bidirectional vortex with sidewall injectionmajdalani.eng.auburn.edu/publications/pdf/2008...the...

1 American Institute of Aeronautics and Astronautics

The Bidirectional Vortex with Sidewall Injection

Joseph Majdalani1 and Erin K. Halpenny2 University of Tennessee Space Institute, Tullahoma, TN 37388

The purpose of this paper is to derive an approximate solution for the bidirectional vortex in a right-cylindrical chamber with sidewall injection. The flowfield may be used to describe the bulk gas motion in a vortex-hybrid rocket chamber as well as in other cyclonic devices that combine circulatory motion with mass transfer. Our mathematical model is based on steady, rotational, axisymmetric, incompressible, and quasi-viscous flow conditions. Two distinctive perturbation parameters are used: the ratio of sidewall-to-tangential injection velocities and the reciprocal of the vortex Reynolds number, which combines the swirl number, chamber aspect ratio, and viscous Reynolds number. First, an Euler-type solution is obtained using variation of parameters and suitable boundary conditions that secure the sidewall mass injection requirement. This enables us to reproduce the two-cell, bipolar motion observed in vortex-hybrid thrust chambers. Second, to capture the viscosity-dominated forced vortex and sidewall boundary layers, the regularized tangential momentum equation is expanded in the reciprocal of the vortex Reynolds number. A uniformly valid, triple-deck approximation for the tangential velocity is then constructed using matched asymptotics. Viscous corrections in the axial and radial directions are also resolved. Additionally, we calculate pressure distributions, axial and radial velocity extrema, vorticity formation, roll torques, and the dynamic mantle location that separates inner and outer vortices. Finally, by relating fundamental variables to the bidirectional swirl number and wall regression rate, essential flow characteristics are captured throughout the chamber. As a windfall, an explicit relation is obtained linking the mantle location to the wall injection rate.

Nomenclature ,a b = chamber radius and outlet radius, aβ iA = inlet area of incoming swirl flow

A = constant parameter, 2csc( ) /πβ κ κ= B = constant parameter, 2 2[1 cos( )]Aβ πβ+ L = chamber aspect ratio, 0 /L a p = normalized pressure, 2/ ( )p Uρ

iQ = inlet volumetric flow rate at the base iQ = normalized flow rate,

1 2 2/ ( ) /i iQ Ua A aσ− = =

inQ = total incoming flow rate, in i wQ Q Q= + wQ = wall injected flow rate, 2 waLUπ

Re = injection Reynolds number, /Ua ν wRe = sidewall injection Reynolds number, /wU a ν

,r z = normalized radial and axial coordinates, / ,r a /z a S = unidirectional swirl number, / iab Aπ πβσ= u = normalized velocity ( ru , zu , uθ )/U U = tangential injection velocity, ( , )u a Lθ

wU = sidewall injection velocity, ( , ) / (2 )r wu a z Q aLπ− = V = vortex Reynolds number defined in Tables 3 and 4

wV = wall vortex Reynolds number defined in Tables 3 and 4 1H. H. Arnold Chair of Excellence in Advanced Propulsion, Department of Mechanical, Aerospace and Biomedical Engineering. Member AIAA. Fellow ASME. 2Graduate Research Assistant, Department of Mechanical, Aerospace and Biomedical Engineering. Member AIAA.

44th AIAA/ASME/SAE/ASEE Joint Propulsion Conference & Exhibit21 - 23 July 2008, Hartford, CT

AIAA 2008-5018

Copyright © 2008 by J. Majdalani. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.


Greek α = pure constant, 216 1 0.644934π − β = normalized discharge radius, /b a δ = reciprocal of the Reynolds number, / ( )Uaν ε = sidewall injection parameter, /wU U η = action variable, 2rπ κ = tangential inflow parameter, 1(2 )Lπσ − κ = modified inflow parameter, 2csc( )κ πβ μ = dynamic viscosity ν = kinematic viscosity, /μ ρ ρ = density σ = modified swirl number, 1 2/ ( ) /i iQ S Ua Qπβ

− = = Ω = mean flow vorticity, ∇ × u Symbols i = inlet property at the base, 0z L= r = radial component or partial derivative w = sidewall property z = axial component or partial derivative θ = azimuthal component or partial derivative

= overbars denote dimensional variables

I. Introduction HE purpose of this article is to present a solution for the bidirectional vortex in a chamber with sidewall injection. The chief motivation for such a study stems from a propulsive application or, more specifically, the

description of the bulk fluid motion in a hybrid vortex engine with transpiring walls.1-5 Our representation also serves to model cyclonic flows in chambers with sidewall mass addition. So before engaging in the subject of cyclonic motion and its effective use in combustion chambers, a brief review of the pertinent literature is in order. Studies of swirling flows, both external and confined, are spurred on by a variety of applications that extend over widely dissimilar length-scales. According to Penner,6 the geophysical sciences attracted some of the earliest interest in the subject, being primarily concerned with naturally occurring swirling phenomena such as the formation of fire-whirls, whirlpools, tornados, dust-devils, hurricanes, typhoons, tropical cyclones, and waterspouts. In astrophysics, the expansion of cosmic jets, galactic pinwheels, and wormholes is another subject that entails large-scale vortex motions (see Königl7). In industrial applications, both unidirectional columnar vortices and multi-directional vortex patterns constitute areas of investigation that are directly relevant to the design and operation of practical devices. By promoting circulatory motion in conjunction with heat or mass transfer, one is able to achieve efficient mixing, heat exchange, chemical dispensation, atomization, or filtration. In this vein, swirl burners, cyclonic furnaces, vortex combustors, counter-swirl heat exchangers, and cyclone separators have been the subject of numerous inquiries. In mass and heat transfer equipment, one often considers not only thermo-gravitational convection but also centrifugal acceleration, which tends to intensify the effect of swirl. As an example, one may cite the classic analysis of the Ranque-Hilsch vortex tube (Hilsch;8 Kurosaka9). Other contributors include, to name a few: Lay,10 Algifri et al.,11 Hirai and Takagi,12 Chang,13 Chang and Dhir,14 Shtern et al.,15,16 Borissov et al.,17 and Martemianov and Okulov.18,19 In chemically reacting flows, swirl and vortex breakdown are often paired to enhance combustion efficiency (Buntine and Saffman;20 Wang and Rusak;21 Levebvre;22 Paschereit et al.;23,24 Santhanam et al.25). In swirl burners, the region of vortex breakdown is deliberately used as a flame-holder. By properly controlling the swirl parameters, flame extinction is mitigated, and combustion is stabilized. In premixed combustion, the breakdown region forms a high temperature zone that is ideally suited for trapping burning particles. Heat exchange with the surrounding spiraling flow helps to stabilize the flame and expedite the completion of ongoing reactions (Sivasegaram and Whitelaw26). When fuel and oxidizer are non-premixed, the hot breakdown region can be effectively used to engulf reacting particles, to the extent of promoting mixing and longer residence times. As will be expounded upon later, a similar effect can be induced by cyclonic motion in a hybrid vortex engine.

T


Various methods have been employed to trigger swirl in cylindrical and conical chambers including tangential fluid injection, inlet swirl vanes, flat or aerodynamically-shaped swirler blades, vortex trippers, twisted tape inserts, propellers, or coiled wires. Among the salient features of the resulting flows, one could set apart vortex breakdown, instability, and reversal as most significant. One of the earliest investigations of columnar vortices may be traced back to the work of Harvey;27 he reported the presence of vortex disruption in rolled-up shear layers above highly-swept lifting surfaces. This breakdown exhibited a distinct stagnation point that was followed by a region of flow reversal. Beyond the stagnation point, an appreciable increase in the vortex core could be noted in addition to flow transition and increased turbulent fluctuations. Two distinct types of breakdowns were reported, and these were dubbed the spiral ‘S-shape’ and the bubble ‘B-shape.’ These two types of disruption modes were sequentially established with successive increases in the Reynolds number and swirl levels. At fixed swirl levels, higher Reynolds numbers caused the breakdown to shift upstream. A third type of breakdown, the double helix, was captured by Sarpkaya28 at low Reynolds numbers. A total of six different modes of vortex disruption were eventually identified and cataloged in a comprehensive flow visualization study by Faler and Leibovich.29 More detail on vortex breakdown and stability can be found in the informative surveys by Leibovich,30,31 Escudier,32 and Rusak.33 Due to the recirculatory patterns associated with vortex propagation and breakdown, the application of swirl has been extensively used as a vehicle for efficient and stable combustion in industrial furnaces, utility boilers, spiral heat exchangers, gas turbines with toroidal zones, turbofans with swirl augmentors, internal combustion engines, and other vortex burners.34 In some devices, swirl is imparted to the primary or secondary jets to enhance their size, entrainment, or pattern development. Due to the swelling that accompanies vortex breakdown, swirling jets are also used as flame-holders with controllable flame characteristics. Generally, coswirl leads to better combustion efficiency while counterswirl is accompanied by a wider recirculation region, a shorter luminous combustion zone, and a larger slip velocity and turbulent intensity along the interjet layer. The coswirling arrangement produces a shorter flame and a weaker sensitivity to changes in hardware and operating conditions (Gupta et al.;35 Durbin and Ballal36). The degree of swirl is quantified by the dimensionless swirl number S , which scales with the ratio of tangential-to-axial momentum forces. For strong swirl ( 0.6S > ), breakdown manifestation begins to develop beyond a critical Reynolds number that marks the transition from the supercritical to the subcritical flow regimes. In the wake of the recirculatory region that accompanies breakdown, a spiraling vortex disturbance is detected. This disturbance is generally unsteady in position, exhibiting random axial excursions. The resulting precession enhances mixing, combustion intensity, and flame length; however, it also leads to such negative effects as combustion oscillations, noise, and pollutant formation, which necessitate active control strategies.24 As noted by Reydon and Gauvin,37 other technological processes in which swirl motion is critical to proper operation include spray dryers, spray coolers, gas scrubbers, and cyclonic separators (Love and Park38). In practice, gas and hydro cyclones are extensively used in the petrochemical, mineral, and powder processing industries (Gupta et al.35). Their bidirectional vortex motion aids in catalyst or product recovery, scrubbing, and dedusting. In the propulsion industry, the implementation of bidirectional swirl also has important uses. It has been recently applied to vortex engines utilizing liquid and hybrid propellants (Chiaverini et al.;39,40 Knuth et al.1-5). The incorporation of cyclonic motion in hybrid thrust chambers leads to a noticeable increase in fuel regression rate and combustion efficiency. This can be achieved, for example, by integrating tangential injection into a Vortex Injection Hybrid Rocket Engine (VIHRE). A schematic of VIHRE is shown in Fig. 1. The main advantage of VIHRE is its ability to trigger a seven-fold increase in the fuel regression rate by comparison to conventional hybrids

Figure 1. Sketch of the bidirectional vortex in a circular port chamber with a weakly injecting sidewall. The bulk motion is an approximate representation of the cyclonic gaseous flow associated with an advanced vortex injection hybrid rocket engine concept.1


