In order to avoid the necessity of having to incorporate a fuel compartment in ramjet design, a solid-fuel combustor that integrates fuel storage into the combustor is usually adopted. However, ramjet rotation (Figure 1(a)) could complicate the complex reacting flow downstream of the combustor entrance (Figure 1(b)) and could give rise to premature combustion extinction. Since spin is common in ramjet operation, it could impact combustor performance and ramjet trajectory in the duration of its flight. Ahmed et al. ref. [1] have examined the effect of spin on the flow alone inside a ramjet combustor in the laboratory. From this experiment, a large decrease in combustion efficiency could be inferred from the fluid dynamic study. The combustion efficiency decrease could be as large as 50% or more depending on the solid fuel used. Therefore, it could be speculated that spin is a major cause of this decrease.

To shed light on this reduction, it is necessary to understand the combusting flow behaviour inside the combustor. Once the oxidant air enters the combustor, it immediately encounters a sudden expansion, thus, creating a recirculation region and a flow reattachment (Figure 1(a)). Downstream of the reattachment

point, a wall jet and a boundary-layer type flow will develop (Figure 1(b)) and they will last until the flow reaches the exit expansion nozzle. The ramjet combustor is normally designed in such a way that the recirculation region serves to anchor the flame (Figure 1(a)) and will probably occupy about 20% of the combustor length, while the wall jet, the boundary-layer and the pipe flow that follows dominate the other 80%. The recirculation region is necessary for stable combustion to occur inside the combustor. Therefore, in order to maintain continuous combustion with good efficiency, a thorough understanding of the flow dynamics inside the combustor is necessary.

Besides the backstep flow, there is also a plane wall jet downstream of the reattachment point, and a developing boundary layer as the flow slowly evolves into a pipe flow in its passage to the exit nozzle (Figure 1(b)). These complexities are further complicated by combustion that creates density stratification, and ramjet spin that creates swirl in the combusting flow. In order to achieve optimal design of the combustor, a good understanding of the isothermal flow complexities inside the ramjet is of utmost importance. If the turbulent flow inside the ramjet were to be simulated, a relatively simple turbulence model, such as a two-equation model, is preferred because it could be easily extended to tackle turbulent combusting flow. Furthermore, if the complex flow were to be correctly modelled by relatively simple turbulence models, such as two-equation models, then an accurate modelling of the individual complex flow by a two-equation model needs to be demonstrated and established. Successful modelling of backstep flow, plane wall jets, and flow with density stratification using two-equation models have been demonstrated by So et al. ref. [2], by Gerodimos and So ref. [3], by Sommer et al. ref. [4], and by Yoo and So ref. [5], respectively, while their suitability for rotating pipe flows has been shown to be viable by Yoo et al. ref. [6], who made comparisons of two-equation simulation results with those obtained using Reynolds-stress models. Furthermore, the Yoo et al. ref. [6] study also showed that two-equation models need improvements, especially in their viability and suitability for modelling swirl and rotating flow in ramjet combustor simulation.

Swirl decay in a straight pipe can be classified into two different types depending on the way swirl is imparted into the flow. The swirl in the first type is created by a rotating pipe while the second type is created by a swirler installed at the entrance of the pipe. A forced vortex or a solid-body rotation is generated in the first type, while the vortex could be a free vortex, a forced vortex, or a combined free/forced vortex in the second flow type. Consequently, the resultant swirling flow behaves differently in the two flow types.

In the creation of a rotating pipe flow, the flow could either pass through a continuously rotating pipe, such as that reported by Murakami and Kikuyama ref. [7] and by Kikuyama et al. ref. [8], or through a rotating pipe into a stationary pipe where the swirl velocity decays as the flow moves downstream, such as the Weske and Sturov ref. [9] experiment, and the investigation of Anwer and So ref. [10]. The counterpart to a concentric annulus has been examined by Hirai et al. ref. [11]. If the term “destabilizing” is used to describe the phenomenon of extra turbulence production created by flow rotation, while the term “stabilizing” is used to describe the rotating flow characteristics without the extra turbulence production present, then flow regions with extra turbulence production present (destabilizing) or absent (stabilizing) could be found in different parts of an axially rotating pipe flow. Near the entrance of a rotating pipe, the wall boundary layer is very thin. Consequently, fluid rotation has to decrease rapidly from pipe rotation at the wall to zero outside of the wall boundary layer. The flow near the wall is influenced by a very high mean tangential shear strain and turbulence production is greatly enhanced.

