THEORETICAL PREDICTION OF REGRESSION RATES IN SWIRL-INJECTION HYBRID ROCKET ENGINES

The authors theoretically and analytically predict what times regression rates of swirl injection hybrid rocket engines increase higher than the axial injection ones by estimating heat §ux from boundary layer combustion to the fuel port. The schematic of engines is assumed as ones whose oxidizer is injected from the opposite side of the nozzle such as ones of Yuasa et al. propose. To simplify the estimation, we assume some hypotheses such as three-dimensional (3D) axisymmetric §ows have been assumed. The results of this prediction method are largely consistent with Yuasa£s experiments data in the range of high swirl numbers.

The authors theoretically and analytically predict what times regression rates of swirl injection hybrid rocket engines increase higher than the axial injection ones by estimating heat §ux from boundary layer combustion to the fuel port. The schematic of engines is assumed as ones whose oxidizer is injected from the opposite side of the nozzle such as ones of Yuasa et al. propose. To simplify the estimation, we assume some hypotheses such as three-dimensional (3D) axisymmetric §ows have been assumed. The results of this prediction method are largely consistent with Yuasa£s experiments data in the range of high swirl numbers. NOMENCLATURE a constant related to regression rate B blowing parameter: B z ≡ (ρu r ) w u ze /τ zw ; B θ ≡ (ρu r ) w u θe /τ θw B t thermodynamic chemical blowing parameter: B t ≡ u ez -h/(u bz h ν ) (in quasi-steady codition, equal to B z ) c f skin-friction coe©cient in axial §ow: c fz = τ rzw /(ρu ze /2); c f θ = τ rθw /(ρu θe /2) c p speci¦c heat at constant pressure C speci¦c heat of solid fuels C H Stanton number D port diameter G o oxidizer mass §ux -h enthalpy di¨erence between §ame sheet and port surface h ν total enthalpy of solid fuel gasi¦cation at ambient temperature k blocking exponent in axial §ows k ′ blocking exponent in swirl §ows l mixing length tensor L port length n mass §ux exponent for hybrids p exponent related to swirl decay in cold §ows p = −0.569Re −0.277 constant used for approximation of the function of blowing parameter in axial §ows q ′ constant used for approximation of the function of blowing parameter in swirl §ows q r radial direction heat §ux ' Q c heat §ux to the wall r radial location ' r regression rate S swirl number t time T temperature u velocity z axial location α z constant related to approximation of C fz β z constant related to approximation of C fz γ z exponent related to approximation of C fz δ boundary layer thickness ε perturbation scale η nondimensional height in boundary layer θ angle κ constant related to mixing length µ average gas viscosity ν average gas kinematic viscosity ξ swirl strength: ξ/2 ≈ S ρ density τ shear stress ϕ nondimensional velocity in boundary layer ω angular velocity ω nondimensional angular velocity: ω = (R − δ θ )ω/u θe

