Analytical viscous flow model and test validation of a swirl injector for rocket engine application

The generally adopted flow model inside a swirl injector, widely used injection concept for propulsive applications, relies upon the hypothesis of ideal flow neglecting the fluid viscosity effects. This model showed significant prediction errors with relatively high viscosity propellants, often leading to the need of an experimental characterization of the injection elements. In this paper, an analytical approach is presented, which includes the effects of viscous diffusion on the injector performance leading to a close form flow solution. The built model is thus experimentally validated testing a liquid oxygen (LOx) and an ethanol injector: the good agreement between the model and the experimental results leads to the construction of the injectors operational maps describing the injector behavior even in the presence of viscous effects.

The generally adopted §ow model inside a swirl injector, widely used injection concept for propulsive applications, relies upon the hypothesis of ideal §ow neglecting the §uid viscosity e¨ects. This model showed sig-ni¦cant prediction errors with relatively high viscosity propellants, often leading to the need of an experimental characterization of the injection elements. In this paper, an analytical approach is presented, which includes the e¨ects of viscous di¨usion on the injector performance leading to a close form §ow solution. The built model is thus experimentally validated testing a liquid oxygen (LOx) and an ethanol injector: the good agreement between the model and the experimental results leads to the construction of the injectors operational maps describing the injector behavior even in the presence of viscous e¨ects.

INTRODUCTION
The use of a swirl injector to feed the combustion chamber of a rocket engine has proven to be a powerful choice in terms of atomization and mixing e©ciencies [1], combustion stability margin, and throttleability [2] together with high mass §ow per element. These features are gained through the tangential introduction of the propellant in the swirl chamber that realizes an angular momentum-driven §ow development. Nevertheless, the high §uid/wall interfacial areas make this injection concept very sensitive to viscosity e¨ects [2] since the Boundary Layer (BL) height can become comparable to the dimensions of the spreading propellant ¦lm thickness. The existing §ow model, developed in the late 1970s by Bazarov et al. [1], relies upon the hypothesis of inviscid §ow: solving the §ow equation in this frame gives, in fact, the possibility to correlate the applied pressure drop -p to the expelled mass §ow ' m by means of a dimensionless coe©cient µ known as mass §ow discharge coe©cient: where R N is the radius of the injector outlet as shown in Fig. 1 and ρ is the density of the driven propellant. De¦ning two other dimensionless groups, the dimensionless circulation C and the ¦lm passage fullness coe©cient ϕ as , the Bazarov inviscid theory is able to correlate the mass §ow coe©cient µ to the injector geometrical features: where R N , R in , and r in are the radii of the nozzle outlet, the injection arm, and the tangential inlets, respectively, geometrical dimensions of the injector as shown in Fig. 1; n is the number of tangential inlets; A ¦lm and A N are the crosssectional areas of the §uid ¦lm and the nozzle outlet; and r gc is the gaseous core radius. Experimental data for this injection concept have been obtained within the SMART Rockets project [3], in which a 500-newton liquid propellant engine is designed with the goal of propel a payload-carrying sounding rocket [4]. This engine in fact features a coaxial swirl injector assembly in which LOx and ethanol are driven to feed an oxide ceramic matrix composite (OCMC) combustion chamber. Experimental data obtained with LOx and ethanol, two §uids with relatively low and high viscosities ( viscosity case while signi¦cant deviations are observed in the second case [7] (see section 4). This prompts viscous di¨usion e¨ects to be the cause of the mismatch between the ideal theory and the experiments and at the same time highlights the need for a model accounting for the §uid viscosity to characterize the injector with a pressure-mass §ow correlation. In this paper, an analytical approach to account for viscosity e¨ects on the injector discharge capability is presented: the BL structure inside the swirl chamber is modeled in terms of the ideal §ow parameters obtaining a close form solution for the viscous discharge coe©cient. The obtained results avail to build the operative map of the swirl injector that is then compared to experimental data obtained via cold §ow testing of the injectors.

VISCOUS FLOW MODEL
The ¦rst step for the inclusion of viscous e¨ects being U and L the characteristic velocity and length of the §ow and µ l the viscosity of the §uid. In order for this parameter to be representative of the §ow viscous behavior, appropriate values should be chosen for the characteristic length and velocity. With reference to Fig. 2, by de¦nition of Reynolds number, what matters is the length across which the §uid is pushed by the inertial forces. In this particular §ow con¦guration, this length is well represented by the ¦lm thickness h sw while, due to the three-dimensional (3D) nature of the problem, the reference velocity should be taken as the total velocity V .