(Knuth et al.4,5). According to the criteria assembled by Casillas et al.,41 this design stands as a feasible propulsion alternative. It overcomes the three principal deficiencies that hybrids have been noted for: low combustion efficiency, low regression rate, and low volumetric loading. The improved performance granted by VIHRE is due to its internal flowfield being dominated by swirling bidirectional motion. The corresponding coaxial, counter-flowing vortex pair increases surface erosion while promoting mixing and turbulence. Another feature of VIHRE that constitutes a departure from conventional hybrid conceptualization is the aft injection of the oxidizer fluid just upstream of the nozzle (Fig. 1). By aligning the injector ports tangentially to the inner circumference, a strong vortex is produced that sweeps the fuel periphery along the entire length of the grain. Fuel particles trapped in this manner are compelled to spiral around the chamber axis while traversing its length twice before exiting. Efficiency is ameliorated due to the markedly increased residence time and the intense mixing between fuel and oxidizer. In addition to the improved regression rate and combustion efficiency, VIHRE utilizes hollow, cylindrical grain cartridges that are simple to manufacture. The corresponding web perforation reduces volumetric loading and precludes the need for large and elaborate case housing. The description of bidirectional swirl over a transpiring surface is certainly an interesting problem in its own right. In fact, former analyses have been mostly carried out in the context of cyclonic motion in vortex separators or vortex engines. Most have relied on experimental and numerical investigations which, until recently, had been conducted under cold-flow conditions. In the treatment of cyclone separators, one may begin with ter Linden,42 whose efforts have focused on determining the influence of geometric parameters on particle-separation efficiency. His experimental work was extended by Kelsall43 and Smith44 who explored hydraulic and gas cyclones, respectively. These were followed by Reydon and Gauvin,37 Lin and Kwok,45 and Ogawa.46 By combining numerical simulations with laboratory measurements, several investigations were later conducted by Hsieh and Rajamani,47 Hoekstra et al.,48 Derksen and Van den Akker,49 and Hu et al.50 These studies frequently relied on laser-doppler velocimetry (LDV) for visualization and the Reynolds Stress Model (RSM) for computation. In a similar context, Fang et al.51,52 and Murray et al.53 resorted to both inert and reactive flow computations in the simulation of bidirectional vortex engines with non-reactive sidewalls. Their work was substantiated by and supplementary to the Particle Image Velocimetry (PIV) measurements of Rom et al.54 The aforementioned studies have established the characteristics of swirling motion to be not only dependent upon the swirl number but also the Reynolds number and the chamber aspect ratio. They have also shown that the swirling intensity is largest near the headwall and that the tangential velocity of the confined fluid changes from free to forced vortex behavior as the flow approaches the axis of rotation. Interestingly, the intersection of the free and forced vortex regions coincides with the point of maximum swirl, which is found to shift inwardly with successive increases in the swirl number. The shift extends nearly uniformly along the chamber length, causing the shape of the vortex core to remain axially invariant; this observation is further confirmed by the visualization experiments of Alekseenko et al.55 Despite the importance of these flow attributes, however, no analytical model had yet been advanced to capture their behavior tacitly. Few studies have been devoted to the mathematical modeling of vortex flows, and the cause of this may be attributed to such complications as the nonlinearities in the Navier-Stokes equations and the uncertainties in the attendant boundary conditions. Closed-form solutions have required the introduction of simplifying assumptions and these have led, at times, to piecewise solutions that are inconsistent with experimental data (Vatistas et al.56). The most well known approximations of vortex flows are those by Rankine,57 Oseen-Lamb,58 and Burgers.59 These models offer piecewise solutions that describe the radial distribution of the tangential velocity; however, they are not concerned with the behavior of the axial or radial motions, which become exceedingly important in cyclonic regimes where flow reversal and the formation of two-cell structures must be accounted for. Aside from the empirical relations by Fontein and Dijksman60 and Smith,44,61 one may cite Sullivan62 and Bloor and Ingham63 who have obtained explicit approximations for bidirectional flows. The latter applied the Polhausen technique to accommodate inlet flow conditions. In principle, their work evolved into the first inviscid, rotational solution for a conical cyclone. As shown by Bloor and Ingham,64 a solution could be arrived at by taking the mean flow vorticity to be inversely proportional to the distance from the axis of rotation. The outcome was a useful approximation which, being strictly inviscid, became naturally unbounded at the center-axis. By considering the bidirectional motion in a cylindrical vortex chamber, Vyas and Majdalani65 have subsequently managed to solve Euler’s equations for the three components of the velocity. Their original model was inviscid and thus bore the same singularity suffered by Bloor and Ingham’s. Since a viscous solution was needed to


handle the core vortex, an asymptotic approximation was pursued and firmly secured. Its predictions were found to be in favorable agreement with existing test measurements and numerical simulations (Vyas and Majdalani66). In the intervening time, the problem was extended to spherical geometry by Majdalani and Rienstra.67 This was realized by solving the vorticity transport equation in spherical coordinates and proving the existence of additional similarity solutions. As a windfall, analytical expressions were obtained for the pressure distribution and core size as function of the geometry and input parameters. In the present article, we extend the analysis of the bidirectional vortex by considering cylindrical chambers with permeable sidewalls. The motivation stems from the need to model the basic core flow in hybrid vortex engines and cyclones with porous walls; the work culminates in the construction of an analytical solution that can be used to describe the bulk gas motion observed in an idealized vortex engine. To this end, a perturbation in the sidewall-to-tangential velocity ratio is first carried out to solve Euler’s equations. A uniformly valid viscous approximation is then derived from the tangential momentum equation. This is arrived at using matched asymptotic expansions to simultaneously capture the dual boundary layers at the core and the sidewall. The same viscous analysis is repeated in the axial and radial directions. The formulation that we develop will be shown to exhibit the key characteristic features of cyclonic motion. It will also constitute a generalization to existing models.

II. Mathematical Model The bidirectional vortex is formed inside a cylindrical chamber of porous length 0L and radius a , with both a closed head end and a partially open downstream end. The exit plane attaches to a straight nozzle of radius b . A sketch is given in Fig. 2 where r and z denote the radial and axial coordinates. The field of interest stretches from the headwall to the base plane in the extent that it remains incompressible. At the base, the fraction of the radius that permits an outflow is given by /b aβ = . Along the remaining portion of the base, an incompressible fluid enters the chamber tangentially to the inner circumference at a prescribed volumetric rate, 2/i iQ Q Ua= . The corresponding tangential velocity U is considered to be sufficiently large to prevent the flow from short-circuiting, a condition by which the injected flow will immediately drift toward and out of the nozzle. Instead, a bidirectional vortex is formed, as in the case of cyclonic separators and furnaces (Bloor and Ingham64). This bidirectional motion is augmented by a secondary flux caused by the radial and uniformly distributed sidewall mass addition. In the hybrid rocket application, the sidewall injection velocity wU may be used to capture the solid fuel regression rate. Practically, wU is appreciably smaller than U . This condition will be evoked in seeking a suitably small parameter. The strong angular momentum carried by the incoming stream causes the formation of a cyclone; this phenomenon subdivides the chamber into two vortex regions: an outer annular section and an inner core region, separated by virtue of a spinning and non-translating cylindrical layer that we call the mantle. The outer vortex occupies the annular region extending from the mantle to the sidewall. It consists of spiraling fluid sweeping up the porous surface while mixing with the wall transpiring mass. At the chamber head end, the outer vortex switches axial polarity, turns inwardly, and continues to spiral toward and out of the nozzle. Our analysis is concerned with the essential features of the ensuing flow field.

A. Equations To characterize the bulk gas motion, a cold-flow model is used. In hybrid rocket analysis, this may be justified by the weak effects of diffusion flames. In solid rocket motors, ignoring the effect of chemical reactions has led to several models that adequately represent the bulk gas motion; one may cite, for example: Culick,68 Majdalani,69

Figure 2. Chamber geometry noted by the presence of a bidirectional vortex in addition to sidewall mass addition.


Griffond et al.,70 Féraille and Casalis,71 and Balachandar et al.72 Along similar lines, the flow is assumed to be (i) steady, (ii) inviscid, (iii) incompressible, (iv) rotational, and (v) axisymmetric. Axisymmetry is warranted by the strong swirl velocity and the absence of friction to decelerate the flow in the tangential direction (Leibovich31). The combination of axisymmetry and frictionless motion leads to another flow attribute of the swirl velocity, namely, axial independence. The weak sensitivity of the swirl velocity to axial variations is corroborated by the work of Leibovich,31 Bloor and Ingham,64 Szeri and Holmes,73 Vatistas et al.,56 and others. Physically, it is granted by the absence of friction between fluid layers and along both the headwall and sidewall. Based on these assumptions, Euler’s equations become

( )1 0r zru u

r r z∂ ∂

+ =∂ ∂

(1)

2 1r r

r zuu u pu u

r z r rθ

ρ∂ ∂ ∂

+ − = −∂ ∂ ∂

(2)

0rru u u

ur rθ θ∂ + =

∂ (3)

1z z

r zu u pu ur z zρ

∂ ∂ ∂+ = −

∂ ∂ ∂ (4)

B. Boundary Conditions The first set of boundary conditions is linked to axisymmetry and headwall impermeability. The second set is due to the inlet configuration and bulk mass conservation. Specifically, one can assume (a) a fully tangential inflow, (b) a zero axial flow at the headwall, (c) a zero radial flow at the centerline, (d) a prescribed radial inflow at the sidewall, and (e) an axial inflow that matches the tangential source. These particular conditions translate into

0

0

, , (tangential injection)0, , 0 (inert head end)0, , 0 (no radial flow across the centerline)

, 0 , (sidewall injection)

, 0 , (inflow at the base)

z

r

r w

i i

r a z L u Uz r ur z ur a z L u U

z L r b Q UA

θ⎧ = = == ∀ == ∀ == ≤ < = −

= ≤ < =

⎪⎪⎪⎨⎪⎪⎪⎩

(5)

C. Normalization In seeking a similarity solution, it is helpful to normalize the principal variables and operators. This can be accomplished by setting

; ; ; =z r bz r aa a a

β= = ∇ = ∇ (6)

; ; ; wr zr zu Uu u

u u uU U U U

θθ ε= = = = (7)

2 2 2 2; ; 2i i w

i wQ A Qpp Q Q L

U Ua a Uaπε

ρ= = = = = (8)

Here, 0( , )U u a Lθ= and ( , )w rU u a z= − represent the average fluid injection velocity at the base and the uniform wall injection velocity along the sidewall, respectively. At this juncture, it may be instructive to highlight the relation that exists between the normalized volumetric flow rate iQ and the unidirectional swirl number S used in the literature.

35 In many studies, such as the one by Hoekstra et al.,48 the swirl number for cyclonic flow is presented as / iS ab Aπ πβσ≡ = (9) where 1iQσ

−≡ refers to the modified swirl number that appears in the analytical solution. Clearly, our modified 1

iQσ−≡ is directly proportional to the classic swirl number S . When 1 / 2β = , 12 2 2.22S π σ σ= .