INTRODUCTION
Hybrid propulsion is expected to be applied for various purposes like space transportation, space tourism, and space education because of its inherent safety and low cost. In conventional hybrid rocket engines, liquid oxidizer is injected into a combustion chamber that contains solid fuel such as hydroxyl-terminated polybutadiene (HTPB) (which is a binder for solid rocket motors). Gasi¦ed oxidizer and fuel combust in the boundary layer over the fuel-port wall surface. Hybrid rockets using HTPB have been developed for many years, but as the regression rate of such rockets is low (up to 1 mm/s at the oxygen mass §ux below 100 kg/(m 2 s)), they seem to have no bright future, i. e., they are at a dead end [1]. That is why multiport systems are required in the practical use of hybrid engines using HTPB. However, such systems decrease launch capability from potential one because of remnants of the fuel in multiport engines and have an anxiety of the drop of them around the end times of combustion.
To solve this problem, swirl injection (or vortex injection) method is proposed as a way to realize higher regression rates without energetic additives or the change of fuels [2,3]. This method is to inject liquid oxidizer that has swirl velocity components. The characteristics of this injection method is that the radial pressure gradient caused by swirl makes the §ame area in the boundary layer closer to the wall of the fuel port. This e¨ect increases the amount of heat transfer from the §ame area to the wall and, therefore, higher regression rates can be achieved. Lab-scale swirl-injection hybrid rocket motors have been developed by several researchers (see, for example, Knuth et al. [2], Yuasa et al. [3]) who proved the increase of regression rates. However, there have been few studies which theoretically and quantitatively predict the increase of regression rates by swirl. They are useful for designing engines of di¨erent scale, for better understanding of various phenomena in engines, and as an alternative way to estimate the e¨ects of swirl strength on the regression rate and other properties. Furthermore, theoretical and analytical studies are §exible in the sence that they are not subject to practical limitations such that a variable is intrinsically independent on other variables, but in actual experiments, such dependence exists because of experimental equipment and conditions. For this reason, theoretical approaches can reveal hidden properties of a phenomenon while experimental approaches cannot. Moreover, current computational §uid dynamics (CFD) approaches do not always accurately simulate actual swirl §ows, and the techniques of swirl §ow simulation in CFD are under development [4]. That is why, it is important to construct the theoretical analytical approach for predicting regression rates in swirl-injection hybrid rockets.
The purpose of this paper is to extend the estimation method of regression rates developed for axial hybrid rocket engines to engines with swirl injection. To predict regression rates, one should estimate the heat §ux to the fuel port based on some already known quantities which represent the §ow ¦eld and other parameters in the combustion chamber using the following equation at quasisteady state: ' where ρ f and h ν depend on fuel species and ambient temperature (h ν includes the amount of speci¦c heat to gasify the fuel at the ambient temperature). Thus, if one has another expression for Q c , the regression rate ' r can be readily estimated. In 1960s, Marxman and Gilbert [5,6] conducted theoretical and analytical studies on boundary layer combustion and evaluated heat transfer to the fuel surface in axial hybrid motors. Their approach starts from the relationship between the heat transfer and the skin-friction suggested by Lees [7]. This relationship means that if the skin-friction with fuel blowing from the wall can be estimated, it is possible to estimate the heat §ux to the wall. Their approach to evaluate the shear stress at the wall with fuel blowing was in expressing it through the simple relationship without fuel blowing given by an empirical formula.
In the present study, the authors ¦rst added some hypotheses needed to simplify the problem and to evaluate the e¨ects of circumferential §ows, and extended the §ow ¦eld to the 3D axisymmetric pattern. However, the concept of the approach to evaluate the heat §ux of swirl-injection hybrids remains the same as that of previous studies in terms of the introduction of Reynolds analogy and the connection with the shear stress at the wall. When Reynolds analogy is considered, one has to keep in mind that the similarity between temperature and velocity ¦elds is applied to the axial component of the velocity vector rather than to the velocity vector itself. Then, the authors made an attempt to evaluate the axial component of the shear stress at the wall in swirl §ows with fuel blowing with those without fuel blowing. On the way of the evaluation, the §ow ¦eld in the boundary layer with swirl and without fuel blowing was used to study the e¨ects of swirl and blowing separately. Eventually, regression rates in swirl-injection hybrids were theoretically derived and their predicted values were compared with the experimental results of Yuasa et al. [3].

