Shallow water experiments to verify a numerical analysis on an aerodynamically thrust-vectored aerospike nozzle

Shallow water experiments are predestined for fast and inexpensive experimental examination of two-dimensional (2D) flow phenomena. In this study, the shallow water analogy is used for the verification of a previous numerical analysis of an aerodynamically thrust-vectored aerospike nozzle. Experiments in a shallow water channel were conducted, using a model of an isentropic spike with a 25 percent truncation and two secondary injection sites. A comparison of the flow phenomena, e. g., shock patterns, shows a wide correspondence of experimental and simulation results, thus verifying the simulation approach and encouraging to continue its improvement. Furthermore, it can be shown that secondary fluid injection is a promising method for active thrust vectoring on aerospike thrusters and gives an objective for future applications.


INTRODUCTION
Aerospike or plug nozzle engines have been developed and tested since the 1950s as an alternative to classical bell nozzles [112]. With their advantageous performance characteristics during the ascent, they have been considered for the upper stage of Saturn V and the Space Shuttle Main Engine [13]. During the 1990s, linear plug nozzle engines were the single-stage-to-orbit propulsion system of choice for the X-33 Mission also known as Venture Star.
Today, well known for their altitude adaptive characteristics up to the designed pressure ratio, this nozzle type is regaining the focus of interest. Besides further advantageous performance characteristics towards comparable bell nozzles, aerospike nozzles provide additional capability of active aerodynamic thrust vectoring. As demonstrated in cold gas §ow tests [13,14], aerodynamic thrust 529 Article available at https://www.eucass-proceedings.eu or https://doi.org/10.1051/eucass/201911529 vectoring provides side force control without mechanical devices, such as gimbals or §aps.
A numerical simulation has been conducted for a parametric analysis concerning thrust vectoring [15]. For the veri¦cation of the observed §ow phenomena, a direct measurement of the hot gas §ow is hardly feasible. The analogy of incompressible liquid §ows with free surfaces on one side and compressible gas §ows on the other side allows the examination of these §ow phenomena in a shallow water channel for the ¦rst approximation. Besides the better accessibility of ¦eld quantities, a better visibility and smaller §uctuation frequencies due to a time scaling e¨ect allow a qualitative analysis of the stream behavior [16].
Within this paper, an experimental §ow phenomena analysis concerning aerodynamic thrust vectoring on aerospike nozzles is presented. It starts with the description of the function principle of aerospike nozzles and thrust vectoring in section 2. In section 3, the underlying relation of hot gas supersonic §ows and shallow water surface waves for the shallow water analogy is summarized. From the nozzle geometry used for the computational §uid dynamics (CFD) simulations, the manufactured water model is derived. Furthermore, this section describes the utilized test bench. The obtained results from the experiments are presented and discussed in section 4. Finally, the §ow phenomena are compared with previous CFD-calculation results.

AEROSPIKE AND THRUST-VECTORING CONCEPTS
Aerospike nozzles can be classi¦ed into linear and annular plug nozzle designs. Both o¨er the possibility of having a single or a clustered combustion chamber arrangement. The clustered combustion chamber design inherently o¨ers thrustvectoring by applying the method of di¨erential throttling of single chambers. Furthermore, it o¨ers a design choice between fully external or partial internal expansion of the exhaust gases by adding an additional divergent nozzle structure behind the nozzle throat [3,4]. The method of di¨erential throttling proposes suitable thrust-vectoring for medium and high thrust rocket engines. Nevertheless, it is not applicable for smaller rocket engines with a single annular combustion chamber. In that case, aerodynamic thrust vectoring is a promising method for side-force generation in order to steer a rocket or satellite. The injection of a secondary §uid with a high radial §ow component into the axial supersonic main exhaust §ow results in a de §ection of the latter. This de §ection creates an additional side-force component to the inherent thrust of secondary §uid §ow. Figure 1 shows the Mach number distribution in which the function principle of aerodynamic thrust-vectoring is visible. The interaction of the secondary §ow with the main §ow causes a bow shock, resulting in an asymmetric §ow and, consequently, pressure distribution around the aerospike nozzle.