D. Basic Formulation Pursuant to Eqs. (6)–(8), rotational axisymmetric mean flow motion is prescribed by 0∇ ⋅ =u ; p⋅∇ = −∇u u (10) After substituting 12 ( )⋅∇ = ∇ ⋅ − × ∇ ×u u u u u u into Eq. (10), one can take the curl of the momentum equation to obtain the steady and inviscid vorticity transport equation 0∇ × × =Ωu ; ≡ ∇ ×Ω u (11) The corresponding boundary conditions become

2in0 0

(1, ) 1; ( ,0) 0; (1, )

ˆ(0, ) 0; ( , ) d d

z r

r

u L u r u z

u z r L r r Q

θ

π β

ε

θ

= = = −⎧⎪⎨

= ⋅ =⎪⎩ ∫ ∫ nu (12)

where ˆ zu⋅ =nu represents the outflow velocity while in i wQ Q Q= + accounts for the injected flow at z L= augmented by the wall-injected fluid wQ . The presence of a small parameter ε in Eq. (12) suggests the possibility of an asymptotic treatment. Specifically, a regular perturbation expansion may be applied to the velocity and its vorticity companion. This can be implemented by letting (0) (1) 2( )ε ε+ +u = u u O ; (0) (1) 2( )ε ε+ +Ω = Ω Ω O (13) These expressions can be substituted into Eq. (11). Immediate expansion of the perturbed vorticity transport equation yields (0) (0) (1) (0) (0) (1) 2( ) 0ε ε⎡ ⎤∇ × × + ∇ × × + ∇ × × + =⎣ ⎦u Ω u Ω u Ω O (14)

The solution to this set is described next.

III. Inviscid Solution Before carrying out the asymptotic treatment, it may be helpful to consider the state of the swirl velocity in light of the foregoing assumptions. Specifically, it may be useful to show that at leading order, the swirl velocity decouples from the momentum equation and reduces the complexity of Eqs. (13)–(14).

A. Free Outer Vortex From the θ − momentum equation given by Eq. (3), one can put

0ru u

ur rθ θ∂⎛ ⎞+ =⎜ ⎟∂⎝ ⎠

(15)

where (1, ) 1u Lθ = . Subsequently, one finds 1/u rθ = (16) Equation (16) confirms the presence of the free vortex motion that is characteristic of swirling inviscid flow. We find that at leading order, both radial and axial components of vorticity vanish identically.

B. Leading Order Approximation At this point, both radial and axial velocity components are still to be determined from the reduced set of equations given by

(0) (0)[ ]1 0r z

ru ur r z

∂ ∂+ =

∂ ∂ (continuity) (17)

(0) (0) (0) (0)[ ] [ ]

0r zu u

r zθ θ∂ Ω ∂ Ω+ =

∂ ∂ (vorticity transport) (18)

(0) (0)

(0)r zu uz r θ

∂ ∂− = Ω

∂ ∂ (vorticity) (19)

Upon realization that the swirl velocity is decoupled from the remaining set (due to axisymmetry), the introduction of the Stokes streamfunction becomes a possibility; thus we let

(0) (0)

(0) (0)1 1; r zu ur z r rψ ψ∂ ∂

= − =∂ ∂

; (20)


where (0) (1) 2( )ψ ψ εψ ε= + +O is a series of diminishing terms. When this transformation is inserted into the vorticity transport equation given by Eq. (18), one obtains, at leading order,

(0) (0)(0) (0)

0z r r r z r

θ θψ ψ⎡ ⎤ ⎡ ⎤Ω Ω∂ ∂ ∂ ∂− + =⎢ ⎥ ⎢ ⎥∂ ∂ ∂ ∂⎣ ⎦ ⎣ ⎦ (21)

and so

(0) (0)

(0) (0)

/

/z z

r r

r

rθ

θ

ψ

ψ

⎡ ⎤ ⎡ ⎤Ω⎣ ⎦ ⎣ ⎦=⎡ ⎤ ⎡ ⎤Ω⎣ ⎦ ⎣ ⎦

(22)

The resulting equality holds for any (0) (0)[ ( , )]rF r zθ ψΩ = (23) This standard form may be substituted into Eq. (22). Then, it can be promptly seen that

{ }{ }

(0)

(0)

(0) (0) (0) (0)

(0) (0)(0) (0)

/ [ ] [ ] [ ][ ] [ ]/ [ ]

zz z z

r rr r

r F F

Fr Fθ ψ

θ ψ

ψ ψ ψψ ψψ

⎡ ⎤Ω⎣ ⎦ = = =⎡ ⎤Ω⎣ ⎦

(24)

According to Eq. (23), F can be a general function of (0)ψ . The two simplest cases correspond to 2F C= and 2 (0)F C ψ= , where C is some constant. It is a simple exercise to verify that the first choice is incongruent with the

boundary conditions; the linear relation, however, proves to be suitable. One can put (0) 2 (0)C rθ ψΩ = (25) One must bear in mind that this linear choice may not be unique, although no other alternatives with the potential of yielding a closed-form solution could be immediately identified. When Eq. (25) is inserted into the vorticity equation, one obtains68

2 (0) 2 (0) (0)

2 2 (0)2 2

1 0C rr rz r

ψ ψ ψ ψ∂ ∂ ∂+ − + =∂∂ ∂

(26)

At this juncture, three of the boundary conditions may be rewritten for the streamfunction. Based on Eq. (12), one now has

(0) (0)

(0) (0)

(0) (0)

0; 0; / 0

0; 0; / 0

1; 0; / 0

z

r

r

z u r

r u z

r u z

ψ

ψ

ψ

⎧ = = ∂ ∂ =⎪

= = ∂ ∂ =⎨⎪ = = ∂ ∂ =⎩

(27)

C. Separation of Variable Solution Equation (26) is clearly separable. One can proceed by setting (0) ( , ) ( ) ( )r z f r g zψ = (28) This decomposes Eq. (26) into

2 2

2 2 22 2

1 d 1 d 1 ddd d

g f f C r fg f r rz r

λ⎛ ⎞

− = − + = ±⎜ ⎟⎝ ⎠

(29)

where λ is a separation constant. For a nonzero λ , the streamfunction exhibits either trigonometric or hyperbolic variations in the axial direction. In a real cyclone, such behavior is unlikely to occur; thus, the possibility of a nonzero separation constant is ruled out for the sake of physicality. The only plausible choice then is 0λ = . On the one hand, this value leads to a linear axial variation of the form 1 2( )g z C z C= + . On the other, it permits retrieving the radial variation of the streamfunction from the Bessel equation

2

2 22

d 1 d 0dd

f f C r fr rr

− + = (30)

and so ( ) ( )2 21 12 2( ) cos sinf r A Cr B Cr= + (31) or ( ) ( ) ( )(0) 2 21 11 2 2 2cos sinC z C A Cr B Crψ ⎡ ⎤= + +⎣ ⎦ (32)


Except for the unknown constants that must be prescribed by the boundary conditions, Eqs. (26) and (32) are identical to those employed by Culick to derive a mean flow approximation for solid rocket motors.68 One could have also arrived at this result through manipulation of the Bragg-Hawthorne equation.

D. Particular Solution Using the constraints associated with Eq. (27), one can evaluate the remaining constants. First, due to the vanishing axial velocity at the head end, one deduces that 2 0C = . This leaves ( ) ( )(0) 2 21 11 2 2cos sinC z A Cr B Crψ ⎡ ⎤= +⎣ ⎦ (33) Second, (0) (0, ) 0ru z = implies that 0A = . Third, as

(0)ru vanishes along the sidewall, one must have

( )11 2sin 0C B C = (34) Realizing that neither 1 0C = nor 0B = are acceptable outcomes, one is left with 12sin( ) 0C = ; forthwith, a fundamental solution may be conceived with 2C π= . Without sacrificing generality, one may set 1 1C = and write (0) 2sin( )Bz rψ π= (35) The velocity field corresponding to Eq. (35) becomes (0) 1 2 1 2sin( ) 2 cos( )r θ zBr r r B z rπ π π

− −= − + +e e eu (36) In order to calculate the last constant, mass balance may be globally applied to account for the radial inflow along the sidewall. Given that in out i wQ Q Q Q= = + (37) one calculates

2i

iA

Qa

= and 2(2 )

2wwU aL

Q LUa

ππε= = (38)

to obtain

in out2 2i

i wA

Q Q Q L Qa

πε= + = + = (39)

At leading order ( 0ε = ), mass conservation requires that (0) (0)

0 0ˆ2 d 2 dz ir r u r r Q

β βπ π⋅ = =∫ ∫nu (40)

and so 2csc( ) / (2 )iB Q Lπβ π= (41) It follows that the leading order velocity field may be expressed by

2

(0)2

sin( ) 12 sin( )

ir θ

Q rrL r

ππ πβ

= − +e eu 22 cos( )sin( )i

zQ z

rL

ππβ

+ e (42)

Thus, by letting

22 csc( )2 sin( )iQ

Lκ κ πβ

π πβ≡ = (43)

one can put (0) 2sin( )z rψ κ π= ; (0) 2 24 sin( )rz rθ π κ πΩ = (44) and (0) 1 2 1 2sin( ) 2 cos( )r θ zr r r z rκ π πκ π

− −= − + +e e eu (45) Other important flow characteristics at leading order include

2 2 2 2 2(0)

3

1 sin ( ) sin(2 )r r rpr r

κ π π π⎡ ⎤+ −∂ ⎣ ⎦=∂

(46)

(0)

2 24p zz

π κ∂ = −∂

(47)

and { }(0) 2 2 2 2 2 21 12 21 8 1 cos(2 )p r r z rκ π π− ⎡ ⎤Δ = − + + −⎣ ⎦ (48)


IV. First Order Equation with Sidewall Mass Addition Before setting up the first order solution, it must be noted that the swirl velocity is not perturbed, and as such, the angular momentum equation remains uncoupled from the axial and radial momentum equations. At ( )O ε , the perturbed mass conservation and vorticity transport equations appear as (1) 0∇ ⋅ =u ; (1) (0) (0) (1) 0∇ × × + ∇ × × =u Ω u Ω (49) The corresponding boundary conditions become

2(1) (1) (1) (1)

0 0ˆ( ,0) 0; (1, ) 1; (0, ) 0; ( , ) d dz r r wu r u z u z Q r L r r

π βθ= = − = = ⋅∫ ∫ u n (50)

As before, one may let

(1) (1)

(1) 1 1r z r r

ψ ψ∂ ∂= − +

∂ ∂r zu e e (51)

At ( )εO , the first and second terms in the linearized vorticity transport equation give

(0) (1)∇ × ×u Ω(0) (1) (0) (1)[ ] ( )r zu u

r zθ θ

θ

⎧ ⎫∂ Ω ∂ Ω= − +⎨ ⎬

∂ ∂⎩ ⎭e (52)

and

(1) (0)∇ × ×u Ω(1) (0) (1) (0)[ ] [ ]r zu u

r zθ θ

θ

⎧ ⎫∂ Ω ∂ Ω= − +⎨ ⎬

∂ ∂⎩ ⎭e (53)

