text{h}}}\). However, larger wall lengths, i.e., \({L}_{{\text{w}}}\ge {2D}_{{\text{h}}},\) caused the overexpanded jet to deflect upward. It can be observed from the figure that irrespective of expansion conditions, larger wall lengths \(({L}_{{\text{w}}}={4D}_{{\text{h}}})\) always produced upward jet deflection. The physics behind the direction of jet deflection is explained in Sect. 3.3 using wall pressure data.

Fig. 9
figure 9

Effect of wall length on jet deflection angle at different expansion conditions of the jet flow

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3.3 Wall pressure distribution for different wall lengths

Since the jet–wall interaction leaves its footprint on the wall surface as pressure distribution, this distribution has been analyzed in this section (see Figs. 10, 11, and 12) to interpret the interaction effects. Wall pressure distribution on the flat wall indicated alternate regions of positive and negative pressure (relative to laboratory conditions), revealing compression and expansion domains prevalent on the wall surface. These domains were the reflection of the wave structure above the wall which consisted of alternating shock and expansion waves. The shock–expansion train was enveloped between the wall boundary layer and the free-boundary shear layer of the supersonic jet grazing past the wall surface. The longitudinal \({p}_{{\text{w}}}\) distributions were obtained at various wall lengths along the mid-width line of the wall. The maximum uncertainty expected in the measurements in \({p}_{{\text{w}}}\) was ± 0.05% of 700 kPa (the maximum gauge pressure in the experiments) which was about ± 0.35 kPa. Representing this uncertainty by error bars with respect to y-axis was difficult. The error bar height would have been less than the symbol size in Figs. 10, 11, and 12. For example, a value of 0.50 on y-axis would have uncertainty ranging from 0.4965 to 0.5035.

Fig. 10
figure 10

Experimental wall pressure distributions on flat walls of various lengths at NPR = 4 (overexpansion)

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Fig. 11
figure 11

Experimental wall pressure distributions on flat walls of various lengths at NPR = 6 (Near optimum expansion)

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Fig. 12
figure 12

Experimental wall pressure distributions on flat walls of various lengths at NPR = 8 (Underexpansion condition)

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The mid-width line \({p}_{w}\) distributions for various wall lengths at NPR = 4 have been shown in Fig. 10. It can be seen from the figure that the mid-width line \({p}_{{\text{w}}}\) distribution was wavy along the wall length, consisting of peaks and troughs. The variation in wall length did not affect this nature. The peak pressures were above atmospheric, and the pressures in the troughs were sub-atmospheric. Further, it can also be noticed that the increase in wall length along the downstream affected the upstream \({p}_{{\text{w}}}\) very little. It shows the insensitivity of the upstream \({p}_{{\text{w}}}\) distribution to the increase in wall length downstream. For example, when the wall length \({L}_{{\text{w}}}\) was increased from \({2D}_{{\text{h}}}\) to \({4D}_{{\text{h}}}\), the \({p}_{{\text{w}}}\) distribution over the upstream length of \({2D}_{{\text{h}}}\) was relatively unaffected by the increase in wall length to \({4D}_{{\text{h}}}\). As NPR = 4 corresponded to overexpansion condition, the wall surface very close to the nozzle-exit plane experienced sub-atmospheric pressure for all wall lengths. Further, it can also be observed from the figure that the magnitude of peak pressure decreased along the wall length indicating a decaying strength of the shock structure, which can also be observed from the schlieren images in Fig. 6.

At near optimum conditions, i.e., NPR = 6 (as per isentropic theory the correct NPR is 5.77), the \({p}_{{\text{w}}}\) distribution on the mid-width line of the wall impressed by the wave structure in the jet flow field (Fig. 11) was somewhat different from the one at NPR = 4 (overexpansion). From Fig. 11, it can be observed that there was a continuous decline in \({p}_{{\text{w}}}\) from the wall leading edge. For \({L}_{{\text{w}}}>{D}_{{\text{h}}}\) the wall surface also experienced negative pressures relative to the ambient conditions. The figure also shows the insensitivity of the upstream \({p}_{{\text{w}}}\) distribution to the increasing wall length, as in the case of NPR = 4. However, the magnitudes of peak pressures were somewhat lower at NPR = 6 when compared to those at NPR = 4. The magnitudes of both positive pressures at peaks and the negative pressure at troughs declined and approached ambient value as the flow proceeded down to the wall trailing edge. It can also be observed that the location of the first minimum wall pressure shifted away from wall leading edge from 0.5Dh to 1.8Dh for Lw \(\ge 2{D}_{{\text{h}}}\)

Figure 12 presents the \({p}_{{\text{w}}}\) distributions on the mid-width line at NPR = 8. There was a smooth drop of \({p}_{{\text{w}}}\) along the mid-width line up to L ≤ \({2D}_{{\text{h}}}\). Further, as before, it was observed that as the length of the wall was increased, the upstream \({p}_{{\text{w}}}\) distribution was noticed to be insensitive to the increase in wall length. At this NPR, a large negative \({p}_{{\text{w}}}\) region existed for \({L}_{{\text{w}}}>{D}_{{\text{h}}}\). The wavy nature of the \({p}_{{\text{w}}}\) distribution persisted for larger wall lengths. However, this distribution at NPR = 8 differed from the other distributions at NPRs 4 and 6, in terms of the magnitude of wall pressure peaks and troughs and their locations on the wall. The recurring presence of positive and negative pressure regions on the wall in an alternating fashion suggests both entrainment and spillage of the flow could be also be repeating phenomena over a long wall (\({L}_{{\text{w}}}\) \(\ge 4{D}_{{\text{h}}}\)).