In the entrance region of the rotating pipe, the near-wall flow is destabilized by rotation. Far downstream, the flow becomes fully developed and the fluid rotates as a solid body. Since the solid-body rotation curve has a constant slope, turbulence production due to the tangential mean strain is essentially zero. In this downstream region, rotation gives rise to a stabilizing effect on the flow. Between the entrance and the fully developed region, the flow is influenced by both destabilizing and stabilizing forces created by the rotating pipe. However, the transition from destabilizing to mixed to stabilizing occurs over a finite length and depends on different factors; among them are inlet flow conditions, the flow Reynolds number, Re, and the swirl number, S. In this case, S can be approximated by the ratio of the pipe rotation velocity to the mean bulk velocity of the pipe, or S = ωD/2U_m. This three-dimensional flow has also been investigated and reported by Murakami and Kikuyama ref. [7], and Kikuyama et al. ref. [8].

In the experiments carried out by Weske and Sturov ref. [9] and by Anwer and So ref. [10], the flow very near the wall was immediately affected by a large mean tangential shear strain due to the sudden relaxation of surface rotation. Therefore, the flow in the entrance region of their rotating pipe flow is influenced by both stabilizing and destabilizing effects. Eventually, the solid-body rotation would decay completely, and a fully developed pipe flow would result. Therefore, the flow characteristics downstream of the rotating section would depend on the inlet conditions, on Re and on S. Weske and Sturov ref. [9] measured the Reynolds stresses and the mean flow for two different S and they found that, even at 150D downstream, the turbulence field has not completely recovered its stationary behavior. On the other hand, the experiments of Anwer and So ref. [9] were carried out to study the effects of S on wall shear only and did not examine the decay phenomenon. Their measurements were obtained at 2D downstream of the entrance. From their data, it is clear that the destabilizing effect, although only present in about 10% of the pipe diameter, dominates over the stabilizing effect, thus promoting turbulence production along the pipe.

Swirling flow, with either a free or a forced vortex generated by swirl generators, in a straight pipe has not been investigated extensively in the past. In this case, the free or forced vortex evolves in the downstream direction into a combined free and forced vortex before decaying. Among the more relevant studies, the work of Kreith and Sonju ref. [12], Backshall and Landis ref. [13], Yajnik and Subbaiah ref. [14], Murakami et al. ref. [15], Padmanabhan and Janek ref. [16], Ito et al. ref. [17], Kito ref. [18], Kito and Kato ref. [19], Algifri et al. ref. [20], Kitoh ref. [21] and Parchen et al. ref. [22] could be mentioned. The earlier studies up to 1984, including the axisymmetric nature of swirling flow examined by Kito ref. [18], were mainly concerned with the mean field, the pressure drop, and the decay of mean swirl in a pipe. Little attention has been paid to the decay of the turbulence field. On the other hand, some measurements of the turbulence field evolution have been provided by Algifri et al. ref. [20], Kitoh ref. [21] and Parchen et al. ref. [22]. As a result, turbulence structure in the core and the outer region is found to be quite different from those in the near wall region. This difference might be difficult to replicate using conventional turbulence models for swirling flow simulations.

The studies mentioned above cover a wide range of S. Depending on the type of swirler used, the free-to-forced portion of the vortex will vary. The flow characteristics near the pipe centerline also differ significantly depending on S. For flows with S ≪ 1, a recirculation region did not exist near the pipe core because the resulting pressure depression is not sufficient to create such a reversed flow. For flows with S ≈ 1, a reversed flow region starts to appear depending on whether other conditions, such as Re, inlet conditions, etc., are favorable to the formation of a recirculation region. For example, the experiments of Kitoh ref. [21] showed a reversed flow region while the measurements of Murakami et al. ref. [15] did not, even though both studies were carried out at an initial S ≥ 1. The decay rate was also found to be dependent on the above conditions. For example, in the experiments of Murakami et al. ref. [15], the mean flow was still evolving even at 190D downstream of the inlet. On the other hand, Weske and Sturov ref. [9] found that the mean flow was quite similar to that of a fully developed turbulent pipe flow at about 150D downstream of the inlet, while the turbulence field was still evolving.