MODELING AND HYPOTHESES OF FLOWS IN SWIRL-INJECTION HYBRID ROCKET ENGINES
First of all, let us consider 10 hypotheses related, mainly, to the §ow ¦eld. These can largely be classi¦ed into two types. The assumptions of the ¦rst type Table 1 The hypotheses on swirl §ows No. Hypotheses 1 Flow in the combustion chamber is axisymmetric 2 Prandtl number in the §ow is 1 3 Flow in the boundary layer is incompressible 4 Axial velocity components are uniform in axial and radial directions except for the boundary layer over the fuel-port wall surface 5 Circumferential velocity distribution is the same as rigid-body rotation in the radial direction except for the boundary layer over the fuel-port wall surface 6 Axial velocity components in the boundary layer obey the power law when there is no blowing from solid fuel 7 Circumferential angular velocity components in the boundary layer obey the power law when there is no blowing from solid fuel 8 Swirl without fuel blowing from the solid fuel decays exponentially to the axial direction 9 Thickness of axial boundary layer is larger than that of the circumferential one 10 Heat §ux to the fuel port by heat convection is much larger than the one by radiation are the assumptions taken from the theory of Marxman et al. to simplify the complex §ow in combustion chambers. The assumptions of the second type are the newly added assumptions adopted to simplify the handling swirl §ows and corresponding to the features observed in experiments related to swirl §ows. All hypotheses assumed in this paper are summarized in Table 1. These hypotheses are adopted for the engines whose schematics are of the same type as proposed by Yuasa et al. (Fig. 1). Among the hypotheses of Table 1, No. 1 is set to simplify the §ow ¦eld. Numbers 24 and 6 are the assumptions adopted in the theory of Marxman et al. Number 2 makes it possible to apply Reynolds analogy. Number 3 is  [9], the §ow ¦elds substantiating these hypotheses are observed downstream where the e¨ects of the swirlers used are small. The reason for applying the power law to the angular velocity with No. 7 is that the de¦nition of shear stress in the radial direction parallel to the circumferential direction is expressed as µr(∂ω/∂r). Hypothesis No. 9 is adopted following the experimental results of Kito et al. [8] and Steenbergen [9] and to simplify the problem. In all their experimental results obtained downstream where the e¨ects of the swirlers are small, the axial component of the boundary layer thickness is larger than the circumferential one. Number 10 is set to simplify the §ow ¦eld and some researchers such as Karabeyoglu and Altman [10] have also adopted this assumption for this reason. This assumption is reasonable in swirl-injection hybrids. In the aspect of heat transfer into solid fuels, convective heat transfer is dominant in regions where turbulence in boundary layer is well developed [11]. The §ow created by swirl injectors seems to be more turbulent than that induced by axial injectors. Moreover, stream lines of swirl §ow are longer than those of axial §ow at the same axial position, and the turbulence in the swirl §ow in the boundary layer seems to be developed at a shorter axial position than that for the axial §ow. However, the present authors think this assumption is not always suitable if metal or carbon powders are added to fuel to increase radiation or absorption of heat during combustion.

DERIVATION OF REGRESSION RATES IN SWIRL-INJECTION HYBRID ROCKET ENGINES
The aim of this section is to theoretically derive the equation for regression rates in swirl-injection hybrid rocket motors on the basis of the hypotheses discussed in the previous section. Because the regression rates are linked to the heat §ux to the wall through Eq. (1) of energy conservation, in order to derive the expression for the regression rate in terms of the variables of swirl §ow, let us express the heat §ux to the wall using §uid dynamics and combustion considerations. Therefore, ¦rst, in the same way as Marxman et al., let us try to relate the axial velocity ¦eld with the temperature ¦eld through Reynolds analogy which is extended from Lees£s model in two-dimensional (2D) coordinates to the model in axisymmetric 3D coordinates. Because each radial partial di¨erential of them is proportional to the axial components of shear stress and heat §ux, respectively, it is possible to express heat §ux through the §ow variables in the combustion chamber (Fig. 2). Next, let us express the axial (1) and the empirical rule for the skin-friction coe©cient in 2D §at-plate boundary layer without fuel blowing makes it possible to derive the regression rates for the swirl-injection engines.