Substitution of Eqs. (52)–(53) into the linearized vorticity transport equation given by Eq. (49) leads to

(0) (1) (1) (0) (0) (1) (1) (0) 0r r z zu u u ur zθ θ θ θ∂ ∂⎡ ⎤ ⎡ ⎤Ω + Ω + Ω + Ω =⎣ ⎦ ⎣ ⎦∂ ∂

(54)

where

(1) (1) (1) (1)

(1) 1 1r zu uz r z r z r r rθ

ψ ψ⎡ ⎤ ⎡ ⎤∂ ∂ ∂ ∂ ∂ ∂Ω = − = − −⎢ ⎥ ⎢ ⎥∂ ∂ ∂ ∂ ∂ ∂⎣ ⎦ ⎣ ⎦

(55)

To be consistent with the similarity transformation of the velocity at zeroth order, the radial velocity must be dependent on the radial coordinate. On the one hand, this requires a streamfunction of the form (1) ( )z h rψ = . On the other, one may let (1) (1) ( )r ru u r= so that Eq. (55) reduces to

(1) (1)

(1) 1zur r r rθ

ψ⎡ ⎤∂ ∂ ∂Ω = − = − ⎢ ⎥∂ ∂ ∂⎣ ⎦

(56)

After inserting (1) ( )z h rψ = and Eqs. (45) and (56) into Eq. (54), one collects

2

2 21 d sin( ) d 1 d 4 sin( )d d d

r h r hr r r r r r

π π π⎡ ⎤⎛ ⎞ −⎢ ⎥⎜ ⎟

⎝ ⎠⎣ ⎦2 2 21 d 1 d 1 d4 cos( ) 8 sin( ) 0

d d dh hr r

r r r r r rπ π π π⎛ ⎞− + =⎜ ⎟

⎝ ⎠ (57)

It is now appropriate to employ 2rη π≡ . After some algebra, Eq. (57) collapses into

3 2

3 2

d d dsin cos sin cos 0dd d

h h h hη η η ηηη η

− + − = (58)

To make further headway, one may formulate a guess as to the solution of the above equation in the form of( ) sinh Cη η= . Using the method of variation of parameters (see Zhou and Majdalani74), the total solution ( )h η

may be compiled by allowing C to vary. Starting with the derivatives, sin cosh C Cη η′ ′= + ; sin 2 cos sinh C C Cη η η′′ ′′ ′= + − (59) and sin 3 cosh C Cη η′′′ ′′′ ′′= + 3 sin cosC Cη η′− − (60) one may substitute Eqs. (59)–(60) into Eq. (58). Many terms cancel except for 2sin sin(2 ) 2 0C C Cη η′′′ ′′ ′+ − = (61) Equation (61) can be readily solved;75 the result is ( )11 2 32( ) cotC C C Cη η η= − + (62) where 1C , 2C , and 3C are pure constants. Recalling that ( ) sinh C η η= , one may put ( )11 2 32sin cosh C C Cη η η= − + (63) Returning to the radial coordinate, the total solution may be represented by


2 2 2 21 11 2 32 4( ) sin( ) cos( ) cos( )h r C r C r C r rπ π π π= − − (64) and so (1) 2 2 2 21 11 2 32 4sin( ) cos( ) cos( )C z r C z r C r z rψ π π π π= − − (65) Similarly, one finds (1) 1 2 1 2 21 11 2 32 4sin( ) cos( ) cos( )ru C r r C r r C r rπ π π π

− −= − + + (66) In order to avoid violating the underlying assumption of symmetry about the axis of rotation, 2C must vanish. Moreover, to comply with the wall injection boundary condition in Eq. (50), one must have 3 4 /C π= . It must be realized that the hardwall boundary condition at 0z = is automatically satisfied, having employed the proper ansatz,

(1) 2 2 21( ) sin( ) cos( )z h r C z r r z rψ π π= = − . Only one constant, 1C , remains undetermined. This appears in the axial

term (1) 2 2 2 212 cos( ) 2 cos( ) 2 sin( )zu C z r z r zr rπ π π π π= − + (67) To fix 1C , global mass balance must be secured at first order. Starting with (1)

02 2 ( , ) dw zQ L u r L r r

βπ π= = ∫ (68)

one finds 2 2 21 [1 cos( )]csc( )C β πβ πβ= + . Forthwith, the first order streamfunction and its velocity components may be updated as (1) 2 2 2 2[1 cos( )]csc( ) sin( )z rψ β πβ πβ π= + 2 2cos( )z r rπ− (69) (1) 2 2 2 2[1 cos( )]csc( )sin( ) /ru r rβ πβ πβ π= − +

2cos( )r rπ+ (70) and (1) 2 2 2 22 [1 cos( )]csc( ) cos( )zu z rπ β πβ πβ π= +

2 2 22 cos( ) 2 sin( )z r zr rπ π π− + (71)

V. Flowfield Characteristics

A. Sidewall Velocity Estimates The wall injection velocity may be estimated from experiments yielding correlations for sr , the solid fuel regression rate.5 The estimates are generally based on the assumption of steady-state regression of propellant grain. To that end, one must recall that simple mass conservation along the pyrolyzing grain surface requires that

,s s g wr Uρ ρ= where subscripts ‘s’ and ‘g’ refer to the solid and gas phases, respectively. The gas density at the regressing surface can be estimated using the ideal gas equation of state. Based on empirical studies by Chiaverini et al.,76 the average surface temperature may be taken to be 1,000 K. The solid phase may be specified to be, for example, HTPB fuel with the corresponding density calculated accordingly. In the same vein, the regression rate,

,sr can be obtained from available literature.5

For the typical hybrid vortex engine, the wall injection velocity wU varies between 0.3 and 2.5 m/s, while the average oxidizer injection velocity U is held at 260 m/s. The wall injection parameter ε may hence range from 0.001 to 0.01.

B. Mantle Sensitivity to Sidewall Velocity The mantle, the fluid layer that separates the outer vortex from the inner one, can rotate about the chamber axis; however, it cannot axially translate. It is defined by the surface along which the axial velocity vanishes, thus the mantle can be located by solving for the root of (0) (1) 0z z zu u uε= + = (72) and so 2 2 2 22 cos( ){ [1 cos( )]csc( ) }z rπ πκ πε β πβ πβ ε+ + − 2 22 sin( ) 0zr rπε π+ = (73) Using *r β= to denote the radius of the mantle, this root can be determined from Eq. (73) for an arbitrary chamber opening β . We are especially interested in the ideal flow that can be achieved when the nozzle radius is coincident with the mantle radius. Thus, by setting *β β= , the radius of the inner vortex (exiting the chamber) will match the radius of the chamber opening. This ideal condition leads to a smooth outflow that effectively mitigates the formation of corner vortices that could be exacerbated by wall collisions in the exit plane. Granted this idealization, Eq. (73) reduces to 2 2 2 2* * * *2 cos( ){ [1 cos( )]csc( ) }z πβ πκ πε β πβ πβ ε+ + − 2 2* *2 sin( ) 0zπε β πβ+ = (74)


For 0.01κ = and 0.001ε = , the mantle location obtained via Eq. (74) is 0.7179. Thereafter, by increasing the wall injection to 0.005ε = , *β shifts to 0.7535, and the mantle is pushed closer to the wall. Explicit roots may be obtained asymptotically using a quadratic polynomial retrieved from a Taylor-series expansion of Eq. (74) about the point * 1 / 2β = ; this particular root corresponds to the limiting physical process for which the solution approaches the case of insignificant sidewall injection ( 0ε → ). After some algebra, one gets

* ( * *) / *;A C Bβ = −2 2

2 2 2 2 2 2 2

* / 2; * 4 2 2* 4 6 6 4 2

A BC

π ε ε πε π ε πκ

ε πε π ε πεκ π εκ π κ

⎧ = = − + −⎪⎨

= − + − + +⎪⎩ (75)

The roots determined using Eq. (75) are 0.7179, 0.7542, and 0.7904 corresponding to 0.01κ = and ε values of 0.001, 0.005, and 0.01, respectively. Unsurprisingly, both the swirl parameter κ and the wall injection parameter ε affect the mantle location *β . This is contrary to the behavior observed in the case of bidirectional flow in a chamber with hard walls, such as an idealized liquid rocket engine.77 In order to assess the mantle sensitivity to wall injection, ε can be varied at constant κ or vice versa. Results are shown in Table 1 where κ is held at 0.01 while ε is varied from 0 to 0.01. The limiting case of 0ε = may be used to describe the flowfield in the liquid vortex engine.77 When wall injection is increased, the mantle is pushed closer to the sidewall; this trend can be attributed to the increased secondary mass flowing into the inner vortex at higher regression rates. The increased mass flux causes the inner vortex to expand by pressing the outer annular region toward the sidewall.

C. Streamlines In order to better visualize the bidirectional motion, streamlines are plotted in Figs. 3(a-c) for 0.01κ = and in Figs. 3(d-f) for 0.001,κ = while the sidewall injection parameter /wU Uε = is permitted to undergo incremental increases from 0 to 0.001. For steady flow, one recalls that streamlines, pathlines, and streaklines coincide in describing the trajectory of fluid particles throughout the chamber. These plots confirm that the mantle location does not vary in the axial direction and that the turning point in the axial velocity approaches the sidewall at higher values of ε or when the swirl is increased (i.e., by lowering κ ). The full asymptotic expression for ψ may be written as (0) (1) 2( )ψ ψ εψ ε= + +O { }2 2 2 2 2 2 2 2sin( ) csc( ) [csc( ) cot( ) cot( )] ( )z r r rπ κ πβ ε πβ β πβ π ε= + + − +O (76)

D. Axial Velocity Distribution The axial velocity is briefly described in Fig. 4a. There, it can be seen that as ε is increased from 0 to 0.01, the centerline velocity is nearly doubled. This appreciable velocity increase can be once more attributed to the role of sidewall mass injection. In this case, it causes the axial velocity magnitude to increase throughout the inner vortex, a

Table 1. Samples of mantle variation

No. κ ε *β * 1/ 2β −

1 0.01 0.000 0.707 0.000 2 0.01 0.001 0.718 0.011 3 0.01 0.005 0.752 0.045 4 0.01 0.010 0.786 0.079

ε = 0a)

ε = 0d)

ε = 0.001b)

ε = 0.001e)

1 2 3 40

1

r

ε = 0.01

c) z 1 2 3 4

ε = 0.01

f) z

Figure 3. Streamlines illustrating bidirectional vortex circulation with increasing sidewall injection and swirl intensity. Here κ = 0.01 (left) and 0.001 (right). Recall that swirl is intensified with successive decreases in .κ


trend that supports the streamline behavior described earlier. The total axial velocity becomes

(0) (1) 2( )z z zu u uε ε= + +O

2 2 2 2 2 2 2 22 cos( ){ csc( ) [csc( ) cot( ) tan( ) 1/ ]} ( )z r r rπ π κ πβ ε πβ β πβ π π ε= + + + − +O (77) In the present solution, zu does not vanish at the sidewall. This deficiency is overcome using a boundary layer treatment in Section VII. Nonetheless, Eq. (77) enables us to precisely calculate the location of the mantle. As shown in Fig. 4b, the mantle draws nearer to the sidewall with successive increases in the blowing parameter or the swirl number. Due to added mass along the wall, we find 1/ 2,β ≥ with the equality being reserved for the impervious sidewall case with 0.wUε = = Thus, by either increasing ε or decreasing κ , the mantle moves closer to the wall. This behavior confirms the shift in streamline curvature depicted in Fig. 3.

