Shock Waves (2018) 28:919–932 https://doi.org/10.1007/s00193-017-0787-8 ORIGINAL ARTICLE Transient response of a liquid injector to a steep-fronted transverse pressure wave 1 1 1 1 D. Lim · S. Heister · D. Stechmann · B. Kan Received: 14 June 2017 / Revised: 17 October 2017 / Accepted: 9 November 2017 / Published online: 4 December 2017 © The Author(s) 2017. This article is an open access publication Abstract Motivated by the dynamic injection environment mon in current rocket or air-breathing engines. For example, posed by unsteady pressure gain combustion processes, the rotating detonation engine (RDE) is currently receiving an experimental apparatus was developed to visualize the substantial attention as a candidate for future applications dynamic response of a transparent liquid injector subjected and serves as a major motivation for the present study. to a single steep-fronted transverse pressure wave. Experi- Under ideal operation, the annular RDE combustion cham- ments were conducted at atmospheric pressure with a variety ber contains one or more azimuthally traveling detonation of acrylic injector passage designs using water as the working waves traversing the annulus at velocities approaching the ﬂuid. High-speed visual observations were made of the injec- Chapman–Jouguet (CJ) value, while regions downstream of tor exit near ﬁeld, and the extent of backﬂow and the time to the wave passage supply fresh propellants to support the per- reﬁll the oriﬁce passage were characterized over a range of petuation of the next incoming wave. Because detonation injection pressures. A companion transient one-dimensional waves can travel at speeds well over 1000 m/s, injectors in model was developed for interpretation of the results and to these devices are subject to dynamic downstream pressures elucidate the trends with regard to the strength of the trans- oscillating at kilohertz frequencies. Theoretically, detonation verse pressure wave. Results from the model were compared waves can produce pressure ratios in excess of 30 depending with the experimental observations. on reactant mixtures, and these strong waves create pres- sures that exceed the injection pressure such that backﬂow Keywords Transient response · Liquid injector · Detona- of propellants is a distinct possibility. Understanding the tran- tion · Pressure wave sient response of the injector under these conditions is crucial since reverse ﬂow of the entire injector passage would bring combustion products into the manifold with potentially dis- 1 Introduction astrous consequences. The transient injection ﬂow recovery subsequent to wave passage must also be understood in order Pressure gain combustion (PGC) research has been rapidly to assess injection and ﬁlling characteristics that prepare gaining attention as a potential means to produce thrust the combustible mixture for the next arrival of a detonation or generate power at higher efﬁciency than conventional wave . constant-pressure combustion technology [1,2]. These tran- Unfortunately, only limited prior work has focused on sient devices rely on the detonative mode of combustion injector response to violent events such as the passage of as opposed to deﬂagration at constant pressure, as is com- a detonation wave. Moreover, most prior research was moti- vated by injector response during combustion instabilities in constant-pressure combustion engines, not detonation Communicated by J. Yang and A. Higgins. engines. Nevertheless, the classic work dates back to the B D. Lim 1950s with Miesse  and Reba and Brosilow  whose email@example.com efforts are also covered in some detail in the NASA Spe- cial Publications (SP Series) from the 1970s [6,7]. The Reba Maurice J. Zucrow Laboratories, Purdue University, 500 Allison Rd, West Lafayette, IN 47907, USA and Brosilow linear analysis describes amplitude and phase 123 920 D. Lim et al. Fig. 1 P&ID of the pre-detonator shift of a plain-oriﬁce atomizer to an imposed pressure oscil- exposed to a detonation event that was initiated at atmo- lation of arbitrary frequency. The relevant frequency here spheric conditions. At atmospheric initial pressure, dynamic (injection frequency) is the rate at which ﬂuid in the passage backﬂow was observed at low/modest injection manifold is replaced, i.e., v/L. These classical studies show that the pressures—typically about 0.07–0.2 atm gauge pressures. injection response to a small amplitude sinusoidal pressure Because the differential pressure created by a detonation perturbation tends to roll off to low amplitudes when fre- event scales linearly with the ambient pressure, we would quencies are, say, an order of magnitude higher than that of expect larger injector pressure drops to display the dynamic the injection frequency. behavior uncovered at the ambient pressure detonation con- MacDonald et al.  extended the Reba and Brosilow ditions explored in this introductory work. analysis to consider nonlinear pressure perturbations using an axisymmetric CFD approach. Their results show that the nonlinear response is less than the linear result except at fre- quencies in excess of 5 v/L. These authors also included the 2 Experimental facility effect of the vena contracta pulsations that are attributed to hydrodynamic instability of this reentrant region near a sharp To meet the objectives of the study, a cold-ﬂow experi- oriﬁce inlet. Simple models have also been used to assess ment was designed to represent a single injector element of the inﬂuence of injector unsteadiness on the jet development an RDE. An oxygen/hydrogen “pre-detonator” (henceforth outside of the oriﬁce exit [9,10]. While this work helped to referred to as “predet”) was developed to drive a detonation through an optically accessible test section, and a high- establish nonlinear effects of a ﬁnite amplitude sinusoidal pressure disturbance, it does not address the nature of steep- speed camera recorded the response of the liquid—water in this case—as the detonation wave passes. Figure 1 provides fronted waveforms consistent with passing detonation events. In the more recent era, substantial efforts have been a schematic arrangement of the plumbing and instrumen- tation diagram (P&ID) for the device. We are indebted devoted to understanding the response of speciﬁc injector types including swirl injectors [11–14], shear coaxial injec- to the Detonation