Reynolds Analogy
Reynolds analogy claims the similarity of the velocity boundary layer to the thermal boundary layer. This analogy holds because the e¨ect of axial partial di¨erentials in both viscosity and pressure terms is much smaller than radial partial di¨erentials and, therefore, can be ignored in boundary layers. Let us extend this analogy to the 3D axisymmetric coordinates.
Compare the momentum equation with the energy conservation equation. The momentum equation for the axial component can be written as where the hypothesis No. 1 was used and it was assumed that the axial gradients of shear stress and pressure are much smaller than the radial ones as an approximation which can commonly be used in the boudary layer. The energy conservation law can be written as where it was assumed that the e¨ect of viscous dissipation can be ignored as compared with the convective heat transfer coming from §ame. Assuming that Prandtl number is 1 (hypothesis No. 2), Reynolds analogy can be used and the temperature distribution can be related to the velocity distribution in the boundary layer. Thus, one can consider the radial distribution of u z to be similar to that of T in the boundary layer. This similarity can be expressed as follows: Then, the heat §ux to the wall can be linked with the shear force stress at the wall because the axial temperature di¨erential at the wall is proportional to the heat §ux to the wall and the velocity axial di¨erential is proportional to the shear stress at the wall. The di¨erence between the scales of the axial and radial di¨erentials yields the following approximations: Equations (2) and (3) yield From the de¦nition of Stanton numbers and Eq. (4), the following equation can be derived: For the sake of convenience in subsequent calculations, the nondimensional axial skin-friction parameter can be de¦ned that is called ¤axial skin-friction coe©-cient¥ where the minus sign is put to show that the shear stress at the wall is always opposit to the §ow direction. Therefore, the heat §ux to the wall can be written as ' Let us focus on the right-hand side of Eq. (6). In this expression, there is no variable related to circumferential components. Considering the fact that the circumferential energy balance is zero because of the axisymmetric §ow, this result is reasonable. Then, one can expect that the axial friction coe©cient should be a¨ected only by swirl because of the hypothesis of Marxman et al. [12]. Therefore, in the next section, an attempt is made to express the axial skinfriction coe©cient through the axial distance, mass §ux, swirl number, scale of the motor, and other variables known in advance.

Turbulent Flow Models
Next, let us express the axial skin-friction coe©cient through other parameters that can be known in advance. The ¦rst of three di¨erent turbulent stress models is the Prandtl£s mixing length theory extended to three dimensions by Czernuszenko and Rylov [13]. If each eigenvector of the mixing length tensor is parallel to each cylinder coordinate axis and all norms of the eigenvectors are the same, the axial component of the shear stress including Reynolds stress is expressed as where the circumferential partial di¨erentials are assumed to be much larger than the axial ones. The second one comes from the expression in the boundary layer theory for the evaporating surface of §at plate in 2D coordinates extended by Dorance and Dore [14] to the 3D axisymmetric §ows: where the boundary layer thickness is assumed to be much thinner than the port radius. The de¦nitions of B z and B θ are Equations (8) and (9) are derived by evaluating the Reynolds stress and blowing factor in the same way as it is made for the §at plate. Note that B z is constant throughout the fuel port while B θ is not. The reason for this treatment is discussed below in subsection 3.6.
So far, two of three turbulent models have been already introduced. From now on, let us use these equations for evaluating the velocity distribution and the axial skin-friction coe©cient. Combining Eq. (7) with Eq. (8) yields Here, the power law in cases of no fuel vaporization (hypotheses Nos. 6 and 7) is written as where n z = n θ = 1/7 is set. Applying Eqs. (11) to the absolute values of the velocity partial di¨erentials in Eq. (10) yields where l = κ(R − r) and κ = 0.4. Equation (12) can be integrated in the radial direction from the edge of the boudary layer to the fuel port wall. This integration yields c fz 2 ≈ κ 2 (n θ u θe /u ze + n z ) ln 1 + κ 2 Re δz (n θ u θe /u ze + n z ) where the approximations R ≫ δ z , δ θ and κ 2 Re δz n θ ≫ 1 are used. The ¦rst term in the right-hand side of Eq. (13) can be approximated in the easier way as where (α z , β z , γ z ) = (0.00769, 0.0233, −0.125) is set. Let us refer ξ ≡ u θe /u ze to as the ¤swirl strength.¥ If ξ = 0, Eq. (14) becomes the same as Marxman£s approximation [6]. Though the axial skin-friction coe©cient has been written as Eq. (14) through the variables descibing the §ow properties, in Eq. (14), there are three variables that cannot be easily determined: ξ, u ze , and δ z . Therefore, three additional constraints are necessary.