Figure 1
Local Mach number distribution for a thrust-vectored aerospike §ow [15] Due to high thermal loads at the nozzle exit as well as aspects of weight optimization and di©culties in manufacturing sharp edges, an ideal isentropic spike usually gets truncated. This truncation is always bound to losses according to the nozzle performance [3]. Additionally, plug nozzles o¨er various advantages for exo-atmospheric §ight missions. Because of their shape, plug nozzles can reach greater expansion ratios and, therefore, a higher speci¦c impulse while having the same footprint size with regard to bell nozzles. The cooling of aerospike nozzles is one of their major disadvantages [17].

EXPERIMENTAL SETUP
A brief summary of the shallow water analogy is presented, followed by a description of the utilized test bed and the nozzle model.

Shallow Water Analogy
For examinations using analogies, a physical process (original) is studied based on the second process (model) which can be described with the same mathematical correlations (¦eld quantities, boundary and initial conditions). The geometry of the model needs to be equal or similar to the original one.
The analogy of shallow water waves (wave length ≥ 20h W ) and 2D gas streams can be derived by using energy and mass conservation equations for both §uids. The evaluation of this set of equations delivers a relation between the water height h W in the shallow water channel and the corresponding gas parameters: temperature T , density ρ, and pressure p. The parameter relations between these two §ow models are Table 1 Parameter relation [16] Gas (κ = 2) Shallow water T /T0 hW /hW,0 ρ/ρ0 hW /hW,0
It is important to note that this analogy is exact only for gas §ows with an isentropic exponent of κ = 2. Therefore, this analogy is used rather for qualitative than for absolute quantitative evaluation in the present case. Furthermore, a correspondence between sonic velocity a and traveling speed of surface waves c = (g h W ) 1/2 (g is the gravity constant) can be found. The Froude number (Fr = w/c, w is the local water speed) can be used correspondingly to the Mach number Ma to classify the §ow: Ma < 1 : subsonic gas §ow ⇐⇒ Fr < 1 : subcritical water §ow; Ma = 1 : sonic/choked gas §ow ⇐⇒ Fr = 1 : critical water §ow; Ma > 1 : supersonic gas §ow ⇐⇒ Fr > 1 : supercritical water §ow.

Test Bed ¡ Shallow Water Channel
For the realization of the experiments, the shallow water channel of the Institute of Fluid Mechanics at the TU Dresden has been used. This test bed was designed in 1976 to visualize §uid §ows [19]. A setup scheme of the test bed is shown in Fig. 2.

Figure 2
Shallow water test bed [19] This test bed can be used for models with a footprint up to 1500 × 1000 mm and a height up to 200 mm. The applied nozzle model uses almost the complete base area of the experimental test bay. A volume §ow up to 350 m 3 /h is realized with a 15-kilowatt pump. The water height can be determined manually with a traversing unit. A vertically adjustable dipstick is then used to measure the height. For the measurement on a single data point, the dipstick is lowered from above until it barely touches the water surface. The local water height is obtained from a vernier scale on the dip stick adjusting mechanism. The last tenths of millimeters are usually overcome by the water surface tension, resulting in a measurement accuracy of around ±1 mm.
To ensure consistent measurements, the water §ow has to enter the model stationary, uniformly, parallel, and without §uctuations. For this purpose, several components were added to the test bed: the sieve di¨user throttles the §ow in a short path without any separations which could cause §uctuations. Sieves and a §ow straightener in the §ow calming segment reduce residual §ow rotations and large-scale §uctuations. The nozzle in front of the experimental model is used to accelerate the §ow and to create a uniform velocity pro¦le with a small boundary layer. The weirs can be used to ensure a stationary feeding water height in front of the test setup and, therefore, within the experimental model.