Engine Research Facility at AFRL for donating a functional predet that creates the desired DDT tors [15,16], and swirl coaxial injectors currently of interest for oxidizer-rich staged combustion cycles [17,18]. Brady event . Major elements in the P&ID include two fast- also recently published a peripherally related study regarding response solenoid valves (LeeCo models IEPA2411241H line priming . The RDE community has recently begun to and IEPA2 411141H), a mixing and ignition chamber, and produce fully coupled simulations in which dynamic injec- a deﬂagration-to-detonation transition (DDT) tube. Ignition was achieved using a remote-control automotive spark plug tion is directly coupled to the detonation passage [20,21]. Results to date show that the very presence of injector and (NGK model ME 8). The DDT tube was a 6.4-mm (0.25 in.) stainless steel tube approximately 88 mm (3.5 in.) in length, plenum dynamics will cause the detonation wave structures to be different from that obtained with ideal injectors. Unfor- with an inner diameter of 4.6 mm (0.18 in.). Its upstream end was tapped internally with 10–32 threads to a depth of tunately, past efforts [20–23] have all focused on injection of gaseous propellants; the challenges of two-phase and liquid approximately 25 mm (1 in.). Feed pressures for both hydro- gen and oxygen were set to 1.38 MPa (200 psia). Due to the injection schemes remain to be addressed. As there are currently no available studies of the response small size of the propellant feed lines (1.6 mm or 1/16 in. tub- of liquid injectors to violent events such as a passing deto- ing), we did not possess the instruments capable of measuring nation, we began by exploring the range of conditions over the ﬂow rates of each propellant. However, we approximated which dynamic behavior was observed in the channel when the equivalence ratio of the mixture to be two based on choked ﬂow calculations. 123 Transient response of a liquid injector to a steep-fronted... 921 Fig. 3 Drawings of tested injector designs. From left to right: L, M, S, Fig. 2 Drawing of the test article (rotated 90 counterclockwise) show- and P ing major features. The hot gas path has a rectangular cross section of 4.6 mm by 13.7 mm (0.18 in. by 0.54 in.) in the injector module block sage to guide the ﬂow. Based on earlier tests with a different design, a water outlet port was added directly across from The predet was interfaced with the end of an expand- the injector exit to prevent the accumulation of water within ing acrylic channel on the test article. The test article also the detonation passage since this could lead to inconsistent included a modular injector insert and pressure instrumenta- pressure conditions during the experiment. tion as highlighted in Fig. 2. The transition channel section The steady-ﬂow discharge coefﬁcients of the injectors was included to diffuse the detonation to the cross-sectional were also determined using the catch-and-weigh method. area required for the injector module, and a low-expansion In the range of manifold pressures tested, all injectors had half-angle of 5 was employed to avert any potential ﬂow discharge coefﬁcients consistent with the developing ﬂow separation; design details are included in . Visual obser- regime such that their values were strongly dependent on the vations of wave structures in the near-oriﬁce-exit region imposed pressure drop. The non-constant discharge coefﬁ- conﬁrmed that the design was sufﬁcient to maintain a pla- cients were a source of uncertainty as discussed in Sect. 4. nar wave at the test article. The transition section was used A chart of the discharge coefﬁcients plotted against injec- for all injector conﬁgurations studied, while the injector mod- tor pressure drop is shown in Fig. 4. Rather modest pressure ules were replaced with the various designs for different tests. drops were employed in the study to assess regions where In all conﬁgurations, the predet efﬂuent entered the transi- injection dynamics were important. Design P shows a signif- tion section through a 6.4-mm (0.25 in.) compression tube icantly lower discharge coefﬁcient than the other injectors, ﬁtting. A channel width of 4.6 mm (0.18 in.) was chosen to presumably due to a second vena contracta in the narrower match the inner diameter of the DDT tube, and the height was plenum. machined to 13.7 mm (0.54 in.) such that a ﬂat rectangular Injector manifold pressure was measured using a 400 kPa proﬁle was obtained. A high-frequency pressure transducer (60 psia) GE Druck pressure transducer located approxi- was located on the channel sidewall at the same axial station mately 8 cm (3 in.) upstream of the injector and was manually as the injection site. controlled using a needle valve 25 cm (10 in.) upstream of We fabricated four different injector designs to assess the the pressure transducer. inﬂuence of injector length and the presence of an upstream The predet was fed from 34.5 MPa (5000 psia) hydrogen plenum (countersink) on the overall response. Dimensions and oxygen K-bottles, while nitrogen purge and deionized of each injector are provided in Table 1, and schematics of water were linked directly to the laboratory’s supply lines. each of the four designs, designated long, medium, short, and A high-speed data acquisition system was used to control plenum (L, M, S, and P, respectively), to represent their key the predet’s solenoid valves and record the proﬁle of the features are shown in Fig. 3. All injector designs featured a pressure wave at a sampling rate of 1 MHz. The pressure 90 conical transition between the plenum and injector pas- transducer used to capture the pressure signal was a Kulite Table 1 Injector module design dimensions Design Oriﬁce diameter D [mm (in.)] Injector length L [mm (in.)] Plenum diameter [mm (in.)] L (Long) 0.84 (0.033) 7.62 (0.30) 6.35 (0.25) M (Medium) 0.84 (0.033) 5.08 (0.20) 6.35 (0.25) S (Short) 0.84 (0.033) 3.81 (0.15) 6.35 (0.25) P (Plenum) 0.84 (0.033) 3.81 (0.15) 2.06 (0.081) 123 922 D. Lim et al. ever, it is no trivial task to simulate 3-D transient two-phase ﬂows and parametric studies become impractical. Instead, we developed an unsteady, one-dimensional, lumped-parameter computational model to help interpret the experimental mea- surements and serve as a preliminary design and analysis tool. The model solves for the dynamic response of a