SOLID AND HYBRID PROPULSION
The ¦rst one is the Boussinesq approximation, which is the third of the three turbulent models: Applying Eqs. (15) and (16) to Eqs. (8) and (9) yields It is assumed here that τ rzw δ z /((µ+ρε)u ze ) consists of the product of two singlevariable functions of η z and B z just in the same way as in Marxman£s [5]. In the cases of no blowing, Eqs. (17) and (18) are equivalent to the derivative-type of Eqs. (11). Moreover, one can use the approximation: because of the hypothesis No. 6. Therefore, Eq. (17) can be expressed as Equation (19) can be now integrated in the radial direction from the edge of the boudary layer to the fuel port wall. Applying the boundary conditions ϕ z (η z = 0) = 0 and ϕ z (η z = 1) = 1 leads to As for the circumferential direction, similar to the derivation of Eq. (20), one can derive where R ≫ δ z , δ θ is assumed. Now, the velocity ¦elds have been evaluated for the radial direction in the boundary layer. These equations are used in two situations. One of them is the case when one compares shear stresses with and without fuel blowing. Another is the case when Eqs. (14), (20), and (21) are combined to obtain the Karman£s momentum integral equation and the axial skin-friction coe©cient is derived as a function of axial position.

Karman£s Momentum Integral Equation
Let us now derive the Karman£s momentum integral equation in the axisymmetric pipe §ow as the second condition for eliminating an unknown variable in Eq. (14). The mass conservation and the momentum conservation laws read: mass conservation law momentum conservation law where axial partial di¨erentials were ignored in viscous terms. Equation (22) multiplied by zu z plus Eq. (23) multiplied by r and the partial integration of Eq. (19) yield the momentum integral equation where (ρu r ) w must be considered as a nonzero variable. Contrary to axial §ows, ∂P/∂r is approximated as 0 at the edge of boundary layer; in swirl §ows, this assumption cannot be used because of Eq. (24). Thus, one should consider how to evaluate the pressure gradient (∂/∂z)(rP/ρ). Now, the pressure in the boundary layer can be expressed as where the last expression is an approximated form of the ¦rst and the second expressions when only the largest scale terms (∂R 2 δ 2 /∂z) are left. Applying Eqs. (28) and (29) to Eq. (25) and nondimensionalizing it, one obtains: Because the order of magnitude of the last two terms in the right-hand side of Eq. (30) is δ 2 /(RL) which is much smaller than the one in the right-hand side (δ/L), one can approximate Eq. (30) as This is the Karman£s momentum integral equation in the 3D axisymmetric coordinates. This equation is the last one of the four models related to boundary layers.

Estimation of the Axial Skin-Friction Coe©cient in Swirl Flows with No Blowing
The aim of this subsection is to show the last of three conditions needed to evaluate the axial skin-friction coe©cient. Also, an attempt will be made to evaluate this coe©cient in swirl and axial §ows without blowing.
Before the last condition related to swirl decay will be introduced, let us de¦ne the indicator which shows the swirl strength. The swirl number is a way to express the swirl strength as: Considering the hypotheses Nos. 4 and 5 and ignoring boundary layers, rough calculation yields Here, ξ is called ¤swirl strength.¥ The last condition is the hypothesis No. 8 with the mathematical manifestation as where p = −0.569Re −0.277 D /D. These equations are empirically derived from Kito et al. [8] and Steenbergen [9]. Now, necessary and su©cient conditions have been assembled which are Eqs. (20), (21), (31), and (32), to rewrite Eq. (14) as a single-variable function of axial direction. Next, let us compare and evaluate c fz for the case without blowing and swirling, namely, evaluate c fz as where (c fz /2)| Bz=0,ξ0=0 is equivalent to c f /2 in the cases of 2D §at plate and c f /2 follows a famous empirical rule: In this subsection, c fz | Bz=0,ξ0 /c fz | Bz=0,ξ0=0 will be evaluated for the initial swirl strength and axial direction. Now, for the cases without blowing, substituting Eqs. (14) and (32) into Eq. (31) and integration of Eq. (31) in the axial direction from 0 to z yield where the bar means that the variable is nondimensionalized by the port diameter. By substituting Eq. (35) into Eq. (31) and dividing Eq. (31) with swirl by the one without swirl, one obtains