Nozzle Model
The contour of the annular full isentropic aerospike nozzle for a 3-kilonewton rocket engine has been determined with the FORTRAN code of C. C. Lee [1,20]. In this code, the simpli¦cation of a one-dimensional isentropic §ow is used to calculate the nozzle contour via the area expansion ratio. The properties of the hot gases have been derived from the combustion of ethanol and liquid oxygen with a fuel mass ratio of 1:1. The same fuel combination is used in the SMART Rockets Project [21]; therefore, this nozzle could potentially be tested under real conditions at the TU Dresden. Table 2 lists the complete set of parameters for the nozzle design, delivering the spike contour shown in Fig. 3. During  performed using CFD-analyses, a truncated spike with 75% of the full isentropic length has been chosen for further evaluation [15]. The main reasons for truncation are: less weight, improved cooling properties, and a better producibility of the spike (no sharp edges) while maintaining almost the §ow pattern of the full isentropic spike. Within this study, the in §uence of the secondary injection position and variable mass §ow rate on obtainable side forces and pressure distribution on the spike were investigated. The spike contour used in this parameter study serves as the basis for the development of the shallow water model (see also Fig. 3). For reasons of better accessibility to measure the water surface, the nozzle for the shallow water channel has been scaled up by the factor of 5.
First estimations showed that the primary diamond shaped high-pressure region contoured by the trailing shock should be well within the dimensions of the model. In order to realize the secondary injection, a water reservoir is implemented within the nozzle contour. Two 28-millimeter wide gate valves have been installed at 60% and 90% of the length of the remaining spike. The third gate valve is used upstream at the nozzle front to regulate the mass §ow of water into this reservoir and, therefore, indirectly, the water height and pressure of the injection, respectively.
A set of inlet gaps provides an ambient §ow simulating a working condition of the rocket engine traveling through the atmosphere. Two sieve rows are used to realize a more uniform ambient §ow in axial direction. Additionally, two channels on each side of the model have been added to purge the major amount of water delivered by the test bench pump. This was necessary due to the limited adjustability of the water §ow provided by the pump. For further possibilities of adjusting the water §ow, one gate valve per channel has been included. The model within the channel is presented in Fig. 4.
Having de¦ned the nozzle model to be tested and described the utilized shallow water channel, this section presents the results of the experiments. At ¦rst, the undisturbed §ow around the center body of the aerospike nozzle is examined. Subsequently, the occurring §ow phenomena are shown, including a measured surface height map. In the second subsection, the §ow phenomena observed during §ow de §ection caused by secondary §uid injection are discussed.

Undisturbed Flow
Like bell nozzles, aerospike nozzles have a speci¦c design pressure ratio p e /p 0 at which they are fully adapted. This §ow state is characterized by an exhaust §ow parallel to the nozzle axis. In case of an overexpanded §ow state, the working pressure ratio is above the design value and the downstream located exhaust §ow bends towards the nozzle axis. For an underexpanded §ow state, the working pressure ratio is below the design value and the gas §ow bends away from the nozzle axis.
The aim of the ¦rst experiments was the reproduction of these three major §ow states of expanding gases: overexpanded, adapted, and underexpanded. All of them can be created by adjusting the ambient §ow and, thereby, ambient pressure by reducing/raising the ambient §ow inlet cross section while keeping the combustion chamber at a constant prede¦ned condition. A twine (indicated by an arrow in Fig. 5) has been used in the pictures to illustrate the free jet boundary. The usual method of adjusting the ambient pressure with the out §ow weir could not be used, since the full open inlets of the ambient §ow already created a water height resulting in an overexpanded nozzle §ow.
For a comparison of the experimental results with the previously conducted simulations [22], the §ow phenomena are identi¦ed in the latter according to the  Figure 6 The CFD-result Mach number distribution (top) and static pressure distribution (bottom) [22] work of Hall [23]. Figure 6 shows the location of all relevant §ow phenomena for the overexpanded §ow in a Mach number (top) and a static pressure distribution (bottom). It is obvious that most phenomena clearly visible in the Mach number plot are barely noticeable in the pressure plot. But as shown below, most of them are detectable on the photographs.
For the undisturbed, overexpanded §ow condition, a water surface map has been measured manually for the interesting §ow area using the traversing. With a horizontal and vertical resolution of 10 mm * , the map shown in Fig. 7 was obtained. During the measurement, the water height within the combustion chamber has been kept constant to 90 ± 4 mm. The height §uctuation is due to the lack of a ¦ne adjustability of the water pump. These §uctuations at the inlet water height resulted in changes of the water surface below the water height measurement accuracy of ±1 mm.
In this surface map, the expanding §ow region and the trailing shock are clearly visible. The aerodynamic spike is visible through the dark blue area at the nozzle base. The weak expansion wave can be identi¦ed by the high gradient above the shear layer of the aerospike. The mentioned §ow phenomena can be clearly identi¦ed in a photograph (Fig. 8). The surface map was not recalculated into a pressure map for hot gases due to the κ = 2 inaccuracy. Nevertheless, the §ow phenomena of the conducted experiment and the numerical analyses correspond closely.