column of liquid with density ρ and length L subjected to a highly transient downstream pressure disturbance as highlighted in Fig. 5. We consider a ﬁxed manifold pressure, P , and an initial chamber pressure of P . Additionally, x represents the column end location for the purposes of tracking its motion along the oriﬁce passage. Fig. 4 Discharge coefﬁcients versus injector pressure drop for injector While injector ﬂow dynamics have been of interest to elements used in the study the combustion stability and water hammer communities for many years, we have not found an analysis comparable to XTEL-190, ﬂush-mounted as per manufacturer speciﬁcation. this simple approach in the existing literature. By and large, Its diaphragm is protected by a built-in perforated screen the combustion stability community has assessed transient that increases the sensor’s rise time to approximately 20 µs, response to sinusoidal waveforms using both linear [5,6] and causing the pressure readings to be spread horizontally (tem- nonlinear  models. With water hammer, ﬁnite wave speeds porally). However, according to the pressure-wave shape are employed as the applications typically stem from “long analytical study presented in the following section, the total pipes.” In this case, linear and nonlinear wave equations are delivered impulse instead of the peak pressure amplitude was employed to assess dynamics . the main driver of injector response. Therefore, we have rea- As an initial step, the column of liquid was modeled as a son to believe that the temporal skewing of the pressure signal ﬁxed mass, and a step change in pressure from P to P was 1 2 had little impact on our analysis to follow. We performed considered. The downstream pressure P can be set to zero ﬁve tests at each test condition to quantify repeatability and without loss of generality, i.e., we measure all pressure dif- measured the backﬂow distance and reﬁll time of the vari- ferences with respect to this initial gauge pressure. Applying ous injectors from the high-speed camera footage. Detailed Newton’s second law to the liquid column with F = PA information of the test hardware and facilities can be found gives in . dv 1 ρLA = A P − P ± ρv . (1) o o m e dt 2 3 One-dimensional numerical model development The nature of the interaction between a transverse pressure The dynamic pressure term appearing in (1) stems from the wave and a cylindrical liquid injection is highly three- fact that the entire column is moving prior to a disturbance in dimensional, and accurate predictions of injector response the downstream pressure. Consequently, the entire manifold will certainly require CFD studies in three dimensions. How- stagnation pressure has to be applied in order to stagnate Fig. 5 Schematic of sequence of events showing a ﬁxed mass responding to a change in pressure differential 123 Transient response of a liquid injector to a steep-fronted... 923 Fig. 7 Plot of dimensionless response time t versus pressure ratio p. Fig. 6 Timestep sensitivity study showing the insensitivity of the solu- r −8 The shaded region indicates the range of typical pressure ratios in a tion to further reﬁnements to timestep size beginning at t = 1 × 10 s realistic system −8 the ﬂuid. The upper sign applies when the ﬂow is moving the interface location. A timestep of 1 × 10 swas shown to the right (positive x-direction), and the lower sign applies to be sufﬁciently small to produce converged results (Fig. 6). during backﬂow conditions. It becomes apparent from the Additional details of the numerical treatment can be found above equation that for a plain oriﬁce, the cross-sectional in . area does not play a role in the problem. During backﬂow, The oriﬁce transit time τ represents a fundamental quan- the entire column of liquid will be pushed upstream. Letting tity in describing the dynamic response of the column: v represent the initial Bernoulli velocity of the ﬂow prior to the disturbance, we have τ = . (5) 2P v = . (2) We can also deﬁne a dimensionless pressure p = P /P that m 2 characterizes the strength of the imposed disturbance relative Similarly, the Bernoulli velocity after the step change, v ,is to the manifold pressure. Consider the case when P = 0at 2 1 t < 0 and p = P /P = constant when t > 0. This rep- m 2 resents a step pressure change which drives the ﬂuid ﬂow to 2|P − P | m 2 v =± . (3) 2 a different steady state. We can deﬁne a response time t as the time taken for the ﬂow to reach 95% of the difference between v and v since an asymptotic behavior is expected 1 2 Here, v takes the positive sign when P < P . When 2 2 m as the ﬂow approaches v and the driving acceleration dimin- P > P , the ﬂow reverses and v takes on a negative value. 2 m 2 ishes. Equation (1) is a nonlinear ordinary differential equation that Figure 7 depicts the behavior of this response time over a is integrated numerically to give instantaneous v and x val- wide range of pressure disturbance amplitudes. When p 1, ues using an explicit second-order accurate in time method the imposed disturbance is a small fraction of the initial man- for computing the liquid–gas interface velocity, v. A second- ifold pressure and the response time tends to asymptote to order backward Euler differencing scheme is utilized to solve t ≈ 3τ under these conditions. For a very strong distur- for the velocity at the current time level i in terms of quan- bance such as that imparted by a detonation wave p 1, tities at time levels i − 1 and i − 2. The resulting difference the most rapid response is attained under these conditions approximation for the velocity at the current time level is with t < 2τ . When very weak disturbances are imposed ( p ≈ 1), the oriﬁce takes the longest to respond since the 1 t (i −1) imposed forces are the smallest under these conditions. For (i ) (i −1) (i −1) v = v + P − P ± ρv 2 ρ L pressure ratios consistent with detonation waves (indicated 1 2 1 2 t by the shaded region in Fig. 7), the results give response times (i −1) (i −2) (i −1) (i −2) + −P + P ± ρv ∓ ρv . e e 2 2 2ρ L in the range 2τ< t < 5τ and the injector does respond on (4) a timescale consistent with the time ﬂuid spends in the pas- sage under nominal injection conditions. While instructive, The scheme is started using ﬁrst-order methodology in these results are of limited use since detonation events are a standard implementation of