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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.
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2 American Institute of Aeronautics and Astronautics
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
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3 American Institute of Aeronautics and Astronautics
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
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4 American Institute of Aeronautics and Astronautics
(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
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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.
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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= .
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7 American Institute of Aeronautics and Astronautics
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)
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8 American Institute of Aeronautics and Astronautics
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)
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9 American Institute of Aeronautics and Astronautics
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)
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10 American Institute of Aeronautics and Astronautics
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
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11 American Institute of Aeronautics and Astronautics
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)
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12 American Institute of Aeronautics and Astronautics
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
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13 American Institute of Aeronautics and Astronautics
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.
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14 American Institute of Aeronautics and Astronautics
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.=
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15 American Institute of Aeronautics and Astronautics
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.
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16 American Institute of Aeronautics and Astronautics
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)
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17 American Institute of Aeronautics and Astronautics
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
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18 American Institute of Aeronautics and Astronautics
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
− + − −
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19 American Institute of Aeronautics and Astronautics
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
− − −
−
⎧ ⎡ ⎤− − <
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20 American Institute of Aeronautics and Astronautics
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
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21 American Institute of Aeronautics and Astronautics
( )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.
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22 American Institute of Aeronautics and Astronautics
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.
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23 American Institute of Aeronautics and Astronautics
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.
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24 American Institute of Aeronautics and Astronautics
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.
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25 American Institute of Aeronautics and Astronautics
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