Due to the complex nature of swirling flow and its decay in a straight pipe, it’s modelling has not been investigated extensively. Among the more notable studies are those of Kreith and Sonju ref. [12], Parchen et al. ref. [22], Kobayashi and Yoda ref. [23], and Khodadadi and Vlachos ref. [24]. Three different approaches to model swirl decay were discussed in these studies. In all but one investigation, wall functions were assumed. An analytical approach was adopted by Kreith and Sonju ref. [12] who assumed a constant eddy viscosity and an axial flow given by the mean axial velocity in a fully developed pipe flow. Thus simplified, the equation governing the tangential velocity, W, can be reduced to an equation with one unknown once the eddy viscosity is specified. Using their own experimental data, they were able to derive an eddy viscosity that was parametric in Re and used it to solve for the evolution of W. They were able to derive an exponential decay law for S. The results thus obtained were in good agreement with their own experimental measurements. Two-equation k-ε models and their variations had been deployed by Kobayashi and Yoda ref. [23] and Khodadadi and Vlachos ref. [24] to simulate swirl decay in a straight pipe. These calculations indicated that the k-ε model was, in general, inadequate and gave poor prediction of the flow in the pipe core region, especially the flow depression along the axis. The coefficients and model constants of the k-ε model could be modified to give better correlations with data as suggested by Kobayashi and Yoda [23]. However, such modifications were rather ad hoc and could not be easily generalized to flows other than those analyzed by Kobayashi and Yoda ref. [23], and Khodadadi and Vlachos ref. [24].

An algebraic stress model plus the standard k and ε equations were adopted by Parchen et al. ref. [22] to simulate swirl decay in a straight pipe. They also compared their results with standard k-ε model calculations and other model predictions obtained by their colleagues. The results showed that, as far as swirl decay was concerned, the standard k-ε model gave a better prediction than the algebraic stress model. However, both models failed to capture the behavior of the mean flow correctly as swirl decayed in the downstream direction. Specifically, the standard k-ε model failed to predict any enhancement of radial momentum exchange as a result of swirl. On the other hand, the algebraic stress model predicted a weak enhancement. Also, for flows with small S, the standard k-ε model performed the best among all models tested. These results were quite puzzling and could be partially explained by attributing the discrepancies to the use of wall functions in the modelling. Consequently, near-wall diffusive transport of all stress components was under-estimated; thus, it could lead to a situation where error cancels out error and give rise to a better prediction by the k-ε model.

These studies led to the realization that not a single k-ε model, without some kind of ad hoc modifications, could correctly track the swirling flow evolution in a straight pipe. Therefore, an alternative approach to model swirling flow is necessary. According to Bardina et al. ref. [25], who studied rotation effects in isotropic turbulence, their findings indicate that the effects of flow rotation and/or swirl should be accounted for in the ε-equation. They suggest modelling the ε-equation to take swirl effects into account directly and explicitly. In addition, Fu et al. ref. [26] and Kim and Rhode ref. [27] showed that using algebraic and second-moment closure models could greatly improve swirl flow simulations. Their model results were in good agreement with rotating flow data. On the other hand, Bernard and Speziale ref. [28] and Speziale and Bernard ref. [29] have found that, if homogeneous shear flow and its asymptotic state were to be calculated correctly, an explicit term representing the effect of vortex stretching should be included in the modified ε-equation.