the backward Euler scheme. highly transient, characterized by a steep-fronted pressure Equation (4) can be numerically integrated in time to obtain spike followed by a period of pressure decay. For this rea- 123 924 D. Lim et al. Fig. 8 Pressure proﬁle of triangular pulses with constant pressure Fig. 10 Plot of non-dimensional displacement versus non-dimensional impulse time for various decay times and peak pressures Fig. 9 Plot of non-dimensional velocity versus non-dimensional time Fig. 11 Plot of non-dimensional recovery time versus non- for various decay times and peak pressures dimensional decay time son, we consider a triangular-shaped pressure disturbance characterized by instantaneous rise to a maximum pressure notion in terms of the location of the end of the column. P followed by a linear decay in pressure. This proﬁle will Once the imposed impulse has been applied, all the cases be more representative of the pressure created by the passage tend to converge toward the same overall system response. of a detonation wave. The results of this study indicate that the horizontal skewing We can deﬁne τ as the duration of linear decay and of pressure proﬁles due to the pressure transducer’s rise time pressure impulse as the area under the triangular-shaped dis- would not result in signiﬁcant error. turbance (I = 0.5τ P ), where P is now the peak pressure. Figure 11 depicts the overall recovery time t , deﬁned c 2 2 rec Here, we consider several P and τ combinations under the as the time required for the injection speed to return to 95% 2 c constraint that I = constant. Keeping manifold pressure and of its original value. Results show only small variations in therefore oriﬁce transit time τ constant, we can use τ to non- recovery time over the range of conditions considered. Thus, dimensionalize τ . Making τ /τ = 1 and P = 10P the the oriﬁce dynamic response will depend almost exclusively c c 2 m baseline case, the value of τ /τ can be decreased while keep- on the impulse generated by the wave. ing I constant to make the pressure proﬁle steeper; the cases While the ﬁxed-mass analysis highlights some top-level considered are shown in Fig. 8. characteristics of importance, we desire a more accurate rep- Figures 9 and 10 show velocity and position histories. resentation of the system to account for additional physical Figure 9 shows varying degrees of initial deceleration since phenomena. For example, as the free surface propagates into the peak pressure is now different for each of the cases. the oriﬁce passage, the gas has less mass in the oriﬁce pas- While there is a more violent velocity excursion for a high- sage to accelerate. We removed the ﬁxed-mass constraint and amplitude short pulse as compared to a low-amplitude long included viscous effects to produce a variable-mass model pulse, the asymptotic behavior and overall response time for the injector. Flow beyond the inlet and exit of the injector vary little for the cases considered. This is a fundamental was neglected by freezing x at 0 or L when it exceeded those result that is important to system dynamics as the shape of values such that only the mass within the injector channel the imposed overpressure is of less concern than the over- was used for calculations. The following diagram shows the all impulse applied to the system. Figure 10 reinforces this control volume used in the calculations (Fig. 12). 123 Transient response of a liquid injector to a steep-fronted... 925 Fig. 13 Macroscopic view of events during a typical test Fig. 12 Diagram showing the computational control volume bounded by the green dashed boundary The net acceleration of the liquid column is the sum of the From (1), the acceleration due to the imposed pressure acceleration due to pressure gradient and the deceleration gradient in the axial direction is due to friction: 1 1 a = P − P ± ρ v . (6) pg m e avg fric ρ L 2 avg a = a − . (11) pg Here, the density is the mass-weighted average of the liquid and gas present in the oriﬁce. It is important to account for the Numerical treatment of the variable-mass model remained combusted gas density here (obtained from NASA CEA ) −8 thesameas(4), and the same timestep of 1 × 10 spro- so that the acceleration remains bounded when the entire duced results that were insensitive to further reﬁnements in injector is ﬁlled with combustion products. timestep . We also used empirical pressure data in this In real ﬂow, frictional loss is expected on the channel wall case to provide a direct comparison between the experimen- and is calculated using the Fanning friction factor, f , accord- tal measurements and analysis predictions. Results from this ing to comparison are included in the following section. f = , Re ≤ 2100 (7) Re 1 ε 1.256 √ =−4.0log + √ , Re > 2100, (8) 4 Results and discussion 3.7D f Re f Figure 13 shows a macroscopic view of events as a detonation where Re = ρv D/μ is the Reynolds number. The Cole- wave travels down the channel. The wave travels from top to brook equation  for the turbulent regime requires f to be bottom in the right half-plane, and water ﬂows from left to solved numerically. After f is obtained and wall shear stress right. The water jet is shattered into a ﬁne mist by the passage is computed, the net frictional force on the domain will be of the high-amplitude detonation front. At the same time, the column of water in the injector is pushed back toward the F = fρv A , (9) plenum by the sudden spike in pressure. This series of images fric wet taken at 12,000 frames per second (fps) and 304 by 512 pixels resolution serves to provide an overall picture of the events where v is the interface velocity and A is the wall area wet during each test run. Subsequently, we reduced the viewing currently wetted by the ﬂuid. Friction of gaseous combustion window signiﬁcantly to enable higher frame rates (more than products is neglected. The mass of liquid in the oriﬁce is 80,000 fps) and thereby capture more detail regarding the then the product of the oriﬁce cross section, liquid column injector’s response. It should be noted that the injector shown length, and liquid density. The mass of the ﬂuid in the injector here was from a prior experiment and was not of the same channel is given by design as those used to generate the data presented in this 2 article. However, the events occurring downstream of the π D xρ avg m = . (10) injector face remain unchanged. 