Specializing the thickness and the total velocity at the injector nozzle exit section, so to earn ease of representation, and rearranging the Reynolds de¦nition with the help of the ideal §ow relations [1], the total velocity can be expressed in terms of the injector mass §ow. In this way, the overall Reynolds number inside the swirl chamber can be de¦ned as a function of the injector mass §owrate: where the thickness of the annular section has been expressed using the thin annulus approximation valid for h sw ≪ R N (if this hypothesis does not hold, the actual thickness is computed without simpli¦cations) and the de¦nition of passage fullness coe©cient ϕ: Hence, As an example, Table 2 shows the values of the de¦ned swirl Reynolds number for different §ows of liquid nitrogen (LN 2 ), LOx, and ethanol, speci¦ed for the LOx and ethanol injectors built within the SMART Rockets project [4].
The values of the Reynolds number con-¦rm what has been observed in the injectors tests: for LOx and LN 2 , the Reynolds number is one order of magnitude higher than the one of the ethanol, no sensible viscosity e¨ects should be noticed so that the inviscid model should be capable of accurately predicting the empirical results. The opposite happens on the ethanol line since the very low Reynolds, justi¦ed by higher viscosity of the §uid, makes the problem di©cult to be realistically described when neglecting viscosity e¨ects that play an important role in the §ow development.
What happens is that viscous di¨usion causes a mass §ow reduction through the injector because of the velocity strati¦cation near the wall. The nonslip condition at the wall forces the particles entrained in the §ow near the swirl chamber wall to have zero velocity but moving away from the injector surface, the §ow gains its nominal velocity since viscosity e¨ects are con¦ned in the vicinity of the wall. This causes a strati¦cation of the velocity pro¦le that can no longer be considered uniform but radially varying. Recalling the concept of the BL theory, it can be stated that being the BL a region near the solid wall where all the viscosity e¨ects are con¦ned, the velocity varies from zero at the wall to the inviscid value at the BL edge since outside the BL, no viscosity e¨ect is felt; thus, the §uid behaves as it would in an inviscid §ow (Fig. 3).
The consequence of the viscosity-induced velocity pro¦le strati¦cation is that in the viscous case, the mass §ow entrained in the BL thickness is actually lower than the one that would be entrained in the same length in an inviscid §ow. Since the velocity outside the BL is uniform and equal in both cases, it can be stated and veri¦ed by direct integration that the overall mass §ow in the viscous case is reduced because of viscous di¨usion e¨ects: The curious fact is that this process of mass §ow reduction takes places in both ethanol and LOx §ows with the same mechanism: however, being the BL height inversely proportional to the square root of the Reynolds number, in the LOx §ow, Re is so high that viscosity e¨ects are con¦ned in an extremely thin region near the wall, so that the decrease in mass §ow becomes negligibly small. This rather simple explanation of the viscous di¨usion in the injector §ow well describes what has been observed experimentally for the two di¨erent §ows.
The focus is, therefore, shifted in ¦nding the velocity distribution in the BL and use (4) to obtain the mass §ow defect. In particular, what matters is the BL pro¦le at the end of the injector duct: the BL thickness increases along the axial coordinate of the injector and the representative ¦gure for the mass defect is the integration of the velocity pro¦le at the injector outlet where the viscosity-dominated region has the highest thickness. A convenient choice that suites this problem could be the use of the de¦nition of the BL displacement thickness δ * . This characteristic length value is the thickness that should be added to the wall (or equivalently deprived to the §ow near the wall) in order to obtain an equivalent uniform §ow, with uniform velocity equal to that outside the BL, whose mass §ow equals the one that would be obtained by integration of the actual velocity pro¦le, as sketched in Fig. 4.
The next step would be to evaluate the displacement thickness at the injector outlet and then treat the §ow inside the injector as uniform like in the inviscid case, but on a ¦ctitious geometry in which the wall is shifted by a δ * length. In this way, the mass §ow §owing through this ¦ctitious geometry is the actual mass §ow through the injector including viscous mass defects.