De §ected Flow
In this subsection, the §ow phenomena which occur due to secondary injection are presented and discussed. This discussion is focused on an overexpanded main §ow. To complete the survey on the conducted experiments, Fig. 9 shows the §ow phenomena for adapted and underexpanded §ows. When the secondary §uid is injected, a bow shock establishes towards the freestream boundary (Figs. 10 and 11). This bow shock is accompanied by a region of higher pressure upstream the injection when compared to the undisturbed §ow. This local pressure increase indicates a stagnation point at and a local subcritical §ow upstream the injection site. Waidmann [24] discovered similar §ow phenomena during the investigation of secondary injection in a supersonic §ow within a conical nozzle. Furthermore, he identi¦ed an oblique shock on front of the injection site. Between oblique shock, nozzle wall, and secondary injection bow shock, he discovered a recirculation area. Such a recirculation area  §ow experiences a necking caused by the bow shock, which itself is re §ected at the free-stream boundary towards the centralized compression shock behind the spike. This shock wave creates an expansion fan after the re §ection, which is no longer veri¦able at the downstream located region behind the ¦rst interference with the trailing shock.
Injection site at 60% of the spike length When using the upstream located injection site, a similar bow shock appears like it is shown in the previous case (Figs. 12 and 13). Having the injection site closer to the narrowest nozzle cross section, the shock wave now interacts  directly with the expanding primary §ow. As a result, a clearly visible unilateral necking is formed at the position where the bow shock meets the shear layer of the main §ow, even at a minimal secondary mass §ow [22]. Consequently, a new expansion region is formed starting from the necking point. This region reaches the high-pressure region behind the trailing shock and proceeds downstream as a new shock wave. This shock wave gets re §ected on the opposing boundary of the free-stream and propagates further within the exhaust gas §ow in the CFD-simulation. In the shallow water experiment, this shock wave is no longer noticeable after the re §ection on the opposing boundary. Overall, the §ow ¦eld gets much more in §uenced by the injection at the 60 percent position injection site compared to the downstream located injection site at 90% considered above.
In this paper, shallow water experiments conducted for the veri¦cation of §ow phenomena found in a previously realized numerical §ow simulation are presented. Ensuing from the §uid model of this simulation, a shallow water model has been derived, manufactured, and tested in the described test bed. An extensive analysis of the §ow phenomena in the water channel, a water surface map of the overexpanded, unde §ected aerospike §ow, and the comparison of the results with the simulation yield the veri¦cation of the simulation and encourage to continue further research. With an automatic traversing and surface measurement, full surface maps of more §ow situations, e. g., adapted and underexpanded but essentially with secondary injection, are mandatory. Furthermore, a full spike or a completely di¨erent aerospike nozzle contour could be analyzed. In the second step, the present authors plan to continue this research in a supersonic wind tunnel, in which the pressure on the spike surface and the obtained side-forces could be measured.