The two-equation models and their improvements discussed above cannot fulfill the need for a general two-equation model that could simutaneously replicate backstep flow, wall jet, thermal boundary-layer flow, developing pipe flow, and swirling and rotating pipe flow, such as those commonly encounter in spinning ramjet combustors. Recent studies on swirling flow simulations reported by Andrew and Cui ref. [30], Sagol et al. ref. [31], Javadi ref. [32], Li et al. ref. [33] and Alhmadi et al. ref. [34], to mention a few recent studies, also did not emphasize their ability to model the highly complex swirling flow found in spinning ramjet combustors. Similar to other swirling flow models, theirs could also accurately mimic swirling flow without the presence of added flow complexities found in spinning ramjet combustor. Therefore, the need to develop a turbulence model that could properly handle these complexities still has not been fulfilled.

Based on the above review on swirling flow modelling, it can be concluded that the time is ripped for the development of a relatively more general swirling flow model based on the two-equation model used in the studies of So et al. ref. [2], Gerodimos and So ref. [3], Sommer et al. ref. [4], Yoo and So ref. [5], and Yoo et al. ref. [6]. Thus configured, the new model could better handle flow complexities found in spinning ramjet combustors. In addition, the swirling flow modelling studies quoted above suggest that the conventional ε-equation also needs modification if the effects of vortex stretching and rotation also were to be simulated. These studies further suggest that the ideas of Bardina et al. ref. [25], Bernard and Speziale ref. [28] and Speziale and Bernard ref. [29] could be used to derive a new near-wall ε-equation. The resultant ε-equation together with a near-wall k-equation, such as that adopted by So et al. ref. [2] to treat backstep flow, by Gerodimos and So ref. [3] to model plane wall jets, by Sommer et al. ref. [4] to treat flow with heat transfer, and by Yoo and So ref. [5] to simulate density stratified flows, could give rise to an improved k-ε model.

The approach used to accomplish this objective is similar to that adopted by So et al. ref. [35] to account for low-Reynolds-number and near-wall turbulence effects in the ε-equation in order to correctly simulate such flows. Therefore, in the present approach, the ε-equation is modified to account for swirl and pipe rotation, while the modified k-equation adopted in previous studies refs. [2] [3] [4] [5] [6] is used without implementing further modelling change to account for swirl effects. Thus formulated, the k-equation and the modified ε-equation together will give an improved k-ε model that is suitable for modelling complex turbulent flows such as those found in spinning solid-fuel ramjet combustor.

The improved k-ε model is validated against swirling and rotating pipe flow measurements and other two-equation k-ε modelling results. In particular, the new model with isotropic eddy viscosity invoked is assessed against the simulations of two other conventional models that adopt the conventional k-ε equations. One model retains the isotropic eddy viscosity assumption, while another invokes anisotropic eddy viscosity. This way, the effects of stress anisotropy on swirling flow simulations can also be assessed; thus allowing the merits or lack thereof of the explicit modelling of swirl effects in the ε-equation to be thoroughly studied in this investigation.

Consider an incompressible turbulent swirling flow in a straight pipe. The mean flow and modeled k-ε equations with gradient transport assumed can be written in Cartesian tensor form as

∂ U i ∂ x i = 0 , (1)

D U i D t = − 1 ρ ∂ P ∂ x i + ν ∂ 2 U i ∂ x i ∂ x j − ∂ u i u j ¯ ∂ x j , (2)

D k D t = ∂ ∂ x i [ ( ν + ν t σ k ) ∂ k ∂ x i ] + Π k − ε , (3)

D ε D t = ∂ ∂ x i [ ( ν + ν t σ ε ) ∂ ε ∂ x i ] + C ε 1 ε k Π k − C ε 2 ε ε ˜ k + ξ , (4)

where D/Dt is the material derivative, U_i is the i^th component of the mean velocity, x_i is the i^th component of the coordinate, Π k = u i u j ¯ ( ∂ U i / ∂ x j ) is the production of k, P is the mean pressure, ρ is fluid density, t is time, ε ˜ = ε − 2 ν ( ∂ k / ∂ x i ) 2 is a reduced ε, ν_t is the turbulent eddy viscosity, − u i u j ¯ is the Reynolds stress tensor, ξ is a near-wall correction function for the ε-equation and σ_k, σ_ε, C_ε1 and C_ε2 are model constants. Here, the overbar is used to denote time average. The near-wall correction function derived for the SASY model is given by So et al. ref. [35] as