123 926 D. Lim et al. Fig. 15 Representative pressure data showing minimal thermal drift Fig. 14 Twenty-three pressure signals plotted on the same graph show- ing consistent strength and proﬁle of detonations in the consistent and repeatable pressure-wave measurements throughout all 95 experiments conducted for Designs L, M, 4.1 Pressure data S, and P . The representative pressure trace chosen is shown in Fig. 15. Over half of the total sets of pressure data that we collected showed unexpected excursions shortly after the passage of the pressure wave, and the occurrence is believed to be the 4.2 Video data result of thermal drift of the pressure sensor. In earlier tests with a different design, the pressure sensor was situated Figures 16 and 17 are sequenced still images extracted from directly across from the injector oriﬁce and was therefore the videos showing the three different types of response continuously covered with a ﬁlm of water. In those tests, pres- observed. Images were backlit with a 500-W halogen lamp sure excursions were not observed in the data. When water and captured at over 80,000 fps, with a 2-µs exposure. was allowed to leave the detonation passage freely, the trans- While the experiments were performed with the injectors ori- ducer was directly exposed to the hydrogen–oxygen ﬂame ented horizontally, the images shown here have been rotated during blowdown and therefore produced an increased out- 90 counterclockwise for formatting reasons. The detonation put voltage. For instances where thermal drift was minimal, channel is at the top of the images, and the injector plenum is it was likely that water droplets had come to rest directly on at the bottom. Injection direction is from bottom to top. The the transducer face due to the splashing resulting from injec- detonation wave traverses the injector face from left to right. tion recovery or passage over-relaxation (where surrounding We classiﬁed the results of the experiments under three gas gets drawn back into the detonation channel). The pres- broad categories: complete backﬂow, partial backﬂow, and sure data containing thermal drift could not be used directly limited backﬂow. Complete backﬂow is deﬁned by gaseous as input for the numerical analysis. However, analysis of all combustion products penetrating the entire length of the ori- the pressure data revealed that the detonations produced by ﬁce and becoming trapped in the plenum. Partial backﬂow the predet were consistent in strength and proﬁle. Figure 14 occurs when the gas–liquid interface propagates upstream shows 23 sets of pressure data in which thermal drift was into the injector passage with the gaseous phase occupying absent or minimal. The peak height, duration, and shape of the entire cross section of the passage. Finally, limited back- the pressure signals were largely similar. ﬂow is characterized by the case where gas occupies just a From the ﬁgure, we see that the pressure ratios of the ﬁrst fraction of the injector cross section near the exit plane. The peaks cluster around the value of three. From later tests with absence of inversion of the liquid–gas interface is also a char- a modiﬁed test article, we determined that the pressure waves acteristic of limited backﬂow. Examples of each category are traveled at speeds between 1200 and 1300 m/s. The pressure shown in the ﬁgures that follow. ratios and wave speeds both indicate that CJ detonation had Under very low-speed liquid injection conditions, com- not been achieved; instead, the reaction zone was decoupled bustion gases propagate up the injector passage all the way from the shock. Presently, we have reason to believe that into the injection plenum as illustrated in Fig. 16 (i.e., a com- the second peaks correspond to the pressure rise due to the plete backﬂow situation). Here, we should point out that decoupled reaction zone. the injection pressure was very low (6.9 kPa or 1 psi), so The next best option was to choose a set of data that was this condition is likely not representative of high-pressure relatively unaffected by thermal drift and had peak values combustion conditions in a pressure gain device. Gas ﬁrst similar to the others, and use it as a single representative enters the injector passage through the boundary layer on input for all computations. The justiﬁcation for doing so lies the upwind side of the oriﬁce and the liquid–gas interface 123 Transient response of a liquid injector to a steep-fronted... 927 Fig. 17 Sequence of images showing partial backﬂow in injector M at p = 6.9 kPa (1 psi). Camera resolution was 240 × 56 pixels at Fig. 16 Sequence of images showing complete backﬂow into the 83,333 fps plenum of injector S at p = 6.9 kPa (1 psi). Dashed lines indicate regions occupied by the gaseous phase. Camera resolution was 208 × 56 pixels at 88,888 fps appears tilted toward the upwind side of the injector passage due to this effect. For the test shown, the interface moves the entire length of the injection passage in about 300 µs with an average velocity of 12.7 m/s. Upon penetration into the oriﬁce plenum, some of the gas remains trapped in the plenum due to buoyancy effects. The period between 450 and 517 µs is presumably a time when there is nearly no liquid in the oriﬁce passage except for a small annular liquid region along the wall of the oriﬁce, evident from the visible distor- tion close to the exit plane. At 517 µs, liquid surrounding the previously continuous column of gas pinches off the column at the oriﬁce entrance and forms a new free surface. The pro- cess is apparent during video playback, but not easily seen in the still images. At 596 µs, the free surface becomes more Fig. 18 Sequence of images showing limited backﬂow in injector S at visible as recovery begins. Note that the liquid–gas interface p = 28 kPa (4 psi). Dashed lines show the approximate edge of the is tilted toward the downwind side of the oriﬁce passage in ejected liquid. Camera resolution was 208 × 56 pixels at 88,888 fps the last frame; presumably, the upwind side of the passage recovers ﬁrst during this highly transient process. Figure 17 depicts partial backﬂow that could presumably ﬁrst pushing into the upwind boundary layer in the oriﬁce occur if a low/intermediate liquid feed pressure (soft injection thereby leading to a tilted free surface. Similarly