The observation from Figs. 10, 11, and 12 that the upstream \({p}_{{\text{w}}}\) distribution was more or less unaffected because of increase in wall length could be due to the characteristic feature of supersonic flows in which the upstream conditions are insensitive to downstream conditions. Hence, the changes in wall length, i.e., an increase in wall length toward downstream direction did not result any noticeable changes in the upstream \({p}_{{\text{w}}}\) distribution. Further, the physics for the upward or downward deflection of the jet at various wall lengths could also be explained by the \({p}_{{\text{w}}}\) distribution pattern in different expansion conditions, i.e., NPR = 4, 6, and 8 shown in Figs. 10, 11, and 12, respectively. The ambient pressure line was taken as the reference line in the wall pressure plots. If the integrated area above this line was more than the one below it, the jet suffered upward deflection, otherwise the opposite. This was verified from the pressure plots analysis and was also consistent with flow physics. When the area was more, the wall experienced a net downward force and the reaction by the wall was upward which acted on the jet flow above wall. This upward reaction on the jet flow caused an upward deflection of the jet. The opposite occurred, when the integrated area above the ambient line was less than the one below it.

3.4 Effect of wall length on normal force and moment coefficients

As jet–wall interaction effects cause unique \({p}_{{\text{w}}}\) distributions corresponding to a set of values of NPR and \({L}_{{\text{w}}}\), the effect has been quantified in terms of normal force and moment coefficients of the wall. As the experimental data on wall pressure along the mid-width line are available, two-dimensional coefficients were calculated from this data.

3.4.1 Two-dimensional normal force and moment coefficients from experimental wall pressure data

Figure 13 illustrates the nomenclature and details for the calculation of the two-dimensional normal force and moment coefficients generated by the flat wall due to interaction with the jet flow field. The top surface wall pressure (\({p}_{{\text{w}}}\)) distribution was experimentally obtained and the bottom surface wall pressure was the ambient pressure (\({p}_{{\text{a}}}\)) which was constant. The difference between the two at any L, i.e., \(\left({p}_{{\text{w}} }-{p}_{{\text{a}}}\right)\left(L\right)\) indicates the local gauge pressure on the top surface of the wall. A positive normal force coefficient (\({C}_{{\text{n}}}\)) indicates a net downward force on the wall, and a positive moment coefficient (\({C}_{{\text{m}}}\)) represents a clockwise moment about the wall leading edge.

Fig. 13
figure 13

Illustration of two-dimensional calculations of coefficients of force and moments from wall pressure data

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In the mid-width plane, the following calculation procedure was followed to compute two-dimensional normal force and moment coefficients. Unit depth was assumed perpendicular to the mid-width plane.

\({L}_{{\text{w}}}\)= Full length of the wall

L = Distance along the wall

\({p}_{{\text{w}}}\) = Wall static pressure

\({p}_{{\text{a}}}\) = Ambient pressure

\(\left({p}_{{\text{w}} }-{p}_{{\text{a}}}\right)\left(L\right)\) = Polynomial equation of experimentally obtained wall pressure distribution

$$\left({p}_{{\rm w} }-{p}_{{\rm a}}\right)\left(L\right)= {a}_{0}+{a}_{1}L+{a}_{2}{L}^{2}+{a}_{3}{L}^{3}+ \dots +{a}_{k}{L}^{k}$$
(1)

where k is the polynomial order

$$\text{Normal force per unit depth}={{F}_{{\text{n}}}}^{\prime}= \underset{0}{\overset{{L}_{{\text{w}}}}{\int }}\left({p}_{{\text{w}} }-{p}_{{\text{a}}}\right)\left(L\right) {\text{d}}L$$
(2)
$$\text{Moment per unit depth }={M}{\prime}\underset{0}{\overset{{L}_{{\text{w}}}}{\int }}\left({p}_{{\text{w}} }-{p}_{{\text{a}}}\right)\left(L\right)L {\text{d}}L$$
(3)
$${\text{Normal force coefficient}\, C}_{{\text{n}}}= {{F}_{{\text{n}}}}^{\prime} / {p}_{0}{L}_{{\text{w}}}$$
(4)
$$\text{Moment coefficient}\, {C}_{{\text{m}}}= {M}^{\prime}/ {p}_{0}{{L}_{{\text{w}}}}^{2}$$
(5)

The two-dimensional calculations to obtain normal force and moment coefficients using the wall pressure data on the mid-width line along the wall length would primarily give an understanding of the bulk effect of flow interaction with the wall when the wall was maintained at different lengths, at the nozzle exit. Though the calculations (from Eqs. 1–5) were two-dimensional, the experimental data pertained to three-dimensional flow. Further, depending on the locations of wall pressure ports and the number of ports, the values of these coefficients may vary. However, this section aims to illustrate the effect of change in wall length on the nature of variation of these coefficients. This is not expected to vary as a result of an increased number of ports from the existing number (15) used in the present investigation.