COMPARISON OF THE REGRESSION RATES OF SWIRL ENGINES WITH EXPERIMENTS
In order to validate the model, one can compare the increase of the regression rate by swirl predicted by Eq. (50) and obtained in experiments by Yuasa et al. [3]. In this section, the prediction of Eq. (50) will be compared with the experiments in two ways. One of them is the comparison between the representative regression rates in the axial direction with space averaged values of the experimental results. Another one is the comparison of the axial distributions of regression rates.

Comparison of Representative and Averaged Regression Rates Along the Axial Direction
The fuel and oxidizer used in Yuasa£s experiments are polymethylmethacrylate (PMMA) and gaseous oxygen. For PMMA [16], B t = 10 and µ = 5.0 · 10 −7 Pa·s were set. To compare the prediction with the averaged data, let us set the representative axial location in Eq. (50) as L/2. The geometric swirl numbers of the injectors are 0, 9.7, and 19.4 and the range of the oxidizer mass §ux is from 10 to 70 kg/(m 2 s). The case with the port length L = 150 mm is shown in Fig. 3 and that with L = 500 mm is shown in Fig. 4.
In Yuasa£s experiments, they claimed it was too di©cult to measure the actual swirl numbers in their motors and when they plotted the regression rates, they used a kind of index called geometric swirl number indicating the strength of the swirl. This index is determined only by the geometry of engines and injectors and Figure 3 The ratio of constant a in swirl hybrid rocket engines for L = 150 mm: §ow and by separate consideration of the e¨ects of swirl §ow and fuel blowing on the skin-friction coe©cient. However, when applying vortex injection to hybrids, the initial swirl numbers will be designed to increase the regression rate and, by authors£ opinion, this estimation method will be useful.