ξ = f w 1 ( − L ε k Π k + M ε ˜ 2 k − N ε ε ˜ k ) , (5)

where f w 1 = exp [ − ( R e t / 40 ) 2 ] is a damping function used to ensure the disappearance of the effects of ξ away from the wall, R e t = k 2 / ν ε is the turbulence Reynolds number and the constants are given as L = 1.5, M = 0.5 and N = 0.57. Another reduced ε is introduced and this is given by ε ¯ = ε − 2 ν k / x 2 2 , where x₂ is the wall normal coordinate. The other model constants are specified as σ_k = 1, σ_ε = 1.45, C_ε1 = 1.50 and C_ε2 = 1.83.

If an isotropic eddy viscosity is assumed, the definitions for − u i u j ¯ and the eddy viscosity, ν_t, are given by

− u i u j ¯ = 2 ν t S i j − 2 3 k δ i j , (6a)

ν t = C μ f μ k 2 ε , (6b)

where S i j = ( ∂ U i / ∂ x j + ∂ U j / ∂ x i ) / 2 is the mean strain rate, C_μ = 0.096 and f_μ is a damping function introduced to render proper asymptotic behavior for the Reynolds shear stress. The specification for f_μ is

f μ = [ 1 + 3.45 R e t ] tanh ( x 2 + 115 ) , (7)

where x 2 + = u τ x 2 / ν and u τ is the friction velocity.

The boundary conditions for swirling flows can be stated as U = V = 0, W = 0 for a stationary pipe and W = ωD/2 for a rotating pipe, k = 0 and

ε w = 2 ν ( ∂ k / ∂ x 2 ) w 2 at x₂ = 0. Here, x₂ is taken to measure positive away from

the wall. Its relation to r, the radial coordinate, is x₂ = D/2 − r (Figure 2). At the symmetry axis, the boundary conditions are zero radial gradient for all quantities except the Reynolds shear stress, which is specified as zero.

The above model assumes gradient transport and isotropic eddy viscosity. Therefore, it cannot account for anisotropy effects. In order to capture this particular characteristic without resorting to a full second-order modelling of the flow, it is proposed to couple an anisotropic eddy viscosity with the above two-equation model to calculate swirling flows. The anisotropic model adopted is that proposed by Speziale ref. [38] and can be written as

− u i u j ¯ = 2 ν t S i j − 2 3 k δ i j − 4 C D ν t 2 k [ S i k S k j − 1 3 S k l S k l δ i j + S ˜ i j − 1 3 S ˜ k k δ i j ] , (8a)

where

S ˜ i j = ∂ S i j ∂ t + U k ∂ S i j ∂ x k − ∂ U i ∂ x k S k j − ∂ U j ∂ x k S k i , (8b)

is the frame indifferent Oldroyd derivative of S_ij and C_D = 1.68 is a dimensionless constant. It should be noted that the isotropic eddy viscosity is recovered in the limit of CD goes to zero. This anisotropic eddy viscosity along with Equations (3) to (5) is designated as the ASAYS model. The objectives here are first to demonstrate the ease with which the two-equation model can be used with an anisotropic eddy viscosity, where no other modifications are required of the modeled equations, and second to assess the anisotropy effects on the calculation of turbulent flows with high swirl. Once the validity of the concept has been demonstrated, the anisotropic eddy viscosity can be used with other two-equation models. Of course, alternative anisotropic eddy viscosities can be used. An example is the proposal of Gatski and Speziale ref. [39].

The improved ε-equation for swirling flows is derived in two steps. For the sake of completeness, a brief description of this model is given below. First, the abilities to model free shear flows and their decay as well as near-wall flows are considered. The model thus derived is the SSGZ model reported in So et al. ref. [36]. This is followed by a second modification where the effects of swirl and their modelling are examined. These requirements, therefore, necessitate a complete re-examination of the e-equation and its modelling. However, the form of the modeled equation should remain simple and should not deviate substantially from the established form of a modeled production and a modeled destruction term. In other words, the e-equation should have the same form as that given in Equation (4), but with different constants and damping functions assumed for the production and destruction terms. Furthermore, the near-wall correction function ξ should again be derived to render the improved equation asymptotically correct near a wall.