Figure 4
Boundary layer displacement thickness [8] Evaluation of the BL mattering thicknesses is, however, not a trivial challenge, especially due to the 3D nature of the §ow con¦guration. Anyway, known solutions for pretty di¨erent §ow con¦gurations can be taken as a reference. The relevant di¨erences in terms of geometric and §uid-dynamic e¨ects between the actual and the reference §ow con¦gurations can thus be discussed to assess the limits of validity and the di¨erences that those e¨ects bring on the growth of the BL thickness that is the primary quantity of interest. The Blasius solution for the BL developing across a §at plate §ow can be taken as a reference: in this simple §ow con¦guration, no pressure gradients are experienced so that the BL grows naturally with the §ow coordinate. The solution for the growth of the displacement thickness as a function of the §ow coordinate is [9]: δ * (x) = 1.7208 x where the x coordinate represents the portion of the wall length wett by the §uid. The di¨erent factors in §uencing the growth of the BL in the actual con¦guration with regard to the reference one come from two e¨ects: (1) presence of a negative axial pressure gradient toward the injector nozzle due to propellant acceleration; and (2) e¨ect of curvature developing centrifugal forces inside the §ow.
The ¦rst e¨ect is usually bene¦cial in terms of growth of the BL since negative pressure gradients along the streamwise direction are stabilizing and retard the BL£s thickness growth. The second e¨ect requires a little bit more attention since centrifugal forces set up due to the convex curvature build up a radial pressure strati¦cation, according to the local equilibrium of a §ow element, the pressure di¨erential in the radial direction is: where R is the local radius of curvature of the §ow, of the order of the chamber radius, and depending on the axial-to-azimuthal velocity ratio. The important fact is that the pressure gradient in the radial direction is positive, meaning that the §ow is pushed against the wall by centrifugal e¨ects (centrifugal pumping), leading to a further decrease of the BL thicknesses with respect to the reference ones.
As a result, inclusion of the e¨ects of the di¨erent §ow con¦gurations with regard to the Blasius one gives that the BL£s properties, in particular, the displacement thickness, grow in a slower fashion with the §ow coordinate meaning that the use of the Blasius pro¦le turns up in a conservative overestimation of the viscosity e¨ects. This fact is really encouraging because if it were the opposite, namely, the BL thickness was underestimated, the Blasius model could no longer have been used because the faster growth of the BL in the injector would have probably lead to separation of the §ow, condition at which the BL behaves in a totally di¨erent fashion with respect to the reference one.
Drawing conclusions, the use of a Blasius pro¦le still well describes the mechanisms that lead to the mass §ow defect bringing an acceptable overestimation of the BL thicknesses with the advantage of a handy analytical solution amenable to be easily accommodated into the swirl injector §ow model. Accordingly, the BL£s displacement thickness for the swirl injector can be expressed in a convenient way using the de¦nition for Re sw (2) and the ¦lm thickness (3): where the coordinate x sw is now the curvilinear abscissa of the helical path of the §ow through the swirl chamber, representing the length of the wet wall£s surface. This is the mattering length for the evaluation of the BL growth along the path of the §uid inside the injector as sketched in Fig. 5. This parameter can be found by geometrical considerations about the helicallike pattern of the §ow as it is intuitive the shorter the pitch of the helix, the longer the wet surface and vice versa, since the axial length of the helix is ¦xed by the injector size. An exact characterization of the swirled path can be done with the help of the inviscid theory using the result for the spread angle or equivalently the axial-to-total velocity ratio. The use of the inviscid theory for the evaluation of the path turns to be helpful in obtaining a ¤linearized¥ viscous result as a correction of the inviscid one that stands as a reference.
In a cylindrical reference frame, the parametric equations of a helical curve of radius R, pitch p, and azimuth ϑ are: x = R cos ϑ ; y = R sin ϑ ; z = p 2π ϑ .  The spread angle β of the helix can be related to the inviscid §ow parameters via cos β = w V = µ ϕ and by geometrical analysis, the relation holding between the pitch and the spread angle (Fig. 6) is: The goal of the geometrical analysis is to ¦nd the variation of the §ow coordinate x sw and, in particular, its value at the end of the nozzle (z = z N ) as a function of the §ow parameters. This length can be found by straight integration of the helix equation; with few calculus, it is obtained: and with the variable substitution the wet path length l sw = x sw (z n ) is obtained: Hence, it is now possible to express the displacement thickness evaluated at the nozzle outlet section. Leading parameter for the expression of the viscous mass defect: Being able to pass into the equivalent uniform §ow in which the wall thickness is shifted by a displacement thickness length and remaining in the thin ¦lm approximation, the defect mass §ow can be found via a simple proportion. In fact, dealing with uniform §ows, the defect mass §ow and the displacement thickness stand in the same relation of the ideal mass §ow and the total thickness ¦lm, the same can be stated for the mass §ow coe©cients: Hence, with the de¦nition (3) of the ¦lm thickness, Finally, using (5), the relation for the viscosity-corrected mass §ow discharge coe©cient writes: This relation ¦nally gives the mass §ow coe©cient of the injector accounting for viscous losses. The new dimensionless parameter µ vis has the same task of the ideal one for performance prediction: relate the pressure drop to the mass §ow established in the injector via (1). This time, the mass §ow coe©cient does not only depend on the injector£s basic geometry like it was in the ideal model. The longitudinal dimension of the injector is considered since it ¦xes the length of the wet surface on which viscous e¨ects act. There is also a mass- §ow dependence through the swirl Reynolds number: the higher the velocity of the §owing §uid, the slower the growth of the BL, meaning that going towards higher mass §ows (increasing Re) the viscosity e¨ects are less and less important. Substitution of the Re sw de¦nition (2) into (6) allows to plot the µ vis / ' m characteristic curve and to compare it to the inviscid theory. Figure 7 shows the trend of the mass §ow coe©cients of the LOx and ethanol injectors compared to the ideal theory.