3.4.2 Variation of normal force and moment coefficients with \({{\varvec{L}}}_{{\varvec{w}}}\)

Figure 14 illustrates the variation of normal force coefficient (\({C}_{{\text{n}}}\)) with wall length at three expansion conditions of the jet, i.e., NPR = 4, 6, and 8. The calculations for \({L}_{{\text{w}}}={0.5D}_{{\text{h}}}\) were avoided as the number of pressure ports was small, and this number was not expected to give fair coefficient values. It can be observed from the figure that changes in wall length resulted in significant changes of \({C}_{{\text{n}}}\) up to \({L}_{{\text{w}}}={4D}_{{\text{h}}}\) at all NPRs, but beyond this, the change effected by \({L}_{{\text{w}}}\) was relatively small. At NPR = 4 (overexpansion), \({C}_{n}\) was negative at \({L}_{{\text{w}}}={D}_{{\text{h}}}\) and positive for other larger values of \({L}_{{\text{w}}}\). At NPR = 6, \({C}_{{\text{n}}}\) was negative at \({L}_{{\text{w}}}={2D}_{{\text{h}}}\). Only at underexpansion conditions (NPR = 8), \({C}_{{\text{n}}}\) remained positive for all values of \({L}_{{\text{w}}}\). Figure 15 gives the variation of \({C}_{m}\) with wall length at different NPR. The behavior of \({C}_{{\text{m}}}\) variation with wall length was similar to \({C}_{{\text{n}}}\) Vs \({L}_{{\text{w}}}\). At NPR = 8, though \({C}_{{\text{m}}}\) was positive at all \({L}_{{\text{w}}}\), varied very little from \({L}_{{\text{w}}}={2D}_{{\text{h}}}\) onwards.

Fig. 14
figure 14

Wall normal force coefficient variation with non-dimensional wall length at different expansion conditions of the jet flow

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Fig. 15
figure 15

Wall moment coefficient variation with non-dimensional wall length at different expansion conditions of jet flow

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Before ending this section, the authors would like to put forth the observation that whenever \({C}_{{\text{n}}}\) was positive, the jet suffered an upward deflection and when negative, a downward deflection. This may be verified with schlieren images in Figs. 6, 7, and 8 corresponding to different expansion conditions. As the magnitude and sign of \({C}_{{\text{n}}}\) depends upon \({p}_{{\text{w}}}\) distribution which is the footprint of shock-expansion wave structure in the jet on the wall, the above observation was consistent with the flow physics that the \({p}_{{\text{w}}}\) distribution, jet shock structure and the jet deflection were intertwined. Further, it would also be reasonable to relate the coefficients to the physics of jet–wall interactions.

4 Conclusions

The interaction of a rectangular supersonic jet with a flat wall was studied experimentally at three different expansion conditions (NPR = 4, 6, and 8). Schlieren flow visualization images and wall pressure measurements were obtained in the experiments. The effect of flat wall length variation at three different expansion conditions on shock cell structure, jet deflection, and force and moment coefficients were studied. The following conclusions were arrived at based on the present investigation:

  1. (i)

    In overexpansion conditions, short wall lengths, i.e., up to \({L}_{{\text{w}}}={D}_{{\text{h}}}\) caused a downward deflection and the deflection being maximum at \({L}_{{\text{w}}}\)=\(0.5{D}_{{\text{h}}}\). This would be advantageous for high-speed aircraft for easy take-off when the propulsion nozzle is rectangular and provided with an aft-deck and designed for overexpansion at sea-level conditions. The net upward force on the bottom surface of the wall would add up for more lift. However, this may also contribute to pitch down moment on the vehicle.

  2. (ii)

    The maximum upward deflection (7.1°) occurred at underexpansion conditions (NPR = 8) at \({L}_{{\text{w}}}={D}_{{\text{h}}}\).

  3. (iii)

    For given expansion conditions, increase in wall length beyond \({2D}_{{\text{h}}}\) had less influence on change in the shock pattern in the upstream of the flow. This shows that the near field shock structure on the wall with respect to the nozzle exit was not affected by the increase in wall length beyond a threshold value of wall length.

  4. (iv)

    The recurring occurrence of alternating positive and negative pressure regions on the wall surface relative to ambient conditions indicated that spillage and entrainment of the flow over the wall could be repeating phenomena at large wall lengths such as Lw \(\ge 4{D}_{{\text{h}}}\).

  5. (v)

    Normal force and moment coefficients (two-dimensional) variations with wall length were similar. Changes in wall length resulted in significant changes of \({C}_{{\text{n}}}\) up to \({L}_{{\text{w}}}={4D}_{{\text{h}}}\) at all NPRs, but beyond this, the change effected by \({L}_{{\text{w}}}\) was relatively small. Positive and negative values of Cn were linked to upward and downward deflection of the jet, respectively.