E. Radial Velocity Distribution The radial velocity is illustrated in Fig. 5 for 0.01κ = and 0.001 with the usual values of the perturbation parameter. It is interesting to note the shift in maximum ru in the direction of the wall with successive increases in sidewall mass addition. This trend is consistent with the movement of the mantle. The radial velocity for the hybrid model may be expressed by (0) (1) 2( )r r ru u uε ε= + +O

2 1 2 2 2 2 2 2 2sin( ) { csc( ) [csc( ) cot( ) cot( )]} ( )r r r rπ κ πβ ε πβ β πβ π ε−= − + + − +O (78)

To accurately predict the maximum radial velocity and its locus, one can set 0ru′ = and solve for mr r= . The outcome is

2 2 2 2 2csc( )sin( ) 2 csc( )cos( )m m mr r rκ πβ π πκ πβ π−

2 2 2 2 2 2[ cos( ) 2 cot( ) cos( )m m m mr r r rε π πβ πβ π+ −

2 2 2 4 22 csc( ) cos( ) 2 sin( )m m m mr r r rπ πβ π π π− −2 2 2 2 2cot( )sin( ) csc( )sin( )] 0m mr rβ πβ π πβ π+ + = (79)

Considering the transcendental nature of Eq. (79), a numerical root finding technique may be employed. Results are cataloged in Table 2, where κ is kept fixed while ε is varied. One finds that the radial velocity maxima occur at

Table 2. Radial velocity maxima

No. κ ε maxr max( )ru

1 0.01 0.000 0.609 -0.015 2 0.01 0.001 0.618 -0.016 3 0.01 0.005 0.650 -0.021 4 0.01 0.010 0.683 -0.027

0 0.2 0.4 0.6 0.8 1-10

-5

0

5

10u z

/(κ z)

a) r

ε 0 0.001 0.005 0.01

0.7170.725

0.750

0.7750.800

0.8250.850

0.8750.900

0.001 0.02 0.04 0.06 0.08 0.100

0.02

0.04

0.06

0.08

0.10

κ

ε

b)

Figure 4. Variation of a) the axial velocity with the wall injection parameter at 0.01κ = and b) the mantle location.

0 0.2 0.4 0.6 0.8 1-4

-3

-2

-1

0

u r/κ ε

0 0.001 0.005 0.01

ra) 0 0.2 0.4 0.6 0.8 1-14-12-10-8-6-4-20

u r/κ ε

0 0.001 0.005 0.01

rb)

Figure 5. Radial velocity distribution at several wall injection parameters and either a) 0.01κ = or b) κ = 0.001.


the normalized radii of 0.61, 0.62, 0.65, and 0.68 for 0ε = , 0.001, 0.005, and 0.01, respectively. At these locations, max| ( ) |ru is rendered equal to 0.015, 0.016, 0.021, and 0.027.

Alternatively, an asymptotic expression for the radial velocity maximum and its location may be determined using the same series expansion approach utilized for the mantle location. One finds ( ) /m m m mr A C B= − (80) where ( ) ( )2 2 2 3 2 2 2 22 csc( ) 16 2 16 2 cot( ) 16 2mA πβ ε π ε κ π κ π ε πβ εβ εβ π⎡ ⎤= + − − − − −⎣ ⎦

(81)

( )3 2 2 26 2 24 cot( ) 24 csc( )mB πε π ε β ε πβ πβ ε κ⎡ ⎤= − − − +⎣ ⎦ (82) ( )2 3 2 2 26 2 24 cot( ) 24csc( )m mC A πε π ε β ε πβ πβ ε κ⎡ ⎤= − + + + +⎣ ⎦

( ) ( )3 2 2 2 2 2 2 2cot( ) 24 4 csc( ) 24 4 24 4πε π ε πβ β ε π β πβ ε π ε κ π κ⎡ ⎤× + + − + − + −⎣ ⎦ (83)

The radial velocity maxima for ε values of 0.001, 0.005, and 0.01 are approximated at 0.6210, 0.6511, and 0.6832, respectively. Their corresponding radial velocities are -0.01635, -0.02121, and -0.02703.

F. Pressure Distribution The pressure gradients in the radial and axial directions can be determined using Eqs. (10), (13), (45), (70), and (71). After some algebra, one finds

2 2 2 2 2 2 2

3

1 csc ( )[sin ( ) sin(2 )]p r r rr r

κ πβ π π π∂ + −=

∂

22 2 2 4

3

2 csc( ) {[ cot( ) csc( ) ]rr

κ πβε β πβ πβ π+ + −

2 2 4 2 2sin ( ) cos ( )r r rπ π π× + 2 2 2 2 2[ cot( ) csc( )]sin(2 )}r rπ β πβ πβ π− + (84) and

2 2 2 2 24 csc ( ) 8 csc( )p z zz

π κ πβ επκ πβ∂ = − −∂

2 2cot( ) csc( ) 1πβ π πβ⎡ ⎤− −⎣ ⎦ (85)

Partial integration of Eqs. (84)–(85) with respect to r and z enables us to calculate the total pressure drop. One finds

( )

2 22 2 2 2

2

csc ( ) 1 1 84

p r zrπβ κ π⎡Δ = − + +⎣

22 2 2

2

csc( )cos(2 ) cos(2 )2

rr

κ πβκ π πβ ε⎤+ + −⎦

( )(2 2 2 2 2 2cot( ) csc( ) 1 8 r zβ πβ πβ π⎡× + +⎣ ) ( )2 2 2 2cos(2 ) 8 sin(2 )r r z rπ π π ⎤− − + ⎦ (86)

In practice, it must be noted that Eqs. (84) and (86) are virtually independent of ε , thus they are well represented by their corresponding curves described by Vyas and Majdalani65 for the no wall injection case. Specifically, they support the presence of an upward flowing outer vortex. Only the axial pressure gradient is affected by sidewall mass addition, and the corresponding behavior is displayed in Fig. 6a where /p z∂ ∂ is shown along the chamber centerline. Clearly, the pressure drop in the axial direction is more pronounced when the mass to be driven out of the chamber is increased. In Fig.6b, the radial pressure gradient is plotted and shown to be dominated by the 31 / r term contributed by the inviscid tangential velocity. In order to overcome the attendant singularity at the origin, a viscous treatment is required; this is later provided in Section VI.

0 1 2 3 4 5-0.06

-0.03

0pz

∂∂

z

ε 0 0.001 0.005 0.01

a) 0 0.2 0.4 0.6 0.8 10

20

40

60

80

100pr

∂∂

r

ε 0 0.001 0.005 0.01

b) Figure 6. Variation of a) the axial pressure gradient along the centerline and b) the radial pressure gradient at κ 0.01.=


So far, an exact closed-form analytical expression for the simulated hybrid vortex has been presented. The solution emerges from the inviscid Navier-Stokes equations and corroborates the existence of a bipolar, coaxial, vortex pair inside a swirl-driven, porous chamber. The present formulation, albeit approximate, exhibits most of the known features of the bidirectional vortex, specifically those that have been reported in numerical simulations5,78 and laboratory tests.2,3 In addition to its ability to predict pressure, velocity, and vorticity distributions away from the regions of nonuniformity, the present solution captures the movement of the mantle due to variations in the regression rate. In short, the inviscid formulation for the hybrid vortex engine supports the existence of a cyclonic circulation based on the fundamental equations of motion and a judicious set of boundary conditions.

VI. Tangential Boundary Layers It has been well established that the free vortex solution presented earlier for the swirl velocity is a suitable approximation only when sufficiently removed from the chamber axis. As the centerline is approached, transition to forced vortex motion must be entertained en route to suppressing the known singularity at 0r = . Physically, the forced vortex is induced by viscous forces. These dominate near the chamber axis to the extent of mitigating further growth in the swirl velocity. The inability of inviscid solutions to display the forced vortex behavior is a known feature of swirling flows.31,64 At the sidewall, another boundary layer emerges as a consequence of the no slip requirement in the wall-tangential direction. This is needed to bring the swirl velocity to zero at the sidewall. The treatment of these boundary layers is an essential feature of this problem.

A. Tangential Boundary Layer Equation From Eq. (78), one can express the radial velocity as 1 2 1 2 2sin( ) sin( ) cos( )ru Ar r Br r r rκ π ε π π

− −⎡ ⎤= − − −⎣ ⎦ (87)

where

2

2 2 2

csc( )csc( )[1 cos( )]

AB

πβ

πβ β πβ

⎧ =⎪⎨

= +⎪⎩ (88)

By retaining the dominant viscous terms in the swirl momentum equation, one can write

( )dd( ) d 1 1;

d d dr ruruu

r r r r r Re Uaθθ νδ δ

⎡ ⎤= ≡ =⎢ ⎥

⎣ ⎦ (89)

where / ( )Uaδ ν≡ is small, being the reciprocal of the tangential-injection Reynolds number. The regularized momentum equation can be recast in the form of

2

2

d d2d drX Xu δ πηη η

= (90)

where 2rη π≡ and X ruθ≡ (91) Using these variable transformations, Eq. (87) can be converted into ( ) sin / / cosru A Bκ ε η π η ε η π η= − + + (92) Then, by substitution into Eq. (90), one gets

2

2

d 1 sin d( ) cos 02 dd

X XA B ηδ π κ ε ε ηπ η ηη

⎡ ⎤+ + − =⎢ ⎥

⎣ ⎦ (93)

η

0 π

cham

ber a

xis

outer regioninner sidewall

s ηδ

= q π ηδ−

=

Figure 7. Diagram depicting inner and wall-tangential boundary layers and the spatial transformations needed to rescale their regions of nonuniformity to a [0,1] interval.


The ensuing relation represents the key boundary layer equation that must be asymptotically manipulated to capture the forced vortex behavior and the sidewall boundary layer. The corresponding regions of nonuniformity are depicted in Fig. 7.