Comparison of the Axial Distribution of Regression Rates
Next, let us compare the prediction of the local regression rates in the swirl injection hybrids with the experiments conducted by Yuasa et al. The geometric swirl numbers of the injectors are 0, 9.7, and 19.4 and the axial location where the local regression rates are measured is from 30 to 500 mm; the oxidizer mass §ux is 56.9 kg/(m 2 s). Figure 5 compares the local regression rate for axial injection obtained by Eq. (47) and in the experiment. In the experimental data, while the local regression rate decreases from the front edge of the fuel port to the middle of the fuel port and increases towards the end, the predicted regression rate by Marx-man£s evaluation gradually decreases along the axial distance. Furthermore, the location where the prediction agrees with the experimental data is only around the local minimun position. In the present authors£ opinion, this disagreement suggests that other e¨ects which increase the regression rate such as radiation and the increase of the mass §ux due to fuel blowing have to be considered.
In Fig. 6, the local regression rates predicted by Eq. (45) are compared with Yuasa£s experiments for the swirl-injection hybrid mortors. Similar to the case of axial injection, while both predicted regression rates and experimental results are of the same order of magnitude for high swirl numbers, their values are not the same. As in the case of comparison of the averaged rates, especially, at low  Fig. 6, one can understand that even in the cases of axial injection where Marxman£s model is thought to be applicable, the predictions of the regression rates do not agree with the experimental data and only the spatially averaged prediction agrees with the corresponding experimental data. However, according to Chiaverini et al. [11], the radiation e¨ect is weakened by developing turbulence in boundary layers. In swirl §ow, because streamlines near the wall are longer than in the axial §ow, turbulence in the boundary layer is supposed to develop earlier and the e¨ects of radiation should be low in swirl §ow. Nevertheless, the discrepancy between theoretical and experimental regression rates does not become narrower for di¨erent swirl numbers. This consideration indicates that there exist a different mechanism of swirl number decline or the e¨ects of fuel mass addition are large.
In view of it, before focusing on improving the evaluation of the e¨ects caused by swirl injection, the present authors plan to use more accurate prediction models than the Marxman£s classical one for predicting the distribution of local regression rates in axial §ows which will consider the e¨ect of the increase in the mass §ow because of fuel vaporization.
As for the accuracy of estimating the e¨ects of swirl, it is necessary to reconsider the relation of actual swirl numbers to the geometric ones. On the one hand, in the axial §ow with swirl strength ξ 0 = 19.4, the predicted result is close to the experiment only around the front of fuel port; on the other hand, at ξ 0 = 38.8, the best agreement shifts to the middle of fuel port. In the authors£ opinion, this is due to the fact that the swirl number of 0.66 is only applied to low swirl §ows. In these studies, the swirl number of 0.66 has been used based on the Motoe and Shimada numerical calculations [17]. However, this number is the result of condition that the geometric swirl number is 5.5 (ξ 0 = 11) and in higher swirl §ows, the actual swirl numbers can decrease. In particular, in cases of high swirl strength, it is reasonable that this swirl number may not be used. Therefore, it is necessary to study the relation between actual and geometric swirl numbers.

CONCLUDING REMARKS
In this paper, the authors have theoretically reconstructed and extended Marx-man£s quasi-steady boundary layer combustion model and the prediction method for regression rates in swirling hybrid rocket motors. This has been made by extending the 2D §at-plate boundary layer theory to the 3D axisymmetric theory. The derived heat §ux equation includes the e¨ect of initial swirl strength and the swirl-strengthened fuel blocking e¨ect. The blocking exponent for strong swirl injection is calculated to be 0.965 in contrast to 0.77 for axial injection. By using this heat §ux, eventually, the equation to evaluate the regression rate in swirling hybrid rocket motors has been derived.
To con¦rm the accuracy of this method, the predicted results were compared with the experiments by Yuasa et al. in two ways. In the ¦rst, the representative increase of the regression rates by swirl in the axial direction were compared with the averaged regression rates from the experiments. Though the assumed §ow ¦eld seems to be di¨erent from the experiments to some extent, the estimated regression rates are of the same order of magnitude at all swirl strengths and ¦t especially well at strong swirls. In the second, the predicted local regression rates were compared with experimental data from Yuasa et al. for both axial and swirl §ows. The prediction of the classical Marx-man£s theory for axial injection motors was compared with the data for axial §ow and the results were of the same order of magnitude, however, not accurate enough to claim the regression rates can be predicted to know the detailed performance. In the authors£ opinion, the reason for this disagreement is that other e¨ects which increase regression rates such as radiation and the increase of the mass §ux by fuel blowing or other mechanisms inherent in swirl injection hybrid rocket engines have to be considered, though the radiation e¨ect can be weakened at high swirl numbers. In swirl injection, because the theory derived in this paper is based on the classical theory of Marxman et al., the accuracy of the prediction is also low. Therefore, to improve the prediction of the local regression rates in swirl injection hybrid motors, one should provide some theoretical correction of Marxman£s boundary layer combustion model. Compared with experimental results, the predicted regression rates of ξ 0 = 38.8 were found to shift higher than the ones at low swirl num-bers. This is because the relation of actual swirl numbers to the geometric ones can decrease due to the increase of geometric swirl numbers and it is necessary to reconsider the value of this relation at higher geometric swirl numbers.