As Fig. 7 shows, there is no sensible shift between ideal and viscous model on the LOx injector, the low viscosity and, in particular, the high Reynolds number realized in the injector make the ideal model an accurate tool for the injector performance prediction. A di¨erent situation is depicted on the ethanol injector, ideal and viscous models give con §icting results leading to an evident shift in the mass §ow coe©cient prediction. Even the trend to increase the discharge capability toward high mass §ows is more pronounced emphasizing the ethanol injector sensitivity to Reynolds number changes. 3 ¡ pressure sensor; 4 ¡ manometer; 5 ¡ pressure control valve with manual pressure release; 6 ¡ pressure control valve; 7 ¡ venting valve; 8 ¡ ¦lter; 9 ¡ gate valve; 10 ¡ main valve ethanol; 11 ¡ main valve LOx; 12 ¡ §owmeter; 13 ¡ venturi tube with di¨erential pressure manometer; 14 ¡ solenoid control valve (2/2-way); 15 ¡ solenoid control valve (3/2-way); 16 ¡ solenoid control valve (2/2-way); and 17 ¡ combustion chamber

EXPERIMENTAL SETUP
To validate the built viscous model, the experimental apparatus realized within the SMART Rockets project [10] has been set for the cold §ow testing of the injectors. A scheme of the test bench is shown in Fig. 8.
The test facility built for the sake of testing a 500-newton LOxethanol propelled engine features two propellant tanks for the LOx (cryogenic proven) and ethanol, respectively, both pressurized by a gaseous nitrogen tank and connected with the injectors. Several sensors are installed on the propellants lines: temperature sensors are mounted in the LOx Figure 9 The CAD section view of the LOx (a) and ethanol (b) injectors [7] tank to control the vapor fraction and to monitor the fueling cycles of the cryogenics, pressure sensors are mounted inside the tank and before the injectors to control the line pressures. Mass §ow sensors on each line measure the §owrates and, in particular, the LOx mass §ow sensor is also able to measure the §uid density accounting for vapor presence in the cryogenic that always works near saturation conditions. The connection between the test bench and the injector has been realized using a copper pipe on the ethanol line and a thermally insulated pipe on the LOx line, insulation of the LOx line and cooling cycles proved in fact to be a critical challenge for obtaining vapor free shots of the cryogenic during the test campaigns. Figure 9 shows the CAD (computer-aided design) section views of the two injectors used for the tests.
Both injectors are fed via three tangential inlets connected to the feeding lines of the test bench and are designed to work in a coaxial fashion. Both injectors have been previously designed using the inviscid theory [3,10] and this gave a substantial shift between predictions and test observations. It is through the validation of the viscous §ow model that a more reliable prediction of the injectors mass §ow coe©cients will be obtained, giving the possibility to readjust the injectors internal geometries and to adapt the coaxial assembly into the propulsion system.
The manufacturer of the pressure sensors claims a measurement error of 0.2 bar while the maximum mass §ow error is 5 g/s, satisfactory values since the tests are realized in the neighborhood of 10-bar pressure and 150 g/s mass §ow, way far from the sensor error range. Attention should be paid to realizing stable and steady §ows since both results of ideal and viscous theory were obtained under the hypothesis of steady §ow ¦eld.