B. Inner and Outer Expansions The outer expansion of Eq. (93) can be swiftly initiated. Using a regularly perturbed series of the form

( ) ( ) ( ) 20 1 ( )

o o oX X Xδ δ= + +O , one collects

( )

( )00 0

d1 sin( ) cos 0; constant2 d

ooXA B X Cηπ κ ε ε η

π η η⎡ ⎤

+ − = = =⎢ ⎥⎣ ⎦

(94)

where the superscript ‘o’ denotes an outer expansion. Note that the leading order outer solution is merely a duplication of the previously assumed free vortex expression, ( ) ( )0 0

o oX ru Cθ= = . The inner equation that underscores the role of viscous stresses may be arrived at by introducing a spatially magnified scale proportionate to the forced vortex region (see Fig. 7). This may be accomplished by stretching the outer variable by means of

msηδ

= (95)

where ‘s’ is the inner scale. The exponent ‘m’ may be determined from the distinguished limit at which consistency in asymptotic orders is achieved. Substitution into Eq. (93) yields

2 ( ) ( )

2 2

d 1 ( ) 1 dsin( ) cos( ) 02 dd

i im m

m m m

X A B Xs sss s

δ π κ ε δ ε δπδ δ δ

+⎡ ⎤+ − =⎢ ⎥⎣ ⎦ (96)

Because we are interested in studying the effects near the chamber axis, we proceed by linearizing all functions near 0η = . This operation entails no loss in generality. Using MacLaurin series expansions for the sine and cosine

terms, we put

2 ( ) ( )

3 21 13! 2!2 2

d 1 ( ) 1 d( ) ... 1 ( ) ... 02 dd

i im m m

m m m

X A B Xs s sss s

δ π κ ε δ δ ε δπδ δ δ

+⎧ ⎫⎡ ⎤ ⎡ ⎤+ − + − − + =⎨ ⎬⎣ ⎦ ⎣ ⎦⎩ ⎭ (97)

which, by virtue of some cancellations, begets

{ }2 ( ) ( )

1 2 21 16 22

d 1 d( ) 1 ( ) ... 1 ( ) ... 02 dd

i im m mX XA B s s

ssδ κ ε π δ ε δ

π− ⎡ ⎤ ⎡ ⎤+ + − + − − + =⎣ ⎦ ⎣ ⎦ (98)

To achieve a balance between diffusive and convective terms, one must have 1m = . This enables us to collapse Eq. (98) into

[ ]2 ( ) ( )

22

d 1 d( ) ( ) 02 dd

i iX XA Bss

π κ ε ε δπ

+ + − + =O (99)

Then, using an inner expansion of the form ( ) ( ) ( )0 1i i iX X Xδ= + +… , one collects, at leading order,

[ ]2 ( ) ( )

0 02

d d1 ( ) 02 dd

i iX XA B

ssπ κ ε ε

π+ + − = (100)

The solution is simple, namely, ( )0 0 1( / ) exp( )

iX A A sφ φ= − − ; ( )12 /A Bφ κ ε ε π≡ + − (101) The emerging integration constants may be merged with the outer solution using Prandtl’s matching principle. Accordingly, the outer limit of the inner solution must equal the inner limit of the outer expansion. Thus, by placing ( ) ( )0 00

lim = lim i os

X Xη→∞ →

or 0 1 00lim ( / ) exp( ) lim s

A A s Cη

φ φ→∞ →

− − = (102)

one deduces the common part to both inner and outer approximations, ( )0 0 0i

oX C A⎡ ⎤ = =⎣ ⎦ . The composite inner (ci)

solution may be arrived at by way of superposition from ( )( ) ( ) ( ) ( )0 0 0 0 0 1( / ) exp /ci o i i oX X X X A A ϕ φη δ⎡ ⎤= + − = − −⎣ ⎦ (103)

C. Nonsingular Swirl Component At this juncture, we may return to the radial coordinate and express the composite inner swirl velocity as

( ) 20 10

1 expciA Au rr Aθ

φ πφ δ

⎡ ⎤⎛ ⎞ ⎛ ⎞= − −⎢ ⎥⎜ ⎟ ⎜ ⎟⎝ ⎠⎢ ⎥⎝ ⎠⎣ ⎦

(104)


The two remaining unknowns may be secured from the problem’s physical constraints. On the one hand, knowing that the azimuthal velocity must vanish along the chamber axis, one must set ( ) (0) 0ciuθ = , thereby retrieving

0 1 /A A φ= . On the other hand, one must equate ( ) (1)ciuθ to the wall-tangential speed. This condition requires that

[ ]1( / ) 1 exp( / ) 1A φ πφ δ− − = or 1 1 exp( / )A φ

πφ δ=

− − (105)

The non-secular approximation for the swirl velocity is now at hand. We find and define

21

42( ) 1 1 exp( / ) 1

1 exp( / )

Vrci r eu

r rθπ φ δπφ δ

−⎡ ⎤− − −= ⎢ ⎥− −⎣ ⎦

; 4V πφδ

≡ (106)

where the hybrid form of the vortex Reynolds number V surfaces.79 When sidewall mass addition is present, this important parameter takes the form

2 2 2 22( ) 2 2 { csc( ) csc( )[1 cos( )]} 2A BV κ ε π ε π κ πβ ε πβ β πβ ε

δ δ+ − + + −

= =

2 2 2 20

2 2csc( ) csc( )[1 cos( )]i w w

A U URe

aL U Uπβ π πβ β πβ

⎧ ⎫= + + −⎨ ⎬

⎩ ⎭

2 2 2 2

0

csc( ) 2 csc( )[1 cos( )] 2i w wQ

Re ReL

πβ π πβ β πβν

= + + − (107)

where /w wRe U a ν= is the sidewall injection Reynolds number.

D. Sidewall Expansion In order to capture the rapid changes near the sidewall, we note that by virtue of 0 η π≤ ≤ , we may introduce the slow coordinate

mqπ ηδ−

= (108)

Here mδ refers to the thickness of the wall-tangential boundary layer in the η variable. Using (w) to denote a wall expansion, Eq. (93) may be written as

2 ( ) ( )

2 2

d 1 ( ) 1 dsin( ) cos( ) 02 dd

w wm m

m m m

X A B Xq qqq q

δ π κ ε π δ ε π δπδ π δ δ

⎡ ⎤+− − − − =⎢ ⎥−⎣ ⎦

(109)

Taylor-series expansions of the trigonometric terms yield

( )2 ( ) ( )

1 2162

d 1 d( ) 1 ( ) 02 dd

w wm mX XA B

qqδ π κ ε π ε δ

π− ⎡ ⎤− + − + + =⎣ ⎦ O (110)

A balance between diffusion and convection may be achieved for 1m = . This distinguished limit enables us to write a wall expansion in the form ( ) ( ) ( ) 20 1 ( )

w w wX X Xδ δ= + +O . At leading order, we collect

( )2 ( ) ( )

20 0162

d d1 ( ) 1 02 dd

w wX XA B

qqπ κ ε π ε

π⎡ ⎤+ + − − =⎣ ⎦ (111)

The solution is nearly at hand. Integration readily gives ( )0 0 1( / )exp( )

wX B B qϕ ϕ= − − ; ( )( )21 12 6 1 /A Bϕ κ ε π ε π⎡ ⎤≡ + − −⎣ ⎦ (112) The two required auxiliary conditions may be established from the no slip at the sidewall and matching with the composite inner solution. These are

( )0

( ) ( )0 00 1

(0) 0

lim = lim

w

w ci

r r

X

X X→ →

⎧ =⎪⎨⎪⎩

(113)

The first condition gives 1 0B Bϕ= . We thus have ( )0 1 / 1 exp /B πϕ δ= − −⎡ ⎤⎣ ⎦ . Through backward substitution, the sidewall approximation emerges. This is

( )

214( )

0 14

1 exp (1 )

1 expww

w

V rX

V

⎡ ⎤− − −⎣ ⎦=− −

or ( )

214( )

14

1 exp (1 )11 exp

ww

w

V ru

r Vθ⎧ ⎫⎡ ⎤− − −⎪ ⎪⎣ ⎦= ⎨ ⎬− −⎪ ⎪⎩ ⎭

(114)

where the wall-characteristic Vortex Reynolds number is given by


2 2 2 2 212 61

6

2 ( 1){ csc( ) csc( )[1 cos( )]} 22 ( 1)( )wV A Bπ π κ πβ ε πβ β πβ ε

π π κ ε εδ δ

− + + −⎡ ⎤= − + − =⎣ ⎦

2 2 2 2 2 21 16 60

2 2( 1)csc( ) ( 1)csc( )[1 cos( )]i w w

A U URe

aL U Uπ πβ π π πβ β πβ

⎧ ⎫= − + − + −⎨ ⎬

⎩ ⎭

2 2 2 2 2 21 106 6( 1)csc( ) / ( ) 2 ( 1) csc( )[1 cos( )] 2i w wQ L Re Reπ πβ ν π π πβ β πβ= − + − + − (115)

Table 3. Model for outlet matching inner vortex, *=β β

Variable Two-term approximation

ru 1 2 2 2 2 2 2 2sin( ){ csc( ) [csc( ) cot( ) cot( )]}r r r rπ κ πβ ε πβ β πβ π−− + + −

uθ 2 21 1

4 4 (1 ) 1[1 ]wVr V re e r− − − −− −

zu 2 2 2 2{ csc( ) [1 cos( )]csc( ) / }κ πβ ε β πβ πβ ε π+ + −

21

4 (1 )2 2 22 [cos( ) sin( )][1 ]wV rz r r r eπ π επ π − −× + − zΩ

2 21 14 4 (1 )1 1

2 2wVr V r

wVe V e− − −−

ψ 2 2 2 2 2 2 2sin( ){ csc( ) [csc( ) cot( ) cot( )]}z r r rπ κ πβ ε πβ β πβ π+ + −

ε wUU

κ 0 0

12 2 2

i iQ AL UaL aLπσ π π

= =

σ 2

i

S UaQπβ

=

wRe wU a Reε

ν=

0V 0

iQLν

V 2 2 2 2

0

csc( ) 2 [csc( ) cot( )] 2i w wQ Re ReL

πβ π πβ β πβν

+ + −

wV 2 2 2 2 2 21 1

6 60

( 1)csc( ) 2 ( 1) [csc( ) cot( )] 2i w wQ Re ReL

π πβ π π πβ β πβν

− + − + −

Table 4. Model for fixed outlet radius, = 1 / 2β

Variable Two-term approximation

ru 2 2 2sin( ) / cos( ) sin( ) /r r r r r rκ π ε π π⎡ ⎤− + −⎣ ⎦

uθ 2 21 1

4 4 (1 ) 1[1 ]wVr V re e r− − − −− −

zu 21

4 (1 )2 2 22 cos( ){ [(1 1/ ) tan( )]}[1 ]wV rz r r r eπ π κ ε π π − −+ − + −

zΩ 2 21 1

4 4 (1 )1 12 2

wVr V rwVe V e

− − −−

ψ 2 2 2sin( ){ [1 cot( )]}z r r rπ κ ε π+ −

σ 2 0.45S S

π

V 00

2( 1) 2( 1)i w wQ Re V ReL

π πν

+ − = + −

wV 2 21 1

6 60

( 1) 2[ ( 1) 1]i wQ ReL

π π πν

− + − −


While the near-wall approximation vanishes at 1r = , it exhibits the free vortex aspect away from the wall.