during the system) is employed. As with the large backﬂow condition in recovery phase, the downwind side of the oriﬁce passage will Fig. 16, the liquid–gas interface is tilted toward the upwind experience higher-pressure gas conditions last and therefore side of the injector and the interface inverts its tilt as ﬂow might cause a delayed recovery relative to the upwind side of recovers and liquid pushes out the two-phase region. This the passage. It is surprising that this multidimensional argu- interesting behavior appears consistently in the results and ment appears to hold even when the free surface is pushed a appears to be a fundamental multidimensional effect. Further substantial distance upstream into the oriﬁce passage. Here, tests performed with the test article rotated 180 such that the the backﬂow duration is of the order of 400–500 µs. Flow pressure wave traveled from bottom to top (not shown) dis- recovery appears to occur over a similar time interval. played identical surface behavior, i.e., the surface tilts upwind Figure 18 shows a limited backﬂow situation that is per- during the backﬂow phase and downwind during recovery haps most relevant to high-pressure operation and injection phase, indicating that gravity has little effect on the shape conditions. Here, the manifold pressure was sufﬁciently high of the free surface. One can imagine the high-pressure gas to prevent the injector from checking off completely; this 123 928 D. Lim et al. Fig. 20 Plot of absolute reﬁll time versus non-dimensional pressure Fig. 19 Plot of absolute backﬂow distance versus non-dimensional drop pressure drop behavior might be characterized as a stiff injection system. tests at each condition, and in general, the data show a very The yellow dashed lines show the approximate edge of con- repeatable performance with the exception of a few outliers. tinued liquid ﬂow from the downwind region of the oriﬁce In Fig. 19, all designs exhibit a nonlinear relation between even while the upwind portions of the oriﬁce were under- backﬂow and pressure drop. As one might expect, the short- going backﬂow. These dynamics tend to be more readily est injector (Design S) had the largest amount of backﬂow apparent in the video playback. Even though liquid ﬂow per- since it had the smallest amount of liquid mass in the pas- sists at the injection plane, the ﬂow rate is very low relative sage to displace. Designs M and L then follow in reducing to the full-ﬂow condition. The third image (middle) shows order. While Designs P and S have the same oriﬁce length the injector in a state of backﬂow, and the fourth shows it in and diameter, the plenum in Design P drastically reduced the the process of recovery. In both of these images, the slope amount of backﬂow. This suggests that at least some of the direction of the liquid–gas interface remained the same. In ﬂuid in the plenum region is interacting with the pressure other words, the free surface tilt inversion does not tend to wave imparted by the detonation. The plenum design may occur under limited backﬂow conditions. therefore provide a mechanism to tune the injector’s resis- tance to back-pressure disturbances, and it may be desirable 4.3 Measurements from video data to consider this design feature for PGC applications. We introduce the term “reﬁll time” as the length of time We analyzed the video data to obtain backﬂow distance and for the injector to backﬂow and completely reﬁll with liquid. time taken between the arrival of the detonation wave and Reﬁll time is an important parameter as it is necessary to reﬁlling of the oriﬁce with liquid. The backﬂow distance was understand when liquid arrives into the chamber to support deﬁned to be the maximum displacement of the liquid–gas the next detonation wave passage. Figure 20 shows how the interface observed along the centerline of the injector, and absolute reﬁll time varies as a function of P/P .For the reﬁll time was deﬁned as the time between the ﬁrst observ- conditions tested, the reﬁll times are all quite long relative able arrival of the pressure wave and complete reﬁlling of to that which might be acceptable in an RDE application. the injector with liquid. Under limited backﬂow conditions, As the oriﬁce pressure drop/velocity is increased, reﬁll time portions of the oriﬁce continue to ﬂow in the positive direc- can be all but eliminated. Nevertheless, we investigated the tion (into the detonation channel) and the free surface may soft injection conditions to assess regions where signiﬁcant never cross the centerline, resulting in zero measured back- backﬂow and reﬁll times exist. In general, the results show a ﬂow distance. For this reason, the measurements are more surprising lack of sensitivity to injector design at low values qualitative, but have value for representing gross trends. of P/P . Designs M and S have reﬁll times that almost Figures 19 and 20 show measured backﬂow distance and overlap with each other, with Design L’s data points also in reﬁll time as a function of dimensionless pressure drop close proximity. At higher P/P , Designs P and L have P/P for all injector types. Note that while the injector noticeably smaller reﬁll times. The plenum appears to serve pressure drop is a dynamic value during tests, the P on a role of isolating the impact of the dynamic response to the the abscissa represents only the initial pressure drop. The near-exit region. interested reader may refer to  for the absolute manifold The relative insensitivity of backﬂow distance and reﬁll pressure values in each test. We conducted a total of ﬁve time to injector design seems to suggest that the injector’s 123 Transient response of a liquid injector to a steep-fronted... 