TEST RESULTS
The test campaign has been conducted using ethanol and the simulant §uid LN 2 in place of LOx for safety reasons. Being LN 2 a cryogenic §uid with similar transport properties of LOx, the results can be reliably scaled. The injectors have been tested in two di¨erent cold §ow fashions, single §ow and combined ethanolLN 2 §ow in order to check if any mutual throttling e¨ect exists when the two injectors work in the coaxial fashion, providing a mixing zone to the §uids before the jet expulsion. Arranged the test bench for the particular test to be run, several shots were obtained with the goal of reaching steady-state conditions for a representative time interval. The cryogenic shots were always preceded by cooling preshots in order to cool down the LOx line obtaining shorter settling time for the §uid to become vapor-free, realizing thus steadier test runs.
For both the prescribed §ow fashions, a pressure range 520 bar has been covered with a corresponding mass §ow range 50200 g/s in order to conveniently simulate the range of interest for the rocket engine the injectors belong to. Steady-state values of pressures, mass §ows, and densities are evaluated and compared to the ideal and viscous predictions and Eq. (1) is used with the measured data to obtain the observed values of the mass §ow coe©cients.
The ¦rst result obtained is that no sensible throttling e¨ect has been observed with the injectors working in combined §ow fashion. This is an important result since the combined §ow fashion is the actual working condition of the injectors in the engine. The absence of mutual throttling e¨ects means that in the design phase, the operative map of each injector can be used by itself in order to set the geometric and pressure parameters, then the injectors can be assembled maintaining the same pressure §owrate relations.
Results in terms of mass §ow coe©cient are shown in Figs. 10 and 11 [7]; for the LOx injector, previous results obtained with LOx cold §ow tests are also displayed: the LOx injector results show a good agreement between empirical observation and theory, once again there is no substantial di¨erence between ideal and viscous theory given the high values of the Reynolds number for this kind of §ow. A certain degree of scattering of the test results is observed due to the density §uctuations of the cryogenics, symptom of a certain vapor fraction in the §uid also con¦rmed by the slightly lower density observed, especially in the lower mass §ow range, index of lower 1 ¡ ideal; 2 ¡ viscous; 3 ¡ Ethanoltests; and 4 ¡ Ethanol-combined injection pressure thus reduced saturation temperature of the cryogenic; and results obtained with the ethanol injector con¦rm with excellent accuracy the viscous theory: in addition to being very close to the predicted values, the test results con¦rm the trend of discharge capability reduction towards decreasing §owrates indexes of lower Reynolds numbers thus higher in §uence of viscous e¨ects in the §ow. In addition, the analytical prediction slightly underestimates the discharge coe©cient (overestimates the viscous e¨ects) and this is in accordance with the hypotheses holding behind the viscous model, in particular, the use of a Blasius BL pro¦le parametrized around the swirl path. This assumption in fact neglects the bene¦cial pres- Figure 12 The LOx injector map: 1 ¡ viscous pressure drop; 2 ¡ inviscid pressure drop; 3 ¡ test LOx; and 4 ¡ test LN2 Figure 13 Ethanol injector map: 1 ¡ viscous pressure drop; 2 ¡ inviscid pressure drop; 3 ¡ test single; and 4 ¡ test combined sure gradients coming from the §uid centrifugation that retard the growth of the BL slightly increasing the discharge capability of the injector.
Using the new de¦nition of the discharge coe©cient (6), it has been possible to build the new pressuremass §ow correlation and to design the operational maps of the injectors. Figures 12 and 13 show the operative maps of both injectors together with the obtained test results highlighting the di¨erence between ideal and viscous maps [7].

CONCLUDING REMARKS
A viscous §ow model has been built for the swirl injector, obtaining an analytical expression for the mass §ow discharge coe©cient, depending both on the injector geometrical features and on the §uid£s transport properties that characterize the §ow BL structure. A cold §ow test campaign on the injector has been conducted and con¦rmed the predictions of the viscous §ow model: test results con¦rmed with excellent accuracy the viscous predictions also highlighting the in §uence of the Reynolds regime on the injector discharge capability. This is observed very clearly in the ethanol injector since no substantial di¨erence between ideal and viscous §ow model holds for the low viscosity case of the LOx injector. Through the de¦nition of the viscous discharge coe©cient, the injectors have been characterized via pressuremass §ow correlations, key aspect for the design phase of the injector themselves and for the assignment of the line pressures in the propulsion assembly, otherwise done in an experimental way to account for the inappropriateness of the ideal model to describe highly viscous §ows. The presented model represents thus a powerful tool for the understanding and design of this injector concept even when propellant viscosity e¨ects cannot be neglected.