E. Uniformly Valid Swirl Velocity By properly combining the composite inner and wall expansions, a solution may be constructed in a manner to incorporate the problem’s key constraints. While still allowing for free vortex motion in the outer region, this composite solution also stands to capture the velocity adherence condition at the wall and the forced vortex behavior at the centerline. This can be achieved by superimposing

2121

4( ) ( ) ( ) ( ) 40 0 0 0 1 1

4 4

1 exp (1 )1 exp( )1

1 exp( ) 1 exp( )wc ci w ci

ww

V rVrX X X X

V V

⎡ ⎤− − −− − ⎣ ⎦⎡ ⎤= + − = + −⎣ ⎦ − − − − (116)

Recalling that wall-tangential injection is permitted at z l= , one may express the swirl velocity in the piecewise form

( )

2 21 14 4

214

(1 )

( )

1 1 ; 0

1 1 ; (tangential injection)

wVr V r

c

Vr

e e z Lru u

e z Lr

θ θ

− − −

−

⎧ ⎡ ⎤− − <


The sensitivity of the tangential velocity to the sidewall injection and vortex Reynolds numbers is illustrated in Fig. 8. This is accomplished by displaying a one-order-of-magnitude variation in 0V at constant wRe or, conversely, a one-order-of-magnitude variation in wRe at constant 0V . For the range under consideration, the maximum tangential speed can, in some cases, exceed several times the average circumferential injection value at entry. It can clearly be seen that increasing the vortex Reynolds number has the largest influence on increasing the maximum tangential speed and decreasing the diameter of the forced vortex core. Increasing the blowing Reynolds number has a similar, albeit less pronounced, effect. In the same vein, the sensitivity of the solution to the blowing Reynolds number appears to diminish at higher vortex Reynolds numbers. The reader is cautioned that when the flow is turbulent, the Reynolds number must be calculated based on the turbulent eddy viscosity, tμ , instead of the standard, molecular viscosity .μ

VII. Axial and Radial Boundary Layers The viscous tangential momentum equation has been solved asymptotically, thus enabling us to capture the forced vortex behavior at the core and the thin tangential layer at the sidewall. The same approach may be extended to the axial and radial momentum equations. Our purpose here is to ensure that the no slip requirement at the sidewall is equally satisfied by all three velocity components.

A. Axial Boundary Layer Equation We begin with the axial momentum equation which, after normalization and the discarding of insignificant axial derivatives, appears as

2

2

1z z zr

u u upur z r rr

δ⎛ ⎞∂ ∂ ∂∂

= − + +⎜ ⎟∂ ∂ ∂∂⎝ ⎠ (120)

The boundary conditions for this particular problem include the no slip constraint at the sidewall, and the merger with the outer solution at the edge of the boundary layer. These requirements translate into

( )

( )

0

1, 0

lim ( , )z

oz zr

u z

u r z u→

=⎧⎪⎨ =⎪⎩

(121)

The radial velocity obtained in Eq. (92) may be written as

One can then make substitutions for 2 2cos( ) sin( )rA Bu r r r

rκ εε π π+= − (122)

Similarly, the pressure gradient given by Eq. (85) may be expressed as

2 2 2 2 2 24 csc ( ) 8 csc( ) cos( ) 1p z z A Az

π κ πβ επκ πβ πβ π∂ ⎡ ⎤= − − − −⎣ ⎦∂ (123)

Substituting 2rη π= , the radial velocity, and the axial pressure gradient into Eq. (120), one obtains

( )2

2

12 cos sinz z zA Bu u uπ κ ε

πδ ε η ηη η η ηη

+⎡ ⎤⎛ ⎞∂ ∂ ∂− + + −⎜ ⎟ ⎢ ⎥∂ ∂∂⎝ ⎠ ⎣ ⎦

{ }22 2 cos( ) 1 zA A A Aπκ πκ ε πβ π η⎡ ⎤= − − −⎣ ⎦ (124) This form represents the reduced axial momentum equation that requires rescaling in the boundary layer region.

B. Sidewall Expansion As in the tangential case, we introduce a slow coordinate in the form of ( ) /q π η δ= − (125) Substitution into Eq. (124) yields an expression for the near-wall velocity representation, ( )wzu :

( ) ( )

( )( ) ( )

2 ( ) ( ) ( )

2 2

1 1 1 12 cos sinw w w

z z zA Bu u uq qq q q qq

π κ επδ ε π δ π δ

δ π δ δ π δδ⎡ ⎤ ⎡ ⎤+∂ ∂ ∂

− + − − − −⎢ ⎥ ⎢ ⎥− ∂ − ∂∂⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

2

2 22 cos( ) 1 AzA A Aq

ε πκπ πβ πκ π δ

⎧ ⎫⎡ ⎤= − − −⎨ ⎬⎣ ⎦ −⎩ ⎭ (126)

Then, taking ( ) 20 1 ( )w

zu Z Zδ δ= + +O to represent the wall expansion for the axial velocity, the above expression may be rearranged and simplified into


( )2

20 0162

1 ( 1) 02

Z ZA B

qqπ κ ε π ε

π∂ ∂

⎡ ⎤+ + − − =⎣ ⎦ ∂∂ or

20 0

2 0Z Z

qqϕ

∂ ∂+ =

∂∂ (127)

where Taylor-series expansions of the trigonometric terms have been used. Note that we are only interested in the leading order approximation, given the small size of δ . In the process, we recover the same form obtained in the tangential analysis with ϕ having already been defined in Eq. (112). Furthermore, our boundary conditions translate into

[ ]0

( )0

(0, ) 0

lim ( , ) 2 ( )cos sinozq

Z z

Z q z u z A Bπκ πε ε η εη η→∞

=⎧⎪⎨ = = + − +⎪⎩

(128)

Unlike the previous case, the complexity of the limiting condition is apparent here due to the variable nature of the outer solution. To overcome this issue, we introduce a dependent variable transformation such as

[ ]

0 2 ( )cos sinz

Zz A B

ζπκ πε ε η εη η

=+ − +

(129)

In the ( , )q z space, this relation collapses into

[ ]

0 ( )2 ( )z

Zz A B

ζ δπκ πε ε

= − ++ −

O (130)

and so, Eq. (127) returns

2

2 0z z

qqζ ζ

ϕ∂ ∂

+ =∂∂

(131)

with new, constant boundary conditions, namely, ( )0, 0z zζ = and lim ( , ) 1zq q zζ→∞ = (132)

A solution to this set can now be achieved in the form of

( ) ( )21 exp 1 exp 1z q rϕζ ϕ πδ⎡ ⎤= − − = − − −⎢ ⎥⎣ ⎦

(133)

so that

214 (1 )2 2 2

0 ( , ) 2 ( )cos( ) sin( ) 1 wV rZ q z z A B r r r eπκ πε ε π πε π − −⎡ ⎤⎡ ⎤= + − + −⎣ ⎦ ⎣ ⎦ (134)

where the wall vortex Reynolds number 4 /wV πϕ δ= re-emerges. A composite velocity expansion may be constructed from the sum

214 (1 )( ) ( ) ( ) ( ) 2 2 22 ( / ) cos( ) sin( ) 1 ( )wV rc o w wz z z z ou u u u z A B r r r eπ κ ε ε π π ε π δ

− −⎡ ⎤⎡ ⎤ ⎡ ⎤= + − = + − + − +⎣ ⎦ ⎣ ⎦ ⎣ ⎦O (135)

This expression provides a uniformly valid solution in the axial direction. It is plotted in Fig. 9 at several representative values of the control parameters. Note that as the Reynolds number is increased, the sidewall boundary layer grows thinner in a manner similar to that of the boundary layer in the tangential direction. A more detailed assessment of the boundary layers is pursued in Section VII(E).

0 0.2 0.4 0.6 0.8 1-10

-5

0

5

10

u z/(κ

z)

a) r

ε 0 0.001 0.005 0.01

Re = 1,000Rew = ε Re

0 0.2 0.4 0.6 0.8 1-10

-5

0

5

10

u z/(κ

z)

b) r

ε 0 0.001 0.005 0.01

Re = 10,000Rew = ε Re

Figure 9. Variation of the axial velocity with the wall injection parameter at κ = 0.01 and a Reynolds number Re of a) 1,000 and b) 10,000.


C. Radial Boundary Layer Equation By neglecting axial derivatives, one can write the radial momentum equation as

2 2

2 2

1r r r rr

uu u u upur r r r rr r

θ δ⎛ ⎞∂ ∂ ∂∂

− = − + + −⎜ ⎟∂ ∂ ∂∂⎝ ⎠ (136)

with boundary conditions

( )

( )

0

1

lim ( )r

or rr

u

u r u

ε

→

⎧ = −⎪⎨ =⎪⎩

(137)

One can then make the following replacements:

( ) ( )2 2cos sin ;r A Bu r r rrκ εε π π+= − 1 ;u

rθ= (138)

( ) ( ) ( ) ( )2 2 2 2 2 2 2 22

3 3 3

sin sin sin sin1r r r r r rupr r r r r

θκ π π π κ π π π⎡ ⎤ ⎡ ⎤− −∂ ⎣ ⎦ ⎣ ⎦= + = +

∂ (139)

After substituting the above values into Eq. (136), we insert η and discard terms of order 2ε ; we are left with

( )2 3/2 2 2 3/2

2 5/2 3/ 2

d d d1 sin sin4 sin 2 cos sin 1d dd

r r rA Bu u uA Aπ κ εκ π η ηδ π η ε η η κ π

η η η η ηη η η⎡ ⎤ +⎡ ⎤⎛ ⎞ ⎛ ⎞

+ + − − = −⎢ ⎥⎜ ⎟ ⎢ ⎥ ⎜ ⎟⎢ ⎥ ⎝ ⎠⎝ ⎠ ⎣ ⎦⎣ ⎦

(140)

D. Sidewall Expansion Once more we apply the slow coordinate transformation ( ) /q π η δ= − and utilize Taylor-series expansions for the trigonometric terms to arrive at

2 ( ) 3/ 2

3162 2 5/ 2

d4 [( ) ( ) ]

d ( )

wru A q q

q qδ δκ ππ π δ π δδ π δ

+ − − − +−

…

( ) ( )31

6

d2 cos( ) [( ) ( ) ]( ) d

wrA B uq q q

q qπ κ ε

ε π δ π δ π δδ π δ

+⎧ ⎫⎪ ⎪+ − − − − − +⎨ ⎬−⎪ ⎪⎩ ⎭

…

3 31 1

2 2 3/2 6 63/2

[( ) ( ) ] [( ) ( ) ]1

( )( )q q q q

Aqq

π δ π δ π δ π δκ π

π δπ δ⎧ ⎫− − − + − − − +⎪ ⎪= −⎨ ⎬

−− ⎪ ⎪⎩ ⎭

… … (141)

where ( )wru denotes a near-wall approximation of the radial velocity. This term can be conveniently expanded in integer powers of ,δ namely, ( ) 20 1 ( ).

wru R Rδ δ= + +O Through substitution into Eq. (141) and the collection of

terms of the same order, we are left with

( )

220 01

62

1 ( 1) 02

R RA B

qqπ κ ε π ε

π∂ ∂

⎡ ⎤+ + − − =⎣ ⎦ ∂∂ or

20 0

2 0R R

qqϕ

∂ ∂+ =

∂∂ (142)

The resulting second order ODE mirrors its counterpart in the axial direction except for the boundary conditions, specifically

( )0

( )0

(0)

lim ( ) cos / sinorq

R

R q u r A B

ε

ε η κ ε π η η→∞

= −⎧⎪⎨ = = − +⎪⎩

(143)

Again, we note that a transformation of the dependent variable is required to make the second boundary condition constant. To this end, we inspect the outer solution and set

( )

0 cos / sinr

Rr A B

ζε η κ ε π η η

=− +

(144)

Forthwith, Eqs. (142) and (143) collapse into

2

2 0r r

qqζ ζ

ϕ∂ ∂

+ =∂∂

with ( )0 1

lim ( ) 1r

rqq

ζζ

→∞

⎧ =⎪⎨ =⎪⎩

(145)

This set can be readily solved to find 1rζ = . This simple result supports the uniformly valid nature of the outer solution for the radial velocity. Due to normal injection at the sidewall, the standard no-slip condition is immediately secured. The boundary layer correction in the radial direction is hence unnecessary.