According to Launder et al. ref. [40], the spreading rate of free shear layers can be predicted correctly if the constants in Equations (3) and (4) are changed to the following set: σ_k = 1, σ_ε = 1.4, C_ε1 = 1.50 and C_ε2 = 1.9. Further, the modelled destruction term ε ε ˜ / k should be replaced by ε 2 / k . In addition, as pointed out by Launder et al. ref. [41], the set of constants chosen should yield a correct von Karman constant κ for the log-linear region of wall-bounded turbulent flows. However, such an ε-equation could not yield correct results for the simulation of homogeneous turbulence decay. On the other hand, Hanjalic and Launder ref. [42] found that the decay of homogeneous free turbulence can be correctly simulated if and only if the destruction term in the dissipation rate equation is modified by a damping function that is dependent on Re_t. They proposed the following damping function f ε = 1 − 0.22 exp [ − ( 6 R e t ) 2 ] . This suggests that C ε 1 ( ε / k ) Π k in Equation (4) should remain the same while C ε 2 ( ε ε ˜ / k ) should be replaced by C ε 2 f ε ( ε 2 / k ) . Thus modified, the near-wall correction function ξ can be derived following the procedure outlined in So et al. ref. [43]. The resultant function that yields asymptotically consistent results for k and ε can be incorporated into the ε-equation to render it valid for both low-Reynolds-number turbulence and near-wall flows. The modified ε-equation is given by

D ε D t = ∂ ∂ x i [ ( ν + ν t σ ε ) ∂ ε ∂ x i ] + C ε 1 f 1 ε k Π k − C ε 2 f 2 ε ε ˜ k (9)

where f 1 = [ 1 − α f w 1 ] and, as before, f w 1 is given by f w 1 = exp [ − ( R e t / 40 ) 2 ] and f 2 = f ε while ε ˜ is given by

ε ˜ = ε − C ε 3 C ε 2 f 2 ν [ ∂ k ∂ x i ] 2 , (10)

while the constants are determined by So et al. ref. [43] to be C ε 3 = 2.955 and α = 10 . The ε boundary condition would lead to ε ˜ = 0 at the wall and thus rendering (9) valid as the wall is approached. Therefore, the SSGZ model consists of solving Equations (3), (6), (9) and (10). However, the damping function f_μ is given by So et al. ref. [44] as

f μ = [ 1 + 80 exp ( − R e t ) ] [ 1 − exp ( − R e ε 43 − R e ε 2 330 ) ] 2 ( 1 + 3 R e t − 3 / 4 ) , (11)

where R e ε = ( ν ε ) 1 / 4 x 2 / ν is a Reynolds number based on the Kolmogorov velocity scale. This model has been validated against a wide variety of flows by So et al. refs. [43] [44] and good correlation with data is obtained for each of the flow case studied.

As will be seen later, the ε-equation given by (9) fails to improve the predictions of swirling flows. The reason is that it does not account for the effects of swirl in an explicit manner. In a study on isotropic turbulence under the influence of a constant angular velocity w, Bardina et al. ref. [25] found that the effects of rotation should appear explicitly in the ε-equation. Furthermore, their simulation results show that the effect on ε is approximately linear in ω. As a result, they proposed a reduced ε-equation, which can be written as,

d ε d t = − C ε 2 ε 2 k − C ε 3 ω ε , (12)

to analyze rotating isotropic turbulence. The last term in this equation was proposed by Bardina et al. ref. [25] to account for constant rotation effects in isotropic turbulence and the calculated results from Equation (12) are in good agreement with data. However, for more general flows, where the angular velocity Ω is a function of position and/or where mean strain might be present, a more appropriate angular velocity or rotation rate such as Ω = [ Ω i j Ω i j / 2 ] 1 / 2 should be used instead. Here, the mean rotation rate tensor is given by Ω i j = [ ∂ U i / ∂ x j − ∂ U j / ∂ x i ] / 2 . In the present analysis, it is obvious that Ω should be adopted rather than ω.