929 Fig. 21 Schematic summarizing observed behavior in near-exit region of the oriﬁce due to a passing strong pressure wave response is a local phenomenon. In general, the static pres- sure at the injector exit is the same regardless of injector length, and the local ﬂow processes appear to be primarily controlled by this parameter. However, the boundary layer thickness is also playing a role as it is evident that gas pen- etration begins in the upstream boundary layer region and can be more pronounced when the boundary layer is thicker. Figure 21 provides a schematic representation of the events based on this assertion and our observations. As a result, the injectors showed similar responses even though their lengths differed by up to a factor of two. 4.4 Comparison of predictions with measurements We used the lumped-parameter model outlined in Sect. 3 to Fig. 22 Sample output of predicted injector response. Upper left: input pressure signal, lower left: net acceleration on liquid, upper right: inter- simulate the response of each injector type. Figure 22 pro- face velocity, lower right: interface location vides a sample output from one of the simulations in response to the measured overpressure in Fig. 15. The lower left plot shows the acceleration on the liquid in the injector oriﬁce caused by the pressure pulse. As expected, it has the same proﬁle as that of the pressure signal. The velocity proﬁle in the upper right shows how quickly the liquid–gas inter- face moves within the injector. Positive values indicate ﬂow toward the detonation channel, and negative values signify backﬂow. The last and most important plot is the time his- tory of the liquid–gas interface location. From this plot, the two parameters of interest are obtained: maximum backﬂow distance and recovery time. These are the two measurable quantities from the experiments and are therefore the bases of comparison. We made comparisons between the lumped-parameter model and the experimental results for Designs L, M, and Fig. 23 Plot of non-dimensional error in backﬂow distance versus ini- S. We chose to exclude design P since the 1-D model did not tial Reynolds number include provisions to consider plenum geometry and Design P deviated signiﬁcantly from a plain oriﬁce. Figure 23 shows model prediction errors for dimensionless backﬂow distance issue here appears to be the non-planar nature of the free sur- (as a percentage of oriﬁce length) as a function of the injection face. Despite the small time scales, the ﬂuid in the passage Reynolds number computed based on the imposed pressure does recognize the direction the wave is traversing and the drop. Errors in predicted backﬂow in Fig. 23 are very large at surface is tilted as such. However, the large errors occur in low injection Reynolds numbers, and the backﬂow distance a domain where the injection velocity is very low, and these is vastly under-predicted by the simple model. The major are not typical conditions that would be expected in a func- 123 930 D. Lim et al. 0.0 may be required to better replicate recovery time peri- ods. -0.5 While the recovery results display longer times than we -1.0 might expect with the simple model, the injection condi- -1.5 tions where dynamic surface behavior is observed occur at very low manifold pressures, and pressure drops more con- -2.0 2500 3500 4500 5500 6500 7500 sistent with operational devices display minimal recovery times from our study. Study of these physics at high ambi- Design L Design L Limited Backﬂow ent pressures is necessary to conﬁrm the expected behavior, Design M Design M Limited Backﬂow i.e., minimal recovery time for “stiffer,” high-pressure injec- Design S Design S Limited Backﬂow tors. Unfortunately, oriﬁce cavitation becomes a concern Fig. 24 Plot of non-dimensional error in reﬁll time versus initial when replicating these high pressure drops with ambient Reynolds number back-pressure and these concerns have limited the range of injection conditions that could reasonably be studied with tioning RDE. The model has a more reasonable agreement at this initial research effort. higher Reynolds number injection conditions, but the overall backﬂow in these cases is also much smaller. The non-dimensional error in reﬁll time comparison is 5 Conclusions shown in Fig. 24. The one-dimensional model consistently under-predicts the reﬁll time by about a factor of 1.75– We studied the transient response of a liquid injector sub- 2.25 (actual reﬁll time 75–125% larger than the prediction). jected to a steep-fronted transverse pressure wave by expos- Clearly, these errors indicate that the 1-D treatment does not ing a single plain-oriﬁce injector to a weak hydrogen–oxygen capture the fundamental physics at work. Three potential detonation in a transparent structure. Water was the injected phenomena could play a role in the disparity: ﬂuid, and injection differential pressures of 6.9–34.5 kPa (1– 5 psi) were used in injectors that varied in length from 3.81 to (i) Multidimensional ﬂow effects The high-speed imaging 7.62 mm (0.15–0.30 in.). High-speed videos and companion displays a surface that is far from planar. The interface high-frequency pressure measurements provided simultane- tilts toward the passing wave during backﬂow and undu- ous pressure and surface shapes during ﬂuid backﬂow within lates toward a planar surface during the recovery phase. the injector. We also created a one-dimensional ﬂow model Clearly, a 1-D model cannot capture this complex mul- to assess abilities in predicting the measured response on this tidimensional motion. basis. (ii) Viscous effects As the injection Reynolds numbers are Results have shown that the behavior of the liquid is far very low in cases that display dynamic free surface from one-dimensional. Instead, the mechanism for backﬂow motion, boundary layers are thick and viscous forces is complex because of the boundary layer dynamics that most play a large role. The dynamic surface shapes that are likely play a major role in gas penetration, especially at low observed are clearly affected by this factor as the inter- injector Reynolds numbers. Speciﬁcally, the detonation wave face pushes furthest into the low-momentum ﬂuid in appears to ﬁrst propagate into the injector along the boundary the boundary layer on the upstream side of the oriﬁce; layer on the upwind side of the oriﬁce. Because the high- the simple model does not properly account for these pressure gas ﬁrst propagates into this region, the free surface physics. is tilted upwind during backﬂow. An interesting reversal of (iii) Dynamic manifold response The model presumes a con- the surface tilt is observed during ﬂow recovery, except in stant manifold pressure, whereas the manifold will also cases when the overall extent of backﬂow is limited. respond as the compression wave from the passing deto- The maximum extent of backﬂow is strongly correlated nation travels forward into this region. Wave reﬂections with the injection pressure drop, and higher P injection within the manifold will create an expansion wave that can limit or eliminate backﬂow. In general, longer oriﬁce will travel back downstream in an attempt to equilibrate passages tend to exhibit less backﬂow since a larger mass pressures after the passage of the detonation. This wave of liquid must be accelerated by the high-pressure gases. We will temporarily decelerate (or even stagnate) the inter- uncovered an interesting behavior with a design that featured face until additional wave reﬂections serve to recover the a small plenum behind the injection oriﬁce. The reduced initial manifold pressure prior to the violent event. For plenum diameter tended to limit backﬂow, and this design these reasons, a compressible treatment of the ﬂow pas- feature might offer a mechanism to control dynamic response sage, with a full consideration of the manifold design, in practical systems. Non-dimensional Error (Reﬁll) Transient response of a liquid injector to a steep-fronted... 