E. Boundary Layer Characterization There are three boundary layers that need to be characterized: the core layer corresponding to the forced vortex region and both tangential and axial boundary layers arising at the sidewall. As we saw in the previous section, the radial boundary layer is not present. 1. Forced Vortex Layer The forced vortex region extends over 0 ,cr δ≤ ≤ where cδ is the dimensionless distance from the centerline to the point where the swirl velocity reaches its peak at max( )uθ (see Fig. 8). To determine cδ we find the appropriate zero of the radial derivative of .uθ Starting with Eq. (117), we get:

( )

( )

121

2

2 2 2 2

2 1 2pln 1, 2.242 2.242

2 { csc( ) [csc( ) cot( )]}c

e

V V Reδ

π κ πβ ε πβ β πβ ε

−⎡ ⎤− − − −⎢ ⎥⎣ ⎦= =+ + −

(146)

where pln( , )x y is the product-log function. In accordance with the laminar theory of swirling flows, the thickness of the viscous core appears to be inversely proportional to the square root of the hybrid vortex Reynolds number. This parameter combines the effects of swirl, viscosity, and sidewall injection. The corresponding peak velocity is given by max( ) 0.319 .u Vθ It should be noted that under high speed conditions, a turbulent eddy viscosity may be used in lieu of the molecular viscosity to avoid overpredicting the maximum velocity in the chamber. This will require dividing the measured vortex Reynolds number by the eddy viscosity ratio. A dual axis plot of cδ and max( )uθ versus V is given in Fig. 10a. Note that the maximum swirl velocity grows with successive increases in V . This may be attributed to the cumulative effects of higher tangential speeds at entry and added mass flux across the sidewall. The axial invariance of the peak velocity and its locus is corroborated by several numerical and experimental investigations. 2. Tangential and Axial Boundary Layers We take wδ to be the non-dimensional thickness of the sidewall boundary layer. This layer denotes the distance from the sidewall to the point where the tangential velocity reaches 99% of its final value. To calculate wδ , we set

( ) ( )0.99w ou uθ θ= and solve for the appropriate radius wr . The boundary layer thickness is then deduced from 1w wrδ = − . Starting with Eq. (114), we put

( )

214

14

1 exp (1 )1 10.991 exp

w w

w ww

V rr rV

⎧ ⎫⎡ ⎤− − − ⎛ ⎞⎪ ⎪⎣ ⎦ =⎨ ⎬ ⎜ ⎟− − ⎝ ⎠⎪ ⎪⎩ ⎭

(147)

We then solve for wr to obtain

( )144 ln 0.01 0.99exp 18.42071 1 1 1ww

w w

V

V Vδ

⎡ ⎤+ −⎣ ⎦= − + − − (148)

The wall approximation is valid for 32.wV > Furthermore, an expansion of the radical may be attempted to obtain

2 4.605174ln10 9.21034ln10 ln10 11 2 8w

ww ww w VV VV Vδ

⎡ ⎤ ⎛ ⎞⎛ ⎞ ⎛ ⎞ + += + + +⎢ ⎥ ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎝ ⎠⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦…… (149)

101 102 10310-2

10-1

1

0.1

1

10

δc (u

θ )max

(uθ )max

δc

Va)

turbulence

101 102 103

10-2

10-1

1

δw, exact δw, 2-term approx. δw, 1-term approx.

δw

Vwb)

Figure 10. Variation of a) the core layer thickness and maximum swirl velocity and b) the sidewall boundary layer.


A similar analysis can be carried out for the axial boundary layer zδ . Based on Eq. (135), one finds z wδ δ= . Physically, this equality may be connected to the axial invariance of the tangential boundary layer and the dominance of the radial pressure gradient. Given that constwδ = at any z location, it follows that the axial boundary layer must also remain constant. Otherwise, any axial growth in zδ would also translate into an increase in

wδ . It can thus be seen that the constancy of the tangential and axial layers in all directions leads to a uniform boundary layer thickness along the entire sidewall. This result supports the hypothesis that vortex-fired hybrid chambers are likely to achieve uniform burning along their grains due to the uniformity of their boundary layer zones. The wall boundary layer is shown in Fig. 10b using the exact representation in addition to one- and two-term approximations. The inverse dependence on the wall vortex Reynolds number is readily apparent. It may be interesting to examine the wall-to-core thickness ratio which may be estimated by

4.6051718.4207 4.10843 10.446068 1 1w

wc w w

VVVV V

δδ

⎛ ⎞ ⎛ ⎞+ +− −⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎝ ⎠⎝ ⎠… (150)

This ratio is clearly a function of both V and wV ; these are, in turn, strict functions of 0V , wRe , and β via Eqs. (107) and (115) for an outflow radius that matches the inner vortex radius, or through Eqs. (118)-(119) for a fixed outflow radius of 1 / 2. Interestingly, their ratio is independent of δ and may be expressed as

2 2 2

2 2 2

/ 1 cos( ) sin( )

/ 1 cos( ) sin( )wV

V

πα κ ε β πβ πβ

π κ ε β πβ πβ

⎡ ⎤+ + −⎣ ⎦=⎡ ⎤+ + −⎣ ⎦

(151)

As illustrated in Fig. 11a, this ratio of Reynolds numbers varies between approximately 0.285 and 0.645, the latter being the limiting value of α . As for the mantle sensitivity to the boundary layers, it is found to be virtually insignificant. The curves shown in Fig. 11b lend support to the mantle being fundamentally unaffected by core or wall corrections. Mathematically, the influence of δ on the positioning of β is negligible.

F. Pressure Distribution The pressure may be evaluated now that the viscous corrections are at hand. Based on Euler’s equations, we begin with the radial momentum equation,

2

( )r rr zu u up u u

r r r zθ δ

∂ ∂∂= − − +

∂ ∂ ∂O (152)

Injecting the uniformly valid representations for the velocity components, we retrieve, after some algebra,

( ) ( ) ( )214 23 2 2 2 2 2 2 1 21 sin sin 2r Vp r e A r A r rr κ π πκ π−− −∂ ⎡ ⎤

= − + −⎢ ⎥∂ ⎣ ⎦

( ) ( ) ( )4 2 2 22 cos 2 2 sin 2A B B r r Br rεκ π π π π⎡ ⎤+ − − −⎣ ⎦ (153) Equation (153) may be tested over the range of physical parameters. At the outset, it may be simplified and collapsed into

( )214 23 1 r Vp r er −−∂

−∂

(154)

0 0.004 0.008 0.012 0.016 0.020.2

0.3

0.4

0.5

0.6

0.7

ε

κ 0.001 0.002 0.005 0.01

V w /

V

a) 0 0.004 0.008 0.012 0.016 0.020.7

0.8

0.9

1.0

ε

κ 0.001 0.002 0.005 0.01

Man

tle lo

catio

n, β

b)

Figure 11. Variation of a) /wV V and b) β with the swirl and blowing parameters.


The axial gradient can be similarly obtained from

( )z zr zu up u u

z r zδ

∂ ∂∂= − − +

∂ ∂ ∂O (155)

Although Eq. (155) is easily manageable, its solution appears at higher order. Its contribution to the overall analysis can be shown to be inconsequential. Therefore, consistent with the asymptotic analysis leading to the viscous corrections, the axial pressure gradient may be discounted without affecting the accuracy of the resulting pressure distribution. Returning to the radial pressure gradient, one may make use of symbolic programming75 to integrate Eq. (154) up to an additive constant. The result can be rearranged and simplified into

( ) ( ) ( )214 22 2 21 1 1 10 2 4 2 41 Ei Eir Vp p r e V r V r V−− ⎡ ⎤= − − − − − −⎣ ⎦ (156) where Ei( )x denotes the second exponential integral function. Interestingly, only the core corrections seem to influence the pressure distribution and its radial gradient. The sidewall corrections, although helpful in securing the no slip requirement, do not play a major role in the pressure balance. The uniformly valid pressure and the radial pressure gradients are illustrated in Fig. 12. By comparing Fig. 12c-d to Fig. 6b, it is clear that the singularity near the centerline has been effectively suppressed through the use of matched-asymptotic expansions. The tempering effect due to viscosity is reduced when the viscous parameter δ is decreased by one order of magnitude (left-to-right). Clearly, viscosity is needed to reduce the steepness in the pressure and its gradient.

G. Vorticity Mean flow vorticity can be directly calculated from

( )

( )

214

214

(1 ) 2 2 2

(1 ) 2 2 2

/ 1/ cos( ) sin( )

4 1 / 2 / sin( ) cos( )

w

w

V rw

zz V r

V e A B r r ru uurz

r r r e A B r r rθ θ

θ θ

κ ε π π πεπ

π κ ε π π π

− −

− −

⎧ ⎫⎡ ⎤+ − +⎣ ⎦∂∂ ⎛ ⎞ ⎪ ⎪= − + + = ⎨ ⎬⎜ ⎟ ⎡ ⎤∂ ∂⎝ ⎠ ⎡ ⎤+ − + − −⎪ ⎪⎣ ⎦⎢ ⎥⎣ ⎦⎩ ⎭

Ω e e e

2 21 1

4 4 (1 )

2wVr V rw

zVV e eV

− − −⎡ ⎤+ −⎢ ⎥⎣ ⎦e (157)

As one may infer from an order of magnitude scaling, the axial vorticity dominates over the azimuthal contribution. This can be attributed to the dominance of swirl in the ( , )r θ plane. Furthermore, given that no vorticity can originate from the free vortex segment of the solution, axial vorticity is most visible in the core region

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1.0

ref

ppΔ

Δδ = 0.001

r

ε 0 0.001 0.005 0.01

a) 0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1.0

ref

ppΔ

Δ

δ = 0.0001

r

ε 0 0.001 0.005 0.01

b)

0 0.2 0.4 0.6 0.8 10

20

40

60δ = 0.001p

r∂∂

r

ε 0 0.001 0.005 0.01

c) 0 0.2 0.4 0.6 0.8 10

1000

2000δ = 0.0001p

r∂∂

r

ε 0 0.001 0.005 0.01

d)

Figure 12. Variation of the pressure distribution (a,b) and th

bidirectional vortex with sidewall injectionmajdalani.eng.auburn.edu/publications/pdf/2008...the...

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