931 While the backﬂow results exhibited signiﬁcant trends 3. Bykovskii, F.A., Zhdan, S.A., Vedernikov, E.F.: Continuous spin detonations. J. Propul. Power 22(6), 1204–1216 (2006). https://doi. with differing injector designs, the reﬁll time (time for the org/10.2514/1.17656 free surface to return to the oriﬁce exit plane) did not dis- 4. Miesse, C.: The effect of ambient pressure oscillations on the disin- play strong inﬂuence from injector design and all concepts tegration and dispersion of a liquid jet. Jet Propul. 25(10), 525–530 that we studied had similar behavior, at least at low injec- (1955). https://doi.org/10.2514/8.6813 5. Reba, I., Brosilow, C.: Combustion instability: liquid stream and tion pressure drops. At higher pressure drops more realistic droplet behavior. Part III: the response of liquid jets to large of actual operating conditions, reﬁll time was shorter for the amplitude sonic oscillations. WADC TR 59-720, Wright Air Devel- longer injection element and the design employing the nar- opment Center, United States Air Force (1960) rower plenum feature. Once again, the plenum appears to 6. Harrje, D., Reardon, F. (eds.): Liquid propellant rocket combustion instability, pp. 373–377. NASA SP-194 (1972) offer features that might be desirable for operational devices. 7. Nurick, W.H., Gill, G.S.: Liquid rocket engine injectors. NASA The experiments carried out at atmospheric pressure indi- SP-8089 (1976) cate that while the relatively low injection pressures were 8. MacDonald, M., Canino, J., Heister, S.D.: Nonlinear response unable to prevent backﬂow from occurring, it only took of plain-oriﬁce injectors to nonacoustic pressure oscillations. J. Propul. Power 23(6), 1204–1213 (2007). https://doi.org/10.2514/ approximately 20.7 kPa (3 psi) of pressure drop to resist the 1.31189 pressure wave to the point where backﬂow was only limited. 9. Rump, K.M., Heister, S.D.: Modeling the effect of unsteady cham- Since it is impractical to completely eliminate backﬂow due ber conditions on atomization processes. J. Propul. Power 14(4), to the scaling of detonation pressure with initial pressure, it 576–578 (1998). https://doi.org/10.2514/2.7645 10. Heister, S.D., Rutz, M., Hilbing, J.: Effect of acoustic perturbations is a likely scenario that we would want to design injectors to on liquid jet atomization. J. Propul. Power 13(1), 82–88 (1997). operate in the limited backﬂow regime. https://doi.org/10.2514/2.5132 Lastly, the 1-D model shows some promise in the predic- 11. Bazarov, V.G., Lyul’ka, L.A.: Nonlinear interactions in liquid pro- tion of backﬂow distance at higher initial Reynolds numbers, pellant rocket engine injectors. In: 34th AIAA/ASME/SAE/ASEE Joint Propulsion Conference and Exhibit, Joint Propulsion Confer- but lacks accuracy in predicting reﬁll time, whose predic- ences, AIAA Paper 1998-4039 (1998). https://doi.org/10.2514/6. tion error did not appear to show any direct dependence on 1998-4039 Reynolds number or injector length. Assessing the model 12. Ismailov, M., Heister, S.: Dynamic response of rocket swirl injec- in the higher Reynolds number/injection velocity conditions tors, part I: wave reﬂection and resonance. J. Propul. Power 27(2), 402–411 (2011). https://doi.org/10.2514/1.B34044 is desirable because oriﬁce cavitation limited the range of 13. Ismailov, M., Heister, S.: Dynamic response of rocket swirl injec- injection velocities in this study. We desire to perform fur- tors, part II: nonlinear dynamic response. J. Propul. Power 27(2), ther investigations at higher-chamber-pressure conditions as 412–421 (2011). https://doi.org/10.2514/1.B34045 it will permit the use of higher injection velocities and thin- 14. Ahn, B., Ismailov, M., Heister, S.: Experimental study swirl injec- tor dynamic response using a hydromechanical pulsator. J. Propul. ner boundary layers that will perhaps make the model more Power 28(3), 585–595 (2012). https://doi.org/10.2514/1.B34261 relevant. 15. Kim, B.-D., Heister, S.D.: Two-phase modeling of hydrodynamic instabilities in coaxial injectors. J. Propul. Power 20(3), 468–479 Acknowledgements We gratefully acknowledge the Air Force Ofﬁce (2004). https://doi.org/10.2514/1.10378 of Scientiﬁc Research (Contract FA9550-14-1-0029) for supporting this 16. Kim, B.-D., Heister, S.D., Collicott, S.H.: Three dimensional ﬂow research. simulations in the recessed region of a coaxial injector. J. Propul. Power 21(4), 728–742 (2005). https://doi.org/10.2514/1.12651 Open Access This article is distributed under the terms of the Creative 17. Canino, J.V., Heister, S.D., Sankaran, V., Zakharov, S.I.: Commons Attribution 4.0 International License (http://creativecomm Unsteady response of recessed-post coaxial injectors. In: ons.org/licenses/by/4.0/), which permits unrestricted use, distribution, 41st AIAA/ASME/SAE/ASEE Joint Propulsion Conference and and reproduction in any medium, provided you give appropriate credit Exhibit, AIAA Paper 2005-4297 (2005). https://doi.org/10.2514/ to the original author(s) and the source, provide a link to the Creative 6.2005-4297 Commons license, and indicate if changes were made. 18. Tsohas, J., Heister, S.D.: Numerical simulations of liquid rocket coaxial injector hydrodynamics. J. Propul. Power 27(4), 793–810 (2011). https://doi.org/10.2514/1.47761 19. Brady, B.: Transient ﬂuid ﬂow in short-pulse operation of bipro- pellant